Vibration Control in Bats using Piezoelectric-based Smart Materials

A thesis submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy.

Jia Long Cao.

MEng (RMIT University)

School of Aerospace, Mechanical and Manufacture Engineering RMIT University , . August 2006 Declaration i

Declaration

I hereby declare that a. The work presented in this thesis is that of my own except where acknowledged to others, and has not been submitted previously, in whole or in part, in respect of any academic award.

b. The work of this research program has been carried out since the official commencement date of the program approved.

Acknowledgements ii

Acknowledgements

I would like to thank my senior supervisor, Associate Professor Sabu John, for his guidance and tremendous support during my PhD study. His generous help, substantial information and regular encouragement always kept me track and was the key to my completion. I felt fortunate to have the opportunity to work under his supervision.

I would also like to thank my second supervisor, Dr. Tom Molyneaux, for his excellent technical advice, invaluable knowledge and great help.

Thirdly, I would like to thank our industry partners, Kevin Davidson from Davidson Measurement and Peter Thompson from Kookaburra Sport, for their financial contribution and great interest in the project, their precious time spent on our regular meetings and materials contributed to the project.

In addition, I would like to thank our lab and workshop technicians, Peter Tkatchyk and Terry Rosewarne, for their generous and friendly help with my experiments and products prototyping, otherwise, I would have had a lot of troubles.

Finally, I would like to thank the Australia Research Council (ARC) for the funding of the project and the scholarship which supported my graduate study. iii Table of Contents

Declaration …...……………………………….………...…………...…...……….…...…... i

Acknowledgements .……………………………………………………………..…...……. ii

Table of Contents ...………………………………….…………………...……….…..…… iii

Abstract …………...………………………………………………………………...…..…. xiv

List of Symbols …………...……………………………………………………………….. xv

Chapter 1 Introduction.…………..……….………..……..……..…………….….…... 1

1.1 Motivation ...…...….………...……………………………………….....…...…. 1

1.2 Vibration control with Smart materials ……...……………..……....………….. 2

1.3 The objectives of this study ...………………....……………………..…..….…. 3

1.4 The organization of the thesis ………..……...………………………….....…... 4

1.5 Publications originated from this research ...………………………….....…….. 6

Chapter 2 Literature Review: Mechanical Vibration Control ...….…..…..…… 7

2.1 Background of the ……………..…………….…....….…...………. 8

2.2 Traditional mechanical vibration control techniques .….....….…....…………... 11 2.2.1 Vibration isolation …...………………………………...……….……….. 11 2.2.2 Vibration absorption …...………………………………...…….……….. 12 2.2.3 Vibration damping …...…………………………………....…….……… 13 2.2.3.1 Viscoelastic damping …...…………………….…………...……. 13 2.2.3.2 Structural damping …...……………………..….…………...…... 16 2.3 Non-conventional vibration control with Smart material components..…...…... 16 2.3.1 Smart material and structure ..……………...……………...…….……… 16 2.3.2 Piezoelectric materials ..…………………....……………...…….……… 18 2.3.2.1 Orientation and notation …...………….…………………...…… 22 2.3.2.2 Piezoelectric constitute equations ………………………………. 23 2.3.2.3 Modelling of piezoelectric transducer ………………………….. 26 2.3.3 Passive piezoelectric vibration shunt control …………...…………….… 26 iv

2.3.4 Active piezoelectric vibration control ……..…………...……….………. 36 2.3.5 Power system for active piezoelectric vibration control ……...………… 40 2.3.6 Advantages and disadvantages of passive and active piezoelectric controls ………………………………………………………………….. 42

2.4 Chapter summary………….……...……………………………………...…….. 43

Chapter 3 Analytical Modelling of Passive Piezoelectric Vibration Shunt on Beams …..…...……………………………………..……...…………...… 45

3.1 Motion equation derivation of composite beam using energy method ………... 46 3.1.1 Euler-Bernoulli beam model …………………………...…………...…... 46 3.1.2 Displacement, strain, stress and velocity relations ……………………… 47 3.1.3 Potential energy ……………………………………………….………… 49 3.1.4 Kinetic energy ………………………...……………………….………... 51 3.1.5 Hamilton’s principle ……………………………………………….……. 52

3.2 Convert partial differential equations (PDE) to ordinary differential equations (ODE) using Galerkin’s method ……….…………………………… 55

3.3 Vibration control with the piezoelectric shunt circuit …………..……….....…. 57 3.3.1 Series R-L shunt ………………………..…………………………...…... 57 3.3.2 Parallel R-L shunt ………………………………….…………….…..… 62

3.4 Numerical examples …………………….………………………...……..…..… 66

3.4.1 Vibration shunt control simulation 1 (PZT placed at L/2).……………… 67

3.4.2 Vibration shunt control simulation 2 (PZT placed at L/4)…….…..…..… 70

3.4.3 Sensitivity test of control frequency mismatch………………………..… 71

3.5 Chapter summary ……………..…….…………………………………..….….. 76

Chapter 4 Composite Beam Analysis and Design ……………………………..…. 77

4.1 Stress and strain in composite beam ...…………………….……………...... …. 77

4.2 Parameters of the two-layer piezoelectric composite beam …………………… 80 4.2.1 Neutral surface location ………………………….……………………... 80 4.2.2 Area moment of inertia ……………….………….……………………... 85 4.2.3 Average normal strain and stress ………….….……….………………... 90

4.3 Discussion of composite beam design for maximizing strain in the PZT ……... 94 v

4.3.1 The relation between electrical field and mechanical strain in the PZT ... 94 4.3.2 Analytical precursor to the strain analysis in the PZT Patch …….…….. 99 4.3.3 Comparison of FEA simulation results of piezoelectric vibration shunt control for beams with different materials ……………….………….…. 104

4.3.4 Application of the PZT-composite beam technology and analysis of the handle of the Cricket bat - Introduction of carbon-fibre composite material ………………………………………………………………….. 113 4.3.5 Proposed design of Cricket bat handle ………………………………….. 115 4.3.6 Other benefits of using carbon-fibre material in Cricket bat …………… 117

4.4 Chapter summary …….…………………..………………………………….… 119

Chapter 5 Numerical Modelling of Passive Piezoelectric Vibration Control using FEA Software ANSYS®…………….……………………...……. 121

5.1 Introduction of ANSYS® ………………….……………….…...……………... 122 5.1.1 General steps of a finite element analysis with ANSYS®….……..…..…. 122 5.1.2 Some elements used for implementation ……………..………..…..…… 123 5.1.3 Couple field analysis ……………..………………………….……...…... 127

5.2 A simple example of the finite element formulation ………………….…….…. 127

5.3 Modelling of passive piezoelectric shunt circuit with PSpice®………….….…. 133 5.3.1 Single mode resistor-inductor (R-L) shunt..…………………………….. 133 5.3.2 Multimode parallel resistor-inductor shunt...………….….…………..…. 136

5.4 Modelling of passive piezoelectric shunt circuit with ANSYS®……....…...….. 139 5.4.1 Single mode resistor-inductor shunt..………………...………….………. 139 5.4.2 Multi-mode resistor-inductor shun.….……………………...……..…….. 141 5.4.3 Shunt effectiveness related to the placement of the PZT patch ………… 142

5.5 Modelling and analysis of the Cricket bat…………………………………...…. 147 5.5.1 Cricket bat with wooden handle …………………………...…………..... 147 5.5.2 Cricket bat with composite hollow tube handle ………..……..………… 148 5.5.3 Cricket bat with sandwiched handle …………………....……..………… 150

5.6 Chapter summary ………………………………………..…………………..… 154

Chapter 6 Experiment and Implementation ….……….………………………..…. 156

6.1 Shunt circuit design .………………………………………………………..….. 157 vi

6.2 Vibration shunt control for beams ….………...……………………………..…. 160

6.3 Modal test of Cricket bats with laser vibrometer …...…………………………. 163 6.3.1 Wooden handle Cricket bat …………………….…………...…………... 164 6.3.2 Carbon-fibre composite tube handle Cricket bat …………………….….. 165 6.3.3 Carbon-fibre composite sandwich handle Cricket bat ………………….. 168 6.3.4 Vibration shunt control for the Cricket bat ……………….…………….. 172

6.4 Field testing of the Smart bat ………………………………………………….. 183

6.5 Chapter summary …...…………………………………………………………. 184

Chapter 7 Frequency Estimation and Tracking ………………………………..…. 185

7.1 Introduction of filters.……………………………………………………..…… 185

7.2 Generic notch filter………………….………...……………………………..… 188

7.3 Adaptive notch filter …...………………………………...……………………. 191 7.3.1 Simplified adaptive notch filter …………………….…………...……..... 191 7.3.2 Notch filter coefficient estimation ……………..……………….……….. 193 7.3.3 Examples of adaptive using cascaded 2nd order notch ANFs……………. 195

7.4 Chapter summary …...…………………………………………………………. 203

Chapter 8 Conclusions and future work ……....………………………………..….. 204

8.1 Conclusions .………………………………………………………………..….. 205 8.1.1 Analytical work on the PZT-based vibration shunt circuit ………….….. 205 8.1.2 Field-Coupled Simulations ………….…………………………………... 205 8.1.3 Cricket bat Innovations ………….………………………………….…... 205 8.1.4 Frequency Tracking Analysis ………….……………………………….. 206

8.1.5 Power amplifier proposal ………….……………………………………. 206

8.1.6 Cricket bat prototyping and field testing ………………………………... 206

8.2 Future work ….………...………………………………………..…………...… 207

vii Appendix 1 Modelling of Vibration Isolation and Absorption of the SDOF Structure………….………….….….……………………………………... 208

Appendix 2 Estimation of System Damping ……………………………………… 215

Appendix 3 Positive Position Feedback (PPF) Control ………………………… 220

Appendix 4 Proposed Power Amplifier using Amplitude Modulation and Piezoelectric Transformer for Active Vibration Control …………. 226

Appendix 5 Beam Modelling and Modal Analysis ………………………………. 232

Appendix 6 Derivation of the Equations in Chapter 3 ..………………………... 247

Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes ….. 255

References ……....………………………………..………………………………………. 273

List of Tables

Table 2-1 Types of fundamental piezoelectric relation ……...….….…………………. 23 Table 6-1 Natural frequency comparison among various Cricket bats ……...….….…. 172 Table A5-1 Common boundary conditions for lateral vibration of uniform beam ….….. 241 List of Figures Figure 2-1 Geometric parameters of Cricket bat ………………….…………………… 9 Figure 2-2 Mechanical structures interconnected by isolation system .………………... 12

Figure 2-3 Visco-elastic material (VEM) models ……...…………………………….… 14 Figure 2-4 Viscoelastic damping treatments……………………………………………. 15

Figure 2-5 Crystal structure of a traditional ceramic..………………………………….. 19 Figure 2-6 Polarizing of a piezoelectric ceramic.………………………..………….….. 20 Figure 2-7 Coordinator axes for piezoceramic …….…………………………...……… 22 Figure 2-8 Piezoceramic under stress ……..………………………………………..….. 25 Figure 2-9 Simplified equivalent circuit for a piezoelectric transducer ……………….. 26 Figure 2-10 Basic classification of passive piezoelectric vibration shunt.…………...….. 27 Figure 2-11 Passive piezoelectric vibration damping.………………………...……...….. 28 Figure 2-12 Inductive piezoelectric shunt …………………………………………...…... 29

viii

Figure 2-13 Model comparison of resonant shunted piezoelectric of a SDOF system….. 30 Figure 2-14 Model comparison of proof mass damper of a SDOF system.………….….. 30 Figure 2-15 Multi-mode vibration shunt using multiple R-L-C shunt branches …..…….. 31 Figure 2-16 Equivalent model for a piezoelectric material with a load resistor ………… 32 Figure 2-17 Inductor-resistor (R-L) parallel shunt.…………...………………………….. 33 Figure 2-18 Multi-mode vibration shunt using series blocking circuits ………………… 34 Figure 2-19 Multi-mode vibration shunt using parallel blocking circuits ………………. 34 Figure 2-20 Block diagram of a basic active control system.…………...……………….. 37 Figure 2-21 Voltage vs. force generation vs. displacement of a piezo actuator ………… 41 Figure 3-1 Composite beam made of a cantilever beam with PZT attached ……….….. 46 Figure 3-2 Beam deformation on x-z plane due to bending ……………………………. 47 Figure 3-3 Equivalent circuit of series piezoelectric R-L shunt ……………………….. 58 Figure 3-4 Equivalent circuit of parallel piezoelectric R-L shunt …………….……….. 63 Figure 3-5 Impulse response of beam (PZT placed at L/2) …...... ………………….….. 68 Figure 3-6 Frequency response of beam (PZT placed at L/2) ……..…………………... 69 Figure 3-7 Impulse response of beam (PZT placed at L/4) ……..………………….….. 70 Figure 3-8 Frequency response of beam (PZT placed at L/4) ……..…………………... 71 Figure 3-9 Sensitivity test (1st mode) ………………………………………………...... 72 Figure 3-10 Sensitivity test (2nd mode) ……………………………………………….…. 74 Figure 3-11 Sensitivity test (3rd mode) ………………………………………………….. 75 Figure 4-1 A two-layer composite beam made of the PZT and uniform beam .……….. 78 Figure 4-2 Cross sectional area of a multi-layer composite beam ……………………... 81 Figure 4-3 Neutral surface location of composite beam …..………………..………….. 83 Figure 4-4 Distance between neutral surface and centeroid of the PZT …….…………. 84 Figure 4-5 Area moment of inertia of the PZT layer about the neutral surface …....…... 88 Figure 4-6 Area moment of inertia of the beam layer about the neutral surface ………. 89 Figure 4-7 Total area moment of inertia of the composite beam about the neutral surface …………...…………………………………………………………. 90 Figure 4-8 Average strain of the PZT layer ……...…………………………………….. 91 Figure 4-9 Average strain of the beam layer …………………………………...……… 92 Figure 4-10 Average stress of the PZT layer …………………..………………...……… 93 Figure 4-11 Average stress of the beam layer …………………………………...……… 94 Figure 4-12 Average strain in the PZT layer (the zoomed plot of Figure 4-8) ………….. 96 Figure 4-13 Average strain in the PZT layer (the zoomed plot of Figure 4-8) ………….. 97 Figure 4-14 Average strain in the PZT layer against the thickness ratio ……………….. 98 ix

Figure 4-15 Strain energy of the PZT (U p ) …………………………..……………..….. 101

Figure 4-16 Strain energy of the beam (Ub ) …………………………………….……… 101

Figure 4-17 Total strain energy of the composite beam (Ub +U p ) …………………….. 102 Figure 4-18 Ratio of strain energy of the PZT over strain energy of the beam

(U p /Ub ) ……....………………………………………………………..….. 102 Figure 4-19 Ratio of strain energy of the PZT over total strain energy of the

composite beam (U p /U ) ………………………………………………….. 103 Figure 4-20 Harmonic analysis of the aluminium beam vibration ……………………… 106 Figure 4-21 Harmonic analysis of the wooden beam vibration …………………………. 109 Figure 4-22 Composite beam with a PZT patch attached ……………………………….. 109 Figure 4-23 Modal analysis of the carbon-fibre beam …………………………….…….. 110 Figure 4-24 Harmonic analysis of the carbon-fibre beam vibration …………………….. 112 Figure 4-25 The Young’s modulus of materials against density ………………….…….. 113 Figure 4-26 Structure of a multi-pile carbon-fibre material …………….………………. 114 Figure 4-27 The variable stiffness lay-up combination for a 12-pile carbon-fibre strip 115 Figure 4-28 Cross sectional of the carbon-fibre and the wood sandwich beam ………… 116 Figure 4-29 Cross sectional area of the sandwich beam with the PZT attached …….….. 116 Figure 4-30 Impulse signal and its corresponding power spectrum …………………….. 119 Figure 5-1 SOLID45 3-D 8-Node structural solid ………….………………………….. 124 Figure 5-2 SOLID95 3-D 20-Node structural solid ………….………………………… 124 Figure 5-3 SOLID5 3-D coupled-field solid ………….……………..…………………. 125

Figure 5-4 CIRCU94 piezoelectric circuit element …..………..……………..…….….. 126 Figure 5-5 Two-node beam element ……………….……………………………….….. 128 Figure 5-6 Hermitian shape functions …….……………………………………………. 129 Figure 5-7 Two-element beam ………….………………………………………..…….. 132 Figure 5-8 PSpice® model of single mode series resistor-inductor piezoelectric shunt …………………………………………………………………….….. 133 Figure 5-9 Transfer function of the single mode resistor-inductor shunt …………..….. 135 Figure 5-10 Multiple parallel single mode shunt circuits in series with “blocking” switch ………………………………………………………………….…… 136 Figure 5-11 Multiple parallel single mode shunt circuits in series with “blocking” circuit ………………………………………………………………….…… 137 Figure 5-12 Equivalent circuit of multi-mode parallel shunt ..………………..………… 138 Figure 5-13 Transfer function of a multi mode parallel R-L shunt …..…………………. 138 Figure 5-14 ANSYS® modelling of the parallel R-L piezoelectric shunt for beam …….. 139 x

Figure 5-15 First four bending modes of the structure (with PZT attached) ……………. 140 Figure 5-16 ANSYS® modelling of the multi-mode piezoelectric shunt for beam ……... 141 Figure 5-17 Harmonic analysis of beam vibration with multi-mode piezoelectric shunt .. 142 Figure 5-18 Beam with PZT patch attached in the middle of the beam (free-free BC) 143 Figure 5-19 First four bending modes of the beam in Figure 5-15 ……………………… 143 Figure 5-20 Harmonic analysis of the beam in Figure 5-17 …………………………….. 144 Figure 5-21 Beam with PZT attached away from the middle of the beam ………….…... 145 Figure 5-22 First four bending modes of the beam in Figure 5-20 ……………………… 145 Figure 5-23 Harmonic analysis of the beam in Figure 5-20………...…………………… 146 Figure 5-24 Cricket bat model built with Solidworks®…………………….……………. 147 Figure 5-25 Bending modes of the Cricket bat with wooden handle (less stiff handle) 148 Figure 5-26 Bending modes of the Cricket bat with hollow composite handle (stiffer handle) ……………………………………………………………… 149 Figure 5-27 Cricket bat mounted with PZT patch ………………………………………. 150 Figure 5-28 Bending modes of hollow composite handle Cricket bat mounted with PZT patch ……………………………………………………………….….. 151 Figure 5-29 Fork like composite beam ………………………………………………….. 151 Figure 5-30 Cricket bat with sandwich handle ………………………………………….. 152 Figure 5-31 Bending modes of Cricket bat with sandwich handle ……………………… 153 Figure 5-32 Preferred model of Cricket bat handle embedded with composite beam and PZT …………..…………………………………………………….…... 153 Figure 5-33 Bending mode of the Cricket bat in Figure 5-31 …………………………… 154 Figure 6-1 Block diagram of cantilever beam shunted with parallel R-L shunt ……….. 156 Figure 6-2 Generalized impedance converter (GIC) ….……………..…………………. 157

Figure 6-3 Parallel R-L piezoelectric shunt circuit with a gyrator-based inductor …….. 158 Figure 6-4 Transfer function of parallel R-L shunt with gyrator-based inductor ……… 159 Figure 6-5 Double adjustable gyrator-based inductors ………………………………… 160

Figure 6-6 Vibration shunt for aluminium beam (test 1) ………………………………. 161 Figure 6-7 Vibration shunt for aluminium beam (test 2) ………………………………. 161 Figure 6-8 Vibration shunt for wooden beam (test 1) ………………………………….. 161 Figure 6-9 Vibration shunt for wooden beam (test 2) ………………………………….. 162 Figure 6-10 Vibration shunt for carbon-fibre composite beam (test 1) …………………. 162 Figure 6-11 Vibration shunt for carbon-fibre composite beam (test 2) …………………. 162 Figure 6-12 Set-up of Cricket bat testing with laser vibrometer ……...………………… 163 Figure 6-13 Wooden handle Cricket bat ………………………………………………… 164 xi

Figure 6-14 Bending modes of the wooden handle Cricket bat ………………………… 165 Figure 6-15 Carbon-fibre composite tube handle Cricket bats #1……………….………. 165 Figure 6-16 Carbon-fibre composite tube handle Cricket bats #2……………….………. 166 Figure 6-17 First three bending modes of the bat in Figure 6-15 .………………………. 166 Figure 6-18 First three bending modes of the bat in Figure 6-16 .………………………. 167 Figure 6-19 Carbon-fibre composite sandwich handle Cricket bat #1 ……………….…. 168 Figure 6-20 Bending modes of the carbon-fibre composite sandwich handle Cricket bat #1 …………..……………………………………………………….…... 169 Figure 6-21 Carbon-fibre composite sandwich handle Cricket bat #2 ……………….…. 169 Figure 6-22 Accelerate spectrum of the #2 sandwich handle Cricket bat …………....…. 170 Figure 6-23 Bending modes of the carbon-fibre composite sandwich handle Cricket bat #2 …………..……………………………………………………….…... 171 Figure 6-24 Blade and pre-made sandwich handle for sandwich handle Cricket bat #3… 171 Figure 6-25 The location of the PZT patches on the handle ……....…………………….. 173 Figure 6-26 Vibration of the sandwich handle Cricket bat #1 (1st mode) .……………… 174 Figure 6-27 Vibration of the sandwich handle Cricket bat #1 (2nd mode) ………………. 174 Figure 6-28 Vibration of the sandwich handle Cricket bat #1 (3rd mode) ……………… 174 Figure 6-29 Vibration of the sandwich handle Cricket bat #1 (4th mode) ……………… 175 Figure 6-30 Vibration of the sandwich handle Cricket bat #1 (1st mode, random excitation ………………………………………………………………….... 175 Figure 6-31 Vibration of the sandwich handle Cricket bat #1 (2nd mode, random excitation ………………………………………………………………….... 175 Figure 6-32 The first prototype of new Cricket bat integrated with vibration shunt circuit ……………………………………………………………………….. 176 Figure 6-33 Exploded view of the handle of the first prototype of new Cricket bat ……. 177 Figure 6-34 Vibration of the first new Cricket bat prototype (1st mode) ...……………… 177 Figure 6-35 Vibration of the first new Cricket bat prototype (2nd mode) ……………….. 178 Figure 6-36 Vibration of the first new Cricket bat prototype (3rd mode) ……………….. 178 Figure 6-37 Vibration of the first new Cricket bat prototype (4th mode) ……………….. 178 Figure 6-38 The influence of power supply to the effectiveness of shunt control ……… 181 Figure 6-39 The second prototype of new Cricket bat integrated with vibration shunt circuit ……………………………………………………………………….. 182 Figure 6-40 Vibration of the second new Cricket bat prototype (1st mode, random excitation) …………………………………………………………………... 183 Figure 7-1 Direct realization of an Nth order non-recursive FIR digital filter …………. 186 Figure 7-2 Direct realization of an Nth order recursive IIR digital filter ………..….….. 187

Figure 7-3 Block diagram of the general adaptive digital filter ……………………….. 187 xii

Figure 7-4 Block diagram of the generic notch filter …………………………….……. 188 Figure 7-5 Frequency response of notch filter ………………….……………………… 190 Figure 7-6 Pole-zero diagram of the 2nd order notch filter ……………….……………. 190 Figure 7-7 Adaptive notch filter by cascading multiple 2nd order ANF sections ……… 191 Figure 7-8 Direct realization of the constrained 2nd order IIR notch filter …………….. 196 Figure 7-9 Input signal and the output signals of each 2nd order ANF (SNR = 0 dB) 197 Figure 7-10 Adaptation history of the ANF (SNR = 0 dB) ……………………………… 198 Figure 7-11 Transfer functions of the ideal and estimated notch filters (SNR = 0 dB) …. 198 Figure 7-12 Spectra of the input signal and the output signals of each 2nd order ANF (SNR = 0 dB) …..……………………………………………………….…... 199 Figure 7-13 Input signal and the output signals of each 2nd order ANF (SNR = -10 dB) …..…………………………………………………………. 199 Figure 7-14 Adaptation history of the ANF (SNR = –10 dB) …………………………… 200 Figure 7-15 Transfer functions of the ideal and estimated notch filter (SNR = -10 dB) … 200 Figure 7-16 Spectra of the input signal and the output signals of each 2nd order ANF (SNR = –10 dB) ..………………………………………………………..….. 201 Figure 7-17 Input and output signals of each 2nd order ANF (decayed sinusoid, SNR = 0 dB) ..………………………………………………………..……………….. 201 Figure 7-18 Adaptation history of the ANF (decayed sinusoid, SNR = 0 dB) …………... 202 Figure 7-19 Transfer functions of the ideal and estimated notch filter (decayed sinusoid, SNR = 0 dB) ..………………………………………………………..……... 202 Figure 7-20 Spectra of the input signal and the output signals of each 2nd order ANF (decayed sinusoid, SNR = 0 dB) ……………………………………………. 203

Figure A1-1 Single-degree-of-freedom vibration isolator ……………………………….. 208 Figure A1-2 Models of vibration absorber ………..….………………………………….. 211 Figure A1-3 Spectrum of structure displacement with damped absorber ……………….. 214 Figure A2-1 Single-degree-of-freedom (SDOF) system ……………………….………... 215 Figure A2-2 Frequency response of a SDOF system ……………………………………. 218 Figure A2-3 Impulse response of a SDOF system ….…………………………………… 219 Figure A3-1 Block diagram of SDOF structure with PPF control ………………………. 220 Figure A3-2 SIMULINK model of SDOF structure with positive position feedback …… 222 Figure A3-3 Impulse/frequency response of SDOF structure with/without PPF control ... 223 Figure A3-4 SIMULINK model of 4-DOF structure with positive position feedback ….. 224 Figure A3-5 Impulse/frequency response of 4-DOF structure with/without PPF control .. 225 Figure A4-1 Rosen type piezoelectric transformer ………………………………………. 226 Figure A4-2 Transoner® piezoelectric transformer ………..….…………………………. 227 xiii

Figure A4-3 Block diagram of an AM power amplifier …………………………………. 228 Figure A4-4 PSpice® functional model of an AM amplifier ……………………….……. 228 Figure A4-5 Simulation result of the modulated signal before being amplified ………… 229 Figure A4-6 Actuation signal before (in red) and after (in green) being amplified ….….. 229 Figure A4-7 Spectrum of actuation signal before (in red) and after (in green) being amplified ..………………..……………………………………………..….. 230 Figure A4-8 History of piezoelectric transformers in terms of power capability ……….. 231 Figure A5-1 Vibration of beam element with rotary inertia and shear deformation …….. 233 Figure A5-2 Bending mode shapes of a clamped-free Euler-Bernoulli beam …..….……. 243 Figure A5-3 Illustration of obtaining mode shapes of a structure from the imaginary part of the FRFs ……..……………………………………………..……….. 244 Figure A5-4 Experimental mode shapes of a clamped-free beam before curve fitting ….. 245 Figure A5-5 Experimental mode shapes of a clamped-free beam after curve fitting ……. 245 Figure A5-6 Comparison between theoretical and experimental mode shapes …….……. 246

Abstract xiv

Abstract

The vibrations of a Cricket bat are traditionally passively damped by the inherent damping properties of wood and flat rubber panels located in the handle of the bat. This sort of passive damping is effective for the high frequency vibrations only and is not effective for the low frequency vibrations. Recently, the use of Smart materials for vibration control has become an alternative to the traditional vibration control techniques which are usually heavy and bulky, especially at low frequencies. In contrast, the vibration controls with Smart materials can target any particular frequency of vibration. This has advantages such as it results in smaller size, lighter weight, portability, and flexibility in the structure. This makes it particularly suitable for traditional techniques which cannot be applied due to weight and size restrictions.

This research is about the study of vibration control with Smart materials with the ultimate goal to reduce the vibration of the Cricket bat upon contact with a . The study focused on the passive piezoelectric vibration shunt control technique. The scope of the study is to understand the nature of piezoelectric materials for converting mechanical energy to electrical energy and vice versa. Physical properties of piezoelectric materials for vibration sensing, actuation and dissipation were evaluated. An analytical study of the resistor-inductor (R-L) passive piezoelectric vibration shunt control of a cantilever beam was undertaken. The modal and strain analyses were performed by varying the material properties and geometric configurations of the piezoelectric transducer in relation to the structure in order to maximize the mechanical strain produced in the piezoelectric transducer. Numerical modelling of structures was performed and field-coupled with the passive piezoelectric vibration shunt control circuitry. The Finite Element Analysis (FEA) was used in order for the analysis, optimal design and for determining the location of piezoelectric transducers. Experiments with the passive piezoelectric vibration shunt control of beam and Cricket bats were carried out to verify the analytical results and numerical simulations. The study demonstrated that the effectiveness of the passive piezoelectric vibration shunt control is largely influenced by the material properties of the structures to be controlled. Based on the results from simple beam evaluations, vibration reduction of up to 42% was obtained with the designed Smart Cricket bat. Finally, for the control circuit to automatically track the frequency shift of structures required in real applications, an adaptive filter protocol was developed for estimating multiple frequency components inherent in noisy systems. This has immediate application prospects in Cricket bats. List of Symbols xv

List of Symbols

1. Nomenclature

SYMBOL QUANTITY UNIT

2 A Acceleration m/s 2 Ap = bhp Cross-sectional area of PZT m 2 Ab = bhb Cross-sectional area of beam m b Width of beam and PZT m c Damping coefficient s⋅N/m cc Critical damping coefficient s⋅N/m E Compliance coefficient field in “ij” direction under N/m2 cij constant electrical

C p Inherent capacitance of PZT F dij Piezoelectric charge constant in “ij” direction m/V (C/N) d(n) Desired signal D Displacement m 2 Di Electrical displacement of PZT in “i” direction C/m e(n) Error signal 2 Eb Elastic modulus of beam N/m (Pa) 2 E p Elastic modulus of PZT N/m (Pa)

Ei Electrical field of PZT in “i” direction V/m 2 EI Equivalent flexural rigidity N⋅m

f p Parallel resonance frequencies of piezoceramic Hz

fs Series resonance frequencies of piezoceramic Hz E Resonant frequency of piezoceramic under short circuit Hz fr f(x, t) Applied force N

FS Input disturbing force of isolator N

FT Output transmitted force of isolator N

gij Piezoelectric voltage constant in “ij” direction V⋅m/N G Shear modulus N/m2 (Pa) List of Symbols xvi hb Thickness of beam m hp Thickness of PZT m

hij Electrical displacement-stress coefficient of PZT N/C in “ij” direction h(n) Finite length impulse response sequence H(x) Heaviside function 4 I b Area moment of inertia of beam about the neutral axis m

1 4 I p Area moment of inertia of PZT about the neutral axis m

2 I p Current source of PZT A

I t Total area moment of inertia

J b Mass moment of inertia kg⋅m k Stiffness N/m Wavenumber m-1 kn

Kij Electromechanical coupling coefficient in “ij” direction L Length of beam m

Leq Equivalent inductor value of gyrator-based inductor m Mass kg M Bending moment N⋅m n Young’s modulus ratio of PZT over beam qr (t) Time function 1Q Electrical charge of PZT C 2 Q Shear force N Ratio of the disturbing force frequency to the natural r frequency of isolation system E Elasticity constant in “ij” direction under constant m2/N sij electrical field D Elasticity constant in “ij” direction under constant m2/N sij dielectric displacement 2 S p = b(x2 − x1) Surface area of PZT m

Tb Kinetic energy of beam J (N⋅m)

Tp Kinetic energy of PZT J (N⋅m) u(x,t) Axial displacements m v(y,t) List of Symbols xvii

Ub Potential energy of beam J (N⋅m)

U p Potential energy of PZT J (N⋅m)

1v Voltage across PZT V 2 v Velocity m/s V Velocity m/s 3 Vb Volume of beam m 3 Vp Volume of PZT m

xd Location on beam of vibration signal detected m x2 − x1 Length of PZT m x(n) Input signal of filter

X S Disturbing motion to isolator m

X T Transmitted motion from isolator m y(n) Output signal of filter w(x, t) Transverse displacement m h z = b Distance from PZT centeroid to origin m b 2 h z = h + p Distance from beam centeroid to origin m p b 2 z Neutral axis of composite beam m

EL Zi Electrical impedance of piezoelectric in “i” direction Ω EL Non-dimensional electrical impedance of piezoelectric Zi Ω

ME Z jj Mechanical impedance of piezoelectric in “j” direction N⋅s/m δW Virtual work J (N⋅m)

δWf Virtual work due to applied force J (N⋅m)

δWp Virtual work due to electrical displacement J (N⋅m)

βij Dielectric impermeability of PZT in “ij” direction m/F (V⋅m/C) δ(x) Dirac Delta Function

φr (x) Mode function η Loss factor 1 κ = Curvature (R is the radius of curvature) m-1 R Wavelength m λn 1 μ Ratio of absorber mass to system mass List of Symbols xviii

2 μ Thickness ratio of PZT over beam th θi i angular notch frequency rad/s ρ Density kg/m3 ω Angular frequency rad/s

ωn Natural Frequency rad/s 2 σ 11 Axial stress in x direction N/m (Pa) σ Damping factor ζ Damping ratio

ψ i (t) Error gradient ψ Phase angle Deg

ε 0 Dielectric constant of free space F/m E Dielectric constant matrix in “ij” direction under F/m εij constant stress

ε 11 Axial strain in x direction Strain of beam ε b Strain of PZT ε p

Where

A – Ampere, C – Coulomb, Deg – Degree, F – Farad, J– Joule, kg – Kilogram, m – Meter, N – Newton, Pa – Pascal, s – Second, V – Volt, W – Watt, Ω – Ohm.

List of Symbols xix 2. Subscripts i Thickness direction of the piezoelectric material j Loading direction k Direction transverse to loading direction ij Loading in j direction and shunt across i direction. The first subscript gives the direction of electrical parameters - the voltage applied or the charge produced. The second subscript gives the direction of mechanical parameters - stress or strain D Measured under constant electrical displacement E Measured under constant electrical field (short circuit) S Measured under constant strain T Measured under constant stress

3. Superscripts

D Measured under constant electrical displacement (open circuit) E Measured under constant electrical field (short circuit) EL Electrical ME Mechanical PMD Proof Mass Damper RSP Resonant Shunted Piezoelectric S Measured under constant strain SU Shunt T Measured under constant stress

List of Symbols xx 4. Acronyms and Abbreviations

AFC Active Fibre Composite AM Amplitude Modulation ANF Adaptive Notch Filter AVC Active Vibration Control CFRP Carbon Fiber Reinforced Plastic COP Centre Of Percussion COT Coefficient Of Restitution DOF Degree-Of-Freedom DSP Digital Signal Processing EMI Electro-Magnetic Interference ER Electro-Rheological FEA Finite Element Analysis FIR Finite Impulse Response FRF Frequency Response Function GIC Generalized Impedance Converter IIR Infinite Impulse Response LQG Linear Quadratic Gaussian MDOF Multi-Degree-Of-Freedom MFC Macro-Fibre Composites MR Magneto-Rheological PDE Partial Differential Equation PMD Proof Mass Damper PPF Positive Position Feedback PT Piezoelectric Transformer PVC Passive Vibration Control

PVDF PolyVinylidine DiFluoride (CH2CF2)

PZT Lead-Zirconium-Titanium (PbZrTiO3) RLS Recursive Least Squares RMS Root Mean Square RSP Resonant Shunted Piezoelectric SDOF Single-Degree-Of-Freedom SMA Shape Memory Alloy List of Symbols xxi SNR Signal-to-Noise Ratio TMD Tuned Mass Damper TVA Tuned Vibration Absorber TVD Tuned Viscoelastic Damper VEM Visco-Elastic Materials HR Handle radius POS Apex position POSF Ridge length SCD Scoop height SCF Scoop length STC Centre spring TH Apex height TOE Toe thickness W Edge thickness

Chapter 1 Introduction 1

Chapter 1 Introduction

1.1 Motivation

Cricket has been around for more than a Century but until now, improvements in the performance of the Cricket bat have been somewhat limited (Bailey, T., 1997, Pratomo, D., 2001). It’s partly due to game regulation restrictions and partly because the application of appropriate high technology to the bat has been virtually non-existent. As anyone who has played the game of Cricket knows, the effect of hitting a ball away from the Sweet Spot can be very uncomfortable due to the transmitted structure-borne torsional and lateral vibration on a player’s hands. There is also a chance for batsman to be injured as a result of the transient vibration. These transient vibrations have up until now, been passively and mechanically damped by the inherent damping properties of wood and flat rubber panels in the handle (Penrose, et al, 1998, Knowles et al, 1996). The understanding is that this sort of passive damping is effective for the high frequency vibrations only while the low frequency vibrations remain uninterrupted.

The high technology has played a significant role in sports competition today. The high technology equipment can give players a more competitive edge as well as greater comfort. As for Cricket, the high technology could potentially result in a simple yet better design of a Cricket bat leading to a marginal advantage over other designs, by offering superior quality and enhanced performance. It can reduce the risk of a Cricket player being physically injured due to the serious vibrations.

The rules of Cricket, although strict, have left open an opportunity to alter the handle design where there is no limitation to the innovative possibilities, either in the form of improved structure, material composition or additional instrument incorporation in the handle. It is possible to incorporate novel state-of-the-art Smart technologies to provide enhanced performance. The use of high stiffness to weight ratio advanced fibre composites and the Chapter 1 Introduction 2 opportunity for innovative design and manufacture using textile composites, present the possibility of significantly altering the dynamic characteristics of the bat by tailoring the design of the handle.

The motivation of this study is to try to bring advanced vibration control technology into the sporting goods industry, seeking and establishing novel applications using modern Digital Signal Processing (DSP) techniques for future sporting goods. Scientifically, the research project gives a theoretical and systematic understanding of the piezoelectric vibration control for Cricket bat and initially links the modern electronics and digital signal processing technology to the traditional Cricket bat. It will have ramifications in the development of even more sophisticated Cricket bats.

1.2 Vibration control with Smart materials

Traditionally, unwanted vibrations are removed or attenuated with a variety of passive treatments including rubber mounts, foam, viscoelastic layers, frication and tuned mass damper (TMD). However, the outcome of these solutions somehow is limited for many applications. In addition, temperature variation and geometric constraints sometimes make them impractical. In the past decade, many researchers have turned their attention to so-called Smart materials technology in conjunction with modern electronic technology as an alternative to traditional mechanical vibration control techniques. The traditional mechanical vibration control techniques are usually vibration isolation which reduces vibration energy transmission from one body to another, by using low stiffness materials such as rubber; vibration absorption which converts some of the vibration energy into another form of energy with viscoelastic materials, usually heat; spring-mass damping which dissipates vibration energy into heat with motion and time by adding mass. The drawbacks of these techniques include: bulkiness and extra weight when being used for low frequency applications. Other drawbacks of traditional vibration suppression methods include: inflexibility in applications due to changing operating field conditions, lack of adaptability and ineffectiveness for high frequency vibration reduction. The Smart materials such as piezoelectric materials integrated with digital signal processing techniques may overcome these drawbacks.

Piezoelectric materials such as lead-zirconate-titanate ceramics (PZT), piezoelectric polymers and Macro-Fiber Composites (MFC) are typical Smart materials suitable for vibration control. They have the ability to convert mechanical energy into electronic energy and vice versa. The piezoelectric materials have a broadband frequency response which can be used to tackle a wide range of vibration frequencies. Compared to the traditional mechanical vibration control Chapter 1 Introduction 3 techniques, the vibration control with Smart materials can result in a smaller size, lighter weight, lower cost, and is more effective in on-going vibration suppression due to changing operating conditions of the structure. They are especially suitable where the traditional techniques cannot be applied due to weight and size restrictions (such as for spaceships, satellites and aircrafts).

The scope of this study is to understand the nature of piezoelectric materials for energy transduction. Physical properties of piezoelectric materials for vibration sensing, actuating and dissipating will be evaluated. The analytical study of vibration energy coupling between beams made of different materials and piezoelectric materials will be undertaken. The numerical study will be carried out using the Finite Element Analysis (FEA) software package ANSYS®; in modeling and designing of a Cricket bat with a composite handle. It will also include modeling piezoelectric vibration shunt control for optimal design and placement of the piezoelectric patches on the Cricket bat. The experimental study of the vibration control will be conducted on beams as well as on Cricket bats to verify the results of analytical and numerical studies. Different types of passive vibration shunt networks (Lesieutre, 1998) will be simulated and tested. An adaptive notch filter algorithms for adaptive vibration control with piezoelectric materials will also be studied and simulations implemented.

1.3 The objectives of this study

The objectives of this study are to:

(a) Bring innovation to vibration attenuation in the Cricket bats, especially 1st ~ 3rd modes.

(b) Reduce the unwanted vibration energy irrespective of the excited frequency.

(c) Achieve a reasonable vibration reduction in a short transient time period for improving the dynamic performance of Cricket bats by using piezoelectric vibration control technique in order to reduce the levels of shock and vibration impact on the batsman.

(d) Ease the risk of being injured by reducing the discomfort of the players due to the vibrations, caused by the Cricket bat-ball impact.

(e) To ensure that the design and implementation should be in line with the international manufacturing standards of Cricket bats and it should conform to the rules of the sport (Oslear, 2000).

Initially, the design may use an external power source such as a battery as a power supply for this study but self-powering will be the ultimate goal in the future. The delivered product is Chapter 1 Introduction 4 expected to have such innovative features that will be perceived as being commercially attractive as long as the rules of the Cricket game are complied with.

The task includes:

(1) Model the piezoelectric vibration shunt control for Cricket bats.

(2) Design, model and prototype new Cricket bat.

(3) Design and implement piezoelectric vibration control shunt technology in Cricket bats.

(4) Design the algorithm for adaptive vibration control to account for the inherent variations in a Cricket bat’s natural frequencies due to the natural variability of the structure of wood.

1.4 The organization of the thesis

This thesis contains seven further chapters:

In chapter 2, a brief overview of Cricket bat history and the research being carried out on Cricket bats is presented. The literature review of traditional vibration control techniques, Smart materials, particularly focusing on piezoelectric materials, passive and active piezoelectric vibration control methods and their issues are presented.

In chapter 3, the analytical study of the passive piezoelectric vibration shunt control of cantilever beam is undertaken. This is undertaken to test the feasibility on the shunt technology on a simple structure such as a uniform-sectioned beam, before applying the technique on the Cricket bat. The equation of motion of a composite beam which consists of a cantilever beam bonded with a PZT patch using Hamilton’s principle and Galerkin’s method was derived.

In chapter 4 the strain, stress analyses and design of composite beams were performed. The influences of geometry sizes and material properties of the composite structure on strain and stress distribution have been discussed. The design of the Smart Cricket bat is proposed.

Piezoelectric shunt circuit modeling in CAD software PSpice®, numerical modeling of the passive piezoelectric vibration shunt control of beams with FEA software ANSYS® are carried out in chapter 5. Different types of Cricket bat handles are designed and modeled with the help of the CAD software - Solidworks®.

In chapter 6 the design and implementation of the shunt circuit using a gyrator-based inductor is presented. Also included, are the modal analyses of Cricket bats using a laser vibrometer, Chapter 1 Introduction 5 the experimental results of the passive piezoelectric vibration shunt control of the beams and Cricket bats.

In chapter 7, the design of an adaptive notch filter using the Recursive Least-Squares (RLS) algorithm for the estimation of frequencies is introduced. This design is verified by a MATLAB® - based simulation. As explained above, this was undertaken to account for the inherent variations in a Cricket bat’s natural frequencies due to the natural variability of the structure of wood.

Conclusions and future work are highlighted in chapter 8.

Chapter 1 Introduction 6 1.5 Publications that originated from this research

[1] Cao, J. L. John, S. and Molyneaux, T., “Finite Element Analysis Modeling of Piezoelectric Shunting for Passive Vibration Control”, International Conference on Computational Intelligence for Modeling Control and Automation – CIMCA'2004, 12- 14 July 2004, Gold Coast, Australia.

[2] Cao, J. L. John, S. and Molyneaux, T., “Piezoelectric-based Vibration Control Optimization Using Impedance Matching Technique”, 2nd Australasian Workshop on Structural Health Monitoring, 16-17December 2004, Melbourne, Australia.

[3] Cao, J. L. John, S. and Molyneaux, T., “Piezoelectric-based Vibration Control Optimization in Nonlinear Composite Structures”, 12th SPIE Annual International Symposium on Smart Structures and Materials, 6-10 March 2005, San Diego, California USA. (SPIE Code Number: 5764-46)

[4] Cao, J. L. John, S. and Molyneaux, T., “Computer Modeling and Simulation of Passive Piezoelectric Vibration Shunt Control”, The 2005 International Multi-Conference in Computer Science & Computer Engineering, June 27-30, 2005, Las Vegas, Nevada, USA.

[5] Cao, J. L. John, S. and Molyneaux, T., “The Effectiveness of Piezoelectric Vibration Shunt Control Based on Energy Transmission considerations”. Accepted for presentation at EMAC 2005 and paper submitted to the Australian and Industrial & Applied Mathematics Division of the Australian Mathematical (ANZIAM) Society Journal. (Under review)

[6] Cao, J. L. John, S. and Molyneaux, T., “Optimisation of Passive Piezoelectric Vibration Shunt Control Based on Strain Energy Transfer and Online Frequency Detection”, 13th SPIE Annual International Symposium on Smart Structures and Materials, 26 February- 2 March 2006, San Diego, California USA. (SPIE Code Number: 6167-66)

[7] Cao, J. L. John, S. and Molyneaux, T., “Strain Effect Modelling of Piezoelectric-based Vibration Control in Composite Material Structures”, submitted to IOP Journals, Smart Materials and Structures. (Under review)

[8] Cao, J. L. John, S. and Molyneaux, T., “Piezoelectric-based Vibration Control in Composite Structures”, The 2006 World Congress in Computer Science, Computer Engineering and Applied Computing, June 26-29, 2006, Las Vegas, Nevada, USA. Chapter 2 Literature Review: Mechanical Vibration Control 7

Chapter 2 Literature Review: Mechanical Vibration Control

Noise and vibration control has long been a subject of engineering research. One can encounter noise and vibration everywhere in daily life, at home and work, during a trip, in sports and recreation etc. In most cases, noise and vibration are undesirable and hazardous. They can harm, damage, and even threaten human health. On the other hand, noises and vibrations are difficult to control, especially with traditional techniques, which are mainly isolation, absorption, and viscoelastic material damping, particularly at low frequencies due to long wavelengths. These methods often result in bulky size, heavy weight and high costs. In addition, temperature variation, geometry constraints and weight limitations sometimes make them impractical for some applications. Noise and vibration in general stem from particle movements in the media which can be represented by wave propagation. Often noises are generated by vibrations. The former are undesired wave disturbances in air and the latter are undesirable wave disturbances in mechanical structures. So the noise control and the vibration control have their similarities. However, this study only focuses on the vibration control and its application on Cricket bats. In this chapter, firstly, an overview of the background of Cricket bat, including the rules restricting the design of Cricket bats and the possibilities of improving Cricket bat performance by using advanced material and electronic technologies within the boundaries of Cricket bat governing rules. Secondly, a review of traditional mechanical vibration control techniques which are the foundation of modern vibration control techniques. This is followed by introduction of Smart materials and structures, focusing on piezoelectric materials and the literature review of piezoelectric vibration control methods which basically fall into two categories: passive piezoelectric vibration control and active Chapter 2 Literature Review: Mechanical Vibration Control 8 piezoelectric vibration control. Finally, discussion of their advantages/disadvantages and their suitability for Cricket bat applications will be presented.

2.1 Background of the Cricket bat

As in many industries, the sporting goods industry faces fierce competition today. In order to win the competition and prevent being phased out, it is important for manufacturers to innovate products and give their goods added functionalities at an acceptable price. These are not only to meet some players’ personal taste, but also to have benefits from a marketing point of view. In sports competition, unwanted vibrations affect the performance, and comfort of players, as well as shorten the life span of products. In recent years, optimising bat/ball impact for ball-hitting games, shock and vibration control on sporting goods, which will suffer severe shock and vibration impact, has come to the fore. The examples are such as aluminium bats (Russell, [I 10]) clubs (Bianchini et al, 1999), rackets (Head_a, [I 5]), mountain bike Smart shock system (Perkins, 1997), skis (Head_b, [I 6]), and Intelligence snowboards (Head_c, [I 7]) which use advanced material technology with/without electronic technology to either enhance performance for the games or reduce the risk of game players being injured, bringing comfort to the game players. These innovative designs have made the sports more attractive, enjoyable and entertaining.

Cricket is a sport with a long history and rich tradition. It is a global sport that uses a Cricket bat to hit a bounce-delivered ball. The Cricket bat consists of basic structures of blade and handle. The former is traditionally made of willow and the latter is made of cane. Due to deep conservatism in the modern rules of the games, formulated in 1774 (Bailey, 1979), restrict the games equipment from using new sport enhancement technologies that have proven to benefit other games of sport. Only a little enhancement has been made to improve the performance of Cricket bat since its inception. The laws of Cricket are a set of rules framed by the Marylebone Cricket Club (MCC) which serve to standardise the format of Cricket matches across the world to ensure uniformity and fairness. The laws of Cricket Law 6 –– The Bat, limits the bat overall shall not be more than 38 inches/96.5cm in length; the blade of the bat shall be made solely of wood and shall not exceed 41⁄4 in/10.8cm at the widest part; the blade may be covered with material for protection, strengthening or repair, such material shall not exceed 1⁄16 in/1.56mm in thickness, and shall not be likely to cause unacceptable damage to the ball (MCC, [I 8]). Within the boundaries of the game rules, several improved bat designs have proceeded to commercial production. These include the introduction of cane and rubber laminates (parallel with the bat length) as handle material aimed to ultimately reduce the Chapter 2 Literature Review: Mechanical Vibration Control 9 transmission of vibration from the blade to the handle on impact and the introduction of improved perimeter weighting in 1960’s on the blade. However, these changes were still far from optimum. The rules of the Cricket game prohibit radical improvement to the blade in which no alteration to the material composition of the blade is permitted, English Willow or its equivalent is mainly used in this regard, which restricts the blade design. The rules do not strictly restrict the handle design, where there is no limitation to the innovative possibilities either in the form of improved structure, material composition, or additional instrument incorporation in the handle. This has left a door open for the possibility to design innovative Cricket bats. These changes are however under the control of the MCC.

Figure 2-1 is the geometric parameters of a Cricket bat (Grant et al, 1996). Technical parameters such as apex position, apex height, ridge length, scoop thickness, etc, are used to describe different basic blade design features.

POS – Apex position, POSF – Ridge length, TH – Apex height, SCF – Scoop length, SCD – Scoop height, TOE – Toe thickness, W – Edge thickness, HR – Handle radius, STC – Centre spring

Figure 2-1 Geometric parameters of Cricket bat (After C. Grant & S. A. Nixon)

With regard to the dynamics of a Cricket bat, the geometric parameters, material properties of the blade and handle such as stiffness, density, governing the bat modal shapes, natural frequencies. In addition, an important feature in terms of Cricket bat performance is the Sweet Spot. The Sweet Spot is where the bat absorbs the least amount of vibrational energy during impact with the ball (Brooks, et al, 2006). Impacts on the Sweet Spot feel best to the hitter and results in imparting velocity to the ball are best. In more precise terms, the Sweet Spot is the impact location where the transfer of energy from the bat to the ball is maximal while the Chapter 2 Literature Review: Mechanical Vibration Control 10 transfer of energy to the hands is minimal (Noble, [I 9]). It is probably due to at this location where the rotation forces of the bat are completely balanced out when hitting a ball in this area. The Sweet Sport is often believed to be the same as Centre of Percussion (COP).

The Centre of Percussion is a point at a distance r below the centre of a object, for example a rod, does not move at the instant the rod is struck (Sears, 1958). The Centre of Percussion may or may not be the Sweet Spot depending on the pivot point chosen. The Centre of Percussion of a cricket bat is the point on a bat, where a perpendicular impact will produce translational and rotational forces which perfectly cancel each other out at some given pivot point while the Sweet Spot of a bat exists partly because bat vibrations are not excited significantly at that spot where the contact between bat and ball and partly because the spot is close to the COP.

The Sweet Spot is essentially an area of a sporting instrument that inflicts maximum velocity of the ball and minimum response to the holder of the instrument (Naruo et al, 1998). On the other hand, when Cricket ball hits outside the Sweet Spot area, less kinetic energy will return back to the ball and the lost energy will remain in the Cricket bat and become vibration wave travelling back and forth. The holy grail of good Cricket bat design is that it should have a Sweet Spot area that is as large as possible.

From the literature of Cricket bat research, researchers mainly focus on improving the dynamic performance of Cricket bats. C. Grant and S. A. Nixon (Grant et al, 1996) used Finite Element Analysis (FEA) to optimise the blade geometric parameters and C. Grant (Grant, 1998) suggested that handle stiffness be strengthened with composite material in order to raise the 3rd flexural vibration mode out of the excitation spectrum, that is, the 3rd mode is not excited during ball/bat impact, results in the 3rd vibration mode being removed completely, leaving only the 1st and 2nd modes to have significant impact on the batsman. R. Brooks and S. Knowles, et al (Knowles et al, 1996) compared impact characteristics and vibrational response between traditional wooden Cricket bats and full composite Cricket bats through impact tests as well as computational analysis. Their findings implied that the composite bat could have higher Coefficient Of Restitution (COR = ball output speed/ball input speed) which means the ball can be bounced back at faster speed. In addition, they implied that the contact time of bat/ball might be optimally tuned with the composite bat, to match the time it takes for the bat to deflect and return to its original position (Brooks et al, 1997). The energy of vibration in the bat can then be returned to the ball giving it a high exit velocity. This contact time matches one of the vibration frequencies of the bat. In similar studies with FEA, J. M. T. Penrose, et al (Penrose et al, 1998), S. John and Z. B. Li (John et Chapter 2 Literature Review: Mechanical Vibration Control 11 al, 2002) had similar conclusions that the bat/ball contact time (or dwell time) appears approximately constant and dictated by the local Hertzian parameters. The ball exit velocity might be increased by optimal tuning of the frequencies of the 2nd or 3rd modes to give a time period of half a cycle approximately equal to the time of contact. They also believed that the 1st flexure mode dominates the impact. These researchers have given an insight and valuable contribution to designing better Cricket bats in future. However, there is little information regarding to the attenuation or reduction of the vibration excited in the Cricket bat. The study of removing or controlling the post-impact vibration in Cricket bats is still in its infancy. The goal of this study is to explore the post-impact vibration control of the Cricket bats. In the next few sections, traditional vibration control methods as well as the emerging new vibration control with Smart material components will be discussed.

2.2 Traditional mechanical vibration control techniques

The collision of two objects will generate vibration in the objects. The impact of the collision excites the vibrations of the natural frequencies of the objects. The natural frequencies, which are decided by the geometry size and material properties of the object, are one of the most important characteristics of the object. Each excited natural frequency results in a single frequency vibration. The vibration of the object is the combination of every single natural frequency vibration of the object. Theoretically, the excited vibration will exist forever if there is no energy dissipation in the object, such as internal damping. Therefore, the mechanical vibration control is to control vibration energy transmission and dissipation. The basic traditional mechanical vibration control techniques are vibration isolation, vibration absorption and vibration damping. They are discussed in following sections respectively.

2.2.1 Vibration isolation

Vibration isolation is achieved by using isolators to reduce the transmission of vibratory force or motion from one mechanical structure to another. The isolator interconnects the source system in which the vibration is generated and the receive system in which the vibration is to be prevented.

The Figure 2-2 shows the block diagram of two mechanical structures interconnected by an isolator where the foundation is considered as rigid and the mass is not considered

(Dimarogonas, 1996). In the figure, m represents mass, ‘ FS ’ and ‘ FT ’ are the input disturbing force and output transmitted force of the isolator respectively; ‘ X S ’ and ‘ X T ’ are the input disturbing motion and output transmitted motion of the isolator respectively. Chapter 2 Literature Review: Mechanical Vibration Control 12 The isolator must be strong enough to hold the source system and the receive system with as smaller dynamic stiffness as possible (Mead, 2000). A single-degree-of-freedom (SDOF) isolator can be modelled as a massless spring with a damper (dashpot).

Fs XT

Source m Receiver m

x

Isolator Isolator

Xs FT

Receiver Foundation Source Foundation

(a) Excited by force acting on mass (b) Excited by foundation motion (Force isolation) (Motion isolation)

Figure 2-2 Mechanical structures interconnected by isolation system (Tse, 1978)

With a SDOF isolator, the transmissibility (TF ), which is used to describe the effectiveness of a vibration isolation system (Dimarogonas, 1996) can be obtained as that in Equation (2.1)

(see Appendix 1). The smaller of TF , the better effectiveness of the isolation system.

2 FT ( jω) 1+[2ζ (ω /ωn )] XT ( jω) TF ( jω) = = 2 2 2 = = TX ( jω) . (2.1) Fs ( jω) [1−(ω /ωn ) ] +[2ζ (ω /ωn )] X s ( jω) where

ω represents frequency, ωn and ζ are the natural frequency and damping ratio of the isolator respectively.

2.2.2 Vibration absorption

Vibration absorption is another traditional method used to reduce vibration. It is achieved by attaching auxiliary mass systems to vibrating systems by springs and damping devices to control the amplitude of the vibration systems. These auxiliary mass systems are known as vibration absorbers or sometimes referred as a tuned mass damper. The vibration absorber Chapter 2 Literature Review: Mechanical Vibration Control 13 absorbs vibration energy by transferring the energy from base structure to absorber. Depending on applications, vibration absorbers fall into two categories. If an absorber has a little damping factor, it is called a dynamic vibration absorber or undamped vibration absorber. Due to the lack of damping, the undamped dynamic vibration absorber has a very high level of vibration itself and is easy to cause fatigue failures. Therefore, it is often necessary to add damping to the absorber (Dimarogonas, 1996). The absorber with damping factor is called a damped vibration absorber or auxiliary mass damper (Harris et al, 2002).

Absorbers are most effective when placed in positions of high displacement or anti-nodes of structures. The absorbers reduce vibrations by absorbing the vibration energy through its mass deflection. The absorber spring must be stiff enough to withstand large forces and deflections.

2 Since absorbers are designed to meet ωa =ωs or ma =ka /ωs , high stiffness ka leads to large

ma, especially for low natural frequencies. This will make the absorber system heavier. Also, the mass-spring structure is not suitable for some applications which have space and location constraints. Details of analytical modelling of vibration absorbers for the SDOF structure can be found in Appendix 1.

2.2.3 Vibration damping

The third traditional passive vibration control technique is vibration damping. Vibration damping refers to a mechanism to dissipate mechanical energy extracted from a vibrating system into another form, usually heat. The vibration damping can be classified in two types: material damping or viscous damping and structural damping or system damping (Ruzicka, 1959). Material damping is the damping inherent in the material to dissipate kinetic and strain energy into heat. Structural (system) damping is the damping caused by discontinuity of mechanical structures, such as at joint, interface, etc.

2.2.3.1 Viscouselastic damping

There are three major types of material used for material damping: (1) viscoelastic materials (VEM), (2) structural metals and non-metals, (3) surface coatings (Harris et al, 2002).

A viscoelastic material such as rubber, plastics and polymers, exhibit characteristics of both viscous fluid and elastic solid. Like elastic materials, viscoelastic materials return to their original shape after being stressed but at lower speed due to viscous damping force. The stress response of viscoelastic materials obeys the superposition principle. The deformation of viscoelastic materials at any instant is a summation of the deformation produced by history stress and the deformation produced by current stress. Viscoelastic materials are sometimes Chapter 2 Literature Review: Mechanical Vibration Control 14 called ‘materials with memory’. Usually, viscoelastic materials can dissipate much more energy per cycle of deformation than steel and aluminium. For conventional structural materials, the energy dissipation per unit-volume per cycle is very small compared to viscoelastic materials.

Generally, the mathematical description of viscoelastic materials is very complicated. However, the simplified version of viscoelastic materials can be modelled by a solid elastic model, that is a spring (with stiffness k) and a viscous element that is a dashpot (with damping coefficient c). Three different models of a viscoelastic material are given in Figure 2-3 (Sun, 1995).

c k c k c 1

F F F F F F k k 2 (a) Maxwell model (b) Voigt model (c) Standard linear model

Figure 2-3 Visco-elastic material (VEM) models (Sun, 1995)

The viscoelastic material modulus property is a complex value,

Er = E1 + iE2 . where

E1 –– Elastic modulus of elasticity which is stress divided by that component of strain in phase with the stress. E1 indicates elastic stiffness property.

E2 –– Loss modulus of elasticity which is stress divided by that component of strain

o 90 out of phase with the strain. E2 indicates the damping energy losses.

The damping property of viscoelastic materials is:

E Loss factor η = loss tangent = tanφ = 2 . E1 where φ is the phase angle that indicates cyclic strain vector, lagging behind cyclic stress vector, during sinusoidal loading of a viscoelastic material.

Viscoelastic materials have been used to enhance the damping in a structure in three different ways: free-layer damping treatment, constrained-layer or sandwich-layer damping treatment and tuned viscoelastic damper. Figure 2-4 shows the three damping treatments (Rao, 2001). Chapter 2 Literature Review: Mechanical Vibration Control 15

Mass

Structure Structure Structure

(a) Free-layer (b) Constrained-layer (c) Tuned damper

–– Damping material –– Constraining layer material

Figure 2-4 Viscoelastic damping treatments (Rao, 2001)

For free-layer damping treatment (as shown in Figure 2-4(a)) the damping material is either sprayed or bonded using a pressure-sensitive adhesive on the surfaces of structure that are primarily vibrating in bending mode. As these surfaces bend, the viscoelastic materials on the surfaces are deformed cyclically in tension-compression. The hysteresis loop of the cyclic stress and strain dissipates the energy. The degree of damping is limited by thickness and weight restrictions. Implicit in this method is that the shear deformations can be ignored. The effect of this damping treatment is because of the large difference in flexural rigidity of the base structure and the viscoelastic material. Therefore, little energy is dissipated from the viscoelastic material. This type of surface will generally not enhance vibration damping of the structure effectively.

For constrained-layer damping treatment (depicted in Figure 2-4(b)), the flexural modulus of the constraining layer is comparable to that of the base structure. During bending, the viscoelastic material is forced to deform in shear due to the excessive difference of the moduli between the viscoelastic material, the base structure, and the constraining layer. Therefore, a considerable amount of energy is dissipated through shear deformation.

The tuned viscoelastic damper (TVD) (illustrated in Figure 2-4(c)) is similar to a dynamic absorber, except that a viscoelastic material is added to dissipate energy. Properly tuned TVD can reduce an unwanted resonance or a narrow band of vibration frequencies.

The viscoelastic material damping treatment is an easy way to reduce vibration but has limited effect especially at low frequency. Usually the stiffness at some part of structures are reduced after viscoelastic materials are embedded. This results in overall structures been weakened. Also, the viscoelastic materials are sensitive to temperature variation results in detuning the resonant frequency.

Chapter 2 Literature Review: Mechanical Vibration Control 16 2.2.3.2 Structural damping

In complex structures, there is energy dissipation during vibration due to Coulomb friction in contact surfaces, riveted joints, and so on. It is not possible to generally predict the damping of a built-up structure. It is extremely difficult to estimate the energy transmitting and exchanging at joints or interfaces of a complex structure because of the uncertainty about their contacts. Testing is the only practical way for determining the damping of most large complex structures. The methods of estimating damping ratio of a SDOF system in frequency domain and time domain can be found in Appendix 2. The methods also can be extended to a multi- degree-of-freedom (MDOF) system.

2.3 Non-conventional vibration control with Smart material components

Vibration control with Smart material is to incorporate Smart material as sensors and/or actuators and electronic system to a structure in a proper way that turns the structure into a Smart structure.

2.3.1 Smart material and structure

Smart systems trace their origin to a field of research that envisioned devices and materials that could mimic human muscular and nervous systems. The essential idea is to produce non- biological systems that will achieve the optimum functionality observed in biological systems through emulation of their adaptive capabilities and integrated design (Akhras, [I 2]).

A Smart structure can be defined as a system which has embedded sensors, actuators and control system, is capable of sensing a stimulus, responding to it, and reverting to its original state after the stimulus is removed (is not damaged). The Smart structure can be used to changing the shape of the structure in a controlled manner, or alternatively, changing the mass, stiffness, or energy dissipation characteristics of the structure in a controlled manner to alter the dynamic response characteristics, such as natural frequencies, mode shapes, settling time of the structure.

Smart materials are the materials that, firstly, have the capability to respond to stimuli and environmental changes, and secondly, to activate their functions according to these changes. The common Smart materials are piezoelectric ceramics, shape memory alloys (SMAs), electro-rheological fluids (ER fluids), magneto-rheological fluids (MR fluids), magneto- restrictives, etc. Following are brief descriptions of the common Smart materials and their applications. Chapter 2 Literature Review: Mechanical Vibration Control 17 Piezoelectric materials –– Piezoelectric materials are widely used Smart materials which can be classified into three types: piezoelectric ceramics, piezoelectric polymers and piezoelectric composites. The piezoelectric ceramics are solid materials which generate charges in response to a mechanical deformation, or alternatively they develop mechanical deformation when subjected to an electrical field. These can be employed either as actuators or sensors in the development of Smart structures. The shape of piezoelectric ceramic materials can be in disc, washer, tube, block, stack, etc. The piezoelectric polymers are thin and soft polymer sheets which usually are used as sensors. The piezoelectric composites are something between ceramic and polymer which can be either used as actuators or sensors.

The piezoelectric materials have a broadband frequency response that can be used in high frequency applications, and they are able to generate large actuation force. For example, stack ceramic actuators with displacements of 100 μm can generate blocked forces equal to several tonnes (Suleman, 2001). Details about piezoelectric materials will be discussed in the next section.

Shape memory alloys (SMA) –– A Shape Memory Alloy (SMA) is an alloy which, when deformed (in the martensitic phase) with the external stresses removed and then heated, will regain its original "memory" shape (in the austenitic phase). Shape memory alloys can thus transform thermal energy directly to mechanical work (Vessonen, 2002). The materials have ability to remember their original shape that have given name of shape memory alloy. Shape memory alloy materials can have shape in washer, tube, strip, wire, etc. In the process, they generate an actuating force. The most popular one is a nickel-titanium alloy known as Nitinol, which has a corrosion resistance similar to stainless steel. However, shape-memory alloys can respond only as quickly as the temperature can shift, which only can be used for very low frequency applications (few Hz).

Electro-rheological fluids (ER fluids) –– ER fluids are suspensions of micron-sized hydrophilic particles suspended in suitable hydrophobic carrier liquids, that undergo significant instanteous reversible changes in their mechanical properties, such as their mass distribution, and energy-dissipation characteristics, when subjected to electric fields (Gandhi et al, 1992). In the neutral state the particles are uniformly distributed throughout the fluid. When the electric field is applied, the relatively high dielectric constant of the particles captures this electric field, redistributes the charge densities within the particles, and cause them to stick together in fibrils. The presence of these fibrils considerably modifies the viscous properties of the fluid (by several orders of magnitude). Chapter 2 Literature Review: Mechanical Vibration Control 18 Magneto-rheological fluids (MR fluids) –– MR fluid is a functional material consisting of carrier liquid, ferro-magnetic particles, plus some additives, which are included to work against article-liquid separation and wear of sealings, for example (Vessonen, 2002). When subjected to a magnetic field, the fluid greatly increases its viscosity, to the point of becoming a viscolelastic solid. The fluid can reversibly and instantaneously change from a free-flowing liquid to a semi-solid with controllable yield strength when exposed to a magnetic field.

ER and MR fluids have applications in areas such as tuneable dampers, vibration isolation systems, and in exercise equipment as clutches, brakes, and resistance controls.

Magnetostrictive materials –– Magnetostriction is observed in a substance when it strains upon application of a magnetic field. Conversely a field is generated when the material is stressed; this field is, however, proportional to the material’s rate of strain (Vessonen, 2002). All magnetic materials exhibit varying degrees of magnetostriction, however only some materials exhibit sufficient magnetostriction for practical use. Early magnetostrictive devices were used Nickel as the magnetostrictive material and this principle found use in sonar applications. Materials such as Terfenol-D are similar to piezoelectric but these respond to magnetic rather than electric fields. Magnetostrictive materials are used to build actuators and sensors. They are typically used in low frequency, high power sonar transducers motors and hydraulic actuators.

2.3.2 Piezoelectric materials

“Piezoelectric effect” was first discovered by Pierre Curie and his brother Jacques in 1880. When a crystal, such as Quartz and Rochelle salt, is under pressure (piezo means pressure in Greek) an electrical potential appears across some of its faces (direct effect). When an electrical field is applied, mechanical deformation of the crystal occurs (converse effect). The essence of “piezoelectric effect” is the relation between electric polarization and mechanical strain in crystals. “Direct effect” indicates that the electric polarization is produced by a mechanical strain in crystals which is proportional to the strain. “Converse effect” indicates that the crystals become strained when electrically polarized and the strain produced is proportional to the polarizing field. The strains produced are about 0.1% at a field of about 10kV/cm. The strain (or displacement) vs. electrical field is nearly linear in the operating region, but the strain saturates at very high fields (Suleman, 2001).

The “Piezoelectric effect” happens to materials with a crystalline structure but with no centre of symmetry. These materials are called non-centrosymmetric materials. A single crystal with a non-centrosymmetric structure has anisotropic characteristics, that is, the properties of the Chapter 2 Literature Review: Mechanical Vibration Control 19 material differ according to the direction of measurement. In material having piezoelectric properties, ions can be moved more easily along some crystal axes than others. Pressure in certain directions results in a displacement of ions. When pressure is released, the ions return to original positions. If a piece of piezoelectric material is heated above a certain temperature, it will lose its piezoelectric properties. This temperature is known as Curie temperature. After cooling below the Curie temperature, the piezoelectric material will not regain its piezoelectric properties.

Figure 2-5 shows a crystal structure of a traditional ceramic (APC, [I 4]). In the Figure 2-5(a), the crystal is under the condition that the temperature is above Curie temperature, the crystal has a centrosymmetric structure and has no piezoelectric characteristics, exhibits a cubic lattice, symmetric arrangement of positive and negative charges with no dipole moment. In the Figure 2-5(b), the crystal is under the condition that the temperature is below Curie temperature, the crystal has a non-centrosymmetric structure and exhibits tetragonal lattice with a net electric dipole moment.

(a) Cubic lattice (b) Tetragonal lattice

–– Pb2+/Ba2+, large divalent metal ion –– O2-, oxygen –– Ti4+/Zr4+, smaller trivalent metal ion

Figure 2-5 Crystal structure of a traditional ceramic (APC, [I 4])

The ceramic material is composed of many randomly oriented crystals or grains, each having one or a few domains, the regions where dipoles are aligned in the same direction. With the dipoles randomly oriented, the ceramic material is isotropic and does not exhibit the “Piezoelectric effect”. By applying electrodes and a strong DC electric field to the ceramic material, a process known as poling, the dipoles will tend to align themselves parallel to the field, so that the ceramic material will have a permanent polarization. Once the ceramic material is polarized, it becomes piezoelectric ceramic with anisotropic. The direction of the Chapter 2 Literature Review: Mechanical Vibration Control 20 poling field is identified as the 3-direction. In the plane perpendicular to the 3-axis, the piezoelectric ceramic is non-directional. The poling process permanently changes the dimensions of the ceramic. The dimensions between the poling electrodes increase and the dimensions parallel to the electrodes decrease.

Figure 2-6 shows the ion polarization direction within a piezoelectric ceramic before and after poling. Before poling, the dipoles are in random polarization, while after poling, the dipoles are locked into a configuration of near alignment, and have a permanent polarization and elongation. The non-centrosymmetric crystalline structure provides a net electric dipole moment within the ceramic unit cell (APC, [I 4]).

(a) before poling (b) during poling (c) after poling

Figure 2-6 Polarizing (poling) of a piezoelectric ceramic (APC, [I 4])

The most common ceramic material to possess the “Piezoelectric Effect” is PZT (Lead-

Zirconium-Titanium – PbZrTiO3). The traditional piezoelectric ceramic is a mass of perovskite crystals, each consisting of a small, tetravalent metal ion usually titanium or zirconium in a lattice of larger, divalent metal ions, usually lead or barium and oxygen O2 ions.

There are some important properties to describe piezoelectric materials. They are electromechanical coupling coefficient K, charge constant d, and voltage constant g, which reflect the fundamental relationships between mechanical energy and electrical energy in piezoceramic.

Electromechanical coupling coefficient K measures the efficiency of converting mechanical energy to electrical energy or vice versa. It is defined as (Crawley, 1987),

mechanical energy convertered to electrical energy K 2 = . (2.2a) input mechanical energy Chapter 2 Literature Review: Mechanical Vibration Control 21

electrical energy convertered to mechanical energy K 2 = . (2.2b) input electrical energy

Electromechanical coupling coefficient K value is mode dependent. For vibration mode ‘33’, coupling coefficient is K33 , which is much larger than that for other directions. For PZT, K33 is about 0.5 ~ 0.7 while K31 and K32 are about 1/3 of the K33 .

π f p ⎛π Δf ⎞ ⋅ ⋅ tan⎜ ⋅ ⎟ 2 fs ⎝ 2 fs ⎠ K31 = K32 = . (2.3) π f p ⎛π Δf ⎞ 1+ ⋅ ⋅ tan⎜ ⋅ ⎟ 2 fs ⎝ 2 fs ⎠

π f ⎛π Δf ⎞ K = ⋅ s ⋅ tan⎜ ⋅ ⎟ . (2.4) 33 ⎜ ⎟ 2 f p ⎝ 2 f p ⎠

where f p and fs are parallel and series resonance frequencies of piezoceramic respectively,

Δf = f p − fs . d constant measures the efficiency of generating strain for a given electrical field that indicates a material’s suitability for actuator applications. It is defined as:

strain developed ⎛ short circuit charge density ⎞ d = ⎜= ⎟ . (2.5) applied field ⎝ applied stress ⎠ g constant measures the efficiency of generating electrical charge for a given stress that indicates a material’s suitability for sensor applications. It is defined as:

open circuit field ⎛ strain developed ⎞ g = ⎜= ⎟ . (2.6) applied stress ⎝ applied charge density ⎠

High d is desirable for piezoceramic to develop motion or vibration. High g is desirable for piezoceramic to generate voltage in response to a mechanical stress. In piezoceramic, there are many K, d, and g donated as, and, which one been taken, depends on the directions of applied electrical field, displacement, stress and strain.

2.3.2.1 Orientation and notation

Because a piezoelectric ceramic is anisotropic, physical constants relate to both the direction of the applied mechanical or electric force and the directions perpendicular to the applied force. Consequently, each constant generally has two subscripts that indicate the directions of the two related quantities, such as stress (force on the ceramic element/surface area of the Chapter 2 Literature Review: Mechanical Vibration Control 22 element) and strain (change in length of element/original length of element) for elasticity. The direction of positive polarization usually is made to coincide with the z-axis of a rectangular system of x, y, and z axes as shown in Figure 2-7. Direction x, y, or z is represented by the subscript 1, 2, or 3, respectively, and shear about one of these axes is represented by the subscript 4, 5, or 6, respectively (APC, [I 4]).

3

Poling direction 6

z y x 2 5 4

1 Figure 2-7 Coordinator axes for piezoceramic

The most frequently used piezoelectric constants are the charge constant (dij), the voltage constant (gij), the permittivity or dielectric constant (εij ), the elastic compliance constant (sii), and the electromechanical coupling coefficient (Kij). The first subscript gives the direction of the electrical parameters — the voltage applied or the charge produced. The second subscript gives direction of the mechanical parameters — stress or strain. Superscripts indicate a constant mechanical or electrical boundary condition. For examples,

d33 represents electrodes are perpendicular to ‘3’ axis and stress is applied in ‘3’ axis.

g31 represents electrodes are perpendicular to ‘3’ axis and strain developed is in ‘1’ axis.

K15 represents electrodes are perpendicular to ‘1’ axis and stress or strain is in shear form around ‘2’ axis.

T ε11 represents permittivity for dielectric displacement and electric field in direction 1 (perpendicular to direction in which ceramic element is polarized), under constant stress.

T E T The relationship among dij , gij , and Kij are: dij = Kij ⋅ ε ii ⋅ s jj and gij = dij /εii ,

T th E where ε ii is the free permittivity of material under constant stress in i direction and s jj is the elastic compliance under constant electrical field in jth direction. Chapter 2 Literature Review: Mechanical Vibration Control 23

2.3.2.2 Piezoelectric Constitutive Equations

The piezoelectric material has the linear constitutive relation (IEEE 1987). There are four types of fundamental piezoelectric relations which are: h-form, d-form, g-form, and e-form listed in the Table 2-1.

Table 2-1 Types of fundamental piezoelectric relation (Ikeda, 1996)

Independent Type Piezoelectric relation Form Variable

⎧T = c D S − hD

S, D Extensive ⎨ S h-form ⎩E = − hS + β D

⎧S = s ET + dE

T, E Intensive ⎨ T d-form ⎩D = dT + ε E

⎧S = s DT + gD

T, D Mixed ⎨ T g-form ⎩E = − gT + β D

⎧T = c E S − eE

S, E Mixed ⎨ S e-form ⎩D =eS + ε E

In the table, variables S and D are named as extensive variable while T and E are named as intensive variable. When the variables used in the relations are solely extensive or intensive, the type is called as extensive or intensive type respectively; otherwise the type is called as mixed type.

The matrix representation of d-form is

⎡S⎤ ⎡sE d ⎤ ⎡T⎤ ⎢ ⎥ = ⎢ T ⎥ ⋅ ⎢ ⎥ . (2.7) ⎣D⎦ ⎣dt ε ⎦ ⎣E⎦ where

D is an electrical displacement vector (charge/area), E is an electrical field vector in a material (volts/meter), S is a material engineering strains vector, T is a material stresses Chapter 2 Literature Review: Mechanical Vibration Control 24

vector (force/area), d is piezoelectric charge-constant matrix, εT is dielectric constant matrix under constant stress, and sE is piezoelectric compliance matrix under constant electrical field.

T ⎡D1 ⎤ ⎡E1 ⎤ ⎡ 0 0 0 0 d15 0⎤ ⎡ε1 0 0 ⎤ ⎢ ⎥ D = ⎢D ⎥ , E = ⎢E ⎥ , d = ⎢ 0 0 0 d 0 0⎥ , εT = 0 ε T 0 , ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 15 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ T ⎥ ⎣D3 ⎦ ⎣E3 ⎦ ⎣d31 d31 d33 0 0 0⎦ ⎣ 0 0 ε3 ⎦

E E E ⎡S1 ⎤ ⎡T1 ⎤ ⎡s11 s12 s13 0 0 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ E E E ⎥ S T s s s 0 0 0 ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 12 11 13 ⎥ ⎢ E E E ⎥ ⎢S3 ⎥ ⎢T3 ⎥ E s13 s13 s33 0 0 0 S = ⎢ ⎥ , T = ⎢ ⎥ and s = ⎢ E ⎥ . ⎢S 4 ⎥ ⎢T4 ⎥ ⎢ 0 0 0 s55 0 0 ⎥ ⎢S ⎥ ⎢T ⎥ ⎢ 0 0 0 0 s E 0 ⎥ 5 5 ⎢ 55 ⎥ ⎢ ⎥ ⎢ ⎥ E ⎣⎢S6 ⎦⎥ ⎣⎢T6 ⎦⎥ ⎣⎢ 0 0 0 0 0 s66 ⎦⎥

where

T E Subscript, ( )t donates the conventional matrix transpose. Superscript ( ) and ( ) on material properties denote that the values are measured under constant stress and constant electrical field (e.g., short circuit) respectively. The “3” direction is associated with the direction of poling and the material is approximately isotropic in other two directions. Note that due to the structure symmetry of the piezoelectric materials, the material properties are

identical in the “1” and “2” directions, i.e. d31 = d32 and d15 = d24 (shear) (Hagood et al, 1991).

The “Piezoelectric Effect” indicates that there is energy conversion from one form to another, happening during the course, which is the fundamental of piezoelectric vibration control. Piezoelectric material can be used either as sensor or actuator. It should be pointed out that usually the mechanical effect of the piezoelectric on structure dynamics is ignored when it is used as a sensor, likewise, the electrical dynamics of the piezoelectric are ignored when it is used as an actuator. Figure 2-8 shows that charges are generated in a piezoceramic and accumulated on the surfaces of electrodes when it is under different kinds of stresses.

Compression stress Extension stress Shear stress

Poling direction Electrodes Chapter 2 Literature Review: Mechanical Vibration Control 25

(a) Longitudinal effect (b) Transverse effect (c) Shear effect

Figure 2-8 Piezoceramic under stress

Although the magnitudes of piezoelectric voltages, movements, or forces are small, and often require amplification, piezoelectric materials have been adapted to an impressive range of applications. Piezoelectric materials have different types. Piezoelectric polymers or soft piezoelectric materials like PVDF (PolyVinylidine DiFluoride – CH2CF2) are light and soft. They are very sensitive to stress and suitable for sensors. Piezoelectric ceramics or hard piezoelectric materials like PZT are very stiff. They can produce high strokes and are suitable for actuators. The piezoelectric composites or flexible piezoelectric materials such as active fibre composites (AFCs) are light, flexible as well as quite stiff under the influence of electrical field. It can be embedded in other materials acting as either a sensor or actuator. AFCs have been developed at MIT to address the most critical properties restricting the use of piezoelectric ceramics: brittleness, limited robustness, limited conformability and in-plane actuation isotropy. AFCs consist of small semi-continuous piezoelectric fibers uniaxially aligned in a polymer matrix (epoxy) between two interdigitated electrodes. Each fiber is surrounded and protected by the polymer matrix (ACI Technology, [I 1]).

2.3.2.3 Modelling of piezoelectric transducer

A piezoelectric transducer can be modelled approximately by an ideal current source in paralleling with inherent inductor, capacitors, and resistor as shown in Figure 2-9(a). Because the capacitor Cs is much smaller than the capacitor Cp, at low frequency, the impedance of Cp can be treated as an open circuit, thus the R-L-Cs branch can be taken away from the model. The model in (a) then can be simplified to that as shown in (b) which is equivalent to that as shown in (c).

Chapter 2 Literature Review: Mechanical Vibration Control 26 Figure 2-9 Simplified equivalent circuit for a piezoelectric transducer

In the Figure 2-9, Ip is an ideal current source and Vp is an ideal voltage source generated due to an external stress, A is the surface area of the piezoelectric transducer where the electrodes are attached. They are expressed as,

I p = Di dA = dijT j dA . (2.8) ∫A ∫A

1 Vp = − I dt = − dijT j dA dt . (2.9) ∫ p ∫∫A C p

The Equation (2.9) shows that the voltage generated is proportional to the product of the d- constant and the external stress.

2.3.3 Passive piezoelectric vibration shunt control

Passive piezoelectric shunt control on a vibration structure is achieved by connecting an electrical shunt circuit in parallel with a piezoceramic (PZT) that is firmly bonded onto the surface of the vibration structure where there is induced vibration strain presented. The mechanical strain energy is then converted into electrical energy and absorbed/damped by the piezoelectric shunt circuit. The shunt circuits as shown in Figure 2-10 are the classification of most common piezoelectric vibration shunt circuits (Lesieutre, 1998).

PZT PZT PZT PZT L SW Cp Cp Cp Cp R C

V V Vp V p p R p Zs

(a) Resistive (b) Inductive (c) Capacitive (d) Switching

Figure 2-10 Basic classification of passive piezoelectric vibration shunt (Lesieutre, 1998)

Figure 2-10(a) is a resistive shunt circuit that dissipates vibration energy through Joule heat. It is analogous to viscoelastic damping but the materials are much stiffer and less prone to temperature variation. It is a wideband control but is not very effective.

Figure 2-10(b) is an inductive shunt circuit that can be tuned to a specific vibration frequency, usually the natural frequency of the structure. The value of inductor is determined by equation Chapter 2 Literature Review: Mechanical Vibration Control 27

2 L = 1/(ω n C p ) , where ωn is the vibration frequency to be controlled. The purpose of the inductor is to cancel the impedance of the inherent capacitor Cp, minimizing the output impedance of the PZT thus maximizing the output of vibration energy from it. It is analogous to a mechanical TMD or Tuned Vibration Absorber (TVA). When the shunt circuit is tuned accurately to the vibration frequency, it can shunt the vibration energy much more efficiently compared to the resistive shunt, however, it is narrowband control and will lose performance dramatically when the targeted frequency is shifted away from the frequency of the shunt circuit.

Figure 2-10(c) is a capacitive shunt circuit. It changes the effective stiffness of the piezoceramic. It is analogous to a TVA.

Figure 2-10(d) is a switching shunt circuit. Zs represents the shunt circuit which can be one of the shunt circuits in (a) ~ (c). The switch is to control the rate of energy transmission.

It was Forward (1979) who first suggested the possibility of using piezoelectric elements with passive electrical shunt for vibration damping and control. He demonstrated vibration damping on optical structures with a passive parallel R-L-C resonant shunt circuit experimentally, which showed that substantial vibration amplitude was reduced.

Severely years later, Uchino et al (1988) experimentally demonstrated resistive piezoelectric shunt and noted a significant vibration damping when the external resistance was changed (Uchino et al, 1988).

At the same time, Hagood and von Flotow (1991) presented a detailed analysis of passive piezoelectric shunt control. They also demonstrated vibration reduction on cantilever beams with resistive and inductive shunt networks experimentally. Their general model is depicted in

SU Figure 2-11(a), where Zi (s) denotes the impedance of the shunt network. Figure 2-11(b) is

ME its equivalent circuit, where Z jj (s) is the mechanical impedance of the piezoelectric for

th EL uniaxial loading in j direction and Zi (s) is the electrical impedance of the piezoelectric with shunt circuit.

SU EL [1/(sC p )]⋅ Zi (s) Zi (s) = SU . (2.10) [1/(sC p )] + Zi (s)

The non-dimensional electrical impedance of the piezoelectric with shunt circuit is,

Z EL Z EL (s) Z SU (s) Z EL (s) = i = i = i . (2.11) i Z 1/(sC ) Z SU (s) +1/(sC ) C p p i p Chapter 2 Literature Review: Mechanical Vibration Control 28 The non-dimensional mechanical impedance of the piezoelectric is,

2 2 ME 1− kij 1− kij 1 Z jj (s) = 2 EL = EL 2 EL = EL . (2.12) 1− kij Zi (s) 1− Zi (s) + (1− kij )Zi (s) EL 1− Zi (s) Zi (s) − 2 1− kij

Ii +

T SU i Zi (s) Ti Vi S i −

(a) Piezoelectric shunted by electrical impedance

F I i + + ) v

ZME(s) C SU EL jj p Zi (s) Zi (s) Vi Velocity ( − − 1:n

(b) Equivalent circuit of piezoelectric shunt of (a)

Figure 2-11 Passive piezoelectric vibration damping (Hagood et al, 1991)

Figure 2-12 illustrates a simple series inductive piezoelectric shunt configuration.

Ii + L Vi R − Figure 2-12 Inductive piezoelectric shunt (Hagood et al, 1991)

where

SU Zi (,s) = Ls + R leads to Chapter 2 Literature Review: Mechanical Vibration Control 29

T EL (Ls + R)/ sC p EL Ls + R Zi (s) = T and Zi (s) = T . (2.13) Ls + R +1/(sC p ) Ls + R +1/(sC p )

2 2 ME 1− kij 1− kij 1 Z jj (s) = 2 EL = EL 2 EL = EL 1− kij Zi (s) 1− Zi (s) + (1− kij )Zi (s) EL 1− Zi (s) Zi (s) + 2 1− kij 1 = T 2 T LC p s + RCp s T 2 T 1− T 2 T LC p s + RCp s LC p s + RCp s +1 T 2 T + 2 LC p s + RCp s +1 1− kij

2 T 2 T S 2 S 2 (1− kij )(LC p s + RC p s +1) LC p s + RC p s + (1− kij ) = T 2 T 2 = S 2 S . (2.14) (LC p s + RC p s)(1− kij ) +1 LC p s + RC p s +1

RSP RSP ME EL Assume Z jj (s) and Zi (s) are Z jj (s) and Zi (s) at resonance,

2 2 RSP 1− kij 2 δ Z jj (s) = 2 RSP =1− kij 2 2 2 . (2.15) 1− kij Zi (s) γ +δ rγ +δ where

S S 2 T E E S E ωe = 1/ LC p , C p = (1− kij )C p , δ = ωe /ωn , γ = s /ωn , r = RC pωn .

They presented the mechanical model and impedance model of Resonant Shunted Piezoelectric (RSP) and mechanical vibration absorber or Proof Mass Damper (PMD) of a SDOF system in Figure 2-13 and Figure 2-14 respectively.

Zmass

k Z spring m ZRSP RSP F v F v − +

(a) Mechanical model of RSP (b) Impedance model of RSP

Figure 2-13 Model comparison of resonant shunted piezoelectric (RSP) of a SDOF system (Hagood et al, 1991)

Chapter 2 Literature Review: Mechanical Vibration Control 30

Zmass

PMD Zspring k

m ZPMD

F v v F + − (a) Mechanical model of PMD (b) Impedance model of PMD

Figure 2-14 Model comparison of proof mass damper (PMD) of a SDOF system (Hagood et al, 1991)

In the impedance model, the mass, spring and RSP (PMD) impedances are connected in parallel, where Zmass (s) = ms and Z spring (s) = k / s . The overall mechanical impedances for the RSP and PMD of the SDOF systems can be expressed as, respectively,

F(s) Z(s) = = ms + k / s + Z RSP (s) . (2.16) v(s) jj

F(s) Z(s) = = ms + k / s + Z PMD (s) . (2.17) v(s) jj where

F(s) and v(s) are force and velocity respectively.

The modelling and derivation of mathematic equations by Hagood et al made a very important contribution to the passive piezoelectric vibration control which has been the foundation of much research work in this field since.

Edberg et al (1992) employed passive vibration shunt to large space truss structures. They developed simulated (synthetic) inductor (known as gyrator) that is an electronic circuit made of operational amplifiers, passive capacitors and resistors to have the characteristics of the passive inductor. The gyrator-based inductor has solved the problem of the large inductor value required for low frequency vibration shunt which is far beyond the range that a passive inductor can normally reach.

Hollkamp (1993) proposed multimode vibration suppression using a single PZT by paralleling several series R-L-C branches, except the very first one being a series R-L circuit. Chapter 2 Literature Review: Mechanical Vibration Control 31 Each R-L-C branch introduces a resonant frequency which targets a particular mode to be suppressed. Additional R-L-C branches can be added to the shunt circuit to increase the amount of electrical resonance. The shunt design for the case of n electrical resonance is shown in Figure 2-15.

Figure 2-15 Multi-mode vibration shunt using multiple R-L-C shunt branches (Hollkamp, 1993)

The non-dimensional electrical impedance can be calculated as,

T EL C p s Zi (s) = N . (2.18) ∑ 1 k =0 Zk (s) where

T Z0 (s) = 1/(sC p ) ,

Z1(s) = L1s + R1 ,

Z2 (s) = L2s + R2 +1/(sC2 ) , …, and

Z N (s) = LN s + RN +1/(sCN ) .

However, it is difficult to tune the circuit since each individual branch of the shunt cannot be considered as an individual single mode shunt. Tuning one electrical resonance to suppress a particular mode will detune the rest of the circuit. He also developed a self-tuning piezoelectric vibration absorber using a motorized potentiometer with a RMS searching algorithm.

C. L. Davis and G. A. Lesieutre (Davis et al, 1995) used the strain energy approach to predict the structural damping added by the use of a resistive piezoelectric shunt. By choosing various Chapter 2 Literature Review: Mechanical Vibration Control 32 materials and structural parameters, a designer may add damping to structural vibration modes.

Wang et al (1996) developed energy-based parametric control using semi-active piezoelectric networks for structural vibration suppression. By analysing the base structure, the parameters of shunt networks are on-line controlled to minimize the energy flowing into the main structure.

Law et al (1996) described the resistive piezoelectric shunt as equivalent to a spring-dashpot model as shown in Figure 2-16.

I

PZT

C p R0 ζ eq k V p

(a) Resistive piezoelectric shunt (b) Equivalent sprint-dashpot model

Figure 2-16 Equivalent model for a piezoelectric material with a load resistor (Law et al, 1996)

In Figure 2-16(b), k and ζ eq are the stiffness and equivalent damping ratio of a piezoelectric ceramic respectively.

EA 1 A k = = . (2.19) L sij L

1 (ωC )2 R K 2 ζ = p 0 ij . (2.20) eq 4π ⋅ f EC 1+ (ωC R )2 ωC R r p p 0 1− p 0 K 2 2 ij 1+ (ωC p R0 )

Where E is the Young’s modulus, sij is the compliance, A is the surface area, L is the length,

E fr is the resonant frequency when the piezoelectric ceramic is short circuited, and Kij is the electromechanical coupling coefficient of the piezoelectric ceramic respectively. Chapter 2 Literature Review: Mechanical Vibration Control 33 When the optimal value of the load resistor matches that of the internal impedance of the piezoelectric ceramic, that is, R0 = 1/ωC p , maximum damping occurs. The maximum damping ratio is,

2 max 1 Kij ζ eq = . (2.21) 4πf E R 2 r 0 1− (1/ 2)Kij

Wu (Wu, 1996, 1998) proposed piezoelectric shunt with parallel R-L circuit as shown in Figure 2-17 for structural damping on cantilever beam. He also employed a “blocking” circuit in series with each parallel R-L shunt circuit for multiple-mode shunt damping of structural vibration, using single PZT which is depicted in Figure 2-18, where Cp denotes the capacitance of the PZT, Ci, and Li, R’i (i = 1, …, n) are denote external capacitor, inductor and resistor.

Ii +

R L Vi −

Figure 2-17 Inductor-resistor (R-L) parallel shunt

The circuit in Figure 2-18 has n branches and shunts n natural frequencies simultaneously. The circuit is designed so that each branch accepts only one particular natural frequency and the rest of the natural frequencies (n-1) are blocked. The blocking circuit is in open circuit when a natural frequency is equal to its resonant frequency, otherwise it is approximately treated as a short circuit. The ith resonant frequency of a blocking circuit is determined by

ωi =1/ LiCi .

(n-1) blocking circuits in series

+

L2 ω2 C2 L3 ω3 C3 Ln−1 ωn−1 Cn−1

L ω C L ω4 C L ω C 3 3 3 4 4 1 1 1

Vi C p

L ω C L ω C L ω C n n n 1 1 1 n−2 n−2 n−2

L1 ' R1 ' L2 ' R2 ' Ln ' Rn ' − Chapter 2 Literature Review: Mechanical Vibration Control 34

Figure 2-18 Multi-mode vibration shunt using series blocking circuits (Wu, 1996)

Recently, Fleming et al (2003) proposed multiple parallel single mode shunt circuits in series using a single PZT. Each parallel single mode shunt circuit also had a “blocking” circuit in parallel as shown in Figure 2-19, which could reduce the inductance requirement (Fleming et al, 2003).

Figure 2-19 Multi-mode vibration shunt using parallel blocking circuits (Fleming et al, 2003)

ˆ ˆ In the figure, Li and Ci are the components of making blocking circuits which act with the same function as the blocking circuit as Wu proposed previously. However, since inductors ˆ Li and Li are in parallel, the overall inductor can be replaced by a smaller inductor Li ',

L ⋅ Lˆ L ' = i i . The values of capacitor and inductor are selected to satisfy the condition i ˆ Li + Li

ˆ ˆ 2 ˆ LiC p = LiCi = 1/ω i , where ω i is the vibration frequency to be shunted. If Ci = 10 ⋅C p and ˆ Li = Li /10 , then Li ' = Li /11 . The inductor value is only one eleventh of that used in a single R-L shunt circuit. Since the capacitance of PZT is very small (usually in the magnitude of 10-8 ~ 10-7 Farads), for low frequency vibration shunt, the value of inductor needed will be at 100s, even 1000s Henrys. It is not possible for a passive inductor to have such a large value, even if for a synthetic inductor. Therefore, the proposed design in Figure 2-19 is useful, though it may lose some shunt performance, compared with the single mode inductive shunt since the Chapter 2 Literature Review: Mechanical Vibration Control 35 blocking circuits cannot be in open/short circuit ideally. However compared with Wu’s proposal, the circuit is much simpler and requires much less electrical components.

In addition to using piezoceramic materials for vibration shunt control mentioned in the previous literature review, another type of piezoelectric materials, active fibre composites (AFCs) are very promising for vibration control. As mentioned before, AFCs originated from the work started at MIT by Bent and Hagood in 1992. AFCs are constructed using piezofibres embedded in an epoxy resin matrix and coated with Kapton skin. Unlike piezoceramic, piezofibres are flexible and soft. However, after the piezofibres being embedded in epoxy resin matrix to form the AFCs, the stiffness of the AFCs increase significantly. AFCs overcome the weakness of monolithic piezoceramic being brittle, they are tough yet flexible, conformable, and have great potential towards integration in complex structures. AFCs operate in the longitudinal mode and thus have significantly higher d and g constants (namely

d33 and g33 ).

Belloli et al (2004) for the first time successfully implemented AFCs in an inductive shunt circuit. They claimed a 17 dB vibration reduction on an aluminium plate during the experiment they carried out. The result is close to that with piezoceramic, however the inductor of the shunt circuit required was a lot higher than that required with piezoceramic due to much smaller inherent capacitance of the AFCs.

Head Sport has employed active fibre composite to produce Smart tennis rackets (Head_a, [I 5]) and Smart skis (Head_b, [I 6]) with energy harvesting actuators made from PZT fibre. In the case of the tennis racquet, piezoelectric composites embedded in the throat of the racquet convert mechanical vibrations caused by the ball impact into electrical energy. The electrical energy is then processed by an electronic circuit located in the racquet’s handle, and within milliseconds the electronic circuit sends the electrical energy back to the piezoelectric composites to reduce the racquet’s vibration. The piezoelectric composites increase power and control authority by reducing the racquet’s vibration. The Smart ski uses similar technology. Piezoelectric fibre composite technology that recycles non-usable, waste energy to perform a function, free of outside control or source of power or batteries. As the ski vibrates during a turn, or on ice, the processed electrical energy of that chatter is harnessed by the piezoelectric fibre composites in the structure of the ski, which then rolls the edge of the ski back into the snow. These self-powered Smart products actively damp the vibration created during a ball strike or edge chatter from a ski turn and use the energy to create electrical force to control the shape of the racket or ski, using PZT fibre’s electrical and Chapter 2 Literature Review: Mechanical Vibration Control 36 mechanical properties to counteract the forces. This active structural control adds up to 15% more power to a ball hit and about 6% more functional edge on a ski (ACI Technology, [I 1]).

2.3.4 Active piezoelectric vibration control

Another vibration treatment is known as Active Vibration Control (AVC) as discussed by Beranek et al (1992). With active vibration control, mechanical force is induced in the piezoelectric materials as the result of electrical energy being applied on the piezoelectric materials to counteract or cancel the existing vibration. Ideally, the force generated by the piezoelectric actuator has the same amplitude as the vibration but in anti-phase. Thus the vibration can be cancelled out known as vibration cancellation. The active vibration controls usually require a control loop or system and a control law.

A typical active vibration control system is an integration of mechanical and electronic components, combining or embedding with signal processing devices such as computer microprocessor/DSP chip acting as a nerve system. Vibration signals detected by sensors from a structure are fed into the nerve system to be analyzed and processed. Based on the sensors’ input, the nerve system generates some kind of control signals with specifically designed control algorithms. The control signals are then amplified to drive electromechanical devices, such as piezoelectric actuators, for producing mechanical force to be applied to the structure.

Figure 2-20 is a block diagram of a basic active control system (Preumont, 1997). When the sensors/actuators as well as electronic components are highly distributed and integrated in the structure, capable to perform functions like self-sensing, self-actuation, self-diagnosing etc. then the structure can be considered as a Smart or Intelligent structure.

High degree of integration

Sensors Structure Actuators

PZT SMA PVDF Control PZT Fiber optics System Magnetostrictive Chapter 2 Literature Review: Mechanical Vibration Control 37

Figure 2-20 Block diagram of a basic active control system (Smart structure) (Preumont, 1997)

In the wake of rapid advances in material technology, high speed electronic devices and sophisticated digital signal processing techniques have made many previously “unthinkable” become possible, and triggered vast interest in active vibration control among researchers and engineers. In the past few decades, various methods of active vibration control have been developed. The common control strategies of the active controls can be classified under two major categories: feedback control such as active damping, and feedforward control such as adaptive filtering. In terms of signal type, the control can also be classified into two types: analogue control and digital control. Analogue control deals with continuous time domain signal while digital control deals with desecrate signal data, usually the samples of analogue signal.

Balas (1978) developed a feedback controller using a position sensor, force actuator and state variable technique for active control of an n modes flexible structure. The flexible structure was a simple supported beam with dynamics modelled on the Euler-Bernoulli partial differential equation. One actuator and one sensor (at different locations) were used to control three modes of the beam effectively. Spillover, due to residual (uncontrolled) modes, was eliminated by prefiltering the sensor outputs to avoid feedback to the controller resulting in instability in close loop.

Bailey et al (1985) designed an active vibration damper for a cantilever beam using a distributed parameter actuator that was a piezoelectric polymer (PVDF) and distributed parameter control theory of Lyapunov’s second method. With the method, if the angular velocity of the tip of the beam is known, all modes of the beam can be controlled simultaneously. This avoids any structural problems with uncontrolled modes. Test results showed that using a constant-gain controller, more than a factor of two was increased in the modal damping with a feedback voltage limit of 200 Vrms. With the same voltage limit, the constant-amplitude controller achieved the same damping as the constant-gain controller for large vibrations, but increased the modal loss factor by more than an order of magnitude to at least 0.04 for small vibration levels.

Crawley (1986) presented the analytical and experimental development of piezoelectric actuators as an element of Intelligent structures that are with highly distributed actuators, Chapter 2 Literature Review: Mechanical Vibration Control 38 sensors, and processing networks. The segmented piezoelectric actuators are either bonded to an elastic substructure or embedded in a laminated composite, which can excite the structure independently. The distributed actuators are used for dynamic vibration and modal shape controlling, predicting the behaviour of flexible structures.

McKinnell (1989) applied active forces to a vibrating beam structure to cancel propagating and evanescent vibration waves by superposition of flexural waves (rather than the superposition of an infinite set of natural modes of vibration) to prevent those waves from carrying the effects of an external disturbance. A system was designed to detect bending waves as they pass a sensor at one point and then inject vibration waves by an active force at another point further downstream, resulting in the effect of the bending waves being cancelled by the injected vibration waves. However, the dispersive nature of the waves made implementation difficult. The difficulties were overcome by including the effects of waves generated by the active forces, and detected at the controller input, in the control system synthesis.

Hagood et al (1990) derived an analytical model of dynamic coupling between a structure and an electrical network through a piezoelectric actuator, using Hamilton’s principle with a Rayleigh-Ritz formulation. Several state space models of driving piezoelectric actuators have been developed, which include direct driven using voltage driven electrodes, charge driven electrodes, mixed voltage-charge driven electrodes, and indirect driven through a passive electrical shunt network.

Tzou et al (1990) used a finite element analysis approach to evaluate the dynamic performance of a structure with self-sensing and control capability. Performance of a thin plate model coupled with the distributed piezoelectric sensor/actuator was studied and evaluated in a free vibration analysis by the FEA method. Two control algorithms, constant- gain negative velocity feedback and constant-amplitude negative velocity feedback, were tested. Both feedback control algorithms showed that in generally damping ratio increases when the feedback voltage/gain increases.

Bai et al (1996) proposed for suppressing the small amplitude random vibrations in flexible beams by using PVDF as sensor, PZT as actuator and Linear Quadratic Gaussian (LQG) as the controller algorithm. The controller is implemented on the basis of a floating-point digital signal processor. The experimental results obtained by using the active vibration control system showed significant attenuation of white noise disturbance in the frequency range of approximately 100 – 800 Hz. Chapter 2 Literature Review: Mechanical Vibration Control 39 Lay et al (1998) designed and implemented a controller by using modal feedback law with collocated arrangement of sensor/actuator pairs for damping vibration in cantilever beams. The benefits of a collocated sensor/actuator first can avoid spillover, which could destabilize the system, and second is possible to implement feedback control system without a reference signal. The controller’s characteristics are determined with the help of pole-zero placement.

Dosch et al (1992) developed an active vibration control technique which allows a single piece of piezoelectric material to concurrently sense and actuate in a closed loop system. Such a self-sensing actuator is truly collocated and suitable for Smart /Intelligent structures. The experimental results showed that active damping vibration in a cantilever beam using a rate feedback control with a single piezoceramic, the settling time of first and second vibration modes were reduced by more than an order of magnitude.

Lee (1990) proposed distributed modal sensors/actuators using PVDF lamina in contrast to conventional point sensors/actuators for active vibration control. Unlike most piezoelectric materials that are fully covered with electrodes, the surfaces of the PVDF laminated modal sensors/actuators were only partially covered with electrodes. The shapes of electrodes were designed to match the specific mode shapes of a structure, thus can measure/excite these modes by those modal sensors/actuators. The concept behind the modal sensors/actuators was to use the charge collecting phenomenon of the surface electrode of piezoelectric materials to perform the modal filtering in the spatial domain rather in the time domain. The surface electrodes and the polarization profile of the distributed sensor/actuator can be used to integrate sensor/actuator signals in the spatial domain. Therefore less intensive signal processing is required. Because of the parallel processing nature presented in these modal sensors/actuators, no phase-lag was generated in a control loop adopting these devices. The sensor signal is proportional to the time derivative of the modal coordinate which is easily used for velocity feedback control. The spillover phenomenon was not presented in such integrated modal sensor/actuator pair. It was shown that critical damping could be achieved using such PVDF sensor/actuator combination.

One of major problems associated with active vibration control is spillover. Flexible structures are modelled as distributed parameter systems and have very large number (theoretically an infinite number) of vibration modes. Some of them are beyond the bandwidth of controllers since sensors/actuators, as well as electronic circuits, have limited bandwidth. These uncontrolled high order modes may destabilize control systems. This is the well-known phenomenon of spillover (Balas 1978). Goh and Caughey (1985) introduced Positive Position Feedback (PPF) control. Since the frequency responses of PPF controllers Chapter 2 Literature Review: Mechanical Vibration Control 40 have sharp roll-off, making closed loop control systems robust and insensitive to spillover, these therefore guarantee the stability of the control systems. The PPF methodology is to couple a highly damped electronic filter to a lightly damped structure through electro- mechanical coupling to suppress the vibration of the structure. With PPF, the structure position information is passed to the electronic filter first then the output of the filter, multiplied by a gain factor, and is positively fed back to the structure (physically, the output signal of the filter after being amplified drives actuators attached on the structure). PPF behaves much like an electronic vibration absorber for structures. The analysis of the PPF control can be found in Appendix 3.

2.3.5 Power system for active piezoelectric vibration control

The piezoelectric actuators have been demonstrated which they can produce high actuation force, have broad frequency bandwidth and fast transient response to stimuli. They are very attractive for many applications. However, some applications such as satellites use DC battery power or self-powered systems and have limitations in power supply due to size and weight constraints. On the other hand, the piezoelectric actuators are required to be driven by high AC voltage (not necessary high current) in order to generate large force and/or displacement. Figure 2-21 shows the force generation vs. displacement of a piezo actuator (displacement 30 µm, stiffness 200 N/µm) at various operating voltages (PI Ceramic, [I 9]). For example, a 333 V voltage can generate 2000 N maximum force with zero displacement or 10 μm maximum displacement with zero force. 333 V is very high voltage though 10 µm displacement is not considerable large. Therefore it is very important to design efficient power suppliers and power amplifiers for driving piezoelectric actuators. Preferably they can be powered by the energy harvested from mechanical energy by piezoelectric materials which become self- powered actuation systems.

Chapter 2 Literature Review: Mechanical Vibration Control 41

Figure 2-21 Voltage vs. force generation vs. displacement of a piezo actuator (PI Ceramic, [I 9])

E According to Equation (2.7), strain produced by a piezoelectric actuator is S = d t E + s T .

When external stress T is zero, the strain S is the product of the charge constant dt

(Meter/Volt) and electrical field E (Volt/Meter). Since d t is a relatively small constant, at a magnitude of 10 −10 , thus it requires a very high voltage to produce a noticeable geometric increment in certain direction, say 1 micrometer, by the piezoelectric actuator. The signals from sensors and controllers are weak signals, and therefore, require amplification before driving the actuator. The amplifiers for driving piezoelectric actuators should not only have a high voltage gain, but also have a large dynamic voltage range and be able to withstand high voltage. The amplifier supplying power to the active vibration control system decides the efficiency and capability of the system. High power amplifiers are usually bulky and consume a lot of power itself. Design of high efficiency, small size power amplifiers is part of ongoing research by many researchers.

Luan et al (1998) developed a DC-DC high frequency switching amplifier for driving Smart material actuators. They used average current mode control techniques to increase the current Chapter 2 Literature Review: Mechanical Vibration Control 42 loop gain at low frequency, which is low due to the reactive (capacitive) load of Smart material actuators.

Janocha et al (1998) presented a switching amplifier for piezoelectric actuators which could transfer the portion of energy exactly necessary to achieve desired output value at the load in every switching cycle. This approach could avoid unnecessary switching cycles which merely heat the transducer and semiconductor components.

Carazo et al (2001) designed a DC-AC piezoelectric-based power supply for driving piezoelectric actuators. The novelty of this approach is using a smaller, lighter, and EMI-free piezoelectric transformer (PT) Transoner® instead of a larger, heavier and EMI-noisy electromagnetic transformer, to convert the low DC voltage of a conventional battery to a high AC voltage for driving piezoelectric actuators. They implemented a circuit that is capable of converting a 24 V DC battery voltage for driving a multiplayer piezoelectric actuator at 100 Vpp within the range of 10 Hz to 500 Hz.

2.3.6 Advantages and disadvantages of passive and active piezoelectric vibration controls

The passive and active piezoelectric vibration controls have their own advantages and disadvantages. The advantages of Passive Vibration Control (PVC) are that the control system is relatively simple, little or no external power is required and is guaranteed to be stable. Therefore, the control system can be made in a compact, portable size and is easier to be embedded in a structure where electronic energy is expensive. However, this method has limited effectiveness particularly when the volume of a structure to be controlled or the magnitude of vibration in the structure is large. The percentage of vibration attenuation in the structure is in inverse proportion to the volume of the structure or the level of vibration in the structure. The effectiveness of the passive vibration control for low frequency vibration is also less than that for high frequency vibration probably due to less energy conversion in unit time. The passive vibration control usually targets at a particular frequency in a narrow band. The control effectiveness is quite sensitive to frequency shifts caused by structure or environment variations, boundary condition change etc. In addition, the large inductor value required for the low frequency passive vibration control also tends to be a problem.

The advantages of the active vibration control are that the control system is robust and flexible, it has broadband effectiveness, capable of suppressing a large level of vibration, has high control authority, and the control effectiveness is less prone to condition variation. Besides, the active vibration control system can be implemented in high-speed DSP chips Chapter 2 Literature Review: Mechanical Vibration Control 43 using digital signal processing techniques with hardware and software, which makes modern advanced control strategies such as adaptive control and Intelligent control to be implemented in the control system possible, as well as allows the control system to be re-programmable resulting in time-saving and lowered costs during design and implementation. Nevertheless, the control system is much more complex, possibly unstable, requires sophisticated control algorithms, and especially requires a large external power source to drive the actuator. The last point is the major stumbling block to applying active vibration control in many applications, since the higher power required for driving actuators means more power consumption, and needs a larger, more complex power system. Many applications, like portable or remote applications which only use battery power are not suitable for applying active vibration control strategies unless some kind of techniques, such as self-powering, can be used to solve the energy problem.

Based on the assumption that the piezoelectric transformer can be used as a voltage step-up device as being suggested by Carazo et al (2001), a novel power amplifier designed by using amplitude modulation technique with piezoelectric transformer for driving the piezoelectric actuator has been proposed by author. The details of the power amplifier can be found in Appendix 4. Functional simulation with PSpice® software suggests that the design works correctly.

2.4 Chapter summary

In this chapter, first, a brief overview has been given of the history of the Cricket bat and the research being carried out on Cricket bats. An interesting idea suggested by C. Grant (Grant, 1998) is to raise the 3rd flexure mode of the Cricket bat out of the excitation spectrum of ball/bat impact by making the Cricket bat handle with composite materials to completely remove the 3rd mode vibration. Then the traditional vibration techniques including vibration isolation, vibration absorption and vibration damping have been discussed. Smart materials particularly piezoelectric materials, their function, properties, mechanical/electrical models, and constitutive equations have been presented. The “Piezoelectric effect” of the materials is the foundation of modern advanced piezoelectric vibration control. The literature review of piezoelectric vibration control has been carried out. It is generally classified as passive vibration control and active vibration control. The passive vibration control including resistive, inductive, capacitive, and switching vibration shunt has been discussed. Among these, the inductive vibration shunt has been found most effective. The passive control system is relatively simple, compact, low cost, creates less demand on electrical power and is Chapter 2 Literature Review: Mechanical Vibration Control 44 guaranteed to be stable, but it has limited effectiveness especially for large objects or high levels of vibration. The active vibration control, on the other hand, is more sophisticated, robust, effective and flexible but is complex and possibly unstable. The major concern is its high demand on electrical power which has limited its use on many portable and remote applications. In the chapter, the work by other researchers on electrical power systems for active piezoelectric vibration control has also been briefly mentioned. The piezoelectric transformer has been considered as a promising alternative to electromagnetic transformer for active piezoelectric vibration control by many researchers. Finally, the advantages and disadvantages of the passive and active piezoelectric vibration control methods have been summarised.

Some of the derivations related to the concepts introduced in this chapter are further elucidated in the Appendices. The analytical analyses of vibration isolation, vibration absorption and damping measurement of the SDOF structure are presented in Appendices 1 & 2. A PPF control example is given in Appendix 3. In Appendix 4, a novel power amplifier is proposed.

Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 45

Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams

In chapter 2, piezoelectric vibration control techniques have been reviewed. There are mainly two categories, one is the active control technique and the other is the passive control technique. For the active piezoelectric vibration control method, it normally requires hundreds of volts to drive the piezoelectric actuators. Although it is more effective than the passive piezoelectric vibration control method, there is a bottleneck problem for practical applications. It is difficult to produce such high voltage with a small electronic circuit at low cost, especially when battery power is used as the power supply.

Due to the deep conservatism in the rules of the Cricket games, there are many restrictions imposed on designing new Cricket bats. The rule only allows the designer to utilize and modify the handle of Cricket bats. Considering the factors of space and weight limitation, power supply issues, portability and cost concerns etc., the active control method seems unfeasible for the vibration control of the Cricket bat, at least for the time being. Therefore, the passive piezoelectric control method, namely using piezoelectric transducers in conjunction with an electrical network to damp the structural vibration becomes an option. The passive piezoelectric vibration control technique requires much less power consumption compared to that of the active piezoelectric vibration control technique. There are several types of passive piezoelectric vibration control configurations that have been briefly described in chapter 2. Among them, the most effective configuration is the resistor-inductor (R-L) shunt control. In terms of connection type between the electrical network and the piezoelectric transducer, it can be further classified as series R-L shunt control, where the R-L shunt circuit is connected in series to the piezoelectric transducer (Figure 2-15) and parallel R-L shunt control which the R-L shunt circuit is connected in parallel to the piezoelectric transducer (Figure 2-20). Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 46 In this chapter, the motion equation of a composite beam that consists of a cantilever beam bonded with a PZT patch using Hamilton’s principle and Galerkin’s method is derived in sections 1 & 2. In section 3, a series R-L and a parallel R-L shunt circuits are coupled into the motion equation respectively by means of the constitutive relation of piezoelectric material and Kirchhoff’s law to control the beam vibration. A numerical example of the parallel R-L piezoelectric vibration shunt control simulated with MATLAB® is presented in section 4.

3.1 Motion equation derivation of composite beam using energy method

The composite beam described in the study is a cantilever beam attached with a piezoelectric ceramic patch (PZT) depicted in Figure 3-1(a). Figure 3-1(b) is the cross-sectional area of the composite beam where the PZT patch is attached. More governing equations regarding beam modeling and modal analysis are derived in Appendix 5.

hp

hb + hp hb 2 b

(a) Composite beam (b) Cross-sectional area of the composite beam

Figure 3-1 Composite beam made of a cantilever beam with a PZT patch attached

3.1.1 Euler-Bernoulli beam model

The model for composite beams used here is based on the Euler-Bernoulli beam model. The Euler-Bernoulli model is a simplified beam model that assumes that the beam deflection is due to the bending moment (a constant moment along the beam) only, and neglects the effects of shear deformation and rotary inertia. It is a widely accepted engineering beam model and has good approximation to the accurate beam model. Following are some assumptions made during the motion equation derivation for the composite beam:

(1) The PZT is perfectly bonded. The PZT patch does not alter the structural dynamics of the cantilever beam.

(2) The piezoelectric ceramic is uniform in geometry size and the strain and stress of the piezoelectric ceramic along the thickness direction are linearly distributed. Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 47 (3) The electrical field is uniform along the surfaces of the piezoelectric ceramic, i.e. E(x,t) = E(t) .

(4) In regard to beam deformation, only bending deformation is considered, shear and rotation deformation are ignored (Euler-Bernoulli assumption).

(5) The beam model is linear and elastic. The material properties of the beam are homogeneous and isotropic. The beam density and the elastic modulus are considered to be constant. The stress and strain lie inside the linear elastic domain.

The geometry sizes of the cantilever beam and the PZT shown in Figure 3-1 are L × b × hb and (x2 − x1) × b × hp respectively. Based on the assumption that the PZT patch attached does not alter the cantilever beam’s structure a great deal, the composite beam can still be approximately considered as a uniform beam. Both beam and PZT are modelled using Euler- Bernoulli model separately. Figure 3-2 illustrates the notation of a beam under bending. In the figure, x-coordinate displacement includes displacement due to axial force (axial displacement) and displacement due to bending (transverse displacement).

M0 ML Deformed beam ∂w ∂x

w z x ∂w u z ∂x Undeformed beam

z Neutral axis (NA)

Figure 3-2 Beam deformation on x-z plane due to bending

3.1.2 Displacement, strain, stress and velocity relations

According to the theory of elementary strength of materials (Davis, 1995), the position vector of an arbitrary point P1(x, y, z) is P1 (x, y, z) = x ⋅ i + y ⋅ j + z ⋅ k . (3.1)

Assume point P2 (x, y, z) is the point P1(x, y, z) after deformation, u0 (x, z,t ) , v0 (y, z,t) ,

w0 (x, y,t) are the neutral axis displacements of the point along the x, y, and z coordinates respectively. The position vector of P2 (x, y, z) can be expressed as Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 48 P2 (x, y, z) = (x + u0 ) ⋅ i + ( y + v0 ) ⋅ j + (z + w0 ) ⋅ k . (3.2) where

∂w(x, y,t) u = u(x,t) − z , (3.3) 0 ∂x ∂w(x, y,t) v = v(y,t) − z , (3.4) 0 ∂y

w0 = w(x, y,t) . (3.5)

The displacement vector is the distance between points P1 and P2 , D(x, y, z) = P2 − P1 = u0 ⋅ i + v0 ⋅ j + w0 ⋅ k . (3.6)

For the uniform Euler-Bernoulli beam under pure bending, if there is deformation only on x-z plane, not on y-z plane which is shown in Figure 3-2, the neutral axis displacements are, respectively,

∂w(x,t) u = u(x,t) − z , (3.7) 0 ∂x

v0 = 0, (3.8)

w0 = w(x,t) . (3.9)

Therefore Equation (3.6) becomes

⎛ ∂w(x,t) ⎞ D(x, z,t) = ⎜u(x,t) − z ⎟ ⋅ i + w(x,t) ⋅ k . (3.10) ⎝ ∂x ⎠ where

∂w(x,t) − z is the horizontal displacement along the x-coordinate due to bending. ∂x

The axial strain along the x-coordinate denoted as ε11 is the differentiating of Equation (3.7) with respect to x, that is,

∂u ∂u ∂ 2 w ε = 0 = − z . (3.11) 11 ∂x ∂x ∂x 2

The axial stress along the x-coordinate denoted as σ11 is the strain times the Young’s modulus E, that is,

σ 11 = Eε11 . (3.12) Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 49 In addition, the velocity of the particle is the differentiating of its displacement with respect to time t. The velocity of the beam element on the x-z plane is the differentiating of Equation (3.10) ∂D(x, z,t) ⎛ ∂u(x,t) ∂ 2 w(x,t) ⎞ ∂w(x,t) ⎜ ⎟ ⋅ ⋅ Vc (x, z,t) = = ⎜ − z ⎟ i + k , ∂t ⎝ ∂t ∂x∂t ⎠ ∂t

The non-dimensional velocity is

2 2 ⎛ ∂u(x,t) ∂ 2w(x,t) ⎞ ⎛ ∂w(x,t) ⎞ ⎜ ⎟ Vc = ⎜ − z ⎟ + ⎜ ⎟ . (3.13) ⎝ ∂t ∂x∂t ⎠ ⎝ ∂t ⎠

3.1.3 Potential energy

The potential energy of the beam due to x-coordinate strain is

2 2 1 1 2 1 ⎛ ∂u ∂ w ⎞ U b = σ 11ε11dVb = Ebε11dVb = Eb ⎜ − z ⎟ dVb ∫∫∫ V ∫∫∫ V ∫∫∫ V ⎜ 2 ⎟ 2 b 2 b 2 b ⎝ ∂x ∂x ⎠

2 2 1 ⎪⎧⎛ ∂u ⎞ ∂u ∂2w ⎛ ∂2w⎞ ⎪⎫ = E − 2z + z2 ⎜ ⎟ dV b ⎨⎜ ⎟ 2 ⎜ 2 ⎟ ⎬ b 2 ∫∫∫ Vb ∂x ∂x ∂x ∂x ⎩⎪⎝ ⎠ ⎝ ⎠ ⎭⎪

⎧ 2 2 2 2 ⎫ L ⎡ ⎤ 1 ⎪ ⎛ ∂u ⎞ ∂u ∂ w 2 ⎛ ∂ w⎞ ⎪ = Eb ⎢⎜ ⎟ − 2z + z ⎜ ⎟ ⎥ dAb dx ∫∫∫ 0 ⎨ A 2 ⎜ 2 ⎟ ⎬ 2 b ⎢⎝ ∂x ⎠ ∂x ∂x ⎝ ∂x ⎠ ⎥ ⎩⎪ ⎣ ⎦ ⎭⎪

⎛ 2 2 2 2 ⎞ 1 L ⎛ ∂u ⎞ hb ∂u ∂ w ⎛ ∂ w⎞ = E ⎜ A − 2b⎛ zdz⎞ + ⎛ z 2dA ⎞⎜ ⎟ ⎟ dx b b ⎜ ⎟ ⎜ ⎟ 2 ⎜ b ⎟⎜ 2 ⎟ 2 ∫∫∫ 0 ⎜ ∂x ⎝∫ 0 ⎠ ∂x ∂x ⎝ Ab ⎠ ∂x ⎟ ⎝ ⎝ ⎠ ⎝ ⎠ ⎠

2 2 2 2 1 L ⎪⎧ ⎛ ∂u ⎞ ∂u ∂ w ⎛ ∂ w ⎞ ⎪⎫ = ⎨Eb Ab ⎜ ⎟ − Ebhb Ab + Eb Ib ⎜ ⎟ ⎬ dx . (3.14) 2 ∫ 0 ∂x ∂x ∂x2 ⎜ ∂x2 ⎟ ⎩⎪ ⎝ ⎠ ⎝ ⎠ ⎭⎪ where

2 Ib = z dz is the area moment of inertia about the neutral axis, u(x,t) and w(x,t) are ∫∫ A b the axial displacement and the transverse displacement of beam respectively, z is the distance from neutral axis, Ab = bhb is the cross sectional area of beam, Eb is the Young’s modulus of beam, and dVb = dAbdx = bdzdx .

The potential energy of the PZT due to the x-coordinate strain is: Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 50

1 1 2 U p = (σ 11ε11 + D3E3 )dVp = [E pε11 + D3 (−h31ε11 + β33D3 )] dVp ∫∫∫ V ∫∫∫ V 2 p 2 p

2 1 ⎪⎧ ⎛ ∂u ∂ 2 w ⎞ ⎡ ⎛ ∂u ∂ 2w ⎞ ⎤⎪⎫ = E ⎜ − z ⎟ + D − h ⎜ − z ⎟ + β D dV ⎨ p ⎜ 2 ⎟ 3 ⎢ 31⎜ 2 ⎟ 33 3 ⎥⎬ p 2 ∫∫∫ V p ∂x ∂x ∂x ∂x ⎩⎪ ⎝ ⎠ ⎣ ⎝ ⎠ ⎦⎭⎪

2 2 2 2 x ⎧ h +h 1 2 ⎪ ⎛ ∂u ⎞ ⎛ b p ⎞ ∂u ∂ w ⎛ 2 ⎞⎛ ∂ w ⎞ = E p Ap ⎜ ⎟ − 2E pb⎜ zdz ⎟ + E p ⎜ z dAp ⎟⎜ ⎟ ∫∫ x ⎨ h 2 ∫∫ A ⎜ 2 ⎟ 2 1 ⎩⎪ ⎝ ∂x ⎠ ⎝ b ⎠ ∂x ∂x ⎝ p ⎠⎝ ∂x ⎠

2 h +h ∂u ⎛ b p ⎞ ∂ w 2 ⎫ − h31 Ap D3 + h31D3b⎜ zdz ⎟ + β 33 Ap D3 dx ∫ h 2 ⎬ ∂x ⎝ b ⎠ ∂x ⎭

⎧ 2 2 2 2 1 x2 ⎪ ⎛ ∂u ⎞ ∂u ∂ w ⎛ ∂ w ⎞ = E A − E h b(2h + h ) + E I ⎜ ⎟ ⎨ p p ⎜ ⎟ p p b p 2 p p ⎜ 2 ⎟ 2 ∫ x1 ∂x ∂x ∂x ∂x ⎩⎪ ⎝ ⎠ ⎝ ⎠

2 ∂u hp ∂ w 2 ⎫ − h31 Ap D3 + h31 D3bh p (hb + ) 2 + β 33 Ap D3 ⎬dx ∂x 2 ∂x ⎭

⎧ 2 2 2 2 1 x2 ⎪ ⎛ ∂u ⎞ ⎛ ∂ w ⎞ ∂u ∂ w = E A + E I ⎜ ⎟ − E A (2h + h ) ⎨ p p ⎜ ⎟ p p ⎜ 2 ⎟ p p b p 2 2 ∫ x1 ∂x ∂x ∂x ∂x ⎩⎪ ⎝ ⎠ ⎝ ⎠

2 ∂u hp ∂ w 2 ⎫ −h31Ap D3 + h31Ap (hb + )D3 2 + Apβ33D3 ⎬dx. (3.15) ∂x 2 ∂x ⎭ where,

2 I p = z dz is the area moment of inertia about the neutral axis, A = bh is the cross ∫∫ A p p p sectional area of PZT, E3 = −h31ε11 + β33D3 (using the h-form piezoelectric relation), and

dVp = dApdx = bdzdx .

The total potential energy of the composite beam is U = U b +U p . If, for simplicity, only ∂u strain caused by bending is considered, the strain caused by axial force is not considered, ∂x ∂u that is = 0 , then total strain can be simplified as: ∂x

2 2 1 L ⎛ ∂ w ⎞ U = U b +U p = Eb Ib ⎜ ⎟ dx ∫ 0 ⎜ 2 ⎟ 2 ⎝ ∂x ⎠

2 2 2 L ⎧ ⎫ 1 ⎪ ⎛ ∂ w⎞ hp ∂ w 2 ⎪ + ⎨Ep I p ⎜ ⎟ + h31Ap (hb + )D3 + Ap β33D3 ⎬[H(x − x1) − H(x − x2 )]dx . (3.16) 2 ∫ 0 ⎜ ∂x2 ⎟ 2 ∂x2 ⎩⎪ ⎝ ⎠ ⎭⎪ where, Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 51

⎧0 for x < x0 H (x) is the Heaviside function, H (x − x0 ) = ⎨ . ⎩1 for x ≥ x0

3.1.4 Kinetic energy

The kinetic energy of the beam is the integration of the product of the mass and velocity of beam elements within the beam volume. From Equation (3.13), the kinetic energy of beam can be calculated as

2 2 2 1 2 1 ⎪⎧⎛ ∂u(x,t) ∂ w(x,t) ⎞ ⎛ ∂w(x,t) ⎞ ⎪⎫ ⎜ ⎟ Tb = Vc dρ b = ρ b ⎨⎜ − z ⎟ + ⎜ ⎟ ⎬dVb 2 ∫∫∫ Vb 2 ∫∫∫ Vb ∂t ∂x∂t ∂t ⎩⎪⎝ ⎠ ⎝ ⎠ ⎭⎪

2 2 2 1 ⎪⎧⎛ ∂u ⎞ ∂u ∂ 2 w ⎛ ∂ 2 w ⎞ ⎛ ∂w ⎞ ⎪⎫ 2 ⎜ ⎟ = ρ b ⎨⎜ ⎟ − 2z + z ⎜ ⎟ + ⎜ ⎟ ⎬dVb 2 ∫∫∫ Vb ∂t ∂t ∂x∂t ∂x∂t ∂t ⎩⎪⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎭⎪

⎧ 2 2 2 2 2 ⎫ 1 L ⎪ ⎛ ∂u ⎞ hb ⎛ ∂u ∂ w ⎞ ⎛ ∂ w ⎞ ⎛ ∂w ⎞ ⎪ ⎜ ⎟ 2 ⎜ ⎟ = ρb ⎨Ab ⎜ ⎟ − 2b zdz ⎜ ⎟ + z dA ⎜ ⎟ + Ab ⎜ ⎟ ⎬dx 2 ∫∫∫ 0 ∂t ∫ 0 ∂t ∂x∂t Ab ∂x∂t ∂t ⎩⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎭⎪

2 2 2 2 2 1 L ⎡ ⎛ ∂u ⎞ ∂u ∂ w ⎛ ∂ w ⎞ ⎛ ∂w ⎞ ⎤ = ρb ⎢ Ab ⎜ ⎟ − hb Ab + I b ⎜ ⎟ + Ab ⎜ ⎟ ⎥ dx . (3.17) 2 ∫ 0 ∂t ∂t ∂x∂t ⎜ ∂x∂t ⎟ ∂t ⎣⎢ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦⎥ where ∂u ∂w ∂ 2w ρ is the density of beam, and are the axial velocities, is the angular b ∂t ∂t ∂x∂t velocity which is caused by the rotation of beam element. Let J b = ρb Ib denotes the mass moment of inertia about neutral axis per unit length of the beam.

Similarly, the kinetic energy of PZT is:

2 2 2 1 2 1 ⎪⎧⎛ ∂u(x,t) ∂ w(x,t) ⎞ ⎛ ∂w(x,t) ⎞ ⎪⎫ ⎜ ⎟ T p = Vc dρ p = ρ p ⎨⎜ − z ⎟ + ⎜ ⎟ ⎬dV p 2 ∫∫∫ V p 2 ∫∫∫ V p ∂t ∂x∂t ∂t ⎩⎪⎝ ⎠ ⎝ ⎠ ⎭⎪

2 2 2 1 ⎪⎧⎛ ∂u ⎞ ∂u ∂ 2 w ⎛ ∂ 2 w ⎞ ⎛ ∂w ⎞ ⎪⎫ 2 ⎜ ⎟ = ρ p ⎨⎜ ⎟ − 2z + z ⎜ ⎟ + ⎜ ⎟ ⎬dV p 2 ∫∫∫ V p ∂t ∂t ∂x∂t ∂x∂t ∂t ⎩⎪⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎭⎪

⎧ 2 2 2 2 2 ⎫ 1 x2 ⎪ ⎛ ∂u ⎞ hb + h p ⎛ ∂u ∂ w ⎞ ⎛ ∂ w ⎞ ⎛ ∂w ⎞ ⎪ ⎜ ⎟ 2 ⎜ ⎟ = ρ p ⎨ Ap ⎜ ⎟ − 2b zdz ⎜ ⎟ + z dA ⎜ ⎟ + Ap ⎜ ⎟ ⎬dx 2 ∫∫∫ x1 ∂t ∫ hb ∂t ∂x∂t A p ∂x∂t ∂t ⎩⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎭⎪

⎡ 2 2 2 2 2 ⎤ 1 x2 ⎛ ∂u ⎞ ∂u ∂ w ⎛ ∂ w ⎞ ⎛ ∂w⎞ ⎜ ⎟ = ρ p ⎢Ap ⎜ ⎟ −(2hb + hp )Ap + I p ⎜ ⎟ + Ap ⎜ ⎟ ⎥ [H(x − x1) − H(x − x2 )]dx. 2 ∫ x1 ∂t ∂t ∂x∂t ∂x∂t ∂t ⎣⎢ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦⎥ (3.18) Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 52

∂u If axial force and beam rotation are ignored, that is axial velocity and the angular velocity ∂t ∂ 2w are disregarded, the total kinetic energy of the composite beam can be simplified as: ∂x∂t

2 2 1 L ⎛ ∂w⎞ 1 L ⎛ ∂w⎞ T = Tb + Tp = ρb Ab ⎜ ⎟ dx + ρ p Ap ⎜ ⎟ []H(x − x1 ) − H(x − x2 ) dx . (3.19) ∫ 0 ∫ 0 2 ⎝ ∂t ⎠ 2 ⎝ ∂t ⎠

3.1.5 Hamilton’s principle

According to the Hamilton’s principle (Fung et al, 2001),

t2 t2 δ (T −U + W) dt = (δT −δU +δWp +δWf )dt = 0 . (3.20) ∫ t ∫ t 1 1

where δW represents virtual work, δWp is the virtual work due to electrical displacement,

δWf is the virtual work due to applied force.

The virtual work due to electrical displacement generated across the electrodes of PZT is:

x2 ⎛ hb +hp ⎞ δWp = E3δD3dVp = ⎜b dz⎟E3δD3dx ∫ V ∫∫ x h p 1 ⎝ b ⎠ L = Ap β33D3[H(x − x1) − H(x − x2 )]δD3dx . (3.21) ∫ 0

Since Equation (3.21) deals with the electrical field only, it does not take into account electromechanical effect, therefore, E3 = β33D3 .

The virtual work due to external force is:

L δWf = f (x,t)δw(x,t)dx . (3.22) ∫ 0 where f (x,t) is the applied force.

Substituting Equations (3.14, 3.19, 3.21, 3.22) into Equation (3.20) obtains:

2 2 t2 1 ⎪⎧ L ⎛ ∂w ⎞ L ⎛ ∂w ⎞ δ ρb Ab ⎜ ⎟ dx + δ ρ p Ap ⎜ ⎟ []H (x − x1) − H (x − x2 ) dx ∫∫ t ⎨ ∫ 0 0 1 2 ⎩⎪ ⎝ ∂t ⎠ ⎝ ∂t ⎠

2 2 2 2 L ⎛ ∂ w ⎞ L ⎛ ∂ w ⎞ −δ Eb Ib ⎜ ⎟ dx −δ E p I p ⎜ ⎟ []H (x − x1) − H (x − x2 ) dx ∫ 0 ⎜ 2 ⎟ ∫ 0 ⎜ 2 ⎟ ⎝ ∂ x ⎠ ⎝ ∂x ⎠

2 L ⎛ hp ∂ w 2 ⎞ −δ Ap ⎜ h31 (hb + )D3 + β 33 D3 ⎟ [H (x − x1 ) − H (x − x2 )] dx ∫ 0 ⎜ 2 ∂x 2 ⎟ ⎝ ⎠ Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 53

LL + Ap β 33 D3[H (x − x1 ) − H (x − x2 )]δD3dx + δ f (x,t)wdx} dt = 0 . (3.23) ∫∫ 00

The terms in Equation (3.23) can be derived respectively as below (see Appendix 6)

(A)

2 2 1 t2 L ⎛ ∂w⎞ t2 L ∂ w δ ρb Ab ⎜ ⎟ dxdt = − ρb Ab δw dx dt . (3.24) ∫∫ t 0 ∫∫ t 0 2 2 1 ⎝ ∂t ⎠ 1 ∂t (B)

2 1 t2 L ⎛ ∂w⎞ δ ρ p Ap ⎜ ⎟ []H(x − x1) − H(x − x2 ) dx dt ∫∫ t 0 2 1 ⎝ ∂t ⎠

2 t2 L ∂ w = − ρ p Ap 2 []H(x − x1) − H(x − x2 ) δw dx dt . (3.25) ∫∫ t1 0 ∂t

(C)

2 2 1 t2 L ⎛ ∂ w⎞ − δ Eb Ib ⎜ ⎟ dxdt ∫∫ t 0 ⎜ 2 ⎟ 2 1 ⎝ ∂x ⎠

L ⎛ 2 L 3 4 ⎞ t2 ⎜ ∂ w ⎛ ∂w ⎞ ∂ w L ∂ w ⎟ = − Eb Ib δ ⎜ ⎟ − δw + δw dx dt . (3.26) ∫∫ t ⎜ 2 3 0 4 ⎟ 1 ⎜ ∂x ⎝ ∂x ⎠ ∂x ∂x ⎟ ⎝ 0 0 ⎠

(D)

2 2 1 t2 L ⎛ ∂ w⎞ − δ Ep I p ⎜ ⎟ []H(x − x1) − H(x − x2 ) dx dt ∫∫ t 0 ⎜ 2 ⎟ 2 1 ⎝ ∂x ⎠

4 3 t2 L ⎧∂ w ∂ w = − E p I p ⎨ 4 [H (x − x1) − H (x − x2 )]+ 2 3 [δ (x − x1) −δ (x − x2 )] ∫ t1 ∫ 0 ∂x ∂x ⎩ ∂2w ⎫ + [δ '(x − x1) −δ '(x − x2 )]⎬δwdxdt . (3.27) ∂x2 ⎭ where

δ (x) is the Dirac-delta function, δ (x) = 0, for x < 0 and x > 0 ,

x0 +Δ f (x)δ (x − x0 )dx = f (x0 ), Δ > 0 is a small number. ∫ x −Δ 0

x0 +Δ f (x)δ '(x − x0 )dx = − f '(x0 ) . ∫ x −Δ 0

Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 54 (E)

2 t L 2 ⎛ hp ∂ w 2 ⎞ − δ Ap ⎜ h31 (hb + )D3 + β 33 D3 ⎟ [H (x − x1 ) − H (x − x2 )] dx ∫∫t 0 ⎜ 2 ⎟ 1 ⎝ 2 ∂x ⎠

1 t2 ⎧ L hp = − h31 Ap (hb + )D3 ⋅[δ '(x − x1) −δ '(x − x2 )]δwdx ∫∫ t ⎨ 0 2 1 ⎩ 2

L ⎫ − Ap β33D3 [H(x − x1) − H(x − x2 )]δD3dx dt . (3.28) ∫ 0 ⎬ ⎭

Substituting Equations (3.24 ~ 3.28) into Equation (3.23) obtains

4 2 4 2 t2 L ⎧ ∂ w ∂ w ⎛ ∂ w ∂ w ⎞ Eb Ib + ρb Ab + ⎜ E p I p + ρ p Ap ⎟⋅[]H (x − x1) − H (x − x2 ) ∫∫ t 0 ⎨ 4 2 ⎜ 4 2 ⎟ 1 ⎩ ∂x ∂t ⎝ ∂x ∂t ⎠

∂3w ⎛ ∂ 2w h ⎞ + 2E I [δ (x − x ) −δ (x − x )]+ ⎜ E I − h A (h + p )D ⎟ [δ '(x − x ) −δ '(x − x )] p p 3 1 2 ⎜ p p 2 31 p b 3 ⎟ 1 2 ∂x ⎝ ∂x 2 ⎠

⎪⎫ + Ap β33D3[H (x − x1) − H (x − x2 )]δD3 − Ap β33D3 [H (x − x1) − H (x − x2 )]δD3 − f (x,t)⎬δwdxdt ⎭⎪

L ⎛ 2 L 3 ⎞ t2 ⎜ ∂ w ⎛ ∂w ⎞ ∂ w ⎟ + Eb Ib δ ⎜ ⎟ − δw dt = 0 , ∫ t ⎜ 2 3 ⎟ 1 ⎜ ∂x ⎝ ∂x ⎠ ∂x ⎟ ⎝ 0 0 ⎠

Or

4 2 4 2 t2 L ⎧ ∂ w ∂ w ⎛ ∂ w ∂ w ⎞ Eb Ib + ρb Ab + ⎜ E p I p + ρ p Ap ⎟ ⋅[]H (x − x1) − H (x − x2 ) ∫∫ t 0 ⎨ 4 2 ⎜ 4 2 ⎟ 1 ⎩ ∂x ∂t ⎝ ∂x ∂t ⎠

∂3w ⎛ ∂ 2w h ⎞ + 2E I [δ (x − x ) −δ (x − x )]+ ⎜ E I − h A (h + p )D ⎟ p p 3 1 2 ⎜ p p 2 31 p b 3 ⎟ ∂x ⎝ ∂x 2 ⎠

L ⎛ 2 L 3 ⎞ ⎪⎫ t2 ⎜ ∂ w ⎛ ∂w ⎞ ∂ w ⎟ [δ '(x − x1) −δ '(x − x2 )] − f (x,t) δwdxdt + Eb Ib δ ⎜ ⎟ − δw dt = 0 . (3.29) ⎬ ∫t ⎜ 2 3 ⎟ 1 ∂x ∂x ∂x ⎭⎪ ⎜ ⎝ ⎠ 0 ⎟ ⎝ 0 ⎠

From Equation (3.29), for an arbitrary δw in 0 < x < L, the motion Equation of the composite beam with PZT attached can be obtained as:

∂ 4w ∂ 4w ∂3w E I + E I []H (x − x ) − H (x − x ) + 2E I [δ (x − x ) −δ (x − x )] b b ∂x4 p p ∂x4 1 2 p p ∂x3 1 2 ∂ 2w ∂2w ∂ 2w + E I [δ '(x − x ) −δ '(x − x )]+ ρ A + ρ A []H (x − x ) − H (x − x ) p p ∂x2 1 2 b b ∂t 2 p p ∂t 2 1 2 Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 55

h − h A (h + p )D [δ '(x − x ) −δ '(x − x )] = f (x,t) . (3.30) 31 p b 2 3 1 2 with the boundary condition:

L ⎧ ∂ 2w ⎛ ∂w ⎞ ∂ 2w ⎛ ∂w ⎞ ⎪Eb Ib 2 δ ⎜ ⎟ = 0 ⇒ 2 = 0 or δ ⎜ ⎟ = 0 ⎪ ∂x ⎝ ∂x ⎠ 0 ∂x ⎝ ∂x ⎠ ⎨ L . (3.31) ⎪ ∂3w ∂3w E I δw = 0 ⇒ = 0 or δw = 0 ⎪ b b ∂x3 ∂x3 ⎩ 0

3.2 Conversion of Partial Differential Equations (PDE) to Ordinary Differential Equations (ODE) using Galerkin’s method

Equation (3.30) is a 4th order, two variables partial different equation (PDE) which describes the distributed parameter model of the composite beam. It has to be solved nonlinearly. In order to solve it linearly, one needs to separate the variables and convert it to an ordinary different equation (ODE). With the mode separation method, the principle modes of the transverse displacement can be expressed as (Fung et al, 2001)

∞ N w(x,t) = ∑φr (x)qr (t) ≈ ∑φr (x)qr (t) . (3.32) r=1 r=1 where φr (x) is the mode function that satisfies all the boundary conditions, qr (t) is a time function.

Substituting Equation (3.32) into Equation (3.30) and using Galerkin’s method, the partial differential Equation (3.30) can be discretized into an ordinary differential equation as shown below

N N (4) (4) Eb Ib ∑φr (x)qr (t) + E p I p ∑φr qr (t)[H (x − x1) − H (x − x2 )] r=1 r=1

N N (3) + 2Ep I p ∑φr qr (t)[δ (x − x1) −δ (x − x2 )] + Ep I p ∑φr ''(x)qr (t)[δ '(x − x1) −δ '(x − x2 )] r=1 r=1

N N + ρb Ab ∑φr (x)qr (t) + ρ p Ap [H (x − x1 ) − H (x − x2 )] ∑φr (x)qr (t) r=1 r=1 h − h A (h + p )D [δ '(x − x ) −δ '(x − x )]− f (x,t) = ε(x,t) . (3.33) 31 p b 2 3 1 2

Minimizing ε by < ε, φs > = 0 (Fung et al, 2001)

L ε(x,t)φs (x) dx = 0 , (s = 1, 2, …, N). (3.34) ∫ 0 Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 56 which means the error is orthogonal to φs (x ).

Substituting Equation (3.33) into Equation (3.34) results in

N L N L (4) (4) Eb Ib φr (x)φs (x)dx qr (t) + E p I p φr (x)φs (x)[H (x − x1) − H (x − x2 )]dxqr (t) ∑∫ 0 ∑∫ 0 r=1 r=1

N L (3) + 2Ep I p φr (x)φs (x)[δ (x − x1) −δ (x − x2 )]dxqr (t) ∑∫ 0 r=1

N L N L + E p I p φr ''(x)φs (x)[δ '(x − x1) − δ '(x − x2 )]dxqr (t) + ρb Ab φr (x)φs (x)dxqr (t) ∑∫ 0 ∑∫ 0 r=1 r=1

N L + ρ p Ap φr (x)φs (x)[H (x − x1) − H (x − x2 )]dxqr (t) ∑∫ 0 r=1

hp L L − h31 Ap (hb + )D3 φs (x)[δ '(x − x1) −δ '(x − x2 )]dx − f (x,t)φs (x)dx = 0 . (3.35) 2 ∫ 0 ∫ 0

Each term in Equation (3.35) can be derived respectively as below (see Appendix 6).

To the 1st term in Equation (3.35),

N L N L (4) ⎛ ⎞ Eb Ib φr (x)φs (x)dx qr (t) = Eb Ib ⎜ φr ''(x)φs ''(x)dx⎟ qr (t) . (3.36) ∑∫ 0 ∑ ∫ 0 r=1 r=1 ⎝ ⎠

To the 3rd term in Equation (3.35)

N L (3) 2E p I p φr (x)φs (x)[δ (x − x1) −δ (x − x2 )]dxqr (t) ∑∫ 0 r=1

N L (4) (3) = −2E p I p []φr (x)φs (x) +φr (x)φs '(x) []H (x − x1) − H (x − x2 ) dxqr (t) . (3.37) ∑∫ 0 r=1

To the 4th term in Equation (3.35)

N ⎛ L ⎞ E p I p ⎜ φr ''(x)φs (x)[δ '(x − x1) −δ '(x − x2 )]dx⎟qr (t) ∑ ∫ 0 r=1 ⎝ ⎠

N L (4) (3) = Ep I p φr (x)φs (x) + 2φr (x)φs '(x) +φr ''(x)φs ''(x)][H(x − x1) − H(x − x2 )]dxqr (t) . (3.38) ∑∫ 0 r=1

To the second last term in Equation (3.35)

hp L hp x2 − h31 Ap (hb + )D3 φs (x)[δ '(x − x1) −δ '(x − x2 )]dx = −h31Ap (hb + )D3 φs ''(x)dx . (3.39) 2 ∫ 0 2 ∫ x1 Substituting Equation (3.36 ~ 3.39) into Equation (3.35) obtains the motion equation of the composite beam in ODE form Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 57

N ⎛ L ⎞ ⎜ []ρb Ab + ρ p Ap [H (x − x1 ) − H (x − x2 )] φr (x)φs (x) dx⎟ qr (t) ∑ ∫ 0 r=1 ⎝ ⎠

N ⎛ L ⎞ + ⎜ []Eb Ib + E p I p[H (x − x1) − H (x − x2 )] φr ''(x)φs ''(x) dx⎟ qr (t) ∑ ∫ 0 r=1 ⎝ ⎠

hp x2 L − h31Ap (hb + )D3 φs ''(x)dx = f (x,t)φs (x)dx . (3.40) 2 ∫ x1 ∫ 0 where:

1 1 ⎛ v(t) ⎞ D = E + h ε = ⎜ + h ε ⎟ . (3.41) 3 ()3 31 11 ⎜ 31 11 ⎟ β33 β33 ⎝ hp ⎠

3.3 Vibration control with the piezoelectric shunt circuit

3.3.1 Series R-L shunt

The electrical current stemmed from electrical charge of PZT is

dQ dD i(t) = − = −S 3 . (3.42) dt p dt where

i(t ) is the output electrical current of PZT which is also the input current of the shunt circuit, Q = S p D3 is the electrical charge of PZT, and S p = b(x2 − x1 ) is the surface area of the PZT plate.

∂ 2 w Substituting Equation (3.41) into Equation (3.42) as well as using ε = −z obtains 11 ∂x 2

S d ⎛ v(t) ⎞ S ⎛ 1 dv(t) ∂3w ⎞ i(t) = − p ⎜ + h ε ⎟ = − p ⎜ − h z ⎟ ⎜ 31 11 ⎟ ⎜ 31 2 ⎟ β33 dt ⎝ hp ⎠ β33 ⎝ hp dt ∂x ∂t ⎠ or

3 dv(t) β33hp ∂ w = − i(t) + h31hp z 2 . (3.43) dt S p ∂x ∂t

By using Kirchhoff’s voltage law, the voltage across the electrodes of the PZT is

di(t) v(t) = L + R i(t) . (3.44) 0 dt 0

Substituting Equation (3.44) into Equation (3.43) obtains, Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 58

d 2i(t) di(t) 1 ∂3w L0 2 + R0 + i(t) = h31hp z 2 , (3.45) dt dt CP ∂x ∂t or

d ⎛ di(t) 1 ⎞ d ⎛ ∂2w ⎞ dV (t) ⎜ L + R i(t) + i(t)dt⎟ = ⎜h h z ⎟ = s . (3.46) ⎜ 0 2 0 ∫ ⎟ ⎜ 31 p 2 ⎟ dt ⎝ dt C p ⎠ dt ⎝ ∂x ⎠ dt where

b(x2 − x1) S p C p = = –– The capacitance of PZT. β33hp β33hp

∂ 2 w V (t) = h h z –– The equivalent voltage source stemmed from mechanical s 31 p ∂x 2 strain.

From Equation (3.46), one can have,

1 L i(t) + R i(t) + i(t)dt = V (t) +V . (3.47) 0 0 ∫ s DC1 C p where VDC1 is a constant representing DC voltage.

If VDC is neglected, Equation (3.47) becomes

1 L i(t) + R i(t) + i(t)dt = V (t) . (3.48) 0 0 ∫ s C p

The equivalent circuit model of Equation (3.48) is depicted in Figure 3-3 that represents the series R-L shunt circuit connected to the piezoelectric transducer.

Figure 3-3 Equivalent circuit of series piezoelectric R-L shunt Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 59

∂ 2 w Integrate both sides of Equation (3.48) with V (t) = h h z from h to h + h to s 31 p ∂x 2 b b p eliminate variable z,

3 h +h ⎛ ⎞ h +h b p ⎜ 1 ⎟ b p ∂ w L0i(t) + R0i(t) + i(t)dt dz = h31hp z dz ∫∫ h ⎜ ⎟ ∫ h 2 b ⎝ Cp ⎠ b ∂x ∂t one can have

di(t) 1 h h (2h + h ) ∂ 2w L + R i(t) + i(t)dt = 31 p b p +V . (3.49) 0 0 ∫ 2 DC 2 dt C p 2 ∂x

If VDC 2 is disregarded, Equation (3.49) can be expressed as

di(t) 1 h h (2h + h ) ∂ 2w L + R i(t) + i(t)dt − 31 p b p = 0 . (3.50) 0 0 ∫ 2 dt C p 2 ∂x

Considering the relation

2 N N ∂ w 1 ⎛ x2 ⎞ = φr ''(x)qr (t) = ⎜ φr ''(x)dx ⎟qr (t) 2 ∑ ∑ ∫ x 1 ∂x r =1 x2 − x1 ⎝ r =1 ⎠ and to substitute it into Equation (3.50) obtains

N di(t) 1 h31hp (2hb + hp ) ⎛ x2 ⎞ L0 + R0i(t) + i(t)dt − ⎜ φr ''(x)dx⎟qr = 0 . (3.51) ∫ ∑∫x dt C p 2(x2 − x1 ) ⎝ r=1 1 ⎠

Equation (3.51) can be simplified as

di(t) R 1 + 0 i(t) + i(t)dt +ψ q (t) = 0 ∫ r r . (3.52) dt L0 L0C p where,

h31hp (2hb + hp ) x2 ψ = − φr ''(x)dx (r = 1, …, N) . (3.53) r ∫x 1 2(x2 − x1 )L0

Since,

1 ∂2w 1 v(t) = V (t) − i(t)dt = h h z − i(t)dt , s ∫ 31 p 2 ∫ C p ∂x C p then,

1 ⎛ v(t) ⎞ 1 ⎛ ∂2w 1 ∂2w⎞ D = ⎜ + h ε ⎟ = ⎜h z − i(t)dt − h z ⎟ 3 ⎜ 31 11 ⎟ ⎜ 31 2 ∫ 31 2 ⎟ β33 ⎝ hp ⎠ β33 ⎝ ∂x hpCp ∂x ⎠ Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 60

1 1 = − ∫i(t)dt = − ∫i(t)dt . (3.54) β33hpCp S p

One can notice that Equation (3.54) is the integration of D3 in Equation (3.42). Substituting Equation (3.54) into Equations (3.40) along with Equation (3.53) obtains

N ⎛ L ⎞ ⎜ []ρb Ab + ρ p Ap[H (x − x1) − H (x − x2 )] φr (x)φs (x) dx⎟ qr (t) ∑ ∫ 0 r=1 ⎝ ⎠

N ⎛ L ⎞ + ⎜ []Eb Ib + E p I p[H (x − x1) − H (x − x2 )] φr ''(x)φs ''(x) dx⎟ qr (t) ∑ ∫ 0 r=1 ⎝ ⎠

L −ψ r L0 i(t)dt = f (x,t)φs (x)dx . (3.55) ∫ ∫ 0

The matrix form of Equation (3.55) is: ~ Mq + Kq + B i = fd . (3.56)

When structure damping is considered, Equation (3.56) can be extended as: ~ Mq + Cq + Kq + B i = fd . (3.57) where

T T q = []q1 q2 ... qN N×1 , i = [i1 i2 ... iN ] N×1 ,

dq d 2q ~ q = , q = , i = idt , dt dt 2 ∫

C = αM +γK –– The Rayleigh damping (or proportional damping) coefficient, α and γ are constants.

⎡m11 m1N ⎤ ⎡k11 k1N ⎤ ⎡c11 c1N ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ M=⎢ ⎥, K=⎢ ⎥, C=⎢ ⎥, ⎣⎢mN1 mNN⎦⎥ ⎣⎢kN1 kNN⎦⎥ ⎣⎢cN1 cNN⎦⎥

⎡ f d1 ⎤ ⎡B1 ⎤ ⎢ ⎥ 0 f = ⎢ ⎥ d ⎢ ⎥ B, = ⎢ ⎥ . ⎢ f ⎥ ⎢ B ⎥ ⎣ dN ⎦ N ×1 ⎣ 0 N ⎦

L msr = []ρb Ab + ρ p Ap [H (x − x1 ) − H (x − x2 )] φr (x)φs (x) dx , ∫0

L ksr = []Eb Ib + Ep I p[H(x − x1) − H(x − x2 )] φ''r (x)φ''s (x) dx , ∫0 Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 61

csr =αmsr +γksr ,

L L fds = f (x,t)φs (x)dx= Fd (t)δ(x− xd )φs (x)dx (Point force applied at x , s = 1, …, N), ∫ 0 ∫ 0 d

h31hp (2hb + hp ) x2 Br = φr ''(x)dx (r = 1, …, N). 2(x − x ) ∫x1 2 1 Equation (3.52) represents piezoelectric vibration control with a single series R-L shunt circuit. In general, vibration control with multiple series R-L shunt branches can be represented as a matrix form (assume shunt branches are mutually non-interfering), ~ i + Γi + Ω i + Ψq = 0 . (3.58) where

R ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡ψ ⎤ ⎢ L 0 ⎥ ⎢C L 0 ⎥ 1 0 ⎢ 1 ⎥ ⎢ p 1 ⎥ ⎢ ⎥ Γ = ⎢ ⎥ , Ω = ⎢ ⎥ , Ψ = . ⎢ RN ⎥ ⎢ 1 ⎥ ⎢ ⎥

⎢ L ⎥ ⎢ C L ⎥ ⎢ ψ ⎥ ⎣ 0 N ⎦ ⎣ 0 p N ⎦ ⎣ 0 N ⎦

Rr , Lr are the resistor and inductor of the parallel R-L shunt circuit respectively, and ψ r is calculated by Equation (3.53).

Now convert the matrix Equations (3.57 & 3.58) into a state space form. Let

x1 = q ,

dx 1 = q = x , dt 2

dx 2 = q = −M -1Kx − M -1Cx - M -1Bx + M −1f , dt 1 2 3 d ~ x3 = i ,

dx 3 = i = x , dt 4

dx 4 = i = −Ψx − Ωx − Γx , dt 1 2 3

y = C1q = C1x1 .

Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 62 where,

π x Nπ x ⎡ m m ⎤ x C1 = ⎢sin sin ⎥ , m is the position where response measured. ⎣ L L ⎦ 1×N

Then the state space form of Equations (3.57 & 3.58) is

x = A x + B u S S . (3.59) y = C S x + D S u where,

T T ~ x = [x1 x2 x3 x4 ] = [q q iL iL ]4N×1 , u = [fd ]N ×1

⎡ 0 I 0 0 ⎤ ⎡ 0 ⎤ ⎢−M−1K −M−1C −M−1B 0 ⎥ ⎢M-1⎥ A =⎢ ⎥ , B = ⎢ ⎥ , S ⎢ 0 0 0 I ⎥ S ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ −Ψ -Ω −Γ 0 ⎦4N × 4N ⎣ 0 ⎦4N×N

π x Nπ x C = C 0 0 0 D = 0 ⎡ m m ⎤ S []1 1×4N , S [ ] 1× N , C1 = ⎢sin sin ⎥ and I is the ⎣ L L ⎦ 1×N unit matrix.

3.3.2 Parallel R-L shunt

From Equations (3.41, 3.42), the electrical current of parallel piezoelectric R-L shunt can be expressed as

dD d ⎧ 1 ⎛ v(t) ⎞⎫ dv dε 3 ⎪ ⎜ ⎟⎪ 11 i(t) = −S p = −S p ⎨ + h31ε11 ⎬ = −C p − h31hpC p = iL + iR . (3.60) dt dt β ⎜ h ⎟ dt dt ⎩⎪ 33 ⎝ p ⎠⎭⎪

According to the Kirchhoff’s current law,

iC + iL + iR = I S . (3.61) where

dv iC = Cp is the electrical current of Cp , iL and iR are electrical currents of the dt dε inductor and resistor of the shunt circuit respectively, I = −h h C 11 is the equivalent S 31 p p dt electrical current source generated due to mechanical stress, v(t) is the voltage across the electrodes of PZT. Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 63 The equivalent circuit model of Equation (3.61) is depicted in Figure 3-4 that represents the parallel piezoelectric R-L shunt. As seen in the Figure, the voltage v(t) can be calculated as below (DC current is disregarded similar to the DC voltage consideration in the series shunt)

1 1 1 ⎛ dε ⎞ v(t) = i dt = I −i −i dt = − h h C 11 +i +i dt ∫∫C ∫()S R L ⎜ 31 p p R L ⎟ Cp Cp Cp ⎝ dt ⎠

1 ⎛ dε L di ⎞ L 1 = − ⎜h h C 11 + 0 L + i ⎟dt = −h h ε − 0 i − i dt . (3.62) ∫⎜ 31 p p L ⎟ 31 p 11 L ∫ L Cp ⎝ dt R0 dt ⎠ R0Cp Cp

Figure 3-4 Equivalent circuit of parallel piezoelectric R-L shunt

Substituting Equation (3.62) into Equation (3.41) obtains D3 for the parallel R-L shunt circuit

1 ⎛ L 1 ⎞ D = − ⎜ 0 i + i dt⎟ , (3.63) 3 ⎜ L ∫ L ⎟ β33hp ⎝ R0Cp Cp ⎠

Then Equation (3.40) becomes:

N ⎛ L ⎞ ⎜ []ρb Ab + ρ p Ap[H(x − x1) − H(x − x2 )] φr (x)φs (x) dx⎟ qr (t) ∑ ∫0 r=1 ⎝ ⎠

N ⎛ L ⎞ + ⎜ []Eb Ib + E p I p[H (x − x1) − H (x − x2 )] φr ''(x)φs ''(x) dx⎟ qr (t) ∑ ∫ 0 r=1 ⎝ ⎠

L + BLiL + BR iLdt = f (x,t)φs (x)dx . (3.64) ∫ ∫0 where

x h31hp (2hb + hp ) L0 2 BL = φs ''(s)dx , (3.65) 2(x − x ) R ∫x1 2 1 0

h31hp (2hb + hp ) x2 BR = φs ''(s)dx . (3.66) ∫x 2(x2 − x1) 1 Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 64

di On the other hand, the voltage of the inductor is v(t) = L L which is equal to the voltage of 0 dt resistor and capacitor in Figure 3-4. Combining with Equation (3.62), one can have

di L 1 L L = −h h ε − 0 i − i dt , 0 31 p 11 L ∫ L dt R0C p C p

Or

di L 1 ∂ 2 w L L + 0 i + i dt = h h z . (3.67) 0 dt R C L C ∫ L 31 p ∂x 2 0 p p

∂ 2 w where axial strain ε has been replaced by − z . 11 ∂x 2

Integrating both sides of Equation (3.67) to eliminate variable z and considering equation ∂ 2 w ⎛ N ⎞ = φ ''(x) q (t) , 2 ⎜ ∑ r ⎟ r ∂x ⎝ r =1 ⎠

2 hb +hp ⎛ di L 1 ⎞ hb +hp ∂ w ⎜ L 0 ⎟ L0 + iL + iLdt dz = h31hp z dz ∫∫ h ⎜ ⎟ ∫ h 2 b ⎝ dt R0C p C p ⎠ b ∂x one can have:

N x diL 1 1 h31hp (2hb + hp ) ⎛ 2 ⎞ + iL + iLdt − ⎜ φr ''(x)dx⎟qr = 0 . (3.68) ∫ ∑∫ x 1 dt R0Cp L0Cp 2(x2 − x1)L0 ⎝ r=1 ⎠

The Equation (3.64) can be written in concise form as:

N N m q + k q + B i + B i dt = f (t) . (3.69) ∑ sr r ∑ sr r L L 2 ∫ R d r=1 r=1

Equation (3.68) can be written as:

di 1 1 N L + i + i dt + ψ q = 0 L ∫ L ∑ r r . (3.70) dt R0C p L0C p r=1

Equation (3.70) represents piezoelectric vibration control with a single parallel R-L shunt circuit. In general, multiple parallel R-L shunt control can be represented in matrix form (assume each control circuit is mutual non-interference),

~ i L + Γi L + Ω i L + Ψq = 0 . (3.71) where Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 65

T ~ d i i = []i i , i = i dt , i = L , L L1 LN N ×1 L ∫ L L dt

⎡ 1 ⎤ ⎡ 1 ⎤ ⎡ψ ⎤ ⎢C R 0 ⎥ ⎢C L 0 ⎥ 1 0 ⎢ p 1 ⎥ ⎢ p 1 ⎥ ⎢ ⎥ Γ = ⎢ ⎥ , Ω = ⎢ ⎥ , Ψ = . ⎢ 1 ⎥ ⎢ 1 ⎥ ⎢ ⎥

⎢ C R ⎥ ⎢ C L ⎥ ⎣ 0 p N ⎦ ⎣ p N ⎦ ⎢ ψN⎥ 0 ⎣ 0 ⎦

Rr , Lr are the resistor and inductor of the parallel R-L shunt circuit respectively, iL is the current of inductor, and ψ r is calculated by Equation (3.53).

Equation (3.71) for parallel shunt has the same form as Equation (3.58) for series shunt but with a different coefficient. In parallel shunt, the current of inductor iL is used as the control variable instead of i in series shunt. Also similar to Equation (3.57), the matrix form of the extended Equation (3.64) for parallel is ~ Mq + Cq + Kq + BLiL + BR i L = fd . (3.72) where

L R ⎡B ⎤ ⎡B ⎤ ⎢ 1 0 ⎥ ⎢ 1 0 ⎥ B L = ⎢ ⎥ , B R = ⎢ ⎥ . L R ⎢ B ⎥ ⎢ B ⎥ ⎣⎢ 0 N ⎦⎥ ⎣⎢ 0 N ⎦⎥

x L h31hp (2hb + hp ) Ls 2 Bs = φs ''(x)dx (s = 1, …, N), 2(x − x ) R ∫x1 2 1 s

x R h31hp (2hb + hp ) 2 Bs = φs ''(x)dx (s = 1, …, N), ∫x 2(x2 − x1) 1

M, C , K, and fd are the same as those in series shunt.

To convert Equations (3.71 & 3.72) to state space form, let

x1 = q ,

dx 1 = q = x , dt 2 Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 66

dx 2 = q = −M -1Kx − M -1Cx - M -1B x − M -1B x + M −1f , dt 1 2 R 3 L 4 d ~ x 3 = i L ,

dx 3 = i = x , dt L 4

dx 4 = i = −Ψx − Γx − Ωx , dt L 1 3 4

y = C1q = C1x1 .

The state space form of Equations (3.71 & 3.72) is the same as Equation (3.59). where: ~ T T x = [x1 x 2 x3 x4 ] 4 N×1 = [q q i L i L ] 4 N×1 , u = [f d ] N ×1 ,

⎡ 0 I 0 0 ⎤ ⎡ 0 ⎤ ⎢−M−1K −M−1C −M−1B −M−1B ⎥ ⎢M-1 ⎥ A =⎢ R L⎥ , B = ⎢ ⎥ , S ⎢ 0 0 0 I ⎥ S ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ −Ψ 0 −Γ −Ω ⎦ 4N × 4N ⎣ 0 ⎦4N×N

⎡ π x Nπ x ⎤ C = C 0 0 0 D 0 m m S []1 1×4N , S = [ ] 1× N , C1 = ⎢sin sin ⎥ . ⎣ L L ⎦ 1×N

So far, we have derived the motion equation of a cantilever beam connected to a piezoelectric shunt network. Solving Equation (3.59) with either series shunt coefficients or parallel shunt coefficients, one can obtain the time domain vibration signal of the composite beam. A numerical example of parallel shunt control is given in the next section.

3.4 Numerical examples

Choose

φ r (x) = sin( rπx / L) , (r = 1, …, N) which satisfies all boundary conditions (Hinged-Hinged), obtains

h31hp (2hb + hp )rπ ⎡ ⎛ rπx1 ⎞ ⎛ rπx2 ⎞⎤ ψ r = ⎢cos⎜ ⎟ − cos⎜ ⎟⎥ , (r = 1, …, N) 2(x2 − x1 )L0 ⎣ ⎝ L ⎠ ⎝ L ⎠⎦

h31hp (2hb + hp )sπLs ⎡ ⎛ sπx1 ⎞ ⎛ sπx2 ⎞⎤ BLs = ⎢cos⎜ ⎟ − cos⎜ ⎟⎥ , (s = 1, …, N) 2(x2 − x1)LRs ⎣ ⎝ L ⎠ ⎝ L ⎠⎦ Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 67

h31hp (2hb + hp )sπ ⎡ ⎛ sπx1 ⎞ ⎛ sπx2 ⎞⎤ BR s = ⎢cos⎜ ⎟ − cos⎜ ⎟⎥ , (s = 1, …, N) 2(x2 − x1 )L ⎣ ⎝ L ⎠ ⎝ L ⎠⎦

For s = r ,

ρb Ab L + ρ p Ap (x2 − x1 ) ρ p Ap L ⎡ ⎛ 2πrx1 ⎞ ⎛ 2πrx2 ⎞⎤ mrr = + ⎢sin⎜ ⎟ − sin⎜ ⎟⎥ , 2 2πr ⎣ ⎝ L ⎠ ⎝ L ⎠⎦

4 ⎛ rπ ⎞ ⎧Eb Ib L + E p I p (x2 − x1) E p I p L ⎡ ⎛ 2πrx1 ⎞ ⎛ 2πrx2 ⎞⎤⎫ krr = ⎜ ⎟ ⎨ + ⎢sin⎜ ⎟ − sin⎜ ⎟⎥⎬ . ⎝ L ⎠ ⎩ 2 2πr ⎣ ⎝ L ⎠ ⎝ L ⎠⎦⎭

For s ≠ r ,

ρ p ApL ⎛ sπx1 rπx1 rπx1 sπx1 sπx2 rπx2 rπx2 sπx2 ⎞ msr = ⎜r sin cos − ssin cos − r sin cos + ssin cos ⎟ , π (r 2 − s2 ) ⎝ L L L L L L L L ⎠

2 2 EpI pL ⎛ rsπ ⎞ ⎛ sπx rπx rπx sπx sπx rπx rπx sπx ⎞ k = ⎜ ⎟ r sin 1 cos 1 − ssin 1 cos 1 −r sin 2 cos 2 + ssin 2 cos 2 . sr 2 2 ⎜ 2 ⎟ ⎜ ⎟ (r − s )π ⎝ L ⎠ ⎝ L L L L L L L L ⎠

3.4.1 Vibration shunt control simulation 1 (PZT placed at L/2)

Given an aluminum beam and a PZT, the parameters are

Aluminum beam:

3 L = 0.5 (m) , b = 0.025 (m) , hb = 0.003 (m) , ρb = 2700 (kg/m ) , Eb = 70 (GPa ) .

PZT:

3 hp = 0.00025 (m) , x1 = 0.22 (m) , x2 =0.28 (m), ρ p = 7600 (kg/m ) , E p = 70 (GPa) ,

8 7 h31 = 7.65×10 (V/m) , β33 = 7.33×10 (m/F) (Wang et al, 1996), Cp =82 (nF) .

Solving the state space Equation (3.59) with MATLAB® obtains the impulse responses of the

−2 T composite beam with a force vector fd = 10 [ 1 1 1 ] (which is equivalent to have equal point force being applied at the anti-node of each mode) and vibration measured at xm = L/ 4 from origin resulting in, C1(r) =sin(rπxm / L) =sin(rπ / 4), r = 1, 2, 3, as shown in Figure 3-5. There are only three frequency components being considered in the vibration (N = 3) with Rayleigh damping coefficient, α = 0.2 and γ =10−7 . One can get the natural frequencies of the composite beam from its eigenvalue λ since λ = ω 2 (where ω is the circular natural Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 68 frequency). The eigenvalue can be obtained by solving the characteristic equation

K − λ M = 0 . For this example, the eigenvalue is

⎡30900 0 0 ⎤ ⎢ ⎥ ⎢ 0 495000 0 ⎥ ⎢ 0 0 2516500 ⎥ ⎣ ⎦ .

Thus the natural frequencies of the composite beam corresponding to the eigenvalue are approximately 28, 112, and 252 Hz respectively.

The optimal value of inductor of the shunt circuit to target the ith mode can be calculated by

1 Li = 2 (i = 1, 2, 3) (2πFi ) C p

th where Fi is the i natural frequency and Cp is the capacitance of the PZT patch.

-3 x 10 Impulse response of beam transverse displacement at xm=L/4 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4 Shunt circuit ON 0.2

0 Shunt circuit -0.2

Displacement (m) Displacement OFF -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

(a) without shunt control (dashed line),

(b) with shunt control (solid line) ( F1 ≈ 28 Hz, F2 ≈112 Hz , F3 ≈ 252 Hz, Cp = 82 nF,

L1 = 400 H , L2 = 24.64 H , L3 = 4.846 H , shunt circuit targets at three modes)

Figure 3-5 Impulse response of transverse displacement of beam measured at xm = L/4 with PZT patch attached in the middle of beam

Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 69

Thus the inductor values to target three natural frequencies can be obtained as L1 = 395 H, L2

= 24.64 H and L3 = 4.846 H respectively. Figure 3-5 shows the impulse response of the transverse displacement of the composite beam with the parallel R-L piezoelectric vibration st shunt circuit ON/OFF. However, the best shunt outcome of the 1 mode seems at L1 = 400 H while other two are at their theoretical values.

Figure 3-6 shows the frequency response of the beam vibration in Figure 3-5. One can see that the vibration amplitudes of the 1st mode and the 3rd mode have been reduced about 21 and 31 dB respectively when the shunt circuit is turned ON. However, the shunt circuit has no affect to the 2nd mode since the PZT patch is placed at the middle of the beam i.e. nd (x2 + x1 ) / 2 = L / 2 = 0.25 (m) where is the node location of the 2 mode (with hinged-hinged boundary condition). Therefore, there is no difference in amplitude of the 2nd mode with or without the shunt circuit being turned ON. However, if the PZT patch is placed away from its node location, it can control the 2nd mode vibration when the shunt circuit is turned on. It can be seen in the next example.

Frequency response of beam transverse displacement at xm=L/4 40 Without PZT shunt With PZT shunt 20

0

-20 Magnitude (dB) -40

-60

-80 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

Figure 3-6 Frequency response of transverse displacement of beam measured at xm = L/4 with PZT patch attached in the middle of beam

Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 70

3.4.2 Vibration shunt control simulation 2 (PZT placed at L/4)

The difference of this one from previous one is that the PZT patch is placed at different location which is x1 =0.095 m, x2 =0.155 m, (x2 + x1 ) / 2 = L / 4 = 0.125 (m) . The point where vibration measured is still at L/4 from origin. Figure 3-7 shows the impulse response of the beam with and without piezoelectric shunt control. With the PZT patch placed at this location, it can control not only the 1st and 3rd modes, but also the 2nd mode since the PZT patch is not at the node location of any mode. The vibration amplitude approaches to zero after a short period of time. Figure 3-8 shows the frequency response of the beam vibration. The vibration amplitudes of the 1st, 2nd and 3rd mode have been reduced about 15 dB, 24 dB and 28 dB respectively but the control efficiency of the 1st and 3rd modes are 6 and 3 dB worse off comparing with the results in previous example because the PZT patch is not attached at anti- node of the 1st and 3rd modes as it was previously. This tells us that the location where the PZT patch placed is very important to control a particular vibration mode. One can notice that the efficiency of vibration control seems better for higher mode frequency though it depends on the location of the PZT.

-3 x 10 Impulse response of beam transverse displacement (xm=L/4) 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) Displacement -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

(c) without shunt control (dashed line),

(d) with shunt control (solid line) ( F1 ≈ 28 Hz, F2 ≈112 Hz , F3 ≈ 252 Hz, Cp = 82 nF,

L1 = 400 H , L2 = 24.64 H , L3 = 4.846 H , shunt circuit targets at three modes)

Figure 3-7 Impulse response of transverse displacement of beam measured at xm = L/4 with

PZT patch attached x1 = 0.19L and x2 = 0.31L

Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 71

Frequency response of beam transverse displacement (xm=L/4) 40 Without PZT shunt With PZT shunt

20

0

-20 Magnitude (dB) -40

-60

-80 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

Figure 3-8 Frequency response of transverse displacement of beam measured at xm = L/4

with PZT patch attached x1 = 0.19L and x2 = 0.31L

3.4.3 Sensitivity test of control frequency mismatch

The R-L shunt control possesses narrowband characteristics, it is quite sensitive to the frequency mismatch between the frequency of shunt circuit and the natural frequency to be controlled. When the shunt circuit is off-tuned, the control will become worse.

-3 Impulse response of 1st mode beam vibration (xm=L/4) x 10 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

st (a) Shunt circuit is tuned to the 1 mode (L1 = 400 H) Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 72

-3 Impulse response of 1st mode beam vibration (xm=L/4) x 10 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) Displacement -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

st (b) Shunt circuit is off-tuned at the1 mode (L1 = 300 H)

-3 Impulse response of 1st mode beam vibration (xm=L/4) x 10 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) Displacement -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

st (c) Shunt circuit is off-tuned at the 1 mode (L1 = 500 H)

Figure 3-9 Sensitivity test of the 1st mode

Figures (3-9 ~ 3-11) are the results of sensitivity test for each mode respectively. Part (a) in the Figures are the results when the shunt circuits are properly tuned, part (b) and (c) in the Figures are the results when the shunt circuits are ill-tuned resulting in control efficiency drop. Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 73

-3 Impulse response of 2nd mode beam vibration (xm=L/4) x 10 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) Displacement -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

nd (a) Shunt circuit is tuned to the 2 mode (L2 = 24.64 H)

-3 Impulse response of 2nd mode beam vibration (xm=L/4) x 10 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

nd (b) Shunt circuit is off-tuned at 2 mode (L2 = 22 H) Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 74

-3 Impulse response of 2nd mode beam vibration (xm=L/4) x 10 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) Displacement -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

nd (c) Shunt circuit is off-tuned at 2 mode (L2 = 28 H)

Figure 3-10 Sensitivity test of the 2nd mode

-3 Impulse response of 3rd mode beam vibration (xm=L/4) x 10 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) Displacement -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

rd (a) Shunt circuit is tuned to the 3 mode (L3 = 4.846 H)

Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 75

-3 Impulse response of 3rd mode beam vibration (xm=L/4) x 10 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) Displacement -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

rd (b) Shunt circuit is off-tuned at the 3 mode (L3 = 4 H)

-3 Impulse response of 3rd mode beam vibration (xm=L/4) x 10 1 Without PZT shunt 0.8 With PZT shunt

0.6

0.4

0.2

0

-0.2 Displacement (m) Displacement -0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec)

rd (c) Shunt circuit is off-tuned at the 3 mode (L3 = 6 H)

Figure 3-11 Sensitivity test of the 3rd mode

One can notice that the shunt control is more sensitive to frequency tuning at higher frequency. The inductor value for controlling higher frequency vibration requires better accuracy. If a structure to be controlled has its natural frequencies constantly varying, which Chapter 3 Analytical Modeling of Passive Piezoelectric Vibration Shunt on Beams 76 is quite possible in reality, it is desirable to have the control frequency of the shunt circuit track the natural frequency of the structure to prevent the frequency mismatch problem. This requires real-time estimation of structure’s excitation frequencies. This is discussed in Chapter 7.

3.5 Chapter summary

In this chapter, an analytical study of resistor-inductor piezoelectric vibration shunt control on a beam structure has been carried out. The motion equation of the composite beam made of a cantilever beam with a PZT patch attached has been derived by using the Hamilton’s principle and Galerkin’s method. The piezoelectric shunt networks have been coupled into the beam equation by means of the constitutive piezoelectric relation and electrical circuit theory as a mechanism to control beam vibration. A numerical example of the parallel R-L piezoelectric vibration shunt control for beam vibration by solving the state space equation has been given. The simulation results with Matlab® have shown that significant vibration reduction can be achieved when the control circuit is tuned to the natural frequency to be controlled. The results have also shown that the control efficiency is sensitive to the location where the PZT patch is placed and the accuracy of the shunt circuit being tuned. Chapter 4 Composite Beam Analysis and Design 77

Chapter 4

Composite Beam Analysis and Design

Composite beams, made of more than one material, have different properties and characteristics from beams made of a single material. Strain and stress distributions in the composite beams are determined by the geometry size, material properties and beam structure. To understand the strain/stress distribution in the composite beams is important for effectively removing vibration energy from the structure using passive piezoelectric vibration control technique. In this chapter, the influence of neutral surface location on strain distribution in the composite beam will be discussed. The equations of neutral surface location, area moment of inertia, strain and stress distribution of a two-layer composite beam made of the PZT and a uniform beam will be derived respectively. Simulations of the derived equations with different thickness ratios and Young’s modulus ratios are presented. Some FEA simulation and experimental results of the piezoelectric vibration shunt control for the beams made of various materials are presented. Carbon fibre composite material and its layout are also briefly introduced. Finally, the design of the Cricket bat handle is proposed in order to maximize the conversion of vibration energy to electrical energy to be dissipated by an electrical network and the benefits of using carbon-fibre materials in Cricket bats will also be described.

4.1 Stress and Strain in Composite Beams

Beams made of more than one material are known as composite beams. The examples of the composite beams are bimetallic beams, sandwich beams and laminated beams, which consist of multi-layer beams of different materials. Figure 4-1(a) shows a simple composite beam that is made of a PZT layer bonded on the outer surface of a uniform cantilever beam. It is assumed that the PZT layer and the beam are firmly bonded together, the PZT and beam models are linear and elastic, and the material properties of the PZT and beam are Chapter 4 Composite Beam Analysis and Design 78 homogeneous and isotropic. Figures 4-1 (b & c) show the cross-sectional area of the composite beam and normal strain distribution in the cross sectional area. In the figure L, b,

hb and hP donate the length, width, thickness of the beam and PZT respectively; Eb and E p are the Young’s moduli, ε b and ε p are normal strains of the beam and PZT respectively.

(a)

(b)

(c)

(a) Composite beam structure (b) Cross-sectional area and normal strain distribution (with high Young’s modulus ratio) (c) Cross-sectional area and normal strain distribution (with low Young’s modulus ratio)

Figure 4-1 A two-layer composite beam made of the PZT and uniform beam

There are different kinds of strains in the composite beams. The strain caused by the bending moment is known as normal strain, the strain caused by shear force is known as shear strain. When a layer of damping material is attached to the outer surface of a vibrating beam under bending, it can be used to dissipate energy by fluctuating normal strain. The normal strain linearly varies through the depths of the PZT and beam layers, in proportion to the local Chapter 4 Composite Beam Analysis and Design 79 bending moment and the distance from the neutral surface of bending. The further the distance of a surface from the neutral surface, the greater the normal strain of the surface. The greatest normal strain occurs along the surface furthest from the neutral surface. In order to dissipate more energy in bending, the damping layer should be attached as far from the neutral surface as possible, or the neutral surface of the composite beam should be located as far from the damping layer as possible. This configuration, known as free/unconstrained layer treatment/damping, can maximize normal strain but is free from shear strain as it is free from shear force. On the contrary, another configuration where the damping layer is sandwiched in the middle of two beams, known as constrained treatment/damping, can maximize the shear strain but produce very little normal strain due to a damping layer close to the neutral surface (Mead, 2000). In order to dissipate one type of strain energy or the other efficiently, the material properties and location of the damping layer should be selected accordingly. In this study, only normal strain is considered.

Although made of different materials, the normal strain distribution through the depth of the composite beams is still linear, same as the normal strain distribution of the beams made of a single material. However, unlike the beams made of a single material, the stress distribution in the composite beams is not linear but piecewise linear. The stress of individual layer is the product of the strain and the Young’s modulus of the material of the layer. The level of normal strain in the layer can be different depending on the location of the neutral surface of the composite beam. The neutral surface is the surface where strain/stress is zero. The strains/stresses below the neutral surface and above the neutral surface have opposite signs. In pure bending, the normal strain ε x and normal stress σ x in x-direction can be expressed as, respectively, z ε = − = −κz , (4.1) x R

σ x = Eiε x = −Eiκz . (4.2) where

1 κ = is the radius of curvature of the composite beam due to bending, R is the radius R of curvature, Ei is the Young’s modulus, and z is the distance on z-coordinate from the neutral surface. For small deflections of the beam compared to the length of the beam, the radius of curvature can be determined from the following equation (assume the composite Chapter 4 Composite Beam Analysis and Design 80

∂w(x,t) beam is under pure bending, that is the x-direction displacement D (x,t) = z and y- x ∂x direction displacement Dy (x,t) = 0 ) and Dz (x, z,t) = w(x,t) ):

∂ 2 w(x,t) 1 2 = − ∂x . (4.3) R 2 3/ 2 ⎡ ⎛ ∂ 2w(x,t) ⎞ ⎤ ⎢1+ ⎜ ⎟ ⎥ ⎜ ∂x 2 ⎟ ⎣⎢ ⎝ ⎠ ⎦⎥

∂2w(x,t) If the composite beam is slightly bent, then << 1, Equation (4.3) can be ∂x2 approximated as: 1 ∂2w(x,t) M κ = = − 2 = − . (4.4) R ∂x Ei I where

2 M = − zσ xdA = −κE z dA = −κEi I , (4.5) ∫∫ A ∫∫ A

I = z 2dA . (4.6) ∫∫ A

M and I are the bending moment and the area moment of inertia of the composite beam respectively.

4.2 Parameters of the two-layer piezoelectric composite beam

In this section, some important parameters of the two-layer piezoelectric composite beam depicted in Figure 4-1, such as the location of neutral surface, area moment of inertia, and normal strain/stress will be derived.

4.2.1 Neutral surface location

The neutral surface location of the composite beams is determined by the geometry sizes and the Young’s moduli of all layers. Considering the cross sectional area of a multi-layer composite beam as shown in Figure 4-2, where Ai and Ei are the cross-sectional area and Young’s modulus of the ith layer respectively. Chapter 4 Composite Beam Analysis and Design 81 Stress is a measure of the intensity of internal forces. If the body is in equilibrium, the sum of

N x-axial forces Fx on the cross sectional area is zero, that is Fx = ∑(Fx )i = 0 , where i=1

(Fx )i = (σ x )i dA. ∫∫ A i

So

Fx = (σ x )1 dA + + (σ x )i dA + + (σ x )N dA = 0 . (4.7a) ∫∫ A ∫∫ A ∫∫ A 1 i N

b (0,0) y h Layer 1, A1, E1 1 h2 Layer 2, A2, E2 hi−1 hi hN −1 hN Layer i, Ai, Ei Layer N, AN, EN

z

Figure 4-2 Cross sectional area of a multi-layer composite beam

Substituting Equations (4.2 & 4.1) to Equation (4.7a) obtain:

E1 (ε x )1 dA + + Ei (ε x )i dA + + EN (ε x ) N dA = 0 , (4.7b) ∫∫ A ∫∫ A ∫∫ A 1 i N

Or

− E1 zdA − − Ei zdA − − EN zdA = 0 . (4.7c) ∫∫ A ∫∫ A ∫∫ A 1 i N where z is the z-coordinate value of the distance from neutral surface.

Let z = zˆ − z , then Equation (4.7c) becomes:

E1 (zˆ − z)dA + + Ei (zˆ − z)dA + + EN (zˆ − z)dA = 0. (4.7d) ∫∫ A ∫∫ A ∫∫ A 1 i N

For the ith term in Equation (4.7d),

∵ z ∈ [hi−1 - z hi − z], Chapter 4 Composite Beam Analysis and Design 82

∴ zˆ ∈ [hi−1 hi ].

hi hi hi E i ( zˆ − z )dA = E i b ( zˆ − z )dzˆ = E i b zˆdzˆ − E i bz dzˆ ∫∫ A ∫ h ∫ h ∫ h i i−1 i−1 i−1 h zˆ 2 i E b = E b − E bz (h − h ) = i (h 2 − h 2 ) − E A z i 2 i i i−1 2 i i−1 i i hi−1 h + h = E b i i−1 (h − h ) − E A z = E A z − E A z . (4.8) i 2 i i−1 i i i i i i i where,

h + h z = i i−1 is the value of the centeroid, and A = b(h − h ) is the area of the ith i 2 i i i−1 layer (i = 1, 2, ..., N) respectively.

Applying Equation (3.8) to each term of Equation (4.7d), Equation (4.7d) can be rewritten as

E1 A1z1 − E1 A1z + + Ei Ai zi − Ei Ai z + + EN AN zN − EN AN z = 0 ,

Or

N N ∑ Ei Ai zi − z∑ Ei Ai = 0 . (4.9) i=1 i=1

From Equation (4.9), the neutral surface location of the multi-layer composite beam can be found by

N ∑ Ai E i z i i =1 z = N . (4.10) ∑ Ai E i i =1

For the composite beam shown in Figure 4-1, we define a thickness ratio μ that is the ratio of the PZT thickness over the beam thickness, and a Young’s modulus ratio n that is the ratio of the Young’s modulus of the PZT over the Young’s modulus of the beam respectively, that is,

h μ = p –– Thickness ratio of PZT over beam, hb

E n = p –– Young’s modulus ratio of PZT over beam. Eb

The composite beams in Figure 4-1 (b & c) have the same thickness ratio but different Young’s modulus ratio. The Young’s modulus ratio in Figure 4-1 (b) is larger than that in Chapter 4 Composite Beam Analysis and Design 83 Figure 4-1 (c). One can see that the strain in the PZT shown in Figure (b) is smaller than that in Figure (c). This is because the smaller Young’s modulus ratio in Figure (c) resulting in the neutral surfaces is further away from the PZT layer. The larger the distance from a surface to the neutral surface, the larger strain the surface has.

From Equation (4.10), the neutral surface location of the composite beam in Figure 4-1 can be obtained as

A E z + A E z h E z + h E z h z + nh z z = p p p b b b = p p p b b b = b b p p Ap Ep + Ab Eb hp Ep + hb Eb hb + nhp

2 2 hb hp hb hp h (h + ) + nh h h + + n 2 b p p b p h (μ n + 2μ +1) = 2 2 = 2 2 = b . (4.11) h + nh h + nh 2(μn +1) b p b p where

h h A = bh , A = bh , z = p , z = h + b are the cross-sectional areas and the p p b b p 2 b p 2 centeroid locations of the PZT and the beam respectively.

Figure 4-3 Neutral surface location of composite beam

Chapter 4 Composite Beam Analysis and Design 84 Figure 4-3 is the MATLAB® simulation of Equation (4.11), the neutral surface location of the composite beam, with four different thickness ratios: μ = 0.1, 0.2, 0.5, and 1 relative to the beam thickness hb = 0.003 (m) respectively. The horizontal axis represents the variable of the Young’s modulus ratio n, the vertical axis represents the location of neutral surface on z- coordinate from the x-y plane.

The location of the neutral surface will approach the centeroid of the PZT layer (hp /2) when the ratio of Young’s modulus n approaches infinity. It can be seen by subtracting the value of centeroid of the PZT from the value of the neutral surface location, that is

hp z − z = z − . (4.12) p 2

Figure 4-4 is the simulation of Equation (4.12). One can see that z − z p monotonously decreases and approaches zero when the ratio of Young’s modulus n approaches infinity.

Figure 4-4 Distance between neutral surface and centeroid of the PZT ( z − z p )

When n = 1, the beam and the PZT have the same Young’s modulus, the location of the neutral surface should be in the middle of the composite beam, that is Chapter 4 Composite Beam Analysis and Design 85

h + h z = b p , 2 Hence,

h h z − z = z − p = b . p 2 2

This has been reflected in the Figure 4-4. One can see that there is an intersection point among all traces in the figure at n = 1, where the distance between centeroid of the PZT and

−3 the location of neutral surface z − z p = hb / 2 = 1.5×10 (m) regardless of their thickness ratios.

4.2.2 Area moment of inertia

The area moment of inertia determines the curvature, hence the strain/stress of the composite beam. The area moment of inertia with respect to an axis in the plane of the area is related to the moment of inertia with respect to a parallel centeroid axis by the parallel axis theorem (Timoshenko, 1973). For a multi-layer composite beam, the area moment of inertia of the ith layer of the composite beam with respect to the neutral surface can be derived according to the definition of the area moment of inertia as below

bh 2 ⎛ i 2 ⎞ I i = (zˆ − z ) dA = ⎜ ( zˆ − z ) dzˆ ⎟dy ∫∫ Ai ∫∫ 0 ⎝ hi−1 ⎠

h h i i 2 b 3 b 3 3 = b (zˆ − z) dzˆ = (zˆ − z) = [(hi − z) − (hi−1 − z) ] ∫ hi−1 3 3 hi−1

2 b 3 ⎡⎛ hi + hi −1 ⎞ ⎤ = (hi − hi −1 ) + b(hi − hi −1 )⎢⎜ ⎟ − z ⎥ 12 ⎣⎝ 2 ⎠ ⎦ = I + A ( z − z ) 2 . (4.13) iLocal i i where

b I = (h − h )3 is the local area moment of inertia with respect to the centeroid of iLocal 12 i i −1 the ith layer.

Equation (4.13) represents the parallel axis theorem for calculating the area moment of inertia of any layer of the multi-layer composite beam.

The bending moment M is related to the sum of the stresses in the composite beam. From Equation (4.5), it can be found by Chapter 4 Composite Beam Analysis and Design 86

M = − (σ x )1 zdA − − (σ x )i zdA − − (σ x ) N zdA ∫∫ A ∫∫ A ∫∫ A 1 i N

2 2 2 = −κE1 z dA − −κEi z dA − −κEN z dA ∫∫ A ∫∫ A ∫∫ A 1 i N

N = −κE1I1 − −κEi Ii − −κEN I N = −κ ∑ Ei Ii = −κ EI . (4.14) i=1 where EI donates the equivalent flexural rigidity of the N-layer composite beam,

N N ⎛ E ⎞ N ⎜ i ⎟ EI = ∑ Ei Ii = E1 ∑⎜ ⎟Ii = E1 ∑ ni Ii . (4.15) i=1 i=1 ⎝ E1 ⎠ i=1

The moment of inertia of the entire cross-sectional area of the N-layer composite beam can be calculated as

N I t = ∑ ni I i . (4.16) i=1 where

I t is the total area moment of inertia of the composite beam, I i and ni are the area

th moment of inertia and the Young’s modulus ratio relative to E1 of the i layer respectively.

E1 is assumed to be the Young’s modulus of the first layer.

For the composite beam in Figure 4-1, one can have

h0 = 0 , h1 = hp , h2 = hp + hb ,

Ap = A1 = bhp , Ab = A2 = bhb ,

h h z = z = p , z = z = h + b . p 1 2 b 2 p 2

The local area moments of inertia of the PZT and beam layers are, respectively,

bh 3 bh 3 I = I = p , I = I = b . p Local 1Local 12 bLocal 2 Local 12

Therefore, from Equation (4.13), the area moments of inertia of the PZT layer and the beam layer about the neutral surface can be derived, respectively,

bh3 I = I + (z − z )2 A = p + (z − z )2 bh , (4.17) p pLocal p p 12 p p Chapter 4 Composite Beam Analysis and Design 87

3 2 2 bhb ⎛ hb ⎞ Ib = Ib + (z − zb ) Ab = + ⎜ z − hp − ⎟ bhb . (4.18) Local 12 ⎝ 2 ⎠ Since

h p E p z p + hb Eb zb hb Eb (zb − z p ) z − z p = − z p = h p E p + hb Eb h p E p + hb Eb

h E (h + h ) h (1 + μ ) = b b b p = b , (4.19) 2(h p E p + hb Eb ) 2(1 + μn)

h E z + h E z h E z − h E z z − z = p p p b b b − z = p p p p p b b h E + h E b h E + h E p p b b p p b b h E (z − z ) ⎛ h E ⎞ h E (h + h ) = p p p b = ⎜ − p p ⎟ b b b p h E + h E ⎜ h E ⎟ 2(h E + h E ) p p b b ⎝ b b ⎠ p p b b h (1+ μ ) = −μn(z − z ) = −μn b . (4.20) p 2(1 + μn)

Substituting Equation (4.19) into Equation (4.17) obtains

bh 3 I = I + (z − z )2 A = p + (z − z )2 bh p pLocal p p 12 p p

3 2 bh p ⎡ hb (1 + μ ) ⎤ = + bh p ⎢ ⎥ 12 ⎣ 2(1 + μn) ⎦ 2 bμ 3 h 3 ⎡ 1 + μ ⎤ = b + bμh 3 12 b ⎢ 2(1 + μn) ⎥ ⎣ ⎦ 3 5 2 4 3 2 bhb μ n + 2μ n + (4μ + 6μ + 3μ ) = 2 2 . (4.21) 12 μ n + 2μn + 1

Figure 4-5 is the plot of Equation (4.21), the area moment of inertia of the PZT layer about the neutral surface with four different thickness ratios.

Chapter 4 Composite Beam Analysis and Design 88

Figure 4-5 Area moment of inertia of the PZT layer about the neutral surface

Similarly, substituting Equation (4.20) into Equation (4.18) obtains the area moments of inertia of the beam layer about the neutral surface

3 2 2 bhb ⎛ hb ⎞ Ib = Ib + (z − zb ) Ab = + ⎜ z − hp − ⎟ bhb Local 12 ⎝ 2 ⎠

3 2 2 bhb ⎡ hb (μ n + 2μ + 1) hb ⎤ = + bhb ⎢ − h p − ⎥ 12 ⎣ 2(μn + 1) 2 ⎦

3 2 2 bhb 3 ⎡ μ n + 2μ + 1 1 ⎤ = + bhb ⎢ − μ − ⎥ 12 ⎣ 2(μn + 1) 2 ⎦ bh 3 (3μ 4 + 6μ 3 + 4μ 2 )n 2 + 2μn + 1 = b . (4.22) 12 μ 2 n 2 + 2μn + 1

Figure 4-6 is the plot of Equation (4.22) with four different thickness ratios.

Chapter 4 Composite Beam Analysis and Design 89

Figure 4-6 Area moment of inertia of the beam layer about the neutral surface

The equivalent flexural rigidity of the composite beam is

EI = Eb Ib + E p I p = Eb (Ib + nI p ) = Eb It . (4.23) where

I t = I b + nI p . (4.24) is the total area moment of inertial of the composite beam.

Substituting Equations (4.21, 4.22) into Equation (4.24) obtains,

bh 3 ⎛ (3μ 4 + 6μ 3 + 4μ 2 )n 2 + 2μn + 1 μ 5 n 2 + 2μ 4 n + (4μ 3 + 6μ 2 + 3μ ) ⎞ I = b ⎜ + n ⎟ t ⎜ 2 2 2 2 ⎟ 12 ⎝ μ n + 2μn + 1 μ n + 2μn + 1 ⎠ bh 3 μ 5 n 3 + (5μ 4 + 6μ 3 + 4μ 2 )n 2 + (4μ 3 + 6μ 2 + 5μ )n + 1 = b . (4.25) 12 μ 2 n 2 + 2μn + 1

Figure 4-7 is the plot of Equation (4.25), the total area moment of inertia of the composite beam about the neutral surface with four different thickness ratios.

Chapter 4 Composite Beam Analysis and Design 90

Figure 4-7 Total area moment of inertia of the composite beam about the neutral surface

4.2.3 Average normal strain and stress

From Equation (4.1) and Equation (4.4), the normal strain along the x-coordinate can be calculated as:

z Mz ε x = − = − . (4.26) R Ei I

For the PZT layer, the strain in the middle (centeroid) surface of the PZT is

M (z p − z) M (z − z p ) (ε p ) Ave = − = Eb Ib + E p I p Eb (Ib + nI p )

⎛ M ⎞ z − z p = ⎜ ⎟ ⎜ ⎟ 3 3 ⎝ Eb ⎠ bh 2 ⎛ bhp 2 ⎞ b + (z − z ) bh + n⎜ + (z − z ) bh ⎟ b b ⎜ p p ⎟ 12 ⎝ 12 ⎠

⎛ M ⎞ z − z = ⎜ ⎟ p ⎜ ⎟ 3 3 . (4.27) ⎝ bE b hb + nh p 2 2 ⎠ + h (z − z ) + nh (z − z ) 12 b b p p

Substituting Equations (4.19, 4.20) into Equation (4.27) obtains: Chapter 4 Composite Beam Analysis and Design 91

hb (1+ μ ) ⎛ M ⎞ 2(1+ μn) (ε ) = ⎜ ⎟ p Ave ⎜ bE ⎟ 3 3 2 2 ⎝ b ⎠ hb + nh p ⎛ hb (1+ μ ) ⎞ ⎛ hb (1+ μ ) ⎞ + hb ⎜ − μn ⎟ + nh p ⎜ ⎟ 12 ⎝ 2(1+ μn) ⎠ ⎝ 2(1+ μn) ⎠

hb (1+ μ ) ⎛ M ⎞ 2(1+ μn) = ⎜ ⎟ ⎜ bh 3 E ⎟ 3 2 ⎝ b b ⎠ 1+ nμ 2 2 ⎛ 1+ μ ⎞ + (μ n + nμ )⎜ ⎟ 12 ⎝ 2(1+ μn) ⎠

⎛ 6M ⎞ μ +1 = ⎜ ⎟ . (4.28) ⎜ bh 2 E ⎟ μ 4 n 2 + (4μ 3 + 6μ 2 + 4μ )n +1 ⎝ b b ⎠

The plot of the average strain in the PZT with four different thickness ratios is shown in Figure 4-8. The Young’s modulus of the beam and bending moment used in the simulation are Eb = 7 (GPa) and M =100 (N ⋅m) respectively.

Figure 4-8 Average strain of the PZT layer (on the middle surface of the PZT)

Similarly, for the beam layer, the strain in the middle (centeroid) surface of the beam is Chapter 4 Composite Beam Analysis and Design 92 M (z − z) M (z − z ) (ε ) = − b = b b Ave E I + E I E (I + nI ) b b p p b b p ⎛ M ⎞ z − z = ⎜ ⎟ b ⎜ ⎟ 3 3 ⎝ Eb ⎠ bh ⎛ bhp ⎞ b + (z − z )2 bh + n⎜ + (z − z )2 bh ⎟ b b ⎜ p p ⎟ 12 ⎝ 12 ⎠ h (1+ μ ) − μn b ⎛ M ⎞ 2(1+ μn) = ⎜ ⎟ ⎜ bE ⎟ 3 3 2 2 ⎝ b ⎠ hb + nh p ⎛ hb (1+ μ ) ⎞ ⎛ hb (1+ μ ) ⎞ + hb ⎜ − μn ⎟ + nh p ⎜ ⎟ 12 ⎝ 2(1+ μn) ⎠ ⎝ 2(1+ μn) ⎠ h (1+ μ ) μn b ⎛ − M ⎞ 2(1+ μn) = ⎜ ⎟ ⎜ 3 ⎟ 3 2 ⎝ bhb Eb ⎠ 1+ nμ ⎛ 1+ μ ⎞ + (μ 2n 2 + nμ )⎜ ⎟ 12 ⎜ 2(1+ μn) ⎟ ⎝ ⎠ ⎛ − 6M ⎞ (μ 2 + μ )n = ⎜ ⎟ ⎜ 2 ⎟ 4 2 3 2 . (4.29) ⎝ bhb Eb ⎠ μ n + (4μ + 6μ + 4μ )n +1

Figure 4-9 Average strain of the beam layer (on the middle surface of the beam)

The plot of average strain in the beam with four different thickness ratios is shown in Figure 4-9. Figure 4-8 and Figure 4-9 show that the strain in the beam and the strain in the PZT have Chapter 4 Composite Beam Analysis and Design 93 opposite sign, and when n is not too large, the average strain of the PZT is much larger than that of the beam.

The stress σ can be obtained from equation σ = Eiε , where Ei is the Young’s modulus (unit: Newton/Meter2 or Pascal).

The average stress of the PZT layer is

⎛ 6M ⎞ (μ +1)n (σ ) = E (ε ) = nE (ε ) = ⎜ ⎟ p Ave p p Ave b p Ave ⎜ 2 ⎟ 4 2 3 2 . (4.30) ⎝ bhb ⎠ μ n + (4μ + 6μ + 4μ)n +1

The average stress of the beam layer

⎛ − 6M ⎞ μ(μ +1)n (σ ) = E (ε ) = ⎜ ⎟ b Ave b b Ave ⎜ 2 ⎟ 4 2 3 2 . (4.31) ⎝ bhb ⎠ μ n + (4μ + 6μ + 4μ)n +1

One can notice that the ratio of average stress in the beam over average stress in the PZT is (σ ) equal to the thickness ratio μ but in opposite sign, i.e. b Ave = −μ . When μ = 1, this (σ p ) Ave means the average stress in the beam and the PZT are the same but with negative sign. Negative sign means that one is in compression and the other is in extension or vice versa. This is consistent with beam theory.

Figure 4-10 Average stress of the PZT layer (on the middle surface of the PZT) Chapter 4 Composite Beam Analysis and Design 94 Figure 4-10 and Figure 4-11 are the plots of the average stress in the PZT and the average stress in the beam respectively.

Figure 4-11 Average stress of the beam layer (on the middle surface of the beam)

4.3 Discussion of composite beam design for maximizing strain in the PZT

In this section, the relationship between the electrical field and the mechanical strain of piezoelectric materials and how to select the Young’s modulus ratio and thickness ratio based on maximizing the strain in the PZT are discussed. Some FEA simulation and experimental results of the piezoelectric vibration shunt control for an aluminium beam, a wooden beam and a carbon-fibre beam are presented. Carbon-fibre composite materials and their layout configuration are briefly introduced. A preferable design of the Cricket bat handle is proposed. The benefits of using carbon-fibre material in the Cricket bat handle are described.

4.3.1 The relation between electrical field and mechanical strain in the PZT

As mentioned in Chapter 2, the essence of “piezoelectric effect” is the relation between electric polarization and mechanical strain in crystals. When mechanically strained, the piezoelectric materials produce electric polarization that is in proportion to the strain. When electrically polarized, the piezoelectric materials produce mechanical strain that is in Chapter 4 Composite Beam Analysis and Design 95 proportion to the polarizing field. For the passive piezoelectric vibration control, the former mechanism is used; for the active vibration control, the latter mechanism is used. The focus here will be on how to maximize the mechanical strain in the PZT for the passive control. The larger strain the PZT has, the larger electrical field it can produce from electric polarization. This can be reflected by the h-form constitutive relation of piezoelectric materials (IEEE 1987)

⎧σ = E pε − hD ⎨ . (4.32) ⎩E = − hε + βD where

E –– Electrical field (V/m) 2 Ep –– Young’s modulus of PZT (N/m or Pa) β –– Dielectric impermeability (m/F or V-m/C) D –– Electrical displacement (C/m2) ε –– Mechanical strain h –– Electrical field-strain coefficient of PZT (V/m or N/C) N –– Newton, m –– Meter, V –– Volt, C –– Coulomb, F –– Farad

Reorganize the first equation of Equation (4.32)

σ = Epε − hD = E'p ε . (4.33)

From Equation (4.33), one can have E − E' D = p p ε . (4.34) h

If D = 0, E'p = Ep , which is the Young’s modulus without considering electrical field or the electrodes of the PZT at short circuit.

Substituting Equation (4.34) into the second equation of Equation (4.32) obtains

Ep − E'p ⎛ β (Ep − E' p ) ⎞ E = −hε + β ε = ⎜ − h⎟ε . (4.35) h ⎝ h ⎠

Equation (4.35) indicates that the electrical field E produced in the PZT is in proportion to its strain ε . Larger strain will produce larger electrical field and lead to a larger output voltage, therefore, more mechanical energy can be converted into electrical energy that can be dissipated by an electrical network. Figure 4-8 has shown that the average strain in the PZT layer monotonously decreases when the ratio of Young’s modulus n increases but the average Chapter 4 Composite Beam Analysis and Design 96 stress in the PZT layer as shown in Figure 4-10 does not have the same trend. This is owing to the distance between the centeroid of the PZT and the neutral surface getting narrower when n increases, therefore the average strain becomes smaller. On the other hand, stress is the product of strain and Young’s modulus. The increase of the Young’s modulus of the PZT compensates for the decrease of strain in the PZT. This is the reason why the average stress in the PZT layer does not follow the same trend as the strain when n increases. However, the average strain in the beam layer shown in Figure 4-9 has the same trend as the average stress in the beam layer as shown in Figure 4-11 since they are related by a constant Young’s modulus. It is important to stress that the electrical field produced by the PZT is in proportion to its strain rather than to its stress. In order to have more strain in the PZT layer, from Figure 4-8, one can see that the Young’s modulus ratio needs to be small.

Figure 4-12 is the zoomed plot of Figure 4-8. For example, looking at a composite beam which has thickness ratio μ = 0. 1, from Figure 4-12, one can see that the average strain for n = 10 is about one-quarter of that for n = 1. The purpose for this analysis is to understand the results of FEA simulation as well as experiments which show that vibration reduction for the wooden beam is much less than the reduction for the aluminium or carbon-fibre beam using the passive piezoelectric vibration shunt control method. With this control method, a PZT patch used as a transducer is firmly attached on a cantilever beam. The whole structure can be treated as a composite beam.

Figure 4-12 Average strain in the PZT layer (the zoomed plot of Figure 4-8) Chapter 4 Composite Beam Analysis and Design 97 This can be explained that because the Young’s modulus of wood is about one-tenth of the Young’s modulus of the PZT (i.e. n = 10) while the Young’s modulus of the aluminium and carbon-fibre beam is close to the Young’s modulus of the PZT (i.e. n ≈ 1), the average strain in the PZT layer is much smaller when the PZT is attached to the wooden beam compared to the average strain in the PZT layer when the PZT is attached to the aluminium or carbon-fibre beam. The FEA simulation and experimental results of the piezoelectric vibration shunt control for the beam with different material are illustrated in the next two subsections. The results have shown that the vibration reduction is only about 8% for the wooden beam and 40% ~ 90% for the aluminium and carbon-fibre beam. The thickness ratio of the composite beam μ used in simulations and experiments is around 0.1 ~ 0.15.

In order to maximize the strain in the PZT layer, the Young’s modulus ratio should be as small as possible, especially for the composite beams with a large thickness ratio. Figure 4-13 is the further zoomed plot of Figure 4-12 (for n ≤1). One can see that in this region, the average strains are much larger than those in other regions but falls sharply with large thickness ratios such as μ =1. The Young’s modulus ratio n ≤ 1 means the base material is stiffer than the PZT material. So the piezoelectric vibration shunt control can have better effect on stiffer structures. If the Young’s modulus ratio is extremely small (say n < 0.04), the composite beam with a large thickness can attract much more strain in the PZT.

Figure 4-13 Average strain in the PZT layer (the zoomed plot of Figure 4-12) Chapter 4 Composite Beam Analysis and Design 98 Figure 4-14 shows the plot of the average strain of the PZT against the thickness ratio, μ , with six different Young’s modulus ratio n. From this Figure, one can see that for very small n, there is an optimal μ to have maximum average strain in the PZT. But for larger n, the average strain of the PZT monotonously decreases when μ increases. This tells us that normally, in order to have a large strain in the PZT, the PZT patch should be much thinner than the substrate structure unless the Young’s modulus ratio is extremely small ( n <0.02). However in theory, when the Young’s modulus ratio is extremely small, which means that the substrate structure is much stiffer than the PZT, then the PZT can acquire very large strain when the substrate structure is thinner than the PZT patch. In such a case, there is an optimum strain value when an optimal thickness of the substrate structure is chosen. In reality, such small Young’s modulus ratio will not be possible unless the Young’s modulus of the PZT is small, such as using soft PZT (known as piezoelectric fibre composite or active fibre composite –– AFC). Otherwise, the substrate structure should always be thicker than the PZT patch.

Figure 4-14 Average strain in the PZT layer against the thickness ratio

Chapter 4 Composite Beam Analysis and Design 99 4.3.2 Analytical precursor to the strain analysis in the PZT Patch

In this study, a few different analysis methods had been tried to understand why the piezoelectric vibration shunt control had worked well for aluminium and carbon-fibre beams but not for wooden beams before the strain analysis method was used. One of them was the strain energy method that focused on the strain energy contained of the PZT. The result was not quite convincing and did not match the simulation and experiment very well. However, as part of this study, it is worthwhile to document it here. The method focused on strain energy rather than strain. The method analyzed the strain energy contained in the PZT relative to the total energy of the composite beam.

With the Euler-Bernoulli beam model, the strain energy of a uniform beam due to bending is

2 2 L L ⎛ ⎞ 1 1 ⎛ 2 ⎞ 1 ⎜ ⎡ ∂ w⎤ ⎟ Ub = εbσ bdVb = ⎜ Ebεb dAb ⎟ dx = Eb ⎢(zˆ − z) 2 ⎥ dAb dx 2 ∫∫∫ Vb 2 ∫∫∫0 ⎝ Ab ⎠ 2 ∫∫∫ 0 ⎜ Ab ∂x ⎟ ⎝ ⎣ ⎦ ⎠

2 2 2 2 L L 1 ⎛ 2 ⎞ ⎛ ∂ w⎞ 1 ⎛ ∂ w⎞ = Eb ⎜ (zˆ − z) dAb ⎟ ⎜ ⎟ dx = Eb Ib ⎜ ⎟ dx . (4.36) ∫∫∫ 0 A ⎜ 2 ⎟ ∫ 0 ⎜ 2 ⎟ 2 ⎝ b ⎠ ⎝ ∂x ⎠ 2 ⎝ ∂x ⎠

According to Equation (4.4), the radius of curvature of the composite beam due to bending can be expressed as

∂2w M M κ = − 2 = − = − . (4.37) ∂x EI EbIb + Ep I p

From Equation (4.37), one can see that κ is independent of x for uniform beams, applying Equations (4.37, 4.22 & 4.25) to Equation (4.36) obtain the strain energy of the beam,

2 2 2 2 2 L 1 ⎛ ∂ w ⎞ 1 M L M L Ib M L Ib U b = Eb Ib ⎜ ⎟ dx = Eb Ib = = ∫0 ⎜ 2 ⎟ 2 2 2 2 ⎝ ∂x ⎠ 2 (Eb Ib + E p I p ) 2Eb (Ib + nI p ) 2Eb It

⎛ 6M 2 L ⎞ [(3μ 4 + 6μ 3 + 4μ 2 )n2 + 2μn +1](μ 2n2 + 2μn +1) = ⎜ ⎟ . (4.38) ⎜ 3 ⎟ 5 3 4 3 2 2 3 2 2 ⎝ bhb Eb ⎠ [μ n + (5μ + 6μ + 4μ )n + (4μ + 6μ + 5μ)n +1]

Similarly, the strain energy of the PZT is

2 2 L L ⎛ ⎞ 1 1 ⎛ 2 ⎞ 1 ⎜ ⎡ ∂ w⎤ ⎟ U p = ε pσ pdVp = ⎜ E pε pdAp ⎟ dx = E p (zˆ − z) dAp dx ∫∫∫ V ∫∫∫0 A ∫∫∫0 A ⎢ 2 ⎥ 2 p 2 ⎝ p ⎠ 2 ⎜ p ∂x ⎟ ⎝ ⎣ ⎦ ⎠ Chapter 4 Composite Beam Analysis and Design 100

2 2 2 2 L L 1 ⎛ 2 ⎞ ⎛ ∂ w⎞ 1 ⎛ ∂ w⎞ = E p ⎜ (zˆ − z) dAp ⎟ ⎜ ⎟ dx = Ep I p ⎜ ⎟ dx ∫∫∫ 0 A ⎜ 2 ⎟ ∫ 0 ⎜ 2 ⎟ 2 ⎝ p ⎠ ⎝ ∂x ⎠ 2 ⎝ ∂x ⎠

2 2 2 1 M L M L nI p M L nI p = E p I p 2 = 2 = 2 2 (Eb Ib + E p I p ) 2Eb (Ib + nI p ) 2Eb It

⎛ 6M 2 L ⎞ [μ 5n 3 + 2μ 4n 2 + (4μ 3 + 6μ 2 + 3μ )n](μ 2n 2 + 2μn +1) = ⎜ ⎟ . (4.39) ⎜ 3 ⎟ 5 3 4 3 2 2 3 2 2 ⎝ bhb Eb ⎠ [μ n + (5μ + 6μ + 4μ )n + (4μ + 6μ + 5μ )n +1]

The total strain energy of the composite beam then is:

2 2 2 2 1 L ⎛ ∂ w ⎞ 1 L ⎛ ∂ w ⎞ U = U b +U p = (Eb Ib + E p I p )⎜ ⎟ dx = (Eb Ib + E p I p ) ⎜ ⎟ dx ∫ 0 ⎜ 2 ⎟ ∫ 0 ⎜ 2 ⎟ 2 ⎝ ∂x ⎠ 2 ⎝ ∂x ⎠

⎛ 6M 2 L ⎞ (μ 2n2 + 2μn +1) = ⎜ ⎟ . (4.40) ⎜ 3 ⎟ 5 3 4 3 2 2 3 2 ⎝ bhb Eb ⎠ [μ n + (5μ + 6μ + 4μ )n + (4μ + 6μ + 5μ)n +1]

From Equations (4.36, 4.37, & 4.38), the ratio of the strain energy of the PZT over the strain energy of the beam and the ratio of the strain energy of the PZT over the total strain energy of the composite beam can be expressed respectively, as:

5 3 4 2 3 2 U p μ n + 2μ n + (4μ + 6μ + 3μ)n = 4 3 2 2 (4.41) Ub (3μ + 6μ + 4μ )n + 2μn +1

U μ 5n3 + 2μ 4n2 + (4μ 3 + 6μ 2 + 3μ)n p = . (4.42) U μ 5n3 + (5μ 4 + 6μ 3 + 4μ 2 )n2 + (4μ 3 + 6μ 2 + 5μ)n +1

Figures (4-15, 4-16 & 4-17) are the plots of the strain energy of the PZT (Equation (4.39)), the strain energy of the beam (Equation (4.38)), and the total strain energy of the composite beam (Equation (4.40)) with Eb = 7 (GPa) and M =100 (N ⋅m) respectively.

Figure 4-18 is the plot of Equation (4.41) and Figure 4-19 is the plot of Equation (4.42) respectively.

Chapter 4 Composite Beam Analysis and Design 101

Figure 4-15 Strain energy of the PZT (U p )

Figure 4-16 Strain energy of the beam (Ub ) Chapter 4 Composite Beam Analysis and Design 102

Figure 4-17 Total strain energy of the composite beam (Ub +U p )

Figure 4-18 Ratio of strain energy of the PZT over strain energy of the beam (U p /)Ub Chapter 4 Composite Beam Analysis and Design 103

Figure 4-19 Ratio of strain energy of the PZT over total strain energy of the composite beam

(U p /)U

Figure 4-15 has shown that for each composite beam with different thickness ratios, there is an optimal value of strain energy contained in the PZT at certain Young’s modulus ratio, which means that if the Young’s moduli of the beam and the PZT are selected properly, the strain energy in the PZT can be optimized. However, one can find that the strain energy in the PZT, for example a composite beam of μ = 0. 1, at n = 1 and that at n = 10 do not differ a great deal. The difference is not big enough to explain the vibration control efficiency mentioned previously. The reason is that the strain energy of the PZT, similar to stress, is not only in proportion to its strain, but also to its Young’s modulus. The reduction of the strain in the PZT layer when n increases is compensated by the increase in the Young’s modulus. Large strain energy of the PZT does not necessarily mean large strain in the PZT, as there is a Young’s modulus influence as seen in Equation 4.35a, for example.

U p /Ub and U p /U as shown in Figure 4-18 and Figure 4-19 will approach infinity and 100% respectively when n approaches infinity. This means that the PZT will ultimately possess all the strain energy when n becomes infinity. This is understandable since when the PZT is much stiffer than the beam, the beam structure can be totally neglected. Certainly the PZT layer will possess all the strain energy although there is little strain in it. Since the electrical Chapter 4 Composite Beam Analysis and Design 104 field produced by the PZT is in proportion to strain only, the strain energy method cannot adequately reflect the efficiency of energy conversion between mechanical energy and electrical energy, therefore, it is unable to properly explain the efficiency of the piezoelectric vibration shunt control for different beams.

4.3.3 Comparison of FEA simulation results of piezoelectric vibration shunt control for beams with different materials

The simulations were carried out using FEA software package ANSYS®. The details of the software and simulation will be presented in Chapter 5. The results shown below are some harmonic analysis of the beam vibration with and without the piezoelectric vibration shunt control.

Figure 4-20 shows the vibration spectrum of the aluminium beam. The boundary condition of the simulation is clamped-free. The point of vibration measured is at tip of the beam. The

Young’s modulus of the aluminium beam is Eb = 70 (GPa) . (a) is the spectrum before shunt, (b) ~ (e) are the spectra of 1st to 4th modes after shunt respectively. The graphs show that the amplitudes of 1st to 4th modes have been reduced by about 84%, 70%, 95%, and 92% (or 16 dB, 10.5 dB, 26 dB, and 22 dB) respectively.

(a) Before shunt

Chapter 4 Composite Beam Analysis and Design 105

(b) Shunt 1st mode

(c) Shunt 2nd mode

Chapter 4 Composite Beam Analysis and Design 106

(d) Shunt 3rd mode

(e) Shunt 4th mode

Figure 4-20 Harmonic analysis of the aluminium beam vibration (simulated using clamped- free boundary condition)

Figure 4-21 shows the vibration spectrum of the wooden beam. The boundary condition of the simulation is free-free. The point of vibration measured is at tip of the beam. The Young’s modulus of the wooden beam is Eb = 6.67 (GPa). (a) is the spectrum before shunt, (b) ~ (e) Chapter 4 Composite Beam Analysis and Design 107 are the spectra of 1st to 4th modes after shunt respectively. One can see that little vibration amplitude drops occur at natural frequencies for the wooden beam after shunt. This is expected from the analysis in the previous section.

(a) Before shunt

(b) Shunt 1st mode Chapter 4 Composite Beam Analysis and Design 108

(c) Shunt 2nd mode

(d) Shunt 3rd mode

Chapter 4 Composite Beam Analysis and Design 109

(e) Shunt 4th mode

Figure 4-21 Harmonic analysis of the wooden beam vibration (simulated using free-free boundary condition)

In order to improve the efficiency of vibration shunt control, a composite beam was constructed as shown in Figure 4-22. The left section is the wooden beam, the right section is the carbon-fibre beam. The Young’s modulus of the carbon-fibre beam is Ec = 66 (GPa) . The PZT patch is attached on the upper surface of the carbon-fibre beam.

PZT Carbon-fibre

Wood

Figure 4-22 Composite beam with a PZT patch attached

Figure 4-23 shows the first four bending modes of the joint beam obtained from modal analysis and Figure 4-24 shows the vibration spectrum of the joint beam obtained from harmonic analysis. (a) shows the spectrum before shunt, (b) ~ (d) show the spectra of 1st to 3rd Chapter 4 Composite Beam Analysis and Design 110 modes after shunt respectively. The boundary condition of both simulation is free-free. The point of vibration measured in Figure 4-24 is at tip of the beam.

Figure 4-23 Modal analysis of the carbon-fibre beam (simulated using free-free boundary condition)

(a) Before shunt Chapter 4 Composite Beam Analysis and Design 111

(b) Shunt 1st mode

(c) Shunt 2nd mode

Chapter 4 Composite Beam Analysis and Design 112

(d) Shunt 3rd mode

Figure 4-24 Harmonic analysis of the carbon-fibre beam vibration (simulated using free-free boundary condition)

The graphs show that the amplitudes of 1st to 3rd modes have been reduced by about 48%, 50%, and 83% (or 5.68 dB, 6 dB, and 15.4 dB) respectively. The reduction of vibration amplitude at natural frequencies is markedly improved from the wooden beam. However, it is not as good as for the aluminium beam since the carbon-fibre beam is only a part of the joint beam, much shorter than the length of the aluminium beam. The carbon-fibre beam section has been bent much less than the aluminium beam has. Nevertheless, the PZT can still get more strain when attached on the carbon-fibre beam than when attached on the wooden beam. The result of vibration reduction for the joint beam is between the reduction for the aluminium beam and the reduction for the wooden beam. During the experiment, the PZT was attached on the carbon-fibre beam that has similar physical size and Young’s modulus as the aluminium beam. The results of vibration reduction for the aluminium beam and for the carbon-fibre beam were comparable which will be shown in Chapter 6.

Chapter 4 Composite Beam Analysis and Design 113 4.3.4 Application of the PZT-composite beam technology and analysis of the handle of the Cricket bat - Introduction of carbon-fibre composite material

Carbon-fibre materials can be an alternative for many materials in structures. The properties of the carbon-fibre materials can be specified by designers. Carbon fibre materials are stiff, light, strong and rust free.

The Young’s modulus of carbon fibre materials, classified as engineering composites in Figure 4-25 (Cebon, et al, 1994), can be from a few GPa to several hundred GPa, even greater than that of steel while the density is much lower than steel (for example, the unidirectional multi-layer CFRP shown in Figure 4-25 has a density about 1600 Kg/m3 in comparison with 7850 Kg/m3 of steel). As the carbon density is low, the specific rigidity is very high.

Figure 4-25 The Young’s modulus of materials against density (Cebon, et al, 1994) Chapter 4 Composite Beam Analysis and Design 114 Carbon fibre materials usually consist of a multi-layer of thin, unidirectional carbon fibre sheets. The unidirectional carbon fibre sheets are a mixture of epoxy and carbon fibres, which are made out of long, thin filaments of carbon similar to graphite. These layers are then compacted together and subjected to a curing process involving raised temperatures and pressures to obtain the rigid and light weight final product

The fibre orientation of the multi-layer carbon fibre sheets can be of different combinations, such as a combination of (0, +45, -45, +90, +90, -45, +45, 0) as shown in Figure 4-26.

Figure 4-26 Structure of a multi-layer carbon-fibre material

The combination is selected based on the stiffness required. 0 degrees is parallel to the long axis of the frame member. Naturally, this gives the largest stiffness. +90 degrees is vertical to the long axis of the frame member. It gives least stiffness. Other degrees give stiffness varying between the two.

In order to remove more vibration energy from Cricket bat handle, the carbon fibre strips were used as a part of the Cricket bat handle. The carbon-fibre strips are unidirectional laminates using a 120 °C curing unidirectional carbon fibre prepreg tape (unidirectional fibres in an epoxy matrix in tape form) manufactured by Advanced Composites Group. Variable stiffness polymeric composite strips were designed and manufactured in order to locate the first anti-node of the Cricket bat in the handle area. The lower 1/3 of the strip (part A in the Figure 4-27, at the blade-handle interface) is one 12-pile with a combination of (0, -45, +45,

+90, +45, -45)s which is softer. The upper 2/3 of the strip (part B in the Figure 4-27, at the Chapter 4 Composite Beam Analysis and Design 115 handle area) is another 12 plies with a stiffer combination of (0, 0, 0, -45, +45, -45)s which is stiffer. 0 degrees fibre orientation is in-line with the handle’s longitudinal axis. The subscript ‘s’ represents a symmetric laminate and the orientation with other 6 plies repeated in symmetric order.

A B

Figure 4-27 The variable stiffness lay-up combination for a 12-ply carbon-fibre strip

It needs to be mentioned here that the ply arrangement described above evolved after numerous experimental trials with various arrangements for obtaining the optimal PZT patch- shunt circuit effectiveness.

4.3.5 Proposed design of Cricket bat handle

Cricket bat vibration control is produced by controlling the vibration at the bat handle. The handle of Cricket bats is traditionally made of wood. As demonstrated in the preceding subsection in this chapter, if the PZT is directly attached to wood, little strain can be transferred into the PZT. The solution is to use the carbon-fibre strip attached to the whole handle area as PZT-shunt circuit effectiveness interfacing medium, and then attach the PZT patch somewhere on the carbon-fibre strip. Since the stiffness of carbon-fibre material can be very large and is specified by the designer, enhanced vibration control effects can be achieved by using the carbon-fibre strip. The results of FEA simulations and experiments have shown remarkable vibration reduction for the carbon-fibre beam using the passive piezoelectric vibration method. However, if the carbon-fibre strip is attached to one side of the wooden beam, there is still the problem of little strain being produced in the carbon-fibre beam such as when the PZT patch is attached to the wooden beam. In order to have high strain in the carbon-fibre layer, two identical carbon-fibre strips were attached at both sides of a wooden beam, which makes a sandwich beam. The cross sectional area of the sandwich beam is shown in Figure 4-28. Wooden material is sandwiched between two carbon-fibre strips attached to both outer side surfaces of the wooden beam.

Chapter 4 Composite Beam Analysis and Design 116

Figure 4-28 Cross sectional of the carbon-fibre and the wood sandwich beam

Since the sandwiched beam is symmetric along the centeroid, the neutral surface is in the middle of the composite beam cross sectional area. From Figure 4-28, it is seen that with the sandwiched beam structure, the strain in carbon-fibre layers is larger than the strain in the wooden layer regardless of large Young’s modulus of carbon-fibre material. One might say why don’t we directly attach the PZT patches to both side of the handle to make a sandwiched structure? The PZT patch is about 70 mm long and can only cover a small part of handle. Since the PZT is much stiffer than wood, the anti-node will not reside in the area where the PZT patches are attached. Therefore the PZT patches will not bend much. This is the reason the PZT patches are not directly attached to both side of the wooden handle.

The PZT patch and the carbon-fibre strip of the sandwiched beam can be treated as another composite beam structure, i.e., the PZT-carbon-fibre beam structure. Since the Young’s modulus ratio of the PZT-carbon-fibre composite beam is much smaller than the Young’s modulus ratio of the PZT-wood composite beam, more strain can be produced in the PZT layer when the PZT patch is attached on the carbon-fibre strip than when it is attached directly on the wooden beam. Figure 4-29 is the cross sectional area of the sandwiched beam attached with the PZT patch.

Figure 4-29 Cross sectional area of the sandwiched beam with the PZT attached

Chapter 4 Composite Beam Analysis and Design 117 If the carbon-fibre strip is made much stiffer and thicker than the PZT patch, the structure alternation of the sandwiched beam due to the PZT patch attached can be neglected. Besides, for the PZT-carbon-fibre composite beam structure, a softer and thinner PZT patch can result in more strain in the PZT layer than that in the carbon-fibre beam layer. In order to maximize the strain in the PZT layer, the analysis above as shown in Figure 4-12 will dictate the following for maximizing the strain in the PZT patch for a given young’s modulus ratio:

1) Small Young’s modulus ratio between the PZT patch and carbon-fibre strip, which means the carbon-fibre strip should be made as stiff as possible.

2) Small thickness ratio between the PZT patch and the carbon-fibre strip, which means the carbon-fibre strip should be made thicker than the PZT patch unless their Young’s modulus ratio is extremely small (in the case of soft PZT being used).

Practically, if the carbon-fibre strip is made too stiff, the anti-nodes of the Cricket bat, especially that of first mode, will not be present in the handle but in the blade area, which means that the Cricket bat handle will barely bend. This will reduce the strain in the PZT patch since the PZT patch is only allowed to be placed in the handle area. This is the reason why it is preferred to use the variable stiffness carbon-fibre strips to control the anti-node positions. There is a compromise between stiffening the carbon-fibre strip and positioning the first anti-node within the handle area. On the other hand, if the carbon-fibre strip is made too thick, it will also shift the first anti-node out of the bat handle. So the design should be made accordingly.

4.3.6 Other benefits of using carbon-fibre material in the Cricket bat

The benefit of using carbon-fibre strips in the Cricket bat handle mentioned in the last section is that it can increase the normal strain in the PZT patch. Another benefit of using the carbon- fibre material is that the carbon-fibre material can help to increase the natural frequencies of the Cricket bats, as a result of increasing the stiffness of the bat handle. Structural vibration is caused by the excitation force that contains frequency components matching the natural frequencies of the structure. When the carbon-fibre strips are stiff enough, the stiffness of the bat handle will be determined mainly by the carbon-fibre strips. The influence of the wooden material in the bat handle can almost be neglected. High frequency vibration usually feels more comfortable than low frequency vibration and is also easier for the materials to absorb.

In terms of shunt circuit technology, high vibration frequency is easier to use than a large inductor which should be avoided. Chapter 4 Composite Beam Analysis and Design 118 Last but not least is that it can help to shift some natural frequencies from the inside of excitation frequency to the outside of excitation frequency and reduce the excitation energy at natural frequencies of the Cricket bat. When a batsman hits a ball, the ball-bat collision produces an excitation impulse of force. The bat vibration is the impulse response of the excitation impulse. The frequency range of the excitation impulse is related to the ball-bat contact time and is in inverse proportion to the contact time. The contact time of an ideal impulse is zero corresponding to infinity frequency range. However, in reality the contact time is nonzero, is about 1 millisecond (Grant, C. 1998). The frequency range of the excitation impulse therefore is about 1 kHz.

Figure 4-30 shows an impulse with width Tw =1 millisecond and the corresponding power spectrum. The power spectrum is the square of the Fourier transform of the impulse. The first zero crossing frequency of the spectrum is fc =1/Tw =1 kHz. The narrower the impulse is, the larger the fc will be.

It is seen that the magnitude of the power spectrum declines when frequency increases in the first lobe (within 1 kHz range). This means the energy of the impulse is lower for higher frequency components. The excitation energy of impulse signals mainly resides in the first lobe. Considering the Cricket bat as a linear system, lower excitation energy certainly causes smaller vibration response. When the energy of the frequency components is lower than a certain threshold, the vibration response caused by these frequency components can be neglected. One can set a threshold, such as 1/3 of maximum energy, for a structure. The frequency at this threshold is defined as an excitation frequency. The vibration outside the excitation frequency can be neglected. So the vibration of the Cricket bat caused by the natural frequencies outside the excitation frequency can be neglected. Lifting up the natural frequencies can firstly, shift some natural frequencies from the inside of the excitation frequency to the outside of the excitation frequency, and secondly, relatively reduce the excitation energy at the natural frequencies, and therefore ease the vibration response.

Chapter 4 Composite Beam Analysis and Design 119

Figure 4-30 Impulse signal and its corresponding power spectrum

4.4 Chapter summary

In this chapter, strain distribution in the composite beams has been analyzed in detail. The average strain in each layer is decided by the physical size of the layers as well as the Young’s modulus of materials used. This is a result of the neutral surface location of the composite beams which plays an important role in the strain distribution. If the neutral surface is close to the centeroid of a layer, the average strain of the layer will be reduced. In order to allocate more strain in a preferred layer, the composite beam should be designed to bring the neutral surface away from this layer. Simulations with different thickness ratio and Young’s modulus ratio of a two-layer composite beam have been conducted. The simulation results help us to understand why the FEA simulation and experimental results of the passive piezoelectric vibration shunt control all showed that the vibration control for the aluminium or carbon-fibre beam is more effective than that for the wooden beam. The analysis provides an analytical tool for applying the passive piezoelectric vibration shunt control method to multi- material- based substrate structures. The FEA simulations and experimental results of piezoelectric vibration shunt control for the aluminium, wooden and carbon-fibre beams are presented. Carbon-fibre materials and the layer structure are briefly introduced. Chapter 4 Composite Beam Analysis and Design 120 Finally, the design of the Cricket bat handle has been proposed in order to maximize the removal of vibration energy from the Cricket bat handle with the passive piezoelectric vibration shunt control method and the benefits of using carbon-fibre strips in the Cricket bat handle are also described. Other advantages of the inherent increase in the natural frequency of the structure when using carbon-fibre strips are presented. This is argued from the drastically reduced power spectrum characteristics for frequencies above a threshold value. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 121

Chapter 5 Numerical Modelling of Passive Piezoelectric Vibration Control using the FEA Software ANSYS®

In previous chapters, analytical modelling and analysis of the structure and vibration control have been conducted. However, the mathematical equations of accurate analytical models of complex geometry structures are usually very difficult to be solved. Instead of solving a complex structure as a whole, numerical methods based on the idea of dividing the complex structure into many simply shaped portions are often used to approximate the accurate model. The small portions are solved individually and then combined. Finite element analysis (FEA) is one of the major numerical solutions. With today’s powerful computer capabilities and sophisticated algorithms, most engineering problems can be solved accurately with the FEA method. There are many commercially available FEA software packages, such as ANSYS® (General purpose on PC and workstations), NASTRAN® (General purpose on mainframes), ABAQUS® (Non-linear and dynamic analyses), PATRAN® (Pre/Post processor), etc.

Efficient vibration control involves obtaining the optimum geometry size, material property of a structure and the effective placement of the piezoelectric transducer. The piezoelectric materials are quite expensive at the moment and are not reusable after being bonded to an object. Therefore, it will be quite costly in both time and resources rely purely on experiment. A computer simulation method of the passive piezoelectric vibration shunt control with ANSYS® has been developed. It is a useful alternative to an experimental approach. The method can help us to analyze, design and optimize the structure and control system prior to actual physical experimentation and implementation. It has many advantages, such as flexibility, re-programmability, time and cost efficiency. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 122 In this chapter, the ANSYS® software package is briefly introduced in section 1. Section 2 gives a simple FEA formulation example. In section 3, the piezoelectric shunt circuits are analyzed with PSpice® program. Section 4 details the modelling of the passive piezoelectric vibration shunt control with the ANSYS®. The simulation of model analysis, spectrum analysis and vibration control for beams are presented. The modal and spectrum analyses of various designs of the proposed Cricket bat are presented in section 5.

5.1 Introduction to ANSYS®

ANSYS® is a general-purpose finite element analysis program with powerful pre- and post- processing capabilities. The program can be used in all disciplines of engineering - structural, mechanical, electrical, electromagnetic, electronic, coupled-field, thermal, fluid, biomedical and so on (Lawrence, 2002, University of Alberta, [I 13]). ANSYS® offers many analysis types such as static, modal, harmonic, transient, spectrum, eigenvalue buckling, and sub structuring. The analysis can be either linear or non-linear. The software is organized into several processors. A typical ANSYS® analysis involves using three processors.

1. Pre-processing (PREP7 processor), where the user provides data such as the geometry, material property, and element types to the program; meshes elements, creates nodes, applies constraints, etc.

2. Solution (SOLUTION processor), where the user defines analysis type and options, applies loads and initiates the finite element solution.

3. Post-processing (POST1 or POST26 processors), where the user reviews the results of the analysis through graphics displays and tabular listings.

There are two methods to interface with ANSYS®. The first one is by means of the graphical user interface (GUI) that is easy to learn and use, but it lacks of flexibility, and is suitable for designing a simple structure. The second one is by means of a command file (batch file) that has a steeper learning curve for users, but has many advantages. An entire analysis can be written and stored in a small text file that is quick to run, portable, and easy to modify. ANSYS® can interface with many CAD software such as Solidworks®, Pro/Engineering®, IDEAS, etc., and import and export many file formats, such as IGES, SAT, and STL.

5.1.1 General steps of a finite element analysis with ANSYS®

Create a finite element model — PREP7 module • Define one or more element types for analysis. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 123

• Assign appropriate physical properties to each type. • Define one or more sets of materials used in the model. • Define mass distribution for dynamic analysis • Create geometry • Mesh elements • Apply loads and boundary conditions

Solution — SOLUTION module • Choose appropriate analysis type and solution algorithm • Initiate solution

Post-processing — POST1 and POST26 modules • View results • Export results

5.1.2 Some elements used for implementation

There are hundreds of elements in the element library of ANSYS®. They are categorized as being LINK, BEAM, PLANE, SHELL, SOLID, MASS, VISCO, CONTACT, FLUID, CIRCU elements and so on, which can be either one-dimensional, two-dimensional or three- dimensional elements.

In general, four characteristic element shapes are possible: point, line, area or volume. A point element is typically defined by one node, e.g., a mass element. A line element is typically represented by a line or arc connecting two or three nodes. Examples are beams, spars, pipes and axisymmetric shells. An area element has a triangular or quadrilateral shape and may be a 2-D solid element or a shell element. A volume element has a tetrahedral or brick shape and is usually a 3-D solid element.

The degrees of freedom of the element determine the discipline for which the element is applicable: structural, thermal, fluid, electric, magnetic or coupled-field. In the piezoelectric vibration shunt control, 3-D structural, 3-D coupled-field and 2-D electric elements are used. They are briefly summarized below. Details can be found in the ANSYS® Element Reference.

SOLID45 — SOLID45 is used for three-dimensional modelling of solid structures such as beams. The element is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions, and orthotropic material properties. The element has plasticity, creep, swelling, stress stiffening, large deflection, and large strain capabilities. The geometry, node locations and the coordinate system for this element are shown in Figure 5-1 (ANSYS 6.1 Documentation, [I 3]). Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 124

Figure 5-1 SOLID45 3-D 8-Node structural solid (ANSYS 6.1 Documentation, [I 3])

SOLID95 — SOLID95 is a higher order version of the 3-D 8-node solid element SOLID45. It is used for three-dimensional modelling of complex solid structures such as Cricket bats, which can tolerate irregular shapes without as much loss of accuracy. The element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal x, y, and z directions. The element may have any spatial orientation. SOLID95 has plasticity, creep, stress stiffening, large deflection, and large strain capabilities. The geometry, node locations and the coordinate system for this element are shown in Figure 5-2 (ANSYS 6.1 Documentation, [I 3]).

Figure 5-2 SOLID95 3-D 20-Node Structural Solid (ANSYS 6.1 Documentation, [I 3])

Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 125 SOLID5 — SOLID5 has a three-dimensional magnetic, thermal, electric, piezoelectric and structural field capability with limited coupling between the fields. The element has eight nodes with up to six degrees of freedom at each node. Various combinations of nodal loading are available for this element depending upon the KEYOPT(1) value used for the degree of freedom selected. KEYOPT (or key option) is a switch, used to turn various element options on or off.

UX, UY, UZ, TEMP, VOLT, MAG if KEYOPT (1) = 0 TEMP, VOLT, MAG if KEYOPT (1) = 1 UX, UY, UZ if KEYOPT (1) = 2 TEMP if KEYOPT (1) = 8 VOLT if KEYOPT (1) = 9 MAG if KEYOPT (1) = 10

The geometry, node locations, and the coordinate system for this element are shown in Figure 5-3.

Figure 5-3 SOLID5 3-D coupled-field solid (ANSYS 6.1 Documentation, [I 3])

CIRCU94 — CIRCU94 is a piezoelectric circuit element and simulates basic linear electric circuit components that can be directly connected to the piezoelectric FEA domain (through interfacing with SOLID5, SOLID98, or PLANE13 coupled-field elements). The element has two or three nodes to define the circuit component and one or two degrees of freedom to model the circuit response. It is applicable to full harmonic and transient analyses. The element model and circuit options, indicating node locations and degrees of freedom are shown in Figure 5-4.

Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 126

(a) Element model

(b) Circuit options

Figure 5-4 CIRCU94 piezoelectric circuit element (ANSYS 6.1 Documentation, [I 3])

Key options of CIRCU94:

KEYOPT(1)

0 – Resistor

1 – Inductor

2 – Capacitor

3 – Independent current source

4 – Independent voltage source

KEYOPT(2)

0 – Constant load (transient) or constant amplitude load (harmonic)

1 – Sinusoidal load

2 – Pulse load

3 – Exponential load

4 – Piecewise Linear load Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 127 For example,

ET, 1, CIRCU94, 0, 0 ! Set up element 1 as a resistor (Constant) ET, 2, CIRCU94, 1, 0 ! Set up element 2 as a inductor (Constant) ET, 3, CIRCU94, 2, 0 ! Set up element 3 as capacitor (Constant) ET, 4, CIRCU94, 3, 1 ! Set up element 4 as independent current source (Sinusoid) ET, 5, CIRCU94, 4, 2 ! Set up element 5 as independent voltage source (Pulse)

5.1.3 Coupled-field analysis

The implementation of the piezoelectric vibration shunt control being made possible in the ANSYS® is thanks to the coupled-field capability of the software. The Coupled-field analysis is an analysis that takes into account the interaction (coupling) between two or more disciplines (fields) of engineering. A piezoelectric analysis, for example, handles the interaction between the structural and electric fields: it solves for the voltage distribution due to applied displacements, or vice versa.

5.2 A Simple example of the finite element formulation

Finite element analysis is to discretize an object of arbitrary geometry into many interconnected small regular shape elements. The matrices of mass, stiffness of these small elements are then superimposed to approximate the characteristics of the whole object. In this section, a simple example of finite element formulation for the Euler-Bernoulli beam is illustrated to give some insight of the FEA.

The Euler-Bernoulli beam motion equation of vibration under pure bending is

∂4w(x,t) ∂2w(x,t) EI + ρ = p(x) . (5.1) ∂x4 ∂t 2 where

w(x,t ) is the transverse displacement of the beam, ρ is the mass density per unit length, EI is the beam flexural rigidity, and p(x ) is the external applied force respectively.

For free vibration, p(x) = 0 , according to the relations (Timoshenko, 1973),

dQ d 2M ∂4w(x,t) p(x) = = = EI = 0 . (5.2) dx dx2 ∂x4 where

Q and M denote the shear force and bending moment respectively. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 128

Then a general polynomial function solution for w(4) (x) = 0 in Equation (5.2) is

3 2 w(x) = c1x + c2 x + c3x + c4 . (5.3)

Differentiating Equation (5.3) with respect to x yields the slope of the beam

dw θ (x) = = 3c x2 + 2c x + c . (5.4) dx 1 2 3

where c1 , c2 , c3 , and c4 are constants.

w

wi w j

θi θ j x

xli = 0 x j =

Figure 5-5 Two-node beam element

Figure 5-5 shows a two-node beam element. The boundary conditions are

w(,0) = wi w '(0) =θ (0) =θi w(,l) = wj w '(l) =θ (l) =θ j .

Evaluation of the displacement and slope at both ends of the element with Equations (5.3 & 5.4) and the boundary conditions obtain,

⎧w(0) = c4 = wi ⎪ ⎪θ (0) = c3 =θi ⎨ 3 2 . (5.5) w(l) = c l + c l + c l + c = w ⎪ 1 2 3 4 j ⎪ 2 ⎩θ (l) = 3c1l + 2c2l + c3 =θ j

Solving Equation (5.5) obtains the constant

⎧ 1 c = (2w + lθ − 2w + lθ ) ⎪ 1 l 3 i i j j ⎪ ⎪ 1 c2 = (−3wi − 2lθi + 3w j − lθ j ) . (5.6) ⎨ l 2 ⎪ ⎪c3 =θi ⎪ ⎩c4 = wi

Substituting Equation (5.6) into Equation (5.3) yields, Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 129

w(x) = N1(x)wi + N2 (x)θi + N3 (x)w j + N4 (x)θ j = N ⋅q . (5.7) where

N = [N1 N2 N3 N4 ] , q = [wi θi w j θ j ] , and

2 3 ⎧ ⎛ x ⎞ ⎛ x ⎞ ⎪N1(x) =1− 3⎜ ⎟ + 2⎜ ⎟ ⎪ ⎝ l ⎠ ⎝ l ⎠ ⎪ 2 3 ⎪ ⎛ x ⎞ ⎛ x ⎞ N2 (x) = x − 2l⎜ ⎟ + l⎜ ⎟ ⎪ ⎝ l ⎠ ⎝ l ⎠ . (5.8) ⎨ 2 3 ⎪ ⎛ x ⎞ ⎛ x ⎞ N (x) = 3⎜ ⎟ − 2⎜ ⎟ ⎪ 3 l l ⎪ ⎝ ⎠ ⎝ ⎠ 2 3 ⎪ ⎛ x ⎞ ⎛ x ⎞ ⎪N4 (x) = −l⎜ ⎟ + l⎜ ⎟ ⎩⎪ ⎝ l ⎠ ⎝ l ⎠

Equation (5.8) represents the shape functions known as Hermitian shape functions as shown in Figure 5-6.

Figure 5-6 Hermitian shape functions

Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 130 The transverse deflection function including time t and spatial x is

T w(x,t) = N1(x)wi (t) + N2 (x)θi (t) + N3 (x)w j (t) + N4 (x)θ j (t) = N(x)⋅q (t) . (5.9) where

N = [N1(x) N2 (x) N3 (x) N4 (x)], q = [wi (t) θi (t) wj (t) θ j (t)].

The kinetic energy of the beam is

2 T l l 1 ⎛ ∂w ⎞ q T 1 T Tb = ρb ⎜ ⎟ dx = ρbN Ndx ⋅q = q mq . (5.10) 2 ∫0 ⎝ ∂t ⎠ 2 ∫0 2 where

l T m = ρbN Ndx , q is the matrix of the differentiating of q with respect to time t. ∫0

Since

2 ⎡ N1 N1N2 N1N3 N1N4 ⎤ ⎢ ⎥ N N N 2 N N N N NTN = [N N N N ]T ⋅[N N N N ] = ⎢ 1 2 2 2 3 2 4 ⎥ , 1 2 3 4 1 2 3 4 ⎢ 2 ⎥ N1N3 N2 N3 N3 N3 N4 ⎢ 2 ⎥ ⎣⎢N1N4 N2 N4 N3 N4 N4 ⎦⎥

From Equation (5.8), the mass matrix can be obtained as

⎡ 156 22l 54 −13l ⎤ ⎢ 2 2 ⎥ l 22l 4l 13l − 3l T ρbl ⎢ ⎥ m = ρbN Ndx = , (5.11) ∫0 420 ⎢ 54 13l 156 − 22l⎥ ⎢ 2 2 ⎥ ⎣−13l − 3l − 22l 4l ⎦

The strain energy of the beam can be expressed as

2 2 T l l 1 ⎛ ∂ w ⎞ q T 1 T U b = EI⎜ ⎟ dx = EI(N'') N''dx ⋅q = q kq . (5.12) ∫0 ⎜ 2 ⎟ ∫0 2 ⎝ ∂x ⎠ 2 2 where

l k = EI(N'')TN''dx . ∫0

From Equation (5.8), one can have Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 131

⎧ 6 12 ⎛ x ⎞ N ''(x) = − + ⎜ ⎟ ⎪ 1 l 2 l 2 l ⎪ ⎝ ⎠ ⎪ 4 6 ⎛ x ⎞ ⎪N2 ''(x) = − + ⎜ ⎟ ⎪ l l ⎝ l ⎠ ⎨ . (5.13) ⎪ 6 12 ⎛ x ⎞ N3 ''(x) = − ⎜ ⎟ ⎪ l 2 l 2 ⎝ l ⎠ ⎪ ⎪ 2 6 ⎛ x ⎞ ⎪N4 ''(x) = − + ⎜ ⎟ ⎩ l l ⎝ l ⎠

Since

T T (N'') N''= [N1'' N2 '' N3 '' N4 ''] ⋅[N1'' N2 '' N3 '' N4 '']

2 ⎡ (N1 '') N1 '' N2 '' N1 '' N3 '' N1'' N4 ''⎤ ⎢ ⎥ N '' N '' (N '')2 N '' N '' N '' N '' = ⎢ 1 2 2 2 3 2 4 ⎥ . ⎢ 2 ⎥ N1'' N3 '' N2 '' N3 '' (N3 '') N3 '' N4 '' ⎢ 2 ⎥ ⎣⎢N1'' N4 '' N2 '' N4 '' N3 '' N4 '' (N4 '') ⎦⎥ therefore the stiffness matrix can be obtained as

⎡ 12 6l −12 6l ⎤ ⎢ 2 2 ⎥ l EI 6l 4l − 6l 2l k = EI(N'')TN''dx = ⎢ ⎥ . (5.14) ∫0 l 3 ⎢−12 − 6l 12 − 6l⎥ ⎢ 2 2 ⎥ ⎣ 6l 2l − 6l 4l ⎦

One also can have the damping matrix as

⎡ 156 22l 54 −13l ⎤ ⎢ 2 2 ⎥ l 22l 4l 13l − 3l T ηbl ⎢ ⎥ c = ηbN Ndx = . (5.15) ∫0 420 ⎢ 54 13l 156 − 22l⎥ ⎢ 2 2 ⎥ ⎣−13l − 3l − 22l 4l ⎦

If the beam is made of s identical elements, the matrix of system mass can be expressed as

s T M = ∑ Ai mi Ai . (5.16) i=1

where Ai is a matrix whose row number is equal to the degrees of freedom of the element and the column number is equal to the degrees of freedom of the system.

n n n n i−1 i i+1 b . (5.17) Ai = [] [0] a×b [I] a×a [0] a×b [0] a×b a×c where Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 132 a, b, and c represent the number of the degrees of freedom of the element, the number of the nodes of the element, and the number of the degrees of freedom of the system respectively; [0] a×b is a zero matrix and [I] a×a is the unit matrix.

n n n 1 2 3 E1 E2

Figure 5-7 Two-element beam

For example, the beam is made of two elements as shown in Figure 5-7. Since the element has two nodes (b = 2) and four degrees of freedom (a = 4). The system has three nodes and six degrees of freedom (c = 6), the matrices of Ai (i = 1, 2) are

⎡1 0 0 0 0 0⎤ ⎡0 0 1 0 0 0⎤ ⎢0 1 0 0 0 0⎥ ⎢0 0 0 1 0 0⎥ A = ⎢ ⎥ , A = ⎢ ⎥ . 1 ⎢0 0 1 0 0 0⎥ 2 ⎢0 0 0 0 1 0⎥ ⎢ ⎥ ⎢ ⎥ ⎣0 0 0 1 0 0⎦ ⎣0 0 0 0 0 1⎦

The matrix of the system mass is

2 T T T M = ∑ Ai mi Ai = A1 m1A1 + A2 m2A2 . i=1 where ⎡ 156 22l 54 −13l ⎤ ⎢ 2 2 ⎥ ρ l 22l 4l 13l − 3l m = m = b ⎢ ⎥ . 1 2 420 ⎢ 54 13l 156 − 22l⎥ ⎢ 2 2 ⎥ ⎣−13l − 3l − 22l 4l ⎦

Similarly, the matrix of the system stiffness is

s T K = ∑ Ai k i Ai , (5.18) i=1 and the matrix of the system damping is

s T C = ∑ Ai ci Ai . (5.19) i=1

For the two element beams

2 T T T K = ∑ Ai k i Ai = A1 k1A1 + A2 k 2A2 , i=1 Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 133 and

2 T T T C = ∑ Ai ci Ai = A1 c1A1 + A2 c2A2 . i=1 where ⎡ 12 6l −12 6l ⎤ ⎡ 156 22l 54 −13l ⎤ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥ EI 6l 4l − 6l 2l η l 22l 4l 13l − 3l k = k = ⎢ ⎥ , c = c = b ⎢ ⎥ . 1 2 l 3 ⎢−12 − 6l 12 − 6l⎥ 1 2 420 ⎢ 54 13l 156 − 22l⎥ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥ ⎣ 6l 2l − 6l 4l ⎦ ⎣−13l − 3l − 22l 4l ⎦

The motion equation of free vibration of the beam is

Mq + Cq + Kq = 0. (5.20) where

T q = [w1(t), θ1(t), w2 (t), θ2 (t), w3 (t), θ3 (t)] .

Solving Equation (5.20), the natural frequencies of the beam can be obtained. This is a simple example of obtaining a system’s characteristics using the finite element method. Details can be found in books of the finite element analysis (Lepi, 1998 and Pilkey et al, 2000, etc.).

5.3 Modelling of passive piezoelectric shunt circuit with PSpice®

5.3.1 Single mode resistor-inductor shunt (R-L shunt)

Resistor-inductor piezoelectric vibration shunt is to control single vibration frequency. A resistor-inductor branch is connected either in series or parallel across the electrodes of PZT.

(a) Series R-L shunt (b) Parallel R-L shunt

Figure 5-8 PSpice® model of single mode series resistor-inductor piezoelectric shunt Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 134 For the series R-L shunt circuit in (a), the transfer function is:

R s R L 2σs H (s) = = = 2 2 . (5.21) 1 2 R 1 s + 2σs + ω + sL + R s + s + n sC p L LC p

j2σω H ( jω ) = 2 2 . (5.22) − ω + j2σω + ω n

When

ω = 0 , H ( jω) = 0 , ω = ωn , H ( jω) =1, and ω → ∞ , H ( jω) = 0 .

The impulse response of Equation (5.21) is,

⎡ σ ⎤ −σ ⋅t h(t) = 2σ ⎢cos(ωd t) − sin(ωd t)⎥ ⋅e U (t) . (5.23) ⎣ ωd ⎦ where

R 1 2 2 2 ⎧0 t < 0 σ = , ωn = , ωd = ωn −σ , U (t) = ⎨ is the step function. 2L LC p ⎩1 t > 0

For the parallel shunt in (b), the transfer function is:

1 1 1 + 2 2 R sL s s H (s) = = = 2 2 . (5.24) 1 1 2 1 1 s + 2σs + ω + s + s + n sC 1 1 RC LC p + p p R sL

− ω 2 H ( jω ) = 2 2 . (5.25) − ω + j2σω + ω n

When ω ω = 0 , H ( jω) = 0 , ω = ω , H ( jω) = n , and ω → ∞ , H ( jω) =1. n 2σ

The impulse response of Equation (5.24) is,

2 2 ⎡ ωd −σ ⎤ −σ ⋅t h(t) = δ (t) − ⎢2σ cos(ωd t) + ⋅sin(ωd t)⎥ ⋅e U (t) . (5.26) ⎣ ωd ⎦ where

1 1 2 2 2 σ = , ωn = , ωd = ωn −σ . 2RC p LC p Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 135 Figure 5-8 shows the PSpice® model of the single mode series and parallel resistor-inductor shunt respectively. Figure 5-9 shows the transfer function of the single mode series and parallel R-L shunt respectively.

1.0V

0.8V

0.6V

0.4V

(V) Amplitude

0.2V

0V 0Hz 0.2KHz 0.4KHz 0.6KHz 0.8KHz 1.0KHz V(R:2) Frequency Frequency (Hz)

(a) Series R-L shunt

10V

8V

6V

4V Amplitude (V) (V) Amplitude

2V

0V 0Hz 0.1KHz 0.2KHz 0.3KHz 0.4KHz 0.5KHz 0.6KHz 0.7KHz 0.8KHz 0.9KHz 1.0KHz V(L:1) Frequency Frequency (Hz)

(b) Parallel R-L shunt

Figure 5-9 Transfer function of the single mode resistor-inductor shunt Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 136 One can see that the R-L shunt has narrow band characteristics. They act as a bandpass filter. The resonant frequency of the filter is determined by

1 fc = . 2π LC p

5.3.2 Multi-mode parallel resistor-inductor shunt

Practically however, there are often more than one vibration modes that need to be controlled simultaneously. In order to control the multi-frequency vibration with a single piezoelectric patch, requires designing an analogue filter-bank. The resonant frequencies of the filter-bank match the natural frequencies of the structure to be controlled. Each frequency is aimed at suppressing one structural vibration mode with the others simultaneously. Fleming et al (2003) proposed multiple parallel single mode shunt circuits in series using single PZT. Each parallel single mode shunt circuit also had a “blocking” circuit in parallel (Fleming et al, 2003). Figure 5-10 shows the concept. The switch in the circuit acts as the “blocking” circuit. If all the switches are “Close” except one, the circuit becomes a single mode parallel R-L shunt circuit.

Figure 5-10 Multiple parallel single mode shunt circuits in series with “blocking” switch Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 137 Any switch in Figure 5-10 usually is in “Close” state until at a specific frequency. Assuming the frequencies of six parallel shunt circuits are: 10.24, 63.37, 176.19, 344.37, 564.7, and

852.9 Hz respectively. The switch F1 is only “Open” when f1 = 10.24 (Hz), … , switch F6 is only “Open” when f6 = 852.9 (Hz). Thus, the switch can be replaced by an ideal parallel R-L resonant circuit, which is in “Open” circuit when at its resonant frequency, otherwise, in “Short” circuit. The values of the capacitor and inductor of the resonant circuit have the relation:

Ci Li1 = C p Li 2 (i = 1, .., 6). where

Ci and Li1 are the capacitor and inductor of the resonant “blocking” circuit respectively; C p is the capacitance of the PZT patch and Li 2 is the inductor of the shunt circuit in parallel with the switch.

Replacing the switches with the R-L resonant “blocking” circuits in Figure 5-10 yields the circuit as shown in Figure 5-11.

Figure 5-11 Multiple parallel single mode shunt circuits in series with “blocking” circuit Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 138 Figure 5-12 is the simplified circuit in Figure 5-11. One can notice that the inductor values used in the simplified circuit are almost one-eleventh of those without “blocking” circuits. However, in reality, the “blocking” circuit is not ideal, it cannot be ideally “Open” and “Close”. Each shunt unit will be affected by others. The vibration shunt effect will be worse off than that of using a single shunt circuit. Figure 5-13 is the PSpice® simulation of the transfer function of the circuit in Figure 5-12 which has shown the characteristics of a filter- bank. The resonant frequencies of the filter-bank are 10.24, 63.37, 176.19, 344.37, 564.7, and 852.9 Hz respectively.

2.0V Figure 5-12 Equivalent circuit of multi-mode parallel shunt

1.5V

1.0V

(V) Amplitude

0.5V

0V 0Hz 0.2KHz 0.4KHz 0.6KHz 0.8KHz 1.0KHz V(L1:1) Frequency Frequency (KHz)

Frequency (KHz)

Figure 5-13 Transfer function of a multi mode parallel R-L shunt Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 139 5.4 Modelling of passive piezoelectric shunt circuit with ANSYS®

The subject used for ANSYS® modelling comprises a 0.5×0.025×0.003 m3 aluminium beam with a 0.05×0.025×0.0005 m3 PZT patch attached. The boundary condition of the beam is “clamped-free”. The PZT is attached at the clamped end. This analysis is linear elastic one without considering internal damping. The element employed for modelling the aluminium beam is a SOLID45, the 3-D structural solid element. The Young’s modulus for the beam is 66 GPa and the density is 2790 kg/m3. The element for modelling the piezoelectric transducer is a SOLID5, the 3-D coupled field solid element. The density of the transducer is 7730 kg/m3. The element for modelling the shunt circuit components (resistor, capacitor and inductor) is CIRCU94, the circuit element.

5.4.1 Single mode resistor-inductor shunt

Figure 5-14 shows the ANSYS® model of the single mode parallel R-L piezoelectric shunt for a cantilever beam. The parallel R-L circuit is connected across the electrodes of PZT. The inherent static capacitance of the PZT measured by the software is about 31.4 nF. The arrow at the tip of the beam indicates where external force applied.

Figure 5-14 ANSYS® modelling of the parallel R-L piezoelectric shunt for beam Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 140 Figure 5-15 gives the first four bending modes of the aluminium beam with the PZT patch attached. The natural frequencies are 10.244, 63.37, 176.187, and 344.374 Hz respectively.

Figure 5-15 First four bending modes of the beam with PZT attached (clamped-free boundary condition)

The harmonic analysis with single mode parallel R-L piezoelectric shunt on aluminium beam with ANSYS® have been shown in Figure 4-20 of Chapter 4. The spectra show that the amplitudes of 1st to 4th modes have been reduced by about 84%, 70%, 95%, and 92% (or 16 dB, 10.5 dB, 26 dB, and 22 dB) respectively. The parallel R-L shunt experiments on a cantilever beam conducted by Wu (Wu, 1996, 1998) reported a 12 ~ 14 dB vibration reduction of resonant response. A. J. Fleming et al claimed that their R-L shunt experiment (using synthetic impedance) exhibited a reduction of up to 22 dB in vibration resonant frequency (Fleming et al, 2003).

The harmonic analysis has shown that the piezoelectric vibration shunt can bring down the amplitude of the target frequency component but may sometimes increase the amplitudes of other frequency components at certain levels. This may be caused by some unwanted positive feedback to those frequency components. This problem is neglected and is not being discussed here. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 141 5.4.2 Multi-mode parallel resistor-inductor shunt

Multi-mode vibration shunt uses a single PZT patch and an analogue filter-bank. The ANSYS® modelling is shown in Figure 5-16. The model is the same as the single mode except it has more electronic components in the shunt circuit, which is similar to the equivalent circuit in Figure 5-14. It has four parallel R-L-C resonant circuits in series designed for shunting the 1st to 4th vibration bending modes of the beam.

Figure 5-17 is the beam vibration spectrum after multi-mode shunting. Comparing with the spectrum of Figure 4-20(a), the 1st to 4th modes were reduced by about 85%, 52%, 86%, and 77% (or 16.5 dB, 6.4 dB, 17 dB, and 12.8 dB) respectively. The experiment of multi-mode shunt with a single PZT patch, by using a “current blocking” circuit in series with shunt circuit, conducted by Wu had a vibration reduction from 7 dB to 15 dB (Wu, 1998). Similar experiments carried out by A. J. Fleming et al by using “current blocking” circuit in parallel with shunt circuit, had vibration reductions of up to18 dB (Fleming et al, 2003). The results obtained from multi-mode shunt to particular frequency is about 3 ~ 6 dB worse than that obtained from the single mode shunt.

Figure 5-16 ANSYS® modelling of the multi-mode piezoelectric shunt for beam (clamped- free boundary condition) Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 142

Figure 5-17 Harmonic analysis of beam vibration with multi-mode piezoelectric shunt (clamped-free boundary condition)

5.4.3 Shunt effectiveness related to the placement of the PZT patch

The location of the PZT patch on the beam will significantly affect the effectiveness of the vibration shunt. The PZT patch is best to be placed at the anti-node area where the beam has the largest bending and strain to achieve optimum effectiveness of the vibration shunt. It can be verified by the simulation below.

Figure 5-18 shows a PZT patch attached in the middle of an aluminium beam with free-free boundary conditions. Figure 5-19 shows the first four bending modes of the beam. One can see that the anti-node of the 1st and 3rd bending modes and the node of the 2nd and 4th bending modes occur in the middle of the beam. Therefore, the 1st and 3rd bending modes have the highest strain while the 2nd and 4th bending modes have the lowest strain in the PZT patch. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 143

Figure 5-18 Beam with PZT patch attached in the middle of the beam (free-free BC)

Figure 5-19 First four bending modes of the beam in Figure 5-16

We can predict that the vibration shunt control has high effectiveness to reduce the 1st and 3rd vibration modes but has little effectiveness to reduce the 2nd and 4th vibration modes. This can be verified by the harmonic analysis in Figure 5-20. The 1st mode in (b) and the 3rd mode in (d) almost disappear, while the 2nd mode in (c) and 4th mode in (e) show almost no reduction compared with the same components in (a). Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 144

(a) Before shunt

(b) Shunt 1st mode (c) Shunt 2nd mode

(d) Shunt 3rd mode (e) Shunt 4th mode

Figure 5-20 Harmonic analysis of the beam in Figure 5-18

Figure 5-21 is the same beam as that in Figure 5-18 but the PZT patch is placed in a different location. Figure 5-22 is the modal analysis giving the first four bending modes. From the Figure, one can see that the anti-node of the 2nd and 4th modes occur at the location of the PZT patch, while the anti-node of the 1st and 3rd modes occur a bit away from the location of the PZT patch. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 145

Figure 5-21 Beam with PZT attached away from the middle of the beam

Figure 5-22 First four bending modes of the beam in Figure 5-21

In this case, the 2nd and 4th vibration modes should be reduced efficiently, while the 1st and 3rd modes also can be reduced, but less efficiently than the former. This can also be verified from the harmonic analysis as shown in Figure 5-23.

The analyses have shown that the location of the PZT patch can significantly influence the outcome of the piezoelectric vibration shunt control. In order to have high vibration shunt Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 146 efficiency, the PZT patch has to be placed at a high strain area. To properly identify or locate the nodes and anti-nodes of the structure to be controlled and the area where strains occur is critical in implementing the vibration control. The geometry sizes and material properties of the structure and PZT patch have also influenced the effectiveness of the vibration shunt control. The simulation results have already been demonstrated in Chapter 4.

(a) Before shunt

(b) Shunt 1st mode (c) Shunt 2nd mode

(d) Shunt 3rd mode (e) Shunt 4th mode

Figure 5-23 Harmonic analysis of the beam in Figure 5-21 Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 147 5.5 Modelling and analysis of the Cricket bat

The major change in the newly designed Cricket bat is the use of high stiffness to weight ratio advanced fibre composites as part of the material variation to the handle. The purposes of this include:

(a) To enhance the strain distribution in the PZT patch, and

(b) To increase the natural frequencies of the bat.

Cricket bats with different handle materials and geometry shapes have been modelled. However, due to the limitation of computer capabilities (very significant memory capacity is needed to perform piezoelectric vibration shunt control for a large object), vibration shunt control for the Cricket bat has not been simulated. The simulations help to identify the anti- nodes, strain of the Cricket bats to make sure that the PZT patch is placed at the high strain area.

5.5.1 Cricket bat with a wooden handle

A Cricket bat model as shown in Figure 5-24 was constructed by using Solidworks®, CAD software. The model then was imported into ANSYS®.

Figure 5-24 Cricket bat model built with Solidworks® Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 148 In the figure, the diameter of the handle is 16 (mm) and the length of the handle is 300 (mm) and the length of the blade is 560 (mm). The blade of the bat is made of willow which density equal to 420 Kg/M3 and the Young’s modulus is equal to 3 GPa. The handle of the bat is made of cane with density equal to 700 Kg/M3 and the Young’s modulus is equal to 6.6 GPa.

Figure 5-25 shows the first four bending modes of the Cricket bat based on strain analysis. The mode frequencies are: 120.657, 404.091, 706.428, and 1219 Hz respectively.

Figure 5-25 Bending modes of the Cricket bat with wooden handle (less stiff handle)

5.5.2 Cricket bat with composite hollow tube handle

The second bat handle is made of a cylinder tube composite. The outer diameter of the cylinder tube is 16 (mm), and the inner diameter of the cylinder tube is 13 (mm). The length of the cylinder tube is 300 (mm). The density and the Young’s modulus of the composite are equal to 1350 Kg/M3 and 70 GPa respectively.

Figure 5-26 displays the first four bending modes of the bat with the cylinder tube composite handle. The natural frequencies of the bat are: 262.938, 720.447, 1368, and 2207 Hz respectively. The natural frequencies of the bat with a composite cylinder handle are much Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 149 higher than those of the bat with the wooden handle, almost doubled. These results were verified by experiments which will be illustrated in Chapter 6.

However, comparing Figure 5-25 with Figure 5-26, one can notice that the strain distribution is different between the two bats. For the wooden handle bat, the high strain resides in the handle section. For the composite handle bat, the high strain shifts to the blade section, especially for the first mode, there is little strain appearing at the joint area of blade and handle (known as shoulder area). This is a result of increasing the handle stiffness.

Figure 5-26 Bending modes of the Cricket bat with hollow composite handle (stiffer handle)

PZT patch cannot be directly placed on the tube handle. In order to place the PZT patch on the tube handle, a small segment is mounted on the handle, then the PZT patch is bonded on the segment as shown in Figure 5-27. The segment adds stiffness to where it is placed. It reduces bending in the area.

Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 150

Figure 5-27 Cricket bat mounted with PZT patch

Figure 5-28 shows the bending modes of the bat. One can see that it further increases the natural frequencies but reduces the strain in the shoulder area. So the design does help to transfer strain into the PZT patch. Such a bat was made and applied with the passive vibration shunt control. The experiment results showed that little reduction in vibration.

5.5.3 Cricket bat with sandwiched handle

The structure of the cylinder tube is highly resistant to bending though it can greatly increase the natural frequency, so another design has been made. Instead of using a composite cylinder tube, a fork like composite beam structure as shown in Figure 5-29 was attached to the outer sides of the wooden handle to make a sandwiched handle as shown in Figure 5-30. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 151

Figure 5-28 Bending modes of hollow composite handle Cricket bat mounted with PZT patch

Figure 5-29 Fork-like composite beam

Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 152

Figure 5-30 Cricket bat with sandwiched handle

Figure 5-31 is the strain analysis of the Cricket bat with the sandwiched handle. It is seen that this design can induce strain to occur at the shoulder while the natural frequencies of the bat are increased.

In order to minimize the change in the appearance of the Cricket bat, the preferred model of the Cricket bat handle should be that as shown in Figure 5-32. The composite beams are embedded in the handle. The PZT patches are then bonded on the composite beams. It should just feel like a normal Cricket bat when the batsman holds it. The strain analysis of the bat in 5-31 is shown in Figure 5-33. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 153

Figure 5-31 Bending modes of Cricket bat with sandwiched handle

Figure 5-32 Preferred model of Cricket bat handle embedded with composite beam and PZT Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 154

Figure 5-33 Bending mode of the Cricket bat in Figure 5-32

Not only will the geometry configuration and material properties of the composite beams affect the natural frequencies of the Cricket bat, but also the distance between the two composite beams. The further apart the two beams, the higher the frequency the bat will be. However, the composite beam has to be wide enough to match the width of the PZT patch, the distance of two beams is limited by the conventional circumferential size range of the handle of the Cricket bat.

5.6 Chapter summary

In this chapter, the finite element analysis has been briefly introduced. The coupled-field capability of ANSYS® has made it possible to simulate the passive piezoelectric vibration shunt control using software. This brings the benefits of flexibility, time and cost saving in designing such a system. The shunt control circuits, especially the parallel resistor-inductor shunt circuit, have been studied by means of the PSpice® and ANSYS® software. Single- mode and multi-mode parallel R-L vibration shunt control for cantilever beams have been simulated with ANSYS®. The results have shown that the vibration amplitude can be greatly reduced when the PZT patch is optimally placed to capture the largest strain fields. Chapter 5 Numerical Modeling of Passive Piezoelectric Vibration Control using FEA Software ANSYS® 155 Several Cricket bat designs were analyzed. A new Cricket bat model was proposed. Unfortunately, the vibration shunt control for the Cricket bat was not preformed because of the high computational demand due to the complex geometric configuration of the Cricket bat. However, the structural analysis of the bat has enabled the identification of the nodes and anti-nodes of (hence largest strain fields) for various modes of vibration. The ANSYS® code of the passive piezoelectric vibration shunt for beams developed in this study can be applied to many large objects including Cricket bats as long as the PC can handle a large amount of elements to perform coupled-field analysis. Extensive simulations with various handle-blade configurations were analysed in this study but due to the brevity requirements of the thesis, not all the simulations have been reported in this thesis. Chapter 6 Experiment and Implementation 156

Chapter 6 Experiment and Implementation

In this section, the experimental results of the passive piezoelectric vibration shunt control for beams and Cricket bats are presented. The goal of the experiments were, firstly, to see whether the piezoelectric vibration shunt control could really reduce the vibration of structures practically as it has been predicted by ANSYS® simulations, and secondly, to work out how to apply the vibration shunt control to Cricket bats. The experiments include vibration shunt control carried out on beams with different materials, modal tests to identify the modal shapes and natural frequencies of Cricket bats with a laser vibrometer, and Cricket bat prototyping with the carbon-fibre composite handle and the vibration shunt control for Cricket bats. Since the experiments were only interested in the vibration amplitude reduction, the vibration test used is a direct test and compares the vibration amplitudes before and after the shunt circuit is turned on. The block diagram of the vibration experiment is shown in Figure 6-1.

Figure 6-1 Block diagram of cantilever beam shunted with parallel R-L shunt Chapter 6 Experiment and Implementation 157 6.1 Shunt circuit design

The shunt circuit is basically made of a resistor and inductor. The inductor is determined by the inherent capacitance C p and the natural frequency to be controlled F, that is,

1 L = 2 . (2πF ) C p

Since the C p is quite small, normally at tens of nano-Farads, the inductor value required to control low frequency vibration will be very large, at tens even hundreds of Henries. The commercially available inductors are normally less than 100 milli-Farads and are not variable. They are not suitable for this application. An active electrical device known as a synthetic inductor or gyrator is often used in vibration shunt control. The synthetic inductor/gyrator is a two-part generalized impedance converter (GIC) with input impedance proportional to the reciprocal of the load impedance, that is expressed as

g 2 Zi (s) = . Z L (s)

If ZL (s) is a resistor, then Zi (s) becomes a conductance or vice versa, or if ZL (s) is a capacitor, then Zi (s) becomes an inductor or vice versa. The generalized impedance converter (Williams et al, 1995, Taylor et al, 1997) is shown in Figure 6-2.

Figure 6-2 Generalized impedance converter (GIC) Chapter 6 Experiment and Implementation 158 Let,

1 Z1 = R1 , Z2 = , Z3 = R3 , Z4 = R4 , and Z5 = R5 , sC2

then the input impedance Zi is

Z1 ⋅ Z3 ⋅ Z5 R1 ⋅ R3 ⋅ R5 ⋅C2 Zi = = s = sLeq . Z2 ⋅ Z4 R4 where

R1 ⋅ R3 ⋅ R5 ⋅C2 Leq = . R4

Figure 6-3 is the PSpice® model of the parallel piezoelectric R-L shunt circuit with a gyrator- based inductor.

Figure 6-3 Parallel R-L piezoelectric shunt circuit with a gyrator-based inductor

The equivalent inductor value in Figure 6-3 is

103 ⋅5×103 ⋅100 ×103 ⋅100 ×10−9 L = = 50 (H ) . eq 103

Then the frequency to be controlled is Chapter 6 Experiment and Implementation 159

1 1 F = = ≈ 100 (Hz) . −9 2π C p Leq 2π 50 ×10 × 50

Figure 6-4 is the simulation of the circuit of Figure 6-3 which shows the same characteristics as the parallel resistor-inductor shunt circuit in Figure 5-7(b). This verifies that the gyrator truly is an inductor.

20V

15V

10V

Amplitude (V) (V) Amplitude

5V

0V 0Hz 0.1KHz 0.2KHz 0.3KHz 0.4KHz 0.5KHz 0.6KHz 0.7KHz 0.8KHz 0.9KHz 1.0KHz V(Cp:1) Frequency Frequency (KHz)

Figure 6-4 Transfer function of parallel R-L shunt with gyrator-based inductor

The advantages of the gyrator-based inductor are its small size and low weight compared with a large value passive inductor. Unlike passive inductors, the gyrator-based inductor is flexible and adjustable. They can be integrated and embedded along with other circuits. It is possible to tune the value of the gyrator-based inductors with a DSP device. The disadvantages are the need for external power, electric noise due to operational amplifiers, and that it possesses inherent non-linearity.

Figure 6-5 is a circuit containing two sets of independent adjustable gyrator-based inductors used in experiments. The IC circuit used is the National Semiconductor LM 324 low power quad operational amplifiers. The power supply range is: 3V ~ 32V (single supply) or ±1.5V ~ ±16V (dual supply). Chapter 6 Experiment and Implementation 160

Figure 6-5 Double adjustable gyrator-based inductors

6.2 Vibration shunt control for beams

The piezoelectric vibration shunt experiments were carried out on the aluminium, wooden and carbon-fibre composite beams respectively. The length, width and thickness of beams are about 500, 25 and 3 mm respectively. The boundary condition is clamped-free. The PZT used is PSI-5A4E piezoelectric sheet (T107-A4E-602) from Piezo Systems. The sensor to measure the vibration is PCB accelerometer (sensitivity: 100 mv/g), the charge amplifier to amplify senor signal is PCB charge amplifier (MOD 462A), miniature vibration shaker and power amplifier (TIRA-BAA 60), signal generator and OROS spectrum analyser.

The main steps of the testing are:

(1) Bond PZT patch close to the clamped end of the beam.

(2) Attach accelerometer near the PZT patch.

(3) Excite the beam with the vibration shaker (sinusoid signal or white noise).

(4) Pre-tune the shunt circuit to the frequency to be controlled.

(5) Adjust the vibration shaker until the vibration amplitude is at an identical level for every beam when the shunt circuit is turned off. Record the amplitude.

(6) Measure the vibration amplitude when the shunt circuit is turned on.

The experiment targets the individual natural frequency of the beam, comparing the vibration amplitudes of the beam between those with and without shunt control. Figures 6-6 to 6-11 show the vibration signals and spectra of the aluminium beam, wooden beam and carbon-fibre composite beam with and without vibration shunt control respectively.

Chapter 6 Experiment and Implementation 161

(a) Before Shunt (b) After shunt

Figure 6-6 Vibration shunt for aluminium beam (test 1)

(a) Before Shunt (b) After shunt

Figure 6-7 Vibration shunt for aluminium (test 2)

(a) Before Shunt (b) After shunt

Figure 6-8 Vibration shunt for wooden beam (test 1) Chapter 6 Experiment and Implementation 162

(a) Before Shunt (b) After shunt

Figure 6-9 Vibration shunt for wooden beam (test 2)

(a) Before Shunt (b) After shunt

Figure 6-10 Vibration shunt for carbon-fibre composite beam (test 1)

(a) Before Shunt (b) After shunt

Figure 6-11 Vibration shunt for carbon-fibre composite beam (test 2) Chapter 6 Experiment and Implementation 163

One can see that the results after shunt are 42 % and 90 % for aluminium beam (Figures 6-6 &

6-7), only 9 % and 8 % for wooden beam (Figures 6-8 & 6-9), and 80 % and 70 % for carbon- fibre composite beam (Figures 6-10 & 6-11) respectively. By using the carbon-fibre composite beam, the vibration reduction efficiency can be greatly improved compared to using the wooden beam (even the aluminium beam in some cases) since the Young’s modulus of the carbon-fibre composite beam selected was much stiffer than that of the wooden beam (even stiffer than the aluminium beam) in the experiments.

6.3 Modal test of Cricket bats with laser vibrometer

The modal testing of Cricket bats has been carried out by using Polytec laser vibrometer. Figure 6-12 shows the testing system and set-up. The testing was done by scanning the whole bat excited with periodic chirp signal. After scanning, the modal shapes, mode frequencies can be obtained.

Figure 6-12 Set-up of Cricket bat testing with laser vibrometer

Chapter 6 Experiment and Implementation 164 Several Cricket bats were tested, one of them is the normal wooden handle bat, two are the carbon-fibre composite tube handle bats, and two are the carbon-fibre composite sandwiched handle bats. The results are given below.

6.3.1 Wooden handle Cricket bat

The wooden handle Cricket bat is the traditional bat commonly used in the Cricket game. Figure 6-13 is the bat to be tested.

Figure 6-13 Wooden handle Cricket bat

Figure 6-14 shows the first three bending modes (from top to bottom) of the bats obtained by the laser vibrometer. Their frequencies are: 125, 395, and 654 Hz respectively. One can notice that the maximum strain of the 1st mode is almost concentrated at the shoulder area.

Chapter 6 Experiment and Implementation 165

Figure 6-14 Bending modes of the wooden handle Cricket bat

6.3.2 Carbon-fibre composite tube handle Cricket bat

Two carbon-fibre composite tube handle Cricket bats as shown in Figures 6-15 and 6-16 have been tested. The handle material of the bat in Figure 6-15 is less stiff than that of the bat in Figure 6-16.

Figure 6-15 Carbon-fibre composite tube handle Cricket bats #1 (less stiff handle)

Chapter 6 Experiment and Implementation 166

Figure 6-16 Carbon-fibre composite tube handle Cricket bat #2 (stiffer handle)

Figure 6-17 shows the first three bending modes of the bat #1 in Figure 6-15. The mode frequencies of the bat are 237, 678, and 1237 Hz respectively.

Figure 6-17 First three bending modes of the bat in Figure 6-15

Chapter 6 Experiment and Implementation 167 Figure 6-18 shows the first three bending modes of the bat #2 in Figure 6-16. The mode frequencies of the bat are 319, 763, and 1431 Hz respectively. The mode frequencies of bat #2 is higher than those of bat #1 since the handle of bat #2 is stiffer than that of bat #1.

Figure 6-18 First three bending modes of the bat in Figure 6-16

One can notice that the strain distributions of the carbon-fibre composite tube handle Cricket bats as shown in Figures 6-17 & 6-18 are different from that of the wooden handle bat shown in Figure 6-14. The maximum strains of the 2nd and 3rd modes are almost concentrated at the shoulder area rather than the 1st mode. This is because the carbon-fibre composite tube handle is too stiff for the 1st mode to be bent at the shoulder area. Also the stiffer handle bat bends less than the less stiff handle bat does. Since the amplitude of the 1st bending mode is the mode with the largest vibration amplitude and is felt worst by batsmen physically, it is the major mode to be controlled (Penrose et al, 1998). Vibration shunt control for the less stiff st handle bat was conducted. However, less than 2 % vibration amplitude reduction for the 1 mode was achieved. In order to reduce the stiffness at the shoulder area, the carbon-fibre composite sandwiched handles were made.

Chapter 6 Experiment and Implementation 168 6.3.3 Carbon-fibre composite sandwiched handle Cricket bat

Figure 6-19 is the first carbon-fibre composite sandwiched handle Cricket bat. The handle is sandwiched by two pieces of carbon-fibre composite beams. The carbon-fibre composite beam is the variable stiffness polymeric composite strip as shown in Figure 4-26 in Chapter 4. The carbon-fibre material at the shoulder area where the PZT patch will be attached, is less stiff although theoretically it should be made stiffer in this area to transfer more strain into the PZT patch. This is actually a compromise in order to shift the 1st bending mode to the shoulder area.

Figure 6-19 Carbon-fibre composite sandwiched handle Cricket bat #1

Figure 6-20 shows the first three bending modes of the sandwiched handle bat #1. The mode frequencies are: 203, 503, and 784 Hz. They are lower than the carbon-fibre tube handle bats but higher than the wooden handle bat. The location of the 1st bending mode is closer to the shoulder area of the bat.

Chapter 6 Experiment and Implementation 169

Figure 6-20 Bending modes of the carbon-fibre composite sandwiched handle Cricket bat #1

Figure 6-21 Carbon-fibre composite sandwiched handle Cricket bat #2 Chapter 6 Experiment and Implementation 170 The bat in Figure 6-21 is the second carbon-fibre composite sandwiched handle Cricket bat. The difference from the first handle is that the wooden material sandwiched between the carbon-fibre composite strips has uniform thickness as shown in the bottom picture of Figure 6-21. Figure 6-22 is the 1st bending mode and the acceleration spectrum of the sandwiched handle bat #2. The 1st bending mode shows it even closer to the shoulder area of the bat.

Figure 6-22 Accelerate spectrum of the #2 sandwiched handle Cricket bat

Figure 6-23 shows the first four bending modes of the sandwiched handle bat #2. It gives clearer picture of where the anti-node locations of the bat are. The mode frequencies are: 156, 438, 655, and 1009 Hz respectively. The frequencies are further dropped and are closer to the frequencies of the wooden handle bat. The frequency drop is partly due to the gap between composite strips of bat #2 handle is narrower (hence lower flexural rigidity) than that of bat #1 handle. However, the frequency drop is not what was desired. It is certainly something that needs to be improved. The constraint here is that due to the laws of the game (MCC, [I 8]), attachments are not allowed anywhere except the handle of the bat. We need to find a break (soft) point along the handle to maximize the bending of the 1st mode. The only likely position is at the shoulder area. It is possible to increase the frequencies as well as have the anti-node of the 1st bending mode close to the shoulder area by optimising the geometry size, Young’s modulus of the carbon-fibre strips and the thickness of the wood material sandwiched between the two strips. It requires more simulations and experiments. A third sandwiched handle Cricket bat was made as shown in Figure 6-24 (before assembly, the blade and pre- made sandwiched handle). Chapter 6 Experiment and Implementation 171

Figure 6-23 Bending modes of the carbon-fibre composite sandwiched handle Cricket bat #2

Slot to hold circuit Slot to hold batteries

Figure 6-24 Blade and pre-made sandwiched handle for sandwiched handle Cricket bat #3

Chapter 6 Experiment and Implementation 172 The sandwiched handle bat #3 was tested after assembly. The mode frequencies, especially the 1st mode, have been improved compared to those of sandwiched handle bat #2. The first three mode frequencies are: 180, 444, and 710 Hz. Table 6-1 lists the natural frequencies of various bats.

Table 6-1 Natural frequency comparison among various Cricket bats

Natural Frequency (Hz)

Bending Carbon - Carbon- Carbon - Carbon- Carbon- Wooden fibre fibre fibre Mode fibre tube fibre tube bat sandwiched sandwiched sandwiched bat #1 bat #2 bat #1 bat #2 bat #3

1st 125 237 317 202 156 180

2nd 395 678 763 503 438 440

3rd 654 1237 1431 784 655 712

4th 960 –– –– 1102 1009 1070

Due to the time and resource limitations, only three carbon-fibre composite sandwiched handle bats were made. The tests have shown that the vibration shunt control efficiency has been improved for the carbon-fibre sandwiched bats compared to wooden bat and composite tube bat. They are discussed in next section.

6.3.4 Vibration shunt control for the Cricket bat

The vibration shunt control was carried out for the bat with the carbon-fibre tube handle but the vibration reduction is very little. It also is a problem to attach the PZT patch on the tube handle. So the carbon-fibre composite sandwiched handle bat was designed and tested. The PZT patches are attached to both sides of the handle as shown in Figure 6-25. The pair at the shoulder area aims to reduce the vibration of the 1st and 3rd modes and the other pair aims to reduce the vibration of the 2nd and 4th modes based on the mode distribution as shown in Figure 6-23. The PZT patches in each pair are connected in parallel then connected to a shunt circuit. There are two parallel R-L shunt circuits for the bat.

Chapter 6 Experiment and Implementation 173

PZT

Figure 6-25 The location of the PZT patches on the handle

Figures 6-26 to 6-29 show the vibration signals and spectra of the 1st to 4th modes of the carbon-fibre composite sandwiched handle bat #1 with and without vibration shunt control respectively. The bat was excited by single frequency sinusoid signal individually. The st nd rd vibration reductions are 20 % for the 1 mode, 43 % for the 2 mode, 30 % for the 3 mode, th st and 25 % for the 4 mode respectively. Figure 6-30 and Figure 6-31 are the results of the 1 and 2nd modes when the sandwiched handle bat was driven by white noise. The figures show a similar amount of vibration amplitude reduction for the bat driven by sinusoidal forcing function and white noise. The results has shown a large improvement on the carbon-fibre tube

bat. According to the relationship between amplitude and energy (by square law), 20 % reduction in vibration amplitude means 36 % reduction in vibration energy.

Chapter 6 Experiment and Implementation 174

(a) Before Shunt (b) After shunt

Figure 6-26 Vibration of the sandwiched handle Cricket bat #1 (1st mode)

(a) Before Shunt (b) After shunt

Figure 6-27 Vibration of the sandwiched handle Cricket bat #1 (2nd mode)

(a) Before Shunt (b) After shunt

Figure 6-28 Vibration of the sandwiched handle Cricket bat #1 (3rd mode) Chapter 6 Experiment and Implementation 175

(a) Before Shunt (b) After shunt

Figure 6-29 Vibration of the sandwiched handle Cricket bat #1 (4th mode)

(a) Before Shunt (b) After shunt

Figure 6-30 Vibration of the sandwiched handle Cricket bat #1 (1st mode, random excitation)

(a) Before Shunt (b) After shunt

Figure 6-31 Vibration of the sandwiched handle Cricket bat #1 (2nd mode, random excitation) Chapter 6 Experiment and Implementation 176

Figure 6-32 The first prototype of new Cricket bat integrated with vibration shunt circuit

The first prototype of new Cricket bat was made from the carbon-fibre sandwiched handle bat

#2 with the piezoelectric shunt circuit together and two 12 V batteries embedded inside. The first prototype of new Cricket bat is shown in Figure 6-32. The dipswitch and a LED are attached at the end of the handle temporarily. More sophisticated switch like vibration/motion switch will be used in future. Figure 6-33 is the exploded view of the handle of the new Cricket bat prototype.

Figure 6-34 to Figure 6-37 shows the vibration signals and spectra of the 1st to 4th modes of the first prototype of new Cricket bat shown in Figure 6-32 with and without vibration shunt st nd control respectively. The vibration reductions are 16 % for the 1 mode, 21 % for the 2 rd th mode, 22 % for the 3 mode, and 16 % for the 4 mode respectively. Since the shunt circuit was difficult to tune after being embedded in the bat and the PZT patches had been attached before the bat was finalized, they might not be in the best positions, therefore, the efficiency of the vibration shunt control was slightly reduced compared to bat #1.

Chapter 6 Experiment and Implementation 177

Piezo-electric Sensors

Switch

LED Light

Batteries

Traditional Cane with Rubber inserts Shunt Circuit board

Variable Stiffness Carbon Fibre 1.5 mm thick laminate.

Figure 6-33 Exploded view of the handle of the first prototype of new Cricket bat

(a) Before Shunt (b) After shunt

Figure 6-34 Vibration of the first new Cricket bat prototype (1st mode) Chapter 6 Experiment and Implementation 178

(a) Before Shunt (b) After shunt

Figure 6-35 Vibration of the first new Cricket bat prototype (2nd mode)

(a) Before Shunt (b) After shunt

Figure 6-36 Vibration of the first new Cricket bat prototype (3rd mode)

(a) Before Shunt (b) After shunt

Figure 6-37 Vibration of the first new Cricket bat prototype (4th mode) Chapter 6 Experiment and Implementation 179 Comparing the vibration shunt results between the Cricket bats and for the beams, the percentage of vibration amplitude reduction of the former is clearly much less than the latter. This is because the volume of the Cricket bats is much larger than the volume of the beams. The Cricket bats certainly contain more vibration energy than the beams do. A single PZT patch can only absorb a limited amount of vibration energy. However, when multiple PZT patches can be placed and connected in parallel, the vibration energy to be absorbed is almost in proportion to the number of PZT patches. It was verified by attaching one PZT patch and two PZT patches to the Cricket bat respectively. The vibration reduction for using two PZT patches is doubled compared with that for using one PZT patch. The other thing that needs to be mentioned is that the shunt efficiency is in inverse proportion to the magnitude of vibration. The larger the vibration, the lower percentage of vibration reduction. This is understandable since the PZT has a finite energy absorption capability.

One phenomenon that was noticed during the experiments is that the efficiency of the shunt control is related to the power supply of the shunt circuit. The efficiency of the shunt control is proportional to the power supply. It is probably because the inductor used is the gyrator- based inductor, which is built with operational amplifiers. The operational amplifiers have a certain dynamic range to operate within. The larger the power supply, the larger the dynamic range of the operational amplifiers. Since the parallel shunt control is an inherent parallel resonant circuit, when the shunt circuit is tuned to the frequency to be controlled, the resonant phenomenon happens and the voltage at the input of the shunt circuit will be very large. The operational amplifiers can be saturated when the input vibration signal exceeds the dynamic range of the operational amplifiers. In such case, the inductor will not be at the right value and the shunt circuit will hence not function correctly.

Figure 6-38 shows an example of the influence of the power supply over the efficiency of the shunt control. Figure 6-38 (a) is the vibration signal before shunt, (b) shows the vibration signal after shunt with 10 voltage power supply to the shunt circuit, (c) show the vibration signal after shunt with 15 voltage power supply to the shunt circuit. Comparing the magnitudes between the signals in Figure 6-38 (b) and Figure 6-38 (c), the vibration reduction

is about 6.5 % in (b) and is about 10 % in (c). The percentage increase in efficiency is about the same as the percentage increase of power supply at this power supply level. Since the power supply of the operational amplifier used is limited to ±16V, no further test has been made. However, the efficiency of the shunt control is expected to rise when the gyrator-based inductor is built with power operational amplifiers and powered with a higher voltage.

Chapter 6 Experiment and Implementation 180

(a) Before shunt

(b) After shunt with ±10V power supply

Chapter 6 Experiment and Implementation 181

(c) After shunt with ±15V power supply

Figure 6-38 The influence of power supply to the effectiveness of shunt control

The vibration reduction shown in Figure 6-38 is not very significant because:

1. The vibration energy was relatively significant.

2. The shunt circuit was slightly out-of-tune at the frequency to be controlled which is evident in Figure 6-40 since the circuit had already been embedded in the Cricket bat. It was thus difficult to tune manually.

Figure 6-39 shows the second prototype of new Cricket bat made from the carbon-fibre sandwiched handle bat #3. Figure 6-40 shows its vibrations under random excitation where: Figure 6-40(a) is the vibration without shunt control and Figures 6-40 (b & c) show the vibration with shunt control. One can see from the vibration spectrum in Figures (b & c), the magnitude dip of the mode 1 is not at the centre of the peak. It means that the shunt circuit was not tuned to the natural frequency accurately therefore the shunt control effectiveness was reduced.

Since the passive R-L piezoelectric shunt control possesses narrow-band characteristics, when the shunt circuit is not tuned appropriately, it will lose performance. The sophisticated control system should be able to automatically tune to the natural frequency to be controlled. It Chapter 6 Experiment and Implementation 182 should firstly have a mechanism to detect the natural frequency of the structure to be controlled. Hence, an adaptive algorithm to detect multiple frequencies embedded in the signal will be discussed in the next chapter.

Figure 6-39 The second prototype of new Cricket bat integrated with vibration shunt circuit

(a) Before Shunt Chapter 6 Experiment and Implementation 183

(b) After shunt

(c) Similar to Figure (b) but with simulated with smaller frequency range

Figure 6-40 Vibration of the second new Cricket bat prototype (1st mode, random excitation)

6.4 Field testing of the Smart bat

The Smart bat shown in Figure 6.39 was field tested by experienced Cricket players both within RMIT University and Kookaburra Sports. The preliminary feedback from these experienced players indicates that the feel of the bat was comfortable for most strokes and there was a perceived enhancement of the Sweet Spot towards the handle. This rather subjective finding could be linked to the higher frequency induced by the carbon-fibre strips in the Smart bat since the displacement is inverse proportion to the mode frequency: gA D = (Space Age Control, [I 11]), 2π 2 F 2

Chapter 6 Experiment and Implementation 184 where,

D, A and F denote displacement, acceleration and mode frequency respectively. g is a constant that is equal to 9.80665 m/s².

Also, it was immediately obvious if the activated shunt circuit provided a ‘better feel’ of the batting stroke compared to when the circuit was not activated. The overall perception however was that the bat did not feel any worse than the traditional Cricket bat. This was a very comforting outcome and it augers well for further development of the Smart bat provided the law makers see it fit to accommodate technological advances in the sport of Cricket.

6.5 Chapter summary

In this chapter, the experimental results of the passive piezoelectric vibration shunt control for various beams and Cricket bats have been shown. Excellent vibration reductions have been achieved for the aluminium and carbon-fibre composite beams, except for the wooden beam, as predicted by the ANSYS® simulation. A few modifications have been made on the traditional wooden Cricket bat. The carbon-fibre sandwiched handle Cricket bats were prototyped after tubular handle designs were extensively trialled. A certain vibration reduction was achieved although the Cricket bat design was restricted by the laws of the game. It is possible to increase the mode frequencies and maintain the anti-node of the 1st bending mode close to the shoulder area by optimising the geometry size, Young’s modulus of the carbon-fibre strips and the thickness of the wood material sandwiched between the two strips. The high power supply of the shunt circuit can help to increase the shunt control efficiency. An important thing is that the natural frequencies of the Cricket bat will shift more or less in reality when boundary conditions change. This will reduce the shunt efficiency due to the narrow-band characteristic of the shunt circuit. Because of this, it was critical to develop a way to online-detect the natural frequency to be controlled and real-time adjust the value of the inductor to track that frequency. The analytical work in this regard is presented in the next chapter.

The field testing of the bat was carried out and the perception of the Smart bat was that it generally felt good to play with. Chapter 7 Frequency Estimation and Tracking 185

Chapter 7 Frequency Estimation and Tracking

As mentioned in previous chapters, the R-L passive piezoelectric vibration shunt control is the narrowband control that targets the natural frequency of structures. It is very sensitive to the shift of the natural frequency to be controlled. When the frequency of a shunt circuit does not match the natural frequency to be controlled, the effectiveness of the shunt control will worsen dramatically. In reality, the natural frequencies of structures often shift when boundary conditions, material properties or environments change. The natural frequencies of Cricket bats will be affected with or without being held by the batsman. Even the hand location and grip of the batsman will affect the natural frequencies. So it is important to have a mechanism to detect and track the natural frequencies in real-time to automatically adjust the parameters of the shunt circuit instantly in order to maintain the effectiveness of the shunt control. In this chapter, a method of detecting multiple sinusoid frequencies buried in noise by using adaptive Infinite Impulse Response (IIR) notch filters is presented. In first section, filters are briefly reviewed. Section 2 gives an example of generic notch filter. The adaptive notch filter for frequency estimation is discussed, and an example is given in section 3. The frequency estimation and tracking is a possible option for future Smart Cricket bats.

7.1 Introduction of filters

Filters are used to remove unwanted frequency components and noise from signals. There are two types of filters, i.e. analogue filters and digital filters. Analogue filters are implemented in electronic hardware, such as resistors, capacitors, and operational amplifiers, etc. Digital filters can be implemented either in hardware or software or both. Since the adaptive algorithms are usually implemented in software, digital filters are often the choice for adaptive filters. The digital filters can be classified as having either finite impulse response (FIR) or infinite impulse response (IIR). FIR filters are non-recursive filters, which means Chapter 7 Frequency Estimation and Tracking 186 they have no feedback paths. IIR filters are recursive filters that have feedback paths. FIR filters have the advantage of stability and linear phase response if desired, whereas IIR filters have the advantage of being very computationally efficient. Figure 7-1 shows the diagram of a direct realization of an Nth order non-recursive FIR digital filter.

x(n −1) x(n − 2) x(n−N+2) x(n−N+1) x(n) z-1 z-1 z-1 z-1

h(0) h(1) h(2) h(N −2) h(N −1)

+ + + + y(n)

Figure 7-1 Direct realization of an Nth order non-recursive FIR digital filter

Given an input sequence, x(n) , and a finite length impulse response sequence, h(n) , the output sequence, y(n) can be obtained according to:

N−1 y(n) = h(n)*x(n) = ∑h(k)⋅ x(n − k) . (7.1) k=0 where * denotes convolution, N is the length of the impulse response, and h(n) = 0 for n < 0 which means that the filter is a causal system.

The transfer function of the above Nth order FIR filter in the z-domain is:

N −1 H (z) = ∑ h(k) ⋅ z −k . (7.2) k =0

Figure 7-2 shows the diagram of a direct realization of an Nth order recursive IIR digital filter. The IIR filter’s output depends not only on the previous M inputs but also on the previous N outputs. The differential equation of an IIR filter can be expressed as:

N M−1 y(n)+∑∑ak ⋅ y(n−k) = bk ⋅x(n−k) . (7.3) k=1 k=0

The transfer function in the z-domain is:

−1 −M +1 b0 + b1 ⋅ z +…+ bM −1 ⋅ z H(z) = −1 −N . (7.4) 1+ a1 ⋅ z +…+ aN ⋅ z Chapter 7 Frequency Estimation and Tracking 187 The IIR filter has zeros as well as poles. The pole positions are determined by the recursive feedback coefficients ak . In order for the filter to be stable, the poles must be located within the unit circle for a causal system.

x(n) + + y(n)

b0 z-1 z-1

+ +

b1 a1 z-1 z-1

+ +

b2 a2

+ +

bM −2 aN −1 z-1 z-1

+ +

bM −1 aN

Figure 7-2 Direct realization of an Nth order recursive IIR digital filter

Adaptive filters are the filters that, in a sense, can self-design their filter coefficients in real- time to optimum values based on certain rules according to the time varying input signal. An adaptive digital filter consists of a programmable digital filter and an adaptive algorithm to update the coefficients of the digital filter. The programmable digital filter can be a FIR or IIR type. Figure 7-3 shows a block diagram of the digital filter.

d(n) + Programmable y(n) − ∑ x(n) A/D Digital Filter e(n)

Adaptive Algorithm Figure 7-3 Block diagram of the general adaptive digital filter Chapter 7 Frequency Estimation and Tracking 188 where

x(n) — The Input signal (such as vibration) of filter,

d(n) — Desired signal, e(n) — Error signal, y(n) — The output signal of filter.

7.2 Generic notch filter

Notch filters are usually used to reject particular frequency component(s) in signals. A notch filter can be simply constructed by an allpass filter. Figure 7-4 is a block diagram of a generic notch filter, where A (z) represents an allpass filter.

1/ 2 X (z) A(z) Y (z)

Figure 7-4 Block diagram of the generic notch filter

The transfer function of allpass filters is unit gain. Their numerator coefficients can be obtained from the denominator coefficients (or vice versa) by reversing the coefficient order and conjugating. Thus the poles and zeros of allpass functions appear in reciprocal conjugate pairs. The general allpass transfer function of an N-stage allpass filter can be represented as,

a* + a* ⋅ z −1 ++ a* ⋅ z −(N −1) + a* ⋅ z − N A(z) = N N −1 1 0 −1 −( N −1) −N . (7.5a) a0 + a1 ⋅ z ++ aN −1 ⋅ z + a ⋅ zN and

A(z) = A(e jω ) =1. (7.5b) where

* ai (for i = 0, 1, …, N) can assume arbitrary values and ai denotes the conjugate of ai . If all the coefficients are real, then the transfer function becomes,

a + a ⋅ z −1 ++ a ⋅ z−(N−1) + a ⋅ z −N A(z) = N N−1 1 0 −1 −(N−1) −N . (7.6) a0 + a1 ⋅ z ++ aN−1 ⋅ z + a ⋅ zN

For the 2nd order real allpass filter, the transfer function can be obtained as Chapter 7 Frequency Estimation and Tracking 189

−1 −2 a2 + a1(1+ a2 )z + z A(z) = −1 −2 . (7.7) 1+ a1(1+ a2 )z + a2 z

Therefore the transfer function of the 2nd order notch filter is,

−1 −2 Y (z) 1 1+ a2 1+ 2a1z + z H (z) = = [1+ A(z)] = −1 −2 . (7.8) X (z) 2 2 1+ a1(1+ a2 )z + a2 z

The zeros of Equation (7.8) are

− 2a ± 4a2 − 4 z = 1 1 = −a ± j 1− a2 ( a ≤1). 1,2 2 1 1 1

One can notice that the radiuses of the zeros are z1,2 = 1 which means the zeros are always on the unit circle. The location of the zeros reflects the notch frequency in frequency domain. The relationship of the notch frequency and the zeros in Equation (7.8) is

ω T = 2πf T = − arccos( a ) n n 1 . where

ωn is the notch normalized angular frequency (ωn = 2π × fn ), T is the sampling interval. If T = 1, then ω n = − arccos(a1 ) .

The poles of Equation (7.8) are

2 − a1 (1+ a2 ) ± [a1 (1+ a2 )] − 4a2 p1,2 = 2 .

The radius of the poles determines the bandwidth of the notch filter. The closer the pole to the unit circle, the narrower the bandwidth and the sharper the notch filter is.

The filter coefficients of the notch filter in Equation (7.8) can be obtained by

⎧a1 = − cos(ω n ) (assume T = 1) ⎪ . (7.9) ⎨ 1− tan( BW / 2) 2 ⎪a2 = = r ⎩ 1+ tan( BW / 2) where BW is 3 dB bandwidth, and r is the pole radius.

For example, ωn = 0.15×(2π ) , r = 0.95, then,

2 a1 = − cos(0.3π ) = −0.587785 , a2 = r = 0.9025,

According to Equation (7.8), the transfer function of the notch filter becomes, Chapter 7 Frequency Estimation and Tracking 190

z 2 −1.17557z +1 H (z) = 0.95125 2 z −1.11826z + 0.9025 .

Figure 7-5 and Figure 7-6 show the plots of the transfer function and pole-zero diagram of above 2nd order notch filter respectively. One can see that the zeros lie on the unit circle. For a notch filter to be stable, the poles must be located inside the unit circle (r < 1).

Figure 7-5 Frequency response of notch filter

Figure 7-6 Pole-zero diagram of the 2nd order notch filter Chapter 7 Frequency Estimation and Tracking 191 7.3 Adaptive notch filter

An adaptive notch filter (ANF) is a notch filter where the notch frequency is determined by a series of observed signal samples to detect and remove sinusoid components buried in the signal. The sinusoid components are usually unknown or time varying. Vibration a signals of flexible structures are often modelled as the combination of harmonic (natural frequency components) signals and additive noise. A second order ANF can be used for detecting single natural frequency. For detecting multiple natural frequencies, the structure of cascaded second order ANFs can be used. When the vibration signal passes though each second order ANF stage, one of the frequency components is detected and removed. The output of preceding 2nd order ANF is the input of succeeding 2nd order ANF.

Figure 7-7 shows a block diagram of a high order adaptive notch filter by cascading multiple 2nd order ANF sections. The selected 2nd order notch filter structure is the IIR filter.

Figure 7-7 Adaptive notch filter by cascading multiple 2nd order ANF sections

7.3.1 Simplified adaptive notch filter

The 2nd order notch filter can be designed by Equation (7.8). However, since there are multiple filter coefficients in Equation (7.8), it requires estimating multiple parameters for the adaptive filter. It increases the complexity of adaptation and the scale of filter stability monitoring. A minimal parameter adaptive notch filter with constrained poles and zeros was proposed by Nehorai (1985) which is an approximation of the generic notch filter. With Nehorai’s method, there are n parameters to be estimated for n sinusoid components rather than 2n for many other adaptive notch filter algorithms. The general transfer function of the 2nth order Nehorai’s notch filter is (Nehorai, 1985),

A(z −1 ) 1+ a z −1 + + a z −n + + a z 2n−1 + z −2n H (z −1 ) = = 1 n 1 (7.10) A(rz −1 ) 1+ ra z −1 + + r n a z −n + + r 2n−1a z 2n−1 + r 2n z −2n 1 n 1 . where r is the pole contraction factor, 0 < r < 1. Chapter 7 Frequency Estimation and Tracking 192 The coefficients of the transfer function in Equation (7.10) has the following relation,

ai = a2n−i (i = 0, 1, …, n-1), and

a0 = a2n = 1.

The ith angular notch frequency given by Equation (7.10) can be obtained as,

θi = arccos( ai / 2) (i = 1, …, n). (7.11a)

or the filter coefficient represented by θi

ai = 2 cos θi ( ai ∈ [-2, 2] ). (7.11b)

The vector of the angular notch frequency is

T θ = [θ1 θn ] . (7.12)

Assume the input signal y(t) is composed of n sinusoid components plus white noise n(t) as below,

n y(t) = ∑Ci cos(θit +φi ) + n(t) . i=1

Let the output signal of the notch filter be

e(t) = H (q−1)y(t) , (7.13a) where q−1 is a unit delay. Hence,

A(q−1) e(t) = y(t) . (7.13b) A(rq−1) where

n−1 −1 −1 −n −2n+1 −2n −i −2n+i −n A(q ) =1+ a1q ++ anq ++ a1q + q = ∑ ai (q + q ) + anq (7.14a) i=0

−1 −1 −1 −1 −n −1 −2n+1 −1 −2n A(rq ) =1+ a1(r q) ++ an (r q) ++ a1(r q) + (r q)

n−1 = a (riq−i + r 2n−iq−2n+i ) + r na q−n ∑ i n . (7.14b) i=0

Equation (7.13b) can be rewritten as,

A(rq−1)e(t) = A(q−1)y(t) , (7.15a) that is, Chapter 7 Frequency Estimation and Tracking 193

n−1 n−1 ⎛ i −i 2n−i −2n+i n −n ⎞ ⎛ −i −2n+i −n ⎞ ⎜∑ ai (r q + r q ) + r anq ⎟e(t) = ⎜∑ ai (q + q ) + anq ⎟y(t) . (7.15b) ⎝ i=0 ⎠ ⎝ i=0 ⎠

The differential equation of Equation (7.15b) is

n−1 n−1 i 2n−i n ∑ ai[r e(t − i) + r e(t − 2n + i)]+ r ane(t − n) = ∑ ai[y(t − i) + y(t − 2n + i)]+ an y(t − n) , i=0 i=0

Or

2n n e(t) = y(t) + y(t − 2n) − r e(t − 2n) + an[y(t − n) − r e(t − n)]

n−1 i 2n−i + ∑ ai[y(t − i) + y(t − 2n + i) − r e(t − i) − r e(t − 2n + i)] i=1 = y(t) + y(t − 2n) − r 2ne(t − 2n) + a ⋅ϕ T (t ) . (7.16) where

a = [a1 an ], ϕ(t) = [ϕ1(t) ϕn (t)] , (7.17)

i 2n−i ⎪⎧y(t − i) + y(t − 2n + i) − r e(t − i) − r e(t − 2n + i) (1 ≤ i ≤ n −1) ϕi (t) = ⎨ . (7.18) n ⎩⎪y(t − n) − r e(t − n) (i = n)

7.3.2 Notch filter coefficient estimation

Since the desired output at notch frequency is zero, that is d(t) = 0, then the prediction error

eˆ(t) at notch frequency is,

eˆ(t) = d (t) − e(t) = −e(t) = −[ y(t) + y(t − 2n) − r 2ne(t − 2n) + a ⋅ϕ T (t)]. (7.19)

Using weighted least-squares criterion (Ljung, 1999 & Chen, 1992), the cost function is

1 N 1 N ˆ 2 2 VN (θ ) = ∑ λt e (t) = ∑ λt e (t) . (7.20) N i=1 N i=1 where N is the number of samples (t = 1, …, N).

The adaptive algorithm is to find the best θ to minimize the argument of the cost function, that is

N ˆ 1 2 θ N = arg min V N (θ ) = arg min ∑ λt e (t) . (7.21) θ θ N t =1

N ∂e(t) Equation (7.21) leads to ∑ = 0 . t =1 ∂θ i Chapter 7 Frequency Estimation and Tracking 194 Let the vector of error gradient be

ψ (t) = [ψ1(t) ψ n (t)]. (7.22a) where

∂e(t) ∂e(t) ∂ai ψ i (t) = − = ⋅ (i = 1,…, n). (7.22b) ∂θi ∂ai ∂θi

From Equation (7.11b), one can have,

∂ai = −sinθi . (7.23) ∂θi ∂e(t) For , differentiating both sides of Equation (7.15a) with respect to ai and considering ∂θ i Equation (7.14), for 1≤ i ≤ n −1,

∂e(t) A(rq −1) + (r iq −i + r 2n−iq−2n+i )e(t) = (q−i + q−2n+i )y(t) . (7.24a) ∂ai for i = n,

∂e(t) A(rq −1) + r nq −ne(t) = q−n y(t) . (7.24b) ∂ai or

⎧y(t − i) + y(t − 2n + i) − r ie(t − i) − r 2n−ie(t − 2n + i) (1≤ i ≤ n −1) −1 ∂e(t) ⎪ A(rq ) = ⎨ ∂a n i ⎩⎪y(t − n) − r e(t − n) (i = n)

= ϕi (t ) . (7.25a)

Hence,

∂e(t) ϕi (t) = −1 . (7.25b) ∂ai A(rq )

Substituting Equations (7.23, 7.25b) into Equation (7.22b) obtains the error gradient with respect to θi is, ϕ (t) ψ (t) = −2 i ⋅sinθ (i = 1,…, n). (7.26) i A(rq−1) i

The adaptive algorithm is the recursive least-squares (RLS) algorithm (Ljung, 1999) that is summarized as Chapter 7 Frequency Estimation and Tracking 195

⎧eˆ(t) = y(t) − ϕ T (t)θˆ(t − 1) ⎪ ⎪ P(t −1)ψ (t) ⎪K (t) = T ⎪ λ (t) +ψ (t)P(t − 1)ψ (t) ⎨ . (7.27) ⎪ 1 T P(t) = []P(t − 1) − K (t)ψ (t)P(t − 1) ⎪ λ (t) ⎪ ⎪ ˆ ˆ ⎩θ (t) = θ (t − 1) + K (t)eˆ(t) where

y(t) is the input signal,

eˆ(t) is the predication error represented by Equation (7.19),

ϕ(t) is determined by Equation (7.18),

θˆ(t) is the optimal frequency vector obtained by Equation (7.11), K(t) and P(t) are filter parameters, λ(t) is the forgetting factor.

7.3.3 Examples of adaptive notch filter using cascaded 2nd order ANFs

The 2nd order notch filter requires estimating only one parameter that is the poles contraction factor r. The transfer function of the 2nd order constrained notch filter can be represented as,

A(z −1 ) 1+ az −1 + z −2 1+ az −1 + z −2 H (z −1 ) = = = . (7.28) A(rz −1 ) 1+ a(r −1z)−1 + (r −1z)−2 1+ arz −1 + r 2 z −2

Comparing Equation (7.28) with Equation (7.8), one can have

1+ a a = 2a , r = 2 , and r 2 = a . 1 2 2

which results in r = a2 =1. This should be a special case for the constrained notch filter to match the generic notch filter. However, the notch frequency is only determined by the coefficients of the numerator of filter transfer function which are the same obtained by Equation (7.8) or Equation (7.28). As long as r →1 and the poles are as close to zeros as possible, the constrained notch filter represented by Equation (7.28) is a good approximation of Equation (7.8).

The error gradient of the 2nd order ANF is

∂eˆ(t) ∂eˆ(t) ∂a ψ = − = − . (7.29) ∂θ ∂a ∂θ Chapter 7 Frequency Estimation and Tracking 196 where a = 2 cos θ ∈ [-2, 2] .

The differential equation of Equation (7.28) is,

e(t) = y(t) + ay(t −1) + y(t − 2) − are(t −1) − r 2e(t − 2) . (7.30)

Figure 7-8 is the direct realization of the constrained 2nd order IIR notch filter represented by Equation (7.30).

y(t −1) y(t − 2) -1 -1 y(t) z z

+ + + e(t) a

z-1 − ra + e(t − 2)

-1 2 z − r e(t −1)

Figure 7-8 Direct realization of the constrained 2nd order IIR notch filter

The predication error is

2 eˆ(t) = d (t) − e(t) = −e(t) = are(t −1) + r e(t − 2) − y(t) − ay(t −1) − y(t − 2) . (7.31)

∂eˆ(t) Let the derivative of Equation (7.31) with respect to “a” be zero, that is = 0 , obtains ∂a

∂e(t) ∂e(t −1) ∂e(t − 2) + ar + r 2 = y(t −1) − re(t −1) . (7.32) ∂a ∂a ∂a

Then let,

∂eˆ(t) ∂e(t) E (t) = = − , ∂a ∂a

Equation (7.32) becomes,

− [1 + arq −1 + r 2 q −2 ]E (t) = y(t − 1) − re(t −1) . (7.33)

Therefore derivative can be represented by, Chapter 7 Frequency Estimation and Tracking 197 ∂eˆ(t) y(t −1) − re(t −1) y(t −1) − re(t −1) E(t) = = − = − . (7.34) ∂a 1+ arq −1 + r 2 q −2 A(rq −1 )

∂a ∂a Since = −2 sin θ , substituting and Equation (7.18) to Equation (7.12) obtain the ∂θ ∂θ gradient as,

y (t − 1) − re (t − 1) ψ = −2 sin θ . (7.35) A(rq −1 )

Figure 7-9 to Figure 7-12 are the simulations of the frequency estimation using the three cascaded 2nd order IIR adaptive notch filter. Figure 7-9 shows the input signal to the ANF that contains three sinusoid frequency components and white noise (top left), and the output signals of three 2nd order ANF stages respectively. The sinusoid frequencies of the input signal are 0.05, 0.1 and 0.2 of sampling frequency respectively. The power of the white Gaussian noise added n(t) , which is independent of the signal component, is 0 dBW. The signal to noise ratio (SNR) of individual frequency component is 0 dB, i.e.

y (t ) = cos( 0.05 × 2π × t ) + cos( 0.1 × 2π × t ) + cos( 0.2 × 2π × t ) + n(t )

Figure 7-9 Input signal and the output signals of each 2nd order ANF (SNR = 0 dB)

Figure 7-10 shows the adaptation progress of the ANF that shows that the estimation converges to three sinusoid frequency components of the input signal after about 100 Chapter 7 Frequency Estimation and Tracking 198 iterations. Figure 7-11 shows the transfer function of the notch filter. One can see that the notch filter has converged to the frequencies of 0.05, 0.1 and 0.2 (Rad/sec) which match the sinusoid frequency components in the input signal.

Figure 7-10 Adaptation history of the ANF (SNR = 0 dB)

Figure 7-11 Transfer functions of the ideal and estimated notch filters (SNR = 0 dB) Chapter 7 Frequency Estimation and Tracking 199 Figure 7-12 shows spectra of the input signal and the output signal of the individual 2nd order ANF. The spectra show that each 2nd order notch filter removes one frequency component.

Figure 7-12 Spectra of the input signal and the output signals of each 2nd order ANF (SNR = 0 dB)

Figure 7-13 Input signal and the output signals of each 2nd order ANF (SNR = –10 dB) Chapter 7 Frequency Estimation and Tracking 200 Figures 7-13 to 7-16 are the simulations which are similar to those in Figures 7-9 to 7-12 respectively but with SNR = –10 dB. Figure 7-13 shows that it takes a longer time to converge due to more noise compared to the result with 0 dB SNR. The simulations have shown that with a larger noise presence, it will make the convergence more difficult.

Figure 7-14 Adaptation history of the ANF (SNR = –10 dB)

Figure 7-15 Transfer functions of the ideal and estimated notch filter (SNR = –10 dB) Chapter 7 Frequency Estimation and Tracking 201

Figure 7-16 Spectra of the input signal and the output signals of each 2nd order ANF (SNR = –10 dB)

Considering the damping factor in structures, the vibrations are a decaying signals rather than constant signals. This will add some difficulties to the estimation. Figures 7-17 to 7-20 are the simulations using decaying sinusoid components in the input signal with SNR = 0 dB. Comparing with the signals in Figure 7-12, the spectra in Figure 7-20 are nosier.

Figure 7-17 Input and output signals of each 2nd order ANF (decaying sinusoid, SNR = 0 dB)

Chapter 7 Frequency Estimation and Tracking 202

Figure 7-18 Adaptation history of the ANF (decaying sinusoid, SNR = 0 dB)

Figure 7-19 Transfer functions of the ideal and estimated notch filter (decaying sinusoid, SNR = 0 dB)

Chapter 7 Frequency Estimation and Tracking 203

Figure 7-20 Spectra of the input signal and the output signals of each 2nd order ANF (decaying sinusoid, SNR = 0 dB)

7.4 Chapter summary

In this chapter, a frequency estimation method using an adaptive notch filter has been presented. The notch filter structure adopted is a constrained poles and zeros IIR structure, which is an approximation of the ideal notch filter but needs only half parameters to be adapted compared with using an ideal one. To estimate n frequency components, n number of 2nd order IIR adaptive notch filter can be cascaded. Each of them is responsible for estimating one frequency component. The adaptive algorithm used is the recursive least-squares (RLS) algorithm. It is a fast and efficient algorithm. An example of three cascaded 2nd order adaptive notch filters was given and simulated. The results show that the notch filter was correctly converged to the frequency components contained in the input signals. With the help of frequency estimation, it is possible to tune the gyrator-based inductor in the passive piezoelectric shunt control automatically with the frequency controlled resistor or capacitor to track the natural frequency shift of the Cricket bat.

This remained an analytical study into frequency tracking for use in a digital control system for a Cricket bat and was hence not implemented in the analogue shunt control system designed and implemented in the prototype Smart Cricket bat described in the previous chapters. Chapter 8 Conclusions and Future work 204

Chapter 8 Conclusions and Future Work

Overview

This research project has explored the possibility of vibration control using Smart materials in combination with electronics and signal processing techniques to reduce the vibration of Cricket bats. The emphasis has been on the passive piezoelectric vibration shunt control method so far. Substantial analytical analyses, numerical simulations and practical experiments have been carried out. The prototype of the newly designed Cricket bat has been designed, evaluated and fabricated.

Although some vibration reduction of the Cricket bat has been achieved, it is the best that could be achieved given the technology used and the restrictions dictated by the rules of the game of Cricket. There are two major reasons which make the task difficult. One is that the governing rules of the Cricket game prevent the radical modification of the Cricket bat. The other is that unlike tennis, the racket or baseball bat, made of thin carbon-fibre or aluminium materials (which are light and have small inherent damping factor), the Cricket bat (made of a thick wooden chunk) has a much larger mass which will acquire much more mechanical energy after the collision with a Cricket ball. Also, the large inherent damping factor of wood makes the external damping less significant as a result. This requires much more control energy being applied to the Cricket bat to achieve a reasonable and notable vibration reduction. In spite of these restrictions, the vibration reductions reported are nevertheless significant when compared to technological improvements in the field of sporting implements.

The work carried out is an initial trial and has paved the way for future work. The conclusions from work undertaken so far and suggestions for future work are described in the following subsections. Chapter 8 Conclusions and Future work 205 8.1 Conclusions

8.1.1 Analytical work on the PZT-based vibration shunt circuit

In chapter 3, an analytical study of resistor-inductor piezoelectric vibration shunt control on a cantilever beam was carried out. The equation of motion of the beam attached with the PZT patch and coupled with electrical shunt network has been derived by using Hamilton’s principle and solved with the help of Galerkin’s method. The analysis has shown the physical interaction between mechanical energy and electrical energy related by the constitutive equations of piezoelectric materials. This augments the theoretical understanding of the passive piezoelectric vibration shunt control.

8.1.2 Field-coupled simulations

Extensive numerical simulations with the FEA software package ANSYS® have been conducted. Significant effort has been spent on developing the ANSYS® program for the passive piezoelectric vibration shunt control which can help to analyse, design and optimise the structure and control system prior to actual physical experimentation and implementation. Some simulation results have been presented in chapters 4 & 5. With the help of the ANSYS® simulations, it was found that the material properties and geometrical sizes of the structure and piezoelectric transducer have significant influence on the effectiveness of the passive piezoelectric vibration shunt control.

8.1.3 Cricket bat innovations

Chapter 4 has been a thorough analysis of the strain generated in the piezoelectric transducer with respect to the geometry sizes and material properties of the transducer and the structure to be controlled. It was found that the neutral axis/surface of the composite structure determines the amount of strain produced in the transducer. Carbon-fibre material has been suggested to be a medium between the structure and the piezoelectric transducer in order to produce more strain in the piezoelectric transducer. The simulation and experimental results have shown that the passive piezoelectric vibration shunt control is markedly effective with respect to the vibration control of carbon-fibre materials.

A comparative experimental study of the passive piezoelectric vibration shunt controls of aluminium, wooden and carbon-fibre composite beams was carried out. The experimental results have shown that excellent vibration reduction can be achieved with the carbon-fibre composite beam that matches the results of analytical analyses and numerical simulations. This finding is significant and can result in several other applications for structural vibration Chapter 8 Conclusions and Future work 206 control. The proposed Cricket bat handle, partially made of carbon-fibre composite material, is one of the possible applications.

A novel sandwiched Cricket bat handle has been designed. The handle, incorporated with specifically designed high stiffness-to-weight ratio advanced carbon-fibre composites strips and a shunted piezoelectric sensor/actuator, has produced a reduction in the transient vibration in the Cricket bat upon impact with a Cricket ball. In addition, another advantage of using the carbon-fibre strips is that the natural frequencies of the Cricket bat have been increased.

The piezoelectric vibration shunt control circuit has been made and used for vibration control for beams and Cricket bats. It has found that the efficiency of the vibration shunt control (vibration reduction) is in proportion to the power supply of the shunt circuit. The higher the supply voltage, the better the vibration reduction. A small circuit, such as this, has been embedded in the Cricket bat handle to control the Cricket bat vibration.

8.1.4 Frequency tracking analysis

To account for the variable natural frequencies of Cricket bats due to the natural variability of wood types and the time dependency of this variation due to the usage of the bat, it become necessary to track and identify these changing frequencies using built-in hardware. The method of fast estimating and tracking multiple frequency components buried in a noisy signal by using cascaded 2nd order adaptive notch filters has been presented in chapter 7. The method can be used for detecting the natural frequency of structures, such as Cricket bats and adjust the control frequency in real-time which is crucial for achieving a high efficiency in the piezoelectric vibration shunt control. It is possible to implement the adaptive notch filter in hardware and embed it in the structure to automatically tune the control frequency of the shunt circuit by adjusting some kind of variable resistors and capacitors to track the natural frequencies of the structure.

8.1.5 Power amplifier proposal

In appendix 4, a novel power amplifier for driving the piezoelectric actuator, by using an amplitude modulation technique and a piezoelectric transformer, has been proposed.

8.1.6 Cricket bat prototyping and field testing

Two Smart Cricket bats have been prototyped and field tested. The preliminary feedback has been positive from these field tests undertaken by experienced Cricket players. However, before the merits of the Smart bat can be declared, more robust batting at the nets and Chapter 8 Conclusions and Future work 207 evaluation will be needed from top-class batsmen. Future work to improve the bat is suggested in the next section.

8.2 Future work

There are many new aspects which could be pursued in the future to improve the Smart Cricket bat.

On Smart Cricket bat technology improvements

¾ To optimise the thickness and the Young’s modulus of the carbon-fibre strips to maximize the strain produced in the PZT patch.

¾ To increase the natural frequencies of the Cricket bat by selecting the material property and geometry size of the carbon-fibre strips but in the meantime maintain the anti-node of the first mode residing in the handle area.

¾ To try to use the soft piezoelectric transducer (piezoelectric fibre composite) instead of the hard piezoelectric transducer for the shunt control.

¾ To implement the adaptive notch filter into a digital-based hardware together with an automatic tuning mechanism to tune the gyrator-based inductor according to the frequency being estimated.

¾ To improve the engineering work in making the bat handle, including finding a proper location to embed the shunt circuit, battery, and using automatic switches such as a motion switch to turn the control circuit OFF and ON.

¾ To increase the supply voltage of the shunt circuit. To replace battery power by a self- powered system that is a challenging task.

¾ If both the self-powered system and miniature power amplifier for driving the piezoelectric actuator can be developed, the active piezoelectric vibration control will be a better alternative to the passive piezoelectric vibration shunt control.

On the developed Smart Cricket bat

¾ To investigate which bending mode does the most damage to cricket players.

¾ More field testing by top class batsmen will be needed on the developed Smart Cricket bat in order to comprehensively evaluate the performance of the bat. Appendix 1 Modeling of Vibration Isolation and Absorption of the SDOF Structure 208

Appendix 1 Modeling of Vibration Isolation and Absorption of the SDOF Structure

This Appendix is linked to Chapter 2 (sections 2.2.1 ~ 2.2.2, pp. 11 ~ 13)

A1.1 Vibration isolation

The classic model of a vibration isolator is a single-degree-of-freedom (SDOF) mass-spring- dashpot system as shown in Figure A1-1. In Figure A1-1, ‘k’ represents the stiffness of a massless spring, ‘c’ represents the viscous damping coefficient of a massless damper and ‘m’ represents a rigid mass respectively. In Figure A1-1a, ‘Fs’ and ‘FT’ are the input disturbing force and output transmitted force of the isolator respectively; in Figure A1-1b, Xs and ‘XT’ are the input disturbing motion and output transmitted motion of the isolator respectively.

Fs XT

Source (m) Receiver (m)

x FT k c k c Xs

Foundation Foundation

(a) Excited by force acting on mass (b) Excited by foundation motion

Figure A1-1 Single-degree-of-freedom vibration isolator

Appendix 1 Modeling of Vibration Isolation and Absorption of the SDOF Structure 209

The purpose of the isolator is to make the amplitude of FT (or X T ) less than the amplitude of

FS (or X S ). The SDOF isolator in Figure A1-1a can be described by differential equations,

mx+ cx + kx(t) = Fs (t) . (A1.1)

FT (t) = cx + kx(t) . (A1.2)

Where dot (·) represents time derivation.

The Laplace transform of the Equations (A1.1) and (A1.2) are

2 (ms + cs + k)X (s) = Fs (s) . (A1.3)

FT (s) = (cs + k)X (s) . (A1.4)

Transmissibility (TF ), which is used to describe the effectiveness of a vibration isolation system, is defined by the ratio of the transmitted force ( FT ) of an isolated system to the input disturbing force (F) (Dimarogonas, 1996). From Equations (A1.3) and (A1.4), one can obtain,

2 2 FT (s) FT (s) X (s) cs + k 2σs + ω n 2ζω ns + ω n TF (s) = = = 2 = 2 2 = 2 2 . (A1.5) Fs (s) X (s) F (s) ms + cs + k s + 2σs + ω n s + 2ζω ns + ω n where

k ω = –– Undamped natural frequency of the isolation system. n m c σ = = ζω –– Damping factor. 2m n c c ζ = = –– Damping ratio. cc 2 km

cc = 2mωn –– Critical damping.

The frequency response of the transmissibility can be expressed as:

2 jζω ω +ω 2 ω ω 2 + (2ζω)2 1+ (2ζr)2 T ( jω) = n n = n n = . (A1.6) F ( jω)2 + j2ζω ω +ω 2 2 2 2 2 (1− r2 )2 + (2ζr)2 n n (ωn −ω ) + (2ζωnω)

ω f Where r = = is the ratio of the disturbing force frequency to the natural frequency of ω n f n the isolation system.

X T (s) Similarly, the transmissibility of the isolation system in Figure A1-1b, TX (s) = , will X s (s) have the same result. Appendix 1 Modeling of Vibration Isolation and Absorption of the SDOF Structure 210

The isolator takes effect only when the transmissibility TF <1. From Equation (A1.6), in order 2 2 to satisfy TF ( jω) < 1, the condition r (r − 2) > 0 must be met. The possible saturations are discussed below.

(1) 0

−2 (2) r =1 (i.e. ω = ωn ) , TF ( jω) = 1+ (2ζ ) , since ζ is relative small, TF ( jω) becomes very large. Instead of isolating the vibration, the isolator amplifies the vibration.

(3) r > 2 (i.e. ω > 2ωn ), TF ( jω) <1 (isolation)

The above discussed results indicate that the vibration isolation method has no effect for vibration frequencies lower than 2 times of the natural frequency of the isolation system, and only has effect for vibration frequencies above 2 times of the natural frequency of the isolation system. The worst scenario happens when ω = ωn , the isolator adds oscillation to the system. The isolator will have better efficiency only for vibrations with frequencies much higher than the natural frequency of the isolator, in other words, the isolator works best when the isolator's natural frequency is small compared to the disturbance frequency. If one wants to isolate low frequency vibration, one has to reduce the natural frequency of the isolation system, which usually leads to increasing the mass of the isolation system to make it larger and heavier.

A1.2 Vibration absorption

Figure A1-2a and Figure A1-2b show the models of the dynamic (undamped) absorber and damped absorber attached on an undamped SDOF vibration system respectively. The differential equations for the damped absorber in Figure A1-2b are,

ms x1 + ca x1 + (ks + ka )x1 = Fs + FT 2 , (A1.7)

ma x2 + ca x2 + ka x2 = FT1 , (A1.8)

FT1 = ca x1 + ka x1 , (A1.9)

FT 2 = ca x2 + ka x2 . (A1.10)

Appendix 1 Modeling of Vibration Isolation and Absorption of the SDOF Structure 211

Absorber x Absorber x 2 2

ma ma

k F k F ca a T 2 a T 2

Fs x1 Fs x1

m s ms

F F ks T1 ks T1

(a) Dynamic (undamped) absorber (b) Damped absorber

Figure A1-2 Models of vibration absorber

Substituting Equation (A1.9) and Equation (A1.10) into Equation (A1.7) and Equation (A1.8) obtains Equation (A1.11) and Equation (A1.12),

ms x1 + ca x1 − ca x2 + (ks + ka )x1 − ka x2 = Fs , (A1.11)

ma x2 − ca x1 + ca x2 − ka x1 + ka x2 = 0 . (A1.12)

Equation (A1.11) and Equation (A1.12) can be written as matrix format in Equation (A1.13),

⎡m11 m12 ⎤⎡x1 ⎤ ⎡c11 c12 ⎤⎡x1 ⎤ ⎡k11 k12 ⎤⎡x1 ⎤ ⎡Fs ⎤ ⎢ ⎥⎢ ⎥ + ⎢ ⎥⎢ ⎥ + ⎢ ⎥⎢ ⎥ = ⎢ ⎥ . (A1.13) ⎣m21 m22 ⎦⎣x2 ⎦ ⎣c21 c22 ⎦⎣x2 ⎦ ⎣k21 k22 ⎦⎣x2 ⎦ ⎣ 0 ⎦ where

m11 = ms , m12 = m21 = 0 , m22 = ma ,

c11 = c22 = ca , c12 = c21 = −ca ,

k11 = ks + ka , k12 = k21 = −ka , k22 = ka .

Equation (A1.13) can be simplified as,

MX + CX + KX = F . (A1.14) where

m m c c k k x x x ⎡ 11 12 ⎤ ⎡ 11 12 ⎤ ⎡ 11 12 ⎤ ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡ 1 ⎤ M = ⎢ ⎥ , C = ⎢ ⎥ , K = ⎢ ⎥ , X = ⎢ ⎥ , X = ⎢ ⎥ , X = ⎢ ⎥ , ⎣m21 m22 ⎦ ⎣c21 c22 ⎦ ⎣k21 k22 ⎦ ⎣x2 ⎦ ⎣x2 ⎦ ⎣x2 ⎦ Appendix 1 Modeling of Vibration Isolation and Absorption of the SDOF Structure 212

⎡Fs ⎤ and F = ⎢ ⎥ . ⎣ 0 ⎦

The frequency domain representation of Equation (A1.11) and Equation (A1.12) are,

2 [(ks + ka − msω ) + jcaω]X1 − (ka + jcaω)X 2 = Fs , (A1.15)

2 − (ka + jcaω)X1 + [(ka − maω ) + jcaω]X 2 = 0 . (A1.16)

The Equation (A1.15) and Equation (A1.16) are linear and nonhomogeneous and can be solved with Cramer’s rule,

Fs − (ka + jcaω) 0 (k − m ω 2 ) + jc ω X (ω) = a a a , (A1.17) 1 2 (ks + ka − msω ) + jcaω − (ka + jcaω) 2 − (ka + jcaω) (ka − maω ) + jcaω

2 (ks + ka − msω ) + jcaω Fs

− (ka + jcaω) 0 X 2 (ω) = 2 . (A1.18) (ks + ka − msω ) + jcaω − (ka + jcaω) 2 − (ka + jcaω) (ka − maω ) + jcaω Or

2 (ka − maω ) + jcaω X1(ω) = 4 2 2 Fs , (A1.19) [mamsω − (kams + ksma + kama )ω + kaks ] + j[ks − (ma + ms )ω ]caω

ka + jcaω X 2 (ω) = 4 2 2 Fs . (A1.20) [mamsω − (kams + ksma + kama )ω + kaks ] + j[ks − (ma + ms )ω ]caω

Let,

2 ka ωa = –– Undamped natural frequency of the absorber. ma

2 ks ωs = –– Undamped natural frequency of the vibration system. ms m μ = a –– Ratio of absorber mass to system mass. ms c c c ζ = a = a = a ––– Absorber damping ratio. cca 2 kama 2maωa

Substituting ωa , ωs , μ , and ζ into Equation (A1.19) and Equation (A1.20) obtains, Appendix 1 Modeling of Vibration Isolation and Absorption of the SDOF Structure 213

2 2 1 (ωa −ω ) + j2ζωaω X1(ω) = 4 2 2 2 2 2 2 2 2 Fs , (A1.21) ms [ω − (ωa + μωa + ωs )ω + ωaωs ] + j2ζωaω[ωs − (1+ μ)ω ]

2 1 ωa + j2ζωaω X 2 (ω) = 4 2 2 2 2 2 2 2 2 Fs . (A1.22) ms [ω − (ωa + μωa + ωs )ω + ωaωs ] + j2ζωaω[ωs − (1+ μ)ω ]

Equation (A1.21) and Equation (A1.22) can be simplified as,

2 2 1 (r − Ωs ) + j2ζrΩs X1(ω) = 4 2 2 2 2 2 Fs , (A1.23) ks [Ωs − (r + μr +1)Ωs + r ] + j2ζrΩs[1− (1+ μ)Ωs ]

2 1 r + j2ζrΩs X 2 (ω) = 4 2 2 2 2 2 Fs . (A1.24) ks [Ωs − (r + μr +1)Ωs + r ] + j2ζrΩs[1− (1+ μ)Ωs ] where

ω r = a –– Ratio of absorber natural frequency to system natural frequency. ωs ω Ωs = –– Frequency normalized to system natural frequency. ωs

The amplitudes of motion without considering phase angle are,

2 2 2 2 X1(ω) (r − Ωs ) + (2ζrΩs ) = 4 2 2 2 2 2 2 2 2 , (A1.25) (Fs / ks ) [Ωs − (r + μr +1)Ωs + r ] + (2ζrΩs ) [1− (1+ μ)Ωs ]

4 2 X 2 (ω) r + (2ζrΩs ) = 4 2 2 2 2 2 2 2 2 . (A1.26) (Fs / ks ) [Ωs − (r + μr +1)Ωs + r ] + (2ζrΩs ) [1− (1+ μ)Ωs ]

The phase angles of X1(ω) and X 2 (ω) are, respectively,

2 −1 2ζrΩs −1 2ζrΩs[1− (1+ μ)Ωs ] ψ X 1(ω) = tan 2 2 − tan 4 2 2 2 2 , (A1.27) r − Ωs Ωs − (r + μr +1)Ωs + r

2 −1 2ζrΩs −1 2ζrΩs[1− (1+ μ)Ωs ] ψ X 2 (ω) = tan 2 − tan 4 2 2 2 2 . (A1.28) r Ωs − (r + μr +1)Ωs + r

When the natural frequency of the absorber is tuned to one of the natural frequencies of the vibration system (i.e. r = 1), it can maximize the energy absorption from the vibration system.

Figure A1-3 is a simulation of Equation (A1.25) with μ = 0.02, ζ = 0.01 and r =1. The displacement amplitude of the vibration system at natural frequencies is dipped due to the Appendix 1 Modeling of Vibration Isolation and Absorption of the SDOF Structure 214 vibration absorber attached. For controlling low frequency vibration, the natural frequency of the absorber has to be low which results in bulky size and heavy weight to the absorber.

Figure A1-3 Spectrum of structure displacement with damped absorber

One can notice that if let the absorber’s damping coefficient ca =0 in Figure A1-2b, the absorber will become equivalent to that in Figure A1-2a. Simply letting the absorber damping ratio ζ = 0 in Equation (A1.25) to Equation (A1.28), these equations then represent the dynamic absorber in Figure A1-2a. Appendix 2 Estimation of System Damping 215

Appendix 2 Estimation of System Damping

This Appendix is linked to Chapter 2 (section 2.2.3, pp. 13 ~ 16)

The system damping can be estimated either in frequency domain or in time domain. In this appendix, the damping estimation of a single degree of freedom (SDOF) structure is described. The methods may work with a bandpass filter to estimate the damping ratio of each individual mode of a MDOF system separately.

A2.1 Frequency domain estimation

The basic model for the SDOF system is shown in Figure A2-1. The system consists of a mass (m), a spring (k) and a viscous dashpot (c), where f (t) and x(t) are general time- varying force and displacement response quantities respectively (Ewins, 2002).

f

m

x (t) k c

Figure A2-1 Single-degree-of-freedom (SDOF) system (Ewins, 2002)

Appendix 2 Estimation of System Damping 216 According to Newton’s law, the differential motion equation of the SDOF system can be derived as

mx(t) + cx(t) + kx(t) = f (t) . (A2.1)

From Equation (A2.1), the Laplace transfer function of the SDOF system can be expressed as,

X (s) 1 1/ m H (s) = = = 2 c k F (s) ms + cs + k s 2 + s + m m 1/ m 1/ m = 2 2 = 2 2 . (A2.2) s + 2σs + ω n s + 2ζω n s + ω n where

k ω = –– Undamped natural frequency of the isolation system. n m

c σ = = ζω –– Damping factor. 2m n

c c ζ = = –– Damping ratio. cc 2 km

cc = 2mωn –– Critical damping.

The frequency domain transfer function of the Equation (A2.2) is

1/ m H ( jω) = 2 2 . (A2.3) (ω −ωn ) + j2ζω nω

The amplitude of the transfer function without considering phase angle is

1/ m H ( jω) = . (A2.4) 2 2 2 2 2 2 (ω − ωn ) + 4ζ ωnω

At the centre frequency ω n ,

1 H ( jω n ) = 2 . (A2.5) 2mζω n

Assume ω1 and ω2 are the 3 dB frequencies and ω1 < ωn < ω 2 . According to the definition that the amplitude at 3 dB frequency points are equal to 1/ 2 times the amplitude at centre frequency which is represented by Equation (A2.5), that is, Appendix 2 Estimation of System Damping 217 1/ m 1 H ( jω ) = = , 1 2 2 2 2 2 2 2 (ω1 − ωn ) + 4ζ ωnω1 2(2mζω n )

1/ m 1 H ( jω ) = = . 2 2 2 2 2 2 2 2 (ω2 − ωn ) + 4ζ ωnω2 2(2mζω n )

Since

H ( jω1) = H ( jω2 ) , that is

2 2 2 2 2 2 2 2 2 2 2 2 (ω1 − ωn ) + 4ζ ωnω1 = (ω2 − ωn ) + 4ζ ωnω2 , or

2 2 2 2 2 2 2 2 2 (2ωn − ω1 − ω2 )(ω2 − ω1 ) = −4ζ ωn (ω2 − ω1 ) , or

2 2 2 2 2 2 − 4ζ ωn = (ωn − ω1 ) + (ωn − ω2 ) = (ωn − ω1)(ωn + ω1) − (ω2 − ωn )(ω2 + ωn ) .

For a small damping system, which has narrow frequency bandwidth, has the following relation,

(ωn − ω1) = (ω2 − ωn ) ≈ ωn , then,

2 2 − 4ζ ωn = (ωn − ω1)[(ωn + ω1) − (ω2 + ωn )] = (ωn − ω1)(ω1 − ω2 ) ≈ ωn (ω1 − ω2 ) .

Therefore, the damping ratio ζ can be expressed as,

ω2 − ω1 f2 − f1 Δf ζ = = = ( f1 < f2 ). (A2.6) 2ωn 2 fn 2 fn

Equation (A2.6) indicates that the damping ratio of a lightly damped SDOF system ζ = c/cc can be estimated in frequency domain by measuring the centre frequency fn and its bandwidth Δf . This method is often used in experiments to determine the damping ratio of vibration modes in frequency domain. Figure A2-2 shows the zoomed frequency response of a SDOF system.

where fn = 50 Hz, Δf = df = 1.158 Hz, using Equation (A2.6), the damping ratio can be estimated as

df 1.158 ζ = = ≈ 0.01 . 2 f n 2 * 50 Appendix 2 Estimation of System Damping 218

Figure A2-2 Frequency response of a SDOF system

A2.2 Time domain estimation

The SDOF system represented by Equation (A2.2) can be rewritten as,

1/ m 1/ m H (s) = 2 2 = 2 2 . (A2.7) s + 2ζω n s + ω n (s + ζω n ) + ω d where

2 ωd = 1− ζ ωn –– Damped natural frequency of the system.

By inverse the Laplace transform, the impulse response of the SDOF system can be obtained as,

1 −ζω nt ~ h(t) = e sin(ω d t) = h (t)sin(ω d t) (t ≥ 0) . (A2.8) mω d where

~ 1 −ζω t ~ −ζω t ~ ~ h (t ) = e n = h (0)e n and h (0) is the value of h (t) at t = 0. m ω d Appendix 2 Estimation of System Damping 219

h(t) decays along with time and the speed of the decay reflects the amount of damping in the ~ ~ system which is only influenced by the h(t). Applying logarithmic on h(t), thus the damping ratio can be estimated by the following equation, ~ 1 ⎛ h(0) ⎞ ⎜ ⎟ ζ = ln⎜ ~ ⎟ . (A2.9) ωnt1 ⎝ h(t1) ⎠ where “ ln ” donates the logarithm to base e. Therefore, in time domain by measuring two ~ ~ ~ amplitudes of h (t) , such as h (0) at t =0 and h (t1 ) at t =t1. Then the damping ratio ζ can be obtained by Equation (A2.9).

The Figure A2-3 is an impulse response of the SDOF system which shows that at t =0, ~ ~ h (0) = 1 , and at t =t1 =1 (s), h (t1 ) = 0. 0432 . By Equation (A2.9), the damping ratio can be estimated as, ~ 1 ⎛ h(0) ⎞ 1 1 ⎜ ⎟ ⎛ ⎞ ζ = ln⎜ ~ ⎟ = ln⎜ ⎟ ≈ 0.01. ωnt1 ⎝ h(t1) ⎠ 2*π *50*1 ⎝ 0.0432⎠ which matches the result estimated in frequency domain previously.

Figure A2-3 Impulse response of a SDOF system Appendix 3 Positive Position Feedback (PPF) Control 220

Appendix 3 Positive Position Feedback (PPF) Control

This Appendix is linked to Chapter 2 (section 2.3.4, p. 36)

In the section below, a brief description about PPF control applied to a structure is given. The homogeneous differential equations representing a SDOF structure and a second order bandpass filter can be represented as,

2 Structure: ξ + 2 ⋅ς s ⋅ω s ⋅ξ + ω s ⋅ξ = 0 , (A3.1)

2 Filter: η+ 2 ⋅ς c ⋅ωc ⋅η + ωc ⋅η = 0 . (A3.2)

where ξ , η , ς s , ς c , ω s , ω c are modal coordinators, damping ratios, resonant frequencies of the structure and electronic filter respectively.

The block diagram of the SDOF structure with a PPF controller applied is as shown in Figure A3-1 (Song et al, 2000).

Structure ξ 2 Disturbance ξ+ 2ς ω ξ +ω ξ = 0 + s s s +

2 2 Gωs ωc Electronic Filter Figure A3-1 Block diagram of SDOFη+ 2 structurς ω ηe+ withω 2η PPF= 0control (Song et al, 2000) η c c c The differential equations to describe the SDOF structure with PPF control shown in Figure A3-1 are, Appendix 3 Positive Position Feedback (PPF) Control 221

2 2 ξ + 2⋅ς s ⋅ω s ⋅ξ +ω s ⋅ξ = G ⋅ω s ⋅η , (A3.3)

2 2 η+ 2⋅ς c ⋅ωc ⋅η +ωc ⋅η = ωc ⋅ξ . (A3.4) where G is a gain factor.

Combine the Fourier transform of Equations (A3.3) and (A3.4) one can have

2 ωc η( jω) = 2 2 ⋅ξ( jω) . (A3.5) ( jω) + 2 ⋅ ( jω) ⋅ς c ⋅ωc + ωc

The unforced vibration of the structure can be expressed as ξ (t) = αe− jω st , the Equation (A3.5) then becomes

α ⋅ω 2 α η( jω) = c δ (ω −ω ) = δ (ω −ω ). (A3.6) ( jω)2 + 2( jω)ς ω +ω 2 s ⎛ ω 2 ⎞ ⎛ω ⎞ s c c c ⎜1− s ⎟ + j2ς ⎜ s ⎟ ⎜ 2 ⎟ c ⎜ ⎟ ⎝ ωs ⎠ ⎝ωc ⎠ where α is a constant and δ(ω) is Dirac’s delta function.

The inverse Fourier transform of Equation (A3.6) is

η(t) = ℑ−1{η( jω)} = β ⋅ e j(ωst+ϕ ) . (A3.7) with

⎛ ωs ⎞ ⎜ 2ς c ⎟ α −1⎜ ω ⎟ β = and ϕ = − tan c .Equation (A3.7) can be 2 2 ⎜ 2 ⎟ 2 ωs ⎛ ω ⎞ 2 ⎛ω ⎞ ⎜ 1− ⎟ ⎜1− s ⎟ + 4ς ⎜ s ⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ c ⎜ ⎟ ⎝ ωc ⎠ ⎝ ωc ⎠ ⎝ωc ⎠ discussed in following three different saturations:

(1) When the structure vibrates at a frequency much lower than the filter’s resonant

frequency (i.e. ω s <<ω c ) the phase angle ϕ approaches to zero according to Equation

(A3.7). Then,η(t) = β ⋅ ei(ω⋅t+ϕ ) ≈ β ⋅ eiω⋅t = ξ (t ) . (A3.8)

Substitute Equation (A3.8) into Equation (A3.3) results in

2 2 ξ + 2⋅ς s ⋅ωs ⋅ξ + (ωs − G ⋅ β ⋅ωs )⋅ξ = 0 . (A3.9)

Equation (A3.9) shows that the PPF controller in this case results in the stiffness term of the structure being decreased, which is called active flexibility. Appendix 3 Positive Position Feedback (PPF) Control 222

(2) When the filter’s resonant frequency is equal to the structure natural frequency (i.e. ω s =

ω c ), the phase angle ϕ in Equation (A3.7) equals to − π / 2 . Then

β β i(ω⋅t−π / 2) i⋅ωs ⋅t i⋅ωs ⋅t η(t) = β ⋅ e ≈ −i ⋅ β ⋅ e = − (i ⋅ωs ⋅ e ) = − ⋅ξ (t) . (A3.10) ωs ωs

Substitute Equation (A3.10) into Equation (A3.3) results in

2 ξ + (2⋅ς s ⋅ωs + G ⋅ β ⋅ωs )⋅ξ +ωs ⋅ξ = 0 . (A3.11)

Equation (A3.11) shows that the PPF controller in this case results in an increase in the damping term of the structure, which is called active damping.

(3) When the structure frequency is much greater than that of the filter (i.e. ω s >> ω c ), the phase angle ϕ approaches − π . Then,

η(t) ≈ β ⋅ ei(ω⋅t−π ) = −β ⋅ ei⋅ω⋅s ⋅t = −ξ (t) . (A3.12)

Substitute Equation (A3.12) into Equation (A3.3) results in

2 2 ξ + 2⋅ς s ⋅ωs ⋅ξ + (ωs + G ⋅ β ⋅ωs )⋅ξ = 0. (A3.13)

Equation (A3.13) shows that the PPF controller in this case results in an increase in the stiffness term of the structure, which is called active stiffness.

Figure A3-2 is the simple SIMULINK model of a SDOF structure applied with PPF control under the condition: f c = f s = 100 Hz , ς s = 0.02 and ς c = 0.48, the input signal is an impulse (created by the difference of two step signals).

Figure A3-2 SIMULINK model of SDOF structure with positive position feedback

Figure A3-3 shows the simulation results of the SIMULINK model in Figure A3-2. The graph in (a) and (b) are the impulse response and frequency response respectively. Blue colour Appendix 3 Positive Position Feedback (PPF) Control 223 traces represent the responses without the controller while red colour traces represent the response with the controller.

(a)

(b)

Figure A3-3 Impulse/frequency response of SDOF structure with/without PPF control

One can see that the amplitude of vibration with PPF control has been significantly reduced. Appendix 3 Positive Position Feedback (PPF) Control 224 Figure A3-4 is the SIMULINK model of a 4-DOF structure applied with PPF control under the condition: fc = f s = 50, 100, 200, and 500 Hz, ς s = 1.6E-3, 8E-4, 4E-4, and 1.6E-4, ς c = 0.32, 0.32, 0.24, and 0.16 respectively.

Figure A3-4 SIMULINK model of 4-DOF structure with positive position feedback

Figure A3-5 is the simulation results of the 4-DOF structure with/without PPF control. The graph in (a) and (b) are the impulse response and frequency response respectively. Blue colour traces represent the responses without PPF control while red colour traces represent the response with PPF control.

The frequency response shows that the magnitudes of the structure’s vibration at natural frequencies have been reduced simultaneously when multiple PPF control is applied. A real complex flexible structure often can be modelled approximately with a finite number degree of freedom. Each of them can be targeted with a second order bandpass filter using PPF method. Appendix 3 Positive Position Feedback (PPF) Control 225

(a)

(b)

Figure A3-5 Impulse/frequency response of 4-DOF structure with/without PPF control Appendix 4 Proposed Power Amplifier using Amplitude Modulation and Piezoelectric Transformer for Active 226 Vibration Control

Appendix 4 Proposed Power Amplifier using Amplitude Modulation and Piezoelectric Transformer for Active Vibration Control

This Appendix is linked to Chapter 2 (sections 2.3.5 ~ 2.3.6, pp. 40 ~ 43)

Since a piezoelectric transformer (PT) is a very promising alternative to the traditional electromagnetic transformer for active vibration control as demonstrated by Carazo et al in (Carazo, 2001), we here propose a novel design of a power amplifier using amplitude modulation (AM) and a piezoelectric transformer for driving the piezoelectric actuator which can be compact and energy efficient. The purpose of using the amplitude modulation technique is to shift the frequency of actuation signal to the resonant frequency of the piezoelectric transformer to achieve maximum voltage step-up.

Driver section Generator section

Figure A4-1 Rosen type piezoelectric transformer Appendix 4 Proposed Power Amplifier using Amplitude Modulation and Piezoelectric Transformer for Active 227 Vibration Control

PT is ultrasonic resonators with two pairs of electrodes provided on the surface of a piezoelectric material, in which electrical energy is carried in the mechanical form. The PT uses both the direct and indirect piezoelectric effects; an applied voltage results in vibration, which in turn generates the output voltage. The input and output electrodes are arranged to provide the impedance transformation, which results in the voltage transformation. PT consists of two piezoelectric material parts, which are driver and generator section with different polarization as shown in Figure A4-1.

Figure A4-2 shows (a) the Transoner® PT with 25 mm diameter, 6 mm total thickness and power capability higher than 70 W, made by FACE Electronics, LC, and (b) the PT characteristics, voltage step-up voltage ratio vs. load (Carazo, 2001).

(a) (b)

Figure A4-2 (a) Transoner® piezoelectric transformer, (b) step-up voltage ratio vs. load

The PTs have many advantages, in addition to being EMI-free, they are generally lighter, smaller, non-flammable, have high efficiency (over 95 %), high power capacity per unit volume leading to smaller size, high Q factor, large voltage step-up at resonant frequency, and a high degree of insulation.

However, PTs have large voltage step-up only at their resonant frequencies, while the electrical actuation signals to drive actuators can have any frequency, usually there are much lower than the resonant frequencies of PTs. The main idea behind the proposed design is to use amplitude modulation (AM) technique to modulate the actuation signal by a carrier which frequency matches the resonant frequency of a PT, thus the AM signal can be largely amplified by the PT. The amplified AM signal must be demodulated to reconstruct the actuation signal before driving the actuator. The demodulation procedure is simply by passing Appendix 4 Proposed Power Amplifier using Amplitude Modulation and Piezoelectric Transformer for Active 228 Vibration Control the amplified AM signal through a rectifier then to a lowpass filter. With a design such as this, the active electronic components may only be needed for the low voltage part of the circuit while the high voltage section can use only passive electronic components. Figure A4-3 is the block diagram of the proposed AM power amplifier.

Figure A4-3 Block diagram of an AM power amplifier

C A B

Figure A4-4 PSpice® functional model of an AM amplifier

Figure A4-4 is the PSpice® functional model of the proposed AM amplifier, where its assumed that the actuation signal to be amplified is 1V/100 Hz, carrier signal is 1V/100 kHz,

Vb is a bias voltage to adjust the percentage of modulation, the transformer used here is the electromagnetic transformer of PSpice® model (TX1) for function verification, the coupling coefficient and ratio of transformation of the transformer are 1 and 1:2500 (i.e. voltage step- Appendix 4 Proposed Power Amplifier using Amplitude Modulation and Piezoelectric Transformer for Active 229 Vibration Control up 1:50) respectively, the rectifier is a full-bridge rectifier (D1 – D4), the lowpass filter is a simple passive R-C filter (R2-Cp), Voff is an offset voltage to remove the DC component in the amplified signal, however, it might not be necessary. This design is also suitable to amplify a multi-frequency actuation signal provided that a proper modulation index (percentage of modulation) is selected.

Figure A4-5 displays the signal at point marked “B” in Figure A4-4, which is the modulated signal before being amplified.

5.0V

0V

-5.0V 0s 10ms 20ms 30ms 40ms 50ms V(R1:2) Time

Figure A4-5 Simulation result of the modulated signal before being amplified

50V

25V

0V

-25V

-50V 0s 10ms 20ms 30ms 40ms 50ms V(R2:2) V(Vs:+) Time

Figure A4-6 Actuation signal before (in red) and after (in green) being amplified

Appendix 4 Proposed Power Amplifier using Amplitude Modulation and Piezoelectric Transformer for Active 230 Vibration Control

Figure A4-6 displays the signals at points marked “A” and “C” in Figure A4-4, which represent the actuation signal before being amplified (in red) and after being amplified (in green) respectively.

Figure A4-7 shows the spectra of the actuation signals of that in Figure A4-6. One can see from Figure A4-6 and A4-7, the actuation signal has been amplified almost 50 times by the proposed AM amplifier.

50V

40V

30V

20V

10V

0V 0Hz 0.1KHz 0.2KHz 0.3KHz 0.4KHz 0.5KHz 0.6KHz 0.7KHz 0.8KHz 0.9KHz 1.0KHz V(R2:2) V(Vs:+) Frequency

Figure A4-7 Spectrum of actuation signal before (in red) and after (in green) being amplified

One of the benefits of the AM amplifier can be to avoid using traditional high power voltage driver made of MOSFETs or transistors at output stage, which generally have low efficiency and high power consumption. Such a voltage driver often requires extra heat sink to dissipate heat to prolong its working life thus making the amplifier bulky and heavy. Another benefit of the AM amplifier is that it amplifies sinusoid signals rather than the PWM (pulse width modulation) amplifiers, which amplify pulse signals which contain infinite frequency components, most of them need to be filtered out later, and therefore, it is much more efficient in terms of energy transmission. However, the power delivery capability of the AM amplifier is dependent on that of the PT. Figure A4-8 shows the history of PTs in terms of power capability. The multistacked disk type PT already exceeds 100 W (at 40 W/cm3) capability (1996 FACE) (Uchino, [I 12]).

Appendix 4 Proposed Power Amplifier using Amplitude Modulation and Piezoelectric Transformer for Active 231 Vibration Control

Figure A4-8 History of piezoelectric transformers in terms of power capability (Uchino, [I 12])

Appendix 5 Beam Modelling and Modal Analysis 232

Appendix 5 Beam Modelling and Modal Analysis

This Appendix is linked to Chapter 3 (Sections 3.1, pp. 46 ~ 55)

Here we first derive the motion equation of a cantilever beam using energy principles and variational methods with the general consideration that the beam is not only affected by the bending moment, but also by shear force and rotary inertia. Then simplify the motion equation gradually to the Euler-Bernoulli modal and give an example of the modal shape of the Euler-Bernoulli beam.

A5.1 Potential energy and kinetic energy of a vibration beam

Two types of energies are usually used to describe the beam vibration, i.e. potential energy and kinetic energy. Part of the potential energy is due to bending moment and part of it is due to shear force. The kinetic energy is partly from the displacement of a beam element parallel to the axis perpendicular to the beam principle plane, and partly from the rotation of the same element. Figure A5-1 depicts the vibration of a beam element with moment inertia, rotary inertia and shear deformation (Tse, 1978).

∂w In the Figure A5-1, is the slope due to bending moment M , θ is the slope due to shear ∂x S ∂w force Q , and θ is the slope combining bending moment and shear force, i.e. =θ −θ . b ∂x b S Neglecting the interaction between the shear and the bending moment, the shear force Q will cause a rectangular element to become a parallelogram (in broken line) without a rotation of the faces (Tse, 1978).

Appendix 5 Beam Modelling and Modal Analysis 233

z Normal to cross- ∂M sectional area M + dx (Shear slope) ∂x θ S ∂w M (Bending slope) ∂x

θb (Total slope) ∂Q Q + dx ∂x w(x,t) Q

x

Figure A5-1 Vibration of beam element with rotary inertia and shear deformation

The potential energy or strain energy for the small neutral axis deflections along z axis due to the bending moment is defined as document

2 L L ⎛ ∂θ (x,t) ⎞ 1 1 2 1 ⎜ ⎛ b ⎞ ⎟ U b = ε xσ xdV = ()Ebε x dA dx = Eb ⎜ z ⎟ dA dx ∫V ∫∫0 A ∫∫0 ⎜ A ⎟ 2 2 2 ⎝ ⎝ ∂x ⎠ ⎠

2 2 L ⎡ ⎤ L 1 ⎛ ∂θb (x,t) ⎞ 2 1 ⎛ ∂θb (x,t) ⎞ = Eb ⎢⎜ ⎟ ()z dA ⎥dx = Eb Ib ⎜ ⎟ dx . (A5.1) ∫∫0 A ∫0 2 ⎣⎢⎝ ∂x ⎠ ⎦⎥ 2 ⎝ ∂x ⎠

where Eb is the Young’s modulus, σ x is the axial stress, L, A, and V are the length, cross- sectional area, and volume of the beam respectively. Ib is the moment of inertia of the beam

2 cross-sectional area with respect to z axis, Ib = z dA. ∫A

The potential energy due to shear force is defined as

2 2 L L 1 ⎛ ∂θ s (x,t) ⎞ 1 ⎛ ∂w(x,t) ⎞ U s = k'GA⎜ ⎟ dx = k'GA⎜θb (x,t) − ⎟ dx . (A5.2) 2 ∫0 ⎝ ∂x ⎠ 2 ∫0 ⎝ ∂x ⎠ where

k' and G are the shear stiffness factor and shear modulus of the beam respectively (Fung, 2001). The shear stiffness factor is the inverse of shape factor that is ratio of the plastic section modulus to the elastic section modulus and depends on the cross-sectional shape (Young et al, 2002).

The potential energy of the beam is then, Appendix 5 Beam Modelling and Modal Analysis 234

2 2 L L 1 ⎛ ∂θb (x,t) ⎞ 1 ⎛ ∂w(x,t) ⎞ U = Ub +U s = Eb Ib ⎜ ⎟ dx + k'GA⎜θb (x,t) − ⎟ dx . (A5.3) 2 ∫0 ⎝ ∂x ⎠ 2 ∫0 ⎝ ∂x ⎠

The kinetic energy from the displacement of the beam element parallel to z coordinate is defined as

2 1 L ⎛ ∂w(x,t) ⎞ Td = ρb A⎜ ⎟ dx , (A5.4) 2 ∫0 ⎝ ∂t ⎠

The kinetic energy from the rotation of the same beam element about an axis perpendicular to the neutral axis is defined as

2 L 1 ⎛ ∂θb (x,t) ⎞ Tr = Jb ⎜ ⎟ dx , (A5.5) 2 ∫0 ⎝ ∂t ⎠

where ρb is the density of the beam, J b = ρb Ib (Dimarogonas, 1996) donates the mass moment of inertia about neutral axis per unit length of the beam, ∂w(x,t) / ∂x is the translational velocity and ∂θb (x,t) / ∂x is the angular velocity of the beam.

The kinetic energy of the beam is then,

2 2 L L 1 ⎛ ∂w(x,t) ⎞ 1 ⎛ ∂θb (x,t) ⎞ T = Td + Tr = ρb A⎜ ⎟ dx + J b ⎜ ⎟ dx . (A5.6) 2 ∫0 ⎝ ∂t ⎠ 2 ∫0 ⎝ ∂t ⎠

The potential energy due to external transverse load p(x,t) , the bending moment and shear force at the beam ends x = 0 and x = L, i.e. M 0 , M L , Q0 , QL , is

L A = − p(x,t)w(x,t)dx − M Lθ L + M 0θ0 + QL wL − Q0w0 . (A5.7) ∫0

A5.2 Motion equation derivation of beam using Hamilton’s principle

The motion equation of beam can be derived by the Hamilton’s principle

t δ 2 U −T + A)dt = 0 (A5.8) ∫ t 1 that is

2 2 2 2 t ⎧ L ⎡ ⎤ 2 ⎪ 1 ⎛ ∂θb ⎞ ksGA⎛ ∂w⎞ 1 ⎛ ∂w⎞ 1 ⎛ ∂θb ⎞ δ ⎢ EbIb ⎜ ⎟ + ⎜θb − ⎟ − ρb A⎜ ⎟ − Jb ⎜ ⎟ − pw⎥dx ∫∫ t ⎨ 0 1 ⎩⎪ ⎣⎢2 ⎝ ∂x ⎠ 2 ⎝ ∂x ⎠ 2 ⎝ ∂t ⎠ 2 ⎝ ∂t ⎠ ⎦⎥

⎫ M LθL + M0θ0 + QLwL − Q0w0 ⎬dt = 0. (A5.9) ⎭ Appendix 5 Beam Modelling and Modal Analysis 235 For the first potential energy term in Equation (A5.9) due to bending moment, there is

2 2 t2 ⎛ L ∂θ ⎞ t2 ⎛ L ∂θ ⎞ t2 L ∂θ ∂(δθ ) ⎜ 1 ⎛ b ⎞ ⎟ 1 ⎜ ⎛ b ⎞ ⎟ ⎛ b b ⎞ δ Eb Ib ⎜ ⎟ dx dt = Eb Ib δ ⎜ ⎟ dx dt = Eb Ib ⎜ dx⎟ dt . ∫∫ t ⎜ 0 ⎟ ∫∫ t ⎜ 0 ⎟ ∫∫ t 0 1 ⎝ 2 ⎝ ∂x ⎠ ⎠ 1 2 ⎝ ⎝ ∂x ⎠ ⎠ 1 ⎝ ∂x ∂x ⎠

Let

∂(δθ ) ∂θ f (x) = b , g'(x) = b , ∂x ∂x

Using integration by parts (i.e. ∫ f (x)g'(x)dx = f (x)g(x) − ∫ g(x) f '(x)dx), we obtain,

L 2 t2 ⎛ L ∂θ ∂(δθ ) ⎞ t2 ⎛ ∂(δθ ) L ∂ (δθ ) ⎞ ⎛ b ⎞⎛ b ⎞ ⎜ b b ⎟ Eb Ib ⎜ ⎜ ⎟⎜ ⎟dx⎟ dt = Eb Ib (θb ) − (θb ) dx dt , ∫∫ t ⎜ 0 ⎟ ∫∫ t ⎜ 0 2 ⎟ 1 ⎝ ⎝ ∂x ⎠⎝ ∂x ⎠ ⎠ 1 ⎝ ∂x 0 ∂x ⎠

Thus,

2 L 2 t2 ⎛ L ∂θ ⎞ t2 ⎛ ∂θ L ∂ θ ⎞ ⎜ 1 ⎛ b ⎞ ⎟ ⎜ b b ⎟ δ Eb Ib ⎜ ⎟ dx dt = Eb Ib (δθb ) − (δθb ) dx dt . (A5.10) ∫∫ t ⎜ 0 ⎟ ∫∫ t ⎜ 0 2 ⎟ 1 ⎝ 2 ⎝ ∂x ⎠ ⎠ 1 ⎝ ∂x 0 ∂x ⎠

For the second potential energy term in Equation (A5.9) due to shear force, there is

2 t2 ⎡ L 1 ⎛ ∂w⎞ ⎤ t2 ⎡ L ⎛ ∂w ⎞⎛ ∂(δw) ⎞ ⎤ δ ⎢ k⎜θb − ⎟ dx⎥dt = k ⎜θb − ⎟⎜δθb − ⎟dx dt ∫∫ t 0 ∫∫ t ⎢ 0 ⎥ 1 ⎣⎢ 2 ⎝ ∂x ⎠ ⎦⎥ 1 ⎣ ⎝ ∂x ⎠⎝ ∂x ⎠ ⎦

t2 ⎡ LL∂w ⎛ ∂(δw) ⎞ ⎛ ∂(δw) ⎞ ⎤ = k ⎜ −δθb ⎟dx − θb ⎜ −δθb ⎟dx dt ∫∫ t ⎢ 00 ∫ ⎥ 1 ⎣ ∂x ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎦ . where k = k'GA.

Let

⎛ ∂(δw) ⎞ ∂w f (x) = ⎜ − δθ b ⎟ , g'(x) = , ⎝ ∂x ⎠ ∂x

Using integrating by parts to the first integration yields

⎡ L 2 ⎤ t2 L ∂w ⎛ ∂(δw) ⎞ t2 ⎛ ∂w ⎞ L ⎛ ∂ w ∂θ ⎞ k −δθ dt = k −θ (δw) − ⎜ − b ⎟(δw)dx dt , ⎜ b ⎟ ⎢⎜ b ⎟ ⎜ 2 ⎟ ⎥ ∫∫ t1 0 ∂x ∂x ∫∫ t1 ∂x 0 ∂x ∂x ⎝ ⎠ ⎣⎢⎝ ⎠ 0 ⎝ ⎠ ⎦⎥

Therefore, the second potential energy term is

2 t2 ⎡ L k ⎛ ∂w ⎞ ⎤ δ ⎢ ⎜ −θb ⎟ dx⎥dt ∫∫ t 0 1 ⎣⎢ 2 ⎝ ∂x ⎠ ⎦⎥ Appendix 5 Beam Modelling and Modal Analysis 236

⎡ L 2 ⎤ t2 ⎛ ∂w ⎞ LL⎛ ∂ w ∂θ ⎞ ⎛ ∂w ⎞ = k −θ (δw) − ⎜ − b ⎟(δw)dx − −θ (δθ )dx dt ⎢⎜ b ⎟ ⎜ 2 ⎟ ⎜ b ⎟ b ⎥ . (A5.11) ∫∫ t1 ∂x 00∂x ∂x ∫ ∂x ⎣⎢⎝ ⎠ 0 ⎝ ⎠ ⎝ ⎠ ⎦⎥

For the first term of the kinetic energy in Equation (A5.9), there is

2 2 t ⎛ ⎞ Lt⎛ ⎞ Lt 2 ⎜ 1 ⎛ ∂w ⎞ ⎟ 1 ⎜ 2 ⎛ ∂w ⎞ ⎟ ⎛ 2 ∂w⎛ ∂(δw) ⎞ ⎞ δ mb ⎜ ⎟ dx dt = mb δ ⎜ ⎟ dt dx = mb ⎜ ⎜ ⎟dt ⎟dx , ∫ t ⎜ ⎟ ∫∫ 0 ⎜ t ⎟ ∫∫ 0 ⎜ t ⎟ 1 ⎝ 2 ⎝ ∂t ⎠ ⎠ 2 ⎝ 1 ⎝ ∂t ⎠ ⎠ ⎝ 1 ∂t ⎝ ∂t ⎠ ⎠

where mb = ρb A .

Let

∂(δw) ∂w f (t) = , g'(t) = ∂t ∂t

Integrating by parts, we obtain

2 t2 2 t ⎛ L ⎞ Lt⎛ ⎞ 2 ⎜ 1 ⎛ ∂w ⎞ ⎟ ⎜ ∂(δw) 2 ⎛ ∂ (δw) ⎞ ⎟ δ mb ⎜ ⎟ dx dt = mb w − w⎜ ⎟dt dx . ∫∫ t 0 ∫∫ 0 t ⎜ 2 ⎟ 1 ⎜ 2 ⎝ ∂t ⎠ ⎟ ⎜ ∂t 1 ∂t ⎟ ⎝ ⎠ ⎝ t1 ⎝ ⎠ ⎠

Since the virtual displacement must be so specified that δw ≡ 0 at t1 and t2 , hence,

∂(δw) / ∂t = δ (∂w / ∂t) ≡ 0 at t1 and t2 (Fung, 2001). Thus,

t L ⎛ 2 ⎞ ⎜ ∂(δw) ⎟ mb w dx = 0 . ∫ 0 ⎜ ∂t ⎟ ⎝ t1 ⎠

Then the first term of kinetic energy is equal to

2 2 t ⎛ L ⎞ t L 2 ⎜ 1 ⎛ ∂w ⎞ ⎟ 2 ⎛ ∂ w ⎞ δ mb ⎜ ⎟ dx dt = − mb ⎜ ⎟δwdxdt . (A5.12) ∫∫ t ⎜ 0 ⎟ ∫∫ t 0 ⎜ 2 ⎟ 1 ⎝ 2 ⎝ ∂t ⎠ ⎠ 1 ⎝ ∂t ⎠

Similar to Equation (A5.12), change variables mb and w by variables J b and θb respectively, the second term of kinetic energy in Equation (A5.9) can be expressed as

2 2 t2 ⎛ L ∂θ ⎞ t2 L ⎛ ∂ θ ⎞ ⎜ 1 ⎛ b ⎞ ⎟ b δ Jb ⎜ ⎟ dx dt = − Jb ⎜ ⎟δθbdxdt . (A5.13) ∫∫ t ⎜ 0 ⎟ ∫∫ t 0 ⎜ 2 ⎟ 1 ⎝ 2 ⎝ ∂t ⎠ ⎠ 1 ⎝ ∂t ⎠

Substituting Equations (A5.10, 11, 12, 13) into Equation (A5.9) yields

2 2 2 2 t ⎧ L ⎛ ⎞ ⎫ 2 ⎪ ⎛ ∂ θ b ⎛ ∂w ⎞ ∂ θ b ⎞ ⎛ ∂ w ∂θ b ⎞ ∂ w ⎪ ⎜ Eb Ib + k⎜ −θ b ⎟ − J b ⎟(δθ b ) + ⎜ k⎜ − ⎟ − mb + p⎟(δw) dxdt ∫∫ t ⎨ 0 ⎜ 2 2 ⎟ ⎜ ⎜ 2 ⎟ 2 ⎟ ⎬ 1 ⎩⎪ ⎝ ∂x ⎝ ∂x ⎠ ∂t ⎠ ⎝ ⎝ ∂x ∂x ⎠ ∂t ⎠ ⎭⎪

Appendix 5 Beam Modelling and Modal Analysis 237

L ⎫ ⎛⎛ ∂θ ⎞ ⎛ ⎛ ∂w ⎞ ⎞ ⎞ ⎪ ⎜ b ( ) Q ( w)⎟ dt 0 . (A5.14) + ⎜⎜ Eb Ib − M ⎟ δθb + ⎜k⎜ −θb ⎟ + ⎟ δ ⎟ ⎬ = ⎝ ∂x ⎠ ⎝ ⎝ ∂x ⎠ ⎠ ⎝ ⎠ 0 ⎭⎪

In order for Equation (A5.14) to be true for any virtual displacements δθb and δw , the following partial differential equations can be obtained

2 2 ∂ θb ⎛ ∂w ⎞ ∂ θb Eb Ib + k⎜ −θb ⎟ − Jb = 0 , (A5.15) ∂x2 ⎝ ∂x ⎠ ∂t 2

⎛ ∂ 2w ∂θ ⎞ ∂ 2w k⎜ − b ⎟ − m + p = 0 , (A5.16) ⎜ 2 ⎟ b 2 ⎝ ∂x ∂x ⎠ ∂t

Reorganize the Equation (A5.16) as

∂θ 1 ⎛ ∂ 2w ∂ 2w ⎞ b = ⎜k − m + p⎟ . (A5.17) ⎜ 2 b 2 ⎟ ∂x k ⎝ ∂x ∂t ⎠

Perform a derivate of x to Equation (A5.15) once and then substitute the derivation ∂θb / ∂x by Equation (A5.17) to eliminate the variable θb as below

E I ∂ 2 ⎛ ∂ 2 w ∂ 2 w ⎞ ⎛ ∂ 2 w 1 ⎛ ∂ 2 w ∂ 2 w ⎞⎞ b b ⎜ k − m + p ⎟ + k⎜ − ⎜ k − m + p ⎟⎟ 2 ⎜ 2 b 2 ⎟ ⎜ 2 ⎜ 2 b 2 ⎟⎟ k ∂x ⎝ ∂x ∂t ⎠ ⎝ ∂x k ⎝ ∂x ∂t ⎠⎠

J ∂ 2 ⎛ ∂ 2 w ∂ 2 w ⎞ – b ⎜ k − m + p ⎟ = 0 . 2 ⎜ 2 b 2 ⎟ k ∂t ⎝ ∂x ∂t ⎠ or

4 4 4 2 2 2 ∂ w mb ∂ w ⎛ Eb Ibmb ⎞ ∂ w ∂ w Eb Ib ∂ p Jb ∂ p Eb Ib 4 + Jb 4 −⎜ Jb + ⎟ 2 2 + mb 2 = p − 2 + 2 . (A5.18) ∂x k ∂t ⎝ k ⎠ ∂x ∂t ∂t k ∂x k ∂t with boundary conditions at beam ends

∂θ either E I b − M = 0 or δθ = 0. (A5.19a) b b ∂x b and

⎛ ∂w ⎞ either k⎜ −θb ⎟ + Q = 0 or δw = 0. (A5.19b) ⎝ ∂x ⎠

Equation (A5.18) is the Timoshenko equation for lateral vibration of beams, which has included not only the displacement due to the bending moment, but also the displacements affected by shear deformation and rotary inertia. However, the effects of shear and rotation Appendix 5 Beam Modelling and Modal Analysis 238 are uncoupled. There are three situations to be discussed below when missing one or both of the shear deformation and rotary inertia.

∂ 2 w ∂4w (1) When only the rotary inertia is excluded which means = 0 and = 0, then the ∂x∂t ∂x2∂t 2 Equation (A5.18) becomes

∂4w m ∂4w ∂2w E I ∂2 p J ∂2 p E I + J b + m = p − b b + b . (A5.20) b b ∂x4 b k ∂t 4 b ∂t 2 k ∂x2 k ∂t 2 with the same boundary conditions as in Equation (A5.19).

(2) When only the shear effect is excluded which means the shear stiffness or shear modulus

(G) is very large, k = ksGA → ∞ , no shear deformation is produced. The Equation (A5.18) becomes

∂4w ∂4w ∂2w E I − J + m = p . (A5.21) b b ∂x4 b ∂x2∂t 2 b ∂t 2

Since the shear deformation is missing, ∂w /,∂x = θb +θ S = θb θb in the first potential energy term in Equation (A5.9) can be replaced by ∂w/ ∂x , that is

2 ⎛ 2 2 ⎞ t2 ⎛ L 1 ∂θ ⎞ t2 L ⎛ ⎞ ⎜ ⎛ b ⎞ ⎟ ⎜ 1 ∂ w ⎟ δ Eb I b ⎜ ⎟ dx dt = δ Eb I b ⎜ ⎟ dx dt ∫∫t 0 t 0 ⎜ 2 ⎟ 1 ⎜ 2 ∂x ⎟ ∫∫1 ⎜ 2 ∂x ⎟ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠

L ⎛ 2 L 3 4 ⎞ t2 ⎜ ∂ w ⎛ ∂w ⎞ ∂ w L ∂ w ⎟ = Eb I b δ ⎜ ⎟ − δw + δw dx dt . ∫∫t ⎜ 2 3 0 4 ⎟ 1 ⎜ ∂x ⎝ ∂x ⎠ ∂x ∂x ⎟ ⎝ 0 0 ⎠

⎛ ∂w⎞ ⎛ ∂w⎞ Also consider replacing − MLθL + M0θ0 terms in Equation (A5.9) by − M L ⎜ ⎟ + M0 ⎜ ⎟ . ⎝ ∂x ⎠L ⎝ ∂x ⎠0

With these changes, the boundary conditions at beam ends can be derived as

∂ 2w ⎛ ∂w ⎞ either Eb Ib − M = 0 or δ ⎜ ⎟ = 0 . (A5.22a) ∂x2 ⎝ ∂x ⎠ and

∂3w either = 0 or δw = 0. (A5.22b) ∂x3

These are equations governing the motion of a beam including the effect of the rotary inertia known as the Rayleigh’s equations (Fung, 2001). Appendix 5 Beam Modelling and Modal Analysis 239 (3) When both the shear deformation and rotary inertia are excluded which means k → ∞ ∂ 2 w and = 0 , the Equation (A5.18) is simplified as ∂x∂t

∂4w ∂2w E I + m = p . (A5.23) b b ∂x4 b ∂t 2 with the same boundary conditions as that in Equation (A5.22).

In this case, the Equation (A5.23) represents the motion of the Euler-Bernoulli beam (Fung, 2001) which considers only the effect of the bending moment, excluding the effects of shear and rotation. The Equation (A5.23) can be rewritten as

4 2 2 ∂ w ∂ w 1 a 4 + 2 = p . (A5.24) ∂x ∂t mb

E I where a = b b , mb with boundary conditions is the same as in Equation (A5.22).

A5.3 Free vibration modals of the Euler-Bernoulli beam

For the free vibration of the beam, let p(x,t) = 0 in Equation (A5.24), the motion equation then becomes

∂ 4w ∂ 2w a 2 + = 0 . (A5.25) ∂x4 ∂t 2

The natural frequencies and mode shapes of the free beam vibration can be solved from Equation (A5.25) with a set of boundary condition. The common boundary conditions are listed in Table A5-1 (Tse, 1978). In the table, m, ks and kt denote mass, shear stiffness and torsional stiffness respectively.

Using the method of separation of variables, the two-variable lateral deflection function

wn (x,t) can be separated into a form of the product of a single space variable function

φn (x) , known as eigenfunction or shape function, by a single time variable function qn (t) , that is

wn (x,t) = φn (x)⋅qn (t) . (A5.26) Appendix 5 Beam Modelling and Modal Analysis 240 Table A5-1 Common boundary conditions for lateral vibration of uniform beam (Tse, 1978)

Deflection Slope Moment Shear End condition ∂w ∂ 2w ∂ 3w w θ = M = EI Q = EI ∂x ∂x2 ∂x 3

Fixed w = 0 θ = 0

Free M = 0 Q = 0

Hinged w = 0 M = 0

∂w Spring load M = −k Q = k w t ∂x s ∂ 2 w Inertia load M = 0 Q = m ∂t 2

Since each separated function contains only a single variable, the partial derivatives thus can be replaced by their differentiations. Substituting Equation (A5.26) into Equation (A5.25) obtains

d 4φ (x) d 2q (t) a2 n q (t) + n φ (x) = 0 . (A5.27) dx4 n dt 2 n

Let

2 4 2 a d φn (x) 1 d qn (t) 2 4 = − 2 = ωn . (A5.28) φn (x) dx qn (t) dt

where ωn is a constant to satisfy the boundary conditions of the system. The problem of finding such values of ωn , which is the natural frequency of the system, is known as eigenvalue problem. From Equation (A5.28), the Equation (A5.25) can be reduced to

d 4φ(x) − k 4φ(x) = 0 , (A5.29) dx4

d 2q (t) n +ω 2q (t) = 0 . (A5.30) dt 2 n n

Substitute Equations (A5.29) and (A5.30) into Equation (A5.27) to obtain

2 4 2 a k n φ n (x)q n (t) − ω n q n (t)φ n (x) = 0 , Appendix 5 Beam Modelling and Modal Analysis 241 results in

2 2 2 a kn = ωn , Thus

1/ 4 1/ 4 ⎛ω 2 ⎞ ⎛ m ⎞ 2π k = ⎜ n ⎟ = ⎜ b ⋅ω 2 ⎟ = n ⎜ 2 ⎟ ⎜ n ⎟ . (A5.31) ⎝ a ⎠ ⎝ Eb Ib ⎠ λn

The parameters kn and λn are the bending wavenumber and wavelength that have the units of (1/Meter) and (Meter) respectively.

The neutral frequency is the function of kn

E I ω = k 2 b b (n = 1, 2, …). (A5.32) n n m b

The solution for the free vibration of the Euler-Bernoulli beam is

φ (x) = C sin( k x) + C cos( k x) + C sh(k x) + C ch(k x) (A5.33) n 1n n 2n n 3n n 4 n n ,

q n (t ) = C 5 n sin( ω n t ) + C 6 n cos( ω n t) . (A5.34)

For example, for the clamped-free boundary conditions given in Table A2-1 ( w(0) = 0 , w'(0) = 0 , w''(L) = 0 , w'''(L) = 0 ), the boundary conditions for Equation (A5.25) can be expressed as

dφ (x) dφ (x) φ (x) = φ (x) = n = n = 0 . (A5.35) n x=0 n x=L dx x=0 dx x=L

Apply Equation (A5.35) to Equation (A5.33) to obtain,

φ (x) = 0 leads to C = −C . n x=0 4n 2n

dφn (x) = 0 leads to C3n = −C1n . dx x=0

d 2φ (x) n = 0 leads to [sin(k L) + sh(k L)]C + [cos(k L) + ch(k L)]C = 0 . dx 2 n n 1n n n 2n x=L

d 3φ (x) n = 0 leads to [cos(k L) + ch(k L)]C −[sin(k L) − sh(k L)]C = 0 . 3 n n 1n n n 2n dx x=L

The matrix representation of the constants is Appendix 5 Beam Modelling and Modal Analysis 242

⎡sin(kn L) + sh(kn L) cos(kn L) + ch(kn L) ⎤ ⎡C1n ⎤ ⎡0⎤ ⎢ ⎥ ⋅ ⎢ ⎥ = ⎢ ⎥ . ⎣cos(kn L) + ch(kn L) − sin(kn L) + sh(kn L)⎦ ⎣C2n ⎦ ⎣0⎦

The non-zero solution of the constants C1n and C2n can be determined by solving the following determinant

sin(k L) + sh(k L) cos(k L) + ch(k L) n n n n = 0 cos(k L) + ch(k L) − sin(k L) + sh(k L) n n n n .

One can obtain the frequency equation

cos( k n L)ch(k n L) = −1 . (A5.36)

Since

[sin(kn L) + sh(kn L)]C1n + [cos(kn L) + ch(kn L)]C2n = 0 and

[cos(kn L) + ch(kn L)]C1n −[sin(kn L) − sh(kn L)]C2n = 0 , thus

C cos(k L) + ch(k L) sin(k L) − sh(k L) 1n = − n n = n n , C2n sin(kn L) + sh(kn L) cos(kn L) + ch(kn L)

From Equation (A5.33) and the coefficients C1n , C 2 n , C3n , C 4 n , the eigenfunction can be obtained as

φ (x) = −C [sh(k x) − sin( k x)] − C [ch(k x) − cos( k x)] n 1n n n 2 n n n

⎡ cos(kn L) + ch(kn L) ⎤ = −C2n ⎢ch(kn x) − cos(kn x) − [sh(kn x) − sin(kn x)]⎥ . (A5.37) ⎣ sin(kn L) + sh(kn L) ⎦

Substitute from Equations (A5.34), (A5.37) to Equation (A5.36)

wn (x,t) = φn (x)qn (t) =

⎛ cos(k L) + ch(k L) ⎞ ⎜ch(k x) − cos(k x) − n n [sh(k x) − sin(k x)]⎟ ⋅[Asin(ω t) + B cos(ω t)] ⎜ n n sin(k L) + sh(k L) n n ⎟ n n ⎝ n n ⎠ . (A5.38)

where A = C2nC5n and B = −C2nC6n .

The roots kn L can be solved numerically from the frequency equation cos(kn L)ch(kn L) =1.

The first six approximate values of kn L are 1.8751, 4.6941, 7.8548, 10.9955, 14.1372, and Appendix 5 Beam Modelling and Modal Analysis 243 17.2788. From Equation (A5.32), the natural frequencies of the clamped-free Euler-Bernoulli beam can be calculated as

2 2 ω n kn Eb Ib (kn L) Eb Ib f n = = = 2 . (A5.39) 2π 2π mb 2π ⋅ L ρb Ab

where Ab is the cross-sectional area of the beam.

When the shear and rotation effects are considered, the value kn will decrease thus the natural frequency ωn will decrease as well. However, the influence is more pronounced for the higher modes (Fung, 2001).

Figure A5-2 shows the first six normalized bending mode shapes of a clamped-free Euler- Bernoulli beam.

Figure A5-2 Bending mode shapes of a clamped-free Euler-Bernoulli beam

A5.4 An example of obtaining mode shapes from experiment

Since the frequency response function (FRF) becomes purely imaginary at the modal frequency (or natural frequency). Its amplitude is proportional to the modal displacement, and its sign is positive if displacement is in phase with the excitation. The mode shapes can be determined by exciting a set of FRFs at selected locations on the structure with one of them is Appendix 5 Beam Modelling and Modal Analysis 244 fixed as a reference to measure the FRF responses. The imaginary parts of the measured the FRFs can be "picked" at the modal frequencies at which they represent the modal displacement for that specific DOF (Døssing, 1988). Figure A5-3 illustrates an example explaining the method. In the example, DOF #2 is used as the response reference. A set of FRFs is then measured by exciting each of the four defined DOFs in turn. The four imaginary parts of the FRF for each modal frequency represent the associated mode shape. If the measurements were made with calibrated instrumentation, the mode shapes could then be scaled (Døssing, 1988).

Figure A5-3 Illustration of obtaining mode shapes of a structure from the imaginary part of the FRFs (Døssing, 1988)

An experiment has been carried out on a steel beam which has size 0.5× 0.025× 0.003 (m3 ) ,

11 3 st th the Young’s modulus Eb = 2.06×10 (Pa) , and the density ρb = 7830 (kg/m ) . The 1 to 6 theoretical natural frequencies are: 9.943, 62.312, 174.475, 341.896, 565.185, and 844.284 Hz respectively. The natural frequencies obtained from the experiment are: 10.0, 61.87, 171.88, 335.0, 561.25, and 834.38 Hz respectively. Appendix 5 Beam Modelling and Modal Analysis 245 Figure A5-4 and A5-5 show the first six bending mode shapes of a clamped-free beam obtained from experiments. Figure A5-4 is that before curve fitting and Figure A5-5 is that after curve fitting.

Figure A5-4 Experimental mode shapes of a clamped-free beam before curve fitting

Figure A5-5 Experimental mode shapes of a clamped-free beam after curve fitting Appendix 5 Beam Modelling and Modal Analysis 246 Figure A5-6 shows the comparison between theoretical and experimental mode shapes. The 1st to 3rd modes are given in Figure (a) and the 4th to 6th modes are given in Figure (b) respectively. The two match reasonably well.

(a)

(b)

Figure A5-6 Comparison between theoretical and experimental mode shapes Appendix 6 Derivation of the Equations in Chapter 3 247

Appendix 6 Derivation of the Equations in Chapter 3

This Appendix is lnked to Chapter 3 (section 3.1.5, p. 52 ~ 53 and section 3.2, p. 55)

A6.1 Derivation of the Equations (3.24 ~ 3.28) in Chapter 3.

Equation (3.24),

2 2 t ⎛ L ⎞ Lt⎛ ⎞ 1 2 ⎜ ⎛ ∂w ⎞ ⎟ 1 ⎜ 2 ⎛ ∂w ⎞ ⎟ δ ρ b Ab ⎜ ⎟ dx dt = ρ b Ab δ ⎜ ⎟ dt dx ∫∫ t ⎜ 0 ⎟ ∫∫ 0 ⎜ t ⎟ 2 1 ⎝ ⎝ ∂t ⎠ ⎠ 2 ⎝ 1 ⎝ ∂t ⎠ ⎠

Lt⎛ 2 ∂w ⎛ ∂w ⎞ ⎞ Lt⎛ 2 ∂w⎛ ∂(δw) ⎞ ⎞ = ρb Ab ⎜ δ ⎜ ⎟dt ⎟dx = ρb Ab ⎜ ⎜ ⎟dt ⎟dx (δ acts on w(x,t) ) ∫∫ 0 ⎜ t ⎟ ∫∫ 0 ⎜ t ⎟ ⎝ 1 ∂t ⎝ ∂t ⎠ ⎠ ⎝ 1 ∂t ⎝ ∂t ⎠ ⎠

Let

∂(δw) ∂w f (t) = , g'(t) = ∂t ∂t By integration by parts (that is ∫ f (t)g'(t)dt = f (t)g(t) − ∫ g(t) f '(t)dt ), one obtains 0 t2 2 Lt Lt⎛ ⎞ ⎛ 2 ∂w ⎛ ∂(δw) ⎞ ⎞ ⎜ ∂(δw) 2 ⎛ ∂ (δw) ⎞ ⎟ ρ b Ab ⎜ ⎜ ⎟dt ⎟dx = ρ b Ab w − w⎜ ⎟dt dx ∫∫ 0 ⎜ t ⎟ ∫∫ 0 t ⎜ 2 ⎟ 1 ∂t ⎝ ∂t ⎠ ⎜ ∂t 1 ∂t ⎟ ⎝ ⎠ ⎝ t1 ⎝ ⎠ ⎠

2 2 Lt⎛ 2 ⎛ ∂ w⎞ ⎞ t2 L ⎛ ∂ w⎞ = ρb Ab ⎜− ⎜ ⎟δwdt⎟dx = − ρb Ab ⎜ ⎟δw dt dx ∫∫ 0 ⎜ t ⎜ 2 ⎟ ⎟ ∫∫ t 0 ⎜ 2 ⎟ ⎝ 1 ⎝ ∂t ⎠ ⎠ 1 ⎝ ∂t ⎠ Appendix 6 Derivation of the Equations in Chapter 3 248

(Since the virtual displacement must be so specified that δw ≡ 0 at t0 and t1 and, hence,

∂(δw) / ∂t = δ (∂w/ ∂t) ≡ 0 at t0 and t1 ) (Fung, 2001).

Equation (3.25)

Similar to Equation (3.24),

2 1 t2 L ⎛ ∂w ⎞ δ ρ p Ap ⎜ ⎟ ⋅[]H (x − x1 ) − H (x − x2 ) dx dt ∫∫ t 0 2 1 ⎝ ∂t ⎠

Lt⎛ 2 ∂w⎛ ∂(δw) ⎞ ⎞ = ρb Ab ⎜ ⎜ ⎟dt ⎟ ⋅[]H (x − x1 ) − H (x − x2 ) dx ∫∫ 0 ⎜ t ⎟ ⎝ 1 ∂t ⎝ ∂t ⎠ ⎠

Let

∂(δw) ∂w f (t) = , g'(t) = ∂t ∂t

Integrating by parts obtains

Lt⎛ 2 ∂w⎛ ∂(δw) ⎞ ⎞ ρ p Ap ⎜ ⎜ ⎟dt ⎟ ⋅[]H (x − x1 ) − H (x − x2 ) dx ∫∫ 0 ⎜ t ⎟ ⎝ 1 ∂t ⎝ ∂t ⎠ ⎠ 0

t2 2 Lt⎛ ∂(δw) 2 ⎛ ∂ (δw) ⎞ ⎞ = ρ A ⎜ w − w⎜ ⎟dt⎟⋅ H (x − x ) − H (x − x ) dx p p ⎜ 2 ⎟ []1 2 ∫∫ 0 ⎜ ∂t t1 ∂t ⎟ ⎝ t1 ⎝ ⎠ ⎠

2 Lt⎛ 2 ⎛ ∂ w⎞ ⎞ = ρ p Ap ⎜− ⎜ ⎟δwdt⎟⋅[]H (x − x1) − H(x − x2 ) dx ∫∫ 0 ⎜ t ⎜ 2 ⎟ ⎟ ⎝ 1 ⎝ ∂t ⎠ ⎠

2 t2 L ∂ w = − ρ p Ap 2 ⋅[]H (x − x1 ) − H (x − x2 ) δw dx dt ∫∫ t1 0 ∂t

Equation (3.26),

⎛ 2 2 ⎞ ⎛ 2 2 ⎞ 1 t2 L ⎛ ∂ w ⎞ 1 t2 L ⎛ ∂ w ⎞ − δ ⎜ E I ⎜ ⎟ dx⎟ dt = − E I ⎜ δ ⎜ ⎟ dx⎟ dt b b ⎜ 2 ⎟ b b ⎜ 2 ⎟ 2 ∫∫t1 ⎜ 0 ∂x ⎟ 2 ∫∫t1 ⎜ 0 ∂x ⎟ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠

2 2 t2 ⎛ L ∂ w ∂ (δw) ⎞ = − Eb Ib ⎜ dx⎟ dt ∫∫t ⎜ 0 2 2 ⎟ 1 ⎝ ∂x ∂x ⎠

Let

∂2 (δw) ∂ ⎛ ∂w⎞ u(x) = , v'(x) = ⎜ ⎟, ∂x2 ∂x ⎝ ∂x ⎠ Appendix 6 Derivation of the Equations in Chapter 3 249 Integrating by parts obtain,

2 2 ⎛ 2 L 3 ⎞ t2 ⎛ L ∂ w ∂ (δw) ⎞ t2 ∂w ∂ (δw) L ∂w ∂ (δw) − E I ⎜ dx⎟ dt = − E I ⎜ − dx⎟ dt b b ⎜ 2 2 ⎟ b b 2 3 ∫∫t1 0 ∂x ∂x ∫∫ t1 ⎜ ∂x ∂x 0 ∂x ∂x ⎟ ⎝ ⎠ ⎝ 0 ⎠

Then let,

∂3 (δw) ∂w f (x) = , g'(x) = , ∂x3 ∂x

Again integrating by parts, one can have Equation (3.26) as

2 2 t2 ⎛ L ∂ w ∂ (δw) ⎞ − Eb Ib ⎜ dx⎟ dtt ∫∫ t ⎜ 0 2 2 ⎟ 1 ⎝ ∂x ∂x ⎠

L ⎛ 2 L 3 4 ⎞ t2 ⎜ ∂w ∂ (δw) ∂ (δw) L ∂ (δw) ⎟ = − Eb Ib − w + w dx d ∫∫ t ⎜ 2 3 0 4 ⎟ 1 ⎜ ∂x ∂x ∂x ∂x ⎟ ⎝ 0 0 ⎠

L ⎛ 2 L 3 4 ⎞ t2 ⎜ ∂ w ⎛ ∂w ⎞ ∂ w L ∂ w ⎟ = − Eb I b δ ⎜ ⎟ − δw + δw dx dt ∫∫ t ⎜ 2 3 0 4 ⎟ 1 ⎜ ∂x ⎝ ∂x ⎠ ∂x ∂x ⎟ ⎝ 0 0 ⎠

Equation (3.27),

2 2 1 t2 L ⎛ ∂ w⎞ − δ Ep I p ⎜ ⎟ []H(x − x1) − H(x − x2 ) dx dt ∫∫ t 0 ⎜ 2 ⎟ 2 1 ⎝ ∂x ⎠

2 2 t2 L ∂ w ⎛ ∂ w ⎞ = − E p I p δ ⎜ ⎟ []H (x − x1 ) − H (x − x2 ) dx dt ∫∫ t 0 2 ⎜ 2 ⎟ 1 ∂x ⎝ ∂x ⎠

Let

∂2w ∂ ⎛ ∂w ⎞ u(x) = δ []H (x − x1) − H (x − x2 ) , v'(x) = ⎜ ⎟ , ∂x2 ∂x ⎝ ∂x ⎠

By integration by parts, one obtains

2 2 t2 L ∂ w ⎛ ∂ w⎞ − Ep I p δ⎜ ⎟ []H(x − x1) − H(x − x2 ) dx dt ∫∫ t 0 2 ⎜ 2 ⎟ 1 ∂x ⎝ ∂x ⎠ 0 L 2 t2 ⎧∂ w ∂(δw) = − E p I p ⎨ 2 [H (x − x1 ) − H (x − x2 )] ∫ t1 ∂x ∂x ⎩ 0 Appendix 6 Derivation of the Equations in Chapter 3 250

3 2 L ⎡∂ (δw) ∂ (δw) ⎤ ∂w⎫ − [H(x − x1) − H(x − x2 )]+ [δ (x − x1) −δ (x − x2 )] dxdt ∫ 0 ⎢ 3 2 ⎥ ⎬ ⎣ ∂x ∂x ⎦ ∂x ⎭

3 2 t2 L ⎧⎡∂ (δw) ∂ (δw) ⎤ ∂w⎫ = Ep I p [H(x − x1) − H(x − x2 )]+ [δ (x − x1) −δ (x − x2 )] dxdt ∫∫ t 0 ⎨⎢ 3 2 ⎥ ⎬ 1 ⎩⎣ ∂x ∂x ⎦ ∂x ⎭

Then let

∂3(δw) ∂2 (δw) f (x) = []H(x − x ) − H(x − x ) + []δ (x − x ) −δ (x − x ) , ∂x3 1 2 ∂x2 1 2 ∂w g'(x) = , ∂x

Again integrating by parts yields

3 2 t2 L ⎡∂ (δw) ∂ (δw) ⎤ ∂w⎫ E p I p [H (x − x1) − H (x − x2 )]+ [δ (x − x1) −δ (x − x2 )] dxdt ∫∫ t 0 ⎢ 3 2 ⎥ ⎬ 1 ⎣ ∂x ∂x ⎦ ∂x ⎭ 0 L 3 2 t2 ⎪⎧⎡∂ w ∂ w ⎤ = EpI p [H(x − x1) − H(x − x2 )]+ [δ (x − x1) −δ (x − x2 )] ∫ t ⎨⎢ 3 2 ⎥ 1 ⎪ ∂x ∂x ⎩⎣ ⎦ 0

4 3 L ⎡∂ w ∂ w − [H(x − x1) − H(x − x2 )]+ [δ (x − x1) −δ (x − x2 )] ∫ 0 ⎢ 4 3 ⎣ ∂x ∂x

∂3w ∂ 2w ⎤⎫ + 3 [δ (x − x1) −δ (x − x2 )] + 2 [δ '(x − x1) −δ '(x − x2 )]⎥⎬δwdxdt ∂x ∂x ⎦⎭

So Equation (3.27) becomes

2 2 1 t2 L ⎛ ∂ w⎞ − δ Ep I p ⎜ ⎟ []H(x − x1) − H(x − x2 ) dx dt ∫∫ t 0 ⎜ 2 ⎟ 2 1 ⎝ ∂x ⎠

4 3 t2 L ⎧∂ w ∂ w = − EpI p ⎨ 4 [H(x− x1)− H(x − x2 )]+2 3 [δ(x− x1)−δ(x− x2 )] ∫ t1 ∫ 0 ∂x ∂x ⎩

∂2w ⎫ + [δ '(x − x1) −δ '(x − x2 )]⎬δwdxdt ∂x2 ⎭

Appendix 6 Derivation of the Equations in Chapter 3 251 Equation (3.28)

To Equation (3.28), there are two parts, the first one is

2 1 t2 L hp ∂ w δ h31Ap (hb + )D3 2 [H(x − x1) − H(x − x2 )]dx 2 ∫∫ t1 0 2 ∂x

2 1 t2 L hp ⎛ ∂ w⎞ = h31Ap (hb + )D3 ⋅[]H(x − x1) − H(x − x2 ) ⋅δ ⎜ ⎟ dxdt ∫∫ t 0 ⎜ 2 ⎟ 2 1 2 ⎝ ∂x ⎠

The second part is

t L 1 2 2 δ Ap β33D3 ⋅[H(x − x1) − H(x − x2 )]dx 2 ∫∫ t1 0

For the first part, let

∂ ⎛ ∂(δw) ⎞ u(x) = [H(x − x1) − H(x − x2 )], v'(x) = ⎜ ⎟ ∂x ⎝ ∂x ⎠

By integration by parts, it becomes

2 1 t2 L h ⎛∂ w⎞ h A (h + p )D ⋅ H(x−x )−H(x−x ) ⋅δ⎜ ⎟ dxdt 31 p b 3 []1 2 ⎜ 2 ⎟ 2∫∫ t1 0 2 ∂x ⎝ ⎠

1 t2 hp ∂ ⎛∂w⎞ = h31Ap (hb + )D3 ⋅[]H(x− x1)− H(x− x2) ⋅δ ⎜ ⎟dxdt ∫ t 2 1 2 ∂x ⎝ ∂x ⎠ 0 L 1 t2 hp ⎪⎧ ∂(δw) L ∂w⎪⎫ = h31Ap (hb + )D3 [H(x − x1) − H(x − x2 )] − [δ (x − x1) −δ (x − x2 )]δ dxdt ∫∫ t ⎨ 0 ⎬ 2 1 2 ⎩⎪ ∂x 0 ∂x ⎭⎪

Then again let,

∂(δw) f (x) = [δ (x − x ) −δ (x − x )], g'(x) = 1 2 ∂x

Again integrating by parts, we obtain

1 t2 hp L ∂w − h31Ap (hb + )D3 [δ (x − x1) −δ (x − x2 )]δ dxdt 2 ∫ t1 2 ∫ 0 ∂x 0

1 t2 h L L = − h A (h + p )D ⎡[δ (x − x ) −δ (x − x )]δw − [δ '(x − x ) −δ '(x − x )]δwdxdt⎤ 31 p b 3 1 2 0 1 2 2 ∫ t1 2 ⎣⎢ ∫ 0 ⎦⎥

1 t2 L hp = h31Ap (hb + )D3 ⋅[δ '(x − x1) −δ '(x − x2 )]δwdxdt 2 ∫∫ t1 0 2 Appendix 6 Derivation of the Equations in Chapter 3 252 So the first part of Equation (3.28) can be expressed as

2 1 t2 L hp ∂ w δ h31Ap (hb + )D3 [H(x − x1) − H(x − x2 )]dx ∫∫ t 0 2 2 1 2 ∂x

1 t2 L hp = h31 Ap (hb + )D3 ⋅[δ '(x − x1 ) − δ '(x − x2 )]δwdxdt 2 ∫∫ t1 0 2

The second part of Equation (3.28) can be obviously expressed as

L L 1 2 δ Ap β33D3 [H(x − x1) − H(x − x2 )]dx = Ap β33D3[H(x − x1) − H(x − x2 )]δD3dx 2 ∫ 0 ∫ 0

Finally obtaining the Equation (3.28)

2 L ⎛ hp ∂ w 2 ⎞ − δ Ap ⎜ h31 (hb + )D3 + β 33 D3 ⎟ [H (x − x1 ) − H (x − x2 )] dx ∫ 0 ⎜ 2 ⎟ ⎝ 2 ∂x ⎠

1 t2 L hp = − h31 Ap (hb + )D3 ⋅[δ '(x − x1) −δ '(x − x2 )]δwdxdt 2 ∫∫ t1 0 2

L − Ap β33D3 [H(x − x1) − H(x − x2 )]δD3dx ∫ 0

Appendix 6 Derivation of the Equations in Chapter 3 253 A6.2 Derivation of the Equations (3.36 ~ 3.39) in Chapter 3.

Equation (3.36)

To the Equation (3.36), using integrating by parts twice, one can have 0 N L N L (4) ⎡ (3) L (3) ⎤ Eb Ib φr (x)φs (x)dx qr (t) = Eb Ib φr (x)φs (x) − φr (x)φs '(x)dx qr (t) ∑∫ 0 ∑ ⎢ 0 ∫ 0 ⎥ r=1 r=1 ⎣ ⎦ 0 N ⎡ L L ⎤ = Eb Ib −φr ''(x)φs '(x) + φr ''(x)φs ''(x)dx qr (t) ∑ ⎢ 0 ∫ 0 ⎥ r=1 ⎣ ⎦

N ⎛ L ⎞ = Eb Ib ⎜ φr ''(x)φs ''(x)dx⎟ qr (t) ∑ ∫ 0 r=1 ⎝ ⎠

Equation (3.37)

To the Equation (3.37), let

(3) f (x) = φr (x)φs (x), g'(x) = [δ (x − x1 ) − δ (x − x2 )]

By integration by parts, one obtains

N L (3) 2E p I p φr (x)φs (x)[δ (x − x1) −δ (x − x2 )]dxqr (t) ∑∫ 0 r=1 0

N (3) L = 2E p I p { φr φs (x)[H(x − x1) − H(x − x2 )] ∑ 0 r=1

L (4) (3) − []φr (x)φs (x) +φr (x)φs '(x) []H(x − x1) − H(x − x2 ) dxqr (t)} ∫ 0

N L (4) (3) = −2E p I p []φr (x)φs (x) +φr (x)φs '(x) []H (x − x1) − H (x − x2 ) dxqr (t) ∑∫ 0 r=1

Equation (3.38)

To the Equation (3.38), let

u(x) = φr ''(x)φs (x), v'(x) = [δ '(x − x1) − δ '(x − x2 )]

By integration by parts, we obtain

N ⎛ L ⎞ E p I p ⎜ φr ''(x)φs (x)[δ '(x − x1) −δ '(x − x2 )]dx⎟qr (t) ∑ ∫ 0 r=1 ⎝ ⎠ Appendix 6 Derivation of the Equations in Chapter 3 254 0 N = E I ⎧ φ ''(x)φ (x)[δ (x − x ) −δ (x − x )] L p p ∑⎨ r s 1 2 0 r=1 ⎩

L (3) ⎫ − [φr (x)φs (x) +φr ''(x)φs '(x)][δ (x − x1) −δ (x − x2 )]dx ⎬ qr (t) ∫ 0 ⎭

N L ⎛ (3) ⎞ = −Ep I p ⎜ [φr (x)φs (x) +φr ''(x)φs '(x)][δ (x − x1) −δ (x − x2 )]dx⎟qr (t) ∑ ∫ 0 r=1 ⎝ ⎠

Again, let

(3) f (x) = φr (x)φs (x) + φr ''(x)φs '(x), g'(x) = [δ (x − x1) − δ (x − x2 )]

Integrating by parts obtains

N L ⎛ (3) ⎞ − Ep I p ⎜ [φr (x)φs (x) +φr ''(x)φs '(x)][δ (x − x1) −δ (x − x2 )]dx⎟qr (t) ∑ ∫ 0 r=1 ⎝ ⎠ 0 N = −E I [φ (3) (x)φ (x) + φ ''(x)φ '(x)][H (x − x ) − H (x − x )] L p p ∑ { r s s 1 2 0 r =1

L (4) (3) ⎫ + [φr (x)φs (x) + 2φr (x)φs '(x) +φr ''(x)φs ''(x)][H(x − x1) − H(x − x2 )]dx⎬ qr (t) ∫ 0 ⎭

Therefore the Equation (3.38) is

N ⎛ L ⎞ Ep I p ⎜ φr ''(x)φs (x)[δ '(x − x1) −δ '(x − x2 )]dx⎟qr (t) ∑ ∫ 0 r=1 ⎝ ⎠

N L (4) (3) = Ep I p φr (x)φs (x) + 2φr (x)φs '(x) +φr ''(x)φs ''(x)][H(x − x1) − H(x − x2 )]dxqr (t) ∑∫ 0 r=1

Equation (3.39)

+∞ To the Equation (3.39), since f (x)δ '(x − x0 )dx = − f '(x0 ) , thus ∫−∞

hp L − h31 Ap (hb + )D3 φs (x)[δ '(x − x1) −δ '(x − x2 )]dx ∫ 0 2

hp hp x2 − h31Ap (hb + )D3[−φs '(x1) +φs '(x2 )] = −h31Ap (hb + )D3 φs ''(x)dx 2 2 ∫ x1

Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 255

Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes

A7.1 ANSYS® code of modal analysis for aluminium beam with PZT attached

! *** Modal analysis of Aluminium beam with PZT attached *** ! Written by Jia Long Cao

/CLEAR /VIEW,1,1,1,1 /VUP,,Z

/PREP7 ! ET,1,5 ! 8-node solid with all DOF, used ! for piezoceramic transducer ET,2,45 ! 8-node solid with UX,UY,UZ DOF, ! used for aluminium beam MP,DENS,1,7730 ! Density of piezoceramic MP,PERX,1,8.7969E-9 ! Permittivity of piezoceramic MP,PERY,1,8.7969E-9 MP,PERZ,1,8.7969E-9

TB,PIEZ,1 ! Piezoelectric "e" matrix of piezoceramic TBDATA,3,-6.1 TBDATA,6,-6.1 TBDATA,9,15.7

TB,ANEL,1 ! Stiffness "c" matrix of piezoceramic TBDATA,1,12.8E10,6.8E10,6.6E10 TBDATA,7,12.8E10,6.6E10 TBDATA,12,11.0E10 TBDATA,16,2.1E10 TBDATA,19,2.1E10 TBDATA,21,2.1E10 Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 256

MP,EX,2,6.6E10 ! Modulus of Elasticity of aluminium MP,NUXY,2,.345 ! Poisson's ratio of aluminium MP,DENS,2,2790 ! Density of aluminium

BTHK= 0.003 ! Beam thickness CVTHK= 0.0005 ! Piezoceramic covering thickness DEPTH= 0.025 ! Depth of assembly BLNGTH= 0.5 ! Beam length CVLNGTH=0.05 ! Length of covering ! BLOCK,,CVLNGTH,,DEPTH,0,BTHK ! Defines covered beam volume BLOCK,CVLNGTH,BLNGTH,,DEPTH,0,BTHK ! Defines uncovered beam volume VATT,2,,2 ! Associates material 2 and type 2 ! with beam volume BLOCK,,CVLNGTH,,DEPTH,BTHK,BTHK+CVTHK ! Defines piezoceramic cover ESIZE,BLNGTH/30 ! Defines default element size NUMMRG,KP ! Merges duplicate solid model ! entities numstr,node,14 ! Set starting node number for the ! solid model VMESH,ALL ! Generates nodes and elements ! within the volumes NSEL,S,LOC,Z, BTHK NSEL,R,LOC,X,,CVLNGTH CP,1,VOLT,ALL ! Couples volt DOF at beam interface CM,INTRFC,NODE ! Creates a component for the ! interface (Face to beam) *GET,BOT_ELECTRODE,NODE,,NUM,MIN NSEL,S,LOC,Z,BTHK+CVTHK NSEL,R,LOC,X,,CVLNGTH CP,2,VOLT,ALL ! Couples volt DOF at piezoceramic ! outer surface CM,OUTSIDE,NODE ! Creates a component for the outer ! surface *GET,TOP_ELECTRODE,NODE,,NUM,MIN !NSEL,S,LOC,X !D,ALL,UX,,,,,UY,UZ ! Fixes nodes at x=0 NSEL,ALL FINISH ! /COM, **** OBTAIN THE MODAL SOLUTION UNDER SHORT-CIRCUIT CONDITIONS **** ! /SOLU ANTYPE,MODAL ! Modal analysis nmodes = 20 modopt,LANB,nmodes ! Block Lanczos solver mxpand,nmodes,,,yes ! Calculate element results and ! reaction forces TOTAL,30,1 ! Requests automatic generation of ! (structural) DOF D,BOT_ELECTRODE,VOLT,0.0 D,TOP_ELECTRODE,VOLT,0.0 ! Add boundary condition will change ! the modal frequency TOTAL,30,1 ! Requests automatic generation of ! (structural) DOF SOLVE FINISH ! /COM, **** REVIEW THE RESULTS OF THE MODAL SOLUTION **** ! /POST1 SET,LIST ! Lists all frequencies /WINDOW,1,LTOP ! Defines four windows /WINDOW,2,RTOP Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 257

/WINDOW,3,LBOT /WINDOW,4,RBOT

/WINDOW,ALL,OFF ! All windows off *DO,I,1,4 /VUP,I,Z /VIEW,I,1,1,1 ! View for all windows /WINDOW,I,ON ! Turns on window i SET,1,I+6 ! Reads in data for mode I+4*K /TITLE,Aluminium Beam with PZT Patch !PLNSOL,U,SUM,0,1 PLNSOL,EPEL,EQV,0,1 ! Plot Von Mises strain /NOERASE ! No screen erase after plot /WINDOW,I,OFF ! Turns off window i *ENDDO FINISH

A7.2 ANSYS® code of harmonic analysis for aluminium beam with PZT attached

! *** Harmonic analysis of aluminium beam *** ! Written by Jia Long Cao

/VIEW,1,1,1,1 /VUP,,Z /PREP7

ET,1,5 ! 8-node solid with all DOF, used for ! piezoceramic transducer ET,2,45 ! 8-node solid with UX,UY,UZ DOF, used ! for aluminium beam MP,DENS,1,7730 ! Density of piezoceramic MP,PERX,1,8.7969E-9 ! Permittivity of piezoceramic MP,PERY,1,8.7969E-9 MP,PERZ,1,8.7969E-9

TB,PIEZ,1 ! Piezoelectric "e" matrix of piezoceramic TBDATA,3,-6.1 TBDATA,6,-6.1 TBDATA,9,15.7

TB,ANEL,1 ! Stiffness "c" matrix of piezoceramic TBDATA,1,12.8E10,6.8E10,6.6E10 TBDATA,7,12.8E10,6.6E10 TBDATA,12,11.0E10 TBDATA,16,2.1E10 TBDATA,19,2.1E10 TBDATA,21,2.1E10

MP,EX,2,6.6E10 ! Modulus of Elasticity of aluminium MP,NUXY,2,.345 ! Poisson's ratio of aluminium MP,DENS,2,2790 ! Density of aluminium ! BTHK= 0.003 ! Beam thickness CVTHK= 0.0005 ! Piezoceramic covering thickness DEPTH= 0.025 ! Depth of assembly BLNGTH= 0.5 ! Beam length CVLNGTH=0.05 ! Length of covering Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 258

BLOCK,,CVLNGTH,,DEPTH,,BTHK ! Defines covered beam volume BLOCK,CVLNGTH,BLNGTH,,DEPTH,,BTHK ! Defines uncovered beam volume VATT,2,,2 ! Associates material 2 and type 2 ! with beam volume BLOCK,,CVLNGTH,,DEPTH,BTHK,BTHK+CVTHK ! Defines piezoceramic cover ESIZE,BLNGTH/30 ! Defines default element size NUMMRG,KP ! Merges duplicate solid model ! entities numstr,node,14 ! Set starting node number for the ! solid model VMESH,ALL ! Generates nodes and elements ! within the volumes NSEL,S,LOC,Z, BTHK NSEL,R,LOC,X,,CVLNGTH CP,1,VOLT,ALL ! Couples volt DOF at beam interface CM,INTRFC,NODE ! Creates a component for the ! interface (Face to beam) *GET,BOT_ELECTRODE,NODE,,NUM,MIN NSEL,S,LOC,Z,BTHK+CVTHK NSEL,R,LOC,X,,CVLNGTH CP,2,VOLT,ALL ! Couples volt DOF at piezoceramic ! outer surface CM,OUTSIDE,NODE ! Creates a component for the outer ! surface *GET,TOP_ELECTRODE,NODE,,NUM,MIN

NSEL,ALL FORCENODE = node(BLNGTH,DEPTH/2,BTHK) ! Node where force applied DISPNODE = node(BLNGTH,DEPTH/2,BTHK) ! Node where deflection detected FINISH

/SOLU ANTYPE,HARM ! Harmonic analysis HARFRQ,0,1000, ! Frequency range of analysis NSUBST,500, ! Number of frequency steps KBC,0 ! Impulse loads F,FORCENODE,FZ,-1e3 ! Applying force

D,BOT_ELECTRODE,VOLT,0.0 ! Assigns zero voltage to bottom ! electrode D,TOP_ELECTRODE,VOLT,0.0 ! Assigns zero voltage to top ! electrode D,2,VOLT,0.0 ! Assigns zero voltage to node 2 CNVTOL,F,,,,1E-5 ! Small convergence value for force CNVTOL,VOLT ! Default convergence value for ! current flow NSEL,S,LOC,X D,ALL,UX,,,,,UY,UZ ! Fixes nodes at x=0 NSEL,ALL SOLVE FINISH

/POST26 NSOL,2,DISPNODE,U,Z ! Get z-direction deflection at node DISPNODE ! and store it into variable 2 /ERASE /axlab,y,Displacement !/YRANGE,0,14 PLVAR,2 ! Plot variable 2 vs. Freq /window,1,off /noerase FINISH Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 259 A7.3 ANSYS® code of single mode, parallel R-L piezoelectric vibration shunt control for aluminium beam

! *** R-L Parallel Shunt Circuit for Vibration Control *** ! Written by Jia Long Cao

!/SHOW,JPEG /VIEW,1,1,1,1 /VUP,,Z /PREP7

*dim,F,array,6 ! Array to store natural frequencies *dim,L,array,6 ! Array to store inductor values *dim,R,array,6 ! Array to store resistor values *set,i,1 ! Index of the modal to be controlled

!Cp = 31.14E-9 ! Capacitor of PZT

F(1) = 57.391 ! First modal frequency F(2) = 161.987 ! Second modal frequency F(3) = 324.152 ! Third modal frequency F(4) = 546.799 ! Fourth modal frequency F(5) = 834.393 ! Fifth modal frequency

L(1) = 229.9 ! Inductor value for F(1) L(2) = 29.5 ! Inductor value for F(2) L(3) = 7.27 ! Inductor value for F(3) L(4) = 2.58 ! Inductor value for F(4) L(5) = 1.11 ! Inductor value for F(5)

R(1) = 1E9 ! Resistor value for F(1) R(2) = 5E7 ! Resistor value for F(2) R(3) = 5E6 ! Resistor value for F(3) R(4) = 4E5 ! Resistor value for F(4) R(5) = 18E4 ! Resistor value for F(5)

ET,1,5 ! 8-node solid with all DOF, used for ! piezoceramic transducer ET,2,45 ! 8-node solid with UX,UY,UZ DOF, ! used for aluminum beam MP,DENS,1,7730 ! Density of piezoceramic MP,PERX,1,8.7969E-9 ! Permittivity of piezoceramic MP,PERY,1,8.7969E-9 MP,PERZ,1,8.7969E-9

TB,PIEZ,1 ! Piezoelectric "e" matrix of ! piezoceramic TBDATA,3,-6.1 TBDATA,6,-6.1 TBDATA,9,15.7

TB,ANEL,1 ! Stiffness "c" matrix of piezoceramic TBDATA,1,12.8E10,6.8E10,6.6E10 TBDATA,7,12.8E10,6.6E10 TBDATA,12,11.0E10 TBDATA,16,2.1E10 TBDATA,19,2.1E10 TBDATA,21,2.1E10

MP,EX,2,6.6E10 ! Modulus of Elasticity of aluminium MP,NUXY,2,.345 ! Poisson's ratio of aluminium Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 260

MP,DENS,2,2790 ! Density of aluminium ! BTHK= 0.003 ! Beam thickness CVTHK= 0.0005 ! Piezoceramic covering thickness DEPTH= 0.025 ! Depth of assembly BLNGTH= 0.5 ! Beam length CVLNGTH=0.05 ! Length of covering

BLOCK,,CVLNGTH,,DEPTH,,BTHK ! Defines covered beam volume BLOCK,CVLNGTH,BLNGTH,,DEPTH,,BTHK ! Defines uncovered beam volume

VATT,2,,2 ! Associates material 2 and type 2 with beam volume BLOCK,,CVLNGTH,,DEPTH,BTHK,BTHK+CVTHK ! Defines piezoceramic cover

ESIZE,BLNGTH/30 ! Defines default element size NUMMRG,KP ! Merges duplicate solid model entities numstr,node,14 ! Set starting node number for the solid model VMESH,ALL ! Generates nodes and elements within the volumes ! NSEL,S,LOC,Z, BTHK NSEL,R,LOC,X,,CVLNGTH CP,1,VOLT,ALL ! Couples volt DOF at beam interface CM,INTRFC,NODE ! Creates a component for the interface (Face to beam) *GET,BOT_ELECTRODE,NODE,,NUM,MIN

NSEL,S,LOC,Z,BTHK+CVTHK NSEL,R,LOC,X,,CVLNGTH CP,2,VOLT,ALL ! Couples volt DOF at piezoceramic outer surface CM,OUTSIDE,NODE ! Creates a component for the outer surface *GET,TOP_ELECTRODE,NODE,,NUM,MIN

N,1,-0.03,DEPTH/2, 0.03 N,2,-0.03,DEPTH/2,-0.03

ET,3,CIRCU94,1,0 ! Set up the inductor R,1,L(i) RMOD,1,15,0,0 ! Subscript of inductor is 0, i.e. L0 TYPE,3 $ REAL,1 E,TOP_ELECTRODE,BOT_ELECTRODE ! Parallel the inductor to the ! electrodes of PZT E,1,2 ! Parallel the inductor between ! node 1 and 2 ET,4,CIRCU94,0,0 ! Set up the resistor R,2,R(i) RMOD,2,15,0,0 ! Subscript of resistor is 0, i.e. R0 TYPE,4 $ REAL,2 E,TOP_ELECTRODE,BOT_ELECTRODE ! Parallel the resistor to the ! electrodes of PZT E,1,2 ! Parallel the resistor between node 1 and 2 NSEL,ALL FORCENODE = node(BLNGTH,DEPTH/2,BTHK) ! Node where force applied DISPNODE = node(BLNGTH,DEPTH/2,BTHK) ! Node where deflection detected FINISH

/SOLU ANTYPE,HARM ! Harmonic analysis HARFRQ,0,1000, ! Frequency range of analysis NSUBST,500, ! Number of frequency steps KBC,0 ! Impulse loads F,FORCENODE,FZ,-1e3 ! Applying force

D,BOT_ELECTRODE,VOLT,0.0 ! Assigns zero voltage to bottom electrode !D,TOP_ELECTRODE,VOLT,0.0 ! Assigns zero voltage to top electrode D,2,VOLT,0.0 ! Assigns zero voltage to node 2 Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 261

CNVTOL,F,,,,1E-5 ! Small convergence value for force CNVTOL,VOLT ! Default convergence value for current flow !NSEL,S,LOC,X !D,ALL,UX,,,,,UY,UZ ! Fixes nodes at x=0 !NSEL,ALL SOLVE FINISH

/POST26 NSOL,2,DISPNODE,U,Z ! Get z-direction deflection at node DISPNODE ! and store it into variable 2 /ERASE /axlab,y,Displacement /YRANGE,0,4 PLVAR,2 ! Plot variable 2 vs. Freq /window,1,off /noerase FINISH

A7.4 ANSYS® code of modal analysis of carbon-fibre composite sandwich cricket bat with rubber layers (blade built in Solidworks® is imported into ANSYS®)

! Modal analysis of carbon-fibre composite sandwich cricket bat with ! rubber layers ! Written by Jia Long Cao

! The handle consists of two beam composite material sandwiched with wood, ! There are two rubber layers in wood part as well ! The triangle part overlapping with blade is solid wood, not composite.

/BATCH /CONFIG,NRES,50000 ! Maximum substep is 50000 /input,menust,tmp,'',,,,,,,,,,,,,,,,1 /GRA,POWER /GST,ON /PLO,INFO,3 /GRO,CURL,ON ~PARAIN,Blade,x_t,,SOLIDS,0,0 ! Access Blade.x_t file built in Solidworks !* ! A triangle volume to be used as a subtracter K,480,0,-0.016,-0.025, K,481,0,0.016,-0.025, K,482,0,0.016,0.025, K,483,0,-0.016,0.025, K,484,0.12,-0.001,-0.025, K,485,0.12,0.001,-0.025, K,486,0.12,-0.001,0.04, K,487,0.12,0.001,0.04, V,480,481,482,483,484,485,487,486

! Subtract the triangle volume from blade FLST,2,2,6,ORDE,2 FITEM,2,1 FITEM,2,2 VSBV,P51X,2

Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 262

! Create a triangle volume to be fit in the blade K,480,0,-0.016,-0.0175, K,481,0,0.016,-0.0175, K,482,0,0.016,0.0145, K,483,0,-0.016,0.0145, K,484,0.12,-0.001,-0.0165, K,485,0.12,0.001,-0.0165, K,486,0.12,-0.001,0.0245, K,487,0.12,0.001,0.0245, V,480,481,482,483,484,485,487,486 wpoff,0,0,-0.0016 ! Shift current centre point by (x0,y0,z0) wpro,,,-90.000000 ! Rotate –90 degrees CYL4,,,,,0.016,,0.3 ! Create CYLINDER1

FLST,2,2,6,ORDE,2 ! Make cylinder1 and triangle volumes as one part FITEM,2,1 ! Pick up volume 1 FITEM,2,-2 ! to 2 VADD,P51X ! Add all

! BLOCK1 & BLOCK2 are used to make space for embedding composite beams BLOCK,0.0095,0.011,-0.0105,0.0105,0,0.3 BLOCK,-0.0095,-0.011,-0.0105,0.0105,0,0.3 ! BLOCKs are used to make space for embedding rubber layers BLOCK,-0.0005,0.0005,-0.015,0.015,,0.3 BLOCK,-0.0045,-0.0055,-0.014,0.014,,0.3 BLOCK,0.0045,0.0055,-0.014,0.014,,0.3

FLST,3,5,6,ORDE,4 FITEM,3,1 FITEM,3,-2 FITEM,3,5 FITEM,3,-7 VSBV,4,P51X

! BLOCKS are used for making composite beams BLOCK,0.0095,0.011,-0.0105,0.0105,0,0.3 BLOCK,-0.0095,-0.011,-0.0105,0.0105,0,0.3

! BLOCKs are used for making rubber layers BLOCK,-0.0005,0.0005,-0.015,0.015,,0.3 BLOCK,-0.0045,-0.0055,-0.014,0.014,,0.3 BLOCK,0.0045,0.0055,-0.014,0.014,,0.3

FLST,2,7,6,ORDE,3 FITEM,2,1 ! Pick up volume 1 FITEM,2,-6 ! to 6 FITEM,2,8 ! Pick up volume 8 VGLUE,P51X ! Glue all

/VIEW,1,1,1,1 VPLOT

ET,1,SOLID95 !* ! Material property of Blade (willow) MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,1,,6.67E9 ! Young's modulus of willow MPDATA,NUXY,1,,.35 MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,DENS,1,,420 ! Density of willow ! Material property of wood cane Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 263

MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,2,,3E9 ! Young's modulus of wood cane MPDATA,NUXY,2,,.32 MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,DENS,2,,700 ! Density of wood cane

! Material property of composite MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,3,,7E10 ! Young's modulus of composite MPDATA,NUXY,3,,.32 MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,DENS,3,,1350 ! Density of composite

! Material property of rubber MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,4,,6E6 ! Young's modulus of rubber MPDATA,NUXY,4,,.42 MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,DENS,4,,1113 ! Density of rubber

ESIZE,0.0098 MSHAPE,1,3D MSHKEY,0

! Select the composite part of the handle CM,_Y,VOLU VSEL,S,VOLU,,1,2 CM,_Y1,VOLU CHKMSH,'VOLU' CMSEL,S,_Y VATT,3,,1 !* VMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* ! Select the rubber part of the handle CM,_Y,VOLU VSEL,S,VOLU,,4,5 VSEL,A,VOLU,,6 CM,_Y1,VOLU CHKMSH,'VOLU' CMSEL,S,_Y VATT,4,,1 !* VMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* ! Select the wood part of the handle CM,_Y,VOLU VSEL,S,VOLU,,9 CM,_Y1,VOLU CHKMSH,'VOLU' Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 264

CMSEL,S,_Y VATT,2,,1 !* VMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* ! Select Blade Volume CM,_Y,VOLU VSEL,S,VOLU,,7 CM,_Y1,VOLU CHKMSH,'VOLU' CMSEL,S,_Y VATT,1,,1 !* VMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 FINI /SOLU ANTYPE,MODAL ! Modal analysis nmodes = 14 modopt,LANB,nmodes ! Block Lanczos solver ! Calculate element results and reaction forces mxpand,nmodes,,,yes ! Requests automatic generation of (structural) DOF TOTAL,14,1 SOLVE FINI ! /COM, **** REVIEW THE RESULTS OF THE MODAL SOLUTION **** /POST1 /Triad,off SET,LIST ! Lists all frequencies /WINDOW,1,LTOP ! Defines four windows /WINDOW,2,RTOP /WINDOW,3,LBOT /WINDOW,4,RBOT

/WINDOW,ALL,OFF ! All windows off /VUP,1,Z /VIEW,1,1,1,1 ! View for window 1 !/VIEW,1,0,-1 ! View for window 1 /WINDOW,1,ON ! Turns on window i SET,1,7 ! Reads in data for mode i

/TITLE,Modal Analysis of Cricket Bat !PLNSOL,U,SUM,0,1 PLNSOL,EPEL,EQV,0,1 ! Plot Von Mises strain /NOERASE ! No screen erase after plot /WINDOW,1,OFF ! Turns off window 1

/VUP,2,Z /VIEW,2,1,1,1 ! View for window 2 !/VIEW,2,0,-1 ! View for window 2 /WINDOW,2,ON ! Turns on window i SET,1,9 ! Reads in data for mode i !PLNSOL,U,SUM,0,1 PLNSOL,EPEL,EQV,0,1 ! Plot Von Mises strain /NOERASE ! No screen erase after plot Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 265

/WINDOW,2,OFF ! Turns off window 2

/VUP,3,Z /VIEW,3,1,1,1 ! View for window 3 !/VIEW,3,0,-1 ! View for window 3 /WINDOW,3,ON ! Turns on window i SET,1,11 ! Reads in data for mode 11 !PLNSOL,U,SUM,0,1 PLNSOL,EPEL,EQV,0,1 ! Plot Von Mises strain /NOERASE ! No screen erase after plot /WINDOW,3,OFF ! Turns off window 3

/VUP,4,Z /VIEW,4,1,1,1 ! View for window 4 !/VIEW,4,0,-1 ! View for window 4 /WINDOW,4,ON ! Turns on window i SET,1,14 ! Reads in data for mode 14 !PLNSOL,U,SUM,0,1 PLNSOL,EPEL,EQV,0,1 ! Plot Von Mises strain /NOERASE ! No screen erase after plot /WINDOW,4,OFF ! Turns off window 4 /ERASE FINISH

A7.5 MATLAB® code of parallel R-L piezoelectric vibration shunt control for Hinged-Hinged aluminium beam

% Parallel R-L shunt of Hinged-Hinged beam % Written by Jia Long Cao clear b=0.025; % Beam & PZT width L=0.5; % Beam length x1=0.22; % Distance of the left edge of PZT x2=0.28; % Distance of the right edge of PZT, (x2-x1) is the length of PZT xd=0.5*L; % Position of force applied Eb=70e9; % Young's modulus of beam Db=2700; % Density of beam hb=3e-3; % Thickness of beam Ep=70e9; % Young's modulus of PZT Dp=7600; % Density of PZT hp=0.25e-3; % thickness of PZT h31=7.65e8; % PZT constant Ab=b*hb; % Area of beam Ap=b*hp; % Area of PZT

Zb=hb/2; % Center location of beam Zp=hb+hp/2; % Center location of PZT Zn=(Ab*Eb*Zb+Ap*Ep*Zp)/(Ab*Eb+Ap*Ep); % Natural axis of composite beam Ib=b*hb^3/12+Ab*(Zn-Zb)^2; % Moment of inertia of beam Ip=b*hp^3/12+Ap*(Zn-Zp)^2; % Moment of inertia of PZT

N=3; % No. of frequencies

% Initialize matrix K=zeros(N); M=zeros(N); Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 266

C=zeros(N); BL=zeros(N,1); BR=zeros(N,1); Fd=zeros(N,1);

Cp=82e-9; % capacitance of PZT %L0=[400 4.788 1]; % shunt inductors %R0=[3e5 1e4 1]; % shunt resistors L0=[400 24.64 4.846]; % shunt inductors R0=[3.4e5 1e5 7e4]; % shunt resistors

% Coefficient of BL %BL=h31*(hb+0.5*hp)*(L0/R0)*(pi/L)*(cos(pi*x2/L)-cos(pi*x1/L)); % Coefficient of BR %BR=h31*(hb+0.5*hp)*(pi/L)*(cos(pi*x2/L)-cos(pi*x1/L)); % Coefficient of BL %BL=h31*(hp/(x2-x1))*(hb+0.5*hp)*(L0/R0)*(pi/L)*... %(cos(pi*x2/L)-cos(pi*x1/L)); % Coefficient of BR %BR=h31*(hp/(x2-x1))*(hb+0.5*hp)*(pi/L)*(cos(pi*x2/L)-cos(pi*x1/L)); CR=inv(diag(R0*Cp)); % Coefficient of CR CL=inv(diag(L0*Cp)); % Coefficient of CL for r=1:N; Q1(r)=-(h31*hp*(hb+0.5*hp)*(r*pi/L)*abs(cos(r*pi*x2/L)-... cos(r*pi*x1/L)))/((x2-x1)); for s=1:N; BL(s)=h31*(hp/(x2-x1))*(hb+0.5*hp)*(L0(s)/R0(s))*... (s*pi/L)*(cos(pi*x2/L)-cos(pi*x1/L)); % Coefficient of BL BR(s)=h31*(hp/(x2-x1))*(hb+0.5*hp)*(s*pi/L)*... (cos(pi*x2/L)-cos(pi*x1/L)); % Coefficient of BR if r == s K(r,s)=(pi*r/L)^4*(Eb*Ib*L/2+Ep*Ip*(x2-x1)/2+... Ep*Ip*L/(4*pi*r)*(sin(2*pi*r*x1/L)-sin(2*pi*r*x2/L))); M(r,s)=(Db*Ab*L/2+Dp*Ap*(x2-x1)/2+Dp*Ap*L/... (4*pi*r)*(sin(2*pi*r*x1/L)-sin(2*pi*r*x2/L))); else K(r,s)=Ep*Ip*L/pi*(pi^2*r*s/L^2)^2*((r*sin(s*pi*x2/L)... *cos(r*pi*x2/L))/(s^2-r^2)+(s*sin(r*pi*x2/L)*... cos(s*pi*x2/L))/(r^2-s^2)-((r*sin(s*pi*x1/L)*... cos(r*pi*x1/L))/(s^2-r^2)+(s*sin(r*pi*x1/L)*... cos(s*pi*x1/L))/(r^2-s^2))); M(r,s)=Dp*Ap*L/pi*((r*sin(s*pi*x2/L)*cos(r*pi*x2/L))/... (s^2-r^2)+(s*sin(r*pi*x2/L)*cos(s*pi*x2/L))/(r^2-s^2)-... ((r*sin(s*pi*x1/L)*cos(r*pi*x1/L))/(s^2-r^2)+... (s*sin(r*pi*x1/L)*cos(s*pi*x1/L))/(r^2-s^2))); end; end; %Fd(r)=1/100*sin(r*pi*xd/L); % due to discrete force with magnitude 1/100 end;

Fd = (1/100)*[1; 1; 1]; % applied force

Cd=0.2*M+1e-7*K; % Internal damping coefficient

Q=Q1*inv(diag(L0)); % Q1/L0

Q=diag(Q); % Make Q as a diagonal matrix BL=diag(BL); BR=diag(BR); % state-space model AK=-inv(M)*K; % -K/M AC=-inv(M)*Cd; % -C/M Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 267

% A matrix without shunt A1=[zeros(N) eye(N) zeros(N) zeros(N); AK AC zeros(N) zeros(N); ... zeros(N) zeros(N) zeros(N) eye(N); -Q zeros(N) -CL -CR];

% A matrix with shunt A2=[zeros(N) eye(N) zeros(N) zeros(N); AK AC -inv(M)*BR -inv(M)*BL; ... zeros(N) zeros(N) zeros(N) eye(N); -Q zeros(N) -CL -CR]; B=[zeros(N,1); inv(M)*Fd; zeros(N,1); zeros(N,1)]; for r=1:N; C1(1,r)=sin(r*pi/4); % displacement w at midpoint (x=L/2) end;

C=[C1 zeros(1,3*N)]; D=[0];

% impulse response t=linspace(0,1,1e5); IU=1; [y,x,t]=impulse(A1,B,C,D,IU,t); % uncontrolled response [yc,x,t]=impulse(A2,B,C,D,IU,t); % controlled response figure(1); plot(t,y,'b--',t,yc,'r') % unit (mm) legend('Without PZT shunt','With PZT shunt'); title('Impulse response of beam transverse displacement at xm=L/4', 'Fontsize', 18, 'Color', 'r') xlabel('Time (sec)', 'Fontsize', 16, 'Color', 'r') ylabel('Displacement (m)', 'Fontsize', 16, 'Color', 'r') axis([0 1 -1e-3 1e-3]) grid; zoom on; figure(2); plot(20*log10(abs(fft(y))),'b--'); hold on; plot(20*log10(abs(fft(yc))), 'r'); legend('Without PZT shunt','With PZT shunt'); title('Frequency response of beam transverse displacement at xm=L/4', 'Fontsize', 20, 'Color', 'r'); xlabel('Frequency (Hz)', 'Fontsize', 16, 'Color', 'r'); ylabel('Magnitude (dB)', 'Fontsize', 16, 'Color', 'r'); axis([0 1000 -80 40]); grid; zoom on; hold off;

Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 268 A7.6 MATLAB® code for analysing neutral axis, moment inertia, strains, stresses, strain energies etc. illustrated in Chapter 4

% Written by Jia Long Cao clear; M = 100; % Bending moment Ew = 7E9; % Young's modulus of wood L = 0.5; b = 0.025; % Width of the composite beam hw = 0.003; % Wood beam thickness hp(1) = 0.1*hw; % PZT thickness hp(2) = 0.2*hw; % PZT thickness hp(3) = 0.5*hw; % PZT thickness hp(4) = hw; % PZT thickness

N = 10000; % Number of samples dN = 20/N; % Sampling resolution for i = 1:4 u(i) = hp(i)/hw; % Thickness ratio of PZT and wood beam for j = 1:N n = j*dN; % n = Ep/Ew, n is from dN to N*dN % Neutral axis of composite beam, the distance from top edge of PZT % (assume that z (z = 0) starts at top edge of PZT and downwords) Zn(i,j) = (hw/2)*((u(i)^2*n+2*u(i)+1)/(1+u(i)*n)); Zp(i) = hp(i)/2; Zb(i) = hp(i)+hw/2; Zn_Minus_Zp(i,j) = (hw/2)*((u(i)^2*n+2*u(i)+1)/(1+u(i)*n))-Zp(i);

% Moment of inertia of wood part of the composite beam %Iw(i,j) = (b*hw^3/12)*(1+3*u(i)^2*n^2*((1+u(i))/(1+u(i)*n))^2); %Iw(i,j) = b*hw^3/12 + b*hw*(Zn(i,j)-Zb(i))^2; Iw(i,j)=(b*hw^3/12)*(((3*u(i)^4+6*u(i)^3+4*u(i)^2)*... n^2+2*u(i)*n+1))/(u(i)^2*n^2+2*u(i)*n+1);

% Moment of inertia of PZT part of the composite beam %Ip(i,j) = (b*hw^3/12)*(u(i)^3+3*u(i)*((1+u(i))/(1+u(i)*n))^2); %Ip(i,j) = b*hp(i)^3/12 + b*hp(i)*(Zn(i,j)-Zp(i))^2; Ip(i,j)=(b*hw^3/12)*((u(i)^5*n^2+2*u(i)^4*n+4*u(i)^3+... 6*u(i)^2+3*u(i)))/(u(i)^2*n^2+2*u(i)*n+1);

% Overall moment of inertia %I(i,j) = Iw(i,j)+n*Ip(i,j); I(i,j)=(b*hw^3/12)*((u(i)^5*n^3+(5*u(i)^4+6*u(i)^3+... 4*u(i)^2)*n^2+(4*u(i)^3+6*u(i)^2+5*u(i))*n+1))/... (u(i)^2*n^2+2*u(i)*n+1);

Iw_over_Ip(i,j) = Iw(i,j)/Ip(i,j); % Ib/Ip Ip_over_Iw(i,j) = Ip(i,j)/Iw(i,j); % Ip/Ib Iw_over_I(i,j) = Iw(i,j)/I(i,j); % Ib/I

% Average strain of PZT PZT_strain(i,j) = (6*M/(b*hw^2*Ew))*(1+u(i))/... ((u(i)^4*n^2+4*u(i)^3*n+6*u(i)^2*n+4*u(i)*n+1));

% Average strain of beam beam_strain(i,j) = (-6*M/(b*hw^2*Ew))*(u(i)^2+u(i))*n/... ((u(i)^4*n^2+4*u(i)^3*n+6*u(i)^2*n+4*u(i)*n+1)); %beam_strain(i,j) = M*(Zn(i,j)-Zb(i))/(Ew*I(i,j));

Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 269

% Average stress of PZT PZT_stress(i,j) = (6*M/(b*hw^2))*(1+u(i))*n/... ((u(i)^4*n^2+4*u(i)^3*n+6*u(i)^2*n+4*u(i)*n+1));

% Average stress of beam beam_stress(i,j) = (-6*M/(b*hw^2))*(u(i)^2+u(i))*n/... ((u(i)^4*n^2+4*u(i)^3*n+6*u(i)^2*n+4*u(i)*n+1)); %beam_strain(i,j) = M*(Zn(i,j)-Zb(i))/(Ew*I(i,j));

EI(i,j) = Ew*I(i,j); % EI = Ew*Iw + Ep*Ip % Maximum strain of PZT (the upper-most edge) %Sp_max(i,j) = (-M*n*Ew)*(0-Zn(i,j))/EI(i,j); % Maximum strain of PZT (the upper-most edge) Sp_max(i,j) = -n*M*(0-Zn(i,j))/I(i,j); %Sp_max(i,j) = (6*M/(b*hw^2))*(u(i)^2*n+2*u(i)+1)*n/... %(u(i)^4*n^2+(4*u(i)^3+6*u(i)^2+4*u(i))*n+1); % Maximum strain of wood (the lower-most edge %Sw_max(i,j) = (M*Ew)*(hp(i)+hw-Zn(i,j))/EI(i,j); % Maximum strain of wood (the lower-most edge Sw_max(i,j) = M*(hp(i)+hw-Zn(i,j))/I(i,j); %Sw_max(i,j) = (6*M/(b*hw^2))*(u(i)^2*n+2*u(i)*n+1)/... %(u(i)^4*n^2+(4*u(i)^3+6*u(i)^2+4*u(i))*n+1); % Strain of PZT at juncture %Sp_int(i,j) = (-M*n*Ew)*(hp(i)-Zn(i,j))/EI(i,j); Sw_int(i,j) = M*(hp(i)-Zn(i,j))/I(i,j); % Strain of PZT at juncture %Sw_int(i,j) = Sp_int(i,j)/n; % Strain of wood at juncture Sp_int(i,j) = -n*Sw_int(i,j); % Strain of wood at juncture % The ratio of strain energy in PZT over total strain % energy of the composite beam %Ub(i,j) = (M^2*L/(2*Ew))*Iw(i,j)./I(i,j)^2; Ub(i,j)=(6*M^2*L/(Ew*b*hw^3))*((3*u(i)^4+6*u(i)^3+... 4*u(i)^2)*n^2+2*u(i)*n+1)*(u(i)^2*n^2+2*u(i)*n+1) ... /(u(i)^5*n^3+(5*u(i)^4+6*u(i)^3+4*u(i)^2)*n^2+... (4*u(i)^3+6*u(i)^2+5*u(i))*n+1)^2; %Up(i,j) = (M^2*L/(2*Ew))*n*Ip(i,j)./I(i,j)^2; Up(i,j)=(6*M^2*L/(Ew*b*hw^3))*(u(i)^5*n^3+2*u(i)^4*n^2+... (4*u(i)^3+6*u(i)^2+3*u(i))*n)*(u(i)^2*n^2+2*u(i)*n+1) ... /(u(i)^5*n^3+(5*u(i)^4+6*u(i)^3+4*u(i)^2)*n^2+(4*u(i)^3+... 6*u(i)^2+5*u(i))*n+1)^2; U(i,j) = Ub(i,j) + Up(i,j); Up_over_Ub(i,j) = Up(i,j)./Ub(i,j); Up_over_U(i,j) = Up(i,j)./U(i,j); Strain_ratio(i,j) = -PZT_strain(i,j)./beam_strain(i,j); end end

% n is the last value of looping or the maximum value of Ep/Ew w = linspace(0, n, N); figure(1); plot(w, Zn(1,:), w, Zn(2,:), 'r', w, Zn(3,:), 'g', w, Zn(4,:), 'm'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Distance from x-y plane (m)', 'Fontsize', 16, 'Color', 'r'); title('The Neutral Axis of Composite Beam', 'Fontsize', 20, 'Color', 'r'); grid; text(81, 34E-4, 'Ep -- Modulus of PZT', 'Fontsize', 10, 'Color', 'r'); text(81, 33E-4, 'Eb -- Modulus of Beam', 'Fontsize', 10, 'Color', 'r'); text(81, 32E-4, 'hp -- Thickness of PZT', 'Fontsize', 10, 'Color', 'r'); text(81, 31E-4, 'hb -- Thickness of Beam', 'Fontsize', 10, 'Color', 'r'); text(81, 29E-4, 'hb = 0.003 (m)', 'Fontsize', 10, 'Color', 'r'); zoom on;

Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 270 figure(2); plot(w, Zn_Minus_Zp(1,:), w, Zn_Minus_Zp(2,:), 'r', ... w, Zn_Minus_Zp(3,:), 'g', w, Zn_Minus_Zp(4,:), 'm'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Distance (m)', 'Fontsize', 16, 'Color', 'r'); title('Distance between Neutral Axis and Centeroid of PZT',... 'Fontsize', 20, 'Color', 'r'); grid; text(81, 24E-4, 'Ep -- Modulus of PZT', 'Fontsize', 10, 'Color', 'r'); text(81, 23E-4, 'Eb -- Modulus of Beam', 'Fontsize', 10, 'Color', 'r'); text(81, 22E-4, 'hp -- Thickness of PZT', 'Fontsize', 10, 'Color', 'r'); text(81, 21E-4, 'hb -- Thickness of Beam', 'Fontsize', 10, 'Color', 'r'); text(81, 19E-4, 'hb = 0.003 (m)', 'Fontsize', 10, 'Color', 'r'); zoom on; figure(3); plot(w, Iw(1,:), w, Iw(2,:), 'r', w, Iw(3,:), 'g', w, Iw(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Ib (m^4)', 'Fontsize', 16, 'Color', 'r'); title('Area Moment of Inertia of Beam about Neutral Surface',... 'Fontsize', 20, 'Color', 'r'); grid; zoom on; figure(4); plot(w, Ip(1,:), w, Ip(2,:), 'r', w, Ip(3,:), 'g', w, Ip(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Ip (m^4)', 'Fontsize', 16, 'Color', 'r'); title('Area Moment of Inertia of PZT about Neutral Surface',... 'Fontsize', 20, 'Color', 'r'); grid; zoom on; figure(5); plot(w, I(1,:), w, I(2,:), 'r', w, I(3,:), 'g', w, I(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('It (m^4)', 'Fontsize', 16, 'Color', 'r'); title('Total Area Moment of Inertia of Composite Beam about Neutral Surface',... 'Fontsize', 17, 'Color', 'r'); grid; zoom on; figure(6); plot(w, PZT_strain(1,:), w, PZT_strain(2,:), 'r',... w, PZT_strain(3,:), 'g', w, PZT_strain(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Strain', 'Fontsize', 16, 'Color', 'r'); title('The Average Strain of PZT (on the middle surface of PZT)',... 'Fontsize', 20, 'Color', 'r'); grid; zoom on; figure(7); plot(w, beam_strain(1,:), w, beam_strain(2,:), 'r',... w, beam_strain(3,:), 'g', w, beam_strain(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Strain', 'Fontsize', 16, 'Color', 'r'); Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 271 title('The Average Strain of Beam (on the middle surface of beam)',... 'Fontsize', 20, 'Color', 'r'); grid; zoom on; figure(8); plot(w, PZT_stress(1,:), w, PZT_stress(2,:), 'r', w,... PZT_stress(3,:), 'g', w, PZT_stress(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Stress (10^9 N/m^2)', 'Fontsize', 16, 'Color', 'r'); title('The Average Stress of PZT (on the middle surface of PZT)',... 'Fontsize', 20, 'Color', 'r'); grid; zoom on; figure(9); plot(w, beam_stress(1,:), w, beam_stress(2,:), 'r',... w, beam_stress(3,:), 'g', w, beam_stress(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Stress (10^8 N/m^2)', 'Fontsize', 16, 'Color', 'r'); title('The Average Stress of Beam (on the middle surface of beam)',... 'Fontsize', 20, 'Color', 'r'); grid; zoom on;

figure(10); %plot(w, Ip_over_Iw(1,:), w, Ip_over_Iw(2,:), 'r',... %w, Ip_over_Iw(3,:), 'g', w, Ip_over_Iw(4,:),'m'); plot(w, Ub(1,:), w, Ub(2,:), 'r', w, Ub(3,:), 'g', w, Ub(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Strain Energy (Newton-meter)', 'Fontsize', 16, 'Color', 'r'); title('Strain Energy of Beam', 'Fontsize', 20, 'Color', 'r'); grid; zoom on; figure(11); %plot(w, Sw_max(1,:), w, Sw_max(2,:), 'r'); %plot(w, Sw_max(1,:), w, Sw_max(2,:), 'r', ... %w, Sw_max(3,:), 'g', w, Sw_max(4,:),'m'); plot(w, Up(1,:), w, Up(2,:), 'r', w, Up(3,:), 'g', w, Up(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Strain Energy (Newton-meter)', 'Fontsize', 16, 'Color', 'r'); title('Strain Energy of PZT', 'Fontsize', 20, 'Color', 'r'); grid; zoom on; figure(12); %plot(w, Sw_max(1,:), w, Sw_max(2,:), 'r'); %plot(w, Sw_max(1,:), w, Sw_max(2,:), 'r', ... %w, Sw_max(3,:), 'g', w, Sw_max(4,:),'m'); plot(w, U(1,:), w, U(2,:), 'r', w, U(3,:), 'g', w, U(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Strain Energy (Newton-meter)', 'Fontsize', 16, 'Color', 'r'); title('Total Strain Energy of Composite Beam', ... 'Fontsize', 20, 'Color', 'r'); grid; zoom on;

Appendix 7 Partial List of ANSYS® and MATLAB® Simulation Codes 272 figure(13); plot(w, Up_over_Ub(1,:), w, Up_over_Ub(2,:), 'r', ... w, Up_over_Ub(3,:), 'g', w, Up_over_Ub(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Up/Ub', 'Fontsize', 16, 'Color', 'r'); title('The Ratio of Strain Energy of PZT over Strain Energy in Beam',... 'Fontsize', 20, 'Color', 'r'); grid; zoom on; figure(14); plot(w, 100*Up_over_U(1,:), w, 100*Up_over_U(2,:), 'r',... w, 100*Up_over_U(3,:), 'g', w, 100*Up_over_U(4,:),'m'); legend('hp/hb = 0.1', 'hp/hb = 0.2', 'hp/hb = 0.5', 'hp/hb = 1', 1); xlabel('Ratio of Youngs Modulus (Ep/Eb)', 'Fontsize', 16, 'Color', 'r'); ylabel('Up/U (Percentage)', 'Fontsize', 16, 'Color', 'r'); title('The Ratio of Strain Energy of PZT over Total Strain Energy',... 'Fontsize', 20, 'Color', 'r'); axis([0 n 0 100]); grid; zoom on; References 273

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[I 1] ACI Technology, “Active Fibre Composite Materials”, http://www.advancedcerametrics.com/acitechnology.htm (2006).

[I 2] Akhras, Georges, “Smart materials and smart systems for the future”, http://www.journal.forces.gc.ca/engraph/Vol1/no3/pdf/25-32_e.pdf (2006).

[I 3] ANSYS 6.1 Documentation, http://www.oulu.fi/atkk/tkpalv/unix/ansys-6.1/content/Hlp_E_SOLID95.html (2006).

[I 4] APC International Ltd, “Piezo theory”, http://www.americanpiezo.com/piezo_theory/index.html (2006).

[I 5] Head_a, Tennis racquet technology, http://www.head.com/tennis/technology.php?region=eu (2006).

[I 6] Head_b, Head Ski & Boots technology, http://www.head.com/ski/technology.php?region=eu (2006).

[I 7] Head_c, Head Intelligent snowboards technology, http://www.ridehead.com/main?UID=intelligence&lang=en (2006).

[I 8] MCC, Law of cricket, http://www.abcofcricket.com/bet/bet.htm (2006).

[I 9] Noble, Larry, “Inertial and vibrational characteristics of and baseball bats: research and design implications”, http://combat.atomicmotion.com/uploads/pdfs/bat%20physics%20- %20larry%20noble.pdf (2006).

[I 10] PI Ceramic “Tutorial”, http://www.piceramic.com/pdf/PIC_Tutorial.pdf (2006).

[I 11] Russell, Dan, “Vibrational Modes of a Baseball Bat”, http://www.kettering.edu/~drussell/Demos/batvibes.html (2006). References 280 [I 12] Space Age Control, “Sinusoidal motion calculator”, http://www.spaceagecontrol.com/calcsinm.htm (2006).

[I 13] Uchino, Kenji and Carazo, A. V., “High Power Piezoelectric Transformers – Their Applications to Smart Actuator Systems”, http://www.mecheng.sun.ac.za/aboutus/damping/Uchino.pdf (2006).

[I 14] University of Alberta, “ANSYS Tutorials”, http://www.mece.ualberta.ca/tutorials/ansys/index.html (2006).