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Voice Modems Are Used to Transmit Signals Through the Internet Vibration Control in Cricket 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 Melbourne, Australia. 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 on 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 Cricket bat ……………..…………….…....….…...………. 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
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