Effect of Sliding Friction on Spur and Helical Gear Dynamics and Vibro-Acoustics

Home , DFT matrix, Involute, Involute gear, Sliding (motion)

EFFECT OF SLIDING FRICTION ON SPUR AND HELICAL GEAR DYNAMICS AND VIBRO-ACOUSTICS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate School

of The Ohio State University

Song He, B.S., M.S.

* * * * *

The Ohio State University

2008

Dissertation Committee: Approved by Dr. Rajendra Singh, Advisor

Dr. Ahmet Kahraman ______

Dr. Ahmet Selamet Adviser

Dr. Marcelo Dapino Graduate Program in Mechanical Engineering

ABSTRACT

This study examines the salient effects of sliding friction on spur and helical gear dynamics and associated vibro-acoustic sources. First, new dynamic formulations are developed for spur and helical gear pairs based on a periodic description of the contact point and realistic mesh stiffness. Difficulty encountered in the existing discontinuous models is overcome by characterizing a smoother transition during the contact. Frictional forces and moments now appear as either excitations or periodically-varying parameters, since the frictional force changes direction at the pitch point/line. These result in a class of periodic ordinary differential equations with multiple and interacting coefficients, which characterize the effect of sliding friction in spur or helical gear dynamics.

Predictions (based on multi-degree-of-freedom analytical models) match well with a benchmark finite element/contact mechanics code and/or experimental results.

Second, new analytical solutions are constructed which provide an efficient evaluation of the frictional effect as well as a more plausible explanation of dynamic interactions in multiple directions. Both single- and multi-term harmonic balance methods are utilized to predict dynamic mesh loads, friction forces and pinion/gear displacements. Such semi-analytical solutions explain the presence of higher harmonics in gear noise and vibration due to exponential modulations of the periodic stiffness,

ii dynamic transmission error and sliding friction. This knowledge also analytically reveals the effect of the tooth profile modification in spur gears on the dynamic transmission error, under the influence of sliding friction. Further, the Floquet theory is applied to obtain closed-form solutions of the dynamic response for a helical gear pair, where the effect of sliding friction is quantified by an effective piecewise stiffness function.

Analytical predictions, under both homogeneous and forced conditions, are validated using numerical simulations. The matrix-based methodology is found to be computationally efficient while leading to a better assessment of the dynamic stability.

Third, an improved source-path-receiver vibro-acoustic model is developed to quantify the effect of sliding friction on structure-borne noise. Interfacial bearing forces are predicted for the spur gear source sub-system given two gear whine excitations (static transmission error and sliding friction). Next, a computational model of the gearbox, with embedded bearing stiffness matrices, is developed to characterize the motilities of structural paths. Radiated sound pressure is then estimated by using two numerical techniques (the Rayleigh integral method and a substitute source technique). Predicted pressures match well with measured noise data over a range of operating torques. In particular, the proposed vibro-acoustic model quantifies the contribution of sliding friction, which could be significant when the transmission error is minimized through tooth modifications.

iii

DEDICATION

Dedicated to my parents and wife

iv ACKNOWLEDGMENT

I would like to express my sincere appreciation to my advisor, Professor Rajendra

Singh, for his time, guidance, and support over the years both in my academic research and personal life. His intellectual insight and encouragement had a huge impact on my

professional growth. I would also like to thank Professor Ahmet Kahraman, Professor

Ahmet Selamet and Professor Marcelo Dapino for their services on the doctoral committee and for providing constructive suggestions.

I sincerely thank Dr. Todd Rook for providing valuable suggestions. I greatly

appreciate Dr. Rajendra Gunda for granting access to the Calyx software and for offering

insightful comments. I gracefully acknowledge the experimental work conducted by

Vivake Asnani and Fred Oswald, as well as the collaboration with Allison Lake. Dr.

Chengwu Duan is thanked for helping me in both my research and personal life. All colleagues in the Acoustics and Dynamic Laboratory are acknowledged for their encouragement and friendship. I thank Professor Goran Pavić, Professor Jean-Louis

Guyader and Corinne Lotto for their advice and kind help during my stay in INSA Lyon.

The financial support from the Army Research Office, EU’s Marie Curie

Fellowship and OSU Presidential Fellowship is gracefully appreciated.

Finally, I would like to thank my parents and my wife, Lihua, for their love and encouragement throughout my pursuit for the doctoral degree.

v VITA

October 24, 1979………………………..…..….Born – Jiangsu, China

2002………………………..………………..….B.S. Instrumentation Engineering Shanghai Jiao Tong University Shanghai, China

2004………………………..………………..….M.S. Mechanical Engineering The Ohio State University

2004 - 2006………………..………………..….Graduate Teaching and Research Associate, Mechanical Engineering The Ohio State University

2007………………………..………………..….Marie Curie Fellow (EU) National Institute of Applied Science Lyon, France

2007 - 2008………………………..……………Presidential Fellow (Graduate School) The Ohio State University

PUBLICATIONS

Research Publications

1. He, S., Gunda, R., and Singh, R., 2007, “Effect of Sliding Friction on the Dynamics of Spur Gear Pair with Realistic Time-Varying Stiffness,” Journal of Sound and Vibration, 301, pp. 927-949.

2. He, S., Gunda, R., and Singh, R., 2007, “Inclusion of Sliding Friction in Contact Dynamics Model for Helical Gears, ASME Journal of Mechanical Design, 129(1), pp. 48-57.

vi 3. He, S., Cho, S., and Singh, R., 2008, “Prediction of Dynamic Friction Forces in Spur Gears using Alternate Sliding Friction Formulations,” Journal of Sound and Vibration, 309(3-5), pp. 843-851.

FIELDS OF STUDY

Major Field: Mechanical Engineering Dynamics of Mechanical Systems Vibro-Acoustics and Noise Control

vii TABLE OF CONTENTS

Page Abstract...... ii Dedication...... iv Acknowledgment ...... v Vita...... vi

List of Tables ...... xii List of Figures...... xiii List of Symbols...... xx

Chapter 1 Introduction ...... 1 1.1 Motivation...... 1 1.2 Literature Review...... 4 1.3 Problem Formulation ...... 8 1.3.1 Key Research Issues ...... 8 1.3.2 Scope, Assumptions and Objectives...... 11 References for Chapter 1 ...... 14

Chatper 2 Spur Gear Dynamics with Sliding Friction and Realistic Mesh Stiffness ...... 18 2.1 Introduction...... 18 2.2 Problem Formulation ...... 20 2.2.1 Objectives and Assumptions...... 20 2.2.2 Timing of Key Meshing Events...... 23 2.2.3 Calculation of Realistic Time-Varying Tooth Stiffness Functions...... 24 2.3 Analytical Multi-Degree-of-Freedom Dynamic Model...... 27 2.3.1 Shaft and Bearing Stiffness Models...... 27 2.3.2 Dynamic Mesh and Friction Forces...... 28 2.3.3 MDOF Model...... 32 2.4 Analytical SDOF Torsional Model...... 33 viii 2.5 Effect of Sliding Friction in Example I...... 35 2.5.1 Validation of Example I Model using the FE/CM Code ...... 35 2.5.2 Effect of Sliding Friction ...... 40 2.5.3 MDOF System Resonances ...... 42 2.6 Effect of Sliding Friction in Example II ...... 43 2.6.1 Empirical Coefficient of Friction...... 43 2.6.2 Effect of Tip Relief on STE and k(t)...... 45 2.6.3 Phase Relationship between Normal Load and Friction Force Excitations... 50 2.6.4 Prediction of the Dynamic Responses ...... 53 2.7 Experimental Validation of Example II Models...... 56 2.8 Conclusion ...... 65

Chapter 3 Prediction of Dynamic Friction Forces Using Alternate Formulations ...... 68 3.1 Introduction...... 68 3.2 MDOF Spur Gear Model ...... 69 3.3 Spur Gear Model with Alternate Sliding Friction Models ...... 74 3.3.1 Model I: Coulomb Model ...... 74 3.3.2 Model II: Benedict and Kelley Model ...... 75 3.3.3 Model III: Formulation Suggested by Xu et al...... 76 3.3.4 Model IV: Smoothened Coulomb Model ...... 78 3.3.5 Model V: Composite Friction Model...... 78 3.4 Comparison of Sliding Friction Models ...... 80 3.5 Validation and Conclusion...... 85 References for Chapter 3 ...... 89

Chapter 4 Construction of Semi-Analytical Solutions to Spur Gear Dynamics...... 91 4.1 Introduction...... 91 4.2 Problem Formulation ...... 96 4.3 Semi-Analytical Solutions to the SDOF Spur Gear Dynamic Formulation ...... 97 4.3.1 Direct Application of Multi-Term Harmonic Balance (MHBM) ...... 99 ix 4.3.2 Semi-Analytical Solutions Based on One-Term HBM...... 100 4.3.3 Iterative MHBM Algorithm...... 103 4.4 Analysis of Sub-Harmonic Response and Dynamic Instability...... 107 4.5 Semi-Analytical Solutions to 6DOF Spur Gear Dynamic Formulation ...... 110 4.6 Conclusion ...... 116 References for Chapter 4 ...... 122

Chapter 5 Effect of Sliding Friction on the Vibro-Acoustics of Spur Gear System...... 124 5.1 Introduction...... 124 5.2 Source Sub-System Model...... 126 5.3 Structural Path with Friction Contribution ...... 130 5.3.1 Bearing and Housing Models...... 130 5.3.2 Experimental Studies and Validation of Structural Model...... 132 5.3.3 Comparison of Structural Paths in LOA and OLOA Directions ...... 138 5.4 Prediction of Noise Radiation and Contribution of Friction...... 140 5.4.1 Prediction using Rayleigh Integral Technique...... 140 5.4.2 Prediction using Substitute Source Method...... 141 5.4.3 Prediction vs. Noise Measurements...... 144 5.5 Conclusion ...... 148 References for Chapter 5 ...... 150

Chapter 6 Inclusion of Sliding Friction in Helical Gear Dynamics...... 152 6.1 Introduction...... 152 6.2 Problem Formulation ...... 153 6.3 Mesh Forces and Moments with Sliding Friction...... 155 6.4 Shaft and Bearing Models...... 169 6.5 Twelve DOF Helical Gear Pair Model ...... 173 6.6 Role of Sliding Friction Illustrated by an Example ...... 176 6.7 Conclusion ...... 183

x References for Chapter 6 ...... 184

Chapter 7 Analysis of Helical Gear Dynamics using Floquet Theory...... 186 7.1 Introduction...... 186 7.2 Linear Time-Varying Formulation ...... 187 7.3 Analytical Solutions by Floquet Theory...... 199 7.3.1 Response to Initial Conditions...... 201 7.3.2 Forced Periodic Response...... 206 7.4 Conclusion ...... 214 References for Chapter 7 ...... 216

Chapter 8 Conclusion...... 218 8.1 Summary...... 218 8.2 Contributions...... 221 8.3 Future Work...... 223 References for Chapter 8 ...... 224

Bibliography ...... 225

xi LIST OF TABLES

Table Page

2.1 Parameters of Example I: NASA-ART spur gear pair (non-unity ratio)...... 21

2.2 Parameters of Example II-A and II-B: NASA spur gear pair (unity ratio). Gear pair with the perfect involute profile is designated as II-A case and the one with tip relief is designated as II-B case ...... 22

2.3 Averaged coefficient of friction predicted over a range of operating conditions for Example II by using Benedict and Kelly’s empirical equation [2.14]...... 44

5.1 Parameters of the example case: NASA spur gear pair with unity ratio (with long tip relief)...... 128

5.2 Comparison of measured natural frequencies and finite element predictions .... 133

7.1 Relationship between Contact Zones and Contact Regions for the NASA-ART helical gear pair...... 197

xii LIST OF FIGURES

Figure Page

1.1 (a) Schematic of the spur gear contact, where LOA and OLOA represent the line- of-action direction and off line-of-action direction, respectively. (b) Directions of the sliding velocity (V), normal mesh load and friction force in spur gears...... 3

1.2 Comparison of periodic mesh stiffness functions for a spur gear pair ...... 5

1.3 MDOF helical gear dynamic model (proposed in Chapter 6) and its contact mechanics with sliding friction...... 10

1.4 Block diagram for the vibro-acoustics of a simplified geared system with two excitations at the gear mesh (as proposed in Chapter 5)...... 14

2.1 Snap shot of contact pattern (at t = 0) in the spur gear pair of Example I...... 24

2.2 Tooth mesh stiffness functions of Example I calculated by using the FE/CM code (in the “static” mode). (a) Individual and combined stiffness functions. (b) Comparison of the combined stiffness functions...... 26

2.3 Schematic of the bearing-shaft model...... 28

2.4 Normal and friction forces of analytical (MDOF) spur gear system model...... 31

2.5 Validation of the analytical (MDOF) model by using the FE/CM code (in the “dynamic” mode). Here, results for Example I are given in terms of δ ()t and its

spectral contents ∆()f with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = 0.2...... 36

2.6 Validation of the analytical (MDOF) model by using the FE/CM code (in the

“dynamic” mode). Here, results for Example I are given in terms of FtpBx () and

its spectral contents FfpBx () with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = 0.2...... 37

xiii 2.7 Validation of the analytical (MDOF) model by using the FE/CM code (in the

“dynamic” mode). Here, results for Example I are given in terms of FtpBy ( ) and

its spectral contents FfpBy () with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = 0.2 ...... 39

2.8 Effect of µ on δ ()t based on the linear time-varying SDOF model for Example I

at Tp = 2000 lb-in. Here, tc = 1 s ...... 41

2.9 Coefficient of friction µ as a function of the roll angle for Example II, as predicted by using Benedict & Kelley’s empirical equation [2.14]. Here, oil temperature is 104 deg F and T = 500 lb-in...... 45

2.10 Mesh harmonics of the static transmission error (STE) calculated by using the FE/CM code (in the “static” mode) for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B) ...... 46

2.11 Tooth stiffness functions of a single mesh tooth pair for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B)...... 48

2.12 Combined tooth stiffness functions for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B) ...... 49

2.13 Dynamic loads predicted for Example II at 500 lb-in, 4875 RPM and 140 °F with tc = 0.44 ms: (a) Normal loads of gear pair with perfect involute profile (II-A); (b) normal loads of gear par with tip relief (II-B); (c) friction forces of gear pair with perfect involute profile (II-A); (d) friction forces of gear pair with tip relief (II-B)...... 51

2.14 Dynamic shaft displacements predicted for Example II at 500 lb-in, 4875 RPM

and 140 °F with tc = 0.44 ms and fm = 2275 Hz: (a)xp (t ) ; (b) X p (f ) ; (c)ytp ( )

and (d) Yfp ( ) ...... 54

2.15 Dynamic transmission error predicted for Example II at 500 lb-in, 4875 RPM and 140 °F with tc = 0.44 ms and fm = 2275 Hz: (a)δ (t ) ; (b) ∆ (f ) ...... 55

2.16 Sensors inside the NASA gearbox (for Example II-B)...... 57

2.17 Mesh harmonic amplitudes of X p as a function of the mean torque at 140 °F. All

values are normalized with respect to the amplitude of Yp at the first mesh harmonic ...... 59 xiv 2.18 Mesh harmonic amplitudes of yp as a function of the mean torque at 140 °F for

Example II-B. All values are normalized with respect to the amplitude of yp at the first mesh harmonic...... 60

2.19 Predicted dynamic transmission errors (DTE) for Example II over a range of torque at 140 °F: (a) gear pair with perfect involute profile (II-A); (b) with tip relief (II-B). All values are normalized with respect to the amplitude ofδ (II-A) at the first mesh harmonic with 100 lb-in...... 62

2.20 Mesh harmonic amplitudes of xp as a function of temperature at 500 lb-in for

Example II-B. All values are normalized with respect to the amplitude of yp at the first mesh harmonic...... 63

2.21 Mesh harmonic amplitudes of yp as a function of temperature at 500 lb-in for

Example II-B. All values are normalized with respect to the amplitude of yp at the first mesh harmonic...... 64

3.1 (a) Snap shot of contact pattern (at t = 0) in the spur gear pair; (b) MDOF spur gear pair system; here kt ( ) is in the LOA direction...... 71

3.2 (a) Comparison of Model II [3.7] and Model III [3.8] given Tp = 22.6 N-m (200

lb-in) and Ω p = 1000 RPM. (b) Averaged magnitude of the coefficient of friction

predicted as a function of speed using the composite Model V with Tp = 22.6 N-

m (200 lb-in). Here, tc is one mesh cycle...... 81

3.3 Comparison of normalized friction models. Note that curve between 0≤ tt /c < 1

is for pair # 1; and the curve between 1≤ tt /c < 2 is for pair # 0...... 83

3.4 Combined normal load and friction force time histories as predicted using

alternate friction models given Tp = 56.5 N-m (500 lb-in) andΩ p = 4875 RPM...... 84

3.5 Predicted LOA and OLOA displacements using alternate friction models given

Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM ...... 86

3.6 Predicted dynamic transmission error (DTE) using alternate friction models given

Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM ...... 87

xv 3.7 Validation of the normal load and sliding friction force predictions: (a) at Tp =

79.1 N-m (700 lb-in) and Ω p = 800 RPM; (b) at Tp = 79.1 N-m (700 lb-in) and

Ω p = 4000 RPM...... 88

4.1 Realistic mesh stiffness functions of the spur gear pair example (with tip relief)

given Tp = 550 lb-in. (b) Periodic frictional functions...... 92

4.2 (a) Normal (mesh) and friction forces of 6DOF analytical spur gear system model. (b) Snap shot of contact pattern (at t = 0) for the example spur gear pair...... 93

4.3 Semi-analytical vs. numerical solutions for the SDOF model, expressed by Eq.

(4.3), given Tp = 550 lb-in, Ω p = 500 RPM, µ = 0.04. (a) Time domain responses; (b) mesh harmonics in frequency domain...... 102

4.4 Semi-analytical vs. numerical solutions for the SDOF model as a function of pinion speed with µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4...... 106

4.5 Normalized determinant of the sub-harmonic matrix Ku as a function of ωnNS / Ω

with µ = 0.04: (a) Tp = 100 lb-in.; (b) Tp = 550 lb-in...... 109

4.6 (a) 6DOF spur gear pair model and its subset of unity gear pair (3DOF model) used to study the natural frequency distribution; (b) Natural frequencies ΩN as a

function of the stiffness ratio KkBm / ...... 111

4.7 Semi-analytical vs. numerical solutions for the 6DOF model as a function of Ω p with µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4...... 113

4.8 Semi-analytical vs. numerical solutions of the LOA displacement xp for the 6DOF

model as a function of Ω p with KB/km = 100, µ = 0.04 (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4 ...... 117

4.9 Semi-analytical vs. numerical solutions of the OLOA displacement yp for the

6DOF model as a function of Ω p with KB/km = 100, µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4...... 118

xvi 4.10 Semi-analytical vs. numerical solutions of the LOA displacement xp for the 6DOF

model as a function of Ω p with KB/km = 0.37, µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4 ...... 120

4.11 Semi-analytical vs. numerical solutions of the OLOA displacement yp for the

6DOF model as a function of Ω p with KB/km = 0.37, µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4...... 121

5.1 Block diagram for the vibro-acoustics of a simplified geared system with two excitations at the gear mesh...... 125

5.2 Bearing forces predicted under varying Tp given Ωp = 4875 RPM and 140 °F. (a): LOA bearing force; (b) OLOA bearing force. Key: m is the mesh frequency index ...... 129

5.3 (a) Schematic of NASA gearbox; (b) Finite element model of NASA gearbox with embedded bearing stiffness matrices ...... 131

5.4 Comparison of the gearbox mode shape at 2962 Hz: (a) modal experiment result [5.12]; (b) finite element prediction...... 134

5.5 (a) Experiment used to measure structural transfer functions; (b) comparison of transfer function magnitudes from gear mesh to the sensor on top plate...... 137

5.6 Magnitudes of the combined transfer mobilities in two directions calculated at the sensor location on the top plate...... 139

5.7 Comparison of normal surface velocity magnitudes and substitute source strength vectors under Tp = 500 lb-in and Ωp = 4875 RPM. (a) Line 1: interpolated surface velocity on top plate; (b) Line 2: simplified 2D gearbox model with 15 substitute source points; (c) Line 3: substitute source strengths in complex plane for 2D gearbox. Column 1: mesh frequency index m = 1; Column 2: m = 2; Column 3: m = 3...... 145

5.8 Normalized sound pressure (ref 1.0 Pa) predicted at the microphone 6 inch above the top plate under varying torque Tp given Ωp = 4875 RPM and 140 °F ...... 147

5.9 Overall sound pressure levels (ref: 2e-5 Pa) and their contributions predicted at the microphone 6 in above the top plate under Ωp = 4875 RPM and 140 °F. (a) Tp = 500 lb-in (optimal load for minimum transmission error); (b) Tp = 800 lb-in. 149

xvii 6.1 Schematic of the helical gear pair system...... 156

6.2 Tooth stiffness function calculated using Eq. (6.1) based on the FE/CM code [6.11]...... 157

6.3 Contact zones at the beginning of a mesh cycle. (a) In the helical gear pair; (b) contact zones within contact plane. Key: PP’ is the pitch line; AA’ is the face width W; AD is the length of contact zone Z...... 166

6.4 Predicted tooth stiffness functions...... 169

6.5 Schematic of the bearing-shaft model. Here, the shaft and bearing stiffness elements are assumed to be in series to each other. Only pure rotational or translational stiffness elements are shown. Coupling stiffness terms K , K are xθ y yθx not shown...... 170

6.6 Time and frequency domain responses of translational pinion displacements

uuuxp, yp, zp at Tp = 2000 lb-in and Ω p = 1000 RPM. All displacements are normalized with respect to 39.37 µinch (1 µm)...... 178

6.7 Time and frequency domain responses of pinion bearing forces FSB, xp , FSB, yp and

FSB, zp at Tp = 2000 lb-in and Ω p =1000 RPM. All forces are normalized with respect to 1 lb...... 179

6.8 Time and frequency domain responses of composite displacements δ xyz ,δδ , and velocity δ z at Tp = 2000 lb-in and Ω p =1000 RPM. All motions are normalized with respect to 39.37µinch (1µm) or 39.37µinch (1µm/s) ...... 181

7.1 Schematic of the helical gear pair system...... 189

7.2 Contact zones at the beginning of a mesh cycle within the contact plane. Key: PP’ is the pitch line; AA’ is the face width W; AD is the length of contact zone Z.. 191

7.3 Individual effective stiffness Ke,i(t) along the contact zone, where Tmesh is one mesh cycle...... 196

7.4 Piece-wise effective stiffness function defined in six regions within one mesh cycle with µ = 0.4 ...... 198

xviii 7.5 (a) Effective stiffness and (b) homogeneous responses predictions within two -6 mesh cycles given x0 = 2×10 in., v0 = 20 in./s at Ωp = 1000 RPM ...... 204

7.6 Predictions of damped homogeneous responses within two mesh cycles given x0 = -6 2×10 in., v0 = 20 in./s, µ = 0.2 at Ωp = 1000 RPM ...... 206

7.7 Predictions of (undamped) forced periodic responses within two mesh cycles -6 given x0 = 2×10 in., v0 = 20 in./s, Tp = 2000 lb-in, µ = 0.2 and Ωp = 1000 RPM ...... 210

-6 7.8 Steady state forced periodic responses given x0 = 2×10 in., v0 = 20 in./s, Tp = 2000 lb-in., µ = 0.1 and Ωp = 1000 RPM: (a) DTE vs. time; (b) DTE spectra... 212

7.9 Predicted mesh harmonics of (undamped) forced periodic responses as a function -6 of µ given x0 = 2×10 in., v0 = 20 in./s, Tp = 2000 lb-in and Ωp = 1000 RPM: (a) DTE; (b) slope of DTE ...... 213

7.10 Mapping of eigenvalue κ (absolute value) maxima as a function of the ratio of time-varying mesh frequency fmesh(t) to the system natural frequency fn ...... 215

xix LIST OF SYMBOLS

List of Symbols for Chapter 1

O pinion/gear center location P pitch point t time (s) V contact point speed (in./s) x line-of-action direction y off line-of-action direction z axial direction ε unloaded static transmission error (µin.)

Subscripts 1 pinion 2 gear

Abbreviations ART Advanced Rotorcraft Transmission DOF degree-of-freedom DTE dynamic transmission error EHL elasto-hydrodynamic lubrication LOA line-of-action MDOF multi-degree-of-freedom OLOA off line-of-action SDOF single degree-of-freedom STE static transmission error

List of Symbols for Chapter 2 a, b shaft distance (in) CR surface roughness constant c viscous damping (lb-s/in) E Young’s modulus (psi) F force (lb) f frequency (Hz) I area moment of inertia (in4) i, j indices of gear tooth xx J polar moment of inertia (lb-s2-in) K stiffness matrix (lb/in) K stiffness (lb/in) k tooth mesh stiffness (lb/in) L geometric length (in) N normal contact force (lb) n mesh index R tooth surface roughness (in) r radius (in) T torque (lb-in) t time (s) Ve entraining velocity (in/s) Vs sliding velocity (in/s) Wn normal load per unit length of face width (lb/in) X moment arm (in) XΓ load sharing factor x motion variable along LOA axis (in) Y frequency spectrum of motion along OLOA axis (in) y motion variable along OLOA axis (in) ∆ frequency spectrum of dynamic transmission error (in) δ dynamic transmission error (in) ε static transmission error (in) γ nominal roll angle (rad) λ base pitch (in) µ coefficient of friction 2 υ0 dynamic viscosity (lb-s/in ) θ vibratory angular displacement (rad) σ contact ratio Ω angular speed (rad/s) ζ viscous damping ratio

Subscripts 0, 1… n indices of meshing teeth/modes avg average B bearing b base c (mesh) cycle e effective f friction g gear i index of gear tooth m mesh P pitch point p pinion

xxi S shaft x LOA direction y OLOA direction z axial direction

Superscripts . first derivative with respect to time .. second derivative with respect to time

Abbreviations ART Advanced Rotorcraft Transmission DOF degree-of-freedom DTE dynamic transmission error FE/CM finite element/contact mechanics HPSTC highest point of single tooth contact LPSTC lowest point of single tooth contact LTV linear time-varying LOA line-of-action MDOF multi-degree-of-freedom OLOA off line-of-action SDOF single degree-of-freedom

Operators ceil ceiling function floor floor function mod modulus function sgn sign function

List of Symbols for Chapter 3 b empirical coefficient bH semi-width of Hertzian contact band c viscous damping (lb-s/in) E Young’s modulus (GPa) F force (lb) G dimensionless material parameter H dimensionless central film thickness J polar moment of inertia (lb-s2-in) K stiffness (lb/in) k tooth mesh stiffness (lb/in) L geometric length (in) m mass (lb⋅s2/in) N normal contact force (lb) xxii Ph maximum Hertzian pressure (GPa) r radius (in) S surface roughness (µm) SR slide-to-roll ratio T torque (lb-in) t time (s) U speed parameter V, v velocity (m/s) W load parameter wn normal load per unit length of face width (N/mm) X moment arm (in) x motion variable along LOA axis (in) y motion variable along OLOA axis (in) Z face width (mm) α pressure angle (rad) 2 ηM dynamic viscosity (N-s/mm ) Λ film parameter λ base pitch (in) µ coefficient of friction ν Poisson’s ratio θ vibratory angular displacement (rad) ρ profile radii of curvature (mm) Φ regularizing factor σ contact ratio Ω angular speed (rad/s) ζ viscous damping ratio

Subscripts 0, 1… n indices of meshing teeth avg average B bearing b base C Coulomb friction c (mesh) cycle comp composite e entraining component f friction g gear i index of gear tooth p pinion r rolling component S smoothened friction model s sliding component X Xu and Kahraman model

xxiii x LOA direction y OLOA direction

Superscripts . first derivative with respect to time .. second derivative with respect to time ’ effective value

Abbreviations DOF degree-of-freedom DTE dynamic transmission error EHL elasto-hydrodynamic lubrication HPSTC highest point of single tooth contact LPSTC lowest point of single tooth contact LTV linear time-varying LOA line-of-action MDOF multi-degree-of-freedom OLOA off line-of-action

Operators floor floor function mod modulus function sgn sign function

List of Symbols for Chapter 4

A, B harmonic balance coefficients C damping parameter (lb-s/in) c viscous damping (lb-s/in) D Fourier differentiation matrix E gear constant F harmonic balance matrix F force (lb) f frictional function i, j indices J polar moment of inertia (lb-s2-in) K harmonic balance matrix K stiffness (lb/in) k tooth mesh stiffness (lb/in) L geometric length (in) M (friction) torque (lb-in) or mass (lb⋅s2/in) m mass (lb⋅s2/in) N normal contact force (lb) or harmonic order xxiv n mesh index R base radius (in) r radius (in) S index T (normalized) period t time (s) X moment arm (in) ∆ Fourier coefficient vector δ dynamic transmission error (in) ε static transmission error (in) ϑ angle (rad) λ base pitch (in) µ coefficient of friction θ vibratory angular displacement (rad) σ contact ratio τ dimensionless time υ sub-harmonic index Ω angular speed (rad/s) ω mesh frequency (rad) ζ viscous damping ratio

Subscripts 0, 1… n indices of meshing teeth B bearing b base e effective f friction h super harmonic matrix g gear i index of gear tooth k stiffness coefficient P pitch point p pinion u sub-harmonic matrix x LOA direction y OLOA direction δ dynamic transmission error coefficient

Superscripts . first derivative with respect to time .. second derivative with respect to time ’ first derivative with respect to dimensionless time ” second derivative with respect to dimensionless time ^ differential operator xxv nominal value ~ iterative harmonic balance parameter + pseudo-inverse

Abbreviations DFT discrete Fourier transform DOF degree-of-freedom DTE dynamic transmission error FFT fast Fourier transform LOA line-of-action LTV linear time-varying MDOF multi-degree-of-freedom MHBM multi-term harmonic balance method OLOA off line-of-action SDOF single degree-of-freedom

Operators floor floor function mod modulus function sgn sign function | | matrix determinant

List of Symbols for Chapter 5 e error F force (lb) f frequency (Hz) H transfer function Hv Hankel function I identity matrix i, j indices of gear tooth J polar moment of inertia (lb-s2-in) L geometric length (in) K stiffness (lb/in) k tooth mesh stiffness (lb/in) k(ω) wave number m mass (lb⋅s2/in) N normal contact force (lb) n natural (frequency) P sound pressure (Pa) Q source strength (Pa-in2) xxvi r radius (in) S surface area (in2) T torque (lb-in) t time (s) V velocity (in/s) w weighting function X moment arm (in) x motion variable along LOA axis (in) Y mobility (in/s/lb) y motion variable along OLOA axis (in) α angle (rad) λ base pitch (in) µ coefficient of friction θ vibratory angular displacement (rad) ρ air density (lb⋅s2/in4) σ contact ratio Ω angular speed (rad/s) ω angular frequency (rad) Ξ velocity error matrix ζ viscous damping ratio

Subscripts B bearing b base e effective parameter f friction g gear i index of gear tooth m mean component P path p pinion S shaft or source x LOA direction y OLOA direction

Superscripts . first derivative with respect to time .. second derivative with respect to time ~G complex value vector -1 matrix inverse * complex conjugate Abbreviations

xxvii DOF degree-of-freedom LOA line-of-action MIMO multi-input multi-output OLOA off line-of-action RMS mean square root STE static transmission error

Operators floor floor function mod modulus function sgn sign function | | absolute value

List of Symbols for Chapter 6

E Young’s modulus (psi) e unit vector along axis F force (lb) K tooth mesh stiffness (lb/in) k tooth mesh stiffness density (lb/in2) I area moment of inertia (in4) J polar moment of inertia (lb-s2-in) L length of contact line (in) l variable along contact line (in) M moment (lb-in) m mass (lb⋅s2/in) N normal contact force (lb) N mesh index r radius (in) T torque (lb-in) Tmesh mesh period (s) t time (s) u translational motion (in) W face width (in) v velocity of contact point (in/s) x LOA coordinate of contact point (in) z axial coordinate of contact point (in) β helical angle ∆ deformation of contact point (in) µ coefficient of friction λ base pitch (in) δ dynamic transmission error (in) φ pressure angle (deg) σ contact ratio xxviii Θ (static) angular deflection (rad) θ vibratory angular displacement (rad) Ω angular speed (rad/s) ζ viscous damping ratio

Subscripts 0, 1… n indices of meshing teeth A (shaft) cross section area (in2) b base c contact point g gear h (coordinate) upper limit i index of gear tooth l (coordinate) lower limit P pitch point p pinion S shaft s sliding component V viscous component x LOA direction y OLOA direction z axial direction

Superscripts . first derivative with respect to time .. second derivative with respect to time − mean component <-1> matrix inverse T matrix transverse

Abbreviations ART Advanced Rotorcraft Transmission DOF degree-of-freedom FE/CM finite element/contact mechanics LOA line-of-action LTV linear time-varying MDOF multi-degree-of-freedom OLOA off line-of-action

Operators × cross product ceil ceiling function floor floor function xxix mod modulus function sgn sign function

List of Symbols for Chapter 7

C viscous damping coefficient (lb-s/in) e unit vector along axis F force (lb) f frequency (Hz) G state matrix H transition matrix J polar moment of inertia (lb-s2-in) K tooth mesh stiffness (lb/in) k tooth mesh stiffness (lb/in) L length of contact line (in) M moment (lb-in) m mass (lb⋅s2/in) r radius (in) T torque (lb-in) t time (s) v velocity (in/s) W face width (in) X state space response x LOA coordinate of contact point (in) Z contact zone z axial coordinate of contact point (in) β helical angle δ dynamic transmission error (in) ε static transmission error (in) Φ state transition matrix φ pressure angle (deg) γ basis solution κ eigenvalue λ base pitch (in) µ coefficient of friction Π Wronskian matrix θ vibratory angular displacement (rad) σ contact ratio τ integration variable Ω angular speed (rad/s) ζ viscous damping ratio

Subscripts

xxx 0, 1… n indices of meshing teeth c contact point e effective parameter g gear m mesh (frequency) p pinion x LOA direction y OLOA direction z axial direction

Superscripts . first derivative with respect to time .. second derivative with respect to time − time average

Abbreviations DOF degree-of-freedom DTE dynamic transmission error FE/CM finite element/contact mechanics LOA line-of-action LTV linear time-varying OLOA off line-of-action SDOF single-degree-of-freedom

Operators ceil ceiling function floor floor function LommelS1 Lommel function mod modulus function sgn sign function | | absolute value

List of Symbols for Chapter 8

µ coefficient of friction

Abbreviations DOF degree-of-freedom DTE dynamic transmission error LTV linear time-varying LOA line-of-action MDOF multi-degree-of-freedom OLOA off line-of-action xxxi SDOF single-degree-of-freedom

xxxii CHAPTER 1

INTRODUCTION

1.1 Motivation

Spur and helical gears are widely used in vehicles and mechanical devices to transmit large torques while maintaining a constant input-to-output speed ratio. One remaining challenge for modern gear engineering is the reduction of gear noise in ground and air vehicles such as heavy duty trucks and helicopters. Typically, steady state gear

(whine) noise is generated by several sources [1.1-1.2]. Virtually all of the prior researchers [1.1-1.6] have assumed that the main source is static transmission error (STE), which is defined as the derivation from the ideal (kinematic) tooth profile induced by manufacturing errors and elastic deformations. Accordingly, design engineers tend to reduce STE, via improved manufacturing processes and tooth modifications [1.7]. Yet, at higher torque loads, noise levels are still relatively high though STE might be somewhat minimal (say at the design loads). In other cases, the trend in sound pressure levels does not necessarily match the STE vs. torque curves [1.2]. Typical examples include experimental data on the Advanced Rotorcraft Transmission (ART) gears tested by

1 NASA Glenn and OSU [1.8-1.10]. These suggest that additional vibro-acoustic sources must be considered.

The relative speed between V2 and V1 of two meshing gear teeth (with centers at

O1 and O2), as depicted in Figure 1.1(a), changes direction at the pitch point P during each contact event, thus providing additional periodic excitations normal to the direction of contact, as shown in Figure 1.1(b). Certain unique characteristics of the gear tooth sliding make it a potentially dominant factor, despite the somewhat lower magnitudes of friction force. First, due to the reversal in the direction during meshing action, friction is associated with a large oscillatory component, which causes both higher magnitudes as well as higher bandwidth in dynamic responses. Furthermore, friction is more significant at higher torque and lower speeds. In reality, frictional source mechanism is associated with surface roughness, lubrication regime properties, time-varying friction forces/torques and mesh interface dynamics. These lead to interesting gear dynamic phenomena, such as super-harmonic response, unstable regimes, sub-harmonic resonance and angular modulation [1.11-1.15]. Clearly, the diverse effects of friction can only be analyzed by adopting an intra-disciplinary approach, wherein the principles of meshing kinematics, contact and tribological characteristics, dynamics and noise propagation mechanisms are integrated into a cohesive model.

(a)

Sliding Velocity

V2 – V1

Normal Load

Friction Force

(b) Figure 1.1 (a) Schematic of the spur gear contact, where LOA and OLOA represent the line-of-action direction and off line-of-action direction, respectively. (b) Directions of the sliding velocity (V), normal mesh load and friction force in spur gears.

3 Historically, the friction between gear teeth and its cyclic nature have been either ignored or incorporated as an equivalent viscous damping term [1.1-1.2]. Such an approach is clearly inadequate since viscous damping is essentially a passive characteristic and it cannot act as the external excitation to the governing system. Neither does it consider the dynamic effects in the off-line-of-action (OLOA) direction. Hence, there is a definite need for new or improved models that could predict the dynamic and vibro-acoustic responses of a geared system and clarify the role of sliding friction. This is the salient focus of this study.

1.2 Literature Review

In a series of recent articles, Vaishya and Singh [1.13-1.15] have provided an extensive review of prior work. They developed a spur gear pair model with sliding friction and rectangular mesh stiffness by assuming that load is equally shared among all the teeth in contact, as shown in Figure 1.2. They also solved the SDOF system equations in terms of the dynamic transmission error (DTE) by using the Floquet theory and the harmonic balance method [1.13-1.15]. While the assumption of equal load sharing yields simplified expressions and analytically tractable solutions, it may not lead to a realistic model (as shown in Figure 1.2 and then Chapter 2). Houser et al. [1.16-1.17] experimentally demonstrated that the friction forces play a pivotal role in determining the load transmitted to the bearings and housing in the OLOA direction; this effect is more pronounced at higher torque and lower speed conditions.

4 Tooth stiffness (lb/in) stiffness Tooth

Figure 1.2 Comparison of periodic mesh stiffness functions for a spur gear pair. Key: , realistic load sharing (proposed in Chapter 2); , equal load sharing assumed by Vaishya and Singh [1.13].

Velex and Cahouet [1.18] described an iterative procedure to evaluate the effects

of sliding friction, tooth shape deviations and time-varying mesh stiffness in spur and

helical gears and compared simulated bearing forces with measurements. They reported

significant oscillatory bearing forces at lower speeds that are induced by the reversal of

friction excitation with alternating tooth sliding direction. In a subsequent study, Velex and Sainsot [1.19] analytically found that the Coulomb friction should be viewed as a

non-negligible excitation source to error-less spur and helical gear pairs, especially for

translational vibrations and in the case of high contact ratio gears. However, their work

was confined to a study of excitations and the effects of tooth modifications were not

5 considered. Lundvall et al. [1.20] considered profile modifications and manufacturing

errors in a multi-degree-of-freedom (MDOF) spur gear model and examined the effect of

sliding friction on the angular dynamic motions. By utilizing a numerical method, they

reported that the profile modification has less influence on the dynamic transmission

error when frictional effects are included. However, incorporation of the time-varying

sliding friction and the realistic mesh stiffness functions into an analytical (MDOF)

formulation and their dynamic interactions remain unsolved.

In all of the work mentioned above and related literature [1.8-1.20], the sliding friction phenomenon has been typically formulated by assuming the Coulomb formulation with a constant coefficient of friction for modeling convenience. This is partially related to the difficulty associated with the measurement of friction force in a gear mesh. In reality, tribological conditions change continuously due to varying mesh properties, dynamic fluctuations and lubricant film thickness as the gears roll through a full cycle [1.21-1.26]. Thus, coefficient of friction varies instantaneously with the spatial position of each tooth and the direction of friction force changes at the pitch point.

Alternate tribological theories, such as elasto-hydrodynamic lubrication (EHL), boundary lubrication or mixed regime, have been employed to explain the sliding friction under varying operating conditions [1.21-1.23]. For instance, Benedict and Kelley [1.21] proposed an empirical dynamic friction coefficient under mixed lubrication regime based on measurements on a roller test machine. Xu et al. [1.22-1.23] recently proposed yet another friction formula that is obtained by using a non-Newtonian, thermal EHL formulation. Duan and Singh [1.27] developed a smoothened Coulomb model for dry friction in torsional dampers; it could be applied to gears to obtain a smooth transition at

6 the pitch point. Hamrock and Dawson [1.28] suggested an empirical equation to predict

the minimum film thickness for two disks in line contact. They calculated the film

parameter, which could lead to a composite, mixed lubrication model for gears. Rebbechi

et al. [1.8] have successfully used root strains to compute friction force under dynamic

conditions. Recently, Vaishya and Houser [1.9] have shown that quasi-static

measurement of friction force is possible by using the technique of digital filtering to

eliminate the dynamic effects. However, no comprehensive work could be found which

critically evaluates the existing lubrication theories in the framework of an actual gear

mesh. Also, no prior work has incorporated the time-varying coefficient of friction into

MDOF gear dynamics or examined its effect.

Sliding friction at gear teeth also manifests as a noise source, as contended by the

dynamic tests by Houser et al. [1.16]. Borner and Houser [1.17] predicted the dynamic

forces due to friction and qualitatively discussed the radiated sound from the housing.

Most studies on gearbox system dynamics [1.2] have relied on a combination of detailed

finite element, boundary element and semi-analytical methods. Van Roosmalen [1.29] formulated a gearbox model including analytical formulations for the vibration at the gears due to tooth deflections and the vibration transfer through the bearings. Lim et al.

[1.30-1.33] developed a lumped parameter model with a rigid casing and a finite element model with a flexible casing for a simple geared system. However, finite and boundary element methods often require extensive computational time. Over the last four decades, some simplified lumped parameter models have been developed though few have incorporated the torsional and translational motions in both the line-of-action (LOA) and

OLOA directions. Steyer [1.34] examined a single mesh geared system with 6 DOF. By

7 assuming the housing mass is much larger than the gears and shafts, an impedance

mismatch was created with a rigid boundary condition at the bearing location. Thus, the

internal geared system was modeled separately and analytical expressions were presented

for a unity gear pair in terms of the resulting force transmissibility curve. Kartik [1.35] developed a frequency-response based model to predict noise radiation from gearbox housings with a multi-mesh gear set. His work showed that the bearing and mesh stiffness significantly affects the sound pressure in the high frequency range while the casing stiffness controls the response in the range below 4 kHz. However, the transfer function relating the bearing forces to the equivalent force at the housing panel was based on limited experiments. Overall, the above mentioned system models fall short of providing a complete vibro-acoustic model.

1.3 Problem Formulation

1.3.1 Key Research Issues

Governing equations for gear dynamics should lead to a class of damped inhomogeneous periodic differential equations [1.36-1.38] with multiple interacting coefficients [1.13-1.20]. Although similar equations may also be found in a variety of

disciplines such as communication networks [1.36] and electrical circuits [1.39], the gear

friction problems, however, significantly differ from existing models such as the classical

Hill’s equations [1.36] in several ways. First, unlike classic friction problems in most

mechanical systems, the direction of gear friction is normal rather than in the direction of 8 nominal motions. Second, the frictional forces and moments emerge on both sides of the

governing equations as either excitations or periodically-varying parameters. Also, the periodic damping should capture not only the kinematic effects but also the energy dissipation due to sliding friction. Third, the periodic mesh stiffness is not confined to a rectangular wave assumed by Manish and Singh [1.13-1.15], or a simple sinusoid as in the Mathieu’s equation [1.36]. Instead, they should describe realistic, yet continuous,

profiles of Figure 1.2 resulting from a detailed finite element/contact mechanics analysis

[1.40]. Lastly, the stiffness and viscous damping terms incorporate combined (but phase correlated) contributions from all (yet changing) tooth pairs in contact.

Historically, such periodic differential equations are seldom investigated and limited prior research efforts, as reported in the literature review [1.1-1.20], are based largely on numerical integration and the Fast Fourier Transform algorithm. Consequently, there is a clear need for closed form analytical (say by using the Floquet theory) and semi-analytical (say by using the multi-term harmonic balance method) solutions to the dynamic responses of spur and helical gear pairs under the influence of sliding friction.

Recently, Velex and Ajmi [1.41] implemented a harmonic analysis to approximate the dynamic factors in helical gears (based on tooth loads and quasi-static transmission errors). Their work, however, does not describe the multi-dimensional system dynamics or include the frictional effect, which may lead to “multiplicative” terms as described earlier. The parametric friction force excitation may have an influence on the stability of the homogenous system.

Further, for a satisfactory understanding of dynamic behaviors of gears, a higher number of degrees-of-freedom are required for analysis, such as the MDOF helical gear

9 model of Figure 1.3. This is essential for additional phenomena like friction force, torsional-flexural coupling, shaft wobble and axial shuttling, which are yet to be fully understood. Also, to represent practical geared systems, a generalized model is required that incorporates different gear design configurations, lubrication conditions and meshing parameters. Existing solution methodology [1.29-1.35] has to be improved to compute the dynamic response of the entire gearbox, for a combined excitation of transmission error, sliding friction, mean torque and other sources. Subsequently, the relative contribution of various parameters and the resulting noise characteristics need to be understood. This requires an improved source-path-receiver model for the entire gearbox system that incorporates competing noise sources.

ε ()t

Figure 1.3 MDOF helical gear dynamic model (proposed in Chapter 6) and its contact mechanics with sliding friction.

10 1.3.2 Scope, Assumptions and Objectives

Chief goal of this research is to improve the earlier work by Vaishya and Singh

[1.13-1.15] by developing improved mathematical models and proposing new analytical solutions that will enhance our understanding of the influence of friction on gear dynamics and vibro-acoustic behavior. Many dynamic phenomena that emerge due to interactions between parametric variations (time-varying mesh stiffness and viscous damping) and sliding friction will be predicted, along with a better understanding of the relative contributions of transmission error versus sliding friction noise to the gear whine noise. The specific objectives of this study are therefore as follows:

Extend Vaishya and Singh’s work [1.13-1.15] by developing improved MDOF

dynamic system for a spur gear pair that incorporates realistic time-varying mesh

stiffness functions, accurate representations of sliding friction and load sharing

between meshing tooth pairs. (Chapter 2)

Comparatively evaluate alternate sliding friction models [1.21-1.28] and predict

the interfacial friction forces and motions in the OLOA direction. Also, validate

dynamic system models and analytical solutions by comparing predictions to

numerical solutions, the benchmark finite element/contact mechanics code as well

as measurements. (Chapters 2 and 3)

Propose a semi-analytical algorithm based on both single- and multi-term

harmonic balance methods to quickly construct frequency responses of multi-

dimensional spur gear dynamics with sliding friction. This should provide new

insights into the dynamic interactions between parametric excitations. (Chapter 4)

11 Propose a refined source-path-receiver model that characterizes the structural

paths in two directions and develop analytical tools to efficiently predict the

whine noise radiated from gearbox panels and quantify the contribution of sliding

friction to the overall whine noise. Analytical predictions of the structural transfer

function and noise radiation will be compared with measurements. (Chapter 5)

Propose a new three-dimensional formulation for helical gears to characterize the

dynamics associated with the contact plane including the reversal at the pitch line

due to sliding friction. A 12 DOF model will be developed which includes the

rotational and translational motions along the LOA, OLOA and axial directions as

well as the bearing/shaft compliances. (Chapter 6)

Develop improved closed form solutions for the linear time-varying helical gear

system in terms of the dynamic transmission error under the effect of sliding

friction by using the Floquet theory. (Chapter 7)

Scope and assumptions include the following: For the internal spur and helical gear pair sub-systems, the pinion and gear are modeled as rigid disks. The elastic deformations of the shaft and bearings are modeled using lumped elements which are connected to a rigid casing. Also, vibratory angular motions are small in comparison to the mean motion, and the mean load is assumed to be high such that the dynamic load is not sufficient to cause tooth separations [1.42]. If these assumptions are not made, the system model would have implicit non-linearities. Consequently, the position of the line of contact and relative sliding velocity depend only on the nominal angular motions; this leads to a linear time-varying system formulation. Note that different mesh stiffness schemes are assumed for spur and helical gears: For the spur gear analysis, the realistic

12 and continuous mesh stiffness is considered based on an accurate finite element/contact mechanics analysis code [1.40]; Thus, the time-varying stiffness is indeed an effective function which may also include the effect of profile modifications. For the helical gear analysis, however, only those gears with perfect involute profiles are considered and the mesh stiffness per unit length along the contact line (or stiffness density) is assumed to be constant [1.19]. This is equivalent to the equal load sharing assumption by Vishya and

Singh [1.13-1.15]. Such limitation may be further examined in future work.

For the structure-borne whine noise model of the gearbox system, a source-path- receiver model of Figure 1.4 is used. All the assumptions as mentioned above are embedded in the modeling of the internal gear pair sub-system. The unloaded static transmission error and sliding friction are considered as the two main excitations to the system; these are assumed to be most dominant in the LOA and OLOA directions, respectively. Hence, only corresponding structural paths in these two directions are considered by neglecting the moment transfer in the bearing matrices. Also, by assuming the housing mass is much larger than the gears and shafts, an impedance mismatch is created with a rigid boundary condition at the bearing location. Thus, the internal geared system could be modeled separately and its resulting force response provides force excitations to the structural paths. Finally, for the NASA gearbox used as the case study, the box plate is assumed to be the main radiator due to its relatively high mobility as well as the way the gearbox was assembled.

Finally, it is worthwhile to mention that all chapters of this thesis are written in a self-contained manner in terms of formulation, literature review, methods and results.

13 SOURCE Transmission LOA bearing forces error 6 DOF linear-time- Coupling at varying spur gear bearings Sliding pair model + shafts OLOA bearing forces friction

RECEIVER PATH

Sound Radiation Housing Housing structure pressure model velocity model

Figure 1.4 Block diagram for the vibro-acoustics of a simplified geared system with two excitations at the gear mesh (as proposed in Chapter 5).

References for Chapter 1

[1.1] Ozguven, H. N., and Houser, D. R., 1988, “Mathematical Models Used in Gear Dynamics - a Review,” Journal of Sound and Vibration, 121, pp. 383-411.

[1.2] Lim, T. C., and Singh, R., 1989, “A Review of Gear Housing Dynamics and Acoustic Literature,” NASA-Technical Memorandum, 89-C-009.

[1.3] Oswald, F. B., Seybert, A. F., Wu, T. W., and Atherton, W., 1992, “Comparison of Analysis and Experiment for Gearbox Noise,” Proceedings of the International Power Transmission and Gearing Conference, Phoenix, pp. 675-679.

[1.4] Baud, S., and Velex, P., 2002, “Static and Dynamic Tooth Loading in Spur and Helical Geared Systems-Experiments and Model Validation,” American Society of Mechanical Engineers, 124, pp. 334-346.

[1.5] Comparin R. J., and Singh, R., 1990, “An Analytical Study of Automotive Neutral Gear Rattle,” ASME Journal of Mechanical Design, 112, pp. 237-245.

[1.6] Mark, W. D., 1978, “Analysis of the Vibratory Excitation of Gear Systems: Basic Theory,” Journal of Acoustical Society of America, 63(5), pp. 1409-1430.

[1.7] Munro, R. G., 1990, “Optimum Profile Relief and Transmission Error in Spur Gears,” Proceedings of IMechE, Cambridge, England, 9-11 Apr., pp. 35-42.

[1.8] Rebbechi, B. and Oswald, F. B., 1991, “Dynamic Measurements of Gear Tooth Friction and Load,” NASA-Technical Memorandum, 103281.

[1.9] Vaishya, M., and Houser, D. R., 1999, “Modeling and Measurement of Sliding Friction for Gear Analysis,” American Gear Manufacturer Association Technical Paper, 99FTMS1, pp. 1-12.

[1.10] Schachinger, T., 2004, “The Effects of Isolated Transmission Error, Force Shuttling, and Frictional Excitations on Gear Noise and Vibration,” MS Thesis, The Ohio State University.

[1.11] Blankenship, G. W., and Singh, R., 1995, “A New Gear Mesh Interface Dynamic Model to Predict Multi-Dimensional Force Coupling and Excitation,” Mechanism and Machine Theory Journal, 30(1), pp. 43-57.

[1.12] Padmanabhan, C., and Singh, R., 1995, “Analysis of Periodically Excited Non- Linear Systems by a Parametric Continuation Technique,” Journal of Sound and Vibration, 184(1), pp. 35-58.

[1.13] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration, 243(3), pp. 525-545.

[1.14] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671- 694.

[1.15] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.

[1.16] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, “Vibro-Acoustic Effects of Friction in Gears: An Experimental Investigation,” SAE Paper # 2001- 01-1516.

[1.17] Borner, J., and Houser, D. R., 1996, “Friction and Bending Moments as Gear Noise Excitations,” SAE Paper # 961816.

[1.18] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of Mechanical Design, 122(4), pp. 515-522. 15 [1.19] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp. 641- 658.

[1.20] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3), pp. 479-499.

[1.21] Benedict, G. H., and Kelley B. W., 1961, “Instantaneous Coefficients of Gear Tooth Friction,” Transactions of the American Society of Lubrication Engineers, 4, pp. 59-70.

[1.22] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, “Prediction of Mechanical Efficiency of Parallel-Axis Gear Pairs,” ASME Journal of Mechanical Design, 129 (1), pp. 58-68.

[1.23] Xu, H., 2005, “Development of a Generalized Mechanical Efficiency Prediction Methodology,” PhD dissertation, The Ohio State University.

[1.24] Seireg, A. A., 1998, Friction and Lubrication in Mechanical Design, Marcel Dekker, Inc., New York.

[1.25] Baranov, V. M., Kudryavtsev, E. M., and Sarychev, G. A., 1997, “Modeling of the Parameters of Acoustic Emission under Sliding Friction of Solids,” Wear, 202, pp. 125- 133.

[1.26] Drozdov, Y. N., and Gavrikov, Y. A., 1968, “Friction and Scoring under the Conditions of Simultaneous Rolling and Sliding of Bodies,” Wear, 11, pp. 291-302.

[1.27] Duan, C., and Singh, R., 2005, “Super-Harmonics in a Torsional System with Dry Friction Path Subject to Harmonic Excitation under a Mean Torque”, Journal of Sound and Vibration, 285(2005), pp. 803-834.

[1.28] Hamrock, B. J., and Dowson, D., 1977, “Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part III – Fully Flooded Results,” Journal of Lubrication Technology, 99(2), pp. 264-276.

[1.29] Van Roosmalen, A., 1994, “Design Tools for Low Noise Gear Transmissions,” PhD Dissertation, Eindhoven University of Technology.

[1.30] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling Element Bearings. Part I: Bearing Stiffness Formulation,” Journal of Sound and Vibration, 139(2), pp. 179-199.

16 [1.31] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling Element Bearings. Part II: System Studies,” Journal of Sound and Vibration, 139(2), pp. 201-225.

[1.32] Lim, T. C., and Singh R., 1991, “Statistical Energy Analysis of a Gearbox with Emphasis on the Bearing Path,” Noise Control Engineering Journal, 37(2), pp. 63-69.

[1.33] Lim, T. C., and Singh, R., 1991, “Vibration Transmission Through Rolling Element Bearings. Part III: Geared Rotor System Studies,” Journal of Sound and Vibration, 151(1), pp. 31-54.

[1.34] Steyer, G., 1987, “Influence of Gear Train Dynamics on Gear Noise,” NOISE- CON 87 proceedings, pp. 53-58.

[1.35] Kartik, V., 2003, “Analytical Prediction of Load Distribution and Transmission Error for Multiple-Mesh Gear-Trains and Dynamic Studies in Gear Noise and Vibration,” MS Thesis, The Ohio State University.

[1.36] Richards, J. A., 1983, Analysis of Periodically Time-varying Systems, New York, Springer.

[1.37] Jordan, D. W., and Smith, P., 2004, Nonlinear Ordinary Differential Equations, 3rd Edition, Oxford University Press.

[1.38] Thomsen, J. J., 2003, Vibrations and Stability, 2nd Edition, Springer.

[1.39] Kenneth S. K., Jacob K. W. and Alberto S-V, 1990, “Steady-State Methods for Simulating Analog and Microwave Circuits,” Kluwer Academic Publishers, Boston.

[1.40] External2D (CALYX software), 2003, “Helical3D User’s Manual,” www.ansol.www, ANSOL Inc., Hilliard, OH.

[1.41] Velex, P., and Ajmi, M., 2007, “Dynamic Tooth Loads and Quasi-Static Transmission Errors in Helical Gears – Approximate Dynamic Factor Formulae,” Mechanism and Machine Theory Journal, 42(11), pp. 1512-1526.

[1.42] Blankenship, G. W., and Kahraman, A., 1995, “Steady State Forced Response of a Mechanical Oscillator with Combined Parametric Excitation and Clearance Type Non- linearity,” Journal of Sound and Vibration, 185(5), pp. 743-765.

17 CHAPTER 2

SPUR GEAR DYNAMICS WITH SLIDING FRICTION AND REALISTIC MESH

STIFFNESS

2.1 Introduction

In a series of recent articles, Vaishya and Singh [2.1-2.3] developed a spur gear pair model with periodic tooth stiffness variations and sliding friction based on the assumption that load is equally shared among all the teeth in contact. Using the simplified rectangular pulse shaped variation in mesh stiffness, they solved the single-degree-of- freedom (SDOF) system equations in terms of the dynamic transmission error (DTE) using the Floquet theory and the harmonic balance method [2.1-2.3]. While the assumption of equal load sharing yields simplified expressions and analytically tractable solutions, it may not lead to a realistic model. This chapter aims to overcome this deficiency by employing realistic time-varying tooth stiffness functions and the sliding friction over a range of operational conditions. New linear time-varying (LTV) formulation will be extended to include multi-degree-of-freedom (MDOF) system dynamics for a spur gear pair.

18 Vaishya and Singh [2.1-2.3] have already provided an extensive review of prior work. In addition, Houser et al. [2.4] experimentally demonstrated that the friction forces play a pivotal role in determining the load transmitted to the bearings and housing in the off-line-of-action (OLOA) direction; this effect is more pronounced at higher torque and lower speed conditions. Velex and Cahouet [2.5] described an iterative procedure to evaluate the effects of sliding friction, tooth shape deviations and time-varying mesh stiffness in spur and helical gears and compared simulated bearing forces with measurements. They reported significant oscillatory bearing forces at lower speeds that are induced by the reversal of friction excitation with alternating tooth sliding direction.

In a subsequent study, Velex and Sainsot [2.6] analytically found that the Coulomb friction should be viewed as a non-negligible excitation source to error-less spur and helical gear pairs, especially for translational vibrations and in the case of high contact ratio gears. However, their work was confined to a study of excitations and the effects of tooth modifications were not considered. Lundvall et al. [2.7] considered profile modifications and manufacturing errors in a MDOF spur gear model and examined the effect of sliding friction on the angular dynamic motions. By utilizing a numerical method, they reported that the profile modification has less influence on the dynamic transmission error when frictional effects are included. Nevertheless, two key questions remain unresolved: How to concurrently incorporate the time-varying sliding friction and the realistic mesh stiffness functions into an analytical (MDOF) formulation? How to quantify dynamic interactions between sliding friction and mesh stiffness terms especially when tip relief is provided to the gears? This chapter will address these issues.

19 2.2 Problem Formulation

2.2.1 Objectives and Assumptions

Chief objective of this chapter is to propose a new method of incorporating the sliding friction and realistic time-varying stiffness into an analytical MDOF spur gear model and to evaluate their interactions. Key assumptions are: (i) pinion and gear are modeled as rigid disks; (ii) shaft-bearings stiffness in the line-of-action (LOA) and

OLOA directions are modeled as lumped elements which are connected to a rigid casing;

(iii) vibratory angular motions are small in comparison to the mean motion; and (iv)

Coulomb friction is assumed with a constant coefficient of friction µ . If assumption (iii) is not made, the system model would have implicit non-linearities. Consequently, the position of the line of contact and relative sliding velocity depend only on the nominal angular motions.

An accurate finite element/contact mechanics (FE/CM) analysis code [2.8] will be employed, in the “static” mode, to compute the mesh stiffness at every time instant under a range of loading conditions. Here, the time-varying stiffness is calculated as an effective function which may also include the effect of profile modifications. The realistic mesh stiffness is then incorporated into the LTV spur gear model with the contributions of sliding friction. The MDOF formulation should describe both the LOA and OLOA dynamics; a simplified SDOF model will also be derived that describes the vibratory motion in the torsional direction. Proposed methods will be illustrated via two spur gear examples (designated as I and II) whose parameters are listed in Table 2.1 and

Table 2.2. The MDOF model of Example I will be validated by using the FE/CM code

20 [2.8] in the “dynamic” mode. Issues related to tip relief will be examined in Example II in the presence of sliding friction. Finally, experimental results of Example II will be used to further validate our method.

Parameter/property Pinion Gear Number of teeth 25 31 Diametral pitch, in-1 8 8 Pressure angle, deg 25 25 Outside diameter, in 3.372 4.117 Root diameter, in 2.811 3.561 Face width, in 1.250 1.250 Tooth thickness, in 0.196 0.196 Gear mass, lb⋅s2⋅in-1 6.72E-03 1.04E-02 Polar moment of inertia, lb⋅s2⋅in 8.48E-03 2.00E-02 Bearing stiffness (LOA and OLOA), lb/in 20E6 Center distance, in 3.5 Profile contact ratio 1.43 Elastic modulus, psi 30E6 Density, lb⋅s-2⋅in-4 7.30E-04 Poisson’s ratio 0.3

Table 2.1 Parameters of Example I: NASA-ART spur gear pair (non-unity ratio)

Parameter/property Pinion/Gear

Number of teeth 28

Diametral pitch, in-1 8

Pressure angle, deg 20

Outside diameter, in 3.738

Root diameter, in 3.139

Face width, in 0.25

Tooth thickness, in 0.191

Roll angle where the tip modification 24.5 starts (for II-B), deg

Straight tip modification (for II-B), in 7E-04

Center distance, in 3.5

Profile contact ratio 1.63

Elastic modulus, psi 30E6

Density, lb⋅s-2⋅in-4 7.30E-04

Poisson’s ratio 0.3

Range of temperatures, °F 104, 122, 140, 158, 176

Range of input torques, lb⋅in 500, 600, 700, 800, 900

Table 2.2 Parameters of Example II-A and II-B: NASA spur gear pair (unity ratio). Gear pair with the perfect involute profile is designated as II-A case and the one with tip relief is designated as II-B case

22 2.2.2 Timing of Key Meshing Events

Analytical formulations for a spur gear pair are derived via Example I (NASA-

ART spur gear pair) with parameters of Table 2.1. For a generic spur gear pair with non- integer contact ratio σ , n = ceil(σ ) meshing tooth pairs need to be considered, where the

“ceil” function rounds the σ element to the nearest integer towards a higher value.

Consequently, two meshing tooth pairs need to be modeled for Example I (σ = 1.43).

First, transitions in key meshing events within a mesh cycle need to be determined

from the undeformed gear geometry for the construction of the stiffness function. Figure

2.1 is a snapshot for Example I at the beginning of the mesh cycle (t = 0). At that time,

pair #1 (defined as the tooth pair rolling along line AC) just comes into mesh at point A

and pair #0 (defined as the tooth pair rolling along line CD) is in contact at point C,

which is the highest point of single tooth contact (HPSTC). As the gears roll, when pair

#1 approaches the lowest point of single tooth contact (LPSTC) of point B at t = tB, pair

#0 leaves contact. At t =tP , pair #1 passes through the pitch point P, and the relative

sliding velocity of the pinion with respect to the gear is reversed, resulting in a reversal of

the friction force. This should provide an impulse excitation to the system. Finally, pair

#1 goes through point C at t = tc, completing one mesh cycle (tc). These key events are

defined below, where Ω p is the nominal pinion speed, rbp is the base radius of the pinion,

length LAC is equal to one base pitch λ.

t λ tb LAB p LAP tc = , = , = . (2.1) Ω pbpr tc λ tc λ

Ωg

Ω p

Figure 2.1 Snap shot of contact pattern (at t = 0) in the spur gear pair of Example I.

2.2.3 Calculation of Realistic Time-Varying Tooth Stiffness Functions

The realistic time-varying stiffness functions are calculated using a FE/CM code,

External2D [2.8]. An input torque Tp is applied to the pinion rotating at Ω p , and the

mean braking torque Tg on the gear and its angular velocity Ωg obey the basic gear

kinematics. Superposed on the nominal motions are oscillatory components denoted as

θ p and θg for the pinion and gear, respectively. The normal contact forces N0(t), N1(t)

and pinion deflection θ p ()t are then computed by performing a static analysis using

FE/CM software [2.8]. The stiffness function of the ith meshing tooth pair for a generic

24 spur gear pair is given by Eq. (2.2), where the “floor” function rounds the contact ratio σ to the nearest integer towards a lower value, i.e. floor(σ ) = 1 for Example I.

Nti () kti ( )=== , i 0, 1, ... , n floor(σ ). (2.2) rtbpθ p ()

The stiffness function k(t) for a single tooth pair rolling through the entire meshing process is obtained by following the contact tooth pair for n = ceil(σ ) number

of mesh cycles. Due to the periodicity of the system, expanded stiffness function kti ( ) of the ith meshing tooth pair is calculated at any time instant t as:

kticc( )=−+ k[] ( n it ) mod( tt , ) , i = 0, 1, ... , n = floor(σ ). (2.3)

Here, “mod” is the modulus function defined as:

mod(xy , )=−⋅ x y floor( x / y ), if y ≠ 0. (2.4)

For Example I, calculated kt0 (), kt1() functions and their combined stiffness are

shown in Figure 2.2(a). Note that kt0 () and kt1() are, in fact, different portions of kt() as described in Eq. (2.3). Figure 2.2(b) compares the continuous kt() of the realistic load sharing model against the rectangular pulse shaped discontinuous kt() based on the equal load sharing formulation proposed earlier by Vaishya and Singh [2.1-2.3].

Figure 2.2 Tooth mesh stiffness functions of Example I calculated by using the FE/CM code (in the “static” mode). (a) Individual and combined stiffness functions. Key: , total stiffness; stiffness of pair #0; , stiffness of pair #1. (b) Comparison of the combined stiffness functions. Key: , realistic load sharing; , equal load sharing as assumed by Vaishya and Singh [2.1-2.3].

26 2.3 Analytical Multi-Degree-of-Freedom Dynamic Model

2.3.1 Shaft and Bearing Stiffness Models

Next, we develop a generic spur gear pair model with 6 DOFs including rotational

motions (θ p and θg ), LOA translations ( xp and xg ) and OLOA translations ( y p and yg ).

The governing equations are derived in the subsequent sections. First, a simplified shaft model, as shown in Figure 2.3, is developed based on the Euler’s beam theory [2.9].

Corresponding to the 6 DOFs mentioned above, only the diagonal term in the shaft stiffness matrix needs to be determined as follows, where E is the Young’s modulus,

4 Ir= π s /4 is the area moment of inertia for the shaft, and a and b are the distances from pinion/gear to the bearings.

ab+ 2 K ==KEIabab3 ⎡() −+⎤ , K = 0 . (2.5) Sx Sy ab33⎣ ⎦ Sθz

The rolling element bearings are modeled using the bearing stiffness matrix K Bm formulation (of dimension 6) as proposed by Lim and Singh [2.10, 2.11]. Assume that each shaft is supported by two identical axially pre-loaded ball bearings with a mean

axial displacement; the mean driving load Tm generates a mean radial force Fxm in the

LOA direction and a moment M ym around the OLOA direction. The time-varying friction force and torque are not included in the mean loads.

KBx /2

KBy /2 b KBx /2 x

θ z KBy /2 z y

Figure 2.3 Schematic of the bearing-shaft model.

Corresponding to the 6 DOFs considered in our spur gear model, only two

significant coefficients, KBxx and K Byy , are considered for K Bm [2.10, 2.11]. The

combined bearing-shaft stiffness ( KBx and K By in the LOA and OLOA directions) are derived by assuming that the bearing and shaft stiffness elements act in parallel.

2.3.2 Dynamic Mesh and Friction Forces

Figure 2.4 shows the mean torque and internal reaction forces acting on the pinion for Example I. For the sake of clarity, forces on the gear are not shown, which are equal in magnitude but opposite in direction to the pinion forces. Based on the Coulomb 28 friction law, the magnitude of friction force ( Ff ) is proportional to the nominal tooth

load (N) as Ff = µN where µ is constant. The direction of Ff is determined by the calculation of nominal relative sliding velocity, which results in the LTV system

formulation. Denote Xtpi () as the moment arm on the pinion for the friction force acting on the ith meshing tooth pair

Xtpi( )=+−+ L XA ( ni )λ mod( Ω p rt bp ,λσ ), i = 0, 1, ... , n = floor( ). (2.6)

The corresponding moment arm for the friction force on the gear is

Xtgi( )=+−Ω L YC iλ mod( g rt bg ,λσ ), i = 0, 1, ... , n = floor( ). (2.7)

Assume time-varying mesh (viscous) damping coefficient and relate it to kti ( ) by

22 a time-invariant damping ratio ζ m as follows, where Jepgpbggbp=+JJ/ ( Jr Jr)

ctimiie( )=⋅== 2ζ kt ( ) J , i 0, 1, ... , n floor(σ ). (2.8)

The normal forces acting on the pinion are

⎡⎤ Ntpi()== Nt gi () ktr i ()⎣⎦ bpθθε p () t − r bg g () t −+−+ p () t xt p () xt g () (2.9) ⎡⎤ ctibppbggp()⎣⎦ rθ () t−−+−= rθε () t () t x p () t x g () t , i 0, 1, ... , n = floor( σ ).

29 Here ε P (t ) is the profile error component of the static transmission error (STE), and xp(t) and xg(t) denote the translational bearing displacements of pinion and gear, respectively.

For a generic spur gear pair whose jth meshing pair passes through the pitch point within the mesh cycle, the friction forces in the ith meshing pair are derived as follows

⎧ µNtpi ( ), i= 0, 1, ... , j− 1, ⎪ ⎡⎤ Fpfi()tNt=Ω+−−=⎨ µλλ pi ()sgnmod(⎣⎦ p rtniLij bp , ) ( ) AP , , (2.10a) ⎪ ⎩⎪−=+=µNtpi ( ), i jj , 1, ... , n floor(σ ),

⎧ µNtgi ( ), i= 0, 1, ... , j− 1, ⎪ ⎡⎤ Fgfi(tNt )=Ω+−−=⎨ µλλ gi ( )sgn⎣⎦ mod( g rtniLij bg , ) ( ) AP , , (2.10b) ⎪ ⎩⎪−=+=µNtgi ( ), i jj , 1, ... , n floor(σ ).

Consequently, the friction forces for Example I of Figure 2.4 are given as:

Fpf00()tNt= µ p (), (2.10c)

⎡ ⎤ Ftpf11()=Ω−µλ Nt p ()sgnmod(⎣ p rt bp , ) L AP ⎦ , (2.10d)

Fgf00()tNt= µ g (), (2.10e)

⎡ ⎤ Ftgfg11()=Ω−µλ Nt ()sgnmod(⎣ gbgAP rt , ) L⎦ . (2.10f)

Figure 2.4 Normal and friction forces of analytical (MDOF) spur gear system model.

31 2.3.3 MDOF Model

The governing equations for the torsional DOFs are

nn==floor(σσ ) floor( ) J ppθ ()tT=+ p∑∑ XtFt pi () pfi () − rNt bppi (), (2.11) ii==00

nn==floor(σσ ) floor( ) J gθ g()tT=− g +∑∑ XtFt gi () gfi () + rNt bg gi () (2.12) ii==00.

The governing equations of the translational DOFs in the LOA direction are

n=floor(σ ) mxp p() t+++= 2ζ pBx K pBx mx p p () t K pBx x g () t∑ N pi () t 0 , (2.13) i=0

n=floor(σ ) mxg g() t+++= 2ζ gBx K gBx mx g g () t K gBx x g () t∑ N gi () t 0. (2.14) i=0

Here, K pBx and K gBx are the effective shaft-bearing stiffness in the LOA direction, and

ζ pBx and ζ gBx are their damping ratios. Similarly, the governing equations of the translational DOFs in the OLOA direction are

n=floor(σ ) myp p() t++−= 2ζ pB y K pBy my p p () t K pBy y p () t∑ F pfi () t 0, (2.15) i=0

n=floor(σ ) myg g() t++−= 2ζ gB y K gBy my g g () t K gBy y g () t∑ F gfi () t 0. (2.16) i=0 32 The composite DTE, which is the relative dynamic displacement of pinion and

gear along the LOA direction, is defined as

δθ()tr=−+−bp p () tr bg θ g () txtxt p () g (). (2.17)

Finally, the dynamic bearing forces are as:

FpBx()t=− K pBx xt p () − 2ζ pBx K pBx mxt p p (), (2.18a)

FpBy()t=− K pBy yt p () − 2ζ pBb K pBy myt p p (), (2.18b)

FgBx()t=− K gBx xt g () − 2ζ gBb K gBx mxt g g (), (2.18c)

FgBy()t=− K gBy yt g () − 2ζ g Bb K gBy myt g g (). (2.18d)

2.4 Analytical SDOF Torsional Model

When only the torsional DOFs of the spur gear pair are of interest, a simplified but equivalent SDOF model can be derived by assuming that the shaft-bearings stiffness is much higher than the mesh stiffness. After eliminating θp(t) and θg(t) in terms of the

DTE, δθ()tr=−bp p () tr bg θ g () t, the governing SDOF model is obtained for a generic spur

gear pair whose jth meshing pair passes through the pitch point within the mesh cycle:

33 n=floor(σ ) Jt ( )⎡⎤ cttktt ( ) ( ) ( ) ( ) eiiδδδ+++∑ ⎣⎦ i=0 ⎧⎫⎡⎤ sgn⎣⎦ mod(Ω+−−⋅pbprt ,λλ ) ( n j ) L AP n=floor(σ ) ⎪⎪ µ ⎨⎬⎡ XtJrXtJr()+ () ⎤ = ∑ pj g bp gj p bg i=0 ⎪⎪⎡⎤ctii()δδ () t+⋅ kt () () t ⎢ ⎥ ⎣⎦Jr22+ Jr ⎩⎭⎪⎪⎣⎢ pbg gbp ⎦⎥ . (2.19) T n=floor(σ ) e +++⎡⎤ct ()εε () t kt () () t 22∑ ⎣⎦ip ip Jrpbg+ Jr gbp i=0 ⎧⎫⎡⎤ sgn⎣⎦ mod(Ω+−−⋅pbprt ,λλ ) ( n j ) L AP n=floor(σ ) ⎪⎪ µ ∑ ⎨⎬⎡ XtJrXtJrpj() g bp+ gj () p bg ⎤ i=0 ⎪⎪⎡⎤ct()εε () t+⋅ kt () () t ⎢ ⎥ ⎣⎦ip ip Jr22+ Jr ⎩⎭⎪⎪⎣⎢ pbg gbp ⎦⎥

Here the effective polar moment of inertia Je is consistent with that defined in Eq. (2.8)

and the effective torque is TTJrTJrepgbpgpbg= ⋅⋅+⋅⋅. The dynamic response δ ()t is

controlled by three excitations: (i) time-varying Te , (ii) ε p ()t and its derivativeεp ()t and

(iii) sliding friction. For Example I, the governing Eq. (2.19) could be simplified as

Jteδδδ ( )++[ ctctt10 ( ) ( )] ( ) ++[ ktktt 10 ( ) ( )] ( ) + ⎡⎤XtJrXtJr()+ () µδ⎡⎤ct() () t+⋅Ω−+ kt () δ () tp11 g bp g p bg sgnmod(⎡⎤ rt , λ ) L ⎣⎦11⎢⎥22 ⎣⎦pbp AP ⎣⎦⎢⎥Jrpbg+ Jr gbp

()XtJrXtJrp00() g bp+ g () p bg µ ⎡⎤c ()ttkttδδ ()+ () () ⎣⎦0 0 22 Jrpbg+ Jr gbp T (2.20) e =+++++22[][]ct10 ( ) ct ( )εεpp ( t ) kt 10 ( ) k ( t ) ( t ) Jrpbg+ Jr gbp ⎡ XtJrXtJr()+ () ⎤ µε⎡⎤ct() () t+ kt () ε () t p11 g bp g p bg ⋅sgn⎡ mod(Ω−+rt ,λ ) L ⎤ ⎣⎦11pp⎢⎥22 ⎣ pbp AP⎦ ⎢⎥⎣ Jrpbg+ Jr gbp ⎦

()XtJrXtJrpgbpgpbg00()+ () µε⎡⎤ct ( ) ( t )+ kt ( ) ε ( t ) ⎣⎦00pp 22 Jrpbg+ Jr gbp

34 2.5 Effect of Sliding Friction in Example I

2.5.1 Validation of Example I Model using the FE/CM Code

The governing equations of either SDOF or MDOF system models are numerically integrated by using a 4th-5th order Runge-Kutta algorithm with fixed time

step. The ε p ()t and εp ()t components are neglected, i.e. no manufacturing errors other than specified profile modifications are considered. Concurrently, the dynamic responses are independently calculated by running the FE/CM code [2.8] using the Newmark method. Predicted and computed results are compared with good correlations in terms of the DTE, and LOA and OLOA forces, as shown in Figure 2.5 to Figure 2.7. Note that time domain comparisons include both transient and steady state responses but the frequency domain results report only the steady state responses. Figure 2.5 shows that the sliding friction introduces additional DTE oscillations when the contact teeth pass through the pitch point. Figure 2.6 illustrates that the sliding friction enhances the dynamic bearing forces in the LOA direction, especially at the second mesh harmonic.

This is because the moments associated with Fpfi ()t and Fgfi ()t are coupled with the moments of Npi(t) and Ngi(t).

(t) (in) (t) (f) (in) (t) (in) (t) (f) (in) (f)

Figure 2.5 Validation of the analytical (MDOF) model by using the FE/CM code (in the “dynamic” mode). Here, results for Example I are given in terms of δ ()t and its spectral contents ∆()f with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = 0.2. Key: , Analytical (MDOF) model; , FE/CM code (in t domain); o, FE/CM code (in f domain).

x 103 x 103

0.2 3

0.1 1

0 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 (a)Normalized time t / t (b) c Mesh order n x 103 x 103

0.2 3

0.1 1

0 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 (c) (d) Normalized time t / t c Mesh order n

Figure 2.6 Validation of the analytical (MDOF) model by using the FE/CM code (in the

“dynamic” mode). Here, results for Example I are given in terms of FpBx ()t and its spectral contents FpBx ()f with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = 0.2. Key: , Analytical (MDOF) model; , FE/CM code (in t domain); o, FE/CM code (in f domain).

37 Further, the normal loads mainly excite the vibration in the LOA direction, as illustrated by Eqs. (2.9), (2.11) and (2.13). The scales of the bearing forces of Figure

2.7(a-b) for µ = 0 case are the same as those of Figure 2.7(c-d) for the sake of comparison.

The bearing forces predicted by the MDOF model for µ = 0 case approach zero (within the numerical error range). This is consistent with the mathematical description of Eqs.

(2.15-2.16). Larger deviations at this point are observed in Figure 2.7(a-b) for the FE/CM analysis. Figure 2.7shows that the OLOA dynamics are more significantly influenced by the sliding friction when compared with the LOA results of Figure 2.6. In order to accurately predict the higher mesh harmonics, refined time steps (say more than 100 increments per mesh cycle) are needed. Consequently, the FE/CM analysis tends to generate an extremely large data file that demands significant computing time and post- processing work. Meanwhile, the lumped model allows much finer time resolution while being computationally more efficient (by at least two orders of magnitude when compared with the FE/CM). Hence, the lumped model could be effectively used to conduct parametric design studies.

3 x 103 x 10 0.2

0 (f) (lb) (f) (t) (lb) (t) 0.1 pBy pBy F F -5

-10 0 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 (a) (b) Normalized time t / tc Mesh order n

x 103 x 103 0.2 5

0 (t) (lb) (f) (lb) 0.1 pBy pBy F

F -5

-10

0 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 (c) Normalized time t / t (d) Mesh order n c

Figure 2.7 Validation of the analytical (MDOF) model by using the FE/CM code (in the

“dynamic” mode). Here, results for Example I are given in terms of FpBy ()t and its spectral contents FpBy ()f with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = 0.2. Key: , Analytical (MDOF) model; , FE/CM code (in t domain); o, FE/CM code (in f domain).

39 2.5.2 Effect of Sliding Friction

Figure 2.8 shows the calculated DTE without any friction is almost identical to

the STE at a very low speed ( Ω p = 2.4 rpm). However, the sliding friction changes the

shape of the DTE curve. During the time intervaltt∈ [0,P ], the friction torque on the pinion opposes the normal load torque as shown in Figure 2.4, resulting in a higher value of the normal load that is needed to maintain the static equilibrium. Also, friction increases the peak-peak value of the DTE as compared with the STE. For the remainder

of the mesh cycle ttt∈ [Pc , ], friction torque acts in the same direction as the normal load torque. Thus a small value of normal load is sufficient to maintain the static equilibrium.

Detailed parametric studies show that the amplitude of second mesh harmonic increases with the effect of sliding friction.

Figure 2.8 Effect of µ on δ ()t based on the linear time-varying SDOF model for

Example I at Tp = 2000 lb-in. Here, tc = 1 s. Key: , µ = 0; , µ = 0.1.

41 2.5.3 MDOF System Resonances

6 For Example I, the nominal bearing stiffness KKKKpBx====× pBy gBx gBy 20 10

lb/in are much higher than the averaged mesh stiffness km . The couplings between the rotational and translational DOFs in the LOA direction are examined by using a simplified 3 DOF model as suggested by Kahraman and Singh [2.12]. Note that the DTE

is defined here as δ =−rrbpθθ xp bg xg , and the undamped equations of motion are

⎡⎤ ⎡⎤⎧⎫ kk− k ⎧⎫ me 00δδ⎢⎥mm m ⎧⎫ 0 ⎢⎥⎪⎪⎢⎥ ⎪⎪⎪⎪ 00mxkkKkxppmmpBxmp⎨⎬ ++−() ⎨⎬⎨⎬ = 0. (2.21) ⎢⎥⎢⎥ ⎢⎥00mx⎪⎪ ⎪⎪⎪⎪ x 0 ⎣⎦gg⎩⎭⎢⎥−−kkkK + ⎩⎭ g⎩⎭ ⎣⎦mmmgBx()

22 Here, the effective mass is defined as mJJrJrJepggppg=+/ ( ) . The eigensolutions of

Eq. (2.21) yield three natural frequencies: Two coupled transverse-torsional modes ( f1

and f3 ) and one purely transverse mode ( f2 ); numerical values are: f1 = 5,130 Hz, f2 =

8,473 Hz and f3 = 11,780 Hz. Predictions of Eq. (2.21) match well with the numerical simulations using the formulations of section 2.3 (though these results are not shown here). A comparative study verifies that one natural frequency of the MDOF model shifts away from that of the SDOF model (6,716 Hz) due to the torsional-translational coupling effects. In the OLOA direction, simulation shows that only one resonance is present at

1 f = Km/ = 9,748 Hz, which is dictated by the bearing-shaft stiffness. pBy2π pBy p

42 2.6 Effect of Sliding Friction in Example II

Next, the proposed model is applied to Example II with the parameters of Table

2.2. The chief goal is to examine the effects of tip relief and sliding friction. Further, analogous experiments were conducted at the NASA Glenn Research Center Gear Noise

Rig [2.13]. Comparisons with measurements will be given in section 2.7.

2.6.1 Empirical Coefficient of Friction

The coefficient of friction varies as the gears travel through mesh, due to constantly changing lubrication conditions between the contact teeth. An empirical equation for the prediction of the dynamic friction variable, µ , under mixed lubrication has been suggested by Benedict and Kelley [2.14] based on a curve-fit of friction measurements on a roller test machine. Rebbechi et al. [2.15] verified this formulation by measuring the dynamic friction forces on the teeth of a spur gear pair. Their measurements seem to be in good agreement with the Benedict and Kelley equation except at the meshing positions close to the pitch point. This empirical equation, when

modified to account for the average gear tooth surface roughness ( Ravg ), is

8 ⎛⎞3.17× 10XWΓ (γ ) n 44.5 µγ( )= 0.0127CRavg log10 ⎜⎟2 , CRavg = . (2.22a,b) ⎝⎠νγosVV() e () γ 44.5− Ravg

43 where CRavg is the surface roughness constant, Wn is the normal load per unit length of

face width, and υo is the dynamic viscosity of the lubricant. Here Vs (γ ) is the sliding velocity, defined as the difference in the tangential velocities of the pinion and gear, and

Ve ()γ is the entraining velocity, defined as the addition of the tangential velocities, for

roll angle γ along the LOA. Further, Ravg in our case was measured with a profilometer

using a standard method [2.13]. Lastly, X Γ (γ ) is the load sharing factor as a function of roll angle, and it was assumed based on the ideal profile of smooth meshing gears. Figure

2.9 shows µ as a function of roll angle calculated using Eq. (2.22). Since µ was assumed to be a constant earlier, an averaged value is found by taking an average over the roll angles between 19.8 and 21.8 degrees. Table 2.3 lists the µ values that were

computed at each mean torque and oil temperature for Example II (with Ravg = 0.132

µ m).

Temperature (°F) Torque (lb-in) 500 600 700 800 900 104 0.032 0.033 0.034 0.035 0.036 122 0.034 0.036 0.037 0.037 0.038 140 0.036 0.037 0.038 0.039 0.040 158 0.038 0.040 0.041 0.041 0.042 176 0.040 0.041 0.042 0.043 0.044

Table 2.3 Averaged coefficient of friction µ predicted over a range of operating conditions for Example II by using Benedict and Kelly’s empirical equation [2.14] 44

Figure 2.9 Coefficient of friction µ as a function of the roll angle for Example II, as predicted by using Benedict & Kelley’s empirical equation [2.14]. Here, oil temperature is 104 deg F and T = 500 lb-in. Key P: Pitch point at 20.85 deg.

2.6.2 Effect of Tip Relief on STE and k(t)

The STE is calculated as a function of mean torque for both the perfect involute gear pair (designated as II-A) and then one with tip relief (designated as II-B) using

FE/CM code. Figure 2.10 compares the amplitudes of STE spectra at mesh harmonics for both cases. (In this and following figures, predictions are shown as continuous lines for the sake of clarity though they are calculated only at discrete torque points.)

45 x 10-4 2.5

1.5

Transmission error (in) error Transmission 0.5

0 100 300 500 700 900 (a) Torque (lb-in) Transmission error (in) error Transmission

Figure 2.10 Mesh harmonics of the static transmission error (STE) calculated by using the FE/CM code (in the “static” mode) for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B). Key: , n = 1; , n = 2; , n = 3. 46 The first two mesh harmonics are most significantly affected by the tip relief and they are minimal at the “optimal” mean torque around 500 lb-in. For both the II-A and II-

B cases, typical kt() functions of a single meshing tooth over two complete mesh cycles are calculated using Eq. (2.2) for various mean torques, as shown in Figure 2.1. Note that kt() is defined as the effective stiffness since it incorporates the effect of profile modification such as the linear tip relief (II-B). Observe that although the maximum stiffness remains the same, application of the tip relief significantly changes the stiffness profile. For the perfect involute profile (II-A), steep slopes are observed in the vicinities near the single or two teeth contact regimes, and a smooth transition is observed in between these steep regimes. Also, kt() is found to be insensitive to a variation in the mean torque. However, with tip relief, an almost constant slope is found throughout the transition profile between single and two teeth contact regimes. Moreover, a smaller profile contact ratio (around 1.1 at 100 lb-in) is observed for the tip relief case when compared with around 1.6 (at all loads) for the perfect involute pair. The realistic kt() function is then incorporated into the lumped MDOF dynamic model.

Figure 2.12 shows the combined kt() with contributions of both meshing tooth pairs over two mesh cycles for Example II. Observe that the profile of case II-A is insensitive to a variation in the mean torque, but the profile of case II-B shows a minimum around 500 lb-in. Frequency domain analysis reveals that the first two mesh harmonics are most significantly affected by the linear tip modification. Overall, it is evident that significant changes take place in the STE, tooth load distribution and mesh stiffness function due to the profile modification (tip relief), which may be explained by an avoidance of the corner contact at an “optimized” mean torque. 47

Figure 2.11 Tooth stiffness functions of a single mesh tooth pair for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B). Key: , 100 lb-in; , 500 lb-in; , 900 lb-in. 48

Figure 2.12 Combined tooth stiffness functions for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B). Key: , 100 lb-in; , 500 lb-in; , 900 lb-in. 49 2.6.3 Phase Relationship between Normal Load and Friction Force Excitations

Using the 6DOF spur gear model with parameters consistent with the experimental conditions, dynamic studies are conducted for Example II. First, a mean torque of 500 lb-in is used corresponding to the “optimal” case with minimal STE.

Equations (2.13-2.16) show that the normal loads ∑ Ni and friction forces ∑ Ffi excite the LOA and OLOA dynamics, respectively. The force profile of a single tooth pair undergoing the entire meshing process is obtained by tooth pairs #0 and #1 for two continuous meshing cycles as shown in Figure 2.13(a-b) and (c-d) for II-A and II-B cases respectively. Observe that the peak-to-peak magnitude of combined pinion normal load

∑ N pi is minimized for the tip relief gear due to reduced STE at 500 lb-in. However, the combine pinion friction force ∑ Ffpi with tip relief has a higher peak-to-peak magnitude when compared with the perfect involute gear. This implies that the tip relief amplifies

∑ Ffi in the OLOA direction while minimizing ∑ Ni in the LOA direction. Such

contradictory effects are examined next using the phase relationship between N pi and

Ffpi .

Figure 2.13 Dynamic loads predicted for Example II at 500 lb-in, 4875 RPM and 140 °F with tc = 0.44 ms: (a) Normal loads of gear pair with perfect involute profile (II-A); (b) normal loads of gear par with tip relief (II-B); (c) friction forces of gear pair with perfect involute profile (II-A); (d) friction forces of gear pair with tip relief (II-B). Key: , combined; , tooth pair #0; , tooth pair #1. 51 At points A, B, C and D, corner contacts are observed for N pi of the perfect involute gear, corresponding to the time instants when meshing tooth pairs come into or out of contact. These introduce discontinuous points in the slope of the ∑ N pi profile.

Note that N p1 and N p2 between A and B (or C and D) are in phase with each other, which should amplify the peak-to-peak variation of ∑ N pi . For the Ffpi profile of Figure

2.13(c), an abrupt change in the direction is observed at the pitch point P in addition to

the corner contacts. Unlike N pi , the profiles of Ffp1 and Ffp2 of Figure 2.13(c) between A and B (or C and D) are out of phase with each other. This should minimize the peak-to- peak variation of ∑ Ffpi . When tip relief is applied in Figure 2.13(b), corner contacts of

N pi are reduced and smoother transitions are observed at points A, B, C and D. Unlike

the perfect involute gear, N p1 and N p2 are now out of phase with each other between A and B (or C and D), which reduces the peak-to-peak variation of ∑ N pi . However, the

profiles of Ffp1 and Ffp2 of Figure 2.13(d) are in phase with each other in the same region, which amplifies the variation of ∑ Ffpi . The out of phase relationship between N pi and

Ffpi explains why the tip relief (designed to minimize the STE) tends to increase the friction force excitations. This relationship is mathematically embedded in Eq. (2.10) and

graphically illustrated in Figure 2.13, where N p1 and N p2 are in phase while Ffp1 and

Ffp2 are out of phase. Consequently, a compromise would be needed to simultaneously address the dynamic responses in both the LOA and OLOA directions.

52 2.6.4 Prediction of the Dynamic Responses

Dynamic responses including xp ()t , ytp (), Fpbx ()t , Fpby ()t and DTE δ ()t are predicted by numerically integrating the governing equations. Predictions from both perfect and tip relief gears are compared to examine the effect of profile modification in

the presence of sliding friction. Figure 2.14 shows that the normalized xp ()t at 500 lb-in is much smaller (over 90% reduction) when the tip relief is applied. This is because that the STE is the most dominant excitation in the LOA direction and it is minimized at 500 lb-in when the tip relief is applied. An alternate explanation is that the peak-to-peak variation of ∑ N pi is minimized with the tip relief as shown in Figure 2.13.

In the OLOA direction, more significant oscillations are observed for ytp () due to increased ∑ Ffpi excitations with tip relief. Despite that the vibratory components of

∑ N pi are larger than that of ∑ Ffpi , predicted ytp () is actually higher than xp ()t . This shows the necessity of including sliding friction when other excitations such as the STE

are minimized. Note that a phase difference is present in simulated ytp () before and after the tip relief is applied. Predicted pinion bearing forces are not shown here since they depict the same features as the displacement responses of Figure 2.14.

Figure 2.15 shows the DTE predictions, as defined by Eq. (2.17), with and without the tip relief. Similarity between Figure 2.14(a-b) and Figure 2.15(a-b) suggests that the relative LOA displacement plays a dominant role in the DTE responses. However, this conclusion is somewhat case specific as the DTE results depend on the mesh stiffness, bearing stiffness, and gear geometry.

Figure 2.14 Dynamic shaft displacements predicted for Example II at 500 lb-in, 4875

RPM and 140 °F with tc = 0.44 ms and fm = 2275 Hz: (a) xp ()t ; (b) Xfp (); (c) ytp () and

(d) Yfp (). Key: , gear pair with perfect involute profile in t domain (II-A); o, with perfect involute profile in f domain (II-A); , with tip relief (II-B).

Dynamic transmission error (in)

Figure 2.15 Dynamic transmission error predicted for Example II at 500 lb-in, 4875 RPM and 140 °F with tc = 0.44 ms and fm = 2275 Hz: (a)δ ()t ; (b) ∆()f . Key: , gear pair with perfect involute profile in t domain (II-A); o, with perfect involute profile in f domain (II-A); , with tip relief (II-B). 55 2.7 Experimental Validation of Example II Models

Experiments corresponding to Example II-B were conducted at the NASA Glenn

Research Center (Gear Noise Rig) to validate the MDOF spur gear pair model and to establish the relative influence of friction force excitation on the system. Figure 2.16 shows the inside of the gearbox, where a bracket was built to hold two shaft displacement probes one inch away from the center of the gear in the LOA and OLOA directions [2.13].

The probes face a steel collar that was machined to fit around the output shaft with minimal eccentricity. Accelerometers were mounted on the bracket, so the motion of the displacement probes could be subtracted from the measurements, if necessary. A thermocouple was installed inside the gearbox to measure the temperature of the oil flinging off the gears as they enter into mesh. The thermocouple position was chosen to be consistent with Benedict and Kelley’s [2.14] experiment. A common shaft speed of

4875 rpm is used in all tests so that the first five harmonics of the gear mesh frequency

(2275, 4550, 6825, 9100, and 11375 Hz) do not excite system resonances. Data of shaft displacement in the LOA and OLOA directions are collected from the proximity sensors under oil inlet temperatures over the range of temperatures (104, 122, 140, 158, and 176

°F). At each temperature the torque is varied from 500 to 900 lb-in increments of 100 lb- in.

56 Thermocouple OLOA

LOA

Proximity Probes Bracket Accelerometer

Figure 2.16 Sensors inside the NASA gearbox (for Example II-B).

Parametric studies are conducted to examine the dynamic responses under varying operational conditions of temperature and nominal torque. Benedict and Kelly’s [2.14] friction model is used to calculate the empirical µ as given in Table 2.3 and the realistic kt()calculated using FE/CM under varying torques are incorporated into the dynamic model. Since the precise parameters of the experimental system are not known [2.13], both simulated and measured data are normalized with respect to the amplitude of their first mesh harmonic of the OLOA displacement (which is then designed as 100%). This

57 facilitates the comparison of trends and allows simulations and measurements to be viewed in the same graphs from 0 to 100%.

Figure 2.17 compares the first five mesh harmonics of the LOA displacement as a function of mean torque. (In this and other figures, predictions are shown as continuous lines for the sake of clarity though they are calculated only at discrete points like the measurements.) It was observed that overall simulation trend matches well with the experiment. Magnitudes of the first two mesh harmonics are most dominant and they have minimum values around the optimized load due to the linear tip relief. Figure 2.17 also shows predicted first two harmonics for the prefect involute gear (II-A). Compared with the tip relief gear, they increase monotonically with the mean torque and have much higher values than the tip relief gear around the “optimal” torque.

Figure 2.18 compares the first five mesh harmonics of the OLOA displacement, on a normalized basis, as a function of mean torque. The overall simulation trend again matches well with the experiment. However, unlike the LOA responses, the first harmonic of OLOA displacement grows monotonically with an increase in the mean torque. This is because the friction forces increase almost proportionally with normal loads as predicted by the Coulomb law, but the frictional contribution of each meshing tooth pair tends to be in phase with each other for the tip relief gear (II-B). Thus it should amplify the combined friction force excitation in the OLOA direction. Consequently it is not reducing the OLOA direction responses induced by the sliding friction, even though the profile modification can be efficiently used to minimize gear vibrations in the LOA direction.

Figure 2.17 Mesh harmonic amplitudes of X p as a function of the mean torque at 140 °F.

All values are normalized with respect to the amplitude ofYp at the first mesh harmonic. Key: , n = 1 (prediction of II-B); , n = 2 (prediction of II-B); , n = 3 (prediction of II-B); , n = 1 (prediction of II-A); , n = 2 (prediction of II-A); , n = 1 (measurement of II-B); , n = 2 (measurement of II-B); O, n = 3 (measurement of II-B).

Figure 2.18 Mesh harmonic amplitudes of Yp as a function of the mean torque at 140 °F for Example II-B. All values are normalized with respect to the amplitude ofYp at the first mesh harmonic. Key: , n = 1 (prediction of II-B); , n = 2 (prediction of II-B); , n = 3 (prediction of II-B); , n = 1 (measurement of II-B); , n = 2 (measurement of II-B); O, n = 3 (measurement of II-B).

60 Figure 2.19 compares first five mesh harmonics of the normalized DTE for II-A and II-B cases over a range mean torques. Observe that the DTE spectral trends are very similar to the STE spectral trends of Figure 2.10. For example, the harmonic amplitudes of the perfect involute gear grow monotonically with mean torque while the harmonic amplitudes of the tip relief gear have minimum values around the “optimal” torque. Also, the DTE spectra show a dominant second harmonic, whose magnitude is comparable to that at the first harmonic. In some cases for the tip relief gear the second harmonic becomes the most dominant component especially when the mean torque is lower than

350 lb-in.

Finally, Figure 2.20 compares the first five mesh harmonics of the normalized

LOA displacement as a function of operational temperature. The changes in temperature are converted into variation in µ of Table 2.3. Compared with the OLOA motions, both predictions and measurements in the LOA direction give almost identical results at all temperatures. Figure 2.21 shows the first five mesh harmonics of the OLOA displacement with a change in temperature. The first harmonic varies quite significantly even though the changes in µ are relatively small. Consequently, the OLOA dynamics tends to be much more sensitive to a variation in µ as compared with the LOA motions.

Measured data of Figure 2.21 show some variations due to the experimental errors [2.13].

Figure 2.19 Predicted dynamic transmission errors (DTE) for Example II over a range of torque at 140 °F: (a) gear pair with perfect involute profile (II-A); (b) with tip relief (II-B). All values are normalized with respect to the amplitude of δ (II-A) at the first mesh harmonic with 100 lb-in. Key: , n = 1 (II-B); , n = 2 (II-B); , n = 3 (II-B). 62

Figure 2.20 Mesh harmonic amplitudes of X p as a function of temperature at 500 lb-in for Example II-B. All values are normalized with respect to the amplitude ofYp at the first mesh harmonic. Key: , n = 1 (prediction of II-B); , n = 2 (prediction of II-B); , n = 3 (prediction of II-B); , n = 1 (measurement of II-B); , n = 2 (measurement of II-B); O, n = 3 (measurement of II-B).

100

0 95 115 135 155 175 Temperature (deg F)

Figure 2.21 Mesh harmonic amplitudes of Yp as a function of temperature at 500 lb-in for

Example II-B. All values are normalized with respect to the amplitude ofYp at the first mesh harmonic. Key: , n = 1 (prediction of II-B); , n = 2 (prediction of II-B); , n = 3 (prediction of II-B); , n = 1 (measurement of II-B); , n = 2 (measurement of II-B); O, n = 3 (measurement of II-B).

64 2.8 Conclusion

Chief contribution of this study is the development of a new multi-degree of freedom, linear time-varying model. This formulation overcomes the deficiency of

Vaishya and Singh’s work [2.1-2.3] by employing realistic tooth stiffness functions and the sliding friction over a range of operational conditions. Refinements include: (1) an accurate representation of tooth contact and spatial variation in tooth mesh stiffness based on a FE/CM code in the “static” mode; (2) Coulomb friction model for sliding resistance with empirical coefficient of friction as a function of operation conditions; (3) a better representation of the coupling between the LOA and OLOA directions including torsional and translational degrees of freedom. Numerical solutions of the MDOF model yield the dynamic transmission error and vibratory motions in the LOA and OLOA directions. The new model has been successfully validated first by using the FE/CM code while running in the “dynamic” mode and then by analogous experiments. Since the lumped model is more computationally efficient when compared with the FE/CM analysis, it could be quickly used to study the effect of a large number of parameters.

One of the main effects of sliding friction is the enhancement of the DTE magnitude at the second gear mesh harmonic. A key question whether the sliding friction is indeed the source of the OLOA motions and forces is then answered by our model. The bearing forces in the LOA direction are influenced by the normal tooth loads, but the sliding frictional forces primarily excite the OLOA motions. Finally, effect of the profile modification on the dynamic transmission error has been analytically examined under the influence of frictional effects. For instance, the tip relief introduces an amplification in the OLOA motions and forces due to an out of phase relationship between the normal 65 load and friction forces. This knowledge should be of significant utility to the designers.

Future modeling work should examine the effects of other profile modifications and find the conditions for minimal dynamic responses when both STE and friction excitations are simultaneously present. Also, the model could be further refined by incorporating alternate friction formulations.

References for Chapter 2

[2.1] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration, 243(3), pp. 525-545.

[2.2] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671- 694.

[2.3] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.

[2.4] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, “Vibro-Acoustic Effects of Friction in Gears: An Experimental Investigation,” SAE Paper # 2001- 01-1516.

[2.5] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of Mechanical Design, 122(4), pp. 515-522.

[2.6] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp. 641-658.

[2.7] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3), pp. 479-499.

[2.8] External2D (CALYX software), 2003, “Helical3D User’s Manual,” www.ansol.www, ANSOL Inc., Hilliard, OH.

[2.9] Vinayak, H., Singh, R., and Padmanabhan, C., 1995, “Linear Dynamic Analysis of Multi-Mesh Transmissions Containing External, Rigid Gears,” Journal of Sound and Vibration, 185(1), pp. 1-32.

[2.10] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling Element Bearings. Part I: Bearing Stiffness Formulation,” Journal of Sound and Vibration, 139(2), pp. 179-199.

[2.11] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling Element Bearings. Part II: System Studies,” Journal of Sound and Vibration, 139(2), pp. 201-225.

[2.12] Kahraman, A., and Singh, R., 1991, “Error Associated with a Reduced Order Linear Model of Spur Gear Pair,” Journal of Sound and Vibration, 149(3), pp. 495-498.

[2.13] Singh, R., 2005, “Dynamic Analysis of Sliding Friction in Rotorcraft Geared Systems,” Technical Report submitted to the Army Research Office, grant number DAAD19-02-1-0334.

[2.14] Benedict, G. H., and Kelley B. W., 1961, “Instantaneous Coefficients of Gear Tooth Friction,” Transactions of the American Society of Lubrication Engineers, 4, pp. 59-70.

[2.15] Rebbechi, B., Oswald, F. B., and Townsend, D. P., 1996, “Measurement of Gear Tooth Dynamic Friction,” ASME Power Transmission and Gearing Conference proceedings, DE-Vol. 88, pp. 355-363.

67 CHAPTER 3

PREDICTION OF DYNAMIC FRICTION FORCES USING ALTERNATE

FORMULATIONS

3.1 Introduction

Gear dynamic researchers [3.1-3.6] have typically modeled sliding friction phenomenon by assuming Coulomb formulation with a constant coefficient (µ) of friction

(it is designated as Model I in this chapter). In reality, tribological conditions change

continuously due to varying mesh properties and lubricant film thickness as the gears roll

through a full cycle [3.7-3.10]. Thus, µ varies instantaneously with the spatial position of

each tooth and the direction of friction force changes at the pitch point. Alternate

tribological theories, such as elasto-hydrodynamic lubrication (EHL), boundary

lubrication or mixed regime, have been employed to explain the interfacial friction in

gears [3.7-3.10]. For instance, Benedict and Kelley [3.7] proposed an empirical dynamic

friction coefficient (designated as Model II) under mixed lubrication regime based on

measurements on a roller test machine. Xu et al. [3.8, 3.9] recently proposed yet another

friction formula (designated as Model III) that is obtained by using a non-Newtonian,

thermal EHL formulation. Duan and Singh [3.11] developed a smoothened Coulomb 68 model for dry friction in torsional dampers; it could be applied to gears to obtain a smooth transition at the pitch point and we designate this as Model IV. Hamrock and

Dawson [3.10] suggested an empirical equation to predict the minimum film thickness for two disks in line contact. They calculated the film parameter Λ, which could lead to a composite, mixed lubrication model for gears (designated as Model V). Overall, no prior work has incorporated either the time-varying µ()t or Models II to V, into multi-degree- of-freedom (MDOF) gear dynamics. To overcome this void in the literature, specific objectives of this chapter are established as follows: 1. Propose an improved MDOF spur gear pair model with time-varying coefficient of friction, µ()t , given realistic mesh stiffness profiles of Chapter 2; 2. Comparatively evaluate alternate sliding friction models and predict the interfacial friction forces and motions in the off-line-of-action (OLOA) direction; and 3. Validate one particular model (III) by comparing predictions to the benchmark gear friction force measurements made by Rebbechi et al. [3.12].

3.2 MDOF Spur Gear Model

Transitions in key meshing events within a mesh cycle are determined from the undeformed gear geometry. Figure 3.1(a) is a snapshot for the example gear set (with a contact ratio σ of about 1.6) at the beginning (t = 0) of the mesh cycle (tc). At that time, pair # 1 (defined as the tooth pair rolling along line AC) just comes into mesh at point A and pair # 0 (defined as the tooth pair rolling along line CD) is in contact at point C, which is the highest point of single tooth contact (HPSTC). When pair # 1 approaches the

69 lowest point of single tooth contact (LPSTC) at point B, pair # 0 leaves contact. Further, when pair #1 passes through the pitch point P, the relative sliding velocity of the pinion with respect to the gear is reversed, resulting in a reversal of the friction force. Beyond point C, pair # 1 will be re-defined as pair # 0 and the incoming meshing tooth pair at point A will be re-defined as pair # 1, resulting in a linear-time-varying (LTV) formulation. The spur gear system model is shown in Figure 3.1(b) and key assumptions for the dynamic analysis include the following: (i) pinion and gear are rigid disks; (ii) shaft-bearings stiffness elements in the line-of-action (LOA) and OLOA directions are modeled as lumped springs which are connected to a rigid casing; (iii) vibratory angular motions are small in comparison to the kinematic motion. Overall, we obtain a LTV system formulation, as explained in Chapter 2 with a constant µ . Refinements to the

MDOF model of Figure 3.1(b) with time-varying sliding friction µ()t are proposed as

follows. The governing equations for the torsional motions θ p ()t and θg ()t are as follows:

nn==floor(σσ ) floor( ) J pθ p()tT=+ p∑∑ XtFt pi () ⋅ pfi () − rNt bp ⋅ pi (), (3.1) ii==00

nn==floor(σσ ) floor( ) J gθ g()tT=− g +∑∑ XtFt gi () ⋅ gfi () + rNt bg ⋅ gi (). (3.2) ii==00

(a)

−Tgg,−Ω

J g ,mg K gBy

K kt() gBx

K pBy J pp,m

x K pBx Ω pp,T θ y

(b)

Figure 3.1 (a) Snap shot of contact pattern (at t = 0) in the spur gear pair; (b) MDOF spur gear pair system; here kt() is in the LOA direction.

71 Here, the “floor” function rounds off the contact ratio σ to the nearest integer

(towards a lower value); J p and J g are the polar moments of inertia for the pinion and

gear; Tp and Tg are the external and braking torques; N pi ()t and Ngi ()t are the normal

loads defined as follows:

⎡⎤ Ntpi()== Nt gi () ktr i ()⎣⎦ bp ⋅−⋅+−+θθ p () t r bg g () t xt p () xt g () (3.3) ⎡⎤ ctibppbggp()⎣⎦ r⋅−⋅+−θ () t rθσ () t x () t x g () t , i = 0, 1, ... , n = floor( ).

where kti ( ) and cti ( ) are the time-varying realistic mesh stiffness and viscous damping

profiles; rbp and rbg are the base radii of the pinion and gear; xp ()t and xg ()t denote the

translational displacements (in the LOA direction) at the bearings. The sliding (interfacial)

th friction forces Fpfi ()t and Fgfi ()t of the i meshing pair are derived as follows; note that

five alternate µ()t models will be described later.

Fpfi()ttNt= µ () pi (), Fgfi()ttNti==µ () gi (), 0, ... , n . (3.4a,b)

th The frictional moment arms Xtpi () and Xtgi () acting on the i tooth pair are:

Xtpi( )=+−+ L XA ( ni )λλ mod( Ω p rt bp , ), i = 0, ... , n, (3.5a)

Xtgi( )=+−Ω L YC iλλ mod( g rt bg , ), i = 0, ... , n. (3.5b)

72 where “mod” is the modulus function defined as: mod(xy , )= x−⋅ y floor( x / y ), if y ≠ 0 ;

“sgn” is the sign function; Ω p and Ωg are the nominal operational speeds (in rad/s); and λ

is the base pitch. Refer to Figure 3.1(a) for length LAP . The governing equations for the

translational motions xp ()t and xg ()t in the LOA direction are:

n=floor(σ ) mxp p() t+⋅⋅++= 2ζ pBx K pBx m p x p () t K pBx x p () t∑ N pi () t 0 , (3.6) i=0

n=floor(σ ) mxg g() t+⋅⋅++= 2ζ gBx K gBx m g x g () t K gBx x g () t∑ N gi () t 0. (3.7) i=0

Here, mp and mg are the masses of the pinion and gear; K pBx and K gBx are the

effective shaft-bearing stiffness values in the LOA direction, and ζ pBx and ζ gBx are their

damping ratios. Likewise, the governing equations for the translational motions ytp ()

and ytg () in the OLOA direction are written as:

n=floor(σ ) myp p() t+⋅⋅+−= 2ζ pB y K pBy m p y p () t K pBy y p () t∑ F pfi () t 0, (3.8) i=0

n=floor(σ ) myg g() t+⋅⋅+−= 2ζ gB y K gBy m g y g () t K gBy y g () t∑ F gfi () t 0. (3.9) i=0

73 3.3 Spur Gear Model with Alternate Sliding Friction Models

Following a similar modeling strategy of Chapter 2, we obtain a LTV system formulation. Refinements to the MDOF model with time-varying sliding friction µ()t are

th proposed as follows. The sliding (interfacial) friction forces Fpfi ()t and Fgfi ()t of the i

meshing pair are

Fpfi()ttNt= µ () pi (), Fgfi()ttNti==µ () gi (), 0, ... , n . (3.10a,b)

Five alternate µ()t models are described as follows:

3.3.1 Model I: Coulomb Model

The Coulomb friction model with time-varying (periodic) coefficient of

th frictionµCi (t ) for the i meshing tooth pair is derived as follows, where µavg is the

magnitude of the time-average.

⎡⎤ µµCi(trtniLin )=⋅ avg sgn⎣⎦ mod( Ω p bp , λ ) +−− ( ) λ AP , = 0, ... , . (3.11)

74 3.3.2 Model II: Benedict and Kelley Model

The instantaneous profile radii of curvature (mm) ρ()t of ith meshing tooth are:

ρ pi(tL )=+−+ XA ( ni )λλ mod( Ω p rt bp , ) , in= 0, ... , . (3.12a)

ρρgi(tL )=− XY pi ( ti ), = 0, ... , n. (3.12b)

th The rolling (tangential) velocities vtr ( )(m/s) of i meshing tooth pair are:

Ω ρ ()t Ω ρ ()t vt()= ppi , vt( )==ggi , i 0, ... , n. (3.13a,b) rpi 1000 rgi 1000

th The sliding velocity vts ( ) and the entraining velocity vte ( ) (m/s) of i meshing tooth pair are:

vtsi()=− vtvt rpi () rgi (), vtei( )=+ vt rpi ( ) vt rgi ( ) , i = 0, ... , n. (3.14a,b)

The unit normal load (N/mm) is wTZrnp=⋅/cos( wpα ) , where α is the pressure

angle, Z is the face width (mm), Tp is torque (N-mm) and rwp is the operating pitch radius of pinion (mm). Our µ()t prediction for the ith meshing tooth pair is based on the

Benedict and Kelley model [3.7], though it is modified to incorporate a reversal in the

direction of friction force at the pitch point. Here, SSSavg=+0.5( ap ag ) is the averaged

75 surface roughness ( µm ), and ηM is the dynamic viscosity of the oil entering the gear contact.

0.0127× 1.13 ⎡⎤29700w µλλ(trtniL )=⋅ logn ⋅Ω+−− sgn⎡ mod( , ) ( ) ⎤ , Bi 10 ⎢⎥2 ⎣ pbp AP⎦ 1.13− Svtvtavg⎣⎦η M si ( ) ei ( )

in= 0, ... , . (3.15)

3.3.3 Model III: Formulation Suggested by Xu et al.

th The composite relative radius of curvature ρr (t ) (mm) of i meshing tooth pair is:

ρ pi()ttρ gi () ρri ()t = , in= 0, ... , (3.16) ρρpi()tt+ gi ()

The effective modulus of elasticity (GPa) of mating surfaces is

22 ⎡⎤11−−ν p ν g E′ =+2/⎢⎥, where E and ν are the Young’s modulus and Poisson’s ratio, ⎣⎦⎢⎥EEpg respectively. The maximum Hertzian pressure (GPa) for the ith meshing tooth pair is:

wEn ′ Pthi ()= , in= 0, ... , . (3.17) 2000πρri (t )

76 Define the dimensionless slide-to-roll ratio SR() t and oil entraining velocity

th Vte () (m/s) of i meshing tooth pair as:

2()vtsi vtei () SRi () t = , Vtei ()= , in= 0, ... , . (3.18a,b) vtei () 2

The empirical sliding friction expression (for the ith meshing tooth pair), as proposed by Xu et al. based on non-Newtonian, thermal EHL theory [3.8-3.9], is modified in our work to incorporate a reversal in the direction of the friction force at the pitch point:

fSRtP()ihiMavg(), (), tη , S b b3 bbb 2 678 ⎡ ⎤ µηλλXi()t=⋅Ω+−− e PSRtV hi i () ei () t M Rt i ()sgnmod(⎣ p rt bp , ) ( ni ) L AP ⎦ ,

− SRihi() t P ()log( t 10 η M ) Savg f( SRtPi(), hi (), tηη M , S avg) =+ b14 bSRtP i () hi ()log( t 10 M ) + be 5 + be 9 ,

in= 0, ... , . (3.19a,b)

Xu [3.9] suggested the following empirical coefficients (in consistent units) for

the above formula: b1 =−8.916465 , b2 =1.03303 , b3 = 1.036077 , b4 =−0.354068 ,

b5 = 2.812084 , b6 =−0.100601, b7 = 0.752755, b8 =−0.390958 , and b9 = 0.620305 .

77 3.3.4 Model IV: Smoothened Coulomb Model

Xu [3.9] conducted a series of friction measurements on a ball-on-disk test machine and measured the µ()t values as a function of SR; these results resemble the smoothening function reported by Duan and Singh [3.11] near the pitch point (SR = 0) especially at very low speeds (boundary lubrication conditions). By denoting the

th periodic displacement of i meshing tooth pair as xipbpAP(trtniL )=Ω mod( ,λλ ) +−− ( ) , a smoothening function could be used in place of the discontinuous Coulomb friction of

Chapter 2. The arc-tangent type function is proposed as follows though one could also use other functions [3.11]:

22µavgµσ avg µSi(txtxt )=Φ⋅+⋅ arctan[] i ( ) i ( ) , in= 0, ... , (3.20) π ⎡ 22 ⎤ π ⎣1()+Φ xi t ⎦

In the above, the regularizing factor Φ is adjusted to suit the need of smoothening requirement. A higher value of Φ corresponds to a steeper slope at the pitch point. In our work, Φ = 50 is used for a comparative study.

3.3.5 Model V: Composite Friction Model

Alternate theories (Models I to IV) seem to be applicable over specific operational conditions. This necessitates a judicious selection of an appropriate lubrication regime as indicated by the film parameter, Λ, that is defined as the ratio of minimum lubrication

78 22 film thickness and composite surface roughness RRRcomp=+ rms,, g rms p measured with a

filter cutoff wave length Lx , where Rrms is the rms gear-tooth surface roughness [3.13].

The film parameter for rotorcraft gears usually lies between 1 and 10. In the mixed lubrication regime the films are sufficiently thin to yield partial asperity contact, while in the EHL regime the lubrication film completely separates the gear surfaces. Accordingly, a composite friction model is proposed as follows:

⎧µC (t ) simplified Coulomb model, computationally efficient (Model I) ⎪ ⎪ µB (t ) 1<Λ <4, mixed lubrication, (Model II) µ()t = ⎨ (3.21) µ (t ) 4≤Λ <10, EHL lubrication, (Model III) ⎪ X ⎪ ⎩µSpp(tT ) low ΩΛ< , high , 1, boundary lubrication (Model IV)

Application of Model II, III or IV would, of course, depend on the operational and tribological conditions though Model I could be easily utilized for computationally

efficient dynamic simulations. Note that the magnitude µavg of Model I or IV should be determined separately. For instance, the averaged coefficient based on Model II was used in Chapter 2. Also, the critical Λ value between different lubrication regimes must be carefully chosen. The film thickness calculation employs the following equation developed by Hamrock and Dowson [3.10, 3.13], based on a large number of numerical solutions that predict the minimum film thickness for two disks in line contact. Here, G is the dimensionless material parameter, W is the load parameter, U is the speed parameter,

H is the dimensionless central film thickness, and bH is the semi-width of Hertzian contact band:

3 Htci()ρ r1 () t× 10 LX Λ=i ()t , in= 0, ... , (3.22a) Rcomp2()bt Hi

8()wtnrρ 1 s btH1()= , Gk= ηMr E, (3.22b-c) π Er

0.56 0.69 GUi () t Htci ( )= 3.06 0.10 , (3.22d) Wti ()

ηMeivt() −6 wn Uti ()=× 10 , Wti ()= . (3.22e.f) 2()Etrriρ Erriρ ()t

3.4 Comparison of Sliding Friction Models

Figure 3.2(a) compares the magnitudes of µ()t as predicted by Model II and III

for the spur gear set of Chapter 2 given Tp = 22.6 N-m (200 lb-in) and Ω p = 1000 RPM.

The LTV formulations for meshing tooth pairs # 0 and 1 result in periodic profiles for both models. Two major differences between these two models are: (1) The averaged magnitude from Model II is much higher compared with that of Model III since friction under mixed lubrication is generally higher than under EHL lubrication; and (2) while

Model III predicts nearly zero friction near the pitch point, Model II predicts the largest µ value due to the entraining velocity term in the denominator.

80 0.1

0.09

0.08

0.07

0.06 µ 0.05

0.04

0.03

0.02

0.01

0 0 0.5 1 1.5 2 Normalized time t/t c (a)

0.11

0.1

0.09

0.08

0.07 (t)

µ 0.06

0.05

0.04

0.03

0.02 0 1000 2000 3000 4000 5000 Ω (RPM) p (b)

Figure 3.2 (a) Comparison of Model II [3.7] and Model III [3.8] given Tp = 22.6 N-m

(200 lb-in) and Ω p = 1000 RPM. Key: , pair # 1 with Model II; , pair # 0 with Model II; , pair # 1 with Model III; , pair # 0 with Model III; (b) Averaged magnitude of the coefficient of friction predicted as a function of speed using the composite Model V with Tp = 22.6 N-m (200 lb-in). Here, tc is one mesh cycle.

81 As explained by Xu [3.9], three different regions could be roughly defined on a µ versus SR curve. When the sliding velocity is zero, there is no sliding friction, and only rolling friction (though very small) exists. Thus, the µ value should be almost zero at the pitch point. When the SR is increased from zero, µ first increases linearly with small values of SR. This region is defined as the linear or isothermal region. When the SR is increased slightly further, µ reaches a maximum value and then decreases as the SR value is increased beyond that point. This region is referred to as non-linear or non-Newtonian region. As the SR is increased further, the friction decreases in an almost linear fashion; this is called as the thermal region. Model II seems to be valid only in the thermal region

[3.8, 3.9]. Figure 3.2(b) shows the averaged magnitude of µavg predicted as a function of

Ω p using the composite formulation (Model V) with Tp = 22.6 N-m (200 lb-in). An abrupt change in magnitude is found around 2500 RPM corresponding to a transition from the EHL to a mixed lubrication regime. Similar results could be obtained by plotting

the composite µ()t as a function of Tp . Though our composite model could be used to predict µ()t over a large range of lubrication conditions, care must be exercised since the calculation of Λ itself is based on an empirical equation [3.10].

Figure 3.3 compares four friction models on a normalized basis. The curves

between 0≤

82 1.5

1

(t) 0.5 µ

0

-0.5 Normalized Normalized

-1

-1.5

0 0.5 1 1.5 Normalized time t/t c

Figure 3.3 Comparison of normalized friction models. Key: , Model I (Coulomb friction with discontinuity); , Model II [3.7]; , Model III [3.8]; , Model

IV (smoothened Coulomb friction). Note that curve between 0≤ tt /c < 1 is for pair # 1; and the curve between 1≤

Figure 3.4 compares the combined normal loads and friction force time histories

as predicted by four friction models given Tp = 56.5 N-m (500 lb-in) and Ω p = 4875

RPM. Note that while Figure 3.3 illustrates µ()t for each meshing tooth pair the friction forces of Figure 3.4 include the contributions from both (all) meshing tooth pairs. Though alternate friction formulations dictate the dynamic friction force profiles, they have negligible effect on the normal loads.

83

1400

1350 (N) p N

1300

0 0.5 1 1.5 2

100

50

(N) 0 fp F -50

-100

0 0.5 1 1.5 2 Normalized time t/t c

Figure 3.4 Combined normal load and friction force time histories as predicted using alternate friction models given Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Key: , Model I; , Model II; , Model III; , Model IV.

84 3.5 Validation and Conclusion

Figure 3.5 compares the predicted LOA and OLOA displacements with alternate

friction models given Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Note that the differences between predicted motions are not significant though friction formulations and friction force excitations differ. This implies that one could still employ the simplified Coulomb formulation (Model I) in place of more realistic time-varying friction models (Models II to IV). Similar trend is observed in Figure 3.6 for the dynamic

transmission errors (DTE), defined as δθ()tr=−bp p () tr bg θ g () t +−xpg()txt (). The most significant variation induced by friction formulation is at the second harmonic, which matches the results reported by Lundvall et al. [3.6].

Finally, predicted normal load and friction force time histories (with Model III) are validated using the benchmark friction measurements made by Rebbechi et al. [3.12].

Results are shown in Figure 3.7. Based on the comparison, µ is found to be about 0.004 since it was not given in the experimental study. Here, we have made the periodic LTV definitions of meshing tooth pairs # 0 and 1 to be consistent with those of measurements, where meshing tooth pairs A and B are labeled in a continuous manner. Predictions

match well with measurements at both low ( Ω p = 800 RPM) and high ( Ω p = 4000 RPM) speeds. Ongoing research focuses on the development of semi-analytical solutions given a specific µ()t model and an examination of the interactions between tooth modifications and sliding friction.

85

0.15 -59.6

0.1 m) -59.8 m) µ µ ( ( p p x x -60 0.05

-60.2 0 0 1 2 1 2 3 4 5

2 1 1 0.8 m) m)

µ 0 µ 0.6 ( ( p p y y 0.4 -1 0.2 -2 0 0 1 2 1 2 3 4 5 Normalized time t/t Mesh order n c

Figure 3.5 Predicted LOA and OLOA displacements using alternate friction models given

Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Key: in time domain: , Model I; , Model II; , Model III; , Model IV; in frequency (mesh order n) domain: + , Model I; , Model II; , , Model III; +, Model IV.

86

24

23.5 m)

µ 23

22.5 DTE ( DTE 22

21.5 0 0.5 1 1.5 2 Normalized time t/t c

0.4

0.3 m) µ 0.2

DTE ( DTE 0.1

0 1 2 3 4 5 Mesh order n

Figure 3.6 Predicted dynamic transmission error (DTE) using alternate friction models given Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Key: in time domain: , Model I; , Model II; , Model III; , Model IV; in frequency (mesh order n) domain: + , Model I; , Model II; , , Model III; +, Model IV.

87 2000

1500

(N) 1000 p N 500

0 0 0.5 1 1.5 2

5

0 (N) fp F -5

0 0.5 1 1.5 2 Normalized time t/t c (a) 2500

2000

1500 (N) p

N 1000

500

0 0 0.5 1 1.5 2

4

2

0 (N) fp

F -2

-4

-6 0 0.5 1 1.5 2 Normalized time t/t c (b)

Figure 3.7 Validation of the normal load and sliding friction force predictions: (a) at Tp =

79.1 N-m (700 lb-in) and Ω p = 800 RPM; (b) at Tp = 79.1 N-m (700 lb-in) and Ω p = 4000 RPM. Key: , prediction of tooth pair A with Model III; , prediction of tooth pair B with Model III; X, measurement of tooth pair A [3.12]; , , measurement of tooth pair B [3.12]. 88 References for Chapter 3

[3.1] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration, 243(3), pp. 525-545.

[3.2] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671- 694.

[3.3] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.

[3.4] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of Mechanical Design, 122(4), pp. 515-522.

[3.5] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp 641-658.

[3.6] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3), pp. 479-499.

[3.7] Benedict, G. H., and Kelley B. W., 1961, “Instantaneous Coefficients of Gear Tooth Friction,” Transactions of the American Society of Lubrication Engineers, 4, pp. 59-70.

[3.8] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, “Prediction of Mechanical Efficiency of Parallel-Axis Gear Pairs,” ASME Journal of Mechanical Design, 129 (1), pp. 58-68.

[3.9] Xu, H., 2005, “Development of a Generalized Mechanical Efficiency Prediction Methodology,” PhD dissertation, The Ohio State University.

[3.10] Hamrock, B. J., and Dowson, D., 1977, “Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part III – Fully Flooded Results,” Journal of Lubrication Technology, 99(2), pp. 264-276.

[3.11] Duan, C., and Singh, R., 2005, “Super-Harmonics in a Torsional System with Dry Friction Path Subject to Harmonic Excitation under a Mean Torque”, Journal of Sound and Vibration, 285(2005), pp. 803-834.

[3.12] Rebbechi, B., Oswald, F. B., and Townsend, D. P., 1996, “Measurement of Gear Tooth Dynamic Friction,” ASME Power Transmission and Gearing Conference proceedings, DE-Vol. 88, pp. 355-363. 89

[3.13] AGMA Information Sheet 925-A03, 2003, “Effect of Lubrication on Gear Surface Distress.”

90 CHAPTER 4

CONSTRUCTION OF SEMI-ANALYTICAL SOLUTIONS TO SPUR GEAR

DYNAMICS

4.1 Introduction

Periodic differential equations [4.1-4.3] are usually needed to describe the gear

dynamics [4.4-4.12] since significant variations in mesh stiffness k(t) and damping c(t)

are observed, within the fundamental period tc (one mesh cycle). Additionally, dynamic

friction force Ff(t) and torque Mf(t) also undergo periodic variations, with the same period

tc, due to changes in normal mesh loads and coefficient of friction µ, as well as a reversal

in the direction of Ff(t) at the pitch point [4.4-4.6], especially in spur and helical gears.

th For the sake of illustration, typical ki(t) profiles and frictional functions fi(t) for the i meshing pair in spur gears are shown in Figure 4.1; derivations of fi(t) will be explained

later along with particulars of the example case. The fundamental nature of the linear

time-varying (LTV) system is illustrated in Figure 4.2(a); the system model is described in Chapter 2.

91 ) (lb/in) t ( i k

(a) f(t)

(b)

Figure 4.1(a) Realistic mesh stiffness functions of the spur gear pair example (with tip relief) given Tp = 550 lb-in. Key: , kt0 (); , kt1(). (b) Periodic frictional functions. Key: , f0 ()t ; , f1()t ; , f2 ()t . 92

(a)

Ωg

Ω p

(b)

Figure 4.2(a) Normal (mesh) and friction forces of 6DOF analytical spur gear system model. (b) Snap shot of contact pattern (at t = 0) for the example spur gear pair. 93 The governing single degree-of-freedom (SDOF) equation in terms of dynamic

transmission error (DTE) δθ()tr=−bp p () tr bg θ g () t is given below, where subscripts p and

g correspond to the pinion and gear, respectively; θ is the vibratory component of the

rotation; and rb is the base radius.

SS JtJ()⎡⎤ cttktt () () () () sgnmod(⎡ rt , ) ( Sj ) L⎤ ebiδδδµλλ+++Ω+−−∑∑⎣⎦ i⎣ pbp AP⎦ ii==00 (4.1a) ⋅+⎡⎤ct ()δδ () t kt () () t⎡⎤ X () tJr + X () tJr =+ T Tt () ⎣⎦ii⎣⎦ pigbpgipbge ,

22 Jepg= JJ, Jbgbppbg=+Jr Jr , TTJrTJrepgbpgpbg=+, (4.1b-d)

Xtpi( )=+−+ L XA ( Si )λ mod( Ω p rt bp ,λσ ), i = 0, ... , S = floor( ) , (4.1e)

Xtgi( )=+−Ω L YC iλ mod( g rt bg ,λσ ), i = 0, ... , S = floor( ) . (4.1f)

Further, J is the moment of inertia; T and Ω are the nominal torque and rotation speed;

and λ is the base pitch. Tooth pairs #1 and #0 are defined as the pairs rolling along line

AC and CD in Figure 4.2(b), respectively. The jth tooth pair passes though the pitch point

P during the meshing event, and the reversal at P is characterized by the sign function

“sgn” with a constant coefficient of Coulomb friction µ [4.4]. The modulus function

(mod(x, y) = x − y⋅floor(x/y), if y ≠ 0) is used to describe the periodic friction force Ff(t) and the moment arm X(t). The “floor” function rounds off the contact ratio σ to the nearest integers towards a lower value, i.e. S = 1 for the example case. Finally, L corresponds to the geometric length in Figure 4.2(b).

94 The chief goal of this chapter is to find semi-analytical solutions to Eq. (4.1) type

periodic systems which significantly differ from the classical Hill’s equation [4.1] in

several ways. First, the periodickti ( ) is not confined to a rectangular wave assumed by

Manish and Singh [4.4-4.6], or a simple sinusoid as in the Mathieu’s equation [4.1].

Instead, Eq. (4.1) should describe realistic, yet continuous, profiles of Figure 4.1(a) resulting from a detailed finite element/contact mechanics analysis [4.7]. Hence, multiple

harmonics of kti ( ) should be considered. Second, the periodic viscous cti ( ) term should

dissipate vibratory energy due to the sliding friction besides its kinematic effect. Third,

S S the ∑δii()tc () t and ∑δii()tk () t terms of Eq. (4.1) incorporate combined (but phase i=1 i=1

correlated) contributions from all (yet changing) tooth pairs in contact. Consequently, the

relative phase between neighboring tooth pairs should play an important role in the

resulting response δ ()t . Fourth, multiplicative effects betweenkti ( ) , cti ( ) ,()Xti and

δi ()t should result in higher mesh harmonics, which poses difficulty in constructing

closed-form solutions. Lastly, Tt() of Eq. (4.1) represents the time-varying component of

the forcing function due to unloaded (manufacturing) static transmission errorε ()t . This indicates that the frictional forces and moments reside on both sides of Eq. (4.1) as either periodically-varying parameters or external excitations, thus posing further mathematical complications.

95 4.2 Problem Formulation

Sliding friction has been found as a non-negligible excitation source in spur and

helical gear dynamics by Houser et al. [4.8], Velex and Cahouet [4.9], Velex and Sainsot

[4.10], and Lundvall et al. [4.11]. Earlier, Vaishya and Singh [4.4-4.6] developed a SDOF spur gear model with rectangular k(t) and sliding friction profiles; they solved the δ(t) response by using the Floquet theory [4.4] and multi-term harmonic balance method

(MHBM) [4.5]. Their work was recently refined and extended to helical gears in our work (refer to Chapters 5 and 6) where closed-form solutions of δ(t) for a SDOF system

are derived under frictional excitations. While the equal load sharing assumption [4.4-4.6]

yields simplified expressions and analytically tractable solutions, they do not describe

realistic conditions. This particular deficiency has been partially overcame in Chapters 2

and 3 where we proposed a multi-degree-of-freedom (MDOF) model with realistic k(t) and sliding friction functions. However, we utilized numerical integration and fast

Fourier transform (FFT) analysis methods in Chapters 2 and 3 that are often computationally sensitive. Hence, a semi-analytical algorithm based on MHBM is highly desirable for quick constructing frequency responses without any loss of generality.

Recently, Velex and Ajmi [4.12] implemented a similar harmonic analysis to approximate the dynamic factors in helical gears (based on tooth loads and quasi-static transmission errors). Their work, however, does not describe the multi-dimensional system dynamics or include the frictional effect, which may lead to “multiplicative” terms as described earlier.

96 The prime objective of this chapter is thus to extend the above mentioned publications [4.4]. In particular, we intend to develop semi-analytical harmonic balance solutions to the 6DOF spur gear model of Chapter 2 with realistic k(t) and sliding friction functions. The example case used for this study is the unity ratio NASA spur gear (with tip relief); refer to Table 2.2 of Chapter 2 for its parameters. Key assumptions include: (i) the pinion and gear are rigid disks; (ii) vibratory motions are small in comparison to the nominal motion; this would lead to a linear time-varying model; (iii) Coulomb friction is assumed with a constant µ though sign is reversed at the pitch point; (iv) when the torsional component is dominant over the translational component of δ(t) for the 6DOF model of Chapter 2, the harmonic solutions of the SDOF system could be extended to predict translational responses in the line-of-action (LOA) and off-line-of-action (OLOA) directions. Note that semi-analytical method analyzes the 6DOF system as a 5DOF model as it calculates the δ(t) and not absolute angular displacements θp(t) and θp(t); all 6 motion terms are determined in the numerical method.

4.3 Semi-Analytical Solutions to the SDOF Spur Gear Dynamic Formulation

Consider the example case with only the mean load Te, i.e. T(t) = 0 including

ε ()t = 0. Equation (4.1) can be rewritten over the mesh cycle 0 ≤ tt≤ c as follows:

97 ⎡⎤ mteδδδ()++⎣⎦ cttktt00 () () () ()[ 1 ++ EEt 13] ⎡⎤L , (4.2a) ++⎡⎤ct ( )δδ ( t ) kt ( ) ( t ) 1 ++⋅−= E Et sgn( tAP ) F ⎣⎦11⎢⎥() 23 e ⎣⎦⎢⎥Ω pbpr

⎡⎤ E1 =++µλ⎣⎦()LJrLJrJXA g bp YC p bg/ b , (4.2b)

⎡⎤ E2 =++µλ⎣⎦LJrXA g bp() L YC Jr p bg/ J b , (4.2c)

E3 =Ωµ prJrJrJbp( g bp − p bg) / b , mJJeeb= / , Feeb= TJ / . (4.2d-f)

Next express Eq. (4.2) in terms of the dimensionless timeτ = tt / c , such that

222 δ ′()τδτδ==dd / ttc () andδ ′′()τδτδ==dd / ttc ():

mtctkEtEfecδτ′′()++[]00 () τδτ ′ () c ()() τδτ[ 1 ++ 130 c () τ] , (4.3a) 2 ++tccc[][]11 (τδ )′ ( τ ) tk ( τδτ ) ( ) 1 ++= Ef 2132 ( τ ) Ef ( τ ) tF ce

f0 (τ )= mod(τ ,1) , ff10(τ )=−=− sgn[ mod(ττ ,1)PP] sgn[ ( ττ ) ] , (4.3b-c)

f201(τ )=−= mod(τττττ ,1)sgn[] mod( ,1)P ff ( ) ( ) , τ PAP= L / λ . (4.3d-e)

Each periodic function, ki (τ ) , ci (τ ) , fi (τ ) or δ ()τ now has a period of T = 1; Figure

4.1(b) shows the typical f0 ()τ , f1()τ and f2 ()τ functions, which describe the periodic moment arm and sliding friction excitations for the example case.

98 4.3.1 Direct Application of Multi-Term Harmonic Balance (MHBM)

Define the Fourier series expansions of the periodic ki (τ ) and ci(t) in Eq. (4.3) up

to N mesh harmonics as follows, where ωn = 2π n (in rad/s) and n is the mesh order.

NN kAikikinn()τ =+0 ∑∑ A cos(ωτ ) + B kinn sin( ωτ ). (4.4a) nn==11

1 1 1 Akd= ()τ τ , Ak= 2cos()ω ττ d, B = 2sin()kdω ττ. (4.4b-d) ki0 ∫0 i kin∫0 in n kin∫0 in n

NN ckIAAi(τ )==++ 2ζ i ( τ ) e ci0 ∑∑ cin cos( ωτ n ) B cin sin( ωτ n ) . (4.5a) nn==11

1 1 1 Acd= ()τ τ , Ac= 2cos()ω ττ d, B = 2sin()cdω ττ. (4.5b-d) di0 ∫0 i cin∫0 in n cin∫0 in n

The fi(τ) functions could be expanded explicitly as shown below:

11N f0 ()τ =−∑ sin(ωτn ), (4.6a) 2 n=1 nπ

NN4cos⎡⎤ωτ − 1 4sin(ωτnP) ⎣⎦( nP) f1()τ =− 1 2τωτωτPn −∑∑ cos( ) + sin( n ), (4.6b) nn==11ωωnn

⎧ N ⎫ ⎪∑ ⎣⎦⎡⎤1sincoscos()−−ωτnP() ωτ nP () ωτ nP ωτ n +⎪ 14⎪ n=1 ⎪ f ()ττ=−2 + . (4.6c) 2 P 2 ⎨ N ⎬ 2 ωn ⎪ ⎪ ∑ ⎣⎦⎡⎤ωnPτωτωτωωτcos()() nP−− sin nP 0.5 n sin( n ) ⎪⎩⎭n=1 ⎪

99 Finally, assume that the periodic dynamic responseδ ()τ is of the following form:

NN δ ()τωτωτ=+AAδδ0 ∑∑nn cos( ) + B δ nn sin( ). (4.7) nn==11

Substitute Fourier series expansions of Eqs. (4.4-4.7) into Eq. (4.3) and balance

the mean and harmonic coefficients of sin(ωnτ ) and cos(ωnτ ) . This converts the linear periodic differential equation into easily solvable linear algebraic equations (as expressed

below) where Kh is a square matrix of dimension (2N+1) consisting of known

coefficients ofki (τ ) , ci (τ ) andfi (τ ) . By calculating the inverse of Kh , the 2N+1 Fourier coefficients of δ ()τ could be computed at any gear mesh harmonic (n).

⎡⎤Aδ 0 ⎡Fe ⎤ ⎢⎥A ⎢ 0 ⎥ ⎢⎥δ1 ⎢ ⎥ ⎢⎥B ⎢ 0 ⎥ ⎡⎤δ1 ⎣⎦Kh ⎢⎥= ⎢ ⎥ . (4.8) ⎢⎥... ⎢... ⎥ ⎢⎥A ⎢ 0 ⎥ ⎢⎥δ N ⎢ ⎥ ⎣⎦⎢⎥Bδ N ⎣ 0 ⎦

4.3.2 Semi-Analytical Solutions Based on One-Term HBM

Next, we construct one-term HBM [4.13] solutions to conceptually illustrate the method. Set the harmonic order N = 1 (only the fundamental mesh, in addition to the mean term) in Eqs. (4.4-4.8) and balance the harmonic terms in Eq. (4.3). This leads to a

100 Kh matrix of dimension 3. Three of its typical coefficients are given as follows and the rest could be found in a similar manner:

1 ⎛⎞tAck11 A f 21 E 3+− B k 11 B f 11 E 2(/)22 tB ck 01 E 3π +++ A k 00 A k 10 A k 11 A f 11 E 2 K = (4.9a) h11 ⎜⎟ 2 ⎝⎠++222AEtAEkckkfckfckf00 1 00 3 + AAE 10 10 2 + tAAE 10 20 3 + tBBE 11 21 3

K =++A A AEAAE + + tAAE + AAE hkk21 01 11 01 1 kfckfkf 10 11 2 10 21 3 11 10 2 (4.9b) ++tAck11 A f 20 E 3() tA ck 01 E 3 / 2

Khkfkckf31=++++++BAE 11 10 2 B 11 tBA 11 20 E 3 ABE kfckf 10 11 2 tABE 10 21 3 BE k 01 1 B k 01 ttBE (4.9c) −+cckAE 01 3 π k 00 3 2

The Fourier series coefficients of δ ()τ are then obtained by inverting Kh . Figure

4.3 shows that one-term HBM solution predicts the overall tendency (mean and first

harmonic) fairly well when compared with numerical simulations at Tp = 550 lb-in and

Ω p = 500 RPM. This confirms that the one-term HBM (and likewise the MHBM) approach coverts the periodic differential Eq. (4.3) with multiple interacting coefficients into simpler algebraic calculations that are computationally more efficient than numerical integrations and subsequent FFT analyses. Thus, the semi-analytical solution provides an effective design tool. Also, most coefficients of Eq. (4.9) show side-band effects that are introduced by k(t) (or c(t)) and the fi(t) functions.

101 x 10-4 10.4

10

(in) 9.6

9.2

0 1 2 t/t c (a) (in)

(b)

Figure 4.3 Semi-analytical vs. numerical solutions for the SDOF model, expressed by Eq.

(4.3), given Tp = 550 lb-in, Ω p = 500 RPM, µ = 0.04. (a) Time domain responses; (b) mesh harmonics in frequency domain. Key: , , numerical simulations; , , semi-analytical solutions using one-term HBM; , , semi-analytical solutions using 5-term HBM. 102 4.3.3 Iterative MHBM Algorithm

When N ≤ 5, we can utilize a symbolic software [4.14] to balance multiple

harmonic terms and calculate Kh . However, the computational cost involved with each

3 element of Kh increases by N due to the triple multiplication of periodic coefficients in

Eq. (4.3). Consequently, for higher N (say >5), a direct computation of Kh becomes inefficient and thus inadvisable. Instead, we apply a matrix-based iterative MHBM algorithm [4.15, 4.16]. First, define variables Ω and ϑ:

Ω=2/π ()υtc , ϑ =Ωt ∈[0, 2π ) , ϑυϑπ= mod( / 2 , 1). (4.10a-c)

where υ is the sub-harmonic index; also define the differential operator “^” as:

dx xˆ ==Ω −1x , x = Ω xˆ . (4.11a,b) dϑ

Equation (4.3) is then converted into the following form:

ˆ 2 ˆˆ⎡⎤⎡⎤ Ω+Ω+mceδϑ() 00() ϑδϑ() k() ϑδϑ()⎣⎦1 ++ EE 14 ϑ ⎣⎦, (4.12) +Ω⎡⎤ ckϑδϑˆ + ϑδϑ⎡⎤1sgn/1 + EELF + ϑ λϑ − = ⎣⎦11()() ()()⎣⎦() 24()AP e

Or, express it more compactly as:

103 ~ 2 ˆ ~ ˆ Ω meδ ()ϑ + ΩC ()()ϑ δ ϑ + K ()()ϑ δ ϑ = Fe , (4.13a)

Ccϑϑ=+++++11sgn/1 EEc ϑϑ⎡ EE ϑλϑ L −⎤ , (4.13b) () 014124()()( ) ⎣ ( ) ( AP )⎦

Kkϑϑ=+++++11sgn/1 EEk ϑϑ⎡ EE ϑλϑ L −⎤ . (4.13c) () 014124()()( ) ⎣ ( ) ( AP )⎦

For the MHBM, a discrete Fourier transform (DFT) matrix is formed as follows, where

ϑi = i2π / M and M ≥ 2N +1:

⎡⎤1sin()ϑϑ11 cos( ) … sin( NN ϑ 1) cos( ϑ 1) ⎢⎥ 1 sin()()ϑϑ cos… sin ()NN ϑ cos () ϑ F = ⎢⎥22 2 2, (4.14a) ⎢⎥ ⎢⎥ ⎣⎦⎢⎥1sin()ϑϑMM cos ()… sin (NN ϑ M ) cos ( ϑ M )

⎧⎫ˆ ⎧⎫δϑˆ δϑ()1 ⎧⎫δϑ()1 ()1 ⎪⎪ ⎪⎪ ⎪⎪ δϑ ˆ ⎪⎪ˆ ⎪⎪()2 ⎪⎪δϑ()2 ˆ ⎪⎪δϑ() 2 δ ≡=∆⎨⎬F , δˆ ≡ ⎨⎬=∆FD , δˆ ≡ ⎨⎬2 =∆FD (4.14b-d) ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ δϑ()M ˆ ˆ ⎩⎭ ⎩⎭δϑ()M ˆ ⎩⎭⎪⎪δϑ()M

Here, ∆ is a vector of 2N +1 Fourier coefficients; and the Fourier differentiation matrix is given as:

104 ⎡⎤00 0… 0 0 ⎡00 0… 0 0⎤ ⎢⎥… ⎢ … ⎥ ⎢⎥00− 1 0 0 ⎢010− 0 0⎥ ⎢⎥01 0… 0 0 2 ⎢00− 1… 0 0⎥ D = ⎢⎥, D = ⎢ ⎥ (4.14e,f) ⎢⎥ 00 ⎢ 00⎥ ⎢⎥00 0… 0 −N ⎢00 0… −N 2 0⎥ ⎢⎥⎢ 2 ⎥ ⎣⎦00 0… N 0 ⎣00 0… 0 −N ⎦

Applying the DFT to the equation of motion yields the following MHBM equations where F+ is the Moore-Penrose or pseudo-inverse of the DFT matrix:

⎡⎤Ω+Ω+2mFDFCFFKF2 ++ ∆=, (4.15a) ⎣⎦e

K ≡ diag(){ K()ϑϑ12 K ()… K ( ϑM )} , (4.15b)

C ≡ diag(){ C()ϑϑ12 C ()… C ( ϑM )} . (4.15c)

Figure 4.3 shows that the five-term HBM solutions compare well with numerical simulations. Likewise, an increase in N captures higher frequency components around the

10th mesh harmonic, as observed in the numerical simulations. The semi-analytical solutions are efficiently used in Figure 4.4 for parametric studies of δ(t) at the gear mesh

harmonics over a range of Ω p ; and, these calculations are indeed achieved with reduced computational cost.

105

Ω Ω p p

Ω p Ω p

Figure 4.4 Semi-analytical vs. numerical solutions for the SDOF model as a function of pinion speed with µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key: , numerical simulations; , semi-analytical solutions using 5-term HBM.

106 4.4 Analysis of Sub-Harmonic Response and Dynamic Instability

Vaishya and Singh [4.5] examined the parametric instability of a spur gear pair

(with equal load sharing) via the sub-harmonic analysis. A similar approach is implemented along with following improvements. First, since realistic rather than rectangular k(t) profile is examined, system stability could now be evaluated as a function of Tp with contributions from profile modifications. Second, the sub-harmonic matrix was constructed earlier [4.5] as an external forcing function without the frictional effects. In the proposed work, sliding friction is characterized as a parametric excitation and then its effect on instability is examined. Re-expand δ ()τ in terms of the first sub-harmonic term

at 0.5ω0 and higher sub-harmonics such as1.5ω0 :

δu (τωτωτωτωτ )=+++ABABδδδδ0.5 cos( 0.5 ) 0.5 sin( 0.5 ) 1.5 cos( 1.5 ) 1.5 sin( 1.5 ) (4.16)

Substitute δu (τ ) into Eq. (4.3) and expand coefficients at sub-harmonic frequencies. By

balancing harmonics at 0.5ω0 and 1.5ω0 , the following equation is obtained

⎡⎤Aδ 0.5 ⎡0⎤ ⎢⎥B ⎢0⎥ ⎡⎤K ⎢⎥δ 0.5 = ⎢ ⎥ . (4.17) ⎣⎦u 44× ⎢⎥Aδ1.5 ⎢0⎥ ⎢⎥⎢ ⎥ ⎣⎦Bδ1.5 ⎣0⎦

107 For a non-trivial (unstable) solution to the above homogeneous equation, matrix

Ku must be singular. Such points correspond to period-doubling instability [4.5, 4.17]

and are found by computing the determinant Ku as a function of the ratio of mesh

frequency ωn and the natural frequency ΩNS of the corresponding linear time-invariant

SDOF system, which could be estimated from Ie and time-averaged mesh stiffness.

Figure 4.5 shows the normalized Ku for the example case given Tp = 100 lb-in (light

load) and Tp = 550 lb-in (design load); each is calculated for four damping ratios ζ . The

zero-crossing points of the Ku curve suggest the onset of instability [4.5]. Note that the

unstable zone depends on the value of Tp , which also influences k(t) and the “effective” contact ratio. For instance, larger period-doubling unstable zone is observed under a light load condition in Figure 4.5(a), as compared with the design loading condition in Figure

4.5(b). Further, instability could be effectively controlled by an increase in ζ. For

example, when Tp = 550 lb-in, 2% value is sufficient to stabilize the system under

period-doubling condition; however, about 8% is needed at Tp = 100 lb-in to achieve stability under the same condition. Also, a variation in µ seems to have negligible

influence on Ku ; however, in reality the energy dissipated by the sliding friction is usually embedded in an equivalent ζ. Thus an increase in µ should enhance the stability.

108

(a) | u K |

(b)

Figure 4.5 Normalized determinant of the sub-harmonic matrix Ku as a function of

ωnNS/ Ω with µ = 0.04: (a) Tp = 100 lb-in.; (b) Tp = 550 lb-in. Key: , ζ = 0; , ζ = 0.01; ,ζ = 0.05; , ζ = 0.1. 109 4.5 Semi-Analytical Solutions to 6DOF Spur Gear Dynamic Formulation

When the excitation (mesh) frequencies do not coincide with any natural

frequency, the semi-analytical solution, δ ()τθτθτ=−rrbp p () bg g (), of the SDOF model

could approximate the DTE, δ ()τθτθττ=−+−rrxxbp p () bg g () p () g () τ, for a 6DOF spur gear system of Chapter 2, where x(τ) and y(τ) are the bearing displacements in the LOA and OLOA directions respectively. First, the distribution of natural frequencies is examined by using a 6DOF linear time-invariant model of Figure 4.6(a). Since the OLOA motions yp(τ) and yg(τ) are decoupled from other DOFs, we will focus on the coupling (in terms of DTE) between transverse and torsional motions in the LOA direction. This leads to a simplified 3DOF system model [4.18] in terms of xp(τ), xg(τ) and δ ()t . Define the

following parameters for the example case: Mass of pinion (gear) mmMpg==; moment

2 of inertia J pg==JJ; basic radius rrRbp= bg = ; the equivalent mass mJRe = /(2 ) ;

time-averaged mesh stiffness km (Tp=550 lb-in.); shaft-bearing stiffness KBp==KK Bg B .

The natural frequencies of the 3DOF system are found as follows [4.18].

2 2 [][]kMmmBemmBeemB++(2 k K ) m ± kM ++ (2 k K ) m − 4 MmkK Ω=NN1, 3 , (4.18a) 2Mme

K Ω=2 B . (4.18b) N 2 M

110 θg

J, M KB

K km B

K B J, M x

KB θ y θ p

(a)

(b)

Figure 4.6 (a) 6DOF spur gear pair model and its subset of unity gear pair (3DOF model) used to study the natural frequency distribution; (b) Natural frequencies ΩN as a function of the stiffness ratio KBm / k . Key: , ΩSN of SDOF system (torsional only, in terms of DTE); , ΩN1 of 3DOF system; , ΩN2 of 3DOF system; , ΩN3 of 3DOF system. 111

Mode 1 ( ΩN1 ) and Mode 3 ( ΩN 3 ) correspond to the first and second coupled

transverse-torsional modes, respectively; and Mode 2 ( ΩN 2 ) is the purely transverse mode. Also, the natural frequency of the corresponding SDOF torsional system can be

estimated as Ω=NSkm m/ e . Figure 4.6(b) compares the natural frequencies ( ΩN ) of the

3DOF and SDOF models as a function of the stiffness ratioKBm / k . For easy comparison

with the excitation (mesh) frequencyωn , which is determined by the nominal pinion RPM, all natural frequencies are converted in Figure 4.6(b) from rad/s into RPM units. Observe

that ΩNS asymptotically approaches ΩN 3 (or ΩN1 ) when KkBm /< 1 (or KkBm /> 10 ).

Moreover, ωn does not excite any resonance in Zone I ( ωnN<<Ω 1 ) and Zone III

( ωnN>> Ω 3 ). Additionally, non-resonant Zone II could be found for both “soft”

(/KkBm< 1, Ω<<<<ΩNnNS2 ω ) and “stiff” (KkBm /> 10 , ΩNS<<ω n << Ω N 2 ) shaft- bearing cases. In these non-resonant zones, the semi-analytical solution δ(t) of the SDOF model could be extended to the 6DOF system. Figure 4.7 compares the 5-term HBM prediction of δ(t) for the SDOF system with numerical simulation for a 6DOF model for

two limiting value of KBm / k . When KkBm /= 100 , prediction matches well with

numerical simulation. However, when KkBm /= 0.37 (i.e. nominal case of Chapter 2),

good correlation is observed only away from system resonances ΩN in Zones I, II and III.

112

Ω p Ω p

Figure 4.7 Semi-analytical vs. numerical solutions for the 6DOF model as a function of

Ω p with µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key: , predictions using five-term HBM; , numerical simulations with nominal KB (KB/km = 0.37); , numerical simulations with stiff KB (KB/km = 100).

113 The dynamic normal (mesh) loads Ni (τ ) of the pinion and gear are equal in

magnitude but opposite in direction. To account for interactions between ki (τ ) andδ ()τ

as well as between ci (τ ) and δ (τ ) , Fourier series is expanded to find the Ni (τ ) terms up to 2N mesh harmonics.

22NN Nk00()τ =+=+ ()()τδτ c 0 ()() τδτ ANNnnNnn 000∑∑ A cos( ωτ ) + B 0 sin( ωτ ), (4.19a) nn==11

22NN Nk11()τ =+=+ ()()τδτ c 1 ()() τδτ ANNnnNnn 101∑∑ A cos( ωτ ) + B 1 sin( ωτ ). (4.19b) nn==11

Similarly, the dynamic friction forces Ffi ()τ are expanded in Eq. (4.20). Since tooth pair #1 is associated with a periodic change of friction force at the pitch point,

Ff 1()τ is expanded up to 3N mesh harmonics due to a multiplication of k1()τ , δ ()τ and

f1()τ .

22NN FNAAfFFnnFnn00000()τ ==+µ () τ∑∑ cos( ωτ ) + B 0 sin( ωτ ), (4.20a) nn==11

33NN FNfAAfFFnnFnn111101()τ ==++µτ () () τ∑∑ cos( ωτ ) B 1 sin( ωτ ). (4.20b) nn==11

In the LOA (or x) direction, the transfer function (at frequency ω a) with Np(τ) as

input and xp ()τ as output is found by using the corresponding linear time-invariant model

as follows, where K pBx and ζ pBx are the shaft-bearing stiffness and damping terms. 114 X t 2 p ()ω = c . (4.21a) 22 N p ()2−+ωωζmKtp pBx c + jt c pBx Km pBx p

th Magnitude M pxn and phase α pxn at the n mesh order, ωn = 2πτn (rad/s), are

t 2 M ()ω = c , (4.21b) px n 222222 ()4−+ωωζnmKt p pBx c + n t c pBx Km pBx p

2ωζtKm −1 nc pBx pBx p αωpx()tan n = 22. (4.21c) ωnpmKt− pBxc

Thus the pinion bearing displacement xp ()τ in the LOA (or x) direction is expanded as

2N ⎡ ⎤ xMAAp(τωωταω )=+++ px (0)( N00 N 10 )∑ M pxnNnNn ( )( AA 0 1 )cos⎣ n + pxn ( )⎦ n=1 . (4.22) 2N ⎡⎤ +++∑ MBBpx (ωωταω n )( N01 n N n )sin⎣⎦ n px ( n ) n=1

In the OLOA (or y) direction, the magnitude and phase of the transfer function

from friction force Ff(τ) to pinion displacement y p ()τ could be found at ωn as

t 2 M ()ω = c , (4.23a) py n 222222 ()4−+ωωζnmKt p pBy c + n t c pBy Km pBy p

2ωζtKm −1 nc pBy pBy p αωpy()tan n = 22. (4.23b) ωnpmKt− pByc 115 Thus the pinion displacement y p ()τ in the OLOA (or y) direction is expanded as:

yMAMAppyFpyF()τ =+ (0)00 (0) 10 22NN ⎡⎤ ⎡⎤ ++++∑∑MApy()ωωταωωωταω n F00 n cos⎣⎦ n py () n MB py () n F n sin ⎣⎦ n py () n . (4.24) nn==11 33NN ⎡⎤ ⎡⎤ ++++∑∑MApy()ωωταωωωταω n F11 n cos⎣⎦ n py () n MB py () n F n sin ⎣⎦ n py () n nn==11

Equations (4.22, 4.24) confirm that multiplications between periodic coefficients

ki ()τ (or ci (τ ) ), δi (τ ) (or ci (τ ) ) and fi (τ ) lead to higher mesh harmonic components which are commonly observed in spur gears [4.8].

4.6 Conclusion

Figure 4.8 compares the semi-analytical xp ()τ with numerical prediction as a

function of Ω p with KB/km = 100. Good correlations are observed up to 16,000 RPM (in

Zone 1 with high KB/km) including the LOA shaft-bearing resonances at n = 3 and n = 4.

Likewise, the semi-analytical y p in the OLOA direction compares well with numerical simulations in Figure 4.9, where the shaft-bearing resonances are also observed at n = 3 and n = 4.

116

Ω p Ω p

Ω Ω p p

Figure 4.8 Semi-analytical vs. numerical solutions of the LOA displacement xp for the

6DOF model as a function of Ω p with KB/km = 100, µ = 0.04 (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key: , numerical simulations; , predictions using five-term HBM.

117

Ω p Ω p

Ω Ω p p

Figure 4.9 Semi-analytical vs. numerical solutions of the OLOA displacement yp for the

6DOF model as a function of Ω p with KB/km = 100, µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key: , numerical simulations; , predictions using five-term HBM.

118 Although good correlations are usually expected for “stiff” shaft-bearings, it is worthwhile to examine the case where KB is comparable or less than km. By using the nominal parameters of Chapter 2 where KB/km = 0.37, semi-analytical predictions are compared with numerical simulations in Figure 4.10 and Figure 4.11 for LOA and OLOA

responses, respectively. Observe that semi-analytical xp matches numerical simulations

only from system resonances ΩN , as explained earlier in Figure 4.6. Nonetheless, good correlation is observed in the OLOA direction over the operating speed range in Figure

4.11 since the bearing resonance dictates the y p dynamics.

Overall, this chapter has successfully developed semi-analytical solutions to periodic differential equations with time-varying parameters of spur gears including realistic mesh stiffness and sliding friction. Proposed one-term and multi-term HBM predictions compare well with numerical simulations; the computational efficiency is achieved by converting the periodic differential equations into easily solvable algebraic equations, while providing more insight into the dynamic behavior. Both super-and sub- harmonic analyses are successfully conducted to examine the higher mesh harmonics due to multiplicative coefficients and the system stability, respectively. Finally, semi- analytical solutions are developed for a 6DOF system model for the predictions of

(normal) mesh loads, friction forces and bearing displacements in the LOA and OLOA directions, under non-resonant conditions. Methods of this work could be extended to multi-mesh spur gear dynamics.

119

x 10-4 x 10-5 1.2 2 Ω ΩN1 ΩN 2 NS (a) (b)

Z Z Z 0.8 1 2 3

1 Ω 0.4 N 3

0 0 0 5 10 15x103 0 5 10 15x103 (RPM) (RPM) x 10-5 Ω p x 10-5 Ω p 6 (c) (d) 0.8 4

0.4 2

0 0 0 5 10 15x103 0 5 10 15x103 (RPM) Ω Ω p (RPM) p

Figure 4.10 Semi-analytical vs. numerical solutions of the LOA displacement xp for the

6DOF model as a function of Ω p with KB/km = 0.37, µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key: , numerical simulations; , predictions using five-term HBM.

120

Ω p Ω p

Ω Ω p p

Figure 4.11 Semi-analytical vs. numerical solutions of the OLOA displacement yp for the

6DOF model as a function of Ω p with KB/km = 0.37, µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key: , numerical simulations; , predictions using five-term HBM.

121 References for Chapter 4

[4.1] Richards, J. A., 1983, Analysis of Periodically Time-varying Systems, New York, Springer.

[4.2] Jordan, D. W., and Smith, P., 2004, Nonlinear Ordinary Differential Equations, 3rd Edition, Oxford University Press.

[4.3] Thomsen, J. J., 2003, Vibrations and Stability, 2nd Edition, Springer.

[4.4] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration, 243(3), pp. 525-545.

[4.5] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671- 694.

[4.6] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.

[4.7] External2D (CALYX software), 2003, “Helical3D User’s Manual,” www.ansol.www, ANSOL Inc., Hilliard, OH.

[4.8] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, “Vibro-Acoustic Effects of Friction in Gears: An Experimental Investigation,” SAE Paper # 2001- 01-1516.

[4.9] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of Mechanical Design, 122(4), pp. 515-522.

[4.10] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp 641- 658.

[4.11] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3), pp. 479-499.

[4.12] Velex, P., and Ajmi, M., 2007, “Dynamic Tooth Loads and Quasi-Static Transmission Errors in Helical Gears – Approximate Dynamic Factor Formulae,” Mechanism and Machine Theory Journal, 42(11), pp. 1512-1526.

[4.13] Kim, T. C., Rook, T. E., and Singh, R., 2005, “Super- and Sub-Harmonic Response Calculations for A Torsional System with Clearance Non-Linearity using Harmonic Balance Method,” Journal of Sound and Vibration, 281(3-5), pp. 965-993.

122 [4.14] Maple 10 (symbolic software), 2005, Waterloo Maple Inc., Waterloo, Ontario.

[4.15] Padmanabhan, C., Barlow, R. C., Rook, T.E., and Singh, R., 1995, “Computational Issues Associated with Gear Rattle Analysis,” ASME Journal of Mechanical Design, 117, pp. 185-192.

[4.16] Duan, C., and Singh, R., 2005, “Super-Harmonics in a Torsional System with Dry Friction Path Subject to Harmonic Excitation under a Mean Torque”, Journal of Sound and Vibration, 285(2005), pp. 803-834.

[4.17] Den Hartog, J. P., 1956, Mechanical Vibrations, New York, Dover Publications.

[4.18] Kahraman, A., and Singh, R., 1991, “Error Associated with A Reduced Order Linear Model of Spur Gear Pair,” Journal of Sound and Vibration, 149(3), pp. 495-498.

[4.19] Singh, R., 2005, “Dynamic Analysis of Sliding Friction in Rotorcraft Geared Systems,” Technical Report submitted to the Army Research Office, grant number DAAD19-02-1-0334.

123 CHAPTER 5

EFFECT OF SLIDING FRICTION ON THE VIBRO-ACOUSTICS OF SPUR

GEAR SYSTEM

5.1 Introduction

Gears are known to be one of the major vibro-acoustic sources in many practical systems including ground and air vehicles such as heavy-duty trucks and helicopters.

Typically, steady state gear (whine) noise is generated by several sources and the reduction of gear noise is often challenging for most products. Virtually all prior researchers [5.1-5.3] have assumed the main exciter to be the static transmission error

(STE) that is defined as the derivation from the ideal tooth profile induced by manufacturing errors and elastic deformations. However, high precision gears are still unacceptably noisy in practice. When the transmission error has been minimized (say via modifying the tooth profile), the sliding friction remains as a potential contributor to gear noise and vibration. Further, most prior research on gear friction [5.4-5.5] has been confined to the dynamic analysis of the gear pair source sub-system and no attempt has been made to examine the friction related structural path and noise radiation issues. To

124 fill in this void, the main objectives of this chapter are thus to: First, propose a refined source-path-receiver model that characterizes the structural paths in two directions and, second, propose analytical tools to efficiently predict the whine noise and quantify the contribution of sliding friction to the overall whine noise. The system model is depicted in Figure 5.1. The source sub-system includes the spur gear pair and shafts inside the gearbox; these are characterized by a 6 degree-of-freedom (DOF) linear-time-varying model of Chapter 2. The transmission error dominated bearing forces in the line-of-action

(LOA) direction and friction dictated bearing forces in the off line-of-action (OLOA) direction are coupled and transmitted to the housing structure. Radiated sound pressure p(ω) from gearbox panels (at gear mesh frequencies) are then received by microphone(s).

Analytical predictions of the structural transfer function and noise radiation will be compared with measurements.

SOURCE Transmission LOA bearing forces error 6 DOF linear-time- Coupling at varying spur gear bearings Sliding pair model + shafts OLOA bearing forces friction

RECEIVER PATH

Sound Radiation Housing Housing structure pressure model velocity model

Figure 5.1 Block diagram for the vibro-acoustics of a simplified geared system with two excitations at the gear mesh. 125 5.2 Source Sub-System Model

The source sub-system is described by the 6DOF, linear time-varying spur gear

pair model of Chapter 2 that incorporates the sliding friction and realistic mesh stiffness,

which is calculated by an accurate finite element/contact mechanical code [5.6]. Rigid bearing is assumed as boundary conditions due to the impedance mismatch at the shaft/bearing interface. Overall, the system formulations are summarized as following.

The governing equations for the torsional motions θp(t) and θg(t) of pinion and gear are:

nn J ppθ ()tT=+ p∑∑ XtFt pi () pfi () − rNt bppi () (5.1) ii==00

nn J ggθ ()tT=− g +∑∑ XtFtrNt gi () gfi () + bggi () (5.2) ii==00

where n = floor(σ) in which the “floor” function rounds off the contact ratio σ to the

nearest integer (towards a lower value); Jp and Jg are the polar moments of inertia of the

pinion and gear; Tp and Tg are the external and braking torques; rbp and rbg are base radii

of the pinion and gear; and, Npi(t) and Ngi(t) are the normal loads defined as follows:

⎡ ⎤ N pi()tNtktrtr== gi () i ()⎣ bpθθ p () − bg g () txtxt +− p () g ()⎦ (5.3) ct ( )⎡⎤ r ( t ) r ( t ) x ( t ) x ( t ) +−+−ibppbggp⎣⎦θθ g

126 where ki(t) and ci(t) are the realistic mesh stiffness and viscous damping profiles; xp(t)

and xg(t) denote the LOA displacements of pinion and gear centers. The sliding friction

th forces Fpfi(t) and Fgfi(t) as well as their moment arms Xpi(t) and Xgi(t) of the i meshing pair are derived as:

Fpfi()ttNt= µ i () pi (), Fgfi()ttNt= µ i () gi () (5.4a,b)

Xtpi( )=+−+ L XA ( ni )λ mod( Ω p rt bp ,λ ), (5.5a)

Xtgi( )=+−Ω L YC iλ mod( g rt bg ,λ ) (5.5b)

where the sliding friction is formulated by ⎡ ⎤ ; λ µµipbpAP(trtniL )=Ω+−−0 sgn⎣ mod( , λ ) ( ) λ⎦ is the base pitch; “sgn” is the sign function; the modulus function mod(x, y) = x – y ·floor(x/y), if y ≠ 0; Ωp and Ωg are the nominal speeds (in rad/s); and, LAP, LXA and LYC are geometric length constants of Chapter 2. The governing equations for xp(t) and xg(t)

motions in the LOA direction are:

n mxp p() t+++= 2ζ pSx K pSx mx p p () t K pSx x p () t∑ N pi () t 0 (5.6) i=0

n mxg g() t+++= 2ζ gSx K gSx mx g g () t K gSx x g () t∑ N gi () t 0 (5.7) i=0

Here, mp and mg are the masses of the pinion and gear; KpSx and KgSx are the effective

shaft stiffness values in the LOA direction, and ζpSx and ζgSx are the damping ratios.

Likewise, the translational motions yp(t) and yg(t) in the OLOA direction are governed by: 127

n myp p() t++−= 2ζ pS y K pSy my p p () t K pSy y p () t∑ F pfi () t 0 (5.8) i=0

n myg g() t++−= 2ζ gS y K gSy my g g () t K gSy y g () t∑ F gfi () t 0 (5.9) i=0

Both LOA and OLOA bearing forces are predicted for the example case (unity-

ratio NASA spur gear pair whose parameters are listed in Table 5.1) and compared in

Figure 5.2 at the first three gear mesh frequencies as a function of pinion torque Tp.

Observe that the friction dominated OLOA dynamic responses are less sensitive to a variation in Tp.

Parameter/property Pinion/Gear Parameter/property Pinion/Gear

Number of teeth 28 Face width, in 0.25

Diametral pitch, in-1 8 Tooth thickness, in 0.191

Pressure angle, º 20 Center distance, in 3.5

Outside diameter, in 3.738 Elastic modulus, psi 30×106

Root diameter, in 3.139 Shaft stiffness, lb/in 1.29×105

Table 5.1 Parameters of the example case: NASA spur gear pair with unity ratio (with long tip relief)

128 6 m = 1 5 m = 2 m = 3 4

| (lb) 3 pBx |F 2

1

0 500 600 700 800 900 Torque (lb-in)

(a)

7

6

5

4 | (lb)

pBy 3 |F

2

1

0 500 600 700 800 900 Torque (lb-in)

(b)

Figure 5.2 Bearing forces predicted under varying Tp given Ωp = 4875 RPM and 140 °F. (a): LOA bearing force; (b) OLOA bearing force. Key: m is the mesh frequency index.

129 5.3 Structural Path with Friction Contribution

5.3.1 Bearing and Housing Models

Predicted bearing forces by the source sub-system provide excitations to the

multi-input multi-output (MIMO) structural paths for the gearbox of Figure 5.3(a). Force excitations are coupled at each bearing via a 6 by 6 stiffness matrix [K]Bm which is calculated by using the algorithm proposed by Lim and Singh [5.7]. Nominal shaft loads and bearing preloads are assumed to ensure a time-invariant [K]Bm. In order to focus on

the transmission error path and frictional path in the LOA and OLOA directions

respectively, [K]Bm is further simplified into a 2 by 2 matrix by neglecting the moment transfer [5.8] and assuming that no axial force is generated by the spur gear sub-system.

6 Calculated nominal bearing stiffness [5.7] are KBx = KBy= 2.8×10 lb/in at mean operating conditions; these are much larger than the shaft stiffness of 1.29×105 lb/in. This is

consistent with the impedance mismatch assumption made at the shaft/bearing interface.

130

(a)

(b)

Figure 5.3 (a) Schematic of NASA gearbox; (b) Finite element model of NASA gearbox with embedded bearing stiffness matrices. 131 The implementation of [K]Bm into the finite element gearbox model of Figure

5.3(b) is given special attention [5.9-5.10]. At high mesh frequencies (say up to 5 kHz), the dimensions of the bearings are comparable to the plate flexural wavelength. Hence the holes may significantly alter the plate dynamics and such effects must be modeled

[5.10]. A rigid (with Young’s modulus 100 times higher than the casing steel) and mass- less (with density 1% of the casing steel) beam element is used to model the interface from shaft to bearing. Its length is chosen to be very short to avoid the introduction of any

beam resonances in the frequency range of interest. The shaft beam element is connected

to the central bearing node though orthogonal foundation stiffness (KBx and KBy) in the

LOA and OLOA directions, respectively. The central node is then connected to the

circumferential bearing nodes by 12 rigid and mass-less beams (one at each rolling

element’s angular position) which form a star configuration, such that the displacement

of the plate around the bearing hole are equal to the “housing node” at the center.

5.3.2 Experimental Studies and Validation of Structural Model

The finite element model of Figure 5.3 is created by using I-DEAS for the NASA

gearbox with bearing holes, embedded stiffness matrices [K]Bm, stiffening plates as well

as clamped boundary conditions at four rigid mounts. Although the gear pair and shafts

are not included, it has been shown [5.7, 5.9] that an "empty" gearbox tends to describe

the dynamics of the entire gearbox system. Table 5.2 confirms that the natural

frequencies predicted by the finite element model correlate well with measurements

132 reported by Oswald et al. [5.11] despite minor modifications made to the gearbox. Mode shape predictions also match well with modal tests, and Figure 5.4 gives a typical

th comparison of structural mode at the 8 natural frequency (fn = 2962 Hz).

Method/mode index Measurements [5.11] Finite element predictions (Hz) (Hz)

1 658 650

2 1049 988

3 1709 1859

4 2000 1940

5 2276 2328

6 2536 2566

7 2722 2762

8 2962 2962

Table 5.2 Comparison of measured natural frequencies and finite element predictions

133

(a)

(b)

Figure 5.4 Comparison of the gearbox mode shape at 2962 Hz: (a) modal experiment result [5.12]; (b) finite element prediction.

134 In order to validate the structural paths, several transfer functions were measured

for the NASA gearbox by assuming that the quasi-static system response is similar to the

response under non-resonant rotating conditions. The gearbox was modified to allow

controlled excitations to be applied to the gear-mesh and measured. Brackets were

welded to the bedplate of the gear-rig to mount shakers in the LOA and OLOA directions outside the gearbox, as shown in Figure 5.5(a). Stinger rods were connected from the

shakers through two small holes in the gearbox and attached to a collar on the input shaft.

Two mini accelerometers were fastened to a block behind the loaded gear tooth to measure the LOA and OLOA mesh accelerations. Band-limited random noise signals were then used as excitation signals and tests were done with only one shaker activated at

a time with a 600 lb-in static preload. Dynamic responses were measured to generate

vibro-acoustic transfer functions. Sensor # 1 of Figure 5.5(a) is a tri-axial accelerometer

mounted on the output shaft bearing cap to measure the LOA, OLOA, and axial

vibrations. Sensors #2 and #3 are unidirectional accelerometers mounted on the top and

back plates, respectively.

The transfer function of the combined source-path sub-systems is predicted as

following:

Y ()ω HHHH()ωωωω=⋅=⋅ () () () plate (5.10) SP− S P S Ybearing ()ω

where H S ()ω is the motion transmissibility from mesh excitation to translational bearing

responses (in LOA or OLOA directions) by using a 6DOF linear time-invariant spur gear

135 model [5.13]. Note that such a lumped model is insufficient to capture the bending and

flexural modes of the gear flanks and shafts. Here, Yplate ()ω and Ybearing (ω ) are the transfer and driving point mobilities for the (top) plate and the bearing; these are derived from the finite element gearbox model by using the modal expansion method with 1% structural damping for all modes. Figure 5.5(b) shows that the predicted motion transmissibility from gear mesh to the top plate correlates reasonably well with measurement given the complexity of the geared system. The highest frequency is chosen such that the shortest wave-length is 4 times larger than the mesh dimension on the top plate. Recall that the interactions between the shaft and bearings/casing were neglected in our model by the impedance mismatch assumption. Consequently, a 10 dB empirical (but uniformly applied) weighting function w is used to “tune” the magnitude of transfer mobility prediction in Figure 5.5(b) for a better comparison. Further work is needed to explain this shift.

136

(a) 40

20

0

-20 |H| top plate (dB) plate top |H| -40

-60 500 1000 1500 2000 2500 3000 3500 Frequency (Hz)

(b)

Figure 5.5 (a) Experiment used to measure structural transfer functions; (b) comparison of transfer function magnitudes from gear mesh to the sensor on top plate. Key: , measurements; , predictions, . 137 5.3.3 Comparison of Structural Paths in LOA and OLOA Directions

First, assume that (i) the bearing forces predicted by the lumped source model

[5.6] are in phase at either bearing end for both the pinion and gear shafts; and (ii) the bearing forces of pinion and gear are same in magnitude but opposite in directions due to the symmetry of unity ratio gear pair. Second, the overall structural paths are derived for the transmission error controlled LOA (or x) path and the friction dominated OLOA (or y)

path in terms of combined effective transfer mobilities Yex, ()ω and Yey, ()ω :

YwYwYex,,,,,,,,,()ω =−∑∑ pxn pxn ()ωω gxn gxn () (5.11a) nn

YwYwYey,,,,,,,,,()ω =−∑∑ pyn pyn ()ωω gyn gyn () (5.11b) nn

where w is the empirical weighting function (10 dB, as discussed in the previous section); and the subscript n is the index of the two ends of pinion/gear shafts. Figure 5.6 compares

the magnitudes of Yex, ()ω and Yey, ()ω at the sensor location on the top plate. Different peaks are observed in the LOA and OLOA paths spectra. This implies that at certain frequencies (e.g. 650 and 1700 Hz), the OLOA path (and thus the frictional effects) could be dominant over the LOA path (and thus the transmission error effects) given comparable force excitation levels. The proposed method thus provides a design tool to quantify and evaluate the relative contribution of structural path due to sliding friction.

The top plate velocity distribution Vtop (ω ) could then be predicted by using Eq. (5.12),

where FpBx,,()ω and FpBy,,()ω are the pinion bearing forces predicted by the lumped 138 source model in the LOA and OLOA directions. Figure 5.7(a) shows the surface interpolated velocity distributions on the top plate at three mesh harmonics (m =1, 2, 3) given Tp = 500 lb-in and Ωp = 4875 RPM.

11 VFYFY()ω =+ ()ωω () () ωω () (5.12) top22 p,, B x e , x p ,, B y e , y

Figure 5.6 Magnitudes of the combined transfer mobilities in two directions calculated at the sensor location on the top plate. Key: , mobility of the OLOA path; , mobility of the LOA path.

139 5.4 Prediction of Noise Radiation and Contribution of Friction

5.4.1 Prediction using Rayleigh Integral Technique

Since the rectangular top plate is the main radiator [5.14] of the gearbox due to its relatively high mobility, Rayleigh integral [5.15] is used to approximate the sound pressure radiation by assuming that the top plate is included in an infinite rigid baffle and each elementary plate surface is an equivalent point source in the rigid wall. The sound

pressure amplitude is given as follows where ρ is the air density, QVSiii (ωω )=∆ ( ) is

th the source strength of i equivalent source with area ∆Si , k()ω is the wave number and

th ri is the distance of i source to the receiving point.

jωρ Q ()ω Pe()ω = ∑ i − jk()ω ri (5.13) 2π i ri

Compared with conventional boundary element analysis, Rayleigh integral approximates sound pressure in a fraction of the required computation time [5.16]. Hence, it is most suitable for parametric design studies. Although some researchers [5.16] have pointed out that Rayleigh integral may give large errors for sound pressure prediction if applied to strongly directional, three dimensional (3D) fields, such errors are not significant here due to the flat (rather than curved) top plate and favorable surroundings

(such as rigid side plates and anechoic chamber).

140 5.4.2 Prediction using Substitute Source Method

As an alternative to Rayleigh integral, a newly developed algorithm based on the substitute source approach [5.17] is used to compute radiated or diffracted sound field. It is conducted by removing the gearbox and introducing acoustic sources within the liberated space which yield the desired boundary conditions at the box surface (Neumann problem). Solutions are obtained in terms of the locations and/or the strengths of the substitute sources by minimizing the error function between original and estimated particle velocity normal to the interface surface [5.17].

Since the surface velocity distributions of gearbox are essentially symmetric along the center lines due to geometric symmetry, velocity distributions along the border lines of EFGH plane in Figure 5.3(b) are chosen to simplify the 3D gearbox into a 2D radiation model for simpler data representation as well as faster computation. Zero (negligible) velocity distribution is assumed along lines EF, FG and HE since the microphone

(receiver) is positioned above the center of major radiator, i.e. the top plate. A 2D line source uniformly pulsating with unit-length volume velocity Q′ is chosen as the substitute source. Its radiation field is the same in any plane perpendicular to the source line. Amplitudes of the sound pressure and radial velocity of such source are given by the

(2) following, where Hv is the Hankel function of second kind and order v.

kc()ω ρ P()ωωω= QHkr′ ()(2) [()], (5.14a) 4 0

k()ω VjQHkr ()ω =− ′ (2) [()]ω (5.14b) r 4 1

141 A “greedy search” algorithm [5.17] is used to search for “optimal” substitute sources: First, a large number of candidate source positions within the vibrating body are defined, e.g. at the vertices of a square grid. Second, a single position is first found which allows the point source to produce the smallest deviation between the original and estimated normal velocity of surface vibration. The estimation is then subtracted from the original velocity to get a velocity residual. Third, among the rest of candidate points, a new position is found which makes the second source acting at it, maximally reduce the velocity residual of the first step. Once found, the source strengths of both sources are adjusted for a best fit of the original surface velocity and a new residual velocity. Each subsequent step defines a new optimum source position among the ones not already used.

The curve fitting of source strengths is done by minimizing the mean square root (RMS) value of the velocity error. The vector of complex source strength Q′ is related (as shown

below) to the vector V n of complex normal surface velocity at control points via the

G G source-velocity transfer matrix T where rrrij= i− j and αij is the angle between vector

GG rri − and the outer normal to the surface.

−1 QTV′()ω = ()ωω n (), (5.15a)

k()ω TjHkr ()ω =− (2) [()]cos()ωα (5.15b) ij4 1 ij ij

To minimize the impact of an ill-conditioned matrix, the number of control points is kept well above that of independent source points. Minimization of the RMS error

142 using pseudo-inverse yields the following, where the asterisk signifies the conjugate transpose:

−1 QTTTV′()⎡⎤ ()** () () () (5.16) ω = ⎣⎦ωω ω n ω

The difference between synthesized and original surface normal velocities is:

∆=ΞVV()ω ()ωω n (), (5.17a)

−1 ** Ξ=()ω TTT ()ωωω⎣⎦⎡⎤ () () T () ω − I () ω (5.17b)

where I()ω is the identity matrix. The matrix Ξ ()ω appears as a velocity error matrix.

The RMS velocity error is normalized by dividing with the RMS value of original velocity as:

E ()ω VV()ω **ΞΞ ()ωωω () () e ()ω ==RMS nn (5. 18) RMS * VVVnRMS,()ωωω n () n ()

143 5.4.3 Prediction vs. Noise Measurements

Figure 5.7(a) shows predictions of surface interpolated velocity distribution on the top plate at the first three mesh harmonics under Tp = 500 lb-in and Ωp = 4875 RPM.

Note that predictions at high frequencies (e.g. mesh index m = 3) are less “reliable” due to the limitation of element dimensions as compared the wave length. The symmetry of surface velocity distribution leads to the simplification into a 2D gearbox model of Figure

5.7(b). To ensure necessary accuracy for the acoustic radiation, the selected central lines of the 2D plane should capture the dominant structural modes of Figure 5.7(a). Also, the structural wavelength along the central line should be higher than the acoustic wavelength of interest to ensure the validity of the 2D approach.

The source points of Figure 5.7(b) are chosen from a mesh grid of candidate points not too close to the boundary to prevent forming steep gradients of surface pressure. Observe that only 15 substitute sources tend to predict well the surface distribution of velocity magnitude at the gear mesh harmonics. Figure 5.7(c) illustrates the predicted source strengths of the substitute sources in the complex plane for evaluation of the acoustic source properties. A single dominant substitute source is observed at the first mesh harmonic (monopole-like acoustic source); however, several dominant substitute sources are present and these are more equally distributed in the complex plane at the higher harmonics (multi-pole acoustic source).

144

(a)

12 6 891 4 14 14 3 12 13 8 11 9 1 5 15 237 15 12 6 2 10 9 5 12 4 13 11 10 78 15 3414 13 67 10 5 11

(b)

90 5e-5 90 1.5e-5 90 1.5e-5 120 60 120 60 120 60 3e-5 1e-5 1e-5 (c) 150 30 150 30 150 30 5e-6 5e-6 1e-5

180 0 180 0 180 0

210 330 210 330 210 330

240 300 240 300 240 300 270 270 270 (c)

Figure 5.7 Comparison of normal surface velocity magnitudes and substitute source strength vectors under Tp = 500 lb-in and Ωp = 4875 RPM. (a) Line 1: interpolated surface velocity on top plate; (b) Line 2: simplified 2D gearbox model with 15 substitute source points; Key: , original surface velocity magnitude; , surface velocity magnitude by substitute sources; , locations of substitute sources. (c) Line 3: substitute source strengths in complex plane for 2D gearbox. Column 1: mesh frequency index m = 1; Column 2: m = 2; Column 3: m = 3. 145 The simplification from 3D into 2D gearbox model requires examining surface modes and a careful selection of representative plane, which poses additional limitations to its application as a universal method. However, once applicable, the substitute source method provides the following benefits. First, it enables an efficient evaluation of the acoustic source characteristics for whine noise. Second, unlike the Rayleigh integral which assumes that the top plate is part of an infinite rigid baffle, it takes the body shape into account and thus reduces the errors especially in the low frequency range. This also allows a straight forward synthesis of the radiation field for all the sources by using simple superposition as diffraction on the sources does not take place. Finally, compared with boundary element analysis, it does not suffer from the problems of singularities or uniqueness of solution. Nonetheless, it is an approximate method.

Figure 5.8 compares sound pressure measured at the microphone 6 inch above the top plate to predictions by using both the Rayleigh integral as well as the substitute source method under varying pinion torque given Ωp = 4875 RPM and 140 °F.

Predictions correlate well with measurements in terms of trends and relative magnitudes at first three gear mesh harmonics.

146

1.8

1.6

1.4

1.2

1 |P| 0.8

0.6

0.4

0.2

0 500 550 600 650 700 750 800 850 900 Torque (lb-in)

Figure 5.8 Normalized sound pressure (ref 1.0 Pa) predicted at the microphone 6 inch above the top plate under varying torque Tp given Ωp = 4875 RPM and 140 °F. Key: , measurements; , Rayleigh integral predictions , substitute source predictions. Blue, mesh frequency index m = 1; red, m = 2; black, m = 3.

147 5.5 Conclusion

A refined source-path-receiver model has been developed which characterizes the sliding friction induced structural path and associated noise radiation. Proposed Rayleigh integral method and substitute source technique are more efficient for calculating the acoustic field than the usual boundary element technique and thus they provide rapid design tools to quantify the frictional noise. Figure 5.9 compares the sound pressure level predicted given Ωp = 4875 RPM and 140 °F under Tp = 500 lb-in (close to the “optimal” load where transmission error is minimized) and under high torque with Tp = 800 lb-in.

At each gear mesh frequency, individual contributions of transmission error (via the LOA path) and frictional effects (via the OLOA path) are compared to the overall whine noise.

Observe in Figure 5.9(a) that near the “optimal” load, friction induced noise is comparable to the transmission error induced noise (especially for the first two mesh harmonics); thus sliding friction should be considered as a significant contributor to whine noise. However, at non-optimal torques in Figure 5.9(b), friction induced noise is overwhelmed by the transmission error noise, thus sliding friction could be negligible under such conditions. Further work is needed to extend the 2D gearbox into 3D model by using multipole substitute source technique [5.18] and fully examine the friction source, especially under varying lubrication conditions. Effects of the tooth surface finish should be examined as well.

148 100 Overall 90 LOA

80 OLOA

70

60 SPL (dB)SPL 50

40

30 123 Mesh index m

(a)

100 Overall 90 LOA

80 OLOA

70

60 SPL (dB) 50

40

30 123 Mesh index m

(b)

Figure 5.9 Overall sound pressure levels (ref: 2e-5 Pa) and their contributions predicted at the microphone 6 in above the top plate under Ωp = 4875 RPM and 140 °F. (a) Tp = 500 lb-in (optimal load for minimum transmission error); (b) Tp = 800 lb-in. 149 References for Chapter 5

[5.1] Houser, D. R., 1994, “Comparison of Transmission Error Predictions with Noise Measurements for Several Spur and Helical Gears,” NASA-Technical Memorandum, 106647, 30th AIAA Joint Propulsion Conference, Indianapolis, IN.

[5.2] Steyer, G., 1987, “Influence of Gear Train Dynamics on Gear Noise,” NOISE-CON 87 proceedings, pp. 53-58.

[5.3] Ajmi, M., and Velex, P., 2005, “A Model for Simulating the Quasi-Static and Dynamic Behavior of Solid Wide-Faced Spur and Helical Gears,” Mechanism and Machine Theory Journal, 40, pp. 173-190.

[5.4] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of Mechanical Design, 122(4), pp. 515-522.

[5.5] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3), pp. 479-499.

[5.6] External2D (CALYX software), 2003, “Helical3D User’s Manual,” www.ansol.www, ANSOL Inc., Hilliard, OH.

[5.7] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling Element Bearings. Part I: Bearing Stiffness Formulation,” Journal of Sound and Vibration, 139(2), pp. 179-199.

[5.8] Rook T. E., and Singh, R., 1996, “Mobility Analysis of Structure-Borne Noise Power Flow Through Bearings in Gearbox-Like Structures,” Noise Control Engineering Journal, 44(2), pp. 69-78.

[5.9] Van Roosmalen, A., 1994, “Design Tools for Low Noise Gear Transmissions,” PhD Dissertation, Eindhoven University of Technology.

[5.10] Rook, T. E., and Singh, R., 1998, “Structural Intensity Calculations for Compliant Plate-Beam Structures Connected by Bearings,” Journal of Sound and Vibration, 211(3), pp. 365-388.

[5.11] Oswald, F. B., Seybert, A. F., Wu, T. W., and Atherton, W., 1992, “Comparison of Analysis and Experiment for Gearbox Noise,” Proceedings of the International Power Transmission and Gearing Conference, Phoenix, pp. 675-679.

150 [5.12] Singh, R., 2005, “Dynamic Analysis of Sliding Friction in Rotorcraft Geared Systems,” Technical Report submitted to the Army Research Office, grant number DAAD19-02-1-0334.

[5.13] Holub, A., 2005, “Mobility Analysis of a Spur Gear Pair and the Examination of Sliding Friction,” MS Thesis, The Ohio State University.

[5.14] Jacobson, M. F., Singh, R., and Oswald, F. B., 1996, “Acoustic Radiation Efficiency Models of a Simple Gearbox,” NASA Technical Memorandum, 107226.

[5.15] Cremer, L., and Heckl, M., 1973, Structure-Borne Sound, Chapter 6, Sound Radiation from Structures, Springer-Verlag, New York.

[5.16] Herrin, D. W., Martinus, F., Wu, T. W., and Seybert, A. F., 2003, “A New Look at the High Frequency Boundary Element and Rayleigh Integral Approximations,” SAE, paper # 03NVC-114.

[5.17] Pavić, G., 2005, “An Engineering Technique for the Computation of Sound Radiation by Vibrating Bodies using Substitute Sources,” Acta Acustica Journal, 91, pp. 1-16.

[5.18] Pavić, G., 2006, “A Technique for the Computation of Sound Radiation by Vibrating Bodies Using Multipole Substitute Sources,” Acta Acustica Journal, 92, pp. 112-126.

151 CHAPTER 6

INCLUSION OF SLIDING FRICTION IN HELICAL GEAR DYNAMICS

6.1 Introduction

Sliding friction is believed to be one of the major sources of gear vibration and noise especially under high toque and low speed conditions since a reversal in the sliding

velocity takes place along the pitch line. Yet few analytical contact dynamics

formulations that incorporate friction are available in the literature [6.1-6.8]. In a series of

recent articles, Vaishya and Singh [6.1-6.3] reviewed various modeling strategies that

have been historically adopted and then illustrated issues for spur gears by assuming

equal load sharing among the contact teeth. Further, Velex and Cahouet [6.4] considered

the effects of sliding friction in their models for spur and helical gears. They found that

the dynamic bearing forces, as caused by friction at lower speeds, can generate

significant time-varying excitations. Velex and Sainsot [6.5] examined friction

excitations in errorless spur and helical gear pairs, and reported that the friction appears

as a non-negligible excitation source especially for translating motions. Lundvall et al.

[6.6] proposed a multi-body model for spur gears and briefly discussed the role of profile

152 modification in the presence of sliding friction. In Chapter 2, we have developed a more accurate model of the spur gears that includes realistic mesh stiffness and sliding friction, while overcoming the deficiency of Vaishya and Singh’s work [6.1-6.3]. In particular, this model shows that the tip relief could even amplify dynamic motions in some cases due to interactions between mesh and friction forces. Based on the literature review, it is clear that sliding friction has not been adequately considered in the dynamic models for helical gears. Work presented in this chapter attempts to fill this void.

6.2 Problem Formulation

In a helical gear pair, the line-of-action (LOA) lies in the tangent plane of the base cylinders, as defined by the base helix angle βb. Consequently, the moment arms for the out of plane moments change constantly. This phenomenon introduces axial and friction force shuttling effects [6.4]. Thus, the friction forces play a pivotal role in the loads transmitted to the bearings, especially in the off-line-of-action (OLOA) direction.

Blankenship and Singh [6.9] have developed a three-dimensional representation of forces and moments generated and transmitted via a gear mesh interface. This model is formulated in terms of the spatially-varying mesh stiffness and transmission errors which are assumed to be available from quasi-static elastic analyses. The vector formulation leads to a multi-body analysis of geared systems and force coupling due to vibratory changes in the contact plane is directly included. Similar strategy will be adopted here.

153 Objectives of this chapter include the following. First, propose a three-

dimensional formulation to characterize the dynamics associated with the contact plane

including the reversal at the pitch line due to sliding friction. Calculation of the contact

forces/moments will be illustrated via an example case (NASA-ART helical gear pair).

The tooth stiffness density along contact lines will then be calculated using a Finite

Element/Contact Mechanics (FE/CM) analysis code [6.10]. Second, develop a multi-

degree-of-freedom (MDOF) helical gear pair model (with 12 DOFs) which includes the

rotational and translational DOFs along the LOA, OLOA and axial directions as well as

the bearing/shaft compliances. Third, illustrate the effect of sliding friction including the

shuttling effect in the radial direction and coupling between the LOA and OLOA

directions. The following assumptions are made to derive the MDOF linear time-varying

system (LTV) model. 1. The position of the contact lines and relative sliding velocity

depend only on the mean angular motions of the gear pair. If this assumption is not made,

the system will have implicit non-linearities. 2. The mesh stiffness per unit length along

the contact line (or stiffness density k) is constant [6.7]. The constant k is estimated from the geometrical calculation of total length of contact lines and the mesh stiffness, which is computed using the finite element model [6.10]; this is equivalent to the equal load sharing assumption in spur gears [6.1-6.3]. 3. Coulomb’s law with a constant coefficient of friction (µ) is employed like previous researchers [6.1-6.3] though mixed lubrication regime exists. 4. The elastic deformations of the shaft and bearings are modeled using lumped parameter representations for their compliances. Also, it is assumed that the mean load is high and the dynamic load is not sufficient to cause tooth separations [6.11]. Thus

vibro-impacts are not considered here.

154 6.3 Mesh Forces and Moments with Sliding Friction

The helical gear system is depicted in Figure 6.1. The pinion and gear are modeled as rigid cylinders linked by a series of independent stiffness elements that

describe the contact plane tangent to the base cylinders. The normal direction at a point of contact lies in the contact plane (due to the involute profile construction) and is perpendicular to the line of contact. The pinion and gear dynamics are formulated in the coordinate systems located at their respective centers as shown in Figure 6.1. The

nominal motions of the pinion and gear are given as −Ω pez and Ω=Ωg eezzpbpbg rr/ , respectively. Here, z axis coincidences with the axial direction, e is the unit directional

vector and rbp and rbg are the base radii of pinion and gear. An (static) input torque Tp is

applied to the pinion and the (static) braking torque Tg on the gear obeys the basic gear

kinematics. Superposed on the kinematic motions are rotational vibratory motions

denoted by θzp and θzg for the pinion and gear. For the sake of illustration, analytical formulations are demonstrated via the following example case with parameters of the pinion (gear parameters are within the parenthesis): number of teeth 25 (31); outside

diameter 3.38 (4.13) in; pitch diameter 3.125 (3.875) in; root diameter 2.811 (3.561) in;

center distance 3.5 in; transverse diametral pitch 8 in-1; transverse pressure angle 25°;

helix angle 21.5°; face width 1.25 in; polar moment of inertia 8.33E-3 (1.64E-2) lb-s2-in; mass 1.26E-2 (1.58E-2) lb-s2/in. Since the overall contact ratio is around 2.7, either two or three tooth pairs are meshing with each other at any time instant.

155 θ

φ θ

βb

Figure 6.1 Schematic of the helical gear pair system.

The three meshing tooth pairs within a mesh cycle are numbered as #0, #1 and #2,

respectively. Calculate the (static) contact loads N0 ()t , N1()t , N2 ()t and (static) pinion

deflection Θzp ()t by performing a static analysis using the FE/CM formulation [6.10].

The stiffness K0 ()t , K1()t and K2 ()t of the three meshing tooth pairs are calculated as follows:

⎡⎤ Ktiibpzp( )=Θ Nt ( ) /⎣⎦ r ( t ) , i = 0,1,2 (6.1)

156 5 x 10 15

10

5 Tooth stiffness (lb/in) stiffness Tooth

t/T mesh 0 0 0.5 1 1.5 2 2.5

t/Tmesh

Figure 6.2 Tooth stiffness function calculated using Eq. (6.1) based on the FE/CM code [6.11]. Key: , tooth pair #0; , tooth pair #1; , tooth pair #2; , combined tooth pair stiffness.

For our example, the single tooth pair stiffness function K(t) are obtained by

following one tooth pair for three complete mesh cycles, as shown in Figure 6.2, where

Tmesh is the period of one mesh cycle. Observe that the stiffness profile has a trapezoidal shape; some discontinuities exist where corner contacts take place. The overall stiffness function, defined as a combination of all meshing tooth pairs, also follows a trapezoidal

pattern. Meanwhile, the sum of the lengths of contact lines can be calculated by using

either the FE/CM code [6.10, 6.12] or a simplified approximation method based on gear

geometry [6.13]. Since the total length of contact lines and combined tooth stiffness

157 follow a similar pattern, we can assume a constant mesh stiffness density (k) along the

contact lines. Consequently, the time-averaged k is estimated as follows, where Li is the length of the ith contact line.

22 kKtL≈ ∑ ii()/∑ (6.2) ii==00

Denote the LOA, OLOA and axial axes as x, y and z axes respectively; the dynamic motions of the pinion and gear centers consist of three translations (u) and three

T T rotations (θ) such that uuuup= { xpypzpxpypzp,,,,,θθθ} , uuuug= { xgygzgxgygzg,,,,,θθθ} . For

a contact point with local coordinates ( x,,rzbp ) in the pinion coordinate system, its global

motion when considered as part of the pinion is derived as:

⎧⎫⎧⎫uuzrtrxpθ xp ⎧ xp++Ω−θθ yp bp p bp zp ⎫ ⎪⎪⎪⎪ ⎪ ⎪ uupc=+×++=−−Ω+⎨⎬⎨⎬ ypθθθ yp() xerezeuzxtx x bp y z ⎨ yp xp p zp ⎬ (6.3a) ⎪⎪⎪⎪ ⎪ ⎪ ⎩⎭⎩⎭uurxzpθθθ zp ⎩ zp+− bp xp yp ⎭

Denote xg as the x coordinate at the gear center in the pinion coordinate system, the global motion of the contact point when it is considered as part of the gear is:

⎧⎫⎧⎫uuzrtrxgθθθ xg ⎧ xg++Ω+ yg bg g bg zg ⎫ ⎪⎪⎪⎪ ⎪ ⎪ ⎡⎤ (6.3b) ugcygygg=+×−−−+=−−−Ω−−⎨⎬⎨⎬ uθ ⎣⎦() xxerezeu xbgyz ⎨ ygxgg zθθ ()() xx gg txx zp ⎬ ⎪⎪⎪⎪ ⎪ ⎪ ⎩⎭⎩⎭uurxxzgθθθ zg ⎩ zg−+− bg xg() g yg ⎭

158 The deformation of the mesh spring is ∆mesh = (u pc − ugc )⋅ (−cos βbex + sin βbez ) , which can be further simplified as

⎡⎤ ∆=−meshcosβθθθθ b⎣⎦ (uu xp − xg ) + z ( yp − yg ) − ( r bpzp + r bgzg ) + (6.4) ⎡⎤ sinβθθθθbzpzgbpxpbgxgypg⎣⎦ (uu−+ ) ( r + r ) − ( x +− ( xx ) yg )

The velocities of the contact point when considered as part of the pinion or gear

(ignoring the vibratory component) are derived as:

vrexepcbppx=Ω −Ω py, vrexxegcbggxg=Ω −() −Ω gy (6.5a,b)

The relative sliding velocity of the pinion with respect to the gear is:

⎡⎤ vxs =−Ω+⎣⎦pg() xxe − Ω gy (6.6)

From Eq. (6.6) it is clear that vs is in the positive y direction at the beginning of

contact (small x). Further, vs becomes zero at the pitch point xP and changes to the

negative y direction when x > xP . Hence, Eq. (6.6) can be used to determine the direction

of the friction forces. The elemental forces on the meshing tooth pairs of the pinion and

the gear are given in Eq. (6.7) assuming only the elastic effects, and the total mesh forces

are derived by integrating the elemental forces over the contact line as given below in Eq.

(6.8): 159 ⎧ −∆k meshcos β b ⎫ ⎪ ⎪ ∆=−∆=∆Fmesh,, pFkxx mesh g⎨µ meshsgn( − P )⎬ (6.7) ⎪ ⎪ ⎩⎭k∆meshsin β b

⎧ ⎫ −∆kdlcos β ⎪ bmesh∫ ⎪ ⎪ l ⎪ ⎪ ⎪ F =−Fk =µ ∆sgn( xxdl − ) (6.8) mesh,, p mesh g⎨ ∫ mesh P ⎬ ⎪ l ⎪ ⎪ kdlsin β ∆ ⎪ ⎪ bmesh∫ ⎪ ⎩⎭l

To facilitate the integration, the contact zone is divided into three regions as

shown in Figure 6.1, where xb and xe are defined as the lower and upper boundaries of the contact zone in the LOA coordinate system of the pinion. In Region 1 (lightly shaded area within the contact zone of Figure 6.1), the lower and upper limits of x are denoted

as xmb= x and x f ≥ xb , respectively. The z coordinate (on the line of contact) is written as a function of the x coordinate as follows, where W is the face width.

zx()=+− 0.5 W⎡⎤ x x /tanβ (6.9a) ⎣⎦()f b

In Region 2 (white area within the contact zone of Figure 6.1), xmb≥ x and x f ≤ xe . The z coordinate of the contact point is derived as:

z()xWxxxxx=−−− 0.5( 2f mfm) /( ) (6.9b)

160 In Region 3 (darkly shaded area within the contact zone of Figure 6.1), xme≤ x

and x f = xe . The z coordinate of a contact point is given as:

zx()=− 0.5 W +⎣⎦⎡⎤() x − xmb /tanβ (6.9c)

Consider the integral in equation (6.8):

⎡⎤⎡⎤ −−+−−++cosβθθθθbxpxgypygbpzpbgzg⎣⎦ (uu ) z ( ) ( r r ) ∆= ∆dl = ⎢⎥ dl (6.10a) ∫∫mesh ⎢⎥sinβθθθθ⎡⎤ (uu−+ ) ( r + r ) − ( x +− ( xx ) ) ll⎣⎦bzpzgbpxpbgxgype⎣⎦ yg

Recognizing that dz=⋅ dl cos βb and dx=⋅ dl sin βb the above integral yields:

⎡⎤22 ∆=−()()(zzf − m⎣⎦ uu xp − xg − r bpθθ zp + r bg zg )0.5()() − z f − z m θθ yp − yg + (6.10b) ⎡⎤22 ()()(xxf −−++−−−−mzpzgbpxpbgxgeygfmypyg⎣⎦ uu rθ rθθ ) x 0.5()() x x θθ

Compare Eqs. (6.8) and (6.10) and represent the contact forces in the LOA and axial directions on the pinion as:

Fmesh, p,x = −k cos β b ∆ , Fmesh, p,z = k sin β b ∆ (6.11a,b)

161 Due to the sliding friction, the Fmeshpy,, involves a discontinuous sign function and it needs to be evaluated separately for three different cases assuming a constant µ . Case 1:

Both limits of the contact line are less than xP , which implies the contact line on the

pinion is completely below the pitch cylinder (approach action) so that sgn(xx−=−P ) 1.

The friction force of the pinion is

Fmesh, p, y = −µk∆ (6.12)

Case 2: The contact line lies on either side of the pitch cylinder. The integral of friction force need to be evaluated in two parts.

Fmesh, p, y = µk(∆2 − ∆1) (6.13a)

⎡⎤22 ∆=−1 ()()(zzP − m⎣⎦ uu xp − xg − r bpθθ zp + r bg zg )0.5()() − z P − z m θθ yp − yg + (6.13b) ⎡⎤22 (xxP−−++−−−− m )⎣⎦ ( uu zp zg ) ( r bpθ xp r bgθθ xg ) x e yg 0.5( xx P m )( θθ yp yg )

⎡⎤22 ∆=−2 ()()(zzf − P⎣⎦ uu xp − xg − r bpθθ zp + r bg zg )0.5()() − z f − z P θθ yp − yg + (6.13c) ⎡⎤22 (xxf −−++−−−−P )⎣⎦ ( uu zp zg ) ( r bpθ xp r bgθθ xg ) x e yg 0.5( xx f P )( θθ yp yg )

Case 3: Both limits are greater than xP , which implies the contact line on the pinion is

completely above the pitch cylinder (recess action). Consequently, sgn(xx−=P ) 1 and the friction force on the pinion is:

162 Fmesh, p, y = µk∆ (6.14)

Hence, in summary the mesh forces are derived as:

⎧ − k cos βb∆ ⎫ ⎪ ⎪ Fmesh, p = −Fmesh,g = ⎨− µk∆ or µk(∆2 − ∆1) or µk∆⎬ (6.15) ⎪ ⎪ ⎩ k sin βb∆ ⎭

The elemental moments on the pinion at a point on the contact line are derived as:

⎧rzxxbpsinβµ b−− sgn( P ) ⎫ ⎪ ⎪ ∆=∆−−Mkmesh, p mesh⎨ xsinββ b z cos b ⎬ (6.16) ⎪ ⎪ ⎩⎭xxxrµ sgn(−+Pbpb ) cos β

Integrating Eq. (6.16) over the contact line yields the total moments on the pinion as:

M =∆Mdl (6.17) mesh,, p∫ mesh p l

To facilitate the calculation of Eq. (6.17), define two integration operations over

the line of contact with lower and higher limits xl and xh , respectively:

163 xh 22 ()()xxzzhlhl+− ( xx hl − ) ()∆⋅x (,xxlh ) = x ∆ mesh dl =−() c12 + cz f + c 3 ∫ 22 xl (6.18a) 33 ()xxhl− 22⎛⎞xxhf−− x f ⎛⎞ xx lf x f −−++−+czz222 -(hf )⎜⎟ c ( zz lf ) ⎜⎟ c 33232⎝⎠ ⎝⎠

xh ()()z22−−zzz 33 ∆⋅zxx(, ) = z ∆ dl = ( − c + c tan)β hl − c hl ()lh∫ mesh13 b 2 x 23 l (6.18b) x tanβ ⎛⎞()zz33− zz (22− z ) −−−−czzcfb (22 ) tan 2β hl fhl 22hl b⎜⎟ 232⎝⎠

c1 = (uxp − uxg ) − (rbpθ zp + rbgθ zg ) , c2 =−()θ ypygθ (6.18c,d)

c3 = (uzp − uzg ) + (rbpθ xp + rbgθ xg ) − xeθ yg (6.18e)

Common integrals in the vector Eq. (6.17) are now evaluated as follows:

x∆ dl = ∆ ⋅ x (x , x ) , z∆=∆⋅dl z(,) x x (6.19a,b) ∫ mesh ()m f ∫ mesh( ) m f l l

xsgn(xx−∆ ) dl =−∆⋅ xxx ( , ) +∆⋅ xxx ( , ) (6.19c) ∫ Pmesh( ) mP( ) Pf l

z sgn(xx−∆ ) dl =−∆⋅ zxx ( , ) +∆⋅ zxx ( , ) (6.19d) ∫ Pmesh( ) mP( ) Pf l

Consequently, the moments of the mesh force acting on the pinion are given as:

⎧⎫rzxxzxxbp∆+∆⋅−∆⋅sinβµ b( ) ( m , P ) µ( ) ( P , f ) ⎪⎪ Mkmeshp, =−⎨⎬sinββ b() ∆⋅ xxxzxx ( mf , ) −∆⋅ () ( mf , )cos b (6.20) ⎪⎪ ⎩⎭−∆⋅µ ()xxx(,)mP +µβ () ∆⋅ xxx (,) P f +∆ r bp cos b

164 Similarly, the moments of the elemental mesh forces on the gear are given as:

⎧rzxxbgsinβµ b+− sgn( P ) ⎫ ⎪ ⎪ ∆=∆−+Mkmesh, g mesh⎨ x esinβββ b x sin b + z cos b ⎬ (6.21) ⎪ ⎪ ⎩⎭xxxxxxrg µ sgn(−−PPbgb )µβ sgn( −+ ) cos

The total moments due to the mesh forces on the gear are derived as:

⎧⎫rzxxzxxbg∆−∆⋅+∆⋅sinβµ b( ) ( m , P ) µ( ) ( P , f ) ⎪⎪ Mkxmeshg, =−∆⎨⎬ gsinββ b + sin b() ∆⋅ xxxzxx ( mf , ) +∆⋅ () ( mf , )cos β b (6.22) ⎪⎪ ⎩⎭xxxxxxxrg ()∆−∆+21µ () ∆⋅ (,)(,)cosmP −µβ () ∆⋅ P f + bg ∆ b

Note that Eqs. (6.15), (6.20) and (6.22) are formulated for a single tooth pair in contact. For multiple tooth pairs in contact, the dynamics of all meshing tooth pair must be considered. Consider an example to demonstrate the modeling strategy. Since the contact ratio is around 2.7, three tooth pairs are considered in a single mesh cycle. For a generic helical gear pair with contact ratio σ , n = ceil(σ ) number of meshing tooth pairs need to be considered by following the same methodology, where the “ceil” function rounds σ to the nearest integers towards infinity. Figure 6.3(a) shows the contact plan of the example case within a helical gear pair, and Figure 6.3(b) illustrates a zoomed-in snapshot of the contact zone at the beginning of a mesh cycle. At this instant, pair #0 just comes into mesh at point A and pair #1 is in contact along line CI. Likewise pair #2 contacts each other along line MN.

165 θzg

θzp

Figure 6.3 Contact zones at the beginning of a mesh cycle. (a) In the helical gear pair; (b) contact zones within contact plane. Key: PP’ is the pitch line; AA’ is the face width W; AD is the length of contact zone Z.

166 As the gears roll, the contact lines move diagonally across the contact zone. When pair #0 reaches the pitch point P, the relative sliding velocity of the pinion with respect to the gear starts to reverse, resulting in a reversal of friction force along the portion of contact line surpass the pitch point. Once pair #0 reaches the CI line (and pair #1 reaches

MN line) at the end of the mesh circle, pair #0 becomes #1 (and pair #1 becomes #2) corresponding to the start of the next mesh cycle. At any time instant t, the x coordinates

of the three pairs are projected along the AR line and denoted as x0 ()t , x1()t and x2 ()t , as defined below in Eq. (6.23). Here, λ is the base pitch, L represents the geometrical distance, and “mod” is the modulus function defined as mod(xy , )=−⋅ x y floor( x / y ), if y ≠ 0 .

x (trtL )=Ω mod( ,λ ) +, x (trtL )=Ω mod( ,λλ ) + + (6.23a,b) 0TApbp 1 1TApbp 1

x (trtL )=Ω mod( ,λλ ) + 2 + (6.23c) 2TApbp 1

To implement the integration algorithm, the contact regions are further divided into eight contact zones as shown in Figure 6.3(b). Zones 1 and 2 correspond to pair #0 before and after reaching the pitch line; Zones 3-5 and Zones 6-8 correspond to pairs #1 and #2, respectively. The zone classifications and their corresponding integration limits

for the calculation of dynamic forces and moments are derived as following, where xm ,

x f , zm and z f denote the lower and upper limits along x and z axes.

167 Zone 1 (LTA≤

Zone 2 (LTP≤< xtL 0 ( ) TC ): xLxxxLzmfpf = TA , = 0 , = TP , = 0.5 W , 111 1 (6.24b) z=−− 0.5 W⎡⎤⎡⎤ ( xL ) / tanβ , z =−− 0.5 W ( xL ) / tan β mbpb⎣⎦⎣⎦0TA11 0TP

Zone 3 (LxtLxLxxxLzTC≤< 1 ( ) TQ ): mfpf = TA , = 1 , = TP , = 0.5 W , 111 1 (6.24c) zWxL=−− 0.5⎡⎤⎡⎤ ( ) / tanβ , zWxL =−− 0.5 ( ) / tan β mbpb⎣⎦⎣⎦1TA11 1TP

Zone 4 (LTQ≤

Zone 5 (LxtLxxWTD≤< 1 ( ) TE ): mbfpm =− 1 tanβ , xLxLz = TD , = TP , =− 0.5 W , 11 11 (6.24e) zWxL=−− 0.5⎡⎤⎡⎤ ( ) / tanββ , zWxL =−− 0.5 ( ) / tan fbpb⎣⎦⎣⎦1TD11 1TP

Zone 6 (LTE≤

Zone 7 (LTH≤

Zone 8 (LxtL≤< ( ) ): No definition (6.24h) TG11 2 TR

Figure 6.4 shows the analytical tooth stiffness functions of each meshing tooth pair and the combined stiffness calculated for the example case by using the integration algorithm. Observe that both magnitude and shape of the stiffness functions in Figure 6.4 correlate well with those in Figure 6.2 which are obtained using a detailed FE/CM code

[6.10]. Note that the stiffness functions in Figure 6.4 are defined similar to Eq. (6.23), which is different from the definition of Eq. (6.1) corresponding to Figure 6.2.

168 x 10 5 16

12

8

4 Tooth Stiffness (lbf/in) Stiffness Tooth

0 0 0.5 1 1.5 2 2.5 t/T mesh

Figure 6.4 Predicted tooth stiffness functions. Key: , tooth pair #0; , tooth pair #1; , tooth pair #2; , combined tooth pair stiffness.

6.4 Shaft and Bearing Models

Consider the simplified shaft model of Figure 6.5 where l1 and l2 are the distances between the pinion/gear center to the bearing springs, E is the Young’s

4 modulus, Ir= π s /4 is the area moment of inertia and A is the cross sectional area

[6.14-6.15]. The shaft stiffness matrix [K ]S corresponding to the displacement vector

T [,xyz ,,θxyz ,θθ , ]is:

169 ⎡⎤KKSxx00 0− Sxθ 0 ⎢⎥y KK000 ⎢⎥Syy Syθx ⎢⎥K 000 K = ⎢⎥Szz (6.25) []S ⎢⎥KSθθ 00 ⎢⎥xx sym.K 0 ⎢⎥Sθθyy ⎢⎥ ⎣⎦0

where K ==KEIllllllll3/()() +⎡⎤ −+23 () is the bending stiffness, K = Sxx Syy 1 2⎣⎦ 1 2 12 12 Sθθxx

K =+3/EI l l l l is the rocking stiffness, K ==KEIllll3/22 − 2 is Sθθyy ()()12 12 Sxθθyx Sy ()12() 12

the rocking-bending coupled stiffness and KSzz = AE/ ( l12+ l ) is the longitudinal stiffness.

K /2 θθxx K /2 K zz /2 xx l K /2 1 θθxx K yy /2 K /2 K /2 l2 xx θθyy K zz /2 θx K yy /2 K /2 θθyy

θy

Figure 6.5 Schematic of the bearing-shaft model. Here, the shaft and bearing stiffness elements are assumed to be in series to each other. Only pure rotational or translational stiffness elements are shown. Coupling stiffness terms K , K are not shown. xθ y yθx

170 The rolling element bearings in Figure 6.5 are modeled by a stiffness matrix [K ]B of dimension six as proposed by Lim and Singh [6.16]. The mean shaft loads and bearing

preloads are assumed constant to ensure a time-invariant [K ]B matrix. Assume that each shaft is supported by two identical axially preloaded high precision deep groove ball bearings with a mean axial displacement. The helical gear pair is driven by a mean load

Tm which also generates mean radial force FBx in the LOA direction, axial force FBz and

moment M ym around OLOA direction. The [K ]B matrix for each bearing under mean loads has significant coefficients K , K , K , K , K , K , K , K and Bxx Byy Bzz Bθxθx Bθ yyθ Byθx Bxθ y Bxz

K . Note that the translational stiffness K , K and K are highly sensitive to the Bxθ y Bxx Byy Bzz axial preload which is quantified by the mean axial displacement. Thus we obtain:

⎡⎤KKBxx000 Bxz K Bxθ ⎢⎥y KK000 ⎢⎥Byy Byθx ⎢⎥KK00 []K = ⎢⎥Bzz Bzθ y (6.26) B ⎢⎥K 00 ⎢⎥Bθθxx ⎢⎥sym.K 0 ⎢⎥Bθθyy ⎣⎦⎢⎥0

The combined shaft-bearing stiffness matrix is derived as follows where <-1> implies term by term inverse.

171 ⎡KKxx000 xθ 0⎤ ⎢ y ⎥ KK000 ⎢ yy yθx ⎥ ⎢ ⎥ <−1 > K 000 ⎡⎤<−11 > <− > ⎢ zz ⎥ []KKKSB=+⎣⎦ [] S [] B = (6.27) ⎢ Kθθ 00⎥ ⎢ xx ⎥ sym.K 0 ⎢ θθyy ⎥ ⎢ ⎥ ⎣ 0⎦

The restoring forces due to the shaft/bearing stiffness cause forces and moments at the centers of pinion and gear. Consider the two springs at both ends of the pinion shaft, their corresponding forces on the pinion are as follows, where j is the bearing index:

⎧⎫−+−−Kul()θθ K Kul () −− θθ K xx,1 p xp p 1 yp xθθyy ,1 p yp xx ,2 p xp p 2 yp x ,2 p yp 2 ⎪⎪ FKulKKulK=−() −θ −θθθ − ( + ) − (6.28) ∑ SB,, p j⎨⎬ yy ,1 p yp p 1 xp yθθxx ,1 p xp yy ,2 p yp p 2 xp y ,2 p xp j=1 ⎪⎪ ⎩⎭−−Kuzz,1 p zp Ku zz ,2 p zp

The moments due to these bearing forces on the pinion are given as:

⎧⎫lKp1 yy ,1 p() u yp−+ l p 1θθ xp lK p 1 yθθ ,1 p xp + lK p 2 yy ,2 p () u yp ++ l p 2 θ xp lK p 2 y ,2 p θ xp 2 ⎪⎪x x MlKullKlKullK=−() +θ −θθθ − () − − (6.29) ∑ SB,, p j⎨⎬ p 1 xx ,1 p xp p 1 yp p 1 xθθy ,1 p yp p 2 xx ,2 p xp p 2 yp p 2 xy ,2 p yp j=1 ⎪⎪ ⎩⎭0

Similarly, the forces and moments due to the two bearings on the gear are:

172 ⎧⎫−+−−Kul()θθ K Kul () −− θθ K xx,1 g xg g 1 yg xθθyy ,1 g yg xx ,2 g xg g 2 yg x ,2 g yg 2 ⎪⎪ FKulKKullK=−() −θθ − − () + θ − θ (6.30) ∑ SB,, g j⎨⎬ yy ,1 g yg g 1 xg yθθx ,1 g xg yy ,2 g yg g 2 xg g 2 xy ,2 g yg j=1 ⎪⎪ ⎩⎭⎪⎪−−Kuzz,1 g zg Ku zz ,2 g zg

⎧⎫lKg1,1 yyg() u yg−+ l g 1θθ xg lK g 1 yθθ ,1 g xg + lK g 2,2 yyg () u yg + l g 2 θ xg + lK g 2 y ,2 g θ xg 2 ⎪⎪x x MlKullKlKullK=−() +θ −θθθ − () − − (6.31) ∑ SB,, g j⎨⎬ g 1 xx ,1 g xg p 1 yg p 1 xθθy ,1 p yp g 2 xx ,2 g xg g 2 yg g 2 xy ,2 g yg j=1 ⎪⎪ ⎩⎭0

6.5 Twelve DOF Helical Gear Pair Model

First, the viscous damping matrix is derived from the modal properties of the components by assuming modal damping ratios. In the 12 DOF model, the nominal external load is treated as excitations and the parametric excitations of tooth stiffness variation and friction effects are incorporated into a time-varying K matrix. Thus, a direct implementation of modal damping ratio will result in complex-valued viscous damping terms. Consequently, only the diagonal viscous damping terms (in the damping matrix) correspond to the directions of motions are considered, i.e. the diagonal viscous damping terms are assumed to be dominant over other coupling terms. More specifically:

(i) For the translational DOFs along x, y and z axes, the mesh damping force on pinion is:

⎧⎫−+⋅2 ζζKm Kmu ⎪⎪( xxp,1 xxp ,1 p xxp ,2 xxp ,2 p) xp 2 ⎪⎪ ∑ FV,, p j=−⎨⎬2()ζζ yy ,1 pKm yy ,1 p p + yy ,2 p Kmu yy ,2 p p ⋅ yp (6.32) j=1 ⎪⎪ ⎪⎪−+⋅2 ζζKm Kmu ⎩⎭()zzp,1 zzp ,1 p zzp ,2 zzp ,2 p zp

173 The mesh damping force on the gear is:

⎧⎫−+⋅2 ζζKm Kmu ⎪⎪( xxg,1 xxg ,1 g xxg ,2 xxg ,2 g) xg 2 ⎪⎪ ∑ FV,, g j=−⎨⎬2()ζζ yy ,1 gKm yy ,1 g g + yy ,2 g Kmu yy ,2 g g ⋅ yg (6.33) j ⎪⎪ ⎪⎪−+⋅2 ζζKm Kmu ⎩⎭()zz,1 g zz ,1 g g zz ,2 g zz ,2 g g zg

(ii) For the rotational DOFs along the x and y directions (rocking motions), the mesh damping moments for the pinion and gear are:

MKJ=−2ζ ⋅θ , MKJ= −⋅2ζ θ (6.34a, b) V,,,θθθθθxxxxx p p p xp xp Vp,,,θθθθθyyyyy p pypyp

MKJ=−2ζ ⋅θ , MKJ= −⋅2ζ θ (6.35a, b) V,,,θθθθθxxxxx g g g xg xg V,,,θθθθθyyyyy g g g yg yg

(iii) For the rotational DOFs along the axial directions, the time-varying tooth mesh damping is dominant. Thus the damping moments are:

n=floor(σ ) MJrKtrr=−2/ζ () ⋅θθ + (6.36) Vp,,,θθzz p zpbp∑ meshpi θ z() rpzpbgzg i=0

n=floor(σ ) MJrKtrr=−2/ζ () ⋅θθ + (6.37) Vg,,,θθzz g zpbg∑ meshgi θ z() rpzpbgzg i=0

where i is the index for contact tooth pairs, and the floor function rounds the contact ratio

σ to the nearest integer towards minus infinity. Here, rrrpθ zp+ bgθ zg is the relative

174 dynamic velocity along the LOA direction; K ()t and K ()t are the time- mesh,θz p mesh,θz g varying dynamic tooth stiffness functions derived earlier but considering only the rotational DOF in the z direction, and they also incorporate the effect of sliding friction.

The equations of motion for the 12 DOF helical gear pair model are derived as follows. The pinion equations in the three translational directions are:

n=floor(σ ) 22 mup xp=++∑∑∑ F meshxpi,, F SBxpj ,, F Vxpj ,, (6.38a) ijj===011

n=floor(σ ) 22 mup yp=++∑∑∑ F mesh,, yp i F SB ,, yp j F V ,, yp j (6.38b) ijj===011

n=floor(σ ) 22 mup zp=++∑∑∑ F mesh,, zp i F SB ,, zp j F V ,, zp j (6.38c) ijj===011

The pinion equations in the three rotational directions along x, y and z axes are:

n=floor(σ ) 2 JMMMθ =++ (6.39a) xp xp∑∑ mesh,,θx p i SB ,,θθxx p j V , p ij==01

n=floor(σ ) 2 JMMMθ =++ (6.39b) ypyp∑∑ meshpi,,θ yyy SBpj ,,θθ V , p ij==01

n=floor(σ ) J θ =++MMT (6.39c) zpzp∑ meshpi,,θθzz V , p p i=0

The gear equations in the three translational directions are:

175 n=floor(σ ) 22 mug xg=++∑∑∑ F mesh,, xg i F SB ,, xg j F V ,, xg j (6.40a) ijj===011

n=floor(σ ) 22 mug yg=++∑∑∑ F mesh,, yg i F SB ,, yg j F V ,, yg j (6.40b) ijj===011

n=floor(σ ) 22 mup zg=++∑∑∑ F mesh,, zg i F SB ,, zg j F V ,, zg j (6.40c) ijj===011

And, finally the gear equations in the three rotational directions along x, y and z axes are:

n=floor(σ ) 2 JMMMθ =++ (6.41a) xg xg∑∑ mesh,,θx g i SB ,,θθxx g j V , g ij==01

n=floor(σ ) 2 JMMMθ =++ (6.41b) ygyg∑∑ meshgi,,θ yyy SBgj ,,θθ V , g ij==01

n=floor(σ ) J θ =+−MMT (6.41c) zgzg∑ meshgi,,θθzz V , g g i=0

6.6 Role of Sliding Friction Illustrated by an Example

The governing equations are numerically solved for the example case. We will examine the following variables for parametric studies: (i) translational displacements

uuuuuuxp,,, yp,, zp xg yg zg ; (ii) composite displacements δθz = rruubp zp++− bg θ zg xp xg ,

δθy =++−rruubp yp bg θ yg zp zg and δ xbpxpbgxg=+rrθθ +uuzpzg− , which are the coupled

176 torsional-translational motions; (iii) dynamic bearing forces for the simplified case with

llp12= p , KKxx,1 p== xx ,2 p0.5 K xx , p , KKyyp,1== yyp ,20.5 K yyp , and

K zzp,1==KK zzp ,20.5 zzp , . The combined bearing force in the LOA direction for the pinion

is: FSB,, xp=−Ku xx p xp −2ζ xp Km xx , p p . Other bearing forces are defined similarly. The role of sliding friction is illustrated in Figure 6.6 by comparing normalized time and

frequency domain responses of uuuxp, yp, zp (at Tp = 2000 lb-in and Ω=p 1000 RPM) for

µ = 0.01 and µ = 0.1. Note that time t is normalized with respect to the mesh periodTmesh , and n is the harmonic number of the gear mesh frequency. Observe that the OLOA

vibratory motion uyp is most significantly affect by the sliding friction. Increasing µ

almost proportionally enhances the magnitude of uyp over the entire frequency range. It is consistent with Eq. (6.8) which shows the magnitude of the friction force is

proportional to µ . The sliding friction has a moderate influence on uxp in the LOA direction. An increase in µ significantly increases the amplitudes at n = 1 and 2 but the

higher harmonics remain almost unchanged. The axial displacements uxp are least sensitive to µ except for the first two harmonics. Nevertheless, Eq. (6.15) shows that the

axial shuttling excitation is proportional tosin βb . Hence, it is implied that larger helical

angle βb will lead to increased shuttling excitations due to the time-varying mesh stiffness. The bearing resonances are around n = 8 to 10 and they could be easily tuned by

varying bearing stiffness. Predicted gear displacements uuuxg, yg, zg share essentially the characteristics of pinion motions.

177

-3 x 10 8 1.82

4 1.78

1.74 0 4.5 5 5.5 6 6.5 1 2 3 4 5 0.06 0.1 0.04 0

-0.1 0.02

0 4.5 5 5.5 6 6.5 1 2 3 4 5 x 10-3 -1.3 6

4 -1.4 2

-1.5 0 4.5 5 5.5 6 6.5 1 2 3 4 5 t/T mesh Mesh order n

Figure 6.6 Time and frequency domain responses of translational pinion displacements uuuxp, yp, zp at Tp = 2000 lb-in and Ω=p 1000 RPM. All displacements are normalized with respect to 39.37 µinch (1 µm ). Key: , µ = 0.01; , µ = 0.1.

178

3 -690 2 -700

-710 1

-720 0 4.5 5 5.5 6 6.5 1 2 3 4 5 100 30

50 20

0 10

-50 0 4.5 5 5.5 6 6.5 1 2 3 4 5 1.5 290 1 280

270 0.5

260 0 4.5 5 5.5 6 6.5 1 2 3 4 5 t/T Mesh order n mesh

Figure 6.7 Time and frequency domain responses of pinion bearing forces FSB, xp , FSB, yp and FSB, zp at Tp = 2000 lb-in and Ω=p 1000 RPM. All forces are normalized with respect to 1 lb. Key: , µ = 0.01; , µ = 0.1.

179 Figure 6.7 shows dynamic bearing forces in x, y and z directions for the pinion.

Characteristics similar to the displacement responses of Figure 6.6 are observed, implying that the elastic force components dominate over the viscous forces (with 5% damping ratios). Compared with the spur gear set [6.6], the bearing forces in the helical gear pair are reduced by more than one order of magnitude due to the gradual approach

and recess motions. Figure 6.8 shows the composite displacements δ xyz,,δδ around the x, y and z axes. Observe that an increase in µ significantly increases the amplitudes at n = 1

and 2 forδ x andδ y , but has only minor influence on δ z . This observation is consistent with the results reported by Velex and Cahouet [6.4]. Nevertheless, one has to note that

the amplitude of δ z is higher than those of δ x and δ y by at least two orders of magnitude.

To examine the effect of sliding friction, Figure 6.8 also shows the time and frequency

domain responses of δ z , which is the relative torsional-translational velocity along the

LOA direction. It is seen that an increase in µ introduces additional oscillations when tooth pairs pass across the pitch point. Despite that the first mesh harmonic dominates the

spectral contents of δ z orδ z , a careful comparative study shows that the second harmonic is most significantly amplified due to friction.

180

Figure 6.8 Time and frequency domain responses of composite displacements δ xyz,,δδ and velocity δ z at Tp = 2000 lb-in and Ω=p 1000 RPM. All motions are normalized with respect to 39.37 µinch (1 µm ) or 39.37 µinch/s (1 µms/ ). Key: , µ = 0.01 ; , µ = 0.1. 181 The effect of sliding friction could be better observed by varying µ from 0 to 0.3

and then by generating the spectral contents of uuuuuuxp,,, yp,, zp xg yg zg and δ xyz,,δδ up to

15 harmonics (n) of the gear mesh frequency [6.17]. Though the resultant figures are not included here due to space constraints, some observations are as follows. 1. In the LOA direction, an in increase in µ significantly enhances amplitudes not only at n = 1 and 2 of

uxp and uxg (due to the reversal of friction force at pitch point), but also at higher harmonics as well, say around the torsional-transverse mode. This implies that friction force acts as a potential source in the LOA direction to excite resonances that are controlled due to shaft/bearing compliances. 2. In the OLOA direction, an increase in µ

efficiently increases the amplitudes of uuypygand over the entire frequency range, especially at n = 1 and 2 and at the torsional-transverse resonance. This clearly shows that the OLOA dynamics are most significantly dictated by the friction effect. 3. The axial

vibratoryuuzpzg and motions are high at the torsional-transverse resonance (controlled by the axial bearing stiffness), but the resonant amplitude does not seem to depend much on

µ . Nonetheless, friction significantly increases the amplitudes at n = 1 and 2. This indicates that the shuttling forces are relatively insensitive to the sliding friction when compared with the LOA and OLOA dynamics. 4. For the composite torsional-transverse

displacements δ x and δ y , an increase in µ increases most harmonic amplitudes, especially at the first two harmonics and at higher bearing stiffness controlled resonances.

5. An increase in µ has negligible influence on δ z except at the second mesh harmonic.

182 This observation is similar to that found in a spur gear pair of Chapter 2. However, the δ z amplitude in the helical gear pair is not as significantly influenced by the sliding friction.

6.7 Conclusion

A new 12 DOF model for helical gears with sliding friction has been developed; it includes rotational motions, translations along the LOA and OLOA directions and axial shuttling motions. Key contributions include the following: Three-dimensional model has been proposed that characterizes the contact plane dynamics and captures the reversal at the pitch line due to sliding friction. Calculation of the contact forces and moments is illustrated by using a sample helical gear pair. A refined method is also suggested to estimate the tooth stiffness density function along the contact lines by using the FE/CM analysis [6.11]. Among the 12 DOFs described above, the rotational (rocking) motions around the LOA and OLOA directions and the axial motions are usually relatively

insignificant. Therefore, a simplified 6DOF model (with coordinates uuuuxp,, yp,, xg ygθ zp

and θzg ) could be easily derived based on Eqs. (6.3) to (6.40) by neglecting the

uuzpzgxpxgyp,,,,θ θθ andθ yg variables. Such a 6DOF model requires less computational efforts though it should yield results comparable to those by the 12 DOF model. Future work should also include validation of the proposed theory, say by running the FE/CM code in the dynamic mode and by conducting analogous experiments on gears with different friction conditions.

183 References for Chapter 6

[6.1] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.

[6.2] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration, 243(3), pp. 525-545.

[6.3] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671- 694.

[6.4] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of Mechanical Design, 122(4), pp. 515-522.

[6.5] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp 641-658.

[6.6] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3), pp. 479-499.

[6.7] Borner, J., and Houser, D. R., 1996, “Friction and Bending Moments as Gear Noise Excitations,” SAE Transaction, 105(6), pp. 1669-1676.

[6.8] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, “Vibro-Acoustic Effects of Friction in Gears: An Experimental Investigation,” SAE Paper # 2001- 01-1516.

[6.9] Blankenship, G. W., and Singh, R., 1995, “A New Gear Mesh Interface Dynamic Model to Predict Multi-Dimensional Force Coupling and Excitation,” Mechanism and Machine Theory Journal, 30(1), pp. 43-57.

[6.10] Helical3D (CALYX software), 2003, “Helical3D User’s Manual,” www.ansol.www, ANSOL Inc., Hilliard, OH.

[6.11] Padmanabhan, C., Barlow, R. C., Rook, T. E., and Singh, R., 1995, “Computational Issues Associated with Gear Rattle Analysis,” ASME Journal of Mechanical Design, 117, pp. 185-192.

[6.12] Perret-Liaudet, J. and Sabot, J, 1992, “Dynamics of a Truck Gearbox,” Proceeding of 5th Internation Power Transmission and Gearing Conference, Phoenix, 1, pp. 249-258.

184 [6.13] Houser, D. R. and Singh, R., 2004-2005, “Basic and Advanced Gear Noise Short Courses,” The Ohio State University.

[6.14] Lim, T. C., and Singh, R., 1991, “Vibration Transmission through Rolling Element Bearings. Part II: System Studies,” Journal of Sound and Vibration, 139(2), pp. 201-225.

[6.15] Vinayak, H., Singh, R., and Padmanabhan, C., 1995, “Linear Dynamic Analysis of Multi-Mesh Transmissions Containing External, Rigid Gears,” Journal of Sound and Vibration, 185(1), pp. 1-32.

[6.16] Lim, T. C., and Singh, R., 1990, “Vibration Transmission through Rolling Element Bearings. Part I: Bearing Stiffness Formulation,” Journal of Sound and Vibration, 139(2), 179-199.

[6.17] Singh, R., 2005, “Dynamic Analysis of Sliding Friction in Rotorcraft Geared Systems,” Technical Report submitted to the Army Research Office, grant number DAAD19-02-1-0334.

[6.18] Umeyama, M., Kato, M., and Inoue, K., 1998, “Effects of Gear Dimensions and Tooth Surface Modifications on the loaded Transmission Error of a Helical Gear Pair,” ASME Journal of Mechanical Design, 120, pp. 119–125.

[6.19] Guilbault, R., Gosselin, C., and Cloutier, L., 2005, “Express Model for Load Sharing and Stress Analysis in Helical Gears,” ASME Journal of Mechanical Design, 127(6), pp. 1161-1172.

[6.20] Guingand, M., de Vaujany, J. P., and Icard, Y., 2004, “Fast Three-Dimensional Quasi-Static Analysis of Helical Gears Using the Finite Prism Method,” ASME Journal of Mechanical Design, 126(6), pp. 1082-1088.

[6.21] Gagnon, P., Gosselin, C., and Cloutier, L., 1996, “Analysis of Spur, Helical and Straight Level Gear Teeth Deflection by the Finite Strip Method,” ASME Journal of Mechanical Design, 119(4), pp. 421–426.

185 CHAPTER 7

ANALYSIS OF HELICAL GEAR DYNAMICS USING FLOQUET THEORY

7.1 Introduction

This work is an extension of Chapter 6 where we proposed a 12 degrees-of- freedom (DOF), linear time-varying (LTV) analytical model for helical gears that characterizes the contact plane dynamics and captures the velocity reversal at the pitch line due to sliding friction. Earlier, Velex and Cahouet [7.1] had found that the dynamic bearing forces (as related to the sliding friction in helical or spur gears) can indeed generate significant time-varying excitations at lower speeds. Velex and Sainsot [7.2] have examined friction excitations in errorless spur and helical gear pairs, and reported that the friction appears as a non-negligible excitation source especially for translating motions. Lundvall et al. [7.3] proposed a multi-body model for spur gears and briefly discussed the role of profile modification in the presence of sliding friction. Vaishya and

Singh [7.4-7.6] had illustrated frictional issues for spur gears by assuming equal load sharing among the contact teeth. This assumption leads to a rectangular variation in stiffness, which is a special case (zero helical angle) of a generic trapezoidal stiffness

186 profile for helical gears. To overcome this deficiency [7.4-7.6], in Chapter 2 we have developed a more accurate model of the spur gears that incorporates realistic mesh stiffness and sliding friction. Many of the models cited above are solved numerically and thus there is a clear need for analytical (closed form) solutions to the dynamic response of a helical gear pair under the influence of sliding friction. In fact, Vaishya and Singh [7.5] had applied the Floquet theory to a simplified spur gear model to predict responses of parametrically-excited system and to assess the system stability. This chapter will enhance Vaishya and Singh’s work [7.5] by applying the Floquet theory to a helical gear pair.

7.2 Linear Time-Varying Formulation

Chief objectives of this chapter are as follows. First, place emphasis on periodic frictional effects at the gear tooth interface by ignoring other directional properties and the auxiliary components of the gearbox. This will allow us to describe the single mesh helical geared system as a simplified single degree-of-freedom (SDOF), linear time- varying (LTV) oscillator with piece-wise linear effective mesh stiffness; frictional forces/moments will be formulated as parametric excitations. Second, derive closed form solutions for the LTV system in terms of the dynamic transmission error (DTE) under both homogeneous and forced conditions by using the Floquet theory. Third, validate the proposed theory by using the numerical integration method. Key assumptions include: (1)

The vibratory motions are small compared with the kinematical motion, so that the

187 position of the contact lines and the relative sliding velocity depend only on the mean

angular motions of the gear pair. (2) The mesh stiffness per unit length along the contact

lines (i.e. stiffness density k) is constant; this is equivalent to the equal load sharing

assumption in spur gears [7.4-7.6]. (3) Coulomb’s law with a constant coefficient of

friction (µ) is employed [7.1-7.6] though mixed lubrication regimes exist [7.7]. (4) The

bearing stiffness is assumed to be much higher than the mesh stiffness and thus the

shaft/bearings could be simplified as rigid connections. Hence, only the torsional DOFs

are considered in terms of DTE. Also, it is assumed that the mean load is high such that

the dynamic load is insufficient to cause any tooth separations [7.8-7.9].

The helical geared system is depicted in Figure 7.1, where the pinion and gear are modeled as rigid cylinders linked by a series of independent stiffness elements that

describe the contact plane tangent to the base cylinders. The pinion and gear dynamics

are formulated in the coordinate systems located at their respective centers; the nominal

motions are given as −Ω pez and Ω=Ωg eezzpbpbg rr/ , respectively. Here, the z axis coincidences with the axial direction, e is the unit directional vector, and rbp and rbg are

the base radii of pinion and gear. An (static) input torque Tp is applied to the pinion, and

the (static) braking torque Tg on the gear obeys the basic gear kinematics. Superposed on

the kinematic motions are rotational vibratory motions denoted by θzp and θzg for the

pinion and gear. Analytical formulations are demonstrated via the following example

case with parameters of the pinion (relevant gear parameters are within the parenthesis):

number of teeth 25 (31); outside diameter 3.38 (4.13) in.; pitch diameter 3.125 (3.875) in.;

root diameter 2.811 (3.56) in.; center distance 3.5 in.; transverse diametral pitch 8 in.-1;

transverse pressure angle 25°; helix angle βb = 21.5°; face width W = 1.25 in.; polar 188 -3 -2 2 -2 -2 moment of inertia Jpz = 8.33×10 (Jgz = 1.64×10 ) lb-s in.; mass 1.26×10 (1.58×10 )

2 -1 lb-s /in. . Since the overall contact ratio σc = 2.7, either two or three tooth pairs are in

contact at any time instant. The three meshing tooth pairs within one mesh cycle are

numbered as #0, #1 and #2, respectively. A constant mesh stiffness density (k) along the contact lines could be estimated via a static analysis by using the finite elements/contact mechanics (FE/CM) formulation [7.10-7.11].

θ

φ θ

βb

Figure 7.1 Schematic of the helical gear pair system.

189 A simplified SDOF model could be derived in items of DTE

δθ()tr=+bp zp () tr bg θ zg () t at the gear mesh by assuming rigid links at the shaft/bearings.

Note that the dynamic mesh forces oriented in other directions (such as the sliding force

in the OLOA direction) still need to be formulated for calculations of the dynamic

moments in the torsional direction. Hence, the effective torsional tooth stiffness should

have contributions from both the sliding friction and the time-varying elastic tooth

stiffness due to Hertzian contact.

For multiple tooth pairs in contact, n = ceil(σc) (n = 3 for the example case) pairs

of meshing teeth need to be formulated, where the “ceil” function rounds σc to the nearest

integers towards infinity. Figure 7.2 illustrates the snapshot at the beginning of a mesh cycle. At this instant, pair #0 (defined as x (trtL )=Ω mod( ,λ ) +, where λ is the base 0TApbp 1

pitch; L represents the geometrical distance; the modulus function is mod(x,

y)=x−y⋅floor(x/y) for y ≠ 0; and the “floor” function rounds x/y to the nearest integer towards minus infinity) just comes into mesh at point A and pair #1 (defined as x (trtL )=Ω mod( ,λλ ) + + ) is in contact along line CI. Likewise, pair #2 (defined 1TApbp 1 as x (trtL )=Ω mod( ,λλ ) + 2 + ) contacts each other along line MN. As the gears roll, 2TApbp 1

the contact lines move diagonally across the contact zone. When pair #0 reaches the pitch

point P, the relative sliding velocity between pinion and gear starts to reverse, resulting in

a reversal of friction force along the portion of contact line surpass the pitch point. Once

pair #0 reaches line CI (and pair #1 reaches MN line) at the end of the mesh circle, pair

#0 becomes #1 (and pair #1 becomes #2) corresponding to the start of the next mesh

cycle. The dynamic tooth stiffness functions Kp,i(t) and Kg,i(t) are defined below where

190 δ ()t is the dynamic transmission error, Mp,i(t) and Mg,i(t) are dynamic moments on the pinion and gears, respectively:

Mtpi,,() Mt gi () Ktpi,,()=== , Kt gi () ( i 0,1,2). (7.1a,b) rtbpδδ() rt bg ()

Figure 7.2 Contact zones at the beginning of a mesh cycle within the contact plane. Key: PP’ is the pitch line; AA’ is the face width W; AD is the length of contact zone Z.

191 For the example case, Kp,i(t) and Kg,i(t) are explicitly derived for each meshing

tooth pair over eight contact zones (Zi) as shown in Figure 7.2. Also, refer to Chapter 6

for details. Zones 1 and 2 correspond to pair #0 before and after reaching the pitch line;

Zones 3, 4 and 5, and Zones 6, 7 and 8 correspond to pairs #1 and #2, respectively. The

stiffness of the two contact zones for the first pair (#0) are derived as follows, where xm, xf, zm and zf denote the lower and upper limits along x and z axes, as shown in Figure 7.1.

⎧ k ⎡⎤µ −++()xtfmbpbfm( ) x r cosβ () z − zt ( ) Z1 ⎪r ⎢⎥2 ⎪ bp ⎣⎦ ⎪ ⎡⎤µ Ktp,0 ()= ⎨ (7.2a) −+()()xpmp x zt() −+ zt m () r bpbfm cosβ ( z − zt () ) k ⎢⎥2 ⎪ Z2 ⎪ ⎢⎥ rbp ⎢⎥µ ⎪ ++−()()xtfpfp() x z zt () ⎩ ⎣⎦⎢⎥2

⎧ k ⎡⎤µ ()()xtfmgbgbfm( )+− x 2 x + r cosβ z − zt ( ) Z1 ⎪r ⎢⎥2 ⎪ bg ⎣⎦ ⎪ Ktg,0 ()= ⎨ ⎡⎤µµ (7.2b) k ⎢⎥()()xxzztpmpm+−−+−()()() xtxzzt f () pf p () ⎪ 22 Z2 ⎪ r ⎢⎥ bg ⎢⎥++−+xzztztrµβ() 2 () cos zzt − () ⎩⎪ ⎣⎦gfm()() p bgbfm

The second meshing tooth pair (#1) is classified into Zones 3, 4 and 5 as shown in Figure

7.2. Its composite torsional stiffness could be derived for the pinion and gear as:

192 ⎧ ⎡⎤µµ −+()()xxztztpmp() −+ m ()()() xtxzzt f () + p f − p () ⎪ k ⎢⎥22 ⎪ ⎢⎥ Z3 rbp ⎪ ⎢⎥+−rzztbpcosβ b() f m ( ) ⎪ ⎣⎦ ⎪ ⎡⎤µµ −+()()xxtztzpm() p () −+ m()() xtxzzt f () + pf − p () ⎪ k ⎢⎥22 Ktp,1 ( )= ⎨ Z4 (7.3a) r ⎢⎥ ⎪ bp ⎢⎥+−rzztcosβ ( ) ⎪ ⎣⎦bp b() f m ⎪ ⎡⎤µµ ⎪ k ⎢⎥−+()()xxtztzpm() p () −++ m()() xxztzt f pf () − p () ⎪ 22 Z5 r ⎢⎥ ⎪ bp ⎢⎥+−rztzcosβ ( ) ⎩ ⎣⎦bp b() f m

⎧ ⎡⎤µµ ()()xxztztpmp+−−+−() m ()()() xtxzzt f () pf p () ⎪ k ⎢⎥22 ⎪ ⎢⎥ Z3 rbg ⎪ ⎢⎥++−+xgfmµβ()() z zt() 2 zt p () r bgbfm cos z − zt () ⎪ ⎣⎦ ⎪ ⎡⎤µµ ()()xxtztzpm+−−+−() p () m()() xtxzzt f () pf p () ⎪ k ⎢⎥22 Ktg,1 ( )= ⎨ Z4 (7.3b) r ⎢⎥ ⎪ bg ⎢⎥++−+xzzµβ2() ztr cos zz − ⎪ ⎣⎦gfmp()() bgbfm ⎪ ⎡⎤µ µ ⎪ k ⎢⎥()()xxp + mpm()tztz ()−−()() xxztzt fpfp + () − () ⎪ 2 2 Z5 r ⎢⎥ ⎪ bg ⎢⎥++−+xztzµβ() 2 ztr () cos ztz () − ⎩ ⎣⎦gf()() m p bgbf m

The third meshing tooth pair (#2) is divided into contact Zones 6, 7 and 8 as shown in

Figure 7.2. Its composite torsional stiffness could be derived for the pinion and gear as:

⎧ ⎡⎤µµ −+()()xxtztzpm() p () −++ m()() xxztzt f pf () − p () ⎪ k ⎢⎥22 ⎪ ⎢⎥ Z6 rbp (7.4a) ⎪ ⎢⎥+−rztzbpcosβ b() f ( ) m ⎪ ⎣⎦ ⎪ k ⎡⎤µ Ktpfmbpbfm,2 ( )=++−⎨ ⎢⎥() xxtr ( ) cosβ () ztz ( ) Z7 ⎪ rbp ⎣⎦2 ⎪ 0 Z8 ⎪

193 ⎧ ⎡⎤µµ ()()xxtztzpm+−−+−() p () m()() xxztzt f pf () p () ⎪ k ⎢⎥22 ⎪ ⎢⎥ Z6 rbg (7.4b) ⎪ ⎢⎥++−+xztzgfµβ()()() m 2 ztr p () bgbf cos ztz () − m ⎪ ⎣⎦ ⎪ k ⎡⎤µ Ktgfmbpbfm,2( )=++−⎨ ⎢⎥() xxtr ( ) cosβ () ztz ( ) Z7 ⎪ rbg ⎣⎦2 ⎪ 0 Z8 ⎪ ⎪

The undamped torsional equations for the pinion and gear are derived as follows:

2 J pzzpθδε()trKtttT+ ∑ bppi, ()[] ()−= () p (7.5) i=0

2 J gzzgθδ()trKttT+ ∑ bggi, () ()=− g (7.6) i=0

Define DTE δ ()t and reduce Eqs. (7.5) and (7.6) into one equation which describes an equivalent translational definite system as follows, where ε(t) is the unloaded static transmission error. A time-varying viscous damping coefficient Ce(t) is also included given (assumed) damping ratio ζe.

mtCtttKtttFδδεδε()+−+−= ()⎡⎤ () () () () () (7.7a) ee⎣⎦ e[ ] e

2222 JJpz gz rJbp gz K p,, i() t+ rJ bg pz K g i () t me = 22, Kteei()==∑∑ K, () t 22 (7.7b-c) rJbg pz+ rJ bp gz ii==00rJbg pz+ rJ bp gz

rTJbp p gz− rTJ bg g pz Ctei, ()= 2ζ e mKt e e (), Fe = 22 (7.7d-e) rJbg pz+ rJ bp gz

194 Here, me is the effective mass defined in the torsional-transverse direction and Fe is the effective external force due to the nominal torques applied at the pinion and gear. The

th periodically time-varying effective stiffness function Ke,i(t) of the i meshing mesh pair is piece-wise linear, and it incorporates contributions from both the mesh tooth stiffness and the sliding friction. The frictional influence on Ke,i(t) is illustrated in Figure 7.3 over eight contact zones, where a generic effective stiffness function is obtained by following a single tooth pair for three complete mesh cycles since ceil(σc) = 3. When µ = 0 (no friction), Ke,i(t) has a symmetric trapezoidal profile, when high sliding friction is introduced with µ = 0.4, additional discontinuities in the slope emerge during the transitions from Zone 1 to Zone 2, as well as from Zone 6 to Zone 7. These correspond to the conditions when the contact line reaches or leaves the pitch line. Note that the stiffness functions are “continuous” in a piece-wise manner due to the gradual approaching and recess motions of the helical gear pair. Compared with the square-wave shaped tooth stiffness function of a spur gear pair [7.4-7.6], this shape should be more favorable as lower vibro-acoustic levels would be expected.

Given the piece-wise stiffness Ke(t) of Eq. (7.7), we denote j as the index for the th j interval (with a constant slope) and define the generic periodic stiffness function Ke,j(t) over m piece-wise intervals within one mesh cycle as follows:

KKej,,1− ej− Kej,,()tKtTK=+=+ ej ( ) ej ,1− () tt − j− 1 (7.8) ttjj− −1

195

(lb/in) e,i K

Figure 7.3 Individual effective stiffness Ke,i(t) along the contact zone, where Tmesh is one mesh cycle. Key: ––, tooth pair #0 (µ = 0); ––, tooth pair #1 (µ = 0); ––, tooth pair #2 (µ = 0); ...., tooth pair #0 (µ = 0.4); - - -, tooth pair #1 (µ = 0.4); - ⋅ -, tooth pair #2 (µ = 0.4).

For the example case with individual Ke,i(t) (i = 0, …, 2) of Figure 7.3, the combined stiffness functions Ke,j(t) (j = 1, …, 6) are calculated over six contact regions within one mesh cycle. Since the slope is constant within each region, only the stiffness values Ke,j at the starting and ending time instants are needed with Ke,6 = Ke,0 due to the periodicity. The time instants ti of each region within one period could be determined

196 based on Figure 7.2 as follows t0 = 0; t1 = (LEH/λ)T; t2 = (LCQ/λ)T; t3 = (LCD/λ)T; t4 =

(LAP/λ)T; t5 = (LEG/λ)T and t6 = T.

Table 7.1 lists the relationship between the six contact regions defined for the combined stiffness functions Ke,j(t) and the eight contact zones defined for individual meshing tooth pairs as given by Ke,i(t). Although the number of contact zones/regions depends on gear geometry, the proposed modeling strategy could be easily applied to other helical geared systems.

Contact Contact zones of Figure 7.2 region Pair #0 Pair #1 Pair #2

1 Z1 Z3 Z6

2 Z1 Z3 Z7

3 Z1 Z4 Z7

4 Z1 Z5 Z7

5 Z2 Z5 Z7

6 Z2 Z5 Z8

Table 7.1 Relationship between Contact Zones and Contact Regions for the NASA-ART helical gear pair

197 Figure 7.4 compares the combined Ke,j(t) and individual Ke,i(t) functions over one period. Observe that the profile of Ke,j(t) resembles those of individual Ke,i(t): Under zero friction, Ke,j(t) follows a symmetric trapezoidal pattern, where 4 piece-wise intervals exist within one mesh cycle. When the sliding friction is included, two additional discontinuities in the slope are introduced at the transitions from Region 1 to Region 2, as well as from Region 4 to Region 5. Hence, six piece-wise regions need to be analyzed for one complete mesh cycle. Note that a high mean component exists for the combined

Ke,j(t), whose values are always positive (non-zero).

Figure 7.4 Piece-wise effective stiffness function defined in six regions within one mesh cycle with µ = 0.4. Key: .... tooth pair #0, - - - tooth pair #1, - ⋅ - tooth pair #2, –– combined stiffness function. 198 7.3 Analytical Solutions by Floquet Theory

Vaishya and Singh [7.5] suggested analytical solutions to a SDOF spur gear system model with periodic square-wave stiffness. For a helical gear pair, Ke,i(t) varies periodically in a trapezoidal pattern; hence, the spur gear could be regarded as a special

(limiting) case of the helical gear model where the slope of stiffness within each interval is zero rather than an arbitrary constant. Assume ε(t) = 0 (perfect involute profile) and

Ce(t) = 0 (undamped condition), the parametrically excited system of Eq. (7.7) under a mean load Fe is simplified as follows:

mtKtteeδδ()+ () ()= F e (7.9)

The Floquet theory [7.12] is then applied to find analytical solutions to both free

(including the case with a constant viscous damping C = ) and forced e 2ζ eeemK responses. For the example case, six contact regions need to be formulated with the influence of sliding friction. Represent the governing equation in the state space form as:

Xt ()=+ GtXt () () Ft (), Gt()()+ T= Gt (7.10a-b)

⎧ Gt11( ) 0 ≤ t< t ⎪ Gt( )=≤<=⎨ Gjj ( t ) t−1 t t j ( i 2...5) , (7.10c) ⎪ ⎩ Gt65( ) t≤ t< T

⎧⎫δ ()t ⎧ 0 ⎫ Xt( )==⎨ ⎬⎨⎬ , Ft ( ) (7.10d-e) ⎩⎭δ ()t ⎩⎭Fee/ m

199 The solution over one complete mesh cycle T is written in the form of a state transition matrix (Φ). For a piecewise periodic system, this matrix may further be decomposed into

Φj over each contact regions [7.5] in Eq. (7.11), where the functions are continuously differentiable and analytical solutions to the homogeneous equation exist.

Φ()TTtttt,0 = Φ( ,5211) ⋅⋅⋅Φ( ,) Φ( ,0) (7.11)

Each Φ(tj, tj-1) is evaluated from the Wronskian matrix (Π) as:

−1 Φ=ΠΠ≤≤(ttjj,()(),−−−111) t j t j t j t t j, (7.12a)

⎡γ γ ⎤ Π=()t 12 (7.12b) ⎢ ⎥ ⎣γ12γ ⎦

Here, γ1 and γ2 are two basis solutions to the homogeneous equation Xt ( )= GtXt ( ) ( ) .

Use the periodic property of Φ, Floquet theory extends solutions to future states of the system that are apart by n mesh cycles. Thus, the state transition matrix Φ(,0)nT over n cycles and the resulting responses X(t) are given by:

Φ=Φ(nT ,0) ( T ,0)n (7.13)

t Xt()=Φ (,0) t X (0) + Φ (, tτ ) F (ττ ) d , (7.14a) ∫0

Xt()+=Φ nTn ( T ,0()) Xt (7.14b)

200 Equations (7.11-7.14) are of significant importance. First, they drastically reduce the computational time since the results calculated for one mesh cycle can be easily extended to other periods by using matrix multiplication, which is computationally effective. Second, it allows an easier inversion of the matrix.

7.3.1 Response to Initial Conditions

Knowledge of the free response to initial conditions is important to assess the dynamic stability property of the helical gear pair. Within each interval tj-1 ≤ t < tj, Eq.

(7.9) is rewritten in the homogeneous form as:

⎡⎤⎛⎞2t δδ(taq )+−⎢⎥ 2⎜⎟ − + 1 ( t ) = 0, t ≤≤= ttj ( 1...6) (7.15a) jj⎜⎟tt− j−1 j ⎣⎦⎢⎥⎝⎠jj−1

KKK1 ⎛⎞− αββ=−jjj−−11t , = (7.15b-c) jjjj−1 ⎜⎟ mmtteejj⎝⎠− −1

ββjj(tt−− j−−11) jj( tt j) aq=+α , = (7.15d-e) jj24 j

With a change of variable z jjj=+α β t , convert Eq. (7.15) into the Stoke’s equation [7.12]:

2 d δ z j 22+ δ = 0 (7.16) dz jjβ

201 A set of basis solutions are known over tj-1 ≤ t < tj:

γ11/3()tzJ= jj (σ ), γ 21/3()tzJ= jj− (σ ) (7.17a-b)

3/2− 1 whereσβjjj= 2(3)z and J±1/3 ()σ j are the Bessel functions of the first kind of order

±1/3. Use the recurrence relation of Bessel functions to find the Wronskian matrix as:

⎡⎤ zJjjj13(σσ) zJ− 13( j) ⎢⎥ Π=j ()t (7.18a) ⎢⎥zJσσ− zJ ⎣⎦jjjj−23() 23()

⎡ ⎤ −−zJjj23(σσ) zJ j− 13( j) −1 2π ⎢ ⎥ Π=−j ()t (7.18b) 33β ⎢−zJσσ zJ ⎥ j ⎣ jjjj−23() 13()⎦

Note than Eqs. (7.17-7.18) are valid only for the conditions with zj > 0. For the cases in which zj are negative (or zero), the Wronskian matrices are derived in terms of the modified Bessel functions of the first kind (or gamma functions), which could be treated in a similar matter. However, for the SDOF helical gear model with a high positive mean component, all zj have positive values so that Eqs. (7.17-7.18) hold. Also, for the intervals with a negative slope βj (such as contact regions 4 and 5 of Figure 7.4), the corresponding σj also has negative values, which lead to complex Πj(t) generated by the

Bessel functions in Eqs. (7.17-7.18). Nevertheless, due to the supplemental phase

−1 −1 relationship between Π(tj) and Π (tj-1), the state transition matrix Φ(tj-1, tj) = Π(tj)Π (tj-1) still assumes real values within each interval tj-1 ≤ t < tj. Thus, responses to initial 202 conditions X (0)= {δδ (0), (0)}T are derived in Eq. (7.19), where Φ(T, 0) is the discrete transition matrix. Here, Φ(t−nT, 0) needs to be evaluated similar to Eq. (7.11) over the last cycle.

Xt()=Φ ( t − nT ,0) Φ() T ,0n X (0),0 ( ≤ t − nTT < ) (7.19)

Figure 7.5 compares the homogeneous responses given initial condition x0 =

-6 2×10 in., v0 = 20 in./s at Ωp = 1000 RPM, as predicted by using the Floquet theory and the numerical solution (based on the Runge Kutta scheme [7.13]). Since the numerical solution completely overlaps with the Floquet theory prediction, only one pair of comparative results are given with µ = 0.2 in Figure 7.5(b). Observe that increasing sliding friction changes the slopes of the effective stiffness function Ke,j(t), while such effect does not seem to be significant in the DTE response. This is because that the undamped responses are dictated by the dynamic components at the system natural frequency fn = Kee/2()π m , where Ke is the averaged stiffness. For the example case, fn is found to be close to 9.5fm, where fm is the mesh frequency at Ωp = 1000 RPM. Side bands around 8.5fm and 10.5 fm may also be present due to frequency modulation effects.

203

(a)

(b)

Figure 7.5(a) Effective stiffness and (b) homogeneous responses predictions within two -6 mesh cycles given x0 = 2×10 in., v0 = 20 in./s at Ωp = 1000 RPM. Key: ––, µ = 0 (Analytical solution by the Floquet theory); - ⋅ -, µ = 0.2 (Analytical); ...., µ = 0.2 (Numerical); - - - , µ = 0.4 (Analytical).

Damped homogenous response could also be derived by assuming

Ctei,0()=≈= 2ζζ e mKt e e () 2 e mKC e e e with a time-averaged viscous damping Ce0.

Thus, Eq. (7.7) is converted into constantly damped homogenous form as follows:

mtCtKttedδδ()+ e0 d ()+= e () δ d () 0 (7.20a)

204 Ct Ke − e0 −ζ e t 2 me By defining the transformation δψd ()ttete== () ψ () , Eq. (7.20) is further converted into the following expression:

⎛⎞2 Ke mteeeψζ()+ ⎜⎟ 1−= Ktt () ψ () 0 (7.20b) ⎝⎠Kte ()

-3 Since Kee/()Kt≈1, for small viscous damping (say ζe = 5%), its square value (2.5×10 ) is negligible compared with 1. Hence, Eq. (7.20b) assumes the same form as the undamped Eq. (7.15), so that it should have the same solution. This implies that for an oscillator with small and constant viscous damping, the damped homogeneous response could be calculated as follows, where δ(t) is the analytical solution to an undamped system.

Ke −ζ e t me δδd ()tte= () (7.21)

Figure 7.6 shows that the analytical prediction of the damped homogeneous

response with a constant CmKeeee0 = 2ζ correlates well the numerical simulation with a time-varying Ce(t) of Eq. (7.7e). Here, the Ke(t) profile is the same as illustrated in Figure

7.5(a) with µ = 0.2. This implies that Eq. (7.21) could be used to approximate the homogeneous response with periodically varying stiffness and viscous damping parameters.

205

Figure 7.6 Predictions of damped homogeneous responses within two mesh cycles given -6 x0 = 2×10 in., v0 = 20 in./s, µ = 0.2 at Ωp = 1000 RPM. Key: ––, Analytical solution by the Floquet theory with Ce0; - ⋅ -, Numerical with Ce(t).

7.3.2 Forced Periodic Response

For the LTV system of Eq. (7.9), Φ could be applied to compute the response under a periodic excitation. The tractability of the solution depends on both the characteristics of the excitation and the nature of Φ. In general, this problem is solved by expanding the forcing function as well as the time-varying parameters in terms of Fourier coefficients. Clearly, this will lead to errors due to truncation of modes and also 206 significantly increase the computations [7.5]. For the example helical gear pair, three (or six) piecewise linear segments need to be considered for each mesh cycle without (or with) the influence of sliding friction. All integrals associated with the Floquet theory could be analytically found under a mean torque excitation. However, as the number of piecewise linear segments increases within the mesh cycle, such as for the realistic stiffness profile, analytical solutions could become computationally expensive. The forced response of this system is formulated as follows:

t Xt()=Π () t Π−−11 () tX (0) + Π () t Π (τ ) F (ττ ) d (7.22) ∫0

Given the initial condition response ΠΠ()ttX−1 () (0) has already been derived in

Eq. (7.19), only the forced response needs to be derived by applying

Φ=ΦΦ(,0)tt (,τ ) (τ ,0) for any τ:

nT t Xt()=Φ ( t,0 )⎡ Φ−−11 (τ ,0 ) F ()ττ d + Φ ( τ ,0 ) F () ττ d⎤ ⎣⎢∫∫0 nT ⎦⎥ (7.23)

=Φ()tHnHt ,0 ⎣⎦⎡⎤12 () + ()

The solution is found in two parts including the integral number of mesh cycles (H1) and the last cycle (H2) [7.5]. For n complete cycles, the expression for H1(t) is found as:

n iT H nFd=Φ−1 τ ,0 ττ (7.24) 1 ()∑ ∫ i ( ) () i=1 ()iT−1

207

Define τ0 = τ + (i − 1)T and apply the Floquet theory such that:

j−1 ⎧⎫⎡⎤Φ⋅−1 T,0 ⎪⎪⎣⎦() ⎪⎪t ⎡⎤1 −1 n ⎪⎪ΠΠ()tFd ()()τττ + ⎪⎪⎢⎥10∫0 1 0 Hn1 ()= ⎨⎬⎢⎥t (7.25) ∑ −−112 j=1 ⎪⎪⎢⎥ΠΠ()ttt () Π () Π ()()τττ Fd + 10 1 1 21∫t 2 0 ⎪⎪⎢⎥1 t ⎪⎪⎢⎥ΠΠ()ttttt−−11 () ΠΠ () () Π ()3 Π − 1 ()(τττ Fd ) +... 1011 2122 32∫t 3 0 ⎩⎭⎪⎪⎣⎦⎢⎥2

For the last time cycle, the value of H2(t) depends upon the time instant t in the whole mesh cycle. Hence, solutions are derived within each piecewise linear segment as follows, where τ0 = τ + nT.

n tnT− H ()tT=Φ⎡⎤−−11 (,0 )⎡ Π ( 0 ) Π (τ ) Fd (ττ ) ⎤ , 2110⎣⎦⎣⎢ ∫0 ⎦⎥

⎪ tnT− −1 ⎪ ΠΠ110()0 ()τττFd ( ) ; ∫t ⎪ 1 ⎪ ()ttnTt≤− < ⎪ 01 ttnT− ⎪⎡⎤1 −−−111 ΠΠ11()00 ()τ Fd (ττ 0111212 ) +ΠΠΠ () () t () t Π () τ Fd ( ττ 0 ) ; ⎪⎣⎦⎢⎥∫∫0 t1 n ⎪ (7.26) ⎡⎤−1 =Φ⎣⎦()TttnTt,0 ⋅⎨ ()12 ≤− < ⎪ ⎪ ...... T ⎪ ⎡⎤−1 ΠΠ11()0 ()()τττFd 0 + ... ⎪ ⎢⎥∫0 ; ⎪ ⎢⎥tnT− ⎪ +Π()0 Π−−11 ()tt Π () Π ⋅⋅⋅Π t Π − 1()()τττ Fd ⎢⎥111212jj()− 1∫t j 0 ⎪ ⎣⎦j−1 ⎪ ttnTttT≤− < , = ⎩ ()jj−16

208 All matrices in Eq. (7.26) have been analytically derived except for the

0 Π−1 ()τ F ()ττd integral, which could be analytically found by using Eq. (7.27). Note ∫0

T that the constant forcing function F()tFm= {} 0ee / could be taken out of the integral.

Here, the LommelS1 in Eq. (7.27) is the Lommel function [7.14].

zJ()σ dt=− zJ ()σ (7.27a) ∫ 2/3− 1/3

4σ −=+z J()()()σσσσdt J L J L (7.27b) ∫ −−−133 1/3 1 4/3 2

zJ()σ dt= zJ ()σ (7.27c) ∫ −2/3 1/3

2σ z J()()()σσσσdt=− J L − J L (7.27d) ∫ 233 1/3 3− 2/3 4

42 3 L =−−LommelS1( 1, ,β z 2 ) (7.27e) 1 33

12 3 L =−LommelS1(0, ,β z 2 ) (7.27f) 2 33

22 3 L =−−LommelS1( 1, ,β z 2 ) (7.27g) 3 33

12 3 L = LommelS1(0, ,β z 2 ) (7.27h) 4 33

Analytical forced responses predicted by using Eqs. (7.22-7.27) compare well

-6 with numerical results in Figure 7.7, given Ce(t) = 0, x0 = 2×10 in., v0 = 20 in./s, Tp =

2000 lb-in., µ = 0.2 and Ωp = 1000 RPM. Here, the Ke(t) profile is the same as Figure

209 7.5(a) with µ = 0.2; thus six piece-wise contact regions need to be considered within each mesh cycle. Similar to the undamped homogeneous response, the forced responses are also dominated by the dynamic component at the system natural frequency and some side bands due to the frequency modulation effect. Such resonances, however, are efficiently controlled by the viscous damping, which might have negligible effects on the mesh harmonics. Consequently, if any mesh harmonic is away from the resonant frequency, one can obtain the dynamic responses (which should share the same features as damped responses) by filtering out the resonant components in frequency domain. In our example case, a low pass filter is used since fn >> fmesh.

Figure 7.7 Predictions of (undamped) forced periodic responses within two mesh cycles -6 given x0 = 2×10 in., v0 = 20 in./s, Tp = 2000 lb-in, µ = 0.2 and Ωp = 1000 RPM. Key: ––, Analytical solution by the Floquet theory; - ⋅ -, Numerical. 210 Analytical predictions of (undamped) forced responses are compared in Figure

7.8 with numerical simulations obtained from a viscously damped SDOF model as well as a 6DOF model, which is similar to the 12DOF model of Chapter 6. Comparison of steady-state time domain responses in Figure 7.8(a) shows that a viscous damping coefficient of 5% tends to “remove” the dominant resonant components (as compared to the mesh harmonic components) from the forced responses. Also, numerical simulations of DTEs predicted from the SDOF model match well with those from the 6DOF model despite some differences in the mean component. The steady-state time responses are converted into frequency domain and Figure 7.8(b) (where the static term is cut off) shows that the predictions at first five mesh harmonics of the undamped system match very well with the spectra of viscously damped responses calculated by numerical integration. This suggests that the analytical solution could be extended to examine the damped dynamic response in frequency domain.

Figure 7.9(a) shows the predicted mesh harmonics of DTE as a function of µ.

Observe that an increase in µ has the most significant effect on the first two mesh harmonics. Figure 7.9 (b) shows predicted DTE harmonics with respect to µ; these are obtained by the finite difference method, i.e. dδ/dµ ≈ [δ(n)−δ(n−1)]/[µ(n) −µ (n−1)].

Observe that the second harmonic has the highest increasing rate followed by the first harmonic. Moreover, since the amplitude at the second harmonic without friction (µ =0) is much smaller than that of the first harmonic, it is implied that sliding friction has more influences on the second harmonic. This is consistent with the results predicted by the 12

DOF formulation of Chapter 6.

211

(a)

(b)

-6 Figure 7.8 Steady state forced periodic responses given x0 = 2×10 in., v0 = 20 in./s, Tp = 2000 lb-in., µ = 0.1 and Ωp = 1000 RPM: (a) DTE vs. time; (b) DTE spectra. Key: , , undamped analytical prediction; , , damped numerical simulation of SDOF system with ζe = 5%; , , damped numerical simulation of a 6DOF model with ζe = 5% (with mean component compensated).

212

(a)

(b)

Figure 7.9 Predicted mesh harmonics of (undamped) forced periodic responses as a -6 function of µ given x0 = 2×10 in., v0 = 20 in./s, Tp = 2000 lb-in and Ωp = 1000 RPM: (a) DTE; (b) slope of DTE. Key: , n = 1; , n = 2; , n = 3; , n = 4.

213 7.4 Conclusion

Work presented in this chapter extends the earlier work by Vaishya and Singh

[7.5] by applying the Floquet theory to a helical gear pair to examine the effect of sliding friction on the DTE. In particular, the LTV formulations (with parametric excitations) have been developed for a SDOF model, and the effect of sliding friction is quantified as parametric excitations of effective mesh stiffness. The Floquet theory has been successfully applied to obtain closed-form DTE solutions given periodic and piece-wise linear stiffness functions. Responses to both initial conditions and forced periodic functions, under a nominal preload, are derived. Analytical models have been validated by comparing predictions with numerical simulations. Although the coefficient of friction

µ is assumed to be high for the example case for illustrative purpose, the same algorithm could be implemented under realistic conditions when a smaller value of µ is expected.

Overall, the sliding friction has a marginal effect on the dynamic transmission error of helical gears, as compared with spur gears, at least in the context of the torsional model.

Finally, parametric instability issues are briefly examined. Asymptotic stability of a homogenous system can be determined from the discrete transition matrix over one complete period of parametric changes. A sufficient condition for stability is that all the eigenvalues κ of the Φ(T, 0) matrix have absolute values less than unity [7.12]. For the sake of illustration, Figure 7.10 shows the mapping of maximum κ (absolute value) as a function of the ratio of time-varying mesh frequency fmesh(t) to the system natural frequency fn, without viscous damping. Observe that the most dominant unstable region emerges when fmesh(t)/fn ≈ 2, such parametric instability is well explained by Den Hartog

214 [7.15]. Other unstable regions are found when the ratio of fmesh(t)/fn is close to 1, 2/3, 1/3, etc. Also, an increase in µ tends to enhance the max{κ } value in the most dominant

unstable region around fmesh(t)/fn ≈ 2; in addition, it decreases the 1/3 peak while enhancing the peak around 1. When the system operates near unstable regions, various stability performances could be observed though these results are not shown here. For instance, when fmesh(t)/fn is close to 2 (say at 18,000 RPM), long term stability performance is observed. On the other hand, when fmesh(t)/fn falls within the unstable region (say at 19,000 RPM), the homogeneous response grows unbounded.

κ

Figure 7.10 Mapping of eigenvalue κ (absolute value) maxima as a function of the ratio of time-varying mesh frequency fmesh(t) to the system natural frequency fn. Key: , µ = 0.01; , µ = 0.1; , µ = 0.2.

215 References for Chapter 7

[7.1] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of Mechanical Design, 122(4), pp. 515-522.

[7.2] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp. 641-658.

[7.3] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3), pp. 479-499.

[7.4] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.

[7.5] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration, 243(3), pp. 525-545.

[7.6] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671- 694.

[7.7] Borner, J., and Houser, D. R., 1996, “Friction and Bending Moments as Gear Noise Excitations,” SAE Transaction, 105(6), pp. 1669-1676.

[7.8] Padmanabhan, C., Barlow, R. C., Rook, T. E., and Singh, R., 1995, “Computational Issues Associated with Gear Rattle Analysis,” ASME Journal of Mechanical Design, 117, pp. 185-192.

[7.9] Xiao, D. Z., Gao Y., Wang, Z. Q., and Liu, D. M., 2005, “Conjugation Criterion for Making Clearance of the Meshed Helical Surfaces,” ASME Journal of Mechanical Design, 127(1), pp. 164-168.

[7.10] Helical3D (CALYX software), 2003, “Helical3D User’s Manual,” www.ansol.www, ANSOL Inc., Hilliard, OH.

[7.11] Tamminana, V. K., Kahraman, A., and Vijayakar, S., 2007, “A Study of the Relationship between the Dynamic Factors and the Dynamic Transmission Error of Spur Gear Pairs,” ASME Journal of Mechanical Design, 129(1), pp. 75-84.

[7.12] Richards, J. A., 1983, Analysis of Periodically time-varying Systems, New York, Springer.

[7.13] Cartwright, J. H. E., and Piro, O., 1992, “The Dynamics of Runge-Kutta Methods,” International Journal of Bifurcations Chaos, 2, pp. 427-449. 216 [7.14] Gray, A., and Mathews, G. B., 1966, A Treatise on Bessel Functions and Their Applications to Physics, New York, Dover Publications.

[7.15] Den Hartog, J. P., 1956, Mechanical Vibrations, New York, Dover Publications.

[7.16] Abousleiman, V., Velex, P., and Becquerelle, S., 2007, “Modeling of Spur and Helical Gear Planetary Drives with Flexible Ring Gears and Planet Carriers,” ASME Journal of Mechanical Design, 129(1), pp. 95-106.

217 CHAPTER 8

CONCLUSION

8.1 Summary

A combination of analytical and numerical techniques has been utilized to fully understand the influence of sliding friction on spur and helical gear dynamics with vibro- acoustic sources. Many dynamic phenomena that emerge due to interactions between parametric variations (time-varying mesh stiffness/damping) and sliding friction are predicted and partially validated through experiments. A summary of the self-contained chapters is described below.

In Chapter 2, a new multi-degree-of-freedom (MDOF), linear time-varying (LTV) spur gear model has been formulated which overcomes the deficiency of Vaishya and

Singh’s work [8.1-8.3] by employing realistic tooth stiffness functions and the sliding friction over a range of operational conditions. Refinements include: (1) an accurate representation of tooth contact and spatial variation in tooth mesh stiffness based on a finite element/contact mechanics code in the “static” mode; (2) Coulomb friction model for sliding resistance with empirical coefficient of friction as a function of operation

218 conditions; (3) a better representation of the coupling between the LOA and OLOA directions including torsional and translational degrees of freedom. Numerical solutions of the MDOF model yield the dynamic transmission error (DTE) and vibratory motions in the LOA and OLOA directions. The new model has been successfully validated first by using the finite element code while running in the “dynamic” mode and then by analogous experiments. Since the lumped model is more computationally efficient when compared with the finite element analysis, it could be quickly used to study the effect of a large number of parameters.

In Chapter 3, the MDOF spur gear pair model (initially proposed in Chapter 2) has been improved with time-varying coefficient of friction, µ()t , given realistic mesh stiffness profiles. Alternate sliding friction models have been comparatively evaluated and the interfacial friction forces and motions in the OLOA direction were successfully predicted. In particular, one model has been validated by comparing predictions to the benchmark gear friction force measurements made by Rebbechi et al. [8.4].

In Chapter 4, semi-analytical solutions have been developed to periodic differential equations with time-varying parameters of spur gears including realistic mesh stiffness and sliding friction. Proposed one-term and multi-term harmonic balance predictions compare well with numerical simulations; the computational efficiency is achieved by converting the periodic differential equations into easily solvable algebraic equations, while providing more insight into the dynamic behavior. Both super-and sub- harmonic analyses are successfully conducted to examine the higher mesh harmonics due to multiplicative coefficients and the system stability, respectively. Semi-analytical solutions are developed for a 6DOF system model for the predictions of (normal) mesh

219 loads, friction forces and bearing displacements in the LOA and OLOA directions, under

non-resonant conditions.

In Chapter 5, a refined source-path-receiver model has been developed which

characterizes the sliding friction induced structural path and associated noise radiation.

Proposed Rayleigh integral method and substitute source technique are more efficient for

calculating the acoustic field than the usual boundary element technique and thus they provide rapid design tools to quantify the frictional noise. Individual contributions of

transmission error (via the LOA path) and frictional effects (via the OLOA path) are

compared to the overall whine noise at gear mesh frequencies.

In Chapter 6, a new 12 DOF model for helical gears with sliding friction has been

developed; it includes rotational motions, translations along the LOA and OLOA

directions and axial shuttling motions. Three-dimensional model has been proposed that

characterizes the contact plane dynamics and captures the reversal at the pitch line due to

sliding friction. Calculation of the contact forces and moments is illustrated by using a

sample helical gear pair. A refined method is also suggested to estimate the tooth

stiffness density function along the contact lines [8.5].

In Chapter 7, the Floquet theory has been applied to a helical gear pair to examine

the effect of sliding friction on the DTE. In particular, the LTV formulations (with

parametric excitations) have been developed for a SDOF model, and the effect of sliding

friction is quantified as parametric excitations of effective mesh stiffness. The Floquet

theory has been successfully applied to obtain closed-form DTE solutions given periodic

and piece-wise linear stiffness functions. Responses to both initial conditions and forced

periodic functions, under a nominal preload, are derived. Analytical models have been

220 validated by comparing predictions with numerical simulations. Although the coefficient

of friction µ is assumed to be high for the example case for illustrative purpose, the same

algorithm could be implemented under realistic conditions when a smaller value of µ is expected. Parametric instability issues are briefly examined.

8.2 Contributions

Chief contribution of this research has been the development of new or refined mathematical models and analysis techniques that enhance our understanding of the influence of friction on the dynamics and vibro-acoustics of spur and helical gears. One of the main effects of sliding friction is the enhancement of the DTE magnitude at the second gear mesh harmonic. A key question whether the sliding friction is indeed the source of the OLOA motions and forces (in spur gears) is then answered by this study.

The bearing forces in the LOA direction are influenced by the normal tooth loads, but the sliding frictional forces primarily excite the OLOA motions. The effect of the profile modification on the dynamic transmission error has been analytically examined under the influence of frictional effects. For instance, the tip relief introduces an amplification in the OLOA motions and forces due to an out of phase relationship between the normal

(mesh) load and friction forces. This knowledge should be of significant utility to the designers.

New analytical solutions are constructed which provide an efficient evaluation of the frictional effect as well as a more plausible explanation of dynamic interactions in

221 multiple directions. Both single- and multi-term harmonic balance methods are utilized to predict dynamic mesh loads, friction forces and pinion/gear displacements. Such semi- analytical solutions explain the presence of higher harmonics in gear noise and vibration due to exponential modulations of the periodic stiffness, dynamic transmission error and sliding friction. This knowledge also analytically reveals the effect of the tooth profile modification in spur gears on the dynamic transmission error, under the influence of sliding friction. Further, the Floquet theory is applied to obtain closed-form solutions of the dynamic response for a helical gear pair, where the effect of sliding friction is quantified by an effective piecewise stiffness function. Analytical predictions, under both homogeneous and forced conditions, are validated using numerical simulations. The matrix-based methodology is found to be computationally efficient while leading to a better assessment of the dynamic stability.

Finally, the improved source-path-receiver model for friction-induced gear whine noise reveals that near the “optimal” load, friction-induced noise is comparable to the transmission error induced noise (especially for the first two mesh harmonics); thus sliding friction should be considered as a significant contributor to whine noise. However, at non-optimal torques, frictional noise is overwhelmed by the transmission error noise, thus sliding friction could be negligible under such conditions. This confirms that the sliding friction should be viewed as a potential contributor to structure-borne noise for high precision, high power density geared systems.

222 8.3 Future Work

Directions for future work are suggested below (as an extension of this work).

1. Develop coupling indices that would quantify the effect of sliding friction on the

gear dynamic and vibro-acoustic responses. For instance, the 6DOF spur gear

dynamics model (of Chapter 2) suggests that while a change in the coefficient of

sliding friction µ has a global effect, the OLOA dynamic displacements y p and

yg affect other degrees-of-freedom somewhat minimally. This implies a one-way

coupling effect and thus the system could be further divided into a 4DOF

“excitation” sub-system and a 2DOF “response” system for the OLOA dynamics.

This issue has been briefly discussed in Chapter 4 where the multi-term harmonic

balance solutions (based on the SDOF model) are extended to predict dynamics in

multiple directions due to the one-way coupling effect.

2. Examine the effect of various modifications to spur gears and propose new

criteria which would simultaneously consider contributions from both static

transmission error and friction excitations. Also, conduct parametric design

studies to seek suitable profile modification schemes to minimize both combined

excitations over a range of operating loads. Use analytical models to examine

interactions between profile modifications and sliding friction. Further, it is

desirable to examine the mesh stiffness concept (associated with the Floquet

theory) and to quantify the role of sliding friction in terms of discontinuities and

generation of higher harmonics.

223 3. Extend the closed form solutions and semi-analytical solutions (as proposed in

this work) to multi-mesh spur gear dynamics. Application of multi-term harmonic

balance method to multi-mesh gear systems with non-commensurable mesh

frequencies and/or backlash elements would lead to periodic, non-linear

formulations. Also, investigate friction-induced instabilities in the presence of

clearance non-linearities.

4. Refine surface mechanics and tribological models and quantify the effects of

normal load, operating speed, lubricant viscosity, gear body materials on the

magnitude and dynamic characteristics of sliding friction forces. Also, incorporate

the effect of gear surface roughness (random components) and surface

undulations (periodic components) in gear vibration and structure-borne noise

models.

5. Improve the system path model by including compliant shaft formulation and

moment transfer through the bearing matrix. Quantify the error induced by the

impedance mismatch assumed between the gear source sub-system and the

gearbox structural sub-system. Likewise, examine the limitations of the simplified

source-path-receiver network adopted in this work.

6. Extend the simplified 2D sound radiation model [8.6] of the gearbox into a 3D

model which should require a more rigorous analysis by using monopoles (or

multipoles [8.7]) as the substitute sound sources. Examine the substitute sound

source property to quantify the frictional effect on the gear whine noise in terms

of acoustic source strength and directivity. Also, compare analytical predictions of

gear noise with the boundary-element solutions

224 7. Describe other acoustic source mechanisms that may contribute to air-borne noise

induced by sliding friction between gear teeth. Potential acoustic sources may

include the oscillatory behavior of contact zone after release from sliding and

acoustic pumping within the cavities formed between mating teeth.

8. Conduct new experimental work (over a range of tribological, thermal and

operating loads and speeds) to validate and refine analytical predictions. Apply

smart materials based sensors to measure in-situ forces and motions. Finally,

apply active and/or passive control methods to concurrently reduce dynamic mesh

and sliding friction force excitations.

References for Chapter 8

[8.1] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration, 243(3), pp. 525-545.

[8.2] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671- 694.

[8.3] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.

[8.4] Rebbechi, B., and Oswald, F. B., 1991, “Dynamic Measurements of Gear Tooth Friction and Load,” NASA-Technical Memorandum, 103281.

[8.5] Helical3D (CALYX software), 2003, “Helical3D User’s Manual,” www.ansol.www, ANSOL Inc., Hilliard, OH.

225 [8.6] Pavić, G., 2005, “An Engineering Technique for the Computation of Sound Radiation by Vibrating Bodies using Substitute Sources,” Acta Acustica Journal, 91, pp. 1-16.

[8.7] Pavić, G., 2006, “A Technique for the Computation of Sound Radiation by Vibrating Bodies Using Multipole Substitute Sources,” Acta Acustica Journal, 92, pp. 112-126.

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233