Analytical Study on Nonlinear Dynamics of Planetary Gears

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Cheon-Jae Bahk, B.S., M.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2012

Dissertation Committee:

Prof. Robert G. Parker, Advisor Prof. Donald R. Houser Prof. Brian D. Harper Prof. Sandeep M. Vijayakar c Copyright by

Cheon-Jae Bahk

2012 Abstract

This work aims to advance the understanding of nonlinear dynamics of planetary gears and the inﬂuence of the key system parameters on dynamic response. Analytical solutions of nonlinear dynamic model are mainly used to conduct investigations on interesting nonlinear dynamic behaviors.

An analytical lumped-parameter model, which is parametrically excited by time-varying mesh stiffness and includes tooth separation, shows nonlinear dynamic response. The accuracy of the model is correlated against a benchmark ﬁnite element analysis over broad mesh frequency ranges.

The nonlinear dynamic model is analytically solved by perturbation analysis. Concise, closed-form expressions of planetary gear dynamic response are obtained for various resonances. The analytical solution is validated by numerical integration and the harmonic balance method. The rapid calculation of dynamic response with acceptable accuracy demonstrates that the analytical solutions are effective for performing parametric studies.

The explicit inclusion of key system parameters in the analytical solution shows the impact of the system parameters on planetary gear nonlinear vibration. Mesh stiffness discontinuity from tooth contact loss is considered for the analytical solution that gives nonlinear vibrations. Correlation between the external torque and vibration amplitude proves that tooth contact loss can occur even under large torque. Resonances at multiple harmonics

ii of the mesh frequency are distinguished by different excitation sources. Nonlinear subharmonic resonance characterized by response jump phenomena on both sides of the mesh frequency range where resonance occurs is examined.

The impact of system parameters on planetary gear vibrations is investigated by using a generalized planetary gear model including bearing stiffness and relative mesh phase. Use of the well-deﬁned modal properties and closed-form expressions of resonant response con-

ﬁrm the existing mesh phasing rules to suppress selected vibration modes of primary resonance. Extended suppression rules for super- and subharmonic resonances are proposed. In addition to the suppression conditions, the analytical solution discovers the dependence of dynamic response on the system parameters and vibration modes, which provides practical guidance for ﬁnding optimal design parameters for vibration reduction.

A nonlinear analytical tooth profile modification model for planetary gear dynamic response is developed. Perturbation analysis gives a closed-form expression for the frequency response relation including the fundamental tooth profile modification parameters of the modification amount and length. Different effects of tooth profile modification on static transmission error and dynamic response are compared in terms of the modification amount. Strong influence of system parameters on the dynamic effect of tooth profile modification is discovered.

iii DEDICATION

Dedicated to my wife, Jeong-Eun Lim

to my lovely daughters, Bokyung and Namee

and to my parents, who have dedicated their lives to supporting me.

iv Acknowledgments

As I am coming to the ﬁnal stage of my long doctoral degree journey, I would like to take a moment to thank many people who helped me prepare my dissertation.

First, I would like to express my sincere appreciation to my advisor, Prof. Robert G.

Parker. He offered me a great opportunity to explore the academic world. He has been dedicated to support my graduate studies and particularly to improve my academic writing skills. I have learned so many things from him and realize that it is hard to express in words how instrumental he has been in pursuing my doctoral degree. He has been warm, patient, generous, and strict with me. He has kept holding me to such a high standard.

His exceptional intelligence and passion for research truly inspired and encouraged me to become a valued researcher like him. There is no doubt that my success at the Ohio State is largely due to his academic guidance, encouragement, endless patience, and I would like to dump a cooler of Gatorade on him.

I am also very grateful to Prof. Donald Houser, Prof. Brian Harper, and Prof. Sandeep

Vijayakar for being part of an important milestone in my life as serving on my Ph.D dissertation committee. It is my honor to have Prof. Donald Houser in my committee because his tremendous work on the gear world really inspired me to set a high expectation of quality dissertation. I acknowledge Prof. Brian Harper for gladly accepting my request for being a committee member when I was busy to prepare my dissertation. I specially thank to Prof.

Sandeep Vijayakar for providing the wonderful (and FREE) ﬁnite element analysis tool

v that has been extensively used in this work. I would like to acknowledge Prof. Martha

Cooper for agreeing to serve on my defense as a Graduate Faculty Representative and for her proofreading.

I want to say thank you to Commander Jung Hyunjin. It was difﬁcult period of time for me to prepare the study abroad while serving in the ROK Navy. He understood my situation and encouraged me to achieve my goal. One day, I was in danger of missing an important English test due to an order limiting use of vacations. He exceptionally allowed me to go take the test. He probably does not know how grateful I was to him. Now, I say

Thank you Commander Jung.

Now it is my second home town, but everything was very strange and new to me when I arrived in Columbus, 2002. There are too many to name, but many friends helped me settle down in Columbus, OH. I wish all of them the very best.

I am thankful to my lab mates in the Dynamics and Vibration Lab. I feel very lucky to have known these brilliant and talented people. All the valuable discussions and sug- gestions from them have been really helpful for my research. I thank you for all the good times.

I want to thank all the members in Columbus Jungto Society. I remember when I ﬁrst stepped in their weekly meeting room. There were Kim Junja, Ha Ilsuk, Ko Changmi,

Kim Youngsuk, and Kim Sunmin, and they gave me very warm welcome. I was, luckily, the only student in the group and exclusively enjoyed their love until another student Kim

Hyosun joined years later. They have been supportive in many ways and made my life in

Columbus fun and joyful.

I would like to acknowledge Kang Maru. During the ﬁnal quarter, I realized that the graduation requires not only a quality dissertation but also much of paper work and small

vi stuffs to do. Being away from the campus makes it even more difﬁcult to deal with. Maru gladly helped me complete all the necessary but troublesome works. Thank you Maru.

Buckeyes Football Fever. I did not like the football at the ﬁrst time, but now I am a big fan of it. I have had an enjoyable weekend watching their games (well, not always enjoyable). I am so proud of OSU football team and being an alumni of OSU. It may sound funny, but they encouraged me to stand up when my knees were down on the road of my doctoral journey and to run again to the touchdown line. Yes, I am now about to cross the line and to celebrate good enough for an excessive celebrations penalty. Who cares. Their knees are down now. It’s my turn to encourage them to stand up, and I believe they will ﬂy again. Go Bucks.

Finally, I need to acknowledge my family. My wife Jeong-Eun Lim, she is so much talented and strong. She gave birth to two babies, fed them, took care of all the housework such as cooking and cleaning, and graduated earlier than me! Nowadays, she has been again overloaded to support my all-day hard work on the dissertation and understanding my stress attacks. There is too much to pay her back, but I will try. Thank you Jeong-Eun.

My lovely two daughters, their smiles are simply an energy drink to me. I promise I will be a better dad. Thank you to my sister Yina and brother Woosung for covering my absence at home. While I have been away from home for a long time, they have taken care of all the problems. There are not enough words to express how much I thank to my parents. They always believe I can do it and give endless love and support. My dad had a trafﬁc accident.

It was so fortunate that he was not injured seriously. I wish he can recover soon and begin weekly video chatting again.

vii Vita

December 9, 1976 ...... Born - Cheongju, Korea

1999 ...... B.S., Naval Architecture and Ocean En- gineering, Seoul National University Seoul, Korea 2008 ...... M.S., Mechanical Engineering, The Ohio State University Columbus, Ohio, USA 2002 - 2008 ...... Graduate Research Associate, The Ohio State University 2008 - present ...... Gearbox Analyst, Caterpillar Inc.

Publications

Research Publications

C.-J. Bahk and R. G. Parker “Suppression of Super- and Subharmonic Resonances for Planetary Gears Through Mesh Phase”. In preparation, 2012.

C.-J. Bahk and R. G. Parker “Analytical Investigation of Tooth Proﬁle Modiﬁcation Effects on Planetary Gear Dynamics”. In preparation, 2012.

C.-J. Bahk and R. G. Parker “Inﬂuence of System Parameters on Planetary Gear Dynamic Response”. In preparation, 2012.

C.-J. Bahk and R. G. Parker “Analytical Solution for Nonlinear Dynamics of Planetary Gears”. Journal of Computational and Nonlinear Dynamics, 06(2):021007, April 2011.

viii C.-J. Bahk and R. G. Parker “A Study on Planetary Gear Dynamics with Tooth Proﬁle Modiﬁcation”. In ASME Design Engineering Technical Conference, PTG, no. DETC2011- 47346, Washington DC, 2011.

C.-J. Bahk and R. G. Parker “Nonlinear Resonant Planetary Gear Vibration with Contact Loss”. 13th Asia Paciﬁc Vibration Conference,, University of Canterbury, New Zealand, 2009.

C.-J. Bahk and R. G. Parker “Nonlinear Dynamics of Planetary Gears with Equal Planet Spacing”. In ASME Design Engineering Technical Conference, PTG, no. DETC2007/VIB- 35790, Las Vegas, Nevada, 2007.

Fields of Study

Major Field: Mechanical Engineering

ix Table of Contents

Page

Abstract ...... ii

Acknowledgments ...... v

Vita ...... viii

List of Tables ...... xiii

List of Figures ...... xiv

1. Introduction ...... 1

1.1 Motivation and Objectives ...... 1 1.2 Literature Review ...... 3 1.3 Scope of Investigation ...... 8

2. Analytical Solutions for the Nonlinear Dynamics of Planetary Gears ...... 11

2.1 Introduction ...... 11 2.2 System Model ...... 14 2.3 Interpretation of Nonlinear Response ...... 17 2.4 Harmonic Balance Method With Arclength Continuation ...... 25 2.5 Analytical Solution by the Method of Multiple Scales ...... 27 2.5.1 Primary Resonance, ω ≈ ωi ...... 29 2.5.2 Subharmonic Resonance, ω ≈ 2ωi ...... 41 2.5.3 Resonance at Multiples of Mesh Frequency ...... 47 2.6 Conclusions ...... 54

x 3. Inﬂuence of System Parameters on Planetary Gear Dynamic Response . . . . . 56

3.1 Introduction ...... 56 3.2 Analytical Expressions of Dynamic Response ...... 58 3.3 Mesh Phase Effect ...... 68 3.3.1 Primary Resonance, Equally Spaced Planets ...... 69 3.3.2 Primary Resonance, Diametrically Opposed Planets ...... 73 3.3.3 Higher Harmonic Excitation ...... 75 3.3.4 Superharmonic Resonance, Equally Spaced Planets ...... 76 3.3.5 Superharmonic Resonance, Diametrically Opposed Planets . . . 80 3.3.6 Subharmonic Resonance ...... 81 3.3.7 Complete Suppression Conditions ...... 82 3.4 Contact Ratio Effect ...... 95 3.5 Fluctuation of Mesh Stiffness Effect ...... 107 3.6 Damping Effect ...... 111 3.7 Number of Planets Effect ...... 114 3.8 Combined Effects of Multiple Parameters ...... 117 3.8.1 Combined Effect of Number of Planets and Mesh Stiffness Fluc- tuation ...... 117 3.8.2 Combined Effect of Damping with Contact Ratio and Mesh phase 118 3.8.3 Combined Effect of Contact Ratio and Fluctuation of Mesh Stiff- ness ...... 119 3.9 Summary and Conclusions ...... 128

4. Analytical Investigation of Tooth Proﬁle Modiﬁcation Effect on Planetary Gear Dynamics ...... 130

4.1 Introduction ...... 130 4.2 Tooth Profile Modification Model ...... 132 4.2.1 Tooth Profile Error Function Model (Model 1) ...... 134 4.2.2 Loaded Static Transmission Error Model (Model 2) ...... 137 4.3 Comparison of Analytical Tooth Profile Modification Models to Finite Element Model ...... 141 4.4 Perturbation Analysis with Tooth Profile Modification ...... 147 4.4.1 Perturbation Solution Using the Method of Multiple Scales . . . 147 4.4.2 Validation of Perturbation Solution ...... 152 4.5 Results and Discussion ...... 156 4.5.1 Optimal TPM for Minimum Dynamic Response ...... 156 4.5.2 Correlation Between Static and Dynamic Response ...... 161 4.5.3 Correlation Between Sun-planet and Ring-planet TPM ...... 163 4.5.4 Sensitivity of TPM Effects to System Parameters ...... 166

xi 4.6 Conclusions ...... 175

5. Conclusions and Future Work ...... 177

5.1 Conclusions ...... 177 5.1.1 Analytical Solutions for the Nonlinear Dynamics of Planetary Gears ...... 177 5.1.2 Influence of System Parameters on Planetary Gear Dynamic Re- sponse ...... 178 5.1.3 Analytical Investigation of the Dynamic Effects of Tooth Profile Modification on Planetary Gears ...... 179 5.2 Future Work ...... 180

Appendices 184

A. Gsn and Grn for a Purely Rotational Model ...... 184

B. Expressions of Eigenvalues for a Purely Rotational Model ...... 186

∂∆ C. Expressions for max ...... 188 ∂N

Bibliography ...... 189

xii List of Tables

Table Page

2.1 System parameters of a three equally spaced planetary gear rotational model. 18

2.2 Damping values ...... 19

2.3 Fourier coefﬁcients of sun-planet and ring-planet mesh stiffnesses . . . . . 19

3.1 System parameters of an example planetary gear...... 83

3.2 Suppression conditions for superharmonic resonance ...... 86

3.3 Suppression conditions for subharmonic resonance (∆˜ ) ...... 86

4.1 System parameters of the example planetary gear models...... 133

4.2 Fourier coefﬁcients of sun-planet and ring-planet mesh stiffnesses . . . . . 153

xiii List of Figures

Figure Page

2.1 Rotational lumped parameter model of planetary gear system...... 12

2.2 Vibration modes of planetary gear rotational model for parameters in Table 2.1...... 22

2.3 Steady state response of a) RMS (mean removed) sun rotational deﬂection and b) sun mean rotation. Both increasing and decreasing speed sweeps are performed. (—) NI; (···) FE analysis...... 23

2.4 Waterfall spectra of sun and planet rotational deﬂection for decreasing speed by ﬁnite element, (a) and (c), and numerical integration, (b) and (d). . 24

2.5 Tooth separation function and mesh deﬂection. (a) In-phase and (b) out-of- phase (by π) sun-planet and ring-planet mesh deﬂection...... 37

2.6 RMS (mean removed) planet rotational deflection from numerical integration for (a) the first and (b) the second distinct primary resonance for full and limited harmonics of the mesh stiffness variations. (—) Full harmonics of stiffness, (- - -) the first and second harmonics of stiffness, (···) The first harmonic of stiffness...... 38

2.7 Waterfall spectra of planet rotational deﬂection for (a) only the ﬁrst, (b) up to the second and (c) full harmonics of the mesh stiffness variations by numerical integration...... 39

2.8 RMS (mean removed) sun and planet rotational deﬂection for primary resonance of the ﬁrst (a), (c) and the second (b), (d) distinct mode by numerical integration (NI), harmonic balance (HB) and method of multiple scales (MMS). Unstable solutions are shown as dashed lines...... 40

xiv 2.9 RMS (mean removed) planet rotational deﬂection for subharmonic resonance of the second distinct mode. The dashed line is the unstable HB solution...... 45

2.10 Waterfall spectra of planet rotational deﬂection for subharmonic resonance by ﬁnite element for decreasing speed...... 45

2.11 RMS (mean removed) sun rotational deﬂection for subharmonic resonance of the second distinct mode by method of multiple scales (MMS) and numerical integration (NI). Both increasing and decreasing speed are performed by numerical integration...... 46

2.12 Sun-planet tooth separation time and points denoting 5% and 10% deviation between perturbation and numerical integration solutions. (O) subharmonic, (•) primary for the second distinct mode...... 46

2.13 RMS (mean removed) sun rotational deﬂection for (a) second harmonic excitation and (b) superharmonic resonance by numerical integration (NI), harmonic balance (HB), and method of multiple scales (MMS)...... 52

2.14 Sun response time history for different resonances at mesh frequency 2200Hz; (•) Second harmonic excitation only, (···) superharmonic resonance (only excitation at mesh frequency), (- - -) sum of second harmonic excitation and superharmonic resonance, and (—) resonances excited by ﬁrst and second harmonics of mesh stiffness...... 53

3.1 A schematic of two-dimensional lumped parameter model of a planetary gear system ...... 67

3.2 Mesh stiffness variation approximated by a trapezoidal wave function. c is the contact ratio. s is the slope coefﬁcient. T is the mesh period...... 67

3.3 Variation of the peak amplitude of primary resonance with mesh phase. Rotational mode for four (a) equally spaced planets and (b) diametrically opposed planets for the example system in Table 3.1. For (a), ( • ) perturbation and (- -- -) numerical solutions. For (b), perturbation solutions for ( ) even and (− − −) odd zs, and numerical solutions for () even and (•) odd zs under the assumption of k˜sn = k˜sm and k˜rn = k˜rm; perturbation solutions for (-.-) even and (···) odd zs, and numerical solutions for ˜ ˜ ˜ ˜ () even and ( ) odd zs under the assumption of ksn 6= ksm and krn 6= krm. . 87

xv 3.4 Variation of the peak sun rotation vibration amplitude with mesh phase for equally spaced planets. Rotational mode of superharmonic resonance for the example system in Table 3.1 with (a) N = 3, (b) N = 4, (c) N = 5, and(d) N = 6. ( • ) Perturbation and (- -- -) numerical solutions are compared...... 88

3.5 Variation of the peak sun translation vibration amplitude with mesh phase for equally spaced planets. Translational mode of superharmonic resonance for the example system in Table 3.1 with (a) N = 3, (b) N = 4, (c) N = 5, (d) N = 6, (e) N = 7, and (f) N = 8. ( • ) Perturbation and (- -- -) numerical solutions are compared...... 89

3.6 Variation of the peak planet rotation vibration amplitude with mesh phase for equally spaced planets. Planet mode of superharmonic resonance for the example system in Table 3.1 with (a) N = 4, (b) N = 5, (c) N = 6, and(d) N = 7. ( • ) Perturbation and (- -- -) numerical solutions are compared...... 90

3.7 Variation of the peak amplitude with mesh phase for four diametrically opposed planets. (a), (b) Rotational and (c), (d) translational modes of superharmonic resonance for the example system in Table 3.1 with (a), (c) even and (b), (d) odd zs.( • ) Perturbation and () numerical solutions are compared...... 91

3.8 Suppressed sun translation vibration for (a) six and (b) eight diametrically opposed planets with odd zs. System parameters are from the example system in Table 3.1. γ4 = 0.5 is used for (b)...... 92

3.9 Waterfall plots of planet rotation for four equally spaced planets of the example system in Table 3.1. Meshes are in-phase. Modal damping of (a) 3% and (b) 5% are used...... 93

3.10 The effects of mesh phase on ∆˜ for four (a) equally spaced and (b) diametrically opposed planets of the example system in Table 3.1. (- - -) odd and (—) even zs in (b) ...... 94

3.11 The effects of contact ratios on the peak sun rotation vibration amplitude for three equally spaced planets for the example system in Table 3.1. Meshes are in-phase. Primary resonance of (a) the ﬁrst and (b) the second rotational modes for a rotational-translational model and of (c) the ﬁrst and (d) second distinct modes for a rotational model...... 102

xvi 3.12 Contour plot of (a) β1 for the ﬁrst distinct mode and (b) βN+1 for the second distinct mode varying with m and k. (—) N = 3 and (- - -) N = 8 are used for the example system in Table 3.1...... 103

3.13 Variation of the peak sun rotation vibration amplitude with cs for primary resonance of the second distinct mode for the example system in Table 3.1. cr is ﬁxed at 1.5. Perturbation solutions for (—) trapezoidal and (- - -) rectangular mesh stiffness variation. Lines with symbols are from numerical solutions...... 104

3.14 Contour plot of (a) γ1 for the ﬁrst distinct mode and (b) γN+1 for the second distinct mode varying with m and k. (—) N = 3 and (- - -) N = 8 are used for the example system in Table 3.1...... 105

3.15 The effects of contact ratios on the width of the parametric instability boundary. (a) The ﬁrst and (b) the second distinct modes for three equally spaced planets with in-phase meshes for the example system in Table 3.1 . . 106

3.16 Variation of the peak planet translation vibration amplitude with (a) k˜r for the ﬁrst rotational mode and (b) k˜s for the second rotational mode. There are four equally spaced planets with in-phase meshes for the example system in Table 3.1. For (a), k˜s is ( ) 100e6 N/m, (− − −) 200e6 Nm, and (···) 300e6 Nm. For (b), k˜r is ( ) 100e6 N/m, (− − −) 200e6 Nm, and (···) 300e6 Nm...... 109

3.17 Root mean square (mean removed) of sun rotation for primary resonance of the ﬁrst rotational mode. There are four equally spaced planets with in- phase meshes for the example system in Table 3.1. The damping ratio is 3%. k˜s is ﬁxed at 200e6 N/m in (b). Unstable solutions are shown as dashed lines in both (a) and (b). In (b), dashed and solid lines are for ωo f f ≤ 0 and dotted lines are for ωo f f ≥ 0 ...... 110

3.18 Variation of (a) the peak planet rotation vibration amplitude of primary resonance for the ﬁrst rotational mode and (b) the width of the parametric instability boundary for the distinct mode with in-phase meshes for the example system in Table 3.1. For (b), the ﬁrst distinct mode with (—) N = 4 and ( ) N = 6 and the second distinct mode with (- - -) N = 4 and (− − −) N = 6 are compared...... 113

xvii 3.19 Variation of ∂δ/∂N with m and k for ((a) and (c)) the ﬁrst and ((b) and (d)) the second distinct modes for the example system in Table 3.1. (a) and (b) for N = 3, (c) and (d) for N = 8...... 116

3.20 Variation of the width of the parametric instability boundary for (a) k˜r = 2k˜s =200e6 N/m and (b) k˜s = 2k˜r =200e6 N/m. System parameters are from the example system in Table 3.1. Planets are equally spaced with in- phase meshes. For (b), the ﬁrst distinct mode with (—) N = 4 and ( ) N = 6 and the second distinct mode with (- - -) N = 4 and (− − −) N = 6 are compared...... 121

3.21 Combined effects of contact ratio and damping ratio on the width of the parametric instability boundary of the ﬁrst rotational mode for the example system in Table 3.1. Four planets are equally spaced with in-phase meshes. 122

3.22 The width of the parametric instability boundary for the ﬁrst rotational mode for six diametrically opposed planets with different damping ratio: (a) 1% and (b) 3%. System parameters are from the example system in Table 3.1...... 123

3.23 The effects of damping on the mesh phase causing the parametric instability for ( ) the ﬁrst and (− − −) the second rotational modes for the example system in Table 3.1. Amp = the ratio of mesh phase area causing parametric instability to total area...... 124

3.24 Variation of the peak planet rotation vibration amplitude for primary resonance of the second distinct mode with contact ratios. Three equally spaced planets with in-phase for the example system in Table 3.1. k˜s =700e6 - 600e6 = 100e6 N/m and k˜r = (a) 650e6 - 550e6 = 100e6 N/m and (b) 750e6 - 450e6 = 300e6 N/m. cr and cs is ﬁxed at 1.5 for (c) and (d), respectively. For (c) and (d), k˜r = (—) 100e6 N/m, (- - -) 200e6 N/m, and (···) 300e6 N/m.125

3.25 Combined effects of contact ratios and mesh stiffness fluctuation (k˜ = k˜s = k˜r) on the width of parametric instability boundary of the first rotational mode for the example system in Table 3.1. Four planets are equally spaced with in-phase meshes. cs in (a) and cr in (b) are fixed at 1.6...... 126

xviii 3.26 Variation of the width of the parametric instability boundary with the mesh stiffness fluctuation for the first rotational mode. Four planets are equally spaced with in-phase meshes for the example system in Table 3.1. For (a), k˜r is fixed at (···) 150e6 N/m, (−.−) 170e6 N/m, (− − −) 190e6 N/m, and (−−−) 210e6 N/m. For (b), k˜r is fixed at (···) 100e6 N/m, (− − −) 200e6 N/m, and (− − −) 300e6 N/m...... 127

4.1 A schematic of two-dimensional lumped parameter model of a planetary gear system ...... 139

4.2 Sun-planet mesh stiffness for example system 1 described in Table 4.1. . . . 139

4.3 Description of tip and root modiﬁcation. SAP stands for start of active proﬁle.140

4.4 Root mean square (mean removed) sun rotational deﬂection without TPM, with TPM A, and with TPM B. Comparison of FE model and analytical models 1 and 2. Gear parameters and input torque are from example system 1 in Table 4.1. (- - -) Model 1, (-.-) model 2, and (···) FE solution. The solid line is the common solution from models 1 and 2...... 144

4.5 Root mean square (mean removed) sun rotational deﬂection without TPM, with TPM C, and with TPM D. Comparison of FE model and analytical models 1 and 2. Gear parameters and input torque are from example system 2 in Table 4.1. (- - -) Model 1, (-.-) model 2, and (···) FE solution. The solid line is the common solution from models 1 and 2...... 145

4.6 Static transmission errors of the sun-planet gear pair with different tooth proﬁle modiﬁcations. Gear parameters are from example system 1 in Table 4.1 with an input torque on the sun gear of 376.7 Nm. (—) No TPM, (- - -) TPM A, and (-.-) TPM B...... 146

4.7 Root mean square (mean removed) planet rotational deflection of model 1 for example system 1 in Table 4.1. For (a), with (···) and without (—) consideration of the partial tooth separation. For (b), with (- - -) and without (—) consideration of the simplification related to the influence of TPM on the tooth separation...... 154

4.8 Comparison of (- - -) numerical and (—) perturbation solutions with TPM A and TPM AA for the ﬁrst and second rotational modes of example system 1 in Table 4.1...... 155

xix 4.9 Variation of sun rotation peak amplitude with the amount of sun TPM for the first rotational mode of example system 1 in Table 4.1. Tip modification starts from HPSTC and root modification starts from LPSTC. (—) Perturbation, (- - -) numerical, and (···) finite element solutions...... 158

4.10 Variation of sun rotation peak amplitude with the length of sun TPM for the first rotational mode of example system 1 in Table 4.1. Amount of modification is 76.2 µm. (—) Perturbation, (- - -) numerical, and (···) finite element solutions...... 159

4.11 Contour plot of sun rotation peak amplitude varying with the amount and the starting roll angle (or length) of sun TPM for the ﬁrst rotational mode of example system 1 in Table 4.1...... 160

4.12 Variation of static and dynamic response with the amount of sun TPM. Tip modification starts from HPSTC and root modification starts from LPSTC. Gear parameters are from example system 1 in Table 4.1. (- - -) Static transmission error of FE model of sun-planet gear pair, (-.-) sun rotational static deflection in complete planetary gear FE model , (···) FE solution of sun rotational dynamic response from complete planetary gear system, and (— ) perturbation solution of sun rotational dynamic response from complete planetary gear system...... 162

4.13 Variation of planet rotation peak amplitude with the amount of ring TPM for the first rotational mode of example system 1 in Table 4.1. Tip modification starts from HPSTC and root modification starts from LPSTC. (—) Perturbation and (- - -) numerical solutions...... 164

4.14 Contour plot of planet rotation amplitude varying with sun-planet and ring- planet mesh TPM for the first rotational mode of example system 1 in Table 4.1. Tip modification starts from HPSTC and root modification starts from LPSTC...... 165

4.15 Comparison of the variation of sun rotation peak amplitude with the amount of sun TPM for the second rotational mode of example system 1 in Table 4.1. Tip modification starts from HPSTC and root modification starts from LPSTC. (—) Perturbation, (- - -) numerical, and (···) finite element solutions.170

4.16 Rectangular wave mesh stiffnesses and mesh phase between the sun-planet and ring-planet mesh stiffnesses...... 171

xx 4.17 Variation of sun rotation peak amplitude with the amount of sun TPM for the (a) ﬁrst and (b) second rotational modes of example system 1 in Ta- ble 4.1. The minimum sun-planet mesh stiffness is ( ) 600e6 N/m and (− − −) 700e6 N/m. Lines with symbols, for 600e6 N/m and for 700e6 N/m, are from numerical solutions...... 172

4.18 Variation of ( ) |Ξ3a| and (− − −) |Ξ3b| with the amount of sun TPM for the ﬁrst rotational mode of example system 1 in Table 4.1. Lines with symbols, for |Ξ3a| and for |Ξ3b|, are for the second rotational mode. . 173

4.19 Variation of sun rotation peak amplitude with the amount of sun TPM for the (a) ﬁrst and (b) second rotational modes of example system 1 in Table 4.1. The minimum sun-planet mesh stiffness is 600e6 N/m. Phase between sun-planet and ring-planet mesh stiffness is ( ) 0 π and (− − −) π/4, (−.−) π/2, and (···) π. Numerical solutions for ( ) 0 π,(•) π/4, () π/2, and () π...... 174

xxi Chapter 1: Introduction

1.1 Motivation and Objectives

Planetary gears are popular as a power transmission component and are extensively used in a variety of industrial ﬁelds including automotive, earthmoving vehicles, aerospace, wind turbines, machining tools, etc. The popularity mainly comes from their substantial advantages such as high power density, compactness, multiple and large gear ratios, and load sharing among planets. Despite the wide application of planetary gears, concerns on their durability and reliability issues resulting from undesirable dynamic behavior are still in the gear designer’s mind. In addition to planetary gear failures, negative impacts on customer perception of quality from noise and vibration are ongoing issue.

Planetary gears have the same sources of noise and vibration with other geared systems, which are dynamic gear mesh forces and transmission errors primarily caused by gear tooth compliance, tooth micro-geometry error, friction, manufacturing error, etc. Among them, gear tooth compliance is measured as a time-varying mesh stiffness that parametrically excites the planetary gears. Gear tooth backlash is necessary to avoid excessive heat gener- ation and to improve lubrication and smooth meshing action. The backlash has a side effect admitting meshing gear teeth to separate each other when they undergo large deﬂections

1 at resonance or rough external loads like an engine excitation are applied, which leads to strong nonlinear effects such as vibration jump phenomena and subharmonic resonance.

The unique configuration of planetary gears generates additional parameters that significantly affect planetary gear dynamics. Relative mesh phase between planets is possible due to multiple planets. In addition, as a planet meshes with a sun and ring simultaneously, two different kinds of meshes exists: sun-planet and ring-planet meshes. Along with other common gear system parameters, these additional system parameters make it more diffi- cult and complicated exploring planetary gear dynamics. Experiment is hardly an option to investigate the impact of key system parameters on dynamic response due to its ex- tremely demanding cost and time. Several analytical and finite element models have been developed and their computational simulations have been capturing what are measured experimentally. Although the numerical solutions are less expensive than the experiments, still the high computational effort required makes it inefficient for dynamic analyses as well.

Tooth profile modification is commonly used to compensate for large tooth deflection and poor contact pattern targeting reduction of gear vibration, noise, and contact stress.

Most research on tooth profile modification deals with the single-mesh gear pairs, and it also provides the engineering foundation for the application of tooth profile modification to planetary gears. Possible influence of the planetary system parameters on the effects of tooth profile modification has not been fully addressed. In most cases, optimal tooth profile modification for minimal vibration is determined to reduce static transmission error based on common practical thinking regarding strong relation between static transmission error and vibration without solid analytical support proven. These research gaps could result

2 in improper application of tooth proﬁle modiﬁcation to planetary gears that may lead to undesirable dynamic effects.

The primary goal of this study is to conduct numerical and analytical investigations to explore the unknowns of planetary gear dynamics and to fill the above-mentioned research gaps. A nonlinear dynamic model of planetary gears is analytically solved to give a closed-form approximation of the frequency response function. The analytical solution is expected to provide the insight into planetary gear nonlinear dynamics and to enhance the understanding of the influence of various system parameters on planetary gear vibrations. An analytical tooth profile modification model for planetary gears is developed to investigate the dynamic effects of tooth profile modification. Given the lack of systematic studies for planetary gears, an analytical approach is crucial to find an optimal modification for reduction of vibration, which also provides guidance for practical design and tolerance standard.

1.2 Literature Review

Most research on planetary gears has been conducted within the last two decades. Plan- etary gears have received much less research attention than other single-mesh gear pairs due to their complicated conﬁguration. Early studies go back to the 1970s. These works fo- cused on prediction of natural frequencies and vibration modes [1, 2] and mesh phasing to reduce or eliminate rotational and translational vibrations [3, 4]. Most experimental research was done during the early period. Hidaka and his co-workers published a series of experimental works on the dynamic behavior of planetary gears [5–11]. Their work includes several topics such as load distribution, displacement of gear bodies and inﬂuence of torque, mesh phasing, and the thickness of ring gears. Toda and Botman [4] tested the

3 ﬁrst planetary gear stage of a PT6 aircraft to show the minimization of vibration coming from spacing or pitch error by selecting proper angular phasing of the various planets.

Botman [12] performed an experiment on the same planetary gear of the PT6. His test results show several vibration phenomena such as unequal sidebands, load sharing, nonlinear jumps, and minimization of gear error that supports Toda and Botman’s results [4]. The softening nonlinearity and vibration jump phenomenon demonstrate that tooth separation occurs in planetary gears as well as in single mesh gears. In the 1980s, Kasuba and Au- gust [13, 14] developed an iterative analysis to determine load sharing and used variable mesh stiffness to investigate the effects of the sun gear condition on the planetary gear vibration. Ma and Botman [15] showed a signiﬁcant relationship between dynamic load sharing and transmission error through numerical simulation.

Since the 1990s, there are extensive analytical studies on single mesh gears addressing several topics. Various analytical models have been proposed to conduct dynamic analysis of gear systems. Ozg¨ uven¨ and Houser [16] comprehensively reviewed mathematical models developed through the mid-1980s. Wang et al. [17] suggested four groups of the dynamic gear models in the combination of linear/nonlinear and time-invariant/time-variant systems. Most analytical models considered gears as a rigid disk and spring connection for mating gears. Cunliffe et al. [1] and Botman [2] established a linear time-invariant stiffness model and conducted modal analysis of epicyclic gears. Seager [3] used a linear time-invariant model to show that particular harmonic components of excitation can be neutralized by proper choice of tooth numbers. He showed that mesh stiffness variation does not affect the neutralization condition. Kahraman [18] included tooth separations and periodic mesh stiffness in his planetary gear model for load sharing analysis. Lin and

Parker [19] introduced gyroscopic terms induced by carrier rotation into their nonlinear

4 time-varying model. Kahraman [20] and Ambarisha and Parker [21] reduced a pre-existing rotational-translational model to a purely rotational one. Eritenel and Parker [22] extended an existing 2-D lumped parameter model to 3-D to examine the three dimensional motion of helical planetary gears. Recently, several analytical models of compound planetary gears have been developed [23–25].

Understanding of the modal properties is essential to reduce planetary gear vibration.

Cunliffe et al. [1] studied excitation sources for several vibration modes. Botman [2] examined the impact of carrier rotation on the vibration modes. Kahraman [20] derived a closed-form expression of rotational natural frequencies for a purely rotational model. Lin and Parker [19,26] analytically investigated and classiﬁed the properties of vibration modes of equally spaced and diametrically opposed planet systems and used the modal properties to study eigen-sensitivity to system parameters [27]. Wu and Parker [28–30] considered elastically deformable ring gears when they conducted modal property analysis. Kiracofe and Parker [24] and Guo and Parker [25] extended the modal analysis to compound planetary gears. Eritenel and Parker [22] proposed the categorized vibration modes of a 3-D planetary gear model.

A ﬁnite element model represents a higher ﬁdelity gear mesh contact mechanism than the idealized analytical lumped-parameter model. Given the rarity of experimental data,

finite element solutions often become a benchmark to validate the accuracy of analytical models. Conventional finite element methods require a very refined mesh. Due to the huge computational effort and time required to evaluate dynamic response, use of a conventional

ﬁnite element model was limited to static analysis [31]. Vijayakar [32] developed a unique

finite element/contact analysis tool that combines a finite element analysis with an analytical technique, so it can overcome the shortcomings of the conventional finite element

5 model. The semi-analytical finite element model uses more coarse finite element mesh and requires much less computing power than previous conventional finite element models. Thus, one can more easily examine dynamic response of planetary gear systems. The validity of the semi-analytical finite element model is demonstrated by excellent comparison with experimental data [33]. Parker et al. [34] applied the unique finite element/contact mechanics tool to a planetary gear system to investigate dynamic response. A similar finite element technique was employed by other researchers to study the effect of ring gear

ﬂexibility [35], thin rims [36] and tooth proﬁle wear [37]. Recently, Abousleiman and

Velex [38] investigated the three-dimensional dynamic behavior of planetary gears with a hybrid ﬁnite element/lumped-parameter model.

Planetary gear dynamic behavior exhibits strong nonlinearity induced by the tooth contact loss, which includes vibration jump phenomena, chaos, subharmonic resonance, etc.

Parker et al. [34] employed a finite element/contact mechanics tool to capture the nonlinear dynamics. Numerical integration [18, 21, 39, 40] and harmonic balance method [41] have been employed to solve the nonlinear equations of motion. A few studies showed periodic solutions of nonlinear oscillators with backlash. Theodossiades and his colleagues [42,43] presented a new method that combines a piecewise linear analysis with constant coefficients and a classical perturbation technique with time-varying coefficients to obtain periodic steady-state response of gear systems with backlash and time varying mesh stiffness.

Maccari [44] used an asymptotic perturbation method to seek periodic solutions of two coupled-oscillators with periodic damping forces. Ambarisha and Parker [21] predicted jump, chaotic motions, and period-doubling bifurcation with good comparison between analytical and ﬁnite element models.

6 A relative mesh phase is possible in planetary gears due to multiple meshes. Early studies have shown that the mesh phase has a signiﬁcant impact on the dynamic response and, therefore, the mesh phase can be used to develop strategies to reduce planetary gear vibration [3,4]. Hidaka et al. [5] experimentally demonstrated the inﬂuence of mesh phase on the dynamic tooth load. Kahraman [45] and Kahraman and Blankenship [41] considered mesh phasing for both static and dynamic analysis. Parker and Lin [46] presented a complete analytical description of the mesh phasing relationships. Recently Parker [47] and Ambarisha and Parker [48] proposed simple rules to suppress certain harmonics of vibration modes using the symmetry of planetary gears and tooth mesh periodicity.

The gear mesh contact mechanism is significantly influenced by tooth micro-geometry error, and consequently it changes the vibration characteristics. Although the tooth error is one of the significant vibration sources, intentional tooth profile modification is common in the gear industry to reduce gear vibration and noise. Since Walker [49] introduced a tip relief and Harris [50] found the relationship between static transmission errors and gear profile tip relief in what are known as Harris maps, a number of works [51–59] have examined the effect of tooth profile errors on single-mesh gear pair systems. Among them,

Tavakoli and Houser [53] developed an optimization algorithm to seek an optimum profile modification that minimizes the static transmission error of spur gears. Kubo and Kiy- ono [52] experimentally showed that composite convex tooth form error reduces gear vibration. Kahraman and Blankenship [51] performed experimental case studies by varying applied torques and starting roll angles. They showed that a particular tip relief can minimize dynamic transmission error for a given external load. Wagaj and Kahraman [59] investigated the impact of tooth profile modifications on the transmission error excitation for both spur and helical gear models. Much less studies are found for planetary gears.

7 Kahraman [18] and Lin and Parker [19] included gear proﬁle errors in nonlinear dynamic models of a planetary gear, but the errors were neglected in the dynamic analysis. Litvin et al. [60] proposed a modiﬁed geometry of gear tooth surface for equal load distribution.

Abousleiman and Velex [61] showed great reduction of dynamic tooth loads and deflections with the tooth profile relief by using their hybrid 3D planetary gear model. The influence of tooth profile modification on the dynamic response of planetary gears has not been fully addressed and needs further research attention.

1.3 Scope of Investigation

Analytical investigation of planetary gear dynamics is performed. This project ad- dresses the nonlinear dynamic behaviors and the influence of key design system parameters on planetary gear dynamic response by improving analytical nonlinear dynamic models and using analytical solutions. A purely rotational lumped-parameter and finite element models are used to examine various nonlinear dynamic response in Chapter 2. The analytical solutions obtained in Chapter 2 provide insight into planetary gear vibrations and becomes the basis for the subsequent dynamic analyses in this study. Chapter 3 discusses the suppression of planetary gear vibrations using mesh phase and examines the impact of the design parameters on vibrations. In Chapter 4, analytical tooth profile modification models for planetary gears are established and the effects of tooth profile modification on planetary gear vibration are investigated. The scope of each chapter and the contributions of the current research are summarized as follows.

Chapter 2 shows nonlinear dynamic response of planetary gears over the broad speed ranges. The time-varying mesh stiffness parametrically excites planetary gear systems. At

8 resonance, large dynamic response triggers tooth separation that causes strong nonlineari- ties such as vibration jump and chaotic response. Good agreement between the nonlinear analytical model and the ﬁnite element model builds strong conﬁdence in the analytical model for dynamic analysis. Perturbation analysis (method of multiple scales) is employed to derive closed-form approximations of nonlinear dynamic response for primary, subharmonic, and second harmonic resonances. The accuracy of the analytical solutions is validated by numerical integration and harmonic balance method. The limitation of the analytical solutions for strong nonlinearity is discussed. The closed-form solutions elucidate the mathematical structure of frequency response relation and its relation to system parameters and operating conditions.

In Chapter 3, the analytical model is generalized by including bearing stiffness and relative mesh phase. The extended equations of motion are solved by perturbation analysis, which yields closed-form analytical expressions of planetary gear vibration amplitude.

By using the analytical expressions and the well-deﬁned modal properties, the suppression rules of certain vibration modes through the mesh phase proposed in prior studies [47, 48] are conﬁrmed. The extended suppression rules for superharmonic and subharmonic resonances are proposed. In addition to mesh phase, impact of other important system parameters on planetary gear vibrations is studied: gear contact ratio, mesh stiffness variation, damping, and number of planets. Combined effects of multiple system parameters are discussed. Different sensitivity of dynamic response to sun-planet mesh and ring-planet mesh related parameters is mathematically proven. It is found that the impact of the parameters varies with vibration modes.

Chapter 4 investigates the effect of tooth profile modification on planetary gear dynamic response. An analytical tooth profile modification model for planetary gear dynamics is

9 developed. Individual tooth pair loads are considered, which allows one to simulate the partial tooth separation. The model is correlated against a benchmark finite element model for different modifications and planetary gear systems. A closed-form approximation for frequency response relation including tooth profile modification parameters is obtained by perturbation analysis. The accuracy of the closed-form solution is numerically validated.

The effect of tooth profile modification on planetary gear vibrations is illustrated in terms of the modification amplitude and length. Different modifications are found to minimize static transmission error and dynamic response. Interaction between sun-planet and ring- planet tooth profile modifications is examined. The fluctuation of mesh stiffness and mesh phase significantly affect the dynamic effects of tooth profile modification. The sensitivity of the tooth profile modification effect to vibration modes is studied.

10 Chapter 2: Analytical Solutions for the Nonlinear Dynamics of Planetary Gears

2.1 Introduction

Planetary gears (Figure 2.1) are widely used power transmission elements because of their compactness, multiple gear ratios achieved by changing the ﬁxed element, high power to weight ratio, and reduced noise. Their vibration leads to noise and increased dynamic load that accelerates damage to gears, bearings, and other hardware. Gear mesh stiffness associated with tooth compliance varies periodically with time as the number of tooth pairs in contact changes. This mesh stiffness variation parametrically excites the planetary gear system, causing severe vibrations when a harmonic of the time-varying stiffness approaches a natural frequency or twice a natural frequency. Under certain near-resonant operating conditions gear systems experience tooth separation that induces nonlinear effects such as jump phenomena and sub- and superharmonic resonances that dramatically affect the dynamic response. Such nonlinear effects were demonstrated in prior gear experiments

[12, 62].

Simulations have yielded numerical solutions of gear trains with relatively large numbers of degrees of freedom [13, 39, 40]. In addition, the method of harmonic balance (HB) with arclength continuation is a common numerical method because it yields both stable

11 Planet 1

kr1 kru

u1 kcu ur

uc ks1 ksu

ks2

u ks3 2 Sun

kr3 u3 us kr2 u3

Carrier

Planet 2 Ring

Planet 3

Figure 2.1: Rotational lumped parameter model of planetary gear system.

and unstable periodic solution branches [63–65]. Although these numerical methods provide accurate solutions, they do not expose the mathematical structure of the frequency response function and its dependence on system parameters the way analytical solutions do. Most analytical studies of nonlinear gear vibration are limited to single mesh gear systems, however, and analytical solutions for nonlinear planetary gear dynamics do not exist.

Multi-degree-of-freedom linear planetary gear models are introduced and analyzed in

[19,21,26,39]. In these models, the gear mesh stiffness is modeled as a linear time-varying

12 spring. Linear system analysis helps to understand parametric instability due to periodically varying gear mesh stiffness [40], mesh phasing [47, 48], and properties of vibration modes [19, 24, 26, 28]. Kahraman [18] includes tooth separations and periodic mesh stiffness in his planetary gear model. Lin and Parker [19] introduce gyroscopic terms induced by carrier rotation. Sun [66] considers multiple clearances and predicts softening nonlinearity by employing HB. Several researchers [21, 33, 34] adopt a unique ﬁnite element/contact analysis method for analysis of planetary gears. Among them, Ambarisha and Parker [21] examine nonlinear dynamic behavior of planetary gears where jump phenomena, chaotic motions and period-doubling bifurcation are predicted. They show good comparison between analytical and FE models. Given the scarcity of experimental data, such FE solutions are a useful benchmark to evaluate the accuracy of analytical models.

A three-dimensional representation is presented by combining ﬁnite element and lumped parameter models [61]. The structured modal properties of three-dimensional helical planetary gears are categorized in [22]. Recently, ﬂexible ring gears are considered in both

ﬁnite element and analytical models [28, 67].

This study investigates the nonlinear dynamics of spur planetary gears having equally spaced planets. The mesh stiffnesses vary periodically at mesh frequency, and the piecewise linear spring generates non-zero mesh force only for positive relative mesh deﬂection

(teeth in contact). The nonlinear dynamics are interpreted using frequency spectra and root mean square (RMS) values of the steady state gear vibrations obtained by numerical and analytical methods over the important mesh frequency ranges. Concise, closed-form approximations for the frequency response functions of nonlinear and parametric resonances are derived using the method of multiple scales. Only ﬁrst order perturbation solutions are sought except for superharmonic resonance where second order perturbation is applied.

13 Furthermore, a HB solution yields accurate periodic response for varying mesh frequency, and Hill’s method establishes the stability of the solution branches. The accuracy of the perturbation and HB solutions is conﬁrmed by FE and numerical integration (NI) simulations.

Concerns may arise about the suitability of perturbation analysis given that planetary gears exhibit strong nonlinearity due to tooth separation and discontinuous mesh stiffness.

The use of perturbation analysis is validated by comparing with numerical integration (val- idating the perturbation solution) and finite element (confirming the model). In addition, limitations of the perturbation analysis as the nonlinearity grows stronger are discussed by investigating the increasing deviations from numerical solutions as tooth separation becomes more significant.

2.2 System Model

Figure 2.1 shows the two-dimensional, lumped-parameter model of a planetary gear with three equally spaced planets. The nonlinear, time-varying model developed by Lin and Parker [19] and Ambarisha and Parker [21] is used. The components are modeled as rigid bodies with moments of inertia Ic,Ir, Is, Ip where the subscripts c,r,s and p denote the carrier, ring, sun and planet. The model includes only rotational degrees of freedom on the assumption that the tooth meshes are more compliant than the bearings. One can expand the model by introducing bearing stiffness [19, 21]. The occurrence of essential nonlinear behaviors such as tooth separation and nonlinear resonances, which are the main topics addressed in this study, are not affected by the rigid bearing simpliﬁcation. The sun-planet and ring-planet meshes are modeled as time-varying springs whose stiffness varies as the number of teeth in contact changes. In order to model tooth contact loss,

14 the mesh springs act only when compressed and exert no force when the mesh separates.

Signiﬁcant nonlinear dynamics were observed in single-mesh spur gear experiments [62],

and the primary cause of tooth contact loss was conﬁrmed via lumped-modeling and FE

[33]. Similar nonlinear behavior has been observed and predicted in planetary [12,21] and

idler [68] gear systems.

The equation of motion with N planets is [21] Mx¨ + h(x,t) = Mx¨ + K(x,t)x = F, (2.1) T x = [uc,ur,us,u1,··· ,uN]      Ic Ir Is Ip Ip  M = diag + Nmp, , , ,··· ,  r2 r2 r2 r2 r2   c r s p p  | {z } number of planets, N   ∑(k˜sn cosαr + k˜rn cosαr) −∑k˜rn −∑k˜sn k˜r1 − k˜s1 k˜r2 − k˜s2 ··· k˜rN − k˜sN  ∑krn 0 −kr1 −kr2 ··· −krN     ∑ksn ks1 ks2 ··· ksN     kr1 + ks1 0 ··· 0  K(x,t) =   (2.2)  kr2 + ks2 ··· 0     .. .   . .  krN + ksN

k˜sn = ksn cosαs, k˜rn = krn cosαr ksn(t) δsn = −uc cosαs + us + un ≥ 0, ksn(x,t) = ksn(t)Θ(δsn) = n = 1,2,··· ,N 0 δsn = −uc cosαs + us + un < 0, krn(t) δrn = −uc cosαr + ur − un ≥ 0, krn(x,t) = krn(t)Θ(δrn) = 0 δrn = −uc cosαr + ur − un < 0, where uc, ur, us, u1, ··· , uN are rotational deﬂections of the gear bodies along the lines of action (e.g., us = rsθs ); rc, rr, rs, rp are base circle radii; and, ksn (t), krn (t) are the nth sun-planet and ring-planet mesh stiffnesses. The tooth separation is represented by the

Heaviside step function Θ(δ) where δ is the compressive tooth mesh deﬂection at a base circle. When δ is positive the teeth are in contact and the mesh stiffness is active (Θ(δ) = 1

); for contact loss (δ < 0 ), Θ(δ) = 0 and the mesh force vanishes. Backside contact is not considered because the amount of backlash in the example system ( > 600 µm) is significantly larger than the normal amplitude of dynamic response ( < 50µm ). Reasons why

15 backside contact rarely occur in planetary gears are also mentioned in [18]. For spur gears

the time-varying mesh stiffness is commonly approximated by a rectangular or trapezoidal

2π wave with mesh period Tm = ω , where ω is the mesh frequency. One can also obtain the mesh stiffness variation by static FE of the meshing teeth at multiple points in a mesh cy-

cle. Appropriate mesh phase relations between the multiple meshes are enforced according

Zsψn Zrψn to [46]. The mesh phases between planets are given by γsn = ± 2π , γrn = ∓ 2π where the upper sign is for clockwise planet rotation and the lower sign is for counter-clockwise

rotation. Zs, Zr are the sun and ring tooth numbers, and ψn are the circumferential angles

of the planets measured positive counter-clockwise (ψ1 = 0 ).

The time-varying stiffnesses are represented in Fourier series as

" ∞ # (l) jlωt ksn(t) = ksp + ∑ cˆsn e + c.c. , l=1 (2.3) " ∞ # (l) jlωt krn(t) = krp + ∑ cˆrn e + c.c. , n = 1,2,··· ,N, l=1

where c.c. denotes the complex conjugate of the preceding term. No relative phase γrs be-

tween sun-planet and ring-planet meshes (see [46]) is needed because t = 0 in both equa-

tions in Eq. (2.3) corresponds to the same instantaneous conﬁguration of the planetary

gear (as opposed to t = 0 corresponding to the pitch points of each mesh as some software

provides, for example, which occur at different times).

The eigenvalue problem of Eq. (2.1) for the linear time-invariant case using average

mesh stiffnesses ksp and krp is

2 K0vi = ωi Mvi. (2.4)

T The vibration modes are normalized as V MV = I with V = [v1,··· ,vN+3]. K0 is the mean stiffness matrix, that is, K(x,t) = K0 + Kd(x,t) and Kd has zero mean over a mesh cycle.

16 The mean stiffness matrix is expressed as N K0 = ∑ (kspKsn + krpKrn), (2.5) n=1 where Ksn(Krn) is a symmetric matrix consisting of the coefﬁcients of ksn(krn) in Eq. (2.1).

For the case of a fixed ring (ur = 0), for example, 2  cos αs 0 −cosαs −cosαs 0 ··· 0   0 0 0 0 ··· 0     1 1 0 ··· 0    K =  1 0 ··· 0  (2.6) s1  . .   .. .   0 .   .   .. 0  0 For an N planet system with any one of the carrier, ring or sun fixed, there exists one rigid body mode (ω1), two modes with distinct natural frequencies (ω2,ωN+2), and N − 1 degenerate modes (ω3 = ··· = ωN+1) [19]. In the distinct modes, all planets have identical motion, that is, un = u1, n = 2,3,··· ,N. In the degenerate modes, the rotations of the two central gears that are not fixed are zero.

2.3 Interpretation of Nonlinear Response

In order to characterize the nonlinear behavior over the important mesh frequency ranges, the dynamic response is simulated by numerical integration (NI) and ﬁnite element

(FE) analysis for the example system in Table 2.1. Refer to Table 2 in [21] for detailed gear parameters. The planet gears are equally spaced. An input torque is applied to the sun gear and power flows from the sun to the carrier with fixed ring, although other combinations are possible. Carrier rotational vibration is constrained to zero on the assumption there is a large output inertia (as is common), so the rigid body mode is removed. This specific power flow configuration is used as the example throughout this paper although the analytical solutions apply to any configuration. All sun-planet meshes are in-phase with each

17 Table 2.1: System parameters of a three equally spaced planetary gear rotational model.

No. of planets, N 3

Mean sun-planet mesh stiffness, ksp (N/m) 0.620e9

Mean ring-planet mesh stiffness, krp (N/m) 0.851e9

Sun-planet mesh phasing angle, γsn,n = 1,2,··· ,N 0

Ring-planet mesh phasing angle, γrn,n = 1,2,··· ,N 0

Circumferential angle of the nth planet, ψn(rad) 2π(n − 1)/N

Pressure angle of sun-planet mesh, αs(deg) 24.6

Pressure angle of ring-planet mesh, αr(deg) 20.2

Input torque to sun Ts (Nm) 1130 Sun number of teeth 27 Planet number of teeth 35 Ring number of teeth 99 2 Sun inertia, Is/rs (kg) 3.11 2 Planet inertia, Ip/rp (kg) 4.89

other, as are all ring-planet meshes (γsn = γrn = 0). This design occurs when the sun and ring tooth numbers Zs, Zr are integer multiples of the number of planets N. The choice of in-phase meshes equalizes the load sharing among the planets and eliminates any net force from the planetary gear on its support structure [47]. In this case, the Fourier coefﬁcients in

Eq. (2.3) are the same for any planet. FE uses a Rayleigh damping model for the FE mesh and concentrated bearing dampers. Modal damping values used in the analytical model are approximated by calculating damping ratios from the frequency response function obtained by numerical impulse tests of the FE model. The damping ratios are calculated from the

18 Table 2.2: Damping values

Model Freq. range Damping values

Rotational bearing damping Low Rayleigh damping: Finite element Sun: 4.519 Nms Planet: 3.389 Nms α=100 s and β=1e-7 s−1, where C = αM + βK High Rotational bearing damping: 12.43 Nms

Low ρ1=2.7%, ρ2 = ρ3=1.1%, ρ4=1.92% Analytical High ρ1=6%, ρ2 = ρ3=1.1%, ρ4=5%

half-power points of the spectra at each resonance peak [69]. Damping values for the FE and analytical models are listed in Table 2.2. The natural frequencies and vibration modes are shown in Figure 2.2 for the parameters in Table 2.1. The mesh stiffness variations are similar to those used in [21] for a US Army helicopter; the Fourier coefﬁcients in (2.3) are in Table 2.3.

Table 2.3: Fourier coefﬁcients of sun-planet and ring-planet mesh stiffnesses

(l) (l) Harmonicsc ˆs (1e6)c ˆr (1e6) 1 -63.16 + 42.31i -0.96 + 11.71i 2 -3.17 + 13.17i 0.75 + 0.11i 3 -3.51 - 15.35i -1.63 + 1.53i 4 -8.15 - 7.02i 2.54 + 2.11i 5 1.96 + 0.20i 2.23 - 2.22i

19 Figure 2.3(a) shows RMS sun rotation (after subtraction of the mean value) by NI of Eq.

(2.1) and FE as the mesh frequency is varied. Results for both increasing and decreasing speed sweeps are plotted. Comparison of NI and FE shows minimal differences. There are jump phenomena at the first (ω = ω1) and the second (ω = ω4) distinct mode resonance peaks. The resonance peaks lean to the left, implying softening nonlinearity induced by tooth separation. There are additional resonance peaks around 8000 Hz, 1800 Hz (the first distinct mode), and below 1600 Hz mesh frequency. These peaks are combined effects of parametric instability from higher harmonics of mesh stiffness variation and nonlinear subharmonic and superharmonic resonances of the first and second distinct modes. Detailed discussion is given later. Kinks at 1140 Hz, 2350 Hz , 4850 Hz, and 9250 Hz indicate the onset of tooth separation. At mesh frequencies higher than these, tooth separation does not occur and the system behaves linearly. Tooth separation occurs along the entire upper branch for mesh frequencies below a kink. Figure 2.3(b) shows the mean value of sun rotation. Similar jump phenomena are observed as for the RMS curves.

The waterfall plots of Figure 2.4 show frequency spectra of the sun and planet gear rotation as mesh frequency decreases. Compared to the FE solution, NI of Eq. (2.1) simu- lates the dynamic response well through all mesh frequency harmonics. Numerous second and higher harmonic responses are captured (e.g., Figure 2.4(a) and Figure 2.4(b)). Figure

2.4(c) and Figure 2.4(d) show a small chaotic region with distributed spectral content just after the jump-down for the ﬁrst distinct mode (ω ≈ 1740 Hz).

In Figure 2.3 all expected resonance peaks (i.e., lω = ω2 = ω3) of the degenerate modes

ω2 = ω3 are absent. These are planet modes that have no central gear rotation [19]. The

20 planet modes (not normalized) are

T v2 = [0,−1,1,0,··· ,0]

T v3 = [0,−1,0,1,··· ,0] (2.7) . .

T vN+1 = [0,−1,0,0,··· ,1]

The absence of these resonances occurs because the planets are equally spaced and equally phased. In this case the net dynamic mesh forces Φ(t) at each planet must be identical because of cyclic symmetry. While Φ(t) is not known, the fact that it is identical for all planets based on equality of spacing and phasing establishes that the dynamic mesh force vector in Eq. (2.1) is orthogonal to the planet modes as shown by the inner products

T v2 h(x,t) = [0,−1,1,0][∗,Φ(t),Φ(t),Φ(t)] = 0 (2.8) T v3 h(x,t) = [0,−1,0,1][∗,Φ(t),Φ(t),Φ(t)] = 0. Consequently, mesh forces can not excite these planet modes. This result generalizes to N equally spaced, equally phased planets.

21 (a) The first distinct mode ω = 1860 Hz (b) Degenerate mode ω = 2760 Hz 1 2 (a) (b)

Figure 2.2: Vibration modes of planetary gear rotational model for parameters in Table 2.1.

22 35 Tooth separation occurs 30

25 m µ 20

15 RMS of sun, 10

ω ω 1 4 0 0 2000 4000 6000 8000 10000 Mesh frequency, Hz (a)

30 m

µ 25

20 Mean of sun,

ω ω 1 4 10 0 2000 4000 6000 8000 10000 Mesh frequency, Hz (b)

Figure 2.3: Steady state response of a) RMS (mean removed) sun rotational deﬂection and b) sun mean rotation. Both increasing and decreasing speed sweeps are performed. (—) NI; (···) FE analysis.

23 m 5 m 5 µ µ

4 4

3 3

2 2

1 1600 1 1600

0 1400 0 1400 0 0 Planet deflection, 1200 Planet deflection, 1200 2 ency, Hz 2 ency, Hz Mesh frequ 1000 Mesh frequ 1000 4 4 ency 6 800 ency 6 800 harmonics harmonics 8 600 Mesh frequ 8 600 Mesh frequ 10 10 (a) (b)

35 35 m m µ 30 µ 30 25 25 20 20 15 15 10 10 2400 5 5 2400 0 0 2200 Sun deflection, Sun deflection, 0 2200 0 Mesh frequency1 Mesh frequ1 2000 2000 ency, Hz ency, Hz 2 2 1800 ency 1800 harm 3 harmonics3 onics Mesh frequ 4 1600 Mesh frequ 4 1600 (c) (d)

Figure 2.4: Waterfall spectra of sun and planet rotational deﬂection for decreasing speed by ﬁnite element, (a) and (c), and numerical integration, (b) and (d).

24 2.4 Harmonic Balance Method With Arclength Continuation

The harmonic balance (HB) method has been applied to analyze the nonlinear dynamics

of many mechanical systems. An arclength continuation technique allows HB to track

solution branches around turning points [64, 70, 71].

To apply the method, the contact loss discontinuity is ﬁrst approximated by the smooth

function 1 1 Θ(δ) ≈ Θ˜ (δ) = + tanh(cδ) (2.9) 2 2 where c controls the sharpness of the step change near δ = 0. In this paper, c = 10,000 is used, which yields negligible errors between the HB solutions using Eq. (2.9) and NI using the Heaviside step function. The response is discretized into nt time intervals per mesh cycle, and nh Fourier harmonics are used to approximate a periodic solution. The discrete time vector q contains the response evaluated at nt points in a mesh cycle such that

T q = [uc(t1),··· ,uc(tnt ),ur(t1),··· ,uN(tnt )] = Luˆ, F = Lf

T T T T T T uˆ = uˆ c uˆ r uˆ s uˆ 1 ··· uˆ N   L0  ..  L =  .  (2.10) L0   1 cos(ωt1) sin(ωt1) ··· cos(nhωt1) sin(nhωt1)  .  L0 =  .  1 cos(ωtnt ) sin(ωtnt ) ··· cos(nhωtnt ) sin(nhωtnt ) where uˆ i, i = c,r,s,1,··· ,N and f are vectors of Fourier coefﬁcients.

A discrete time mesh force vector H(q) that contains the values h(x,t) in Eq. (2.1) evaluated at nt points in a mesh cycle is introduced. A damping matrix is computed from

−1 T −1 modal damping ratios ζi by C = (V ) diag(2ζiωi)V . Including the damping matrix,

25 Eq. (2.1) is rewritten to contain the response at nt time points in a mesh cycle as

M˜ q¨ + C˜ q˙ + H(q) = Lf (2.11) where M˜ and C˜ are block diagonal matrices of inertia and damping, respectively [68, 71].

Introducing constant matrices A and B as in [64,71], substitution of q = Luˆ into Eq. (2.11) yields

M˜ L¨ uˆ + C˜ L˙ uˆ + H(Luˆ) = Lf (2.12)

ω2MAˆ uˆ + ωCBˆ uˆ + GH(Luˆ) = f (2.13) where G is the pseudo-inverse of L. The vector uˆ that determines the response q is found from vanishing of the norm of the residual r of Eq. (2.13),

2 krk = ω MAˆ uˆ + ωCBˆ uˆ + GH(Luˆ) − f = 0 (2.14)

It is desirable to trace both stable and unstable periodic solution branches for varying mesh frequency ω. Applying arclength continuation [72] for this purpose, an independent arclength variable denoted by is introduced. The augmented solution vector a is

h iT a = uˆ T(s) ω(s) (2.15)

Starting from an initial guess, a point on a solution branch is sought by Newton-Raphson iteration. The iteration continues until krk is less than a user-speciﬁed tolerance. Conver- gence of the iteration depends highly on appropriate selection of the initial guess for a(s).

The initial guess for the current solution is obtained from the previous solution. Detailed explanation can be found in several papers [63, 71–73].

Stability of the solution branches is analyzed in the frequency domain using Hill’s method [63,74]. A linearized equation is formulated by introducing a perturbation q = q∗ + υ

26 around a periodic solution q∗ that satisﬁes Eq. (2.11). Stability of the solution is established by the resulting eigenvalues: a solution is stable if Re(λ) ≤ 0 for all eigenvalues λ and unstable if at least one eigenvalue has Re(λ) > 0.

2.5 Analytical Solution by the Method of Multiple Scales

The method of multiple scales [75] is used to obtain analytical frequency response functions for primary, sub- and superharmonic resonances. Two small parameters are defined as the ratios of the first Fourier coefficients of the sun-planet and ring planet mesh stiffnesses (1) (1) cˆs cˆr to their mean values, ε = and µ = , where the Fourier coefficients from Eq. (2.3) ksp krp no longer contain the planet index subscript n because they are the same for any planet

for the case of equal planet spacing and in-phase meshes. Utilizing ε and µ, the stiffness

functions in Eq. (2.3) are rewritten as

∞ ! (l) jlωt ks(t) = ksp + εksp ∑ cs e + c.c. , l=1 ∞ ! (l) jlωt kr(t) = krp + µkrp ∑ cr e + c.c. , (2.16) l=1 (l) (l) (l) cˆs (l) cˆr cs = = 0(1),cr = = 0(1). (1) (1) cˆs cˆr With Eqs. (2.16) and (2.5), Eq. (2.1) is

" N N # ˆ ˆ 2 Mx¨ + ∑ ksp 1 + εQs KsnΘs(δs) + ∑ krp 1 + µQr KrnΘr(δr) x + O(ε ) = F n=1 n=1 ∞ ∞ ˆ (l) jlωt ˆ (l) jlωt Qs = ∑ cs e + c.c., Qr = ∑ cr e + c.c. l=1 l=1 (2.17) The governing equation (2.16) is transformed to modal coordinates by x = Vz, giving

" N N # ˆ ˆ T z¨ + ∑ ksp 1 + εQs GsnΘ((z)) + ∑ krp 1 + µQr GrnΘ(z) z = V F = f n=1 n=1 (2.18) T T Gsn = V KsnV, Grn = V KrnV

27 n Introducing the multiple time scales tn = ε t, the qth modal response is asymptotically

approximated as

2 zq(t0,t1) = zq0(t0,t1) + εzq1(t0,t1) + 0(ε ), q = 1,2,··· ,N + 1. (2.19)

Accordingly, the tooth separation functions Θs,r(zq) depend on the multiple time scales t0 and t1. FE and NI simulations show only one tooth separation per response period near resonance peaks. Furthermore, the time of tooth separation is small compared to the response period, that is, ξ/T = 0(ε) where ξ is the time of tooth separation and is the response period. With these stipulations, the tooth separation functions are expressed as

Θs = 1 + εθs1,Θrn = 1 + µθr1. (2.20)

The limit of tooth separation time for acceptable accuracy is discussed later.

Incorporating Eq. (2.20) into Eq. (2.18), the qth modal equation is N+1 N 2 z¨q + ελqz˙q + ωq zq + ∑ ∑ εkspQsGsnqw + µkrpQrGrnqw zw = fq, q = 1,2,··· ,N + 1 w=1 n=1 " ∞ # " ∞ # (l) jlωt0 (l) jlωt0 Qs = ∑ cs e + θs1 + c.c., Qr = ∑ cr e + θr1 + c.c., l=1 l=1 (2.21) where Gsnqw, Grnqw are the (q,w) elements of Gsn,Grn in Eq. (2.18). Small modal damping

2ζqωq has been introduced and reformulated as ελq = 2ζqωq. Nonlinear coupling of the modes is evident.

Application of the method of multiple scales [68] using Eq. (2.19) gives the perturbation equations for q = 1,2,··· ,N + 1

2 ∂ zq0 2 2 + ωq zq0 = fq (2.22) ∂t0 2 2 N+1 N ∂ zq1 2 ∂ zq0 ∂zq0 2 + ωq zq1 = −2 − λq − ∑ ∑ kspQsGsnqw + gkrpQrGrnqw zw0, (2.23) ∂t0 ∂t0∂t1 ∂t0 w=1 n=1 (1) (1) where g = cˆr /krp / cˆs /ksp relates µ and ε such that µ = εg. 28 2.5.1 Primary Resonance, ω ≈ ωi

Near primary resonance, the mesh frequency is represented as ω = ωi +εσ where σ is a detuning parameter. The solutions of Eq. (2.22) are

f jωqt0 q zq0 = Aq(t1)e + c.c. + 2 , q = 1,2,··· ,N + 1. (2.24) ωq

Assuming ωq is not an integer multiple of ωi , substitution of all ﬁrst order solutions into the kth modal equation of Eq. (2.23) (k 6= i) gives the solvability condition

N ∂Ak j2ωk + jωkλkAk + ∑ (kspθs1Gsnkk + gkrpθr1Grnkk)Ak = 0. (2.25) ∂t1 n=1

The steady state dynamic response of the nonlinear equation (2.25) near a hyperbolic ﬁxed point follows the same behavior as its linearization about that point [76], so the asymptotically stable steady state solution of (2.25) for all k 6= i is Ak = 0. Thus, dynamic response is conﬁned to the resonant mode q = i. The solution (2.24) for q = i is

f f j(ω−εσ)t0 i iωt0 i zi0 = Ai(t1)e + c.c. + 2 = Bi (t1)e + c.c. + 2 , (2.26) ωi ωi

− jσt where Bi (t1) = Ai(t1)e 1 . Thus, the leading order response is periodic at the mesh frequency. Tooth separation occurs only once per response period, as mentioned earlier in this section. The tooth separation function, as well as the response, is periodic at the mesh frequency with amplitude and phase modulated by t1. Because the amplitude of response

zi0 is modulated by Bi(t1) in Eq. (2.26), the Fourier coefﬁcients of the tooth separation

functions (2.20) also vary with t1, giving

" ( ∞ )# (0) (m) jm(ωt0−φs(t1)) Θs = 1 + ε θs + ∑ θs (t1)e + c.c. (2.27) m=1

with analogous results for the ring-planet mesh. The phases φs and φr will be chosen

subsequently such that the tooth separation is in-phase with the mesh deﬂection.

29 1 jβi(t1) Substitution of Ai(t1) = 2 ai(t1)e into zi0 in Eq. (2.26) gives f f j(ω−εσ)t0 i i zi0 = Ai(t1)e + c.c. + 2 = ai (ti)cos[ωt0 − γi (t1)] + 2 , (2.28) ωi ωi

where γi(t1) = σt1 −βi(t1). From x = Vz and Eq. (1), the sun and ring mesh deﬂections δs

and δr are N+1 fw δs = As cos(ωt0 − γi) + ∑ (vsw + vpw − cosαsvcw) 2 ωw w (2.29) N+1 fw δr = Ar cos(ωt0 − γi) + ∑ (vrw − vpw − cosαrvcw) 2 , w ωw

where As = ai(vsi + vpi − cosαsvci) and Ar = ai(vri − vpi − cosαrvci). Considering that

the tooth separation is in-phase with the mesh deﬂection, φs (t1) = γi (t1) in Eqs. (2.27) and (2.29). For the ring-planet tooth separation function, φr (t1) = φs (t1) + π when the amplitudes of mesh deﬂection As and Ar have opposite sign, otherwise φr (t1) = φs (t1) .

Figure 2.5 illustrates the relationship between mesh deﬂections and tooth separation functions. Based on Eq. (2.29), all of these are even functions of ωt0 − γi. The sun-planet and ring-planet mesh deﬂections are in-phase in Figure 2.5(a) and out-of-phase by π in Figure

2.5(b). Because the tooth separation functions Θs and Θr are even functions of ωt0 −γi, all (m) (m) of the Fourier coefﬁcients θs and θr in Eq. (2.27) are real quantities. After substitution of Eqs. (2.24), (2.27), and ω = ωi + εσ into Eq. (2.23) for q = i, the solvability condition is N ∂Ai ( ) ( ) 0 2 j2(σt1−γi) ¯ j2ωi + jωiλiAi + ∑ ksp θs Ai + θs e Ai Gsnii ∂t1 n=1 N+1 N f (1) j(σt1−γi) w + ∑ ∑ kspθs e 2 Gsniw + (s → r) (2.30) w=1 n=1 ωw N N+1 N ( ) ( ) fw 2 j2σt1 ¯ 1 jσt1 + ∑ kspcs e AiGsnii + ∑ ∑ kspcs e 2 Gsniw + (s → r) = 0, n=1 w=1 n=1 ωw where (s → r) denotes corresponding terms for the ring-planet mesh and all occurrences

1 jβi(t1) of ksp become gkrp. With Ai(t1) = 2 ai(t1)e , separation into real and imaginary parts, 30 and taking only the ﬁrst order mesh stiffness harmonic terms, the autonomous amplitude and phase modulation equations are

∂ai 1 ωi = − ωiaiλi − |χ2|sin(γi + ψ) (2.31) ∂t1 2

∂γi ωiai = ωiaiσ − χ1 − |χ2|cos(γi + ψ), (2.32) ∂t1 N N+1 N ksp (0) (2) (1) fw χ1 = ∑ θs ai + θs ai Gsnii + ∑ ∑ kspθs 2 Gsniw + (s → r) 2 ωw n=1 w=1 n=1 (2.33) N+1 N fw (1) (1) χ2 = ∑ ∑ 2 kspcs Gsniw + gkrpcr Grniw , w=1 n=1 ωw where ψ is the phase angle of χ2. The validity of ignoring the second and higher order mesh stiffness harmonic terms is examined later.

The steady state motions are obtained by letting ∂ai/∂t1 = ∂γi/∂t1 = 0 in Eqs. (2.31) and (2.32), which yields a quadratic polynomial in σ such that q 1 2 2 2 ω = ωi + Ξ1ai + 2Ξ2 ± 2 |Ξ3| − ωi aiζi 2ωiai N h i (1) (0) (2) (1) (0) (2) Ξ1 = ∑ cˆs θs + θs Gsnii + cˆr θr + θr Grnii n=1 (2.34) N+1 N f Ξ = w cˆ(1) (1)G + cˆ(1) (1)G 2 ∑ ∑ 2 s θs sniw r θr rniw w=1 n=1 ωw N+1 N fw (1) (1) Ξ3 = ∑ ∑ 2 cˆs Gsniw + cˆr Grniw . w=1 n=1 ωw

From Eqs. (2.31) and (2.32), the steady state phase γi is −1 ωiaiλi γi = tan − ψ. (2.35) 2(χ1 − ωiaiσ)

From the assumption of ∂ai/∂t1 = ∂γi/∂t1 = 0, the Fourier coefﬁcients of the tooth separation function in Eq. (2.27) and Figure 2.5 are constant and determined as

(0) ξs θs = − = O(1) εT h i (2.36) sin mπ 1 − ξs (m) T θs = = O(1),m = 1,2, ··· εmπ 31 (0) (m) (0) (0) For φr = φs, θr and θr are the same with subscript s → r. For φr = φs + π, θr = θs and h i sin mπ ξr (m) T θr = − = O(1),m = 1,2, ··· (2.37) εmπ

Employing the conditions δs = 0 and δr = 0 in Eq. (2.29), the tooth separation times are   N+1 f  (v + v − cosα v ) w   ∑ sw pw s cw 2  T −1  w=1 ωw  ξs = cos π  ai vsi + vpi − cosαsvci        (2.38) N+1 f  (v − v − cosα v ) w   ∑ rw pw r cw 2  T −1  w=1 ωw  ξr = cos , π  ai vri − vpi − cosαsvci     

where vbw is the (b,w) element of the modal matrix V, and b denotes the corresponding

terms of carrier (c), ring (r), sun (s) and planet (p).

To generate frequency response curves from perturbation, one varies ai and, for each (m) ai, computes the Fourier coefﬁcients θs,r of the tooth separation function from Eqs. (2.36)

and (2.37) followed by calculation of the corresponding frequencies from Eq. (2.34). ai is

converted to physical coordinates using Eq. (2.24) and x = Vz.

Stability of the solution branches is determined from the eigenvalues of (2.31) and

(2.32) linearized about an equilibrium. The ﬁrst two terms Ξ1ai + 2Ξ2 on the right-hand

side of Eq. (2.34) represent the backbone curve of the resonance peak, and the square root

represents the deviation from the backbone.

The peak resonant amplitude for the ith mode is computed from vanishing of the devi-

ation from the backbone in (2.34), that is,

peak |Ξ3| ai = 2 . (2.39) ωi ζi

32 This important quantity depends only on known parameters and the linear system modes

(0,1,2) (0,1,2) and is immediately calculable. Once calculated, the θs and θr are found from

Eqs. (2.36) and (2.37) so that Ξ1 and Ξ2 can be computed, from which the frequency ω

of maximum amplitude is found using Eq. (2.34). From ξ = 0, the amplitudes of the

kink points marking the onset of contact loss (Figure 2.3) at the sun-planet and ring-planet

meshes are

N+1 fw ∑ (vsw + vpw − cosαsvcw) 2 kink w=1 ωw ai,s = vsi + vpi − cosαsvci (2.40) N+1 fw ∑ (vrw − vpw − cosαrvcw) 2 kink w=1 ωw ai,r = . vri − vpi − cosαsvci The corresponding frequencies bounding the range where contact loss occurs are found

from Eq. (2.34).

The Ξk in Eq. (2.34) show how system parameters affect the response. All quantities

are known except the coefﬁcients θ (l) of the tooth separation functions, which are implicit

(l) functions of the amplitude of the response. The θ appear only in Ξ1 and Ξ2. From Ξ3

in Eqs. (2.34) and (2.39) one can see that the peak resonant amplitude is proportional to

the applied torque and mesh stiffness ﬂuctuation, and inversely proportional to the damp-

ing ratio and natural frequency to the power of four. The peak amplitude is calculable and

applicable for cases with or without contact loss. Expression (2.40) shows that the am-

plitude of the contact loss initiation point is also calculated from the known linear system

quantities of vibration modes, pressure angles, applied torque, and natural frequencies.

The readily calculated peak and contact loss initiation point amplitudes determine whether

or not contact loss occurs. When the peak amplitude exceeds the lesser of the sun-planet

and ring-planet contact loss initiation amplitudes contact loss occurs at one

33 peak peak peak peak peak peak (ai >min(ai,s ,ai,r )) or both (ai >max(ai,s ,ai,r )) meshes; otherwise contact loss does not occur and the system behaves linearly.

Common practical thinking holds that large torque prevents contact loss because the vibration must exceed a large static mesh deﬂection to cause contact loss. As one increases the steady external torque, static deﬂection increases. At the same time, however, the present analysis shows that the larger steady torque simultaneously increases the amplitude of vibration for parametrically excited planetary gear system. This causes the contact loss to persist at higher torque.

This conclusion differs significantly from conventional wisdom. The perturbation results giving this mathematical conclusion are, however, supported by independent experiments [62] and finite element simulations [33]. Numerical solutions in the present work also confirm this analytical conclusion. For the sake of simplicity to explain this, we consider only static analysis without inertia. Then, the equation of motion (1) is

K(t)x(t) = F ⇒ x(t) = F/K(t), where K(t) is the time-varying mesh stiffness (i.e., parametric excitation), x(t) is the static response (static transmission error), and F is the constant external force (torque). The static response amplitude, which fluctuates over a mesh cycle, is ∆x = (xmax − xmin) = F((Kmax − Kmin)/(KminKmax)). Thus, ∆x is proportional to the torque. While higher torque increases the static deflection, it simultaneously increases the amplitude of fluctuation by the same factor such that the fluctuation amplitude increases proportionally to the larger static deflection. When inertia is included, the amplitude of dynamic response is similarly proportional to the torque, as the perturbation analysis shows.

Thus, the steady static deﬂection and vibration amplitude increase by the same factor and contact loss then persists at higher torque. This matches the common use of the static response amplitude as the dynamic excitation for gear dynamics.

34 Analytical results conﬁrm this point. Mathematically, both the peak amplitude and the

contact loss initiation points increase linearly with static torque because all the fw in Ξ3

(see Eqs. (2.34) and (2.39)) and Eq. (2.40) are linear in the applied torque. Thus, if contact

loss occurs for any particular torque, it will occur for all other torques. Therefore, large

torque does not prevent contact loss, and the occurrence of contact loss is independent of torque (both low and high torques).

For three equally spaced planets with in-phase meshes, the Ξk in Eq. (2.34) simplify signiﬁcantly. For the ﬁrst distinct primary resonance they are h i (1) (0) (2) 2 (1) (0) (2) 2 Ξ1 =3 cˆs θs + θs (v11 + v21) + cˆr θr + θr v21 f n o =3 1 cˆ(1) (1) (v + v )2 + cˆ(1) (1)v2 Ξ2 2 s θs 11 21 r θr 21 ω1 f n o + 4 cˆ(1) (1) (v + v )(v + v ) + cˆ(1) (1)v v 2 s θs 11 21 14 24 r θr 21 24 ω4 3 f1 h (1) 2 (1) 2 i 3 f4 h (1) (1) i Ξ3 = 2 cˆs (v11 + v21) + cˆr v21 + 2 cˆs (v11 + v21)(v14 + v24) + cˆr v21v24 . ω1 ω4 (2.41)

Similar reductions are readily derived for the second distinct primary resonance. The generalization to N planets is straightforward. Equation (2.39) is unchanged. From Eq. (2.40), the amplitude of the points above which contact loss occurs become

f1 f4 f1 f4 2 (v11 + v21) + 2 (v14 + v24) 2 v21 + 2 v24 kink ω1 ω4 kink ω1 ω4 a1,s = , a1,r = . (2.42) |v11 + v21| |v21|

The frequency response relation (2.34) includes Gsn and Grn from Eq. (2.18), which are

products of the modal matrix and stiffness coefﬁcient matrices. The simpliﬁed expressions

(2.41) explicitly expose that the peak amplitude depends heavily on the vibration modes. If

the sun and planet modal deﬂections are in opposite directions with similar magnitudes (i.e.,

v11 ≈ v21 as in Figure 2.2(a)), the peak amplitude for resonance in that mode is reduced.

Conditions that increase the peak amplitude are also readily observed. In addition, v14 and

35 v24 in the second bracket of Ξ3 show that the peak amplitude of the ﬁrst distinct mode is affected by the second distinct mode (i.e., mode 4) and vice versa. This results from nonlinear modal coupling.

Only the first harmonics of the mesh stiffness variation functions are considered in the first order perturbation analysis. The error due to this truncation is estimated by NI including only selected harmonics of the mesh stiffness. Figure 2.6 compares the RMS planet response for the actual time-varying stiffness (full harmonics) and limited harmonic approximations. As more harmonics of mesh stiffness are considered, slightly larger amplitudes and wider resonance regions are predicted. The additional peak around 1750 Hz occurs from the broadband chaotic response induced by the higher harmonics of mesh stiffness as shown in Figure 2.7. To evaluate the error in the perturbation solution from neglecting higher harmonics of mesh stiffness, only the first harmonic in Figure 2.7 (the one predicted by perturbation) should be considered.

The RMS planet and sun amplitudes for primary resonance from perturbation are compared with NI and HB in Figure 2.8. The HB solution branches follow NI with negligible errors. The perturbation solutions agree well with these solutions. The predicted amplitudes at which contact loss begins match the numerical results, although some deviation occurs for the sun response in the first distinct mode. Unstable solution branches from perturbation compare well with HB except for, once again, the sun response in the first distinct mode. Notice, however, the complexity of the true solution branches from HB at this resonance, so deviations from first order perturbation are expected. A closed loop solution branch exists near this peak, which does not happen for any other resonances or degrees of freedom.

36 1 ξs Θs

Θr ξr

(ωt0 - γ) 0 -T -T/2 δr T/2 T

δs

(a)

ξr ξs

(ωt0 - γ)

-T -T/2 0 T/2 T

(b)

Figure 2.5: Tooth separation function and mesh deﬂection. (a) In-phase and (b) out-of- phase (by π) sun-planet and ring-planet mesh deﬂection.

37 18

m 12 µ

6 RMS of planet,

0 1600 1700 1800 1900 2000 2100 2200 2300 2400 Mesh frequency, Hz (a)

m 7 µ

4 RMS of planet,

1 3000 3500 4000 4500 5000 Mesh frequency, Hz (b)

Figure 2.6: RMS (mean removed) planet rotational deflection from numerical integration for (a) the first and (b) the second distinct primary resonance for full and limited harmonics of the mesh stiffness variations. (—) Full harmonics of stiffness, (- - -) the first and second harmonics of stiffness, (···) The first harmonic of stiffness.

38 (a) (b) 25 25

20 20

15 15

10 10

5 5

Planet deflection,

0 0 0 0 Mesh frequency harmonics Mesh frequency harmonics 1 1

2 2400 2 2400 2200 2200 3 2000 3 2000 1800 1800 4 1600 Mesh frequency, Hz 4 1600 Mesh frequency, Hz

(a) (b)

Planet deflection,

0 0 Mesh frequency harmonics 1

2 2400 2200 3 2000 1800 4 1600 Mesh frequency, Hz

(c)

Figure 2.7: Waterfall spectra of planet rotational deﬂection for (a) only the ﬁrst, (b) up to the second and (c) full harmonics of the mesh stiffness variations by numerical integration.

39 16 7

14 NI 6 start of contact loss 12 MMS m

m 5 µ HB µ 10 4 8

RMS of planet, 3 6 RMS of planet,

4 2

2 1 1600 1700 1800 1900 2000 2100 2200 3000 3500 4000 4500 5000 Mesh frequency, Hz Mesh frequency, Hz (a) (b)

20 25

16 20 14 m m

µ 12 µ 15 10

RMS of sun, RMS of sun, 10 6

4 5 2

0 1600 1700 1800 1900 2000 2100 2200 3000 3500 4000 4500 5000 Mesh frequency, Hz Mesh frequency, Hz (c) (d)

Figure 2.8: RMS (mean removed) sun and planet rotational deﬂection for primary resonance of the ﬁrst (a), (c) and the second (b), (d) distinct mode by numerical integration (NI), harmonic balance (HB) and method of multiple scales (MMS). Unstable solutions are shown as dashed lines.

40 2.5.2 Subharmonic Resonance, ω ≈ 2ωi

Subharmonic resonances where ω ≈ 2ωi have been observed in high-speed aircraft engine planetary gears, an aircraft engine idler gear train, and single gear pair experiments

[62]. Each of these cases showed distinct jump phenomena.

Figure 2.9 compares subharmonic resonance of the example system simulated by FE,

NI and HB. The FE model predicts subharmonic resonance when ω ≈ 2ω4 for ω4=4393 Hz even though there is nothing in the formulation to impose this behavior, i.e., it occurs nat- urally in a FE model that closely mimics real gear operation in its contact algorithm. The analytical model (both NI and HB solutions) accurately matches the FE model with two small deviations. For increasing speed, the analytical model predicts a slightly wider frequency range for the instability region bounded by the nearly vertical branches (about 500

Hz versus 400 Hz for FE). Compared to the subharmonic resonance region width from FE

(2500 Hz), this 100 Hz difference is minor. The center mesh frequencies of the analytical and FE models’ instability region are 8800 Hz and 9150 Hz, respectively. Considering the high mesh frequency (ω ≈ 9000 Hz) and expected small differences in natural frequency between the two models, this 4% difference is minimal. The differences in NI and FE peak amplitudes and jump-down frequencies near 6700 Hz are negligible. These comparisons expand the similar good comparisons between the analytical and FE models of planetary gears for primary and superharmonic resonance presented in [21] to include high-speed subharmonic resonance. This builds additional conﬁdence in the analytical model for this high-speed nonlinear behavior.

In Figure 2.9, jumps bounding both sides of the resonant region are evident between

8500 Hz and 9400 Hz. This phenomenon is unique for subharmonic resonance and does not occur for other resonances where jumps occur only once for increasing or decreasing

41 speed. For decreasing speed, the response initially follows the lower period-Tm solution. It

jumps up to the period-2Tm solution branch when the period-Tm solution becomes unstable

at 9050 Hz (see Figure 2.9 inset). This period-2Tm solution meets a turning point at a

coalescence with a lower unstable branch near 6700 Hz, and the response jumps down

to the period-Tm solution. For increasing speed, the period-Tm solution becomes unstable

and jumps up to the period-2Tm solution at 8500 Hz. The response jumps down to the

stable period-Tm solution at 9050 Hz. Figure 2.10 shows the period-2Tm solution branch

terminating sharply with jumps at high and low frequency for decreasing speed.

FE simulations show that the assumption of one tooth separation per period of response

is still valid for subharmonic resonance. With this stipulation, substituting ω = 2ωi + εσ

into Eq. (2.23), and following a similar procedure as for primary resonance, the frequency

response relation is q 1 Ξ2 2 2 2 ω = 2ωi + Ξ1 + 2 ± 2 |Ξ3| − ωi ζi ωi ai N h i (1) (0) (2) (1) (0) (2) Ξ1 = ∑ cˆs θs + θs Gsnii + cˆr θr + θr Grnii n=1 (2.43) N+1 N f Ξ = w cˆ(1) (1)G + cˆ(1) (1)G 2 ∑ ∑ 2 s θs sniw r θr rniw w=1 n=1 ωw N 1 (1) (1) Ξ3 = ∑ cˆs Gsniw + cˆr Grniw . n=1 2

To more explicitly see the parameter dependencies, simpliﬁed expressions of the Ξk for three equally spaced planets and in-phase meshes can be obtained in a form similar to Eq.

42 (2.41). In this case, the Ξk for subharmonic resonance of the second distinct mode are h i (1) (0) (2) 2 (1) (0) (2) 2 Ξ1 =3 cˆs θs + θs (v14 + v24) + cˆr θr + θr v24 f1 n (1) (1) 2 (1) (1) 2 o Ξ2 =3 cˆs θs (v14 + v24) + cˆr θr v ω2 24 1 (2.44) f n o + 4 cˆ(1) (1) (v + v )(v + v ) + cˆ(1) (1)v v 2 s θs 11 21 14 24 r θr 21 24 ω4 1 h i Ξ = cˆ(1) (v + v )2 + cˆ(1)v2 . 3 2 s 14 24 r 24 Similar results for the ﬁrst distinct mode and the generalization to N planets are straightforward.

The amplitude of the contact loss initiation points delineating contact loss for subharmonic resonance are again calculated from Eq. (2.40), which applies for all resonances.

In contrast to Eq. (2.34), the amplitude ai in Eq. (2.43) is absent in the square root. This leads to open solution branches (Figure 2.11), and the jump-down peak amplitude is not predicted in a ﬁrst order perturbation. Instead, the width of the parametric instability region can be calculated using the deviation from the backbone calculated from Eq. (2.43),

q 4 2 2 2 ∆ = |Ξ3| − ωi ζi . (2.45) ωi

Figure 2.11 compares the RMS sun amplitude for subharmonic resonance by perturbation and NI. The vertical branches mark the boundary of the linear system parametric instability region [40]. These solution branches are vertical until tooth separation occurs, which gives the jump-up and jump-down phenomena mentioned earlier in this section. Ξ3 in Eq. (2.45) does not include the applied torque fw, so the width of the instability region, and thus the peak amplitude, do not change with the applied torque, in contrast to primary resonance.

After the onset of tooth separation the vertical solution branches exhibit softening nonlinearity. Stability analysis similar to primary resonance shows the lower branch is unstable while others are stable.

43 Differences with numerical integration in Figure 2.11 emerge for large amplitude as per-

turbation’s upper stable solution bends upward while NI holds nearly a straight line. This

deviation is expected because the assumption of small tooth separation time as a fraction

of response period weakens as amplitude increases. Figure 2.12 illustrates the relationship

between the tooth separation time and mesh frequency as well as the error in perturbation

relative to NI. Two data plots share the vertical tooth separation time axis, while the top

and bottom mesh frequency axes are for subharmonic (O) and primary (•) resonances of the second distinct mode, respectively. The mesh frequency axis allows comparison with corresponding resonance plots (Figure 2.11 and Figure 2.8(d)). Results are for decreasing mesh frequency, and mesh deﬂection amplitude increases in the left-upper direction.

Only the upper solution branches are considered; no tooth separation occurs on the lower branches. Clearly the tooth separation time increases as mesh deﬂection increases. The 5% and 10% deviation points are where the perturbation solutions differ from the numerical solution by 5% and 10%. As one allows a greater percentage for the deviation point, the corresponding tooth separation time increases. For example, for subharmonic resonance the tooth separation times for 5% and 10% deviation are 0.28 and 0.32, respectively. Each resonance shows a slightly different tooth separation time for the same percentage deviation. For both resonances, however, one can conclude that perturbation works very well for tooth separation times as large as 0.3. Although the plots in Figure 2.12 are for the sun-planet mesh, similar results with comparable tooth separation time limits are obtained for the ring-planet mesh.

44 10 1 by HB

m Stable Unstable Stable 0.5 9

Period T 8.2 8.4 8.6 8.8 9.0 9.2 8 Mesh frequency, kHz

7 m µ 6 finite element

5 harmonic balance

numerical integration 4

RMS of planet, 3 Period 2T solution by HB m 2 Period T solution by HB m

0 6000 6500 7000 7500 8000 8500 9000 9500 Mesh frequency, Hz

Figure 2.9: RMS (mean removed) planet rotational deﬂection for subharmonic resonance of the second distinct mode. The dashed line is the unstable HB solution.

m 10 µ

4 3 Planet deflection,

2 2

0 9500 1 9000 8500 Mesh frequency 8000 7500 harmonics 7000 0 6500 Mesh frequency, Hz

Figure 2.10: Waterfall spectra of planet rotational deﬂection for subharmonic resonance by ﬁnite element for decreasing speed.

45 30

25 NI

m MMS µ

RMS of sun, 10

0 6500 7000 7500 8000 8500 9000 9500 Mesh frequency, Hz

Figure 2.11: RMS (mean removed) sun rotational deﬂection for subharmonic resonance of the second distinct mode by method of multiple scales (MMS) and numerical integration (NI). Both increasing and decreasing speed are performed by numerical integration.

Mesh frequency for , Hz 6500 7000 7500 8000 8500 9000

0.5 Increasing /T

s mesh deflection ξ

0.4

0.3

10% deviation point 5% deviation point 0.2

0.1 Sun-planet tooth separation time, Sun-planet tooth separation time,

0 3600 3800 4000 4200 4400 4600 4800 Mesh frequency for , Hz

Figure 2.12: Sun-planet tooth separation time and points denoting 5% and 10% deviation between perturbation and numerical integration solutions. (O) subharmonic, (•) primary for the second distinct mode.

46 2.5.3 Resonance at Multiples of Mesh Frequency

When mesh stiffness is expressed as a series of Fourier terms as in Eqs. (2.3) and (2.16),

each harmonic can excite resonance. Resonances excited by higher harmonics of mesh

stiffness, where the resonant response frequency is an integer multiple of the fundamental

mesh frequency, are observed in the response spectra of Figure 2.4. Such resonances occur

even for the linear system without contact loss. In addition, nonlinear systems can exhibit

superharmonic resonance where again the response frequency is an integer multiple of

the mesh frequency. In Figure 2.7(a), for example, the resonance in the second distinct

mode (ω4=4393 Hz) at twice the mesh frequency (ωm ≈ 2200 Hz) occurs exclusively from nonlinear superharmonic resonance because only excitation at mesh frequency is included.

Higher harmonic response can also result from a combination of superharmonic resonance and second or higher harmonics of mesh stiffness excitation.

It may be difﬁcult to say by which of these sources a resonance at a multiple of mesh frequency is excited. For example, in Figure 2.4(c) and Figure 2.4(d) the sun exhibits resonance with response at twice mesh frequency, but it is not clear which of second harmonic excitation or superharmonic resonance excites the resonance more strongly. Figure 2.7 shows the possibilities more clearly. One can clearly see nonlinear superharmonic resonance at twice the mesh frequency in Figure 2.7(a) (no other possibility exists). As the second harmonic of mesh stiffness is added in Figure 2.7(b), the resonances from second harmonic excitation and superhamonic resonance overlap and are indistinguishable. We examine both resonant possibilities analytically.

47 Second harmonic excitation, ω ≈ ωi/2

Substituting ω = (ωi/2 + εσ) into Eq. (2.23) and proceeding as for primary resonance

but including the second harmonic of mesh stiffness yields the frequency response relation

for second harmonic excitation as q ωi 1 2 2 2 ω = + Ξ1ai + 2Ξ2 ± 2 |Ξ3| − ωi aiζi 2 4ωiai N h i (1) (0) (2) (1) (0) (2) Ξ1 = ∑ cˆs θs + θs Gsnii + cˆr θr + θr Grnii n=1 (2.46) N+1 N f Ξ = w cˆ(1) (1)G + cˆ(1) (1)G 2 ∑ ∑ 2 s θs sniw r θr rniw w=1 n=1 ωw N+1 N fw (2) (2) Ξ3 = ∑ ∑ 2 cˆs Gsniw + cˆr Grniw . w=1 n=1 ωw

The simpliﬁed Ξk expressions for three equally spaced, in-phase planets are exactly the (1) (1) (2) (2) same as in (2.41) except thatc ˆs andc ˆr in Xi3 are nowc ˆs andc ˆr , indicating the

resonance is excited by the second harmonics of mesh stiffness.

Superharmonic Resonance, ω ≈ ωi/2

In order to investigate superharmonic resonance, ω = (ωi/2 + εσ) is substituted into

Eq. (2.23), but only the ﬁrst harmonic of mesh stiffness is included. Following the same

ﬁrst order perturbation procedure as for primary resonance, the solutions of Eq. (2.22) now

become

fq zq0 = 2 , q = 1,2,··· ,N + 1 (2.47) ωq where Aq in Eq. (2.24) vanishes. The periodic solutions of Eq. (2.23) are

jωqt0 jωt0 zq1 = Mq (t1)e + Nqe + cc, q = 1,2, ··· ,N + 1 N+1 N 4 1 (1) (1) fw (2.48) Nq = − 2 ∑ ∑ kspcs Gsnqw + gkrpcr Grnqw 2 , q = 1,2, ··· ,N + 1. 3 ωq w=1 n=1 ωw

48 Extending the solution to second order perturbation gives

2 2 2 2 ∂ zq2 2 ∂ ∂ ∂ ∂ ∂ 2 + ωq zq2 = − ( 2 + 2 + λq )zq0 − (2 + λq )zq1 ∂t0 ∂t1 ∂t0∂t2 ∂t1 ∂t0∂t1 ∂t0 N+1 N − ∑ ∑ kspQsGsnqw + gkrpQrGrnqw zw1 (2.49) w=1 n=1 N+1 N ˆ 2 ˆ − ∑ ∑ kspQsθs1Gsnqw + g krpQrθr1Grnqw zw0. w=1 n=1

Proceeding to eliminate secular terms in Eq. (2.49) with ω = (ωi/2 + εσ), second order

perturbation gives the frequency response relation for superharmonic resonance as q ωi 1 2 2 2 ω = + Ξ1ai ± 2 |Ξ2| − ωi aiζi 2 4ωiai N h i (1) (0) (2) (1) (0) (2) Ξ1 = ∑ cˆs θs + θs Gsnii + cˆr θr + θr Grnii (2.50) n=1 N+1 N (1) (1) Ξ2 = ∑ ∑ Mw cˆs Gsniw + cˆr Grniw , w=1 n=1

where Mw = εNw. Note that the amplitude for superharmonic resonance is O(ε), in contrast

to all other cases considered where the response is O(1). The simpliﬁed Ξk for the ﬁrst

distinct mode for three equally spaced, in-phase planets are h i (1) (0) (2) 2 (1) (0) (2) 2 Ξ1 = 3 cˆs θs + θs (v11 + v21) + cˆr θr + θr v21

h (1) 2 (1) 2 i h (1) (1) i Ξ2 = N1 cˆs (v11 + v21) + cˆr v21 + N4 cˆs (v11 + v21)(v14 + v24) + cˆr v21v24 4 f1 n (1) 2 (1) 2 o f4 n (1) (1) o M1 = − 2 2 cˆs (v11 + v21) + cˆr v21 + 2 cˆs (v11 + v21)(v14 + v24) + cˆr v21v24 ω1 ω1 ω4 4 f1 n (1) (1) o f4 n (1) 2 (1) 2 o M4 = − 2 2 cˆs (v11 + v21)(v14 + v24) + cˆr v21v24 + 2 cˆs (v14 + v24) + cˆr v24 . ω4 ω1 ω4 (2.51)

Perturbation response amplitudes for superharmonic resonance and second harmonic exci-

tation are compared with NI and HB solutions in Figure 2.13. For the NI and HB solutions

in Figure 2.13, the response component at twice mesh frequency is extracted and used to

calculate the RMS response. The second harmonic of mesh stiffness is included for the NI

49 and HB solutions in Figure 2.13(a) while only the ﬁrst harmonic of mesh stiffness is considered in Figure 2.13(b), so the response depicts resonances from different sources. For second harmonic excitation (Figure 2.13(a)), tooth separation is predicted by perturbation,

NI, and HB around 2200 Hz. HB captures the narrow unstable branch at the top of the second distinct mode resonance, which indicates tooth separation. The perturbation solution matches the response amplitude for both modes. Neither tooth separation nor an unstable branch exists for the ﬁrst distinct mode.

Although perturbation gives a reasonable approximation for superharmonic resonance in Figure 2.13(b), the deviation of perturbation from other solutions is noticeable here, while it is negligible for the second harmonic excitation (Figure 2.13(a)). Compared to the second harmonic excitation (Figure 2.13(a)), lower amplitude is predicted. This observa- tion suggests that the second harmonic excitation contributes more strongly to resonance at the second harmonic of mesh frequency than nonlinear superharmonic resonance.

Figure 2.14 compares resonant time history response excited by mesh frequency only

(nonlinear superharmonic), second harmonic excitation only, the sum of these two cases, and both excitations simultaneously. All response frequencies are at twice the mesh frequency. The mean values are identical as the same external torque is applied. No phasing between the responses is shown. Larger amplitude of response from second harmonic excitation than from superharmonic resonance is evident. Figure 2.14 illustrates how the response for simultaneous mesh frequency and second harmonic excitation (shown by the solid line) matches the sum of the two individual resonant responses (dashed line) with negligible error. This indicates a lack of nonlinear interaction between these two excitation sources and their respective responses. A superposition principle is reasonably accurate for

50 these two response. Therefore, the two resonant behaviors can be analyzed independently and added in this case, which is much simpler than analyzing both simultaneously.

51 12

11 10 HB unstable solution branch

10 8 m µ

6 9

RMS of sun, 4 8 2170 2190 2210 2230

0 800 1000 1200 1400 1600 1800 2000 2200 2400 Mesh frequency, Hz (a)

Second distinct mode ( ω = 2ω = 4390 Hz) 10 4

8 MMS

m µ HB 6

RMS of sun, 4 First distinct mode ( ω = 2ω = 1860 Hz) 1

0 800 1000 1200 1400 1600 1800 2000 2200 2400 Mesh freqeuncy, Hz

(b)

Figure 2.13: RMS (mean removed) sun rotational deﬂection for (a) second harmonic excitation and (b) superharmonic resonance by numerical integration (NI), harmonic balance (HB), and method of multiple scales (MMS).

52 60

50 m µ 40

20 Sun rotational deflection, 10

0 0 1 2 Mesh cycle

Figure 2.14: Sun response time history for different resonances at mesh frequency 2200Hz; (•) Second harmonic excitation only, (···) superharmonic resonance (only excitation at mesh frequency), (- - -) sum of second harmonic excitation and superharmonic resonance, and (—) resonances excited by ﬁrst and second harmonics of mesh stiffness.

53 2.6 Conclusions

This paper examines the nonlinear behavior of a parametrically excited planetary gear with equally spaced planets. Perturbation analysis gives closed-form approximations for primary, sub- and superharmonic, and second harmonic resonance. The analytical results compare well with ﬁnite element, numerical integration, and harmonic balance solutions.

Key system parameters such as mean values and dynamic ﬂuctuations of mesh stiffness, damping, load, and vibration modes are explicitly included in the closed-form solutions.

This helps designers understand how parameter choices can reduce dynamic response. The main results are summarized below.

• The degenerate planet modes can not experience parametric instability for equally

spaced planets and in-phase meshes.

• The frequency response curves at all resonances lean to the left from softening non-

linearity induced by tooth separation. The onset of tooth separation is marked by kinks

in the frequency response curves, and tooth separation occurs for all amplitudes above

these kinks. These contact loss initiation points as well as the important peak ampli-

tude at resonance are expressed analytically as simple forms in terms of known system

parameters. Proportional relationships between the applied torque and both the contact

loss initiation points and the peak amplitude analytically show that higher torque does

not prevent contact loss.

• The analytical and ﬁnite element models predict subharmonic resonance with good

agreement. This extends the reliability of the analytical model over a broader range of

resonant phenomena than prior research [21]. Subharmonic resonance exhibits jump

54 phenomena both above and below the resonant frequency (for both increasing and decreasing mesh frequency). The analytical solution predicts the parametric instability region, and the solution shows that the width of the instability region is not sensitive to the applied torque but to other system parameters such as damping, natural frequency, harmonics of mesh stiffness, etc. This jump behavior is consistent with that observed in certain industrial planetary gear systems.

• As the ratio of tooth separation time to response period increases, the perturbation solution loses its accuracy. Nevertheless, tooth separation times as large as 30% of a mesh cycle still yield good results with errors of 10% or less. This ratio is valid for both sun-planet and ring-planet meshes. This value may differ somewhat for other systems.

• Resonance at multiples of mesh frequency can be excited from higher harmonics of mesh stiffness variation and nonlinear superharmonic resonance. Separate perturbation analyses clarify the contribution from both superharmonic and second harmonic excitation resonance. For the present system, the response is predicted accurately by directly adding the response amplitudes from each resonance analyzed separately.

55 Chapter 3: Inﬂuence of System Parameters on Planetary Gear Dynamic Response

3.1 Introduction

Various analytical and ﬁnite element models for planetary gears have been developed

[1, 19, 34, 45, 61, 77]. Computational simulations of these models have shown that planetary gears exhibit nonlinear dynamic response [21, 34, 45, 66], which is also shown experimentally by Botman [12]. Numerical solutions are relatively easy to obtain compared to experimental measurements. Each solution is, however, valid only for a specific system configuration and often slow because of high computational requirements, so numerical solutions provide limited understanding of the influence of system parameters and their interactions. The influence of planetary gear system parameters on dynamic response has received relatively little research attention analytically; most studies emphasize numerical simulation. Some prior studies deal with key system parameters such as mesh phase, gear contact ratio, and mesh stiffness.

The relative mesh phase occurs because of the multiple meshes in planetary gears. Sev- eral early studies [3,5,78] show a strong inﬂuence of mesh phase on the dynamic loads and the possibility of neutralizing particular excitation by mesh phase. Kahraman [45] considers mesh phase in a dynamic model of planetary gears with four equally spaced planets.

56 Kahraman and Blankenship [41] conclude that no superior mesh phasing condition is possible to reduce dynamic response for all vibration modes. Instead, they claim that only particular operating speed range needs to be speciﬁed to determine an optimal phasing condition. Parker [47] and Ambarisha and Parker [48] propose rules to suppress certain harmonics of mesh frequency exciting the three different types [19] of planetary gear vibration modes. The rules give conditions for whether or not particular vibration modes are suppressed but do not address the dynamic response. Some of these mesh phase effects are evident in the computational simulations in [34].

Contact ratio strongly affects the dynamics. High contact ratios are generally preferred to reduce the bending and contact stresses and the noise. Anderson [79] shows that tooth separation can be avoided by proper selection of contact ratio. Amabili and Rivola [58] conclude that the stability of single gear pairs is affected by the contact ratio. Kahraman and Blankenship [80] experimentally examine the influence of contact ratio on dynamic response. They find that a specific contact ratio minimizes the amplitude of vibration. Liou et al. [81] compare the effect of low and high contact ratio on dynamic load. Lin and

Parker [82] show that contact ratio affects dynamic response and parametric instability for multi-mesh gear systems. Considering planetary gears, they demonstrate that the instability boundaries are sensitive to sun-planet and ring-planet contact ratios and particular instabilities can be suppressed by proper selection of these contact ratios [40]. Vangipuram-Canchi and Parker [83] consider the ring gear ﬂexibility to show the suppression of parametric instabilities by suitable contact ratio conditions. Abousleiman et al. [67] show that the planet radial position errors tend to lower the sun-planet contact ratio and to increase the ring-planet contact ratio. Kim et al. [84] ﬁnd increased frequency components in dynamic response when time-varying contact ratio is considered.

57 Mesh stiffness variation is a primary excitation source and becomes a critical part of time-varying dynamic models. Most studies dealing with mesh stiffness variation are fo- cused on single gear pairs. Among those involving planetary gears, Velex and Flamand [77] show that mesh stiffness has a signiﬁcant inﬂuence on natural frequencies and tooth loads.

Lin and Parker [40] discuss the impact of mesh stiffness variation on parametric instability. Wu and Parker [85] extended the study of parametric instability for planetary gears by considering an elastic continuum ring gear.

Most prior planetary gear parameter studies have been numerical. The large number of system parameters and their interaction effects on vibration make numerical analysis inef-

ﬁcient and limited as a way to develop vibration reduction design knowledge. This paper analytically investigates the inﬂuence of selected system parameters on planetary gear dynamic response. Analytical expressions of the peak amplitude of resonant response and the width of the parametric instability region are derived from perturbation solutions. Rules to suppress resonances and parametric instabilities are presented. The proper selection of parameters and their grouping in certain combinations are shown to reduce dynamic response.

The different inﬂuence of sun-planet and ring-planet mesh parameters on dynamic response is shown mathematically. Altered sensitivity to system parameters of dynamic response at different vibration modes is discussed. Interactions between the system parameters are examined through their combined effects on dynamic response.

3.2 Analytical Expressions of Dynamic Response

Experimental investigation of how the system parameters impact the dynamic response requires large effort, time and expense. The demands are less but still large with computational analysis. In practice, these methods are used to validate a system for selected cases

58 and give limited insight into dynamic behavior due to the small size of the data set. In

contrast, an analytical solution is effective when one explores the parameter dependencies

because it reveals the mathematical structure of the dynamic response and its dependence

on the system parameters. Bahk and Parker [86] obtained an analytical solution of plane-

tary gear dynamic response for a purely rotational degree of freedom model with in-phase

meshes by using perturbation analysis. The perturbation solution of a more generalized

planetary gear model is developed in this study. Bearing stiffness is added, which allows

gear translational motions, and there are no restrictions on the mesh phasing.

The governing equation for the planetary gear shown in Figure 3.1 is

Mx¨ + Kx = F (3.1) T x = [xc,yc,uc,xr,yr,ur,xs,ys,us,ζ1,η1,u1,··· ,ζN,ηN,uN] , where M is the inertia matrix, K = Km(x,t)+Kb is the stiffness matrix including the mesh stiffness Km and the bearing stiffness Kb, and F is the external force (refer to [19] detailed matrices).

The nonlinear, time-varying sun-planet (ksn) and ring-planet (krn) mesh stiffnesses included in Km(x,t) are

ksn(x,t) = ksn(t)Θsn krn(x,t) = krn(t)Θrn n = 1, 2,··· , N 1 δ ≥ 0 1 δ ≥ 0 Θ = rn Θ = rn sn 0 δ < 0 rn 0 δ < 0 sn rn (3.2) δsn = ys cosψsn − xs sinψsn − ζn sinαs − ηn cosαs + us + un

δrn = yr cosψsn − xr sinψrn + ζn sinαr − ηn cosαr + ur − un, where αs and αr are the pressure angles of the sun-planet and ring-planet meshes, ψsn =

ψn − αs and ψrn = ψn + αr, and ψn is the circumferential position angle of the nth planet.

In what follows, sub or superscript s and r denote sun and ring related quantities.

59 The mesh stiffnesses are periodic at the mesh frequency ω and are Fourier expanded as

" ∞ # ¯ ˆ(l) jlωt ksn (t) = ksn + ∑ ksn e + cc l=1 (3.3) " ∞ # ¯ ˆ(l) jlωt krn (t) = krn + ∑ krn e + cc , n = 1, 2,··· , N, l=1

Critical system parameters such as the gear contact ratio, relative mesh phase, and the amplitude of mesh stiffness ﬂuctuation are considered in the mesh stiffness. The time- varying mesh stiffness is approximated by trapezoidal wave forms as shown in Figure 3.2.

The Fourier coefficients are k˜ (l) sn − jlωγsnT kˆsn = sin[lπ(cs − ss)]sin(lπss)e l2π2s s (3.4) k˜ ˆ(l) rn − jlω(γsr+γrn)T krn = 2 2 sin[lπ(cr − sr)]sin(lπsr)e , l π sr where T = 2π/ω is the mesh period. γsn(rn) is the relative mesh phase between the nth sun-planet (ring-planet) mesh and the arbitrarily chosen first sun-planet (ring-planet) mesh, and γsr is the relative mesh phase between the sun-planet and ring-planet meshes at a given planet [46]. k˜sn and k˜rn are the peak-peak amplitudes of the mesh stiffness fluctuations. cs and cr are the contact ratios. ss and sr are the coefficients that determine the slope of the non-horizontal side of the trapezoidal wave function in Figure 3.2.

For the model in (3.1) with the time-invariant mean stiffness matrix K0, the eigenvalue problem is

2 K0vi = ωi Mvi. (3.5)

The modal matrix V = [v1,··· ,vΨ] excluding the rigid body mode is normalized such that

VT MV = I. The modal coordinate vector z is introduced as x = Vz.

60 Application of the method of multiple scales expands the ith modal response and yields

[86, 87] 1 f j[ωt0−γi(t1)] i 2 zi (t0,t1) = ai(t1)e + cc + 2 + εzi1 (t0,t1) + O ε 2 ωi (3.6) n tn = ε t,

ˆ(l) |ks1 | where ε = is the small perturbation parameter less than unity. fi is the ith component k¯s1 T of the modal force vector V F. γi is the phase angle of the response. ai is the amplitude of

the leading order solution zi0 in (3.6). It is the primary output from the analysis.

For primary resonance, which occurs when the mesh frequency is near a natural fre-

quency, the mesh frequency ω is expressed as ω = ωi + εσ where σ is a detuning pa-

rameter. Perturbation analysis [87] gives the frequency-response relation. The addition

of the translational degrees of freedom and the use of trapezoidal mesh stiffness does not

change some of the basic mathematical expressions of the perturbation solution in [86].

The frequency response approximation for the ith mode primary resonance is q 1 2 2 2 ω = ωi + Π1ai + 2Π2 ± 2 |Πpp| − ωi aiζi 2ωiai N h i ˆ(1) (0) (2) ˆ(1) (0) (2) Π1 = ∑ ks1 θsn + θsn Gsnii + kr1 θrn + θrn Grnii n=1 (3.7) Ψ N f Π = w kˆ(1) (1)G + kˆ(1) (1)G 2 ∑ ∑ 2 s1 θsn sniw r1 θrn rniw w=1 n=1 ωw Ψ N fw ˆ(1) ˆ(1) Πpp = ∑ ∑ 2 ksn Gsniw + krn Grniw . w=1 n=1 ωw

(l) (l) N is the number of planets. θsn and θrn are the lth Fourier harmonics of the tooth separa-

tion functions Θsn and Θrn in (3.2). They depend on the amplitude ai and the steady deﬂec-

T tion caused by the applied torque. Gsniw and Grniw are the (i,w) elements of Gsn = V KsnV

T and Grn = V KrnV, where Ksn and Krn are matrices consisting of the coefﬁcients of ksn and krn in Km. The notations for the variables in (3.7) and the rest of this paper follow those

61 used in [86] except that kˆsn and kˆrn replacec ˆsn andc ˆrn to avoid confusion with the notation

for contact ratio, cs and cr.

The peak amplitude of resonant response is an important measure of vibration. For

the ith mode primary resonance, the peak amplitude is calculated from the condition of

2 2 2 |Πpp| − ωi aiζi = 0 in (3.7), giving

peak Πpp ai = 2 . (3.8) ωi ζi (l) (l) With the insertion of kˆsn and kˆrn in (3.4) into (3.7), Πpp becomes

Ψ N f k˜ w sn − jωγsnT Πpp = ∑ ∑ 2 2 sin[π(cs − ss)]sin(πss)Gsniwe ω π ss w=1 n=1 w (3.9) k˜ rn − jω(γrn+γsr)T + 2 sin[π(cr − sr)]sin(πsr)Grniwe . π sr Equally spaced planets have an identical shape of the mesh stiffness for each mesh with a possible phase difference. Small variations of the mesh stiffness for each planet occurs for diametrically opposed planet pair conﬁgurations due to the different mesh loads because the mesh stiffness is a function of the mesh load. This difference is small, however, so it is neglected. Thus, the subscript n is removed from k˜sn and k˜rn in what follows. The validity of this assumption for diametrically opposed planets will be discussed later. In addition, use of γn = γsn = γrn from [46] simpliﬁes (3.9), giving

N peak (n) − jωγnT ai = ∑ Λpp e n=1 " 1 Ψ f k˜ Λ(n) = w s sin[ (c − s )]sin( s )G (3.10) pp 2 2 ∑ 2 π s s π s sniw ωi ζiπ w=1 ωw ss k˜ r − jωγsrT + sin[π(cr − sr)]sin(πsr)Grniwe . sr

(m) (n) (n) In general, Gsniw 6= Gsmiw and Grniw 6= Grmiw, which makes Λpp 6= Λpp . Λpp is independent of n for particular cases, however. Discussion of the cases and similar issues

62 that follow requires understanding of the characteristics of planetary gear vibration modes.

As shown in [19, 20, 25, 26, 28, 40], both equally spaced and diametrically opposed planet

systems have distinct modal properties. For a purely rotational model with rigid bearings,

all modes fall into two categories:

- Distinct modes where all the planets have the same rotation.

- Degenerate modes where the sun, carrier, and ring do not deﬂect.

For a rotational-translational model including bearing stiffness, all modes fall into three

categories:

- Rotational modes where the translations of the sun, carrier, and ring are zero and all

planets have identical motions.

- Planet modes where only the planets have deﬂections while the sun, carrier, and ring

do not move.

- Translational modes where the sun, carrier, and ring have only translation but not rota-

tion.

The eigenvalue problem gives one rigid body mode, which exists only when the input and

output components are rotationally unconstrained. The rigid body mode is not excited by

any operating mesh frequency, so it is neglected.

Considering the ith mode to be a distinct mode of a purely rotational model, Gsnii =

Gsmii and Grnii = Grmii as shown in Appendix A. As an example, for the ﬁrst distinct mode,

2 2 i = 1, Gsn11 = v¯1 = (−v11 cosαs + v31 + v41) in (A.3) is independent of n. When only the central gears are loaded, which is the usual case in practice,

F = [ f¯1, f¯2, f¯3,0,··· ,0]. (3.11)

63 Ψ T ¯ The absence of central gear rotations for a degenerate mode gives fw = vwF = ∑ v jw f j = 0, j=1 so fw 6= 0 only for the distinct modes. When the ith mode is a distinct mode, the combina- (n) (m) tion of fw = 0 for w 6= i and the invariance of Gsnii and Grnii with n yields Λpp = Λpp .

For the rotational-translational model, the conditions discussed in the prior paragraph can be found for the rotational modes. When the ith mode is a rotational mode, Gsnii and

Grnii are independent of n. Similarly for (3.11), the case of only the central gears loaded gives

F = [0,0, f¯3,0,0, f¯6,0,0, f¯9,0,··· ,0]. (3.12)

T The central gears do not rotate for translational and planet modes, which gives fw = vwF = Ψ ¯ ∑ v jw f j = 0. Therefore, fw is zero for w 6= i with only the central gears loaded. As a j=1 (n) (n) (m) result, Λpp becomes independent of n for the rotational modes. Considering Λpp = Λpp , the expression of the peak amplitude in (3.10) for the rotational modes is re-written as

N peak − jωγnT ai = Λpp ∑ e n=1 " 1 Ψ f k˜ Λ = w s sin[ (c − s )]sin( s )G (3.13) pp 2 2 ∑ 2 π s s π s siw ωi ζiπ w=1 ωw ss k˜ r − jωγsrT + sin[π(cr − sr)]sin(πsr)Griwe , sr

(n) where the planet index n is dropped from Λpp , Gsniw, and Grniw because they are indepen-

dent of n.

Resonances other than the primary resonance occur. When the lth harmonic of the

mesh stiffness variations in (3.3) are close to a natural frequency, resonance occurs with

the resonant response frequency ωi ≈ lω. The peak amplitude of this higher harmonic

excitation condition is obtained by a similar perturbation analysis as for (3.8). For second

64 harmonic excitation as an example, the peak amplitude is

peak Πp2 ai = 2 ω ζi i (3.14) Ψ N fw ˆ(2) ˆ(2) Πp2 = ∑ ∑ 2 ksn Gsniw + krn Grniw . w=1 n=1 ωw

(2) (2) The appearance of kˆsn and kˆrn in (3.14) indicates that the resonance is excited by the second harmonic of the mesh stiffnesses.

In a nonlinear phenomenon called superharmonic resonance associated with the contact loss nonlinearity, resonance with the same response frequency as in (3.14) of ωi ≈ 2ω can be excited by the ﬁrst harmonic of mesh stiffness (l = 1) [87]. Second order perturbation is used to obtain the frequency-amplitude relation for superharmonic resonance [68,86]. The peak amplitude for the superharmonic resonance at ω = ωi/2 + εσ is

Πps ai = 2 ωi ζi Ψ N ˆ(1) ˆ(1) Πps = ∑ ∑ Mw ksn Gsniw + krn Grniw (3.15) w=1 n=1 Ψ N 4 1 ˆ(1) ˆ(1) fk Mw = − 2 ∑ ∑ ksn Gsmwk + krn Grmwk 2 . 3 ωw k=1 m=1 ωk Unlike the second harmonic excitation case in (3.14), (3.15) includes only the ﬁrst harmon-

(1) (1) ics of mesh stiffness kˆsn and kˆrn , although the resonant response occurs at twice the mesh frequency for both resonances. With the use of (3.4), γn = γsn = γrn, k˜sn = k˜s, and k˜rn = k˜r,

65 (3.15) is re-written as

Ψ " N Πps k˜ a = = (w) s sin[ (c − s )]sin( s )G i 2 ∑ Λps ∑ π s s π s sniw ωi ζi w=1 n=1 ss k˜ r − jωγsrT − jωγnT + sin[π(cr − sr)]sin(πsr)Grniwe e sr 4 1 1 N (w) ¯ (w) − jωγqT Λps = − 2 2 4 ∑ Λps e 3 ωi ζi ωwπ q=1 Ψ fp k˜ k˜ ¯ (w) s r − jωγsrT Λps = ∑ 2 sin[π(cs − ss)]sin(πss)Gsqwp + sin[π(cr − sr)]sin(πsr)Grqwpe . p=1 ωp ss sr (3.16)

When the mesh frequency approaches twice a natural frequency, ω ≈ 2ωi, parametric instability occurs, and the system becomes unstable. The large response triggers nonlinear tooth contact loss that bounds the parametric instability response to be resonant response with period-2T. This is called a subharmonic resonance [87]. Subharmonic resonance is characterized by jump phenomena on both sides of the mesh frequency range where resonance occurs; jump up and jump down phenomena occur for increasing and decreasing speeds [12, 68, 86, 88]. The width of the frequency interval where parametric instability occurs can be calculated by perturbation [86]. This width correlates with the amplitude of response, which is not easily captured in a mathematical expression [68, 86]. The expression for the width ∆ is q 4 2 2 2 ∆ = |Πws| − ωi ζi ωi N (3.17) 1 ˆ(1) ˆ(1) Πws = ∑ ksn Gsnii + krn Grnii . n=1 2

With the substitutions of (3.4), γn = γsn = γrn, k˜rn = k˜r, and k˜sn = k˜s, Πws becomes N 1 k˜s Πws = ∑ 2 sin[π(cs − ss)]sin(πss)Gsnii 2π ss n=1 (3.18) k˜ r − jωγsrT − jωγnT + sin[π(cr − sr)]sin(πsr)Grniie e . sr

66 ys ,yc ,yr

k ζ r 2 Planet 2 k ru kr2 η2 kp kcu u1 u k r ksu s2 uc Planet 3 Sun ψ2 kc kr xs ,xc ,xr ks3 u3 ks η1 ζ kp 3 ks1 k u kp r1 kr3 s u1 η3 Carrier Ring ζ1 kc

Planet 1

Figure 3.1: A schematic of two-dimensional lumped parameter model of a planetary gear system

(c - 1)T

sT sT

k max

Mesh Stiffness

kmin

t t0 t0 +T

Figure 3.2: Mesh stiffness variation approximated by a trapezoidal wave function. c is the contact ratio. s is the slope coefﬁcient. T is the mesh period.

67 3.3 Mesh Phase Effect

This section studies the impact of mesh phase on dynamic response with special fo-

cus on the suppression of the particular vibration modes by mesh phase for both equally

spaced and diametrically opposed planets. First, the peak amplitude expression in (3.8)

and well-deﬁned modal properties of a planetary gear system are used to conﬁrm the com-

plete suppression rules for primary resonance introduced in [47, 48] where the symmetry

of planetary gears and the periodicity of the gear tooth mesh are mainly used. In addition,

the suppression rules are extended for superharmonic and subharmonic resonances. For a

vibration mode that is not suppressed by any mesh phase, optimal mesh phase that reduce

the resonant response will be examined by using perturbation solutions.

Equally spaced planet systems have N different mesh phase possibilities, including in-

phase. The nth mesh phase γn relative to the arbitrarily chosen ﬁrst mesh [41, 45, 46, 48]

zsψn zs 2π (n − 1)zs (n − 1)K γn = = (n − 1) = = , K = 0,··· ,N − 1, (3.19) 2π 2π N N N where zs is the sun tooth number and K = mod(zs/N). For example, the possible combinations of mesh phase for N = 4 are [0 0 0 0], [0 1/4 2/4 3/4], [0 2/4 0 2/4], and

[0 3/4 2/4 1/4].

For diametrically opposed planets, the expression of the mesh phase is

zsψn zs 2π zs γn = = kn = kn, (3.20) 2π 2π zs + zr zs + zr

where zr is the ring tooth number and kn is an integer. For diametrically opposed planets,

the condition ψn+N/2 = ψn + π requires that both zr and zr are either even or odd [48].

68 From the mesh phase relation between diametrically opposed planets:

zsψn+N/2 zs(ψn + π) zs γ = = = γn + , (3.21) n+N/2 2π 2π 2

the following two different cases are possible,

γ(n+N/2) = γn for even zs 1 (3.22) γ = γ + for odd z . (n+N/2) n 2 s In short, repeated pattern of mesh phase is examined with possible offset of phase 1/2 only for odd zs. For N = 4 as an example, γ = [0 γ2 0 γ2] for even zs and γ = [0 γ2 1/2 γ2 +1/2] for odd zs. For N = 6, γ = [0 γ2 γ3 0 γ2 γ3] for even zs and γ = [0 γ2 γ3 1/2 γ2 +

1/2 γ3 + 1/2] for odd zs.

3.3.1 Primary Resonance, Equally Spaced Planets

Two meaningful vibration modes exist for a rotational model: distinct and degenerate modes. For the distinct mode, it is shown in prior section that Λpp in (3.13) is independent of n. Because Λpp is out of the summation over n as shown in (3.13), the suppression of N − jωγnTm the modes by mesh phase is solely determined by ∑ e . With use of Tm = 2π/ω n=1 and (3.19), Euler’s formula gives

2π(n − 1)K 2π(n − 1)K e− jωγnTm = e− j2πγn = cos2πγ − isin2πγ = cos − isin . (3.23) n n N N

From the trigonometric identities

N 2π(n − 1)K N 2π(n − 1)K K ∑ sin = 0, ∑ cos = 0 for 6= integer, (3.24) n=1 N n=1 N N

N N 2π(n − 1)K 2π(n − 1)K ∑ e− j2πγn = ∑ cos − isin = 0 for K = 1,··· ,N −1, which leads n=1 n=1 N N to the suppression of the distinct mode for out-of-phase.

69 (n) For the degenerate mode, Gsniw 6= Gsmiw and Grniw 6= Grmiw and Λpp in (3.10) is de- (n) (n) pendent of n. Λpp is calculated from the summation over w. When fw in Λpp is zero, wth

component in the summation becomes zero regardless of Gsniw and Grniw. So, only Gsniw∗

∗ and Grniw∗ are considered for all the vibration modes, where w will be used to refer to the

distinct and rotational mode w that makes fw 6= 0 throughout this paper. When the planets

are in-phase, the expression of the peak amplitude is

N peak (n) ai = ∑ Λpp . (3.25) n=1

(n) Gsniw∗ and Grniw∗ in Λpp are calculated from Appendix A as

Gsniw∗ = (−v1w∗ cosαs + v3w∗ + v4w∗ )vn+3,i (3.26) Grniw∗ = (v1w∗ cosαr − v2w∗ + v4w∗ )vn+3,i.

N N N The degenerate mode property ∑ vn+3,i = 0 yields ∑ Gsniw∗ = 0 and ∑ Grniw∗ = 0. As n=1 n=1 n=1 N (n) a result, ∑ Λpp reduces to zero and the degenerate mode is suppressed when the mesh is n=1 in-phase.

The next is the suppression of vibration modes by mesh phase for a rotational-translational

(n) model: rotational, planet and translational modes. Like the distinct mode, Λpp is indepen- N dent of n for the rotational mode, and the peak amplitude is proportional to ∑ e− j2πγn . n=1 Use of (3.23) and (3.24) yields the same suppression condition with the distinct mode: The rotational mode is suppressed for out-of-phase.

To examine the mesh phase effect on the planet mode, use of following planet modal properties is critical. Planet deﬂections for the planet modes are expressed as Pn = ynP1,

T where Pn = [v3n+7,i,v3n+8,i,v3n+9,i] and the yn are scalars that satisfy [19, 26]

N N N ∑ yn sinψn = 0 ∑ yn cosψn = 0 ∑ yn = 0. (3.27) n=1 n=1 n=1 70 Gsniw∗ and Grniw∗ for the planet modes are calculated as

Gsniw∗ = ynv¯psvˆps, Grniw∗ = ynv¯prvˆpr

v¯ps = v10,i sinαs + v11,i cosαs − v12,i, vˆps = v10,w∗ sinαs + v11,w∗ cosαs − v12,w∗ (3.28)

v¯pr = v10,i sinαr − v11,i cosαr − v12,i, vˆpr = v10,w∗ sinαr + v11,w∗ cosαr − v12,w∗ . N N N N (n) ∑ Gsniw∗ and ∑n=1 Grniw∗ in (3.28) are zero from ∑ yn = 0 in (3.27), which gives ∑ Λpp = n=1 n=1 n=1 N (n) 0. When the planets are in-phase, from (3.25) and ∑ Λpp = 0, the suppression of the planet n=1 mode is concluded.

For a case of out-of-phase, γn 6= 0, with use of Tm = 2π/ω, (3.10) can be rewritten as ! Ω f N N peak 1 w − j2πγn − j2πγn a = Ps Gsniwe + Pr Grniwe i ω2ζ π2 ∑ ω2 ∑ ∑ i i w=1 w n=1 n=1 (3.29) k˜ k˜ s r − jωγsrTm Ps = sinπ(cs − ss)sinπss Pr = e sinπ(cr − sr)sinπsr. ss sr N ∗ peak − j2πγn Considering fw = 0 for w 6= w , ai = 0 is achieved only when both ∑ Gsniw∗ e and n=1 N N − j2πγn − j2πγn ∑ Grniw∗ e are zero. From (3.28), ∑ Gsniw∗ e is expressed as n=1 n=1 N " N # − j2πγn ∑ Gsniw∗ e = v¯psvˆps ∑ yn (cos2πγn − isin2πγn) n=1 n=1 " # (3.30) N 2π(n − 1)K 2π(n − 1)K = v¯psvˆps ∑ yn cos − isin . n=1 N N

2π(n − 1)K/N in (3.30) becomes equivalent to ψn = 2π(n − 1)/N for K = 1 and N − 1,

giving N " N N # − j2πγn ∑ Gsniw∗ e = v¯psvˆps ∑ yn cosψn + i ∑ yn sinψn . (3.31) n=1 n=1 n=1 N N N − j2πγn From ∑ yn sinψn = 0 and ∑ yn cosψn = 0 in (3.27), ∑ Gsniw∗ e = 0 is clear. Sim- n=1 n=1 n=1 N − j2πγn ilarly, one can show ∑ Grniw∗ e = 0 for K = 1 and N − 1, which concludes the sup- n=1 pression of the planet mode.

71 For the translational modes, Gsniw∗ and Grniw∗ are (n) (n) (n) Gsniw∗ = v¯ts (v˜ts − v9w) + vˆts v˜ts, Grniw∗ = v¯tr vˆtr

(n) (n) v¯ts = v7i sin(ψsn) − v8i cos(ψsn), vˆts = v3n+7,i sinαs + v3n+8,i cosαs − v3n+9,i

v˜ts = v10,w∗ sinαs + v11,w∗ cosαs − v12,w∗

(n) v¯tr = v3n+7,i sinαr − v3n+8,i cosαr − v3n+9,i, vˆtr = v10,w∗ sinαr − v11,w∗ cosαr − v12,w∗ . (3.32) N − j2πγn Similarly for the planet mode, the translational mode is suppressed when ∑ Gsniw∗ e = n=1 N N N − j2πγn (n) − j2πγn (n) − j2πγn 0 and ∑ Grniw∗ e = 0 from (3.29), which requires each of ∑ v¯ts e , ∑ vˆts e , n=1 n=1 n=1 N (n) − j2πγn and ∑ v¯tr e being zero. n=1 (n) After substitution of ψsn = ψn − αs,v ¯ts in (3.32) is expressed as (n) ¯ v¯ts = α¯ts sinψn + βts cosψn (3.33) ¯ α¯ts = v7i cosαs − v8i sinαs, βts = −v7i sinαs − v8i cosαs. N (n) − j2πγn From (3.23) and γn = zsψn/2π, ∑ v¯ts e becomes n=1 " N N ¯ ¯ (n) − j2πγn αts βts ∑ v¯ts e = ∑ {sinψn(1 + zs) + sinψn(1 − zs)} + {cosψn(1 + zs) + cosψn(1 − zs)} n=1 n=1 2 2 # α¯ β¯ − i ts {cosψ (1 − z ) − cosψ (1 + z )} + i ts {sinψ (1 + z ) − sinψ (1 − z )} . 2 n s n s 2 n s n s (3.34)

Substitution of ψn = 2π(n − 1)/N into (3.34) and use of K= mod (zs/N) gives " N N ¯ (n − )( + K) (n − )( − K) (n) − j2πγn αts 2π 1 1 2π 1 1 ∑ v¯ts e = ∑ sin + sin n=1 n=1 2 N N β¯ 2π(n − 1)(1 + K) 2π(n − 1)(1 − K) + ts cos + cos 2 N N (3.35) α¯ 2π(n − 1)(1 − K) 2π(n − 1)(1 + K) − i ts cos − cos 2 N N # β¯ 2π(n − 1)(1 + K) 2π(n − 1)(1 − K) + i ts sin − sin . 2 N N

72 N (n) − j2πγn All terms in (3.35) become zero for K 6= 1 and N −1 from (3.24), which leads to ∑ v¯ts e = n=1 0.

For the translational modes, there is a pair of vibration modes with a natural frequency of multiplicity two. The planet deﬂections of the pair of vibration modes are related as

[19, 26]

Pn = cos(ψn)P1 + sin(ψn)P¯ 1 (3.36) P¯ n = −sin(ψn)P1 + cos(ψn)P¯ 1,

T where Pn and P¯ n = [v¯3n+7,i,v¯3n+8,i,v¯3n+9,i] are the pair of vibration modes. With use of (n) (n) the relation of the planet deﬂections in (3.36),v ˆts andv ¯tr become (n) ˆ (n) ¯ vˆts = αˆts cosψn + βts sinψn, v¯tr = α¯tr cosψn + βtr sinψn ˆ αˆts = v10,i sinαs + v11,i cosαs − v12,i, βts = v¯10,i sinαs + v¯11,i cosαs − v¯12,i (3.37)

¯ α¯tr = v10,i sinαs − v11,i cosαs − v12,i, βtr = v¯10,i sinαs − v¯11,i cosαs − v¯12,i.

(n) (n) (n) From the analogous expression ofv ˆts andv ¯tr in (3.37) tov ¯ts in (3.33), one can show that N N (n) − j2πγn (n) − j2πγn ∑ vˆts e and ∑ v¯tr e are cast into the similar expression of (3.35) by replacing n=1 n=1 N ¯ ˆ (n) ¯ (n) (n) − j2πγn α¯ts and βts with βts and αˆts forv ˆts and with βtr and α¯tr forv ¯tr . Therefore, ∑ vˆts e n=1 N N (n) − j2πγn (n) − j2πγn and ∑ v¯tr e vanish for the same condition with ∑ v¯ts e , and consequently the n=1 n=1 translational modes are suppressed for K 6= 1 and N − 1.

3.3.2 Primary Resonance, Diametrically Opposed Planets

(n) (m) Similarly for the equally spaced planets, from Λpp = Λpp condition for the distinct and N rotational modes, ∑ e− j2πγn determine the mesh phase suppression rule for the vibration n=1 modes of diametrically opposed planets. For odd zs, use of γ(n+N/2) = γn +1/2 from (3.22)

73 and the trigonometric relationship

sin2πγn+N/2 = sin2π (γn + 1/2) = −sin2πγn (3.38) cos2πγn+N/2 = cos2π (γn + 1/2) = −cos2πγn

N N − j2πγn gives ∑ e = ∑ (cos2πγn − isin2πγn) = 0 for any γn, and this leads to the suppres- n=1 n=1 sion of the distinct and the rotational modes.

For the planet modes to be suppressed, following conditions are required from (3.28):

N N N − j2πγn − jzsψn ∑ Gsniw∗ e = v¯psvˆps ∑ yne = v¯psvˆps ∑ yn(coszsψn − isinzsψn) = 0 n=1 n=1 n=1 (3.39) N N N − j2πγn − jzsψn ∑ Grniw∗ e = v¯prvˆpr ∑ yne = v¯prvˆpr ∑ yn(coszsψn − isinzsψn) = 0. n=1 n=1 n=1

N N N − j2πγn − j2πγn Both ∑ Gsniwe and ∑ Grniwe include ∑ yn(coszsψn − isinzsψn). Use of n=1 n=1 n=1 ψN/2+n = ψn + π condition for diametrically opposed planets gives

N N/2 ∑ yn(coszsψn − isinzsψn) = ∑ (yn ± yN/2+n)coszsψn − i(yn ± yN/2+n)sinzsψn , n=1 n=1 (3.40) where the upper sign is for even zs and the lower sign is odd for zs. It was shown in [48] that yn − yN/2+n = 0 only for N = 4. Therefore, the planet modes of a diametrically opposed planets are suppressed for N = 4 with odd zs.

From Gsniw∗ and Grniw∗ in (3.32), translational modes are suppressed when

N N N (n) − j2πγn (n) − j2πγn (n) − j2πγn ∑ v¯ts e = ∑ vˆts e = ∑ v¯tr e = 0. (3.41) n=1 n=1 n=1

74 N N N (n) − j2πγn (n) − j2πγn (n) − j2πγn (3.33) through (3.37) show that all terms in ∑ v¯ts e , ∑ vˆts e , and ∑ v¯tr e n=1 n=1 n=1 N N have both ∑ cosψn(1 ± zs) and ∑ sinψn(1 ± zs). Use of ψN/2+n = ψn + π gives n=1 n=1 N N/2 ∑ cosψn(1 ± zs) = ∑ cosψn(1 ± zs)[1 + cos(1 ± zs)π] n=1 n=1 (3.42) N N/2 ∑ sinψn(1 ± zs) = ∑ sinψn(1 ± zs)[1 + cos(1 ± zs)π]. n=1 n=1

1 + cos(1 ± zs)π becomes zero for even zs, and consequently the translational modes are suppressed.

3.3.3 Higher Harmonic Excitation

The peak amplitude of the second or higher harmonic excitation can be obtained with

(l) (l) use of corresponding lth harmonic of the mesh stiffness kˆsn and kˆrn in (3.3) and (3.4) as (l) (l) discussed in Section 3.2. Accordingly, lγn in kˆsn and kˆrn alters the mesh phase suppression rules as K = mod(zs/N) and zs are replaced with K = mod(lzs/N) and lzs for equally spaced and diametrically opposed planets, respectively.

For example, (3.14) shows the second harmonic excitation excited by the second harmonic of the mesh stiffness with l = 2. Recalling that in-phase is the only mesh phase that excites the distinct mode and the rotational mode and suppresses the rest of the vibration modes for primary resonance of equally spaced planets, one can ﬁnd additional mesh phase equivalent to in-phase for second harmonic excitation. Because the mesh phase is multi- plied by l = 2, out-of-phase by 1/2 becomes an integer, that is, in-phase. This occurs only for K = N/2 with even number of planets. For example, when the four planets (N = 4) are phased by γn = [0 2/4 0 2/4], where K = 2, the distinct and rotational modes of the second harmonic excitation are excited because the effective mesh phase with l = 2 becomes in-phase as 2γn = 2[0 2/4 0 2/4] = [0 0 0 0].

75 Different suppression rules by mesh phase is also examined for diametrically opposed

planets. For example, odd zs and l = 2 yields the following trigonometric identities:

sin2π(2γn+N/2) = sin2π (2γn + 1) = sin2π(2γn) (3.43) cos2π(2γn+N/2) = cos2π (2γn + 1) = cos2π(2γn). Unlike (3.38), the summation of the trigonometric functions over n does not vanish, and N ∑ e− j2πγn 6= 0; No suppression is achieved for the distinct and rotational modes while they n=1 are suppressed for primary resonance.

3.3.4 Superharmonic Resonance, Equally Spaced Planets

With use of Ps and Pr in (3.29), the peak amplitude of superharmonic resonance (3.16)

is expressed as

Ω (w) (w) ai = ∑ Λps κ w=1 Ω N N ! 4 1 1 fp (w) − j2πγq − j2πγq (3.44) Λps = − 2 2 4 ∑ 2 Ps ∑ Gsqwpe + Pr ∑ Grqwpe 3 ωi ζi ωwπ p=1 ωp q=1 q=1 N N (w) − j2πγn − j2πγn κ = Ps ∑ Gsniwe + Pr ∑ Grniwe . n=1 n=1

(w) (w) Because the mesh phase is included in both Λps and κ in (3.44), it becomes more complicated for one to explore the impact of the mesh phase on the superharmonic resonance than the primary resonance. To determine the suppression of superharmonic resonance

(w) (w) with mesh phase, Λps and κ need to be individually examined. (w) From the analogous expression of Λps in (3.44) to (3.29), one can ﬁnd that the mesh (w) phase suppression rules for primary resonance can be similarly applied to Λps for both

equally spaced and diametrically opposed planets. For equally spaced planets with in-

(w) phase as an example, Λps vanishes for the planet, translational, and degenerate modes w. (w) When w is corresponding to the distinct or rotational modes, Λps = 0 for K = 1,··· ,N −1.

76 (w) For the planet and translational mode w, Λps = 0 for K = 1 and N − 1 and for K 6= 1 and

N − 1, respectively.

(w) ∗ Because Λps 6= 0 with in-phase for the rotational modes w or w = w , the suppression

∗ of the vibration modes for superharmonic resonance is determined by κ(w ). It is shown in N N (w∗) Section 3.3.1 that ∑ Gsniw∗ and ∑ Grniw∗ are zero, and consequently κ = 0, for degen- n=1 n=1 erate, planet, and translational modes. As a result, superharmonic and primary resonances

share the same suppression rules for in-phase. Out-of-phase, K 6= 0, is considered from the

next paragraph.

(w) For rotational modes, Gsniw and Grniw in κ are (n) (n) (n) Gsniw = v¯ss v9i + v˜ssvˆss , Grniw = v¯srv¯sr

(n) (n) v¯ss = −v7w sinψsn + v8w cosψsn + v9w, vˆss = v3n+7,w sinαs + v3n+8,w cosαs − v3n+9,w

v˜ss = −v9i + v10,i sinαs + v11,i cosαs − v12,i

(n) v¯sr = v10,i sinαr − v11,i cosαr − v12,i, v¯sr = v3n+7,w sinαr − v3n+8,w cosαr − v3n+9,w. (3.45)

(w) Only planet and translational modes w are considered for Gsniw and Gsniw because Λps is zero for rotational mode w. With the application of the modal deﬂection properties for planet mode, Gsniw and Grniw in (3.45) for planet mode w are

Gsniw = ynv˜ss(v10,w sinαs + v11,w cosαs − v12,w), Grniw = ynv¯sr(v10,w sinαr − v11,w cosαr − v12,w). (3.46)

From the appearance of yn in both Gsniw and Gsniw, similarly for (3.31), use of (3.27) yields N N − j2πγn − j2πγn (w) ∑ Gsniwe = ∑ Grniwe = 0, and consequently κ = 0, for K = 1 and N − 1. n=1 n=1 (n) For translational mode w, with use of v9w = 0 and substitution of ψsn = ψn −αs,v ¯ss in (n) (3.45) becomes v¯ss , which is (n) ¯ v¯ss = −v7w sinψsn + v8w cosψsn = α¯ ss sinψn + βss cosψn (3.47) ¯ α¯ ss = −v7w cosαs + v8w sinαs, βss = v7w sinαs + v8w cosαs.

77 (n) (n) (n) (n) Because the expressions of v¯ss in (3.47),v ˆss , andv ¯sr in (3.45) are analogous tov ¯ts , N N N (n) (n) ¯(n) − j2πγn (n) − j2πγn (n) − j2πγn vˆts , andv ¯tr in (3.32), ∑ v¯ss e , ∑ vˆss e , and ∑ v¯sr e are cast into the n=1 n=1 n=1 similar form of (3.35) with different coefﬁcients of trigonometric functions, and they vanish for K 6= 1 and N − 1. N (n) − j2πγn Evaluating only ∑ v¯ss e for K = 1 and N − 1 gives n=1

N N N (n) − j2πγn ¯ ∑ v¯ss e = − (−βss ±iα¯ ss) = [−(v7w sinαs + v8w cosαs) ± i(v7w cosαs − v8w sinαs)], n=1 2 2 (3.48) where the upper sign is for K = 1 and the lower sign is for K = N − 1. Use of the relation for translations of central gears in translational modes [19, 26]:

T T Pz = [xz, yz, 0] and P¯ z = [−yz, xz, 0] , z = c, r, s, (3.49)

N (n) − jωγnTm gives the expression of ∑ v¯ss e for the other translational mode of the pair as n=1

N N N (n) − j2πγn ¯ ∑ v¯ss e = − (α¯ ss ± iβss) = [v7w cosαs − v8w sinαs ± i(v7w sinαs + v8w cosαs)]. n=1 2 2 (3.50) N (n) − j2πγn Real and imaginary parts of (3.48) and (3.50) are switched with the sign change. ∑ vˆss e n=1 N (n) − j2πγn and ∑ v¯sr e show the same pattern for K = 1 and N − 1. n=1 (w) (n) (n) (n) Because Λps consists of the analogous terms tov ¯ts ,v ˆts , andv ¯tr , the pattern for (3.48) (w) (w) and (3.50) can be found for Λps . Then, for the pair of translational modes w, both Λps and κ(w), which is non-zero only for K = 1 and N − 1, can be similarly expressed as

(w) T (w) T Λps = [a + bi,b − ai] , κ = [c + di,d − ci] , (3.51)

(w) (w) (w) (w) and ∑Λps κ = (a + bi)(c + di) + (b − ai)(d − ci) = 0. Although each of Λps and κ are not zero for K = 1 and N −1, sum of their product for the pair of the translational mode

78 (w) (w) w reduces to zero. In conjunction with the vanishment of Λps and κ for planet mode w,

the suppression of the rotational mode for K = 1 and N − 1 is concluded.

(w) For translational modes, Gsniw and Grniw in κ are

(n) (n) (n) Gsniw = v¯ts (v7w sinψsn − v8w cosψsn − v9w) + (v¯ts + vˆts )(v3n+7,w sinαs + v3n+8,w cosαs − v3n+9,w)

(n) Grniw = v¯tr (v3n+7,w sinαr − v3n+8,w cosαr − v3n+9,w), (3.52)

(n) (n) (n) wherev ¯ts ,v ˆts , andv ¯tr are deﬁned in (3.32). For planet mode w, use of (3.33) and (3.37) for (3.52) gives

(n) (n) h ˆ ¯ i Gsniw = (v¯ts + vˆts )ynv˜ts = (α¯ts + βts)sinψn + (αˆts − βts)cosψn ynv˜ts (3.53) (n) ¯ Grniw = v¯tr ynvˆtr = (α¯tr cosψn + βtr sinψn)ynvˆtr.

Both Gsniw and Grniw in (3.53) can be expressed with the same form of C1yn sinψn + ˆ ¯ C2yn cosψn, where C1 = (α¯ts +βts)v˜ts and C2 = (αˆts −βts)v˜ts for Gsniw, and C1 = α¯trvˆtr and ¯ (n) C2 = βtrvˆtr for Grniw. From the analogous expression of Gsniw and Grniw tov ¯ts in (3.33), N N N − j2πγn − j2πγn (n) − j2πγn both ∑ Gsniwe and ∑ Gsniwe are cast into the same form of ∑ v¯ts e n=1 n=1 n=1 ¯ in (3.35) with replacement of α¯ts and βts with C1yn and C2yn, respectively. Unlike (3.35), N N − j2πγn − j2πγn ∑ Gsniwe and ∑ Gsniwe do not vanish by any K except K = 0 because of the n=1 n=1 appearance of yn.

From (3.36) and v9w = 0 for translational mode w, (3.52) becomes

(n) (n) 2 (n) (n) Gsniw = (v¯ts + vˆts ) − v¯ts vˆts (3.54) (n) (n) Grniw = v¯tr vˆts .

79 N N − j2πγn − j2πγn With use of (3.33) and (3.37), both ∑ Gsniwe and ∑ Grniwe are expressed as n=1 n=1 the same form:

N N + − j2πγn − j2πγn α β − j2πγn ∑ Gsniwe and Grniwe = ∑ e n=1 n=1 2 | {z } s1 N − γ α β − j2πγn + ∑ sin2ψn − cos2ψn e n=1 2 2 | {z } (3.55) s2 2 2 α = (α¯ts + βˆts) − α¯tsβˆts, β = (αˆts − β¯ts) + αˆtsβ¯ts,

γ = 2(α¯ts + βˆts)(αˆts − β¯ts) − α¯tsαˆts + β¯tsβˆts for Gsniw

¯ 2 2 ¯ α = βtr, β = α¯tr, γ = 2α¯trβtr for Grniw. s1 in (3.55) vanishes from the trigonometric identities (3.24). s2 in (3.55) can be expressed in the similar form of (3.35) with the replacement of 2π(n − 1)(1 ± K)/N with

2π(n − 1)(2 ± K)/N, which leads to κ(w) = 0 for K 6= 2 and N −2 when N ≥ 4. For N = 3,

(w) (w) neither K = 1 nor K = 2 does not satisfy (3.24), and κ 6= 0. With Λps = 0 for K 6= 1 and

(w) (w) (w) (w) N −1, Λps κ becomes zero for any K when N ≥ 4. Considering Λps κ = 0 for planet

mode w with K = 1 and N − 1, it is concluded that the peak amplitude of the translational

mode vanishes at least for K = 1 and N −1 when N ≥ 4 and no suppression rule with mesh phase is found for N = 3.

3.3.5 Superharmonic Resonance, Diametrically Opposed Planets

− j2πγ − j2πγ For translational mode, the expression of Gsniwe n and Grniwe n for translational

mode w is shown in (3.55). When zs is odd, s1 in (3.55) becomes zero from (3.38). Taking

the same procedure for (3.34), one can show that s2 in (3.55) includes both cosψn(2 ± zs)

80 and sinψn(2 ± zs). Similar to (3.42), use of ψN/2+n = ψn + π gives N N/2 ∑ cosψn(2 ± zs) = ∑ cosψn(2 ± zs)[1 + cos(2 ± zs)π] n=1 n=1 (3.56) N N/2 ∑ sinψn(2 ± zs) = ∑ sinψn(2 ± zs)[1 + cos(2 ± zs)π]. n=1 n=1 (w) 1 + cos(2 ± zs)π becomes zero with odd zs, and consequently κ vanishes. For rotatinoal mode w, Gsniw and Grniw are (n) (n) (n) Gsniw = −v¯ts v9w + (v¯ts + vˆts )(v10,w sinαs + v11,w cosαs − v12,w) (3.57) (n) Grniw = v¯tr (v10,w sinαr − v11,w cosαr − v12,w). (n) (n) (n) From the appearance ofv ¯ts ,v ˆts , andv ¯tr , the suppression condition for even zs can be shown by using (3.42). For planet mode w, cosψn and sinψn are included in the expression of Gsniw and Grniw as shown in (3.53), and one can use also (3.42) to show the suppression

− j2πγ − j2πγ Gsniwe n and Grniwe n for even zs. (w) Similarly for equally spaced planets, Λps follows the suppression rules for primary resonance: odd zs for rotational mode w, even zs for translational mode w, and odd zs with

(w) (w) N = 4 for planet mode w. Combination of the suppression conditions of Λps and κ yields that the translational mode of diametrically opposed planets is suppressed for any zs with N = 4.

3.3.6 Subharmonic Resonance

Subharmonic resonance can occur if following condition is met:

2 2 2 |Πws| > ωi ζi (3.58) from (3.17), which yields non-zero ∆; Πws = 0 ensures the suppression of subharmonic resonance. With use of Ps and Pr in (3.29), Πws in (3.17) becomes P N P N s − j2πγn r − j2πγn Πws = 2 ∑ Gsniie + 2 ∑ Grniie . (3.59) 2π n=1 2π n=1 81 (w) Analogous expression of Πws in (3.59) to κ in (3.44) with w → i shows that some of

(w) κ suppression condition can be applied to Πws.

For distinct and rotational modes, Gsnii and Grnii are independent of n, and the use of the

trigonometric identities (3.24) yields Πws = 0 for K 6= 0, which leads to the suppression of

subharmonic resonance. (3.55) can be used to show the suppression of Πws for translational

mode, and the subharmonic is suppressed for K 6= 2 and N − 2 when N ≥ 4.

From (3.38), odd zs suppresses distinct and rotational modes for diametrically opposed

planets. For planet mode, the expression of Gsnii and Grnii is 2 ˆ 2 ˆ Gsnii = ynv¯psvˆps, Grnii = ynv¯prvˆpr (3.60) vˆps = v10,i sinαs + v11,i cosαs − v12,i, vˆpr = v10,i sinαr + v11,i cosαr − v12,i.

From the similarity of Gsnii and Grnii to (3.28), one can obtain analogous expressions to

2 2 (3.39) and (3.40) with yn → yn and yN/2+n → yN/2+n, which yields the suppression for odd zs with N = 4. As shown by (3.55) and (3.56), the translational mode is suppressed with odd zs.

3.3.7 Complete Suppression Conditions

Section 3.3.1 and 3.3.2 mathematically derive all the suppression rules of primary resonance using the expressions of the peak amplitude, which were predicted and numerically validated in prior studies [46,48]. As a selected case, Figure 3.3 shows the peak amplitude of rotational mode for primary resonance of example system in Table 3.1. The ring is fixed, which is common in practice to maximize the gear ratio, and the carrier rotation is constrained not to vibrate with large output inertia. An input load is applied to sun, and carrier takes an output load. Other power flows are also possible. This boundary condition and the basic system parameters given in Table 3.1 will be used for the rest of simulations unless specified. For 4 equally spaced planets in Figure 3.3(a), an excitation of the resonance only

82 Table 3.1: System parameters of an example planetary gear.

2 Sun inertia, Ms = Is/rs (kg) 1.761 2 Planet inertia, Mp = Ip/rp (kg) 2.899 Sun mass (kg) 2.269 Planet mass (kg) 1.387 Carrier mass (kg) 43.68 Maximum sun-planet mesh stiffness (N/m) 8e9 Minimum sun-planet mesh stiffness (N/m) 6e9 Maximum ring-planet mesh stiffness (N/m) 7e9 Minimum ring-planet mesh stiffness (N/m) 5e9 Bearing stiffness (N/m) 2e9

Pressure angle of sun-planet mesh, αs(deg) 20

Pressure angle of ring-planet mesh, αr(deg) 20

Contact ratio, cs = cr 1.6

Slope coefﬁcient of trapezoidal mesh stiffness, ss = sr 0.1 Modal damping ratio (%) 5

Input torque to sun Ts (N-m) 1130

for in-phase (K=0) is clear. Figure 3.3(b) conﬁrms that odd zs suppresses the rotational

mode of diametrically opposed planets. Numerical solutions match perturbation solutions

well. For numerical simulations of the 4 diametrically opposed planets, (zs + zr) = 126 is kept and ψ2 = (32/63)π is considered. Given this condition, below listed even and odd zs

83 are used to create 4 different mesh phases:

even zs : [8 40 10 38] ⇒ γ2 = [0.032 0.159 0.540 0.651] (3.61) odd zs : [9 37 11 39] ⇒ γ2 = [0.286 0.397 0.794 0.905]. As predicted, no suppression condition with even zs for diametrically opposed planets is shown in Figure 3.3(b). The variation of the peak amplitude, however, illustrates that the dynamic response is minimized at the mesh phase γ2=0.5, which demonstrates that perturbation solution provides practical gear design guidance beyond the monotonic suppression condition.

Figure 3.3(b) compares the cases with and without the assumption of the same mesh stiffness variation for each mesh. To examine the realistic mesh stiffness for diametrically opposed planets, the maximum sun-planet mesh stiffness 850e6 N/m and the maximum ring-planet mesh stiffness 750e6 N/m are used for the second and the fourth mesh instead of those given in Table 3.1. For the realistic mesh stiffness, the peak amplitude with even zs is slightly deviated from the case with the assumption and very similar variation of the peak amplitude with the mesh phase is seen for both cases; γ2=0.5 minimizes the dynamic response. For odd zs, the peak amplitude is suppressed regardless of the mesh stiffness assumption. Therefore, the suppression rules for diametrically opposed planets still hold with the assumption of the same mesh stiffness variation.

Suppression rules of superharmonic resonance with particular mesh phase are proposed in section 3.3.4 and 3.3.5. Figure 3.4, 3.5, 3.6, and 3.7 conﬁrm the suppression rules and show additional rules as well. For example, Figure 3.4(a) and Figure 3.4(b) conﬁrm the suppression of the rotational mode for K = 1 and N − 1 as predicted; however, Figure

3.4(c) and Figure 3.4(d) shows that the resonance vanishes for other K. When the number of planets is even, the rotational mode is excited for K = 0 and N/2. Only K = 0 excites the rotational mode with odd N. Numerical solutions conﬁrm the perturbation solutions.

84 Similarly, additional suppression conditions of the translational and planet modes can be

found in Figure 3.5 and Figure 3.6, respectively. For diametrically opposed planets, no

suppression condition is found except for the translational mode. Similarly for the primary

resonance, one can readily examine how the mesh phase affect dynamic response using

the perturbation solution. For example, Figure 3.7(a) and Figure 3.7(b) plot the sun rota-

tional deﬂection of the rotational mode with even zs and odd zs. Minimal vibration can be achieved near γ2=0.35 and 0.65 for even zs and γ2=0.25 and 0.75 for odd zs. The mesh phase minimizing the vibration changes with the different planetary gear systems. Figure

3.7(c) and Figure 3.7(d) shows the translational mode is suppressed for both even and odd zs when the number of planets is 4. Although Suppression of the translational mode with odd zs is analytically derived only for N = 4, Figure 3.8 illustrates that the suppression condition with odd zs extends for N > 4. That is, it is concluded that the translational mode

is not excited for superharmonic resonance. The complete suppression conditions for the

superharmonic resonance are summarized in Table 3.2.

Figure 3.9 shows the waterfall plots of the planet rotation for 4 equally spaced planets.

Planets are in-phase and, as predicted in section 3.3.6, the presence of subharmonic reso-

nance for rotational mode (ωi = 1792Hz) is conﬁrmed in Figure 3.9(a). When the damping is increased to 5%, however, subharmonic resonance is disappeared in Figure 3.10(b) be-

2 2 2 cause |Πws| becomes less than ωi ζi , and the condition (3.58) is not satisﬁed. The combined effect of damping and multi-parameters will be further discussed later. In order to exclude the effect of damping, ∆˜ is introduced as q 2 ∆˜ = 4 |Πws| /ωi. (3.62)

The variation of ∆˜ for rotational mode with mesh phase is plotted in Figure 3.10. Suppres-

sion condition of K 6= 0 for equally spaced planets is clearly illustrated in Figure 3.10(a).

85 Figure 3.10(b) conﬁrms that odd zs suppresses subharmonic resonance of diametrically opposed planets. Similarly for Figure 3.3(b), Figure 3.10(b) shows ∆˜ with even zs is minimized at γ2 = 0.5. Table 3.3 lists the complete suppression condition of ∆˜ . Again, subharmonic can be suppressed even with the conditions other than those in Table 3.3 due to the existence of damping.

Table 3.2: Suppression conditions for superharmonic resonance

Rot-Translational system Rotational system

Planet Rotational Translational Degenerate Distinct

K=0, N/2 for K6= 0, N/2 for all for even N K6= 0, N/2 for even N Equally even N even N spaced K=0 planets K=0, (N±1)/2 K6= 0 for odd K6=(N±1)/2 K6= 0 for odd N for odd N N for odd N≥5

Diametrically opposed no no all no no planets

Table 3.3: Suppression conditions for subharmonic resonance (∆˜ )

Rot-Translational system Rotational system

Planet Rotational Translational Degenerate Distinct

Equally K6=0 for N=4, K6= 0 spaced K=2,3 for N=5, K6= 0 K6=(N±1)/2 for N ≥4 K=0 planets K=3 for N=6

Diametrically opposed odd zs for N = 4 odd zs odd zs no odd zs planets

86 25

20 m µ 15

10 Planet rotation,

0 0 1 2 3 K (a)

25 m µ 20

15 Sun rotation,

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mesh phase, γ 2 (b)

Figure 3.3: Variation of the peak amplitude of primary resonance with mesh phase. Rota- tional mode for four (a) equally spaced planets and (b) diametrically opposed planets for the example system in Table 3.1. For (a), ( • ) perturbation and (- -- -) numerical solutions. For (b), perturbation solutions for ( ) even and (− − −) odd zs, and numerical ˜ ˜ ˜ ˜ solutions for () even and (•) odd zs under the assumption of ksn = ksm and krn = krm; perturbation solutions for (-.-) even and (···) odd zs, and numerical solutions for () even and ( ) odd zs under the assumption of k˜sn 6= k˜sm and k˜rn 6= k˜rm.

87 6 4

3.5 5

3 4 m

m 2.5 µ µ

3 2

Sun rotation, 1.5 2 Sun rotation,

1 1 0.5

0 0 1 2 0 0 1 2 3 K K (a) (b)

3 2.5

2.5 2

2 m m

µ µ 1.5

1.5

1 Sun rotation, Sun rotation, 1

0.5 0.5

0 0 0 1 2 3 4 0 1 2 3 4 5 K K (c) (d)

Figure 3.4: Variation of the peak sun rotation vibration amplitude with mesh phase for equally spaced planets. Rotational mode of superharmonic resonance for the example system in Table 3.1 with (a) N = 3, (b) N = 4, (c) N = 5, and(d) N = 6. ( • ) Perturbation and (- -- -) numerical solutions are compared.

88 0.2 0.2 m m µ µ

0.1 0.1 Sun translation, Sun translation,

0 0 1 2 0 0 1 2 3 K K (a) (b)

0.1 0.1 m m µ µ

0.05 0.05 Sun translation, Sun translation,

0 0 0 1 2 3 4 0 1 2 3 4 5 K K (c) (d)

0.1 0.1 m m µ µ

0.05 0.05 Sun translation, Sun translation,

0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 K K (e) (f)

Figure 3.5: Variation of the peak sun translation vibration amplitude with mesh phase for equally spaced planets. Translational mode of superharmonic resonance for the example system in Table 3.1 with (a) N = 3, (b) N = 4, (c) N = 5, (d) N = 6, (e) N = 7, and (f) N = 8. ( • ) Perturbation and (- -- -) numerical solutions are compared.

89 0.35 0.25

0.3

0.2

0.25 m m µ µ 0.2 0.15

0.15 0.1 Planet rotation, Planet rotation, 0.1

0.05 0.05

0 0 0 1 2 3 0 1 2 3 4 K K (a) (b)

0.2 0.2

0.15 0.15 m m µ µ

0.1 0.1 Planet rotation, Planet rotation,

0.05 0.05

0 0 0 1 2 3 4 5 0 1 2 3 4 5 6 K K (c) (d)

Figure 3.6: Variation of the peak planet rotation vibration amplitude with mesh phase for equally spaced planets. Planet mode of superharmonic resonance for the example system in Table 3.1 with (a) N = 4, (b) N = 5, (c) N = 6, and(d) N = 7. ( • ) Perturbation and (- -- -) numerical solutions are compared.

90 2.5 1.5

1 m m

µ 1.5 µ

1 Sun rotation, Sun rotation, 0.5

0.5

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mesh phase, γ Mesh phase, γ 2 2 (a) (b)

1 1 m m µ µ Planet rotation, Planet rotation,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mesh phase, γ Mesh phase, γ 2 2 (c) (d)

Figure 3.7: Variation of the peak amplitude with mesh phase for four diametrically opposed planets. (a), (b) Rotational and (c), (d) translational modes of superharmonic resonance for the example system in Table 3.1 with (a), (c) even and (b), (d) odd zs.( • ) Perturbation and () numerical solutions are compared.

91 1 m µ

0 Sun translation, -1 1

0.5 γ3 1

0.5 0 γ 0 2 (a)

1 m µ

0 Sun translation, -1 1

0.5 γ3 1

0.5 0 0 γ2

(b)

Figure 3.8: Suppressed sun translation vibration for (a) six and (b) eight diametrically opposed planets with odd zs. System parameters are from the example system in Table 3.1. γ4 = 0.5 is used for (b).

92 25

20 m µ 15

Planet rotation, 5

0 3700 2 3600 1.5 Mesh frequency, Hz 3500 1 0.5 3400 0 Mesh frequency harmonics (a)

20 m µ 15

Planet rotation, 5

0 3700 2 3600 1.5 Mesh frequency, Hz 3500 1 0.5 3400 0 Mesh frequency harmonics (b)

Figure 3.9: Waterfall plots of planet rotation for four equally spaced planets of the example system in Table 3.1. Meshes are in-phase. Modal damping of (a) 3% and (b) 5% are used.

93 300

250

200 ~ ∆ 150

100

0 0 1 2 3 K (a)

300

250

200

150

∆ ~

100

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mesh phase, γ 2 (b)

Figure 3.10: The effects of mesh phase on ∆˜ for four (a) equally spaced and (b) diametrically opposed planets of the example system in Table 3.1. (- - -) odd and (—) even zs in (b)

94 3.4 Contact Ratio Effect

Contact ratio depends on many other design parameters, including the outer diameter,

pressure angle, number of teeth, center distance, etc. Its importance for dynamic response

has been shown in past studies, as mentioned earlier. The appearance of contact ratios in

the expressions for the peak amplitudes in (3.10) and (3.16) and the instability width ∆ for

subharmonic resonance in (3.17) and (3.18) conﬁrm that contact ratio affects the planetary

gear dynamics.

The two contact ratios cs and cr are associated with their own mesh related parameters.

Each one may impact the dynamics differently. In other words, cs may affect the dynamic behavior more signiﬁcantly than cr or vice versa. Because the modal deﬂections determine the values of Gsniw and Grniw, which are linked to cs and cr, respectively, one can expect that the effects of contact ratios on dynamic response vary for each vibration mode.

Figure 3.11 illustrates the different effects of contact ratios at the different vibration modes. The three planets are equally spaced and the meshes are in-phase. Other system parameters are listed in Table 3.1. For the first rotational mode in Figure 3.11(a), the sun rotation peak amplitude is more sensitive to cr than cs; the variation of the peak amplitude with cs is small compared to the variation with cr. The opposite pattern of sensitivity is observed for the second rotational mode in Figure 3.11(b) where cr has less impact than cs. This different sensitivity of the peak amplitude to contact ratios and to the different vibration modes is similarly examined for the purely rotational model in Figures 3.11(c) and 3.11(d). The distinct modes behave similarly to the rotational modes. The peak amplitude changes with cr more significantly than cs for the first distinct mode in Figure 3.11(c), while cs has stronger influence than cr for the second distinct mode in Figure 3.11(d).

95 The peak amplitude in (3.10) is re-written as 1 apeak = ϒ¯ sin[ (c − s )] + ϒ¯ sin[ (c − s )] i 2 2 sn π s s rn π r r ωi ζiπ Ψ N f k˜ sin(πs ) Ψ N f k˜ sin(πs ) ¯ w s s − jωγnT ¯ w r r − jω(γn+γsr)T ϒsn = ∑ ∑ 2 Gsniw e ϒrn = ∑ ∑ 2 Grniw e . w=1 n=1 ωw ss w=1 n=1 ωw sr (3.63)

The relative impact of contact ratios cs and cr on dynamic response is affected by the weighting factors ϒ¯ sn and ϒ¯ rn. The peak amplitude changes with cs more signiﬁcantly for Ψ N ¯ ¯ fw larger ϒsn, which is also true for cr and ϒrn. Analytical expressions for ϒsn = ∑ ∑ 2 Gsniw w=1 n=1 ωw Ψ N fw and ϒrn = ∑ ∑ 2 Grniw are derived and compared for the rotational model to explore w=1 n=1 ωw how cs and cr affect the response differently. System parameters related with the mesh

stiffness and mesh phase in ϒ¯ sn and ϒ¯ rn are excluded to focus on the effects of contact

ratios. The combined effects of contact ratios and other system parameters are discussed

later.

The same power ﬂow condition described in section 3.3.7 is considered for the purely

rotational model with N + 1 degrees of freedom. ϒsn and ϒrn for the ﬁrst distinct mode

(i = 1) are " # f1 2 fN+1 ϒsn = N 2 (v11 + v21) + 2 (v11 + v21)(v1,N+1 + v2,N+1) ω1 ωN+1 " # (3.64) f1 2 fN+1 ϒrn = N 2 v21 + 2 v21v2,N+1 . ω1 ωN+1

where vxy is the (x,y) element of V in Appendix B. When only the central gears are loaded,

f1 and fN+1 are

f1 = v11 f¯1, fN+1 = v1,N+1 f¯1. (3.65)

With use of (3.65), (B.2), and (B.7) in Appendix B, (3.64) is re-written as

ϒsn = N f¯1 [αv1,N+1v21v2,N+1]β1 ϒrn = N f¯1 [αv1,N+1v21v2,N+1] 2 2 2 ω − ω k¯ k¯r − ω Mp (3.66) α = 1 N+1 β = r 1 . 2 2 1 ¯ ¯ ω1 ωN+1 ks ks 96 In a similar process, ϒsn and ϒrn for the second distinct mode (i = N + 1) are ¯ 2 ¯ 2 ϒsn = N f1 αv1,N+1v2,N+1 βN+1 ϒrn = N f1 αv1,N+1v2,N+1 ¯ ¯ 2 (3.67) kr kr − ωN+1Mp βN+1 = . k¯s k¯s

Equations (3.66) and (3.67) show that ϒsn and ϒrn are differentiated by β1 and βN+1, that is, β1 and βN+1 determine which contact ratio, cs or cr, more strongly affects dynamic response. If |β1| and |βN+1| are greater than 1, cs has a stronger impact than cr and vice versa.

2 ¯ ¯ With ωi = λi from (B.4) in Appendix B and the parameters m = Ms/Mp and k = ks/kr,

β1 and βN+1 become

1 q β , β = −kN − m(k − 1) ± N2k2 + 2Nmk(k − 1) + m2 (k + 1)2 , (3.68) 1 N+1 2mk2

where the upper sign in ± is for β1 and the lower sign is for βN+1; this sign convention applies throughout what follows. The relative impact of cs and cr therefore depends on the two ratios of m and k.

For k = 1 (k¯s = k¯r), (3.68) reduces to

1 p β , β = −N ± N2 + 4m2 . (3.69) 1 N+1 2m

For any m, |β1| < 1 and |βN+1| > 1, which shows that the impact of cr dominates over cs for the ﬁrst distinct mode and vice versa for the second distinct mode.

To examine β1 and βN+1 for parameters close to m = 1 and k = 1, small parameters

ε 1 and δ 1 are introduced as

m = 1 + ε k = 1 + δ. (3.70)

97 With substitution of (3.70) into (3.68) and retaining up to the order of ε and δ, (3.68) is

approximated as

1 n p p h p i o β , β ≈ −N ± N2 + 4 + N ± 2 ∓ N2 + 4 ε + N − 1 ± (N + 1) ∓ 2 N2 + 4 δ . 1 N+1 2 (3.71)

The coefficient of ε is always positive, so both β1 and βN+1 increase for larger m. For m = 1 and k = 1, 0 < β1 < 1 and βN+1 < −1 for any positve integer N. Thus, the effect of cs increases relative to cr as Ms increases relative to Mp for the first distinct mode and vice versa for the second distinct mode. The coefficient of δ is negative for β1 and positive for

βN+1. The effect of cs decreases relative to cr as k¯s increases relative to k¯r for both distinct modes.

Figure 3.12 shows the variation of β1 and βN+1 from (3.68) with m and k for selected N.

The predictions from (3.69) and (3.71) are conﬁrmed, and they are valid for different values of m and k. The contours of β1 = 1 and βN+1 = −1 delineate the boundaries where the relative impacts of cs and cr change for the two distinct modes. For most of the parameter ranges shown, |β1| < 1 so cr is more important than cs for the ﬁrst distinct mode, while

|βN+1| > 1 indicates greater importance of cs. β1 > 1 is found only for k < 1 with both

N = 3 and 8. The impact of cs is more dominant than cr for the ﬁrst distinct mode when k¯s < k¯r, while cr has a stronger impact than cs when k¯s > k¯r. For ﬁxed m and k, a slight decrease of β1 occurs for larger N. Larger |β1| and smaller |βN+1| are calculated for larger m, which indicates the opposing effects of m on the impact of contact ratios for the two distinct modes. The effect of m becomes small for k > 1, however. Unlike β1, larger |βN+1| occurs for more planets and smaller m.

Different impact of cs and cr is examined for the ﬁrst and second distinct modes in

Figures 3.11(c) and 3.11(d). With use of k = 1.167 and m = 0.6076 for the example system

98 in Table 3.1 with N = 3, (3.68) calculates β1 = 0.1401 and βN+1 = −4.495. |β1| < 1 and

|βN+1| > 1 means the stronger impact of cr than cs for the ﬁrst distinct mode and stronger

impact of cs than cr for the second distinct mode, which conﬁrms the opposite effects of contact ratios shown in Figures 3.11(c) and 3.11(d).

Figure 3.13 plots the variation of the peak amplitude for the second distinct mode in

Figure 3.11(d), where cr is ﬁxed at 1.5. The perturbation and numerical solutions agree well. For the trapezoidal stiffness, the peak amplitude is minimized near cs = 1.1 and maximized near cs = 1.55. This can be explained analytically. Considering (3.13), the peak amplitude is proportional to |Λp|, where Λp is, after use of (3.67),

N+1 N fw Λp = [βN+1 sin(cs − ss)π + sin(cr − sr)π] ∑ ∑ 2 Grniw. (3.72) w=1 n=1 ωw

With cr = 1.5 and ss = sr = 0.1 for the example system in Table 3.1, (3.72) becomes

N+1 N fw Λp = [βN+1 sin(cs − 0.1)π + sin1.4π] ∑ ∑ 2 Grniw. (3.73) w=1 n=1 ωw

As cs increases from 1 to 1.1, positive sin(cs − 0.1)π decreases and so does |Λp| because

both βN+1 = −4.495 and sin1.4π are negative. When cs > 1.1, βN+1 sin(cs − 0.1)π becomes positive and |Λp| vanishes when

βN+1 sin(cs − 0.1)π + sin1.4π = 0 (3.74) or cs = 1.12 in this case. Similarly, the bracketed expression in (3.73) is maximized at cs = 1.6. In fact, |Λp| is maximized at a slightly lower cs because βN+1 varies slightly with

cs.

Figure 3.13 also shows the response of the system with the rectangular mesh stiffness

where ss = sr = 0. Because ss reduces from 0.1 to 0, the maximum and minimum of

|Λp| and consequently the peak amplitude, occur at smaller cs by roughly 0.1. Thus, the

99 variation of the peak amplitude is shifted to the left compared to the case with trapezoidal

mesh stiffness.

Like the peak amplitude above, different sensitivity of the instability width, which is

the frequency interval where parametric instability occurs, to the sun-planet and ring-planet

parameters is expected because they are associated with different terms in |Πws| in (3.18)

as N ˜ ˜ 1 ks N ks 2 ∑ 2 sin(cs − ss)π sin(ssπ)Gsnii = 2 sin(cs − ss)π sin(ssπ)v2i γi n=1 2π ss 2π ss N ˜ ˜ 1 kr N kr 2 (3.75) ∑ 2 sin(cr − sr)π sin(srπ)Grnii = 2 sin(cr − sr)π sin(srπ)v2i n=1 2π sr 2π sr 2 ¯ ¯ 2 2 γi = ωi Mp − kr /ks = (kβi) , i = 1, N + 1.

In this case, like βi, γi determines which contact ratio more strongly affects the instability

2 width (and so the response amplitude). Because γi = (kβi) for i = 1 and N + 1, the trends for the dependence of γi on m and N are as discussed above in relation to βi for primary

resonance. The variation of γ1 and γN+1 with m and k is plotted in Figure 3.14 for N = 3

and 8. Similar to β1, γ1 decreases as k increases. γ1 < 1 for most cases, which indicates

that the instability width is more sensitive to cr for the ﬁrst distinct mode. γN+1 increases

as k grows but at a slower rate for large k. γN+1 > 1 is observed for all the cases, showing

greater sensitivity to cs than cr for the second distinct mode. For ﬁxed m and k, γ1 decreases

and γN+1 increases for larger N, which indicates opposite effects of N on the sensitivity of

the instability width to cs and cr for two distinct modes.

Figure 3.15 shows the impact of contact ratio on subharmonic resonance. Different

sensitivity of the instability width to cs and cr (as derived above) is clearly evident for the

different vibration modes. For the example system in Table 3.1 with N = 3, γ1 = 0.0267 < 1

and γN+1 = 27.50 > 1. As predicted from these values of γ1 and γN+1, the impact of cr

on the width is dominant for the ﬁrst distinct mode in Figure 3.15(a) while cs has much

100 stronger inﬂuence for the second distinct mode in Figure 3.15(b). The maximum width is calculated near contact ratio 1.5 for both cs and cr near 1.5, which can be derived from

(3.75). The widths vanish for certain ranges of cs and cr, which means that the parametric instability does not occur. Comparison of Figures 3.15(a) and 3.15(b) shows that certain combinations of cs and cr yield zero width for both modes, a deriable result.

The effects of contact ratio are affected by other parameters such as ﬂuctuation of mesh stiffness and damping, and the combined effect will be discussed later.

101 2 2 15 15 15 20 20 5 5 25 1 25 1 1 1.8 25 1.8 3 3 3 30 30

35 1

1.6 1.6 1

1 3

5 r r C C

1 3 1 1.4 1.4 35 3 3 30 30 25 5 25 1 25 1 20 20 1 1.2 1.2 15 15 5 7 5 10 10 10 15 15 7 1 1 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 Cs Cs (a) (b)

2 2 10 10 15 15 14

20 2 8 20 8

20 4 12

4 25 12 1.8 1.8 10 25 2 10 6 30 6 30 30 2 1.6 1.6

8 2 35 8 r

r 35 C C 12

4 1.4 30 1.4 30 2

10 12 25 25 4 10 25 8 6 20 20 2 6 1.2 15 15 14 15 1.2 10 8 14 10 10 10 15 1 1 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 Cs Cs (c) (d)

Figure 3.11: The effects of contact ratios on the peak sun rotation vibration amplitude for three equally spaced planets for the example system in Table 3.1. Meshes are in-phase. Primary resonance of (a) the ﬁrst and (b) the second rotational modes for a rotational- translational model and of (c) the ﬁrst and (d) second distinct modes for a rotational model.

102 3 3

0.1

2 0.1 2

k 0.1

0.1 0.1

1 1

1 1 1 2 0.5 2 0.5 3 2 3 0 1 2 3 4 m (a) 3 3 -1 -2

-1 -10

2 2

-10

k -1 -2 -2 -5

-5 1 1 -2

-10 -5 -10 -5 0.5 0.5

0 1 2 3 4 m (b)

Figure 3.12: Contour plot of (a) β1 for the ﬁrst distinct mode and (b) βN+1 for the second distinct mode varying with m and k. (—) N = 3 and (- - -) N = 8 are used for the example system in Table 3.1.

103 14

10 m µ 8

6 Sun rotation,

0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Cs

Figure 3.13: Variation of the peak sun rotation vibration amplitude with cs for primary resonance of the second distinct mode for the example system in Table 3.1. cr is ﬁxed at 1.5. Perturbation solutions for (—) trapezoidal and (- - -) rectangular mesh stiffness variation. Lines with symbols are from numerical solutions.

104 3

0.01

2 0.1

k 0.01 0.1

0.1 0.1 1 0.01

0.1 1 1 0.5 1 3 0 1 2 3 4 m (a) 3

2 10

100 k

100

10 10 20

1 5

1000

2 0.5 20 20 5 2 5 1.5 0 1 2 3 4 m (b)

Figure 3.14: Contour plot of (a) γ1 for the ﬁrst distinct mode and (b) γN+1 for the second distinct mode varying with m and k. (—) N = 3 and (- - -) N = 8 are used for the example system in Table 3.1.

105 350

300

250

200

150

100

Width of instability, Hz 50

0 2 1.8 2 1.6 1.8 1.4 1.6 1.4 Cr 1.2 1.2 Cs 1 1 (a)

900 800 700 600 500 400 300 200 Width of instability, Hz 100 0 2 1.8 2 1.6 1.8 1.4 1.6 1.4 Cr 1.2 1.2 Cs 1 1 (b)

Figure 3.15: The effects of contact ratios on the width of the parametric instability boundary. (a) The ﬁrst and (b) the second distinct modes for three equally spaced planets with in-phase meshes for the example system in Table 3.1

106 3.5 Fluctuation of Mesh Stiffness Effect

Equation (3.10) pedicts the linear relationship between the peak amplitude and the ﬂuc- tuation of mesh stiffness shown in Figure 3.16. Figure 3.16(a) shows that the planet translational deﬂection is linearly proportional to both k˜r and k˜s, but the impact of k˜s is smaller than k˜r. Like for contact ratio, the impact of k˜s and k˜r on dynamic response is sensitive to the vibration modes. Unlike Figure 3.16(a), the peak amplitude decreases to a minimum as k˜s increases for the second rotational mode in Figure 3.16(b). The minimum of the peak amplitude occurs for larger k˜s as k˜r increases. For the second rotational mode in Figure

3.16(b), when k˜s is ﬁxed at 50e6 N/m, the response increases for larger k˜r. For k˜s = 250e6

N/m in Figure 3.16(b), however, smaller response can occur for larger k˜r. This ﬁnding is opposed to the common view that mesh stiffness ﬂuctuation acts as an excitation source such that larger k˜s and k˜r increase the dynamic response.

Figure 3.17 illustrates how the response amplitude varies with mesh frequency for fixed mesh stiffness fluctuations and with the mesh stiffness fluctuation k˜r for a fixed excitation frequency ω = ωi + ωo f f , where ωo f f represents the deviation from the natural frequency,

ωi = 1792 Hz. From Figure 3.17(a), multiple response amplitudes are possible at ω =

1642 Hz: two stable solutions A1 and A3 and one unstable solution A2. Only one stable solution occurs for ωo f f = 0 Hz and ωo f f = 150 Hz. Keeping the mesh frequency and mean ring-planet mesh stiffness k¯r ﬁxed, Figure 3.17(b) shows the effects of different ring mesh stiffness ﬂuctuation. Larger response occurs for larger k˜r. Similar to the speed sweep simulation in Figure 3.17(a), jump phenomenon is observed for the k˜r sweep in Figure

3.17(b). For a ﬁxed ω = 1642 Hz with ωo f f = −150 Hz as an example, the response increases gradually until k˜r reaches 237e6 N/m. The response jumps from 19 µm to 49

µm when k˜r exceeds 237e6 N/m. For a down-sweep of k˜r, the response jumps down from

107 38 µm to 13 µm when k˜r < 170e6 N/m. For a geared system, which shows a softening nonlinearity due to gear tooth contact loss, the jump phenomenon related with the mesh stiffness fluctuation occurs for ω < ωi, or ωo f f < 0. This jump phenomena demonstrate that a slight change of the mesh stiffness fluctuation can lead to a significant decrease or increase of the dynamic response for a fixed speed.

108 8

6 m µ 5

3 Planet translation,

0 0 50 100 150 200 250 300 ~ kr , 1e6 N/m (a)

2.5

m 2 µ

1.5

1 Planet translation,

0.5

0 0 50 100 150 200 250 300 ~ ks , 1e6 N/m (b)

Figure 3.16: Variation of the peak planet translation vibration amplitude with (a) k˜r for the ﬁrst rotational mode and (b) k˜s for the second rotational mode. There are four equally spaced planets with in-phase meshes for the example system in Table 3.1. For (a), k˜s is ( ) 100e6 N/m, (− − −) 200e6 Nm, and (···) 300e6 Nm. For (b), k˜r is ( ) 100e6 N/m, (− − −) 200e6 Nm, and (···) 300e6 Nm.

109 60

ω = -150 Hz off ω = 1792 Hz ω = 150 Hz 50 i off A1

40 m µ

30 A2 Sun rotation, 20 A3

0 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 Mesh frequency, Hz (a)

ωoff = -150 Hz

50 = -100 Hz A1 ωoff

40 m µ

ωoff = 0 Hz 30 A2 ω = 100 Hz off Sun rotation, 20

ωoff = 150 Hz 10

0 0 50 100 150 200 250 300 ~ kr , 1e6 N/m (b)

Figure 3.17: Root mean square (mean removed) of sun rotation for primary resonance of the ﬁrst rotational mode. There are four equally spaced planets with in-phase meshes for the example system in Table 3.1. The damping ratio is 3%. k˜s is ﬁxed at 200e6 N/m in (b). Unstable solutions are shown as dashed lines in both (a) and (b). In (b), dashed and solid lines are for ωo f f ≤ 0 and dotted lines are for ωo f f ≥ 0

110 3.6 Damping Effect

The expression of the peak amplitude (3.8) shows that the dynamic response is inversely proportional to damping, ai ∝ 1/ζi, and Figure 3.18(a) shows that higher damping reduces the peak amplitude, which conﬁrms the common thinking about the damping effect. For subharmonic resonance, higher damping reduces the terms of the radical sign in (3.17), and consequently the width of the parametric instability boundary is decreased. Re-writing

(3.17) gives 2 2 2 2 |Πws| ∆ + 16ωi ζi = 16 . (3.76) ωi

(3.76) plots an ellipse curve of ∆ and ζi, and the maximum value of the width and damping is calculated as

4|Πws| |Πws| ∆max = , ζimax = 2 . (3.77) ωi ωi 2 ∆max is inversely proportional to ωi while ζimax is inversely proportional to ωi . Figure 3.18(b) shows the damping effect on the width for the distinct modes. The ellipse plot is clear from (3.76), and the instability width rapidly drops for higher damping. Negligible impact of the number of planets on the first distinct mode is examined while the change of the number of planets makes noticeable difference of the instability width for the second distinct mode. The number of planets does not change the maximum damping for both modes where the resonance is completely suppressed. The second distinct mode has the wider maximum width and the smaller maximum damping than those for the first distinct mode. Considering that the maximum width is inversely proportional to ωi from (3.77), one would predict the narrower maximum width for the second distinct mode than the one for the first distinct mode. Figure 3.18(b), however, contradicts the foregoing prediction. This is because other parameters such as the fluctuation of mesh stiffness and modal deflections

111 in |Πws| also affect the response. The combined effects of multiple parameters including damping, contact ratio, mesh stiffnesses, and number of planets will be discussed in the following section.

112 110

100

80 m µ

Planet rotation, 50

20 1 2 3 4 5 Damping ratio, % (a)

1500

1000

500 Width of the instability, Hz

0 0 1 2 3 4 5 6 Damping ratio,% (b)

Figure 3.18: Variation of (a) the peak planet rotation vibration amplitude of primary resonance for the ﬁrst rotational mode and (b) the width of the parametric instability boundary for the distinct mode with in-phase meshes for the example system in Table 3.1. For (b), the ﬁrst distinct mode with (—) N = 4 and ( ) N = 6 and the second distinct mode with (- - -) N = 4 and (− − −) N = 6 are compared.

113 3.7 Number of Planets Effect

Bahk and Parker [86] analytically discover that the applied torque does not change the

width of the parametric instability by showing the absence of the forcing terms in the ex-

pression for the width. Instead, other parameters such as the number of planets N determine

the width.

The maximum width of the parametric instability region is calculated by ∆max = 4|Πws|/ωi from (3.77). With the assumption of k˜ = k˜s = k˜r, c = cs = cr, and s = ss = sr, use of the 2 ¯ ¯ ¯ relationship v1i = v2i ωi Mp − kr + ks /ks in Appendix B gives ∆max for the ﬁrst distinct mode as " 2 2 2 # sinπ(c − s)sinπs ω Mp − k¯r + k¯ ∆ = 1 s 2Nkv˜ 2 . (3.78) max 2 ¯2 21 π s ω1ks

In-phase, γsn = γrn = 0, is considered as out-of-phase suppresses the subharmonic reso-

2 nance. After replacing ω1 and v21 with the known parameters using (B.4) and (B.6) in

Appendix B, taking a derivative of ∆max with respect to N gives

˜ v 2 ∂∆max sinπ(c − s)sinπs kB1 u 32(Nk) = u , (3.79) ∂N π2s C2 t h √ i3 1 Msk¯r Nk + m(k + 1) − A

where m = Ms/Mp and k = k¯s/k¯r. Refer to Appendix C for A, B1, and C1. Similarly,

∂∆max/∂N for the second distinct mode is

˜ v 2 ∂∆max sinπ(c − s)sinπs kBN+1 u 32(Nk) = u . (3.80) ∂N π2s C2 t h √ i3 N+1 Msk¯r Nk + m(k + 1) + A

The variation of ∂∆max/∂N is plotted in Figure 3.19. For the ﬁrst distinct mode in Figure

3.19(a) and Figure 3.19(c), ∂∆max/∂N is positive for most cases, which means that more

number of planets increase the instability region. Negative ∂∆max/∂N is calculated when m < 1.2 and k < 0.5 for N = 3 and m < 3.2 and k < 0.5 for N = 8. The range of m where

114 ∂∆max/∂N is negative becomes wider for N = 8 than N = 3. The variation of ∂∆max/∂N

with k is small for k > 1, while there is signiﬁcant variation for k < 1. For k < 1, ∂∆max/∂N is minimized at certain m, which depends on N, while ∂∆max/∂N continuously increases with large m for k > 1. For the second distinct mode in Figure 3.19(b) and Figure 3.19(d), positive ∂∆max/∂N is observed for all combinations of m and k. Unlike the ﬁrst distinct mode, ∂∆max/∂N decreases with m for all k. Overall, larger ∂∆max/∂N is calculated for

the second distinct mode except for m > 2 and N = 3. This implies the stronger impact of the number of planets on the subharmonic resonance for the second distinct mode than for the ﬁrst distinct mode.

115 3 3

200 100

100

500 200

1000 2 2 230 100

100 200

500 k k 50 200

230

1 1 200 200

100 230 2000 500 1000 230 0.5 0 0 0.5 -100-200 1 2 3 4 1 2 3 4 m m (a) (b)

3 3

100

200

500 40

20 2 2 k k

1000

500 200

1 1

60 40

20 2000 1000 0.5 0 0 0.5 500 -60 -60 -120 1 2 3 4 1 2 3 4 m m (c) (d)

Figure 3.19: Variation of ∂δ/∂N with m and k for ((a) and (c)) the ﬁrst and ((b) and (d)) the second distinct modes for the example system in Table 3.1. (a) and (b) for N = 3, (c) and (d) for N = 8.

116 3.8 Combined Effects of Multiple Parameters

Appearance of system parameters in the perturbation solutions indicates that the pa-

rameters affect the resonant response as they interact each other at the same time. In this

section, combined effects of multiple parameters are discussed to explore the relationship

between multiple system parameters and dynamic response beyond the single parameter

effect.

3.8.1 Combined Effect of Number of Planets and Mesh Stiffness Fluc- tuation

Wider instability width of subharmonic resonance with more number of planets is predicted from ∂∆/∂N > 0 for the most cases in section 3.7. Larger ∂∆/∂N indicates

more signiﬁcant change of the width with the number of planets for the second distinct

mode than the ﬁrst distinct mode. Keeping the mean mesh stiffness the same, the in-

stability width with different k˜s and k˜r are plotted in Figure 3.20. For m = 0.6067 and

k = 1.1667, γ1 = 0.0157 < 1 and γN+1 = 46.9395 > 1 for N = 4 and γ1 = 0.0072 < 1

and γN+1 = 102.0629 > 1 for N = 6 are calculated; one can expect that the change of k˜r

dominates over k˜s for the ﬁrst distinct mode and vice versa for the second distinct mode.

Compared to Figure 3.18(b), Figure 3.20(a) shows that half reduced k˜s = 100e6 N/m con-

siderably reduce both the maximum width and damping only for the second distinct mode

while the variation of the width with damping ratio for the ﬁrst distinct mode changes neg-

ligibly. Similarly, signiﬁcant reduction of the maximum width and damping is seen for the

ﬁrst distinct mode in Figure 3.20(a) where k˜r is reduced by half, 100e6 N/m. As the small

variation of γ1 and γN+1 with N is predicted in Figure 3.14, the different parameter effects

depending on the modes is seen for both N = 4 and 6.

117 3.8.2 Combined Effect of Damping with Contact Ratio and Mesh phase

As discussed in section 3.6, the peak amplitude is inversely proportional to damping

without being affected by any other system parameters. The width of the parametric insta-

bility region is plotted as a function of contact ratio and damping in Figure 3.21. c = cs = cr is considered. As damping increases, the range of contact ratio that excites the subharmonic resonance narrows down to c ≈ 1.6. After certain damping value, here around 3.7%, no parametric instability occurs for any contact ratio. Figure 3.22 illustrates the combined effect of mesh phase and damping on subharmonic resonance for the ﬁrst rotational mode of 6 diametrically opposed planets. Two different damping values are compared: 1% in

Figure 3.22(a) and 3% in Figure 3.22(b). Even zs is considered as subharmonic resonance vanishes for an odd zs. Dominant impact of γ2 over γ3 is clear. The maximum width oc-

curs when γ2 is either 0 or 1 regardless of γ3; γ2 = 0.5 minimizes the instability region.

Higher damping reduces the width and simultaneously the range of mesh phase causing the parametric instability is decreased. The ratio of the area of mesh phase range where the parametric instability occurs (non-zero width) to the total area of both γ2 and γ3 of

[0 1], Amp, is introduced and plotted in Figure 3.23. For the ﬁrst rotational mode , Amp remains 1 until damping ratio reaches around 1.2%; subharmonic resonance does not vanish with any of mesh phase γ2 and γ3, and the minimum width becomes zero for γ2 = 0.5

when damping is around 1.2 %. For damping ratio > 1.2%, Amp gradually decreases as

damping increases. Damping ratio higher than 3.7% suppresses subharmonic resonance

for any mesh phase. Subharmonic resonance for the second rotational mode is more easily

suppressed by damping ratio, here 0.8%.

118 3.8.3 Combined Effect of Contact Ratio and Fluctuation of Mesh Stiff- ness

Figure 3.24 illustrates the effect of different k˜r. For larger k˜r, the ring-planet mesh term

where cr is included in Λpp of (3.13) also gets large, and consequently the impact of cr

becomes strong. So, as can be seen from Figure 3.24(a) and 3.24(b), one can see more

variation of the response with cr for k˜r=0.3e9 N/m than k˜r=0.1e9 N/m. Figure 3.24(c)

shows that the response is minimized at cs ≈ 1.3 for k˜r=0.3e9 N/m while the minimum

response occurs at cs ≈ 1.1 for k˜r=0.1e9 N/m. From (3.13), larger sin(cs − ss)π is needed

to minimize the response when k˜r is increased. From Figure 3.24(c), one can ﬁnd that

the maximized peak amplitude at cs ≈ 1.5 is reduced for larger k˜r. For k˜r = 0.3e9 N/m, the peak amplitude calculated at near cs = 1.5 is smaller than the peak amplitude with small cs < 1.2; this implies that contact ratio 1.5 does not always maximize the response.

Instead, particularly when k˜r/k˜s is greater than 1, contact ratio ≈ 1.5 can yield smaller peak amplitude than contact ratio ≈ 1 or 2. This demonstrates that the ﬁnding from the past studies [80, 81, 89] showing the large dynamic loads and vibrations with the contact ratio ≈ 1.5 for a single gear-pair is not necessarily valid for the planetary gear systems.

Figure 3.25 illustrates the impact of contact ratio and fluctuation of mesh stiffness on subharmonic resonance of the first rotational mode. One can see the fluctuation of mesh stiffness triggering the parametric instability for each value of contact ratio. Sensitivity of cs and cr is examined by fixing cr = 1.6 in Figure 3.25(a) and cs = 1.6 in Figure 3.25(b).

The variation of the width with cr is clearly examined in Figure 3.25(a) while negligible variation of the width with cs is seen in Figure 3.25(b), which indicates that cr has the dominating inﬂuence on the width of the parametric instability over cs for the ﬁrst rotational mode. This different sensitivity of the width on cs and cr is discussed in section 3.4; Figure

119 3.15 describes the different impact of contact ratio depending on the vibration modes. cr ≈

1.6 requires the least amount of k˜ ≈ 170e6 N/m to excite the parametric instability. The parametric instability will be excited by any cr with the similar k˜ ≈ 170e6 N/m. Figure

3.26 shows the sensitivity of the width to k˜s and k˜r with the ﬁxed cs = cr = 1.6. The change of the width over the broad range of k˜s is small compared to k˜r. In Figure 3.26(a), the parametric instability is not excited for any k˜s with k˜r = 150e6 N/m. Slight increase of k˜r to 190e6 N/m results in the parametric instability even when k˜s is zero. Figure 3.26(b) also shows that k˜r ≈170e6 N/m triggers the parametric instability and the sensitivity of the width on k˜s is small.

120 800

700

600

500

400

300 Width of instability, Hz

200

100

0 0 1 2 3 4 5 6 Damping ratio,% (a)

1500

1000

500 Width of instability, Hz

0 0 1 2 3 4 5 6 Damping ratio,% (b)

Figure 3.20: Variation of the width of the parametric instability boundary for (a) k˜r = 2k˜s =200e6 N/m and (b) k˜s = 2k˜r =200e6 N/m. System parameters are from the example system in Table 3.1. Planets are equally spaced with in-phase meshes. For (b), the ﬁrst distinct mode with (—) N = 4 and ( ) N = 6 and the second distinct mode with (- - -) N = 4 and (− − −) N = 6 are compared.

121 300

250

200

150

100

Width of instability, Hz 50

0 1 1.2 5 1.4 4 3 Contact ratio1.6 2 1.8 1 2 0 Damping ratio, %

Figure 3.21: Combined effects of contact ratio and damping ratio on the width of the parametric instability boundary of the ﬁrst rotational mode for the example system in Table 3.1. Four planets are equally spaced with in-phase meshes.

122 300

250

200

150

100 Width of instability, Hz

50 1 0.8 1 0.6 0.8 0.4 0.6 0.4 0.2 γ 0.2 γ 3 0 0 2 (a)

300

250

200

150

100

Width of instability, Hz 50

0 1 0.8 1 0.6 0.8 0.4 0.6 0.4 γ 0.2 3 0.2 γ 0 0 2 (b)

Figure 3.22: The width of the parametric instability boundary for the ﬁrst rotational mode for six diametrically opposed planets with different damping ratio: (a) 1% and (b) 3%. System parameters are from the example system in Table 3.1.

123 1

0.9

0.8

0.7

0.6

mp 0.5 A

0.4

0.3

0.2

0.1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Damping ratio,%

Figure 3.23: The effects of damping on the mesh phase causing the parametric instability for ( ) the ﬁrst and (− − −) the second rotational modes for the example system in Table 3.1. Amp = the ratio of mesh phase area causing parametric instability to total area.

124 2.5 2

m m 1.5 µ µ 1.5 1 1

0.5

Planet rotation, 0.5 Planet rotation,

0 0 2 2 1.8 2 1.8 2 1.6 1.8 1.6 1.8 1.4 1.6 1.4 1.6 1.4 1.4 Cr 1.2 Cr 1.2 1.2 Cs 1.2 Cs 1 1 1 1 (a) (b)

1.8 2.2

1.6 2

1.4 1.8

1.2 m m

µ µ 1.6

1 1.4 0.8

1.2

Planet rotation, 0.6 Planet rotation,

1 0.4

0.2 0.8

0 0.6 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Cs Cr (c) (d)

Figure 3.24: Variation of the peak planet rotation vibration amplitude for primary resonance of the second distinct mode with contact ratios. Three equally spaced planets with in-phase for the example system in Table 3.1. k˜s =700e6 - 600e6 = 100e6 N/m and k˜r = (a) 650e6 - 550e6 = 100e6 N/m and (b) 750e6 - 450e6 = 300e6 N/m. cr and cs is ﬁxed at 1.5 for (c) and (d), respectively. For (c) and (d), k˜r = (—) 100e6 N/m, (- - -) 200e6 N/m, and (···) 300e6 N/m.

125 350

300

250

200

150

100

Width of instability, Hz 50

0 2 1.8 300 1.6 200 c 1.4 r 100 ~ 1.2 k, 1e6 N/m 1 0

(a)

350

300

250

200

150

100 The width of instability, Hz 50

0 2 1.8 300 1.6 200 1.4 c s 1.2 100 ~ k, 1e6 N/m 1 0

(b)

Figure 3.25: Combined effects of contact ratios and mesh stiffness fluctuation (k˜ = k˜s = k˜r) on the width of parametric instability boundary of the first rotational mode for the example system in Table 3.1. Four planets are equally spaced with in-phase meshes. cs in (a) and cr in (b) are fixed at 1.6.

126 200

150

100 Width of instability, Hz

0 0 100 ~ 200 300 ks , 1e6 N/m

(a)

350

300

250

200

150 Width of instability, Hz 100

0 0 100 ~ 200 300 kr, 1e6 N/m

(b)

Figure 3.26: Variation of the width of the parametric instability boundary with the mesh stiffness fluctuation for the first rotational mode. Four planets are equally spaced with in- phase meshes for the example system in Table 3.1. For (a), k˜r is fixed at (···) 150e6 N/m, (−.−) 170e6 N/m, (− − −) 190e6 N/m, and (− − −) 210e6 N/m. For (b), k˜r is fixed at (···) 100e6 N/m, (− − −) 200e6 N/m, and (− − −) 300e6 N/m.

127 3.9 Summary and Conclusions

This chapter analytically investigates how system parameters affect planetary gear dy-

namic response. Peak amplitude of the resonant response and the width of the parametric

instability region are chosen to represent dynamic response. Perturbation solution is uti-

lized in this study because it reveals the mathematical structure of dynamic response as a

function of system parameters. The explicit appearance of the system parameters in the

analytical expressions gives an insight into the interaction between the parameters and dy-

namic response. Resonances are suppressed by proper selection of individual parameters

or by combination of multiple parameters. When no suppression condition is available, one

can still ﬁnd a desired parameter values to achieve minimal vibration using perturbation so-

lutions. Different sensitivity of dynamic response on the system parameters depending on

the vibration modes is examined. This study demonstrates the effectiveness of perturbation

solution in terms of fundamental design guidance tool by providing a deep understanding

of the inﬂuence of system parameters on planetary gear dynamics. The main ﬁndings are

summarized below.

• Perturbation solution conﬁrms the suppression rules with mesh phase introduced by

independent methods in [47, 48]. In addition, suppression conditions for superharmonic

and subharmonic resonances are found and validated by numerical solutions.

• The suppression conditions holds when planetary gears experience tooth separation nonlinearity because the analytical expression of dynamic response does not include the tooth separation related terms. This can be physically explained as the nonlinear response can not be excited when the resonance is suppressed.

• Different impact of sun-planet mesh and ring-planet mesh related parameters on dynamic response is examined. Combination of key system parameters such as the mean

128 mesh stiffness, gear inertia, and the number of planets determines which mesh parameter

more strongly affect dynamic response. Vibration mode changes the sensitivity of dynamic

response to the sun-planet and ring-planet meshes.

• For contact ratio < 2, contact ratio whose value is close to integer does not necessarily

reduce the vibration. This ﬁnding contradicts the past studies about single gear pairs that do

not experience the interaction between sun-planet and ring-planet meshes like a planetary

gear system. Instead, minimum response can be achieved at contact ratio ≈ 1.5 for a

particular combination of the system parameters.

• Opposed to common thinking, reduction of vibration for larger ﬂuctuation of the

mesh stiffness is predicted for particular cases. Vibration measured at off-resonance can

sharply jump with the slight change of the mesh stiffness ﬂuctuation.

• The equation of an ellipse expresses the relationship between the width of the para-

metric instability region and damping. The instability region is exponentially reduced for

higher damping.

• More number of planets can widen the parametric instability region while the reso-

nant peak amplitude is decreased due to an increased load sharing for a given external load.

The sensitivity of the width to the number of planets varies with vibration modes.

• An analytical expression of the parametric instability width shows a condition when subharmonic resonance occurs; triggering value of an each parameter for the parametric instability is determined by other system parameters.

129 Chapter 4: Analytical Investigation of Tooth Proﬁle Modiﬁcation Effect on Planetary Gear Dynamics

4.1 Introduction

Excessive gear tooth deflection due to applied torque causes undesirable tooth contact patterns and increases gear vibration, noise, and contact stress. In more severe cases, tooth surface damage like pitting or gear failures can occur. Tooth profile modification (TPM) is extensively used to compensate for the elastic gear and tooth deflection from the applied torque. The amount of modification is small (µm scale) and depends on the gear tooth de-

ﬂection. Its signiﬁcant impact on the tooth deformation and contact pattern can be utilized to increase the gear durability and reduce gear dynamic response, noise, and contact stress.

Since Harris [50] developed maps showing the relationship between static transmission error and tip relief, numerous works (e.g., [51, 53, 90, 91]) can be found regarding

TPM. Kahraman and Blankenship [56] conducted gear dynamics experiments for a single spur gear pair and showed that there exists a particular tip relief to minimize dynamic transmission error for a given load. Mathematical models of single mesh gear pairs including tooth proﬁle errors were developed to study the impact of TPM on gear dynamics [1, 54, 55, 57, 58, 92]. Liu and Parker [93] introduced a dynamic model of a multi-mesh

130 idler gear system with tooth profile modification. Meanwhile, the effects of TPM on planetary gear dynamics have received much less research attention. Litvin et al. [60] proposed tooth surface modifications to improve load distribution in planetary gears. Kahraman [18] and Lin and Parker [19] included tooth profile modifications in their analytical models of planetary gears, but these were neglected in the dynamic analysis. Abousleiman and

Velex [61] showed that tip relief greatly reduces dynamic tooth loads and displacement amplitudes by numerical simulations of their three-dimensional planetary gear model.

A nonlinear dynamic model of a planetary gear with tooth profile modification is proposed in this paper. The accuracy of the TPM model is evaluated by comparisons with finite element software specially formulated to analyze contact of high-precision surfaces like gear teeth. Perturbation analysis is employed to obtain a closed-form approximation of the dynamic response. This solution gives insight on the parameter dependencies and quickly yields the peak vibration amplitude at resonance. The influence of TPM on planetary gear dynamic response is studied in terms of the amount and the length of modification. The minimum static transmission error and dynamic response are achieved by different amount of modifications. This discrepancy between static and dynamic response emphasizes the importance of dynamic analysis when seeking an optimum TPM to reduce planetary gear vibration. Contrary to the expectation of further vibration reduction, increased dynamic response is observed when the sun-planet and ring-planet TPMs that give minimum response when applied individually are combined. The effect of TPM on planetary gear dynamic response is greatly influenced by system parameters such as mesh stiffness fluctuation and relative mesh phase between the sun-planet and ring-planet meshes.

131 4.2 Tooth Proﬁle Modiﬁcation Model

The two-dimensional lumped-parameter model of a spur planetary gear developed in

[19] and [21] is adopted as the basic model. This is augmented with a tooth proﬁle modiﬁ-

cation (TPM) model. Figure 4.1, from [21], shows a schematic of the planetary gear model.

The matrix equation of motion with N planets is

Mx¨ + K(x,t)x = F (4.1) T x = [xc,yc,uc,xr,yr,ur,xs,ys,us,ζ1,η1,u1,··· ,ζN,ηN,uN] .

K = Km(x,t) + Kb is the stiffness matrix including the nonlinear, time-varying mesh stiffness Km(x,t) and the bearing stiffness Kb (refer to [19] for the detailed system matrices). ksn(t) and krn(t) in [19] are replaced with the nonlinear, time-varying stiffnesses ksn(t) δsn ≥ 0, ksn(x,t) = ksn(t)Θ(δsn) = n = 1,2,··· ,N 0 δsn < 0, krn(t) δrn ≥ 0, krn(x,t) = krn(t)Θ(δrn) = 0 δrn < 0, (4.2)

δsn = ys cosψsn − xs sinψsn − ζn sinαs − ηn cosαs + us + un

δrn = yr cosψsn − xr sinψrn + ζn sinαr − ηn cosαr + ur − un, where ψsn = ψn −αs and ψrn = ψn +αr, ψn is the circumferential position angle of the nth planet, and αs and αr are the pressure angles of the sun-planet and ring-planet meshes.

Figure 4.2 illustrates the sun-planet mesh stiffness for the planetary gear system 1 in

Table 4.1 calculated by quasi-static ﬁnite element analysis at multiple points in a mesh cycle. The transition between single tooth contact and double teeth contact occurs at the highest point of single tooth contact (HPSTC) and the lowest point of single tooth contact

(LPSTC), which are 0.185 and 0.765 in Figure 4.2, respectively. When a geared system is loaded, the elastic tooth deﬂection causes undesirable contact behavior also known as corner contact. In practice, TPM is employed to eliminate this corner contact. Because TPM

132 Table 4.1: System parameters of the example planetary gear models. Parameter System 1 System 2 N 3 6

ksp (N/m) 698e6 683e6

krp (N/m) 691e6 842e6 2π(n−1) [0,24,49,73,97,122]π ψn (rad) N 73

αs and αr (deg) 25 25 Sun number of teeth 27 40 Planet number of teeth 34 33 Ring number of teeth 96 106 2 Sun inertia, Is/rs (kg) 1.76 0.478 2 Planet inertia, Ip/rp (kg) 2.90 0.330 Sun mass (kg) 2.27 8.30 Planet mass (kg) 1.39 3.65 Carrier mass (kg) 43.7 90.5 Sun bearing stiffness (N/m) 2e9 23.3e6 Planet bearing stiffness (N/m) 2e9 2.49e9 Carrier bearing stiffness (N/m) 2e9 11.1e9 Input torque to sun (N-m) 1130 11300

changes the gear tooth contact pattern, the mesh stiffness function adjusts accordingly as

seen in Figure 4.2. In order to take account of the proﬁle modiﬁcation in the dynamic

model, an additional excitation E(x,t) will be derived. E(x,t) acts in addition to the exter- nally applied torque/force vector F(t). Two different TPM models are introduced, and the accuracy of each model is evaluated by comparisons with ﬁnite element solutions.

133 4.2.1 Tooth Proﬁle Error Function Model (Model 1)

In the ﬁrst model, the designed geometric deviations from the involute tooth proﬁle are

directly included in the mesh deﬂection. These deviations or TPM functions are available

from a tooth proﬁle chart measured by a gear inspection or coordinate measuring machine.

Figure 4.3 illustrates a typical linear TPM function. The excitation force generated by

the TPM is mathematically expressed as the product of the mesh stiffness and the TPM

function. The basic concept of this TPM modeling was introduced in prior works [18,

19,54,55,57,93–95]. This model agreed well with dynamic analysis of a benchmark ﬁnite

element solution for a multi-mesh idler gear system [93], and this encourages its application

to planetary gears.

One or more tooth pairs may be in contact at each mesh as the gears rotate. Each

tooth pair has its own stiffness and TPM functions. At a given instant, the resultant TPM

excitation force at a mesh is the sum of the product of the individual time-varying tooth

pair stiffness and TPM functions. Considering the individual tooth pair loads allows one

to model the partial tooth separation that occurs when only some of the tooth pairs lose

contact [93].

With tooth profile modification, the nth sun-planet and ring-planet mesh deflections are

¯ b δsn = δsn − hsn (4.3) ¯ b δrn = δrn − hrn,

b b where hsn and hrn are the sun-planet and ring-planet TPM functions of the bth tooth pair. They are functions of the roll angle and are positive when material is removed (Figure 4.3).

These δ¯sn and δ¯rn replace the δsn and δrn given in (4.2) in the equations of motion for each

gear. For example, after inclusion of TPM, the equations of motion for the sun from [19]

134 are

N Bs 2 b b ¯ ¯ ms(x¨s − 2Ωcy˙s − Ωcxs) − ∑ ∑ ksnΘ (δsn)δsn sinψsn + ksxs = 0 n=1 b=1 N Bs 2 b b ¯ ¯ ms(y¨s + 2Ωcx˙s − Ωcys) + ∑ ∑ ksnΘ (δsn)δsn cosψsn + ksys = 0 (4.4) n=1 b=1 N Bs 2 b b ¯ ¯ (Is/rs )u¨s + ∑ ∑ ksnΘ (δsn)δsn + ksuus = Ts/rs, n=1 b=1 where Ωc is carrier rotation speed, Is is the sun moment of inertia, rs is the sun base radius, us = rsθs, Ts is the external torque on the sun, and Bs is the total number of tooth contact pairs in each sun-planet mesh, which is the nearest integer greater than the contact ratio.

b ksn is the sun-planet mesh stiffness of the bth tooth pair. The mesh stiffnesses of the two contacting tooth pairs for the example system in Table 4.1 calculated by finite element analysis are illustrated in Figure 4.2. The first pair mesh stiffness is active from 0 to 0.765 in a mesh cycle under no load conditions, but the engagement of the tooth pair extends to 0.87 due to the elastic tooth deflection. Assuming that TPM eliminates such extended contact, the first pair mesh stiffness is adjusted to end at 0.765 when TPM is applied, as shown in Figure 4.2. Similar adjustment is applied to the second pair mesh stiffness.

b By moving hsn to the right side of the equation, equation (4.4) is re-written as

N Bs N Bs 2 b b ¯ b b b ¯ ms(x¨s − 2Ωcy˙s − Ωcxs) − ∑ ∑ ksnΘ (δsn)δsn sinψsn + ksxs = − ∑ ∑ ksnhsnΘ (δsn)sinψsn n=1 b=1 n=1 b=1 N Bs N Bs 2 b b ¯ b b b ¯ ms(y¨s + 2Ωcx˙s − Ωcys) + ∑ ∑ ksnΘ (δsn)δsn cosψsn + ksys = ∑ ∑ ksnhsnΘ (δsn)cosψsn n=1 b=1 n=1 b=1 N Bs N Bs 2 b b ¯ b b b ¯ (Is/rs )u¨s + ∑ ∑ ksnΘ (δsn)δsn + ksuus = Ts/rs + ∑ ∑ ksnhsnΘ (δsn). n=1 b=1 n=1 b=1 (4.5) The left sides of equation (4.5) are identical to the equations of motion for the sun in

b [19] except for the tooth separation function Θ (δ¯sn), and this is true for the equations of motion for the other components. Additional excitation terms generated by the tooth proﬁle

135 b modiﬁcation hsn are evident on the right sides of the equations. The excitation for the sun can be expressed as

N Bs b b b ¯ T Es(x,t) = ∑ ∑ ksnhsnΘ (δsn)[−sinψsn,cosψsn,1] . (4.6) n=1 b=1

Applying similar steps for the other gear components, the TPM excitation corresponding to the vector x in (4.1) is deﬁned as

T E(x,t) =[0,Er(x,t),Es(x,t),E1(x,t),··· ,EN(x,t)]

N Br b b b ¯ T Er(x,t) = ∑ ∑ krnhrnΘ (δrn)[−sinψrn,cosψrn,1] n=1 b=1 N ( Bs b b b ¯ T (4.7) En(x,t) = ∑ ∑ ksnhsnΘ (δsn)[−sinαs,−cosαs,1] n=1 b=1

Br ) b b b ¯ T + ∑ krnhrnΘ (δrn)[sinαr,−cosαr,−1] , n = 1, 2,··· , N, b=1

b where krn is the ring-planet mesh stiffness of the bth tooth pair and Br is the total number of tooth pairs in contact at each ring-planet mesh.

Examination of the additional TPM excitation E(x,t) in (4.7) shows that the excitation

is formed by the additional mesh force from TPM that acts along the gear mesh lines of Bs b b b action. For the example of the sun excitation Es(x,t) in (4.6), ∑ ksnhsnΘ is the additional b=1 mesh excitation along the line of action of the nth mesh. This additional mesh force generates an additional torque on the sun (and the planet), as is evident from the unit value of the third element in the vector in (4.6) and noting that the rotational motion coordinate is us = rsθs. The translational force excitation components of Es(x,t) are calculated by

projecting this mesh force to the corresponding xs and ys coordinate directions. The TPM

excitation of the other gears can be interpreted similarly as shown in (4.7). This examina-

tion of E(x,t) suggests that one can have another TPM excitation model by replacing the

additional mesh force excitation terms, which are obtained in model 1 by using the mesh

136 stiffness and the TPM function, with those calculated in different ways, as done in model

4.2.2 Loaded Static Transmission Error Model (Model 2)

Tooth proﬁle modiﬁcation affects the gear meshing action, causing different tooth contact patterns and loaded static transmission error. The change of loaded static transmission error indicates that the mesh force equilibrium without TPM is altered. Additional excitation terms are needed to correct the unbalanced forcing terms when TPM is applied, and these additional excitations are another candidate of the mesh excitation term for E(x,t).

Geared system models using loaded static transmission error as an internal excitation to the system have been developed [92, 96–99]. When the time-invariant average mesh stiffness is used in the model, poor comparison to a benchmark FE model is shown when

TPM is considered [93]. Thus, model 2 retains the nonlinear, time-varying mesh stiffnesses on the left-hand side of (4.1) and uses loaded static transmission errors only to calculate the TPM mesh excitations.

The planetary gear is decoupled into the sun-planet and ring-planet pairs to calculate loaded static transmission errors. The applied torque is chosen such that the nominal mesh force is applied to the individual sun-planet and ring-planet gear pairs. For the example of the sun gear paired with the nth planet gear, the loaded static transmission error δˆsn of the gears without TPM is determined by the equilibrium condition

0 = fˆn − ksnδˆsn, n = 1, 2,··· , N, (4.8)

where fˆn is the nominal mesh force. Because TPM changes the loaded static transmission error δˆsn in (4.8), the loaded static transmission error with TPM becomes δ˜sn. An additional

137 mesh force terme ˜sn is required for force equilibrium in the gear pair system according to

0 = fˆn − ksnδ˜sn + e˜sn, n = 1, 2,··· , N. (4.9)

Incorporating the tooth separation nonlinearity of mesh stiffness that is needed in the dynamic model, the mesh excitatione ˜sn is calculated from (4.9) as

e˜sn = ksnΘ(δ¯sn)δ˜sn − fˆn, n = 1, 2,··· , N. (4.10)

Note that individual tooth pairs at a given tooth mesh are not modeled separately, so the

contact loss of only one tooth pair out of multiple contacting pairs is not considered in this

model. For the ring-planet pair,

e˜rn = krnΘ(δ¯rn)δ˜rn − fˆn, n = 1, 2,··· , N. (4.11)

The additional mesh excitation forcese ˜sn ande ˜rn are generated to satisfy the mesh

force equilibrium when TPM is applied. They can be calculated from (4.10) and (4.11)

once the loaded static transmission errors δ˜sn and δ˜rn are measured or calculated by, for

example, ﬁnite element analysis of the individual sun-planet and ring-planet gear pairs. In

b b contrast to explicit inclusion of the TPM functions hsn and hrn as in model 1, the mesh excitation forces in this model are expressed with the loaded static transmission error as

seen in (4.10) and (4.11). Substitution of these additional mesh excitation forces into (4.7)

yields the alternative formulation of TPM excitation T E(x,t) = [0,Er(x,t),Es(x,t),E1(x,t),··· ,EN(x,t)] N T Er(x,t) = ∑ e˜rn [−sinψrn,cosψrn,1] n=1 (4.12) N T Es(x,t) = ∑ e˜sn [−sinψsn,cosψsn,1] n=1 T T En(x,t) = e˜sn [−sinαs,−cosαs,1] + e˜rn [sinαr,−cosαr,−1] . This E(x,t) is added to the right-hand side of (4.1) exactly as done in model 1.

138 ys ,yc ,yr

k ζ r 2 Planet 2 k ru kr2 η2 kp kcu u1 u k r ksu s2 uc Planet 3 Sun ψ2 kc kr xs ,xc ,xr ks3 u3 ks η1 ζ kp 3 ks1 k u kp r1 kr3 s u1 η3 Carrier Ring ζ1 kc

Planet 1

Figure 4.1: A schematic of two-dimensional lumped parameter model of a planetary gear system

8 x 10 9 Total mesh stiffness without TPM 8

7 Total mesh stiffness with TPM

4 2nd pair stiffness without TPM 1st pair stiffness without TPM

Mesh stiffness, N/m 3

2nd pair stiffness with TPM 2 1st pair stiffness with TPM 1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mesh cycle

Figure 4.2: Sun-planet mesh stiffness for example system 1 described in Table 4.1.

139 Root modification Tip modification

Start of root modification

Start of tip modification Amount of tooth profile modification

Tip SAP Roll angle

Figure 4.3: Description of tip and root modiﬁcation. SAP stands for start of active proﬁle.

140 4.3 Comparison of Analytical Tooth Proﬁle Modiﬁcation Models to Finite Element Model

The two analytical TPM models are compared to a benchmark finite element (FE) model for cases with and without TPM. A unique commercial finite element/contact analysis tool named Calyx [32, 100] is used for the FE analysis. The tool precisely calculates the gear tooth deformation and the contact load with a relatively coarse FE mesh, which reduces simulation time and makes FE dynamic simulation of geared systems viable. The accuracy and efficiency of this FE code have been shown in past studies [21, 33, 34, 93].

Example systems 1 and 2 described in Table 4.1 are used. The ring gear is fixed as is common in practice to achieve the maximum gear ratio. Power flows from the sun input to the carrier output. While the carrier rotates, its rotational deviation from nominal kinematics is constrained to be zero, as occurs with large output inertia. Other power flows are also possible.

Profile modifications are applied simultaneously to the sun gear tip and root areas. Un- less otherwise stated, sun tip modification starts from the highest point of single tooth contact (HPSTC), and sun root modification starts from the lowest point of single tooth contact (LPSTC). Throughout the paper, two different modifications are used for each example system. For example system 1, the modification is 25.4 µm for TPM A and 50.8 µm for TPM B. For example system 2, the modification is 27.94 µm for TPM C and 38.1 µm for TPM D. All the modifications vary linearly with roll angle (Figure 4.3).

For a rotational mode, the sun, carrier, and ring rotate but have no translational deﬂec- tion, and all planets have identical motion [19, 26]. A decreasing speed sweep covering the ﬁrst rotational mode natural frequency at 1815 Hz is conducted for system 1 with and without TPM. Both the analytical and FE models use time domain numerical integration.

141 The root mean square (RMS) of sun rotational deﬂection is plotted in Figure 4.4 (through-

out the paper, rotational deﬂection is us = rsθs for the sun and ui = rpθi for the planets).

Without TPM, the analytical model, which is identical for models 1 and 2 in this case, agrees with the FE solution. Numerical solutions of both TPM models 1 and 2 match the

FE solution reasonably well for TPM A and B. The difference between models 1 and 2 in their deviation from the FE solution is small, although model 1 is slightly more accurate.

Figure 4.5 compares the analytical and FE models for example system 2 in Table 4.1. The analytical model predicts that the peak amplitude occurs near 1700 Hz while the peak amplitude occurs at 1740 Hz with the FE model. This 2% difference is minor. Except for this small difference of the peak resonant response frequency, both models 1 and 2 agree with the FE solution for TPM C and D.

Good agreement with the benchmark FE model for two different planetary gears and two different TPMs for each planetary gear builds conﬁdence in the analytical TPM models.

While the accuracy of both models 1 and 2 is acceptable, model 1 is simpler. It only requires the deﬁnition of TPM parameters such as the amount and starting point of the modiﬁcation.

In contrast, one needs the extra effort to calculate static transmission error with each TPM for model 2. Model 1 will be used throughout the rest of this paper.

Figure 4.4 shows the positive effect of TPM on reducing dynamic response and tooth contact loss. From the FE solution without TPM, tooth contact loss begins at 1950 Hz for decreasing speed and softening nonlinear response occurs. Nonlinear jump down occurs near 1650 Hz. The frequency range with tooth contact loss is 300 Hz. This range is shortened to 100 Hz with TPM A. Contact loss is eliminated completely with TPM B.

Compared to the peak amplitude with no modiﬁcation, the peak amplitude of the response from the FE solution is reduced by 30% and 58% for TPM A and B, respectively.

142 Figure 4.6 compares the static transmission error (STE) of the example system 1 sun- planet pair calculated from the FE model for a torque equal to the nominal sun torque divided by the number of planets, 376.7 N-m = 1130/3 N-m. TPM A reduces the peak- peak STE compared to the case without modiﬁcation, while TPM B increases the peak-peak

STE. The reduction of dynamic response for TPM B as shown in Figure 4.6 is opposite to the increased STE ﬂuctuation for TPM B in Figure 4.6. There exists common thinking [92,

96, 97] that STE can be viewed as an excitation source and that there is therefore a strong correlation between the amplitude of STE fluctuation and dynamic response. Consequently, tooth modifications are often designed to minimize the fluctuation of STE. This thinking is based on single pair gear systems. With such thinking, one would expect, based on

Figure 4.6, the largest dynamic response with TPM B. The response with TPM A is larger, however, indicating a need for better insight and modeling of TPM in planetary gears. The discrepancy between static and dynamic response in terms of TPM will be discussed later.

143 40

35 No TPM

30 m

µ TPM A 25

20 TPM B

15 RMS of sun rotation, 10

0 1500 1600 1700 1800 1900 2000 2100 2200 Mesh frequency, Hz

Figure 4.4: Root mean square (mean removed) sun rotational deﬂection without TPM, with TPM A, and with TPM B. Comparison of FE model and analytical models 1 and 2. Gear parameters and input torque are from example system 1 in Table 4.1. (- - -) Model 1, (-.-) model 2, and (···) FE solution. The solid line is the common solution from models 1 and 2.

144 35

30 No TPM

25 m µ

20 TPM C

TPM D 15

RMS of sun rotation, 10

0 1600 1650 1700 1750 1800 1850 1900 Mesh frequency, Hz

Figure 4.5: Root mean square (mean removed) sun rotational deﬂection without TPM, with TPM C, and with TPM D. Comparison of FE model and analytical models 1 and 2. Gear parameters and input torque are from example system 2 in Table 4.1. (- - -) Model 1, (-.-) model 2, and (···) FE solution. The solid line is the common solution from models 1 and 2.

145 40

35 m µ 30

20 Static transmission error,

10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mesh cycle

Figure 4.6: Static transmission errors of the sun-planet gear pair with different tooth proﬁle modiﬁcations. Gear parameters are from example system 1 in Table 4.1 with an input torque on the sun gear of 376.7 Nm. (—) No TPM, (- - -) TPM A, and (-.-) TPM B.

146 4.4 Perturbation Analysis with Tooth Proﬁle Modiﬁcation

Although numerical solutions of the analytical TPM model are accurate, the numerical

solution is valid only for a speciﬁc case. Numerical simulation is not efﬁcient to investi-

gate the effect of TPM on planetary gear dynamic response. For instance, when one needs

to examine the variation of dynamic response with many different combinations of TPM

parameters to ﬁnd an optimal TPM that minimizes gear vibration, numerical simulation

requires running many cases and a great deal of computational effort. A closed-form ap-

proximation of dynamic response is desirable. Perturbation analysis has proven effective

for investigation of TPM effects on idler gearsets [93] and for planetary gear dynamics [86].

This paper uses perturbation analysis to obtain an analytical solution of the TPM model and

utilizes the solution to study planetary gear dynamics with TPM.

4.4.1 Perturbation Solution Using the Method of Multiple Scales

The total mesh stiffnesses as illustrated in Figure 4.2 are used for perturbation analysis Bs Br b b such that each of the total mesh stiffnesses ksn(t) = ∑ ksn and krn(t) = ∑ krn is a sum of b=1 b=1 the individual tooth pair stiffnesses for the nth planet meshes. The total mesh stiffnesses with TPM are truncated at the left and right sides as shown in Figure 4.2 and discussed earlier. When these total mesh stiffnesses are used, the partial tooth separation is not considered. Figure 4.7(a) compares the numerical solutions with and without consideration of the partial tooth separation for the two TPM cases of example system 1 in Table 4.1. Ignor- ing the partial tooth separation only slightly changes the dynamic response. Although an excessive amount of modification may increase the difference, such a tooth profile modification could result in the increase of dynamic response and is of no interest from a practical point of view.

147 The time-varying mesh stiffnesses, which are presumed known from ﬁnite element

analyses of the individual meshing gear pairs or approximated as trapezoidal functions

over a mesh period, are Fourier expanded as

" ∞ # (l) jlωt ksn(t) = ksp + ∑ cˆsn e + cc l=1 (4.13) " ∞ # (l) jlωt krn(t) = krp + ∑ cˆrn e + cc , n = 1,2,··· ,N, l=1 where ω is the mesh frequency, ksp and krp are the average sun-planet and ring-planet mesh stiffnesses, and cc denotes the complex conjugate of the previous term. The averages and

Fourier coefﬁcients in (4.13) of example systems 1 and 2 are given in Tables 4.1 and 4.2.

With these Fourier expansions, the governing equations of model 1 including the TPM excitations in (4.7) are re-written as

" N N # ˆ ˆ 2 Mx¨ + ∑ ksp 1 + εQsn KsnΘsn(δsn) + ∑ krp 1 + µQrn KrnΘrn(δrn) x + O(ε ) = F + E n=1 n=1 (1) jωt (1) jωt Qˆsn = csn e + cc Qˆrn = crn e + cc 1 δ jn > 0 Θ jn = , j = s, r, 0 δ jn ≤ 0 (4.14)

(1) (1) (l) (l) (1) where ε = |cˆs1 |/ksp 1 and µ = |cˆr1 |/krp 1 are small parameters, csn = cˆsn / cˆs1 =

(l) (l) (1) O(ε) for l ≥ 2, crn = cˆrn / cˆr1 = O(ε) for l ≥ 2, and Ksn and Krn are matrices consisting of the coefﬁcients of ksn and krn in Km [86].

A simplification related to the tooth separation functions is used in (4.14). The possible influence of TPM on the onset of tooth separation is not considered; the nominal mesh deflections δsn and δrn determine tooth separation in Θsn and Θrn. Figure 4.7(b) compares the numerical solutions with and without the simplification for the two TPM cases of example system 1 in Table 4.1. The difference of dynamic response due to the simplification is small, demonstrating the validity of the simplification.

148 Both numerical and FE simulations reveal that only one tooth separation is observed

during the mesh cycle, and the separation interval is small, that is O(ε), compared to the mesh period. As a result, the tooth separation functions are

Θsn = 1 + εθsn Θrn = 1 + µθrn, (4.15)

where both θsn and θrn are O(1). These tooth separation functions are not known a priori.

Each element of the excitation vector E in (4.7) is periodic at the mesh frequency, and their Fourier expansion gives

N E = ∑ [FsnEsnΘsn + FrnErnΘrn] n=1 Bs " ∞ # b b (l) jlωt Fsn = ∑ ksnhsn = e¯sp + ε ∑ e˜sn e + cc (4.16) b=1 l=1

Br " ∞ # b b (l) jlωt Frn = ∑ krnhrn = e¯rp + ε ∑ e˜rn e + cc , b=1 l=1

Bs Br ! b b b b b b where Esn(Ern) is the constant coefﬁcient vector of ∑ ksnhsnΘ ∑ krnhrnΘ in (4.7). b=1 b=1 The vibration modes are calculated from the eigenvalue problem of (4.1) as

2 K0vi = ωi Mvi, (4.17)

where K0 is the mean stiffness matrix of K(x,t) in (4.2). The modal matrix V = [v1,··· ,v3N+9]

is normalized as VT MV = I.

149 With (4.15), (4.16), and introduction of modal coordinates as x = Vz, the qth modal equation of (4.14) is

3N+9 N 2 z¨q + ελqz˙q + ωq zq + ∑ ∑ εkspQsnGsnqw + µkrpQrnGrnqw zw w=1 n=1 3N+9 N − ε ∑ ∑ vwq (PsnEsnw + PrnErnw) = fq + eq, q = 1,2,··· ,3N + 9 w=1 n=1 (4.18) h (1) jωt i h (1) jωt i Qsn = csn e + θsn + cc Qrn = crn e + θrn + cc

" ∞ # " ∞ # (l) jlωt (l) jlωt Psn = e¯spθsn + ∑ e˜sn e + cc Prn = e¯rpθrn + ∑ e˜rn e + cc, l=1 l=1 T T where Gsnqw and Grnqw are the (q,w) elements of Gsn = V KsnV and Grn = V KrnV, vwq

is the (w,q) element of V, Esnw and Ernw are the wth elements of Esn and Ern, fq is the qth 3N+9 N T elements of V F, and eq = ∑ ∑ vwq (e¯spEsnw + e¯rpErnw), and these are constants. Small w=1 n=1 modal damping is introduced as ελq = 2ζqωq.

n With the multiple time scales tn = ε t, the asymptotically approximated qth modal

coordinate is

2 zq (t0,t1) = zq0 (t0,t1) + εzq1 (t0,t1) + O ε , q = 1,2,··· ,3N + 9. (4.19)

Insertion of (4.19) into (4.18) gives the perturbation equations for order ε0 and ε1 as

2 ∂ zq0 2 2 + ωq zq0 = fq + eq (4.20) ∂t0 2 2 3N+9 N ∂ zq1 2 ∂ zq1 ∂zq0 2 + ωq zq1 = −2 − λq − ∑ ∑ kspQsnGsnqw + gkrpQrnGrnqw zw0 ∂t0 ∂t0∂t1 ∂t0 w=1 n=1 3N+9 N + ∑ ∑ vwq (PsnEsnw + PrnEsnw), w=1 n=1 (4.21) (1) (1) where g = cˆr1 /krp / cˆs1 /ksp = µ/ε = O(1) . For the case where the mesh frequency is close to a natural frequency, the mesh fre-

quency is expressed as ω = ωi +εσ, where σ = O(1). The leading order solution of (4.20)

150 for q = i is f + e j(ω−εσ)t0 i i zi0 = Ai(t1)e + cc + 2 . (4.22) ωi

1 jβi(t1) After substitution of Ai(t1) = 2 ai(t1)e , (4.22) becomes

1 f + e j[ωt0−γi(t1)] i i zi0 = ai(t1)e + cc + 2 , (4.23) 2 ωi where γi(t1) = σt1 −βi(t1). The tooth separation function is periodic at the mesh frequency and its Fourier expansion gives

( " ∞ #) (0) (m) jm[ωt0−φs(t1)] Θsn = 1 + ε θsn + ∑ θsn (t1)e + cc (4.24) m=1 and s → r for the ring-planet mesh. The onset of tooth separation is determined by the mesh deﬂection, so the phases φs and φr are chosen such that the tooth separation is in-phase with the mesh deﬂection. Details are discussed in [86]. Substitution of (4.22) and (4.24) into

(4.21) yields the solvability condition for bounded zq1 as

N ∂Ai h ( ) ( ) i 0 2 j2(σt1−γi) ¯ j2ωi + jωiλiAi + ∑ ksp θsn Ai + θsn e Ai Gsnii ∂t1 n=1 3N+9 N h i (1) j(σt −γi) (1) jσt − ∑ ∑ vwi e¯spθsn e 1 + e˜sn e 1 Esnw + (s → r) w=1 n=1 (4.25) 3N+9 N f + e (1) j(σt1−γi) w w + ∑ ∑ kspθsn e 2 Gsniw + (s → r) w=1 n=1 ωw 3N+9 N f + e (1) jσt1 w w + ∑ ∑ kspcsn e 2 Gsniw + (s → r) = 0, w=1 n=1 ωw where all occurrences of ksp become gkrp for the terms abbreviated by (s → r). With

1 jβi(t1) substitution of Ai(t1) = 2 ai(t1)e , the real and imaginary parts of (4.25) are

∂ai 1 ωi = − ωiaiλi − |χ |sin(γi + ψ) ∂t 2 2 1 (4.26) ∂γi ωiai = ωiaiσ − χ1 − |χ2|cos(γi + ψ), ∂t1

151 N 3N+9 N 3N+9 N ksp h (0) (2) i (1) fw + ew (1) χ1 = ∑ θsn ai + θsn ai Gsnii + ∑ ∑ kspθsn 2 Gsniw + ∑ ∑ vwie¯spθsn Esnw + (s → r) n=1 2 w=1 n=1 ωw w=1 n=1 3N+9 N 3N+9 N fw + ew h (1) (1) i h (1) (1) i χ2 = ∑ ∑ 2 kspcsn Gsniw + gkrpcrn Grniw − ∑ ∑ vwi e˜sn Esnw + e˜rn Ernw , w=1 n=1 ωw w=1 n=1 (4.27) where ψ is the phase angle of χ2. For steady state periodic response, the conditions

∂ai/∂ti = ∂γi/∂ti = 0 in (4.26) give the amplitude-frequency approximation as q 1 2 2 2 ω = ωi + Ξ1ai + 2Ξ2 ± 2 |Ξ3| − ωi aiζi 2ωiai N h i (1) (0) (2) (1) (0) (2) Ξ1 = ∑ cˆs1 θsn + θsn Gsnii + cˆr1 θrn + θrn Grnii n=1 3N+9 N f + e Ξ = w w cˆ(1) (1)G + cˆ(1) (1)G 2 ∑ ∑ 2 s1 θsn sniw r1 θrn rniw w=1 n=1 ωw (1) 3N+9 N cˆ (4.28) s1 h (1) (1) i − ∑ ∑ vwi e¯spθsn Esnw + e¯rpθrn Ernw w=1 n=1 ksp 3N+9 N fw + ew (1) (1) Ξ3 = ∑ ∑ 2 cˆsn Gsniw + cˆrn Grniw w=1 n=1 ωw (1) 3N+9 N cˆ s1 h (1) (1) i − ∑ ∑ vwi e˜sn Esnw + e˜rn Ernw . w=1 n=1 ksp

4.4.2 Validation of Perturbation Solution

Figure 4.8 compares the perturbation and numerical solutions from model 1 for example system 1 in Table 4.1 with TPM A and TPM AA (12.7 µm modiﬁcation). The resonant response of the ﬁrst (ωi = 1815 Hz) and second (ωi = 4863 Hz) rotational modes are con-

sidered. Figure 4.8(a) plots sun rotational deﬂection. Both perturbation and numerical

solutions capture the softening nonlinear response. A nonlinear jump phenomenon occurs

at the ﬁrst rotational mode while linear response occurs at the second rotational mode. For

both TPM A and TPM AA, the perturbation solution compares well with the numerical

152 solution for both vibration modes. Good agreement between the perturbation and numerical solutions with TPM A and TPM AA is also evident for the planet translation in Figure

4.8(b).

Note that the peak amplitude with TPM AA is higher than the peak amplitude with

TPM A at the ﬁrst rotational mode for both sun rotation and planet translation, while the peak amplitude with TPM A is higher at the second rotational mode. Both the perturbation and numerical solutions conﬁrm this opposite TPM effect. This indicates that the TPM effect can change for the different vibration modes. The sensitivity of the TPM effect to the vibration mode is discussed later.

Table 4.2: Fourier coefﬁcients of sun-planet and ring-planet mesh stiffnesses Example system 1 Example system 2 Harmonics without TPM with TPM without TPM with TPM 1 -68.91 - 12.06i -84.40 - 13.74i -81.76 - 1.876i -92.65 + 2.998i 2 -28.74 - 9.660i -10.31 - 2.927i -26.07 - 1.612i 4.178 + 0.365i (l) cˆsn , 1e6 3 -4.494 - 1.988i 22.81 + 11.86i 6.001 + 0.123i 31.66 - 3.006i 4 7.627 + 5.854i 14.47 + 10.03i 14.04 + 1.426i 1.500 - 0.665i 5 7.092 + 7.330i -5.803 - 6.242i 6.885 + 1.137i -19.18 + 2.883i 1 -76.03 - 22.21i -30.14 - 3.379i -70.38 - 7.587i -108.8 - 13.89i 2 3.789 + 8.179i 12.31 + 43.86i -37.21 - 1.363i -9.660 + 0.74i (l) cˆrn , 1e6 3 2.856 + 9.019i 19.87 - 36.34i -14.71 + 0.500i 35.19 + 12.54i 4 -0.796 + 2.751i -27.70 + 7.617i -2.650 + 0.883i 16.26 + 5.680i 5 1.546 - 0.572i 14.73 + 8.313i -1.187 + 0.210i -14.55 - 9.790i

153 25

TPM A 20 m µ

TPM B 10 RMS of planet rotation,

0 1500 1600 1700 1800 1900 2000 2100 2200 Mesh frequency, Hz (a)

TPM A 20 m µ

TPM B 10 RMS of planet rotation,

0 1500 1600 1700 1800 1900 2000 2100 2200 Mesh frequency, Hz (b)

Figure 4.7: Root mean square (mean removed) planet rotational deflection of model 1 for example system 1 in Table 4.1. For (a), with (···) and without (—) consideration of the partial tooth separation. For (b), with (- - -) and without (—) consideration of the simplification related to the influence of TPM on the tooth separation.

154 45 First rotational mode

35 Second rotational mode m

µ 30

25 TPM A

TPM AA 15 RMS of sun rotation,

0 1500 2000 2500 3000 3500 4000 4500 5000 5500 Mesh frequency, Hz (a)

10 First rotational mode Second rotational mode

8 2.5 m µ 7

2 TPM A 6

1.5 5

1 4 1600 1700 1800 1900 2000

3 TPM AA RMS of planet translation, 2

0 1500 2000 2500 3000 3500 4000 4500 5000 5500 Mesh frequency, Hz (b)

Figure 4.8: Comparison of (- - -) numerical and (—) perturbation solutions with TPM A and TPM AA for the ﬁrst and second rotational modes of example system 1 in Table 4.1.

155 4.5 Results and Discussion

With the condition of zero deviation from the backbone curve in (4.28), i.e., vanishing

of the square root, the peak amplitude of the resonant response is

peak |Ξ3| ai = 2 . (4.29) ωi ζi

All parameters in Ξ3 in (4.29) and (4.28) are known, so the variation of the peak amplitude (1) (1) can be readily examined. The appearance of the TPM excitation termse ˜sn ande ˜rn in Ξ3

illustrates that the peak amplitude is affected by TPM, so there is an opportunity to reduce

the vibration by applying proper TPM.

4.5.1 Optimal TPM for Minimum Dynamic Response

The variation of the peak sun rotation vibration amplitude with the amount of modiﬁca-

tion is plotted in Figure 4.9. The ﬁrst rotational mode with natural frequency 1815 Hz for

example system 1 in Table 4.1 is considered here and for the rest of the discussions unless

otherwise stated. The perturbation solution is compared to the numerical and ﬁnite element

(FE) solutions. All three solutions agree. The perturbation and numerical solutions of TPM

model 1 give the minimum peak amplitude near 75 µm modiﬁcation, while the FE solution has a broader range of minimized response.

Figure 4.10 illustrates the effect of TPM length. The amount of modification is fixed at 76.2 µm. The tip roll angle is 37.9 deg, and the starting roll angle of tip modification varies from 28 deg to 35 deg. The starting roll angle of root modification is adjusted so that the root modification has the same length as the tip modification. The perturbation solution shows that the peak amplitude is minimized near 30 deg, which is at the HPSTC. The minimum peak amplitude is achieved between 29 deg and 31 deg for the numerical and FE

156 solutions. The perturbation solution shows that the slope of the peak amplitude variation is different before and after the roll angle. The peak amplitude gradually decreases as the roll angle approaches where the response is minimized (30 deg) and increases sharply for roll angles above 30 deg. Such results give design guidance to set tolerances. While it is desirable in the present case to apply TPM starting at 30 deg roll angle, it would be more acceptable to allow longer (i.e., lower starting roll angle) modiﬁcation rather than shorter

(i.e., higher starting roll angle) modiﬁcation. For example, TPM with starting roll angle of 28 deg (error of -2 deg) gives 10 µm peak amplitude while 32 deg (error of +2 deg) gives greater than 20 µm peak amplitude. Both the numerical and FE solutions conﬁrm this pattern.

The combined effect of the amount and the length of TPM can be examined quickly using the perturbation solution. Figure 4.11 shows the contour plot of the sun rotational dynamic response varying with these two parameters for example system 1 in Table 4.1.

The minimum peak amplitude is achieved when the amount of TPM is near 80 µm and the starting roll angle is near 30 deg. This contour plot shows where the response is minimized, and it shows how sensitive the response is to the TPM parameters. The variation of the peak amplitude with the starting roll angle (i.e., the length of modification) becomes more noticeable for large modification. For small TPM such as 20 µm, the peak amplitude remains almost constant until the starting roll angle of 32 deg. When larger modification such as 80 µm is applied, the dynamic response varies more significantly as the length of modification changes.

157 45

m 35 µ

10 Peak amplitude of sun rotation,

0 0 20 40 60 80 100 120 140 Amount of modification, µm

Figure 4.9: Variation of sun rotation peak amplitude with the amount of sun TPM for the first rotational mode of example system 1 in Table 4.1. Tip modification starts from HPSTC and root modification starts from LPSTC. (—) Perturbation, (- - -) numerical, and (···) finite element solutions.

158 50

40 m µ

Peak amplitude of sun rotation, Peak amplitude of sun rotation, 10

0 28 29 30 31 32 33 34 35 Starting roll angle of TPM, deg

Figure 4.10: Variation of sun rotation peak amplitude with the length of sun TPM for the first rotational mode of example system 1 in Table 4.1. Amount of modification is 76.2 µm. (—) Perturbation, (- - -) numerical, and (···) finite element solutions.

159 120 50 20 15 25 40 100 10 30

5 m

µ 80

5 25

10 15 60 15

20 Amount of TPM, 40 25

30 20

40 40 0 28 29 30 31 32 33 Starting roll angle of TPM, deg

Figure 4.11: Contour plot of sun rotation peak amplitude varying with the amount and the starting roll angle (or length) of sun TPM for the ﬁrst rotational mode of example system 1 in Table 4.1.

160 4.5.2 Correlation Between Static and Dynamic Response

It is widely believed that there is strong correlation between loaded static transmission

error (STE) and dynamic transmission error (DTE). From the gear vibration modeling point

of view, STE is often used as an excitation source and as a relative measure of expected

gear vibration. This leads to the conclusion that DTE can be controlled by STE, or, in other

words, that vibration reduction can be achieved by decreasing STE ﬂuctuation. Therefore,

when TPM is used to reduce gear dynamic response, gear designers seek the TPM that

minimizes STE ﬂuctuation. In addition, it is common to analyze sun-planet and ring-

planet pairs separately for planetary gear designs, partially due to the limited availability

of complete planetary gear analysis tools. The desired TPM is determined from these two

individual gear pair analyses, and the two calculated TPM are then applied to the sun-planet

and ring-planet meshes of the full planetary gear. This process assumes that the beneﬁcial

effects of TPM for the individual sun-planet and ring-planet gear pairs remain valid for the

full planetary gear. These ideas are examined below.

In order to examine the effect of TPM on individual gear pairs, the FE model of the sun-

planet pair separated from example system 1 in Table 4.1 is analyzed. The torque yielding

the same sun-planet mesh load as in the full planetary gear is applied to the sun-planet pair.

The variation in RMS of the sun-planet pair STE of the FE model is plotted in Figure 4.12.

The RMS of STE is minimized near 15 µm, while the peak amplitude of sun rotational dynamic response is minimized near 75 µm for perturbation and between 80-100 µm for

FE analysis. This shows that the STE may not be a good indicator predicting the reduction

of dynamic response. The variation of sun rotational static deﬂection in the FE model

of the full planetary gear is also plotted in Figure 4.12. The 30 µm modiﬁcation giving

the minimum RMS of static sun rotational deﬂection is far from the 75-100 µm range

161 where sun dynamic response is minimized. One can still reduce the peak amplitude of the resonant response with either 15 or 30 µm modification (compared to no modification), but this misses the opportunity of large additional vibration reduction that can be achieved by larger TPM. This demonstrates that one can not rely on STE of an individual gear pair nor static deflection in the complete planetary gear to find an optimal TPM that minimizes

DTE. Instead, it is necessary to conduct dynamic analysis of the full system to determine

TPM for reduction of vibration.

10 45

9 40

8 m 35 µ m

µ 7 30 6 25 5 20 4 15 3 RMS of static deflection, RMS of static deflection, 10

2 Peak amplitude of sun rotation,

1 5

0 0 0 20 40 60 80 100 120 Amount of TPM, µm

Figure 4.12: Variation of static and dynamic response with the amount of sun TPM. Tip modification starts from HPSTC and root modification starts from LPSTC. Gear parameters are from example system 1 in Table 4.1. (- - -) Static transmission error of FE model of sun- planet gear pair, (-.-) sun rotational static deflection in complete planetary gear FE model , (···) FE solution of sun rotational dynamic response from complete planetary gear system, and (—) perturbation solution of sun rotational dynamic response from complete planetary gear system.

162 4.5.3 Correlation Between Sun-planet and Ring-planet TPM

One might want to apply TPM at both the sun-planet and ring-planet meshes to maxi-

mize the vibration reduction. Figure 4.13 shows that 5 µm modiﬁcation on the ring-planet mesh yields minimum response. This corresponds to point A in Figure 4.14. From the fact that the response is minimized at 75 µm sun-planet TPM when there is no ring-planet

TPM, as discussed in relation to Figure 4.9, one may think that simultaneously applying both 75 µm sun-planet TPM and 5 µm ring-planet TPM will optimally reduce the dynamic response (point C in Figure 4.14). Figure 4.14 illustrates contrary results, however. The response at point C is larger than the response when either of these two TPM are applied individually. Instead, the response is minimized at point D where 3 µm of ring-planet

TPM and 25 µm of sun-planet TPM are applied. This illustrates that the combination of the optimal (when applied individually) sun-planet and ring-planet modiﬁcations does not necessarily yield the optimal (or even an effective) solution for vibration reduction.

Point D is located roughly along the diagonal line connecting 75 µm sun-planet TPM and 5 µm ring-planet TPM. While the response amplitude varies along this line, the response at point D is similar to the response at either 75 µm sun-planet TPM or 5 µm ring-planet TPM. In this case, no significant benefit is obtained from the combination of sun-planet and ring-planet modifications. In practical situations, such a conclusion offers significant manufacturing cost savings.

163 80

70 m µ 60

20 Peak amplitude of planet rotation, 10

0 0 5 10 15 20 25 30 Amplitude of modification, µm

Figure 4.13: Variation of planet rotation peak amplitude with the amount of ring TPM for the first rotational mode of example system 1 in Table 4.1. Tip modification starts from HPSTC and root modification starts from LPSTC. (—) Perturbation and (- - -) numerical solutions.

164 120 80

m 60

µ 70 100 30 15 35 j i 80

40 50 60 20 7 30 60 10 12 15 k 35 20 40 10 7 10 40 12 h 30

Amount of sun- planet mesh tip relief, Amount of sun- 20 3 5 7 15 15 5 25 7 3 0 0 5 10 15 Amount of ring-planet mesh tip relief, µm

Figure 4.14: Contour plot of planet rotation amplitude varying with sun-planet and ring- planet mesh TPM for the first rotational mode of example system 1 in Table 4.1. Tip modification starts from HPSTC and root modification starts from LPSTC.

165 4.5.4 Sensitivity of TPM Effects to System Parameters

Figure 4.15 shows the variation of sun rotation peak amplitude with the amount of

TPM for the second rotational mode at natural frequency 4863 Hz. Perturbation analysis

predicts the minimum peak amplitude at around 9 µm. Numerical and ﬁnite element solu-

tions show reasonably good agreement. Compared to the case of the ﬁrst rotational mode

as shown in Figure 4.9, the difference between the modiﬁcations minimizing the peak am-

plitude is obvious. If one chooses 75 µm modification to minimize the vibration at the first rotational mode, the resonant response at the second rotational mode will be larger than the response without TPM. This opposite effect of TPM at different vibration modes was seen in Figure 4.8: smaller dynamic response with the larger modification (TPM A) at the

ﬁrst rotational mode in contrast to smaller response with the smaller modiﬁcation (TPM

AA) at the second rotational mode. Figure 4.15 explains why the response with TPM A is larger than the response with TPM AA at the second rotational mode. Both TPM A and

AA have larger modiﬁcation than the optimal amount of 9 µm. TPM A, the larger of the

two, yields higher response than TPM AA because the response increases continuously for

modiﬁcations larger than 9 µm.

Equation (4.29) shows that the peak amplitude is proportional to |Ξ3|. Ξ3 includes TPM

excitation terms as well as system parameters and modal deﬂections. Ξ3 is divided into two

parts Ξ3a and Ξ3b as

(1) 3N+9 N 3N+9 N cˆ fw + ew (1) (1) s1 h (1) (1) i Ξ3 = ∑ ∑ 2 cˆsn Gsniw + cˆrn Grniw − ∑ ∑ vwi e˜sn Esnw + e˜rn Ernw . w=1 n=1 ωw w=1 n=1 ksp | {z } | {z } Ξ3a Ξ3b (4.30)

Depending on the values of Ξ3a and Ξ3b, |Ξ3| can be either decreased or increased by

the TPM contribution in Ξ3b. Both Ξ3a and Ξ3b are complex and can be expressed as

166 Ξ3a = a + bi and Ξ3b = c + di. If the signs of a, b, c, and d are all the same, |Ξ3| will be minimized by an optimal certain TPM. Otherwise, depending on the values of a, b, c, and d, |Ξ3| could be increased for any TPM.

In order to study the relation between the system parameters and TPM, example system 1 is used for the following analysis after replacing the actual mesh stiffnesses (Figure

4.2 and Table 4.2) with rectangular wave mesh stiffness functions. The rectangular mesh stiffness functions allow one to manipulate the stiffness ﬂuctuation and relative mesh phase between the sun-planet and ring-planet meshes, so their impact on TPM effect can be examined. Figure 4.16 illustrates the conceptual rectangular mesh stiffness functions and their relative mesh phase. In this study, the ring-planet mesh stiffness ﬂuctuates between

600e6 N/m and 800e6 N/m with contact ratio 1.7. The maximum value of the sun-planet mesh stiffness is ﬁxed at 800e6 N/m, and two different minimum stiffnesses of 600e6 N/m and 700e6 N/m are considered. The sun-planet contact ratio is 1.5. In order to deﬁne the relative mesh phase γsr [46] between the sun-planet and ring-planet meshes as shown in

Figure 4.16, a reference point is selected to be at the center of the maximum mesh stiffness zone. The mesh phase is determined by the difference between the reference points of each mesh stiffness function.

Figure 4.17 shows how the effects of TPM change with the mesh stiffness and vibration mode. Figure 4.17(a) compares the variation of the sun peak amplitude with the amount of TPM for the ﬁrst rotational mode. For the minimum sun-planet mesh stiffness of 600e6

N/m, the sun response is minimized at 40 µm modiﬁcation. A minimum sun-planet mesh stiffness of 700e6 N/m yields the minimum response at around 30 µm. Smaller modiﬁca- tion is needed to minimize the dynamic response for the reduced sun-planet mesh stiffness

ﬂuctuation. Figure 4.17(b) shows the similar comparison for the second rotational mode.

167 The peak amplitude with the minimum stiffness of 600e6 N/m is minimized at 5 µm (which

is signiﬁcantly smaller than the 40 µm for the ﬁrst rotational mode in Figure 4.17(a)). For

a minimum stiffness of 700e6 N/m, the peak amplitude grows continuously for any modi-

ﬁcation greater than zero.

Figure 4.18 compares the variation of |Ξ3a| and |Ξ3b| for the ﬁrst and second vibration

modes. The minimum sun-planet stiffness of 600e6 N/m is considered. Both |Ξ3a| and

|Ξ3b| increase linearly. |Ξ3a| and |Ξ3b| cross at 40 µm modiﬁcation for the ﬁrst rotational

mode, where the response is minimized in Figure 4.17(a). For the second rotational mode,

the slope of |Ξ3b| is much greater than the slope for the ﬁrst rotational mode while the slope

of |Ξ3a| changes less. The crossing point of |Ξ3a| and |Ξ3b| that minimizes response occurs

at the smaller modiﬁcation because of the steeper slope of |Ξ3b|.

The impact of relative mesh phase γsr between the sun-planet and ring-planet meshes

on the TPM effect is shown in Figure 4.19. Figure 4.19(a) compares the variation of the

peak sun rotation vibration amplitude with the amount of modiﬁcation for different relative

mesh phase γsr. The ﬁrst rotational mode of example system 1 in Table 4.1 is considered.

As the phase increases, the minimum peak amplitude increases (TPM is less effective) and

smaller modiﬁcation yields this minimized peak amplitude. For γsr = π, the peak ampli-

tude is minimized by having no TPM; the response continuously increases with increasing

modification. Such a pattern of the response with the relative mesh phase γsr does not occur for all vibration modes. Figure 4.19(b) shows the impact of the relative mesh phase on the TPM effect for the second vibration mode. The minimum peak amplitude increases as the relative mesh phase increases until γsr = π/2. Unlike the case for the first rotational mode, the peak amplitude is minimized by larger modification for the increased phase.

For γsr = π, the response is minimized but at the largest amount of modiﬁcation, which

168 contrasts sharply with the γsr = π case for the ﬁrst rotational mode that has continuously growing response by TPM as seen in Figure 4.19(a).

169 18

m 14 µ

4 Peak amplitude of sun rotation,

0 0 5 10 15 20 25 30 Amount of modification, µm

Figure 4.15: Comparison of the variation of sun rotation peak amplitude with the amount of sun TPM for the second rotational mode of example system 1 in Table 4.1. Tip modification starts from HPSTC and root modification starts from LPSTC. (—) Perturbation, (- - -) numerical, and (···) finite element solutions.

170 Ring-planet mesh stiffness Mesh stiffness

Sun-planet mesh stiffness T γsr 2π

0 Mesh cycle T

Figure 4.16: Rectangular wave mesh stiffnesses and mesh phase between the sun-planet and ring-planet mesh stiffnesses.

171 30

25 m µ

10 Peak amplitude of sun rotation, 5

0 0 10 20 30 40 50 60 70 80 Amount of modification, µm (a)

16 m µ 14

Peak amplitude of sun rotation, 4

0 0 5 10 15 20 25 Amount of modification, µm

(b)

Figure 4.17: Variation of sun rotation peak amplitude with the amount of sun TPM for the (a) ﬁrst and (b) second rotational modes of example system 1 in Table 4.1. The minimum sun-planet mesh stiffness is ( ) 600e6 N/m and (− − −) 700e6 N/m. Lines with symbols, for 600e6 N/m and for 700e6 N/m, are from numerical solutions.

172 3000

2500

2000

1500 Absolute value 1000

500

0 0 10 20 30 40 50 60 70 80 Amount of modification, µm

Figure 4.18: Variation of ( ) |Ξ3a| and (− − −) |Ξ3b| with the amount of sun TPM for the ﬁrst rotational mode of example system 1 in Table 4.1. Lines with symbols, for |Ξ3a| and for |Ξ3b|, are for the second rotational mode.

173 70

60 m µ 50

20 Peak amplitude of sun rotation,

0 0 10 20 30 40 50 60 70 80 Amount of modification, µm (a)

14 m

µ 12

4 Peak amplitude of sun rotation,

0 0 5 10 15 20 25 Amount of modification, µm (b)

Figure 4.19: Variation of sun rotation peak amplitude with the amount of sun TPM for the (a) ﬁrst and (b) second rotational modes of example system 1 in Table 4.1. The minimum sun-planet mesh stiffness is 600e6 N/m. Phase between sun-planet and ring-planet mesh stiffness is ( ) 0 π and (− − −) π/4, (−.−) π/2, and (···) π. Numerical solutions for ( ) 0 π,(•) π/4, () π/2, and () π.

174 4.6 Conclusions

An analytical tooth profile modification (TPM) model for planetary gears has been developed, and its accuracy for dynamic analysis is evaluated by comparisons with a benchmark finite element analysis. Perturbation analysis yields a closed-form approximation of the planetary gear dynamic response with TPM. The perturbation solution is used to study the impact of TPM on planetary gear dynamics.

Using the analytical response solution, an optimal TPM that minimizes dynamic response is easily and quickly identiﬁed in terms of the amount and length of modiﬁcation.

Sensitivity analysis of TPM parameters can help gear designers set an appropriate design tolerance.

It is common to use the STE of individual gear pairs or the static deflection of the full planetary gear as response metrics when seeking an optimal TPM to minimize the planetary gear vibration. The peak amplitude of dynamic response, however, is not minimized at the amount of modification that minimizes either of these STE or static deflection measures.

This discrepancy contradicts the common thinking about the strong correlation between

STE and DTE and emphasizes the importance of dynamic analysis for ﬁnding an optimum

TPM that minimizes gear vibration.

Combined use of the optimum sun-planet and ring-planet mesh TPM, which are determined by minimizing response when applied individually, is expected to further reduce the dynamic response. Contrary to this expectation, however, dynamic simulations show increased response can occur. The minimal dynamic response is achieved at a much different combination of sun-planet and ring-planet mesh TPM.

System parameters such as the mesh stiffness ﬂuctuation amplitude and the relative mesh phase between the sun-planet and ring-planet meshes change how TPM affects the

175 dynamic response. Different TPMs are required to minimize the gear vibration depending on the amount of mesh stiffness ﬂuctuation and the mesh phase. Dynamic response is not guaranteed to be minimized by TPM. Instead it may continuously grow for larger TPM for certain mesh phase choices.

Different TPM minimizes the vibration at different vibration modes. In other words, the TPM that reduces dynamic response for a given vibration mode can increase dynamic response at other modes. Therefore, one needs to consider the operating speed and which modes are most active when designing the optimal TPM for minimal gear vibration.

176 Chapter 5: Conclusions and Future Work

5.1 Conclusions

This work studies nonlinear dynamics of parametrically excited planetary gear systems.

Nonlinear dynamic behaviors emerge from tooth contact loss caused by large gear vibration

at resonant conditions. Impact of important system parameters on dynamic response is

analytically investigated. Signiﬁcant reduction of planetary gear vibration can be achieved

by proper application of tooth proﬁle modiﬁcation.

5.1.1 Analytical Solutions for the Nonlinear Dynamics of Planetary Gears

Softening nonlinearity induced by the tooth disengagement action is inspected by using an analytical and finite element models. Closed-form expressions of the periodic steady state response for primary, subharmonic, and second harmonic resonances are derived by employing perturbation analysis. The analytical solutions match numerical integration, harmonic balance, and finite element solutions well. Key system parameters are explicitly included in the analytical solutions, which exposes the influence of the parameters on the nonlinear dynamic response.

The closed-form solutions show the proportional relation between the external forcing term and the amplitude of dynamic response, and this demonstrates that larger torque does

177 not necessarily prevent tooth separation. Subharmonic resonance is characterized by jump

phenomena occurring at both above and below resonant frequency. The analytical solutions

show that the frequency interval, which deﬁnes the parametric instability region, where the

vibration jump occurs for subharmonic resonance is not affected by the applied torque.

Perturbation analysis clariﬁes the excitation sources to resonances at multiples of mesh

frequency. It is numerically inspected that a superposition principle is valid for the second

harmonic resonance with different excitation sources.

5.1.2 Inﬂuence of System Parameters on Planetary Gear Dynamic Re- sponse

Perturbation analysis finds closed-form expressions for the peak amplitude of the resonant response and the width of parametric instability region. Use of the well-defined modal properties of planetary gears and the analytical solutions give the complete mesh phasing rules to suppress certain vibration modes of various resonances for equally spaced and diametrically opposed planets systems. The suppression rules through mesh phase introduced in past studies [47, 48] are confirmed. The extended mesh phasing suppression rules are proposed for super- and subharmonic resonances and are validated by numerical solutions.

The absence of tooth separation terms in the closed-form expression of dynamic response proves that the suppression conditions are independent of the tooth separation nonlinearity.

Beyond the suppression rules, the variation of dynamic response with mesh phase and other system parameters illustrates the optimal value of each parameter that reduces the planetary gear vibration. It is mathematically shown that the sun-planet mesh and ring- planet mesh related parameters have different inﬂuences on dynamic response. The different sensitivity of dynamic response to sun-planet and ring-planet meshes is quantiﬁed by simple expressions composed of known system parameters. Vibration modes change the

178 impact of system parameters on dynamic response. Vibration jumps are predicted with the

variation of the mesh stiffness ﬂuctuation. The relation between the width of parametric

instability and damping is expressed by the equation of ellipse that indicates exponentially

decreasing instability region for higher damping. More number of planets can widen the

parametric instability region while the resonant response is reduced due to improved load

sharing. Interactions between multiple parameters at different vibration modes yields sev-

eral interesting ﬁndings opposed to common thinking: Increasing contact ratio for low

contact ratio gears where the contact ratio is less than 2 does not necessarily give minimal

vibration, and the reduction of vibration for larger mesh stiffness variation is observed for

a particular combination of the system parameters.

5.1.3 Analytical Investigation of the Dynamic Effects of Tooth Proﬁle Modiﬁcation on Planetary Gears

An analytical nonlinear dynamic tooth profile modification model is proposed for planetary gears. Individual tooth pairs are considered, so the model can capture the partial tooth separation. The accuracy of the model is validated by a benchmark finite element solutions for different modifications and planetary gear systems.

The tooth profile modification model is analytically solved by perturbation analysis, and the closed-form solution of dynamic response is derived. The analytical solution rapidly calculates the variation of dynamic response with the fundamental tooth profile modification design parameters such as the amount and length of the modification, which clearly quantifies the impact of tooth profile modification. The optimal parameters that minimize the planetary gear vibration can be readily determined.

Static transmission error and dynamic response are minimized at different amounts of tooth profile modification. This finding is different from the common practical thinking

179 regarding the strong correlation between static transmission error and dynamic response based on sing-mesh gear pair studies and demonstrates that dynamic analysis is needed when an optimum tooth proﬁle modiﬁcation is sought to reduce planetary gear vibration.

Contrary to expectations, it is inspected that the combination of the optimal sun-planet and ring-planet tooth profile modifications that minimize response when applied individually increases dynamic response. Analytical investigation discovers that the system parameters such as mesh stiffness and mesh phase have significant influence on the effects of tooth profile modification. Different tooth profile modification minimizes the vibration of different vibration modes, which indicates that the operating speed and vibration modes need to be considered when determining optimal tooth profile modification for the reduction of vibration.

5.2 Future Work

This work can be extended by conducting further investigation of several interesting problems identiﬁed. This section discusses the directions for the future work in the chal- lenging areas.

1. There exist sun-planet and ring-planet meshes in planetary gears because planet gears

simultaneously mesh with a sun and ring gear. Chapter 3 discusses different impact

of sun-planet mesh and ring-planet mesh related parameters on dynamic response.

Chapter 4 shows that the combination of optimal sun-planet and ring-planet tooth

proﬁle modiﬁcations does not necessarily further reduce the planetary gear vibra-

tions. These ﬁndings imply there is dynamic interactions between sun-planet and

ring-planet meshes.

180 Similar to the relative mesh phase between planets, the relative mesh phase γsr be-

tween the sun-planet and ring-planet meshes exists. The expression of γsr is intro-

duced in [47]. γsr is included in the mesh stiffness variations in Chapter 3, but its

inﬂuence on dynamic response is not investigated. To the author’s knowledge, the

study on the dynamic effects of γsr is not found. Like other relative mesh phase, γsr

can have a signiﬁcant impact on dynamic response and interact with other system

parameters. Therefore, the systematic investigation on γsr needs to be performed.

2. Mesh phasing rules to suppress super- and subharmonic resonances are proposed and

validated numerically. Only part of the suppression rules, however, are mathemat-

ically derived. Although the rules are conﬁrmed for many number of planets even

beyond the practical application range, the lack of mathematical support limits the

general conclusions. Along with the use of closed-form solutions and properties of

vibration modes, the symmetry of planetary gears and tooth mesh periodicity used

in [47, 48] can be applied to attack the problem.

3. Tooth proﬁle modiﬁcation alters the tooth contact pattern and the mesh stiffness vari-

ation is subject to be changed. In the proposed analytical tooth proﬁle modiﬁcation

model, only one adjusted mesh stiffness variation is used. Use of one ﬁxed mesh

stiffness variation limits the reliability of the tooth proﬁle modiﬁcation model be-

cause the mesh stiffness variation changes with the amount and length of the mod-

iﬁcation. Figure 4.9 and Figure 4.10 in Chapter 4 show that the deviation of both

analytical and numerical solutions from the benchmark ﬁnite element solution is in-

creased when the amount and the length of modiﬁcation are away from the certain

181 range. For individual tooth proﬁle modiﬁcation to be analyzed, the mesh stiffness

variation needs to be accordingly adjusted for the reliable results.

Large dynamic loads signiﬁcantly changes the adjusted mesh stiffness variation for

the applied of tooth proﬁle modiﬁcation. The accuracy of the analytical model can be

improved by properly adjusting the mesh stiffness variation. A real time adjustment

algorithm is a good candidate. The dynamic mesh load is evaluated at every single

step during the analysis, and the mesh stiffness variation is adjusted corresponding

to the current mesh load. The relation between the mesh load and the mesh stiffness

should be given prior to the dynamic analysis, which can be obtained from static

analysis under various mesh loads.

4. It is inspected that the minimum static transmission and dynamic response is found

at different amount of modiﬁcations. Further analytical investigation is needed to

answer the questions, for example, why does it occur?, under what conditions does it

occur?, and how much different amount of modiﬁcation is expected?

5. Besides the amount and the length of modiﬁcation, there is another tooth proﬁle

modiﬁcation parameter: the shape. Application of the linear modiﬁcation seems to be

dominant in the gear industry, but the parabolic shape modiﬁcation is possible. Past

studies [101, 102] compared the effects of the linear and parabolic modiﬁcations for

the single-mesh gear pair but gave contradictory conclusions. Further investigation

is needed to clarify the confusion in the results of past studies and to thoroughly

understand the difference between the linear and parabolic modiﬁcations in terms of

their impact on the planetary gear dynamic response.

182 6. Although the analytical solutions have been useful and efﬁcient for the comprehen-

sive investigation of nonlinear planetary gear dynamics, experimental validation is

essential to confirm the analytical findings. The unique finite element/contact analy-

sis tool [32] has been extensively used as a benchmark for the analytical and numer-

ical analyses, but the most of its dynamic analysis is limited to the two dimensional

motions. Three dimensional gear motions and the realistic gear connections and

boundary conditions such as splines, bearings, and clutches need to be investigated

experimentally to build solid foundation for reliable analytical models of planetary

gears.

183 Appendix A: Gsn and Grn for a Purely Rotational Model

For a rotational model of a planetary gear with N planets, mesh stiffness matrix Km is   ∑(k˜sn cosαs + k˜rn cosαr) −∑k˜rn −∑k˜sn k˜r1 − k˜s1 k˜r2 − k˜s2 ··· k˜rN − k˜sN  ∑k˜rn 0 −k˜r −k˜r ··· −k˜rN   1 2   ∑k˜sn k˜ k˜ ··· k˜sN   s1 s2   k˜ + k˜ 0 ··· 0  Km =  r1 s1   symmetric k˜ + k˜ ··· 0   r2 s2   .. .   . .  k˜rN + k˜sN k˜sn = ksn cosαs, k˜rn = krn cosαr. (A.1)

Ksn and Krn are the matrices consisting of the coefﬁcients of ksn and krn in Km. With the use of well-structured modal properties for equally spaced or diametrically opposed planets [19, 21], the modal matrix V without a rigid body mode becomes   v11 0 ··· 0 v1,N+2  v 0 ··· 0 v   21 2,N+2   v 0 ··· 0 v   31 3,N+2   v v ··· v v  V = [v1,··· ,vN+2] =  41 42 4,N+1 4,N+2 , (A.2)  v v ··· v v   41 52 5,N+1 4,N+2   . . . .   . . . .  v41 vN+3,2 ··· vN+3,N+1 v4,N+2

184 where v1 and vN+2 are the distinct modes and v2 through vN+1 are the degenerate modes.

Gsn and Grn are

 2  v¯1 v¯1vn+3,2 ··· v¯1vn+3,N+1 v¯1v¯2 2  v ··· vn+3,2vn+3,N+1 v¯2vn+3,2   n+3,2  T  .. . .  Gsn = V KsnV =  . . .  (A.3)  2   symmetric vn+3,N+1 v¯2vn+3,N+1  2 v¯2

 2  vˆ1 −vˆ1vn+3,2 ··· −vˆ1vn+3,N+1 vˆ1vˆ2 2  v ··· vn+3,2vn+3,N+1 −vˆ2vn+3,2   n+3,2  T  .. . .  Grn = V KrnV =  . . . , (A.4)  2   symmetric vn+3,N+1 −vˆ2vn+3,N+1  2 vˆ2 wherev ¯1 = −v11 cosαs +v31 +v41,v ¯2 = −v1,N+2 cosαs +v3,N+2 +v4,N+2,v ˆ1 = −v11 cosαr + v21 − v41, andv ˆ2 = −v1,N+2 cosαr + v2,N+2 − v4,N+2.

185 Appendix B: Expressions of Eigenvalues for a Purely Rotational Model

For a rotational system with a ﬁxed ring and constrained carrier, the eigen value problem

gives N 2 ˜ ˜ ˜ ωi Msv1i = ∑ ksnv1i + ks1v2i + ··· + ksNvN+1,i n+1 2 ˜ ωi Mpv2i = ks1v1i + (kr1 + ks1)v2i (B.1) . .

2 ˜ ωi MpvN+1,i = ksNv1i + (krN + ksN)vN+1,i, 2 2 ¯ ¯ where Ms = Is/rs and Mp = Ip/rp. Considering ks = ks1 = ··· = ksN and kr = kr1 = ··· =

krN, for distinct modes where planet deﬂections are identical, (B.1) gives 2 ¯ Nks − ωi Ms v1i + Nksv2i = 0 (B.2) ¯ ¯ ¯ 2 ksv1i + kr + ks − ωi Mp v2i = 0.

After relating v1i and v2i, (B.2) can be rewritten as ¯ ¯ ¯ ¯2 [λiMs − Nks][λiMp − (kr + ks)] − Nks = 0 (B.3) 2 ¯ ¯ ¯ ¯ ¯ λi MsMp − [Ms(kr + ks)) + NksMp]λi + Nkskr = 0, 2 where λi = ωi . From (B.3), λi is calculated as q 2 Nk¯sMp + Ms k¯r + k¯s ± Nk¯sMp + Ms k¯r + k¯s − 4MsMp Nk¯sk¯r λi = (B.4) 2MsMp

T 2 2 2 2 From V MV = I, v11Ms + Nv21Mp = 1 and v1,N+1Ms + Nv2,N+1Mp = 1, or alternatively

2 2 v1iMs + Nv2iMp = 1, for i = 1 and N + 1. (B.5)

186 Combination of (B.2) and (B.5) gives

¯2 2 ks v2i = . (B.6) ¯ ¯ 2 ¯2 λiMp − kr + ks Ms + NMpks

2 ¯ ¯ ωi Mp − kr + ks Insertion of v1i = v2i from (B.2) into v11v21 + v1,N+1v2,N+1 and the use k¯s of (B.6) and (B.3) gives

v11v21 + v1,N+1v2,N+1 = 0. (B.7)

187 ∂∆ Appendix C: Expressions for max ∂N

A = (Nk − m)2 + mk(2Nk + mk + 2m) (C.1) √ B = B A + B 1 a √ b (C.2) BN+1 = Ba A − Bb 4 4 3 3 2 2 2 3 2 3 4 3 2 Ba = N k +N mk (3k − 4)+N m k (6 − k)+Nm k 5k − 2k − 3k − 4 +m k + k + 2

(C.3) B = N5k5 + N4mk4 (4k − 5) + N3m2k3 3k2 − 6k + 10 + N2m3k2 8k2 − 2k3 b (C.4) −2k − 10) + Nm4k−2k4 + 4k3 − k2 + 6k + 5 − m5 k4 + 2k3 + 2k2 + k + 1 √ 2 2 2 2 C1 = N k + 2Nmk(k − 1) + m (k + 1) + [Nk − m(k + 1)] A (C.5) √ 2 2 2 2 CN+1 = N k + 2Nmk(k − 1) + m (k + 1) − [Nk − m(k + 1)] A (C.6)

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