EFFECT OF SLIDING FRICTION ON SPUR AND HELICAL GEAR DYNAMICS AND VIBRO-ACOUSTICS DISSERTATION. the Degree Doctor of Philosophy in the Graduate School

Transcription

1 EFFECT OF SLIDING FRICTION ON SPUR AND HELICAL GEAR DYNAMICS AND VIBRO-ACOUSTICS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Song He, B.S., M.S. * * * * * The Ohio State University 2008 Dissertation Committee: Dr. Rajendra Singh, Advisor Dr. Ahmet Kahraman Dr. Ahmet Selamet Dr. Marcelo Dapino Approved by Adviser Graduate Program in Mechanical Engineering

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3 ABSTRACT This study examines the salient effects of sliding friction on spur and helical gear dynamics and associated vibro-acoustic sources. First, new dynamic formulations are developed for spur and helical gear pairs based on a periodic description of the contact point and realistic mesh stiffness. Difficulty encountered in the existing discontinuous models is overcome by characterizing a smoother transition during the contact. Frictional forces and moments now appear as either excitations or periodically-varying parameters, since the frictional force changes direction at the pitch point/line. These result in a class of periodic ordinary differential equations with multiple and interacting coefficients, which characterize the effect of sliding friction in spur or helical gear dynamics. Predictions (based on multi-degree-of-freedom analytical models) match well with a benchmark finite element/contact mechanics code and/or experimental results. Second, new analytical solutions are constructed which provide an efficient evaluation of the frictional effect as well as a more plausible explanation of dynamic interactions in multiple directions. Both single- and multi-term harmonic balance methods are utilized to predict dynamic mesh loads, friction forces and pinion/gear displacements. Such semi-analytical solutions explain the presence of higher harmonics in gear noise and vibration due to exponential modulations of the periodic stiffness, ii

4 dynamic transmission error and sliding friction. This knowledge also analytically reveals the effect of the tooth profile modification in spur gears on the dynamic transmission error, under the influence of sliding friction. Further, the Floquet theory is applied to obtain closed-form solutions of the dynamic response for a helical gear pair, where the effect of sliding friction is quantified by an effective piecewise stiffness function. Analytical predictions, under both homogeneous and forced conditions, are validated using numerical simulations. The matrix-based methodology is found to be computationally efficient while leading to a better assessment of the dynamic stability. Third, an improved source-path-receiver vibro-acoustic model is developed to quantify the effect of sliding friction on structure-borne noise. Interfacial bearing forces are predicted for the spur gear source sub-system given two gear whine excitations (static transmission error and sliding friction). Next, a computational model of the gearbox, with embedded bearing stiffness matrices, is developed to characterize the motilities of structural paths. Radiated sound pressure is then estimated by using two numerical techniques (the Rayleigh integral method and a substitute source technique). Predicted pressures match well with measured noise data over a range of operating torques. In particular, the proposed vibro-acoustic model quantifies the contribution of sliding friction, which could be significant when the transmission error is minimized through tooth modifications. iii

5 DEDICATION Dedicated to my parents and wife iv

6 ACKNOWLEDGMENT I would like to express my sincere appreciation to my advisor, Professor Rajendra Singh, for his time, guidance, and support over the years both in my academic research and personal life. His intellectual insight and encouragement had a huge impact on my professional growth. I would also like to thank Professor Ahmet Kahraman, Professor Ahmet Selamet and Professor Marcelo Dapino for their services on the doctoral committee and for providing constructive suggestions. I sincerely thank Dr. Todd Rook for providing valuable suggestions. I greatly appreciate Dr. Rajendra Gunda for granting access to the Calyx software and for offering insightful comments. I gracefully acknowledge the experimental work conducted by Vivake Asnani and Fred Oswald, as well as the collaboration with Allison Lake. Dr. Chengwu Duan is thanked for helping me in both my research and personal life. All colleagues in the Acoustics and Dynamic Laboratory are acknowledged for their encouragement and friendship. I thank Professor Goran Pavić, Professor Jean-Louis Guyader and Corinne Lotto for their advice and kind help during my stay in INSA Lyon. The financial support from the Army Research Office, EU s Marie Curie Fellowship and OSU Presidential Fellowship is gracefully appreciated. Finally, I would like to thank my parents and my wife, Lihua, for their love and encouragement throughout my pursuit for the doctoral degree. v

7 VITA October 24, Born Jiangsu, China B.S. Instrumentation Engineering Shanghai Jiao Tong University Shanghai, China M.S. Mechanical Engineering The Ohio State University Graduate Teaching and Research Associate, Mechanical Engineering The Ohio State University Marie Curie Fellow (EU) National Institute of Applied Science Lyon, France Presidential Fellow (Graduate School) The Ohio State University Research Publications PUBLICATIONS 1. He, S., Gunda, R., and Singh, R., 2007, Effect of Sliding Friction on the Dynamics of Spur Gear Pair with Realistic Time-Varying Stiffness, Journal of Sound and Vibration, 301, pp He, S., Gunda, R., and Singh, R., 2007, Inclusion of Sliding Friction in Contact Dynamics Model for Helical Gears, ASME Journal of Mechanical Design, 129(1), pp vi

8 3. He, S., Cho, S., and Singh, R., 2008, Prediction of Dynamic Friction Forces in Spur Gears using Alternate Sliding Friction Formulations, Journal of Sound and Vibration, 309(3-5), pp FIELDS OF STUDY Major Field: Mechanical Engineering Dynamics of Mechanical Systems Vibro-Acoustics and Noise Control vii

9 TABLE OF CONTENTS Page Abstract... ii Dedication... iv Acknowledgment... v Vita... vi List of Tables... xii List of Figures...xiii List of Symbols... xx Chapter 1 Introduction Motivation Literature Review Problem Formulation Key Research Issues Scope, Assumptions and Objectives References for Chapter Chatper 2 Spur Gear Dynamics with Sliding Friction and Realistic Mesh Stiffness Introduction Problem Formulation Objectives and Assumptions Timing of Key Meshing Events Calculation of Realistic Time-Varying Tooth Stiffness Functions Analytical Multi-Degree-of-Freedom Dynamic Model Shaft and Bearing Stiffness Models Dynamic Mesh and Friction Forces MDOF Model Analytical SDOF Torsional Model viii

10 2.5 Effect of Sliding Friction in Example I Validation of Example I Model using the FE/CM Code Effect of Sliding Friction MDOF System Resonances Effect of Sliding Friction in Example II Empirical Coefficient of Friction Effect of Tip Relief on STE and k(t) Phase Relationship between Normal Load and Friction Force Excitations Prediction of the Dynamic Responses Experimental Validation of Example II Models Conclusion Chapter 3 Prediction of Dynamic Friction Forces Using Alternate Formulations Introduction MDOF Spur Gear Model Spur Gear Model with Alternate Sliding Friction Models Model I: Coulomb Model Model II: Benedict and Kelley Model Model III: Formulation Suggested by Xu et al Model IV: Smoothened Coulomb Model Model V: Composite Friction Model Comparison of Sliding Friction Models Validation and Conclusion References for Chapter Chapter 4 Construction of Semi-Analytical Solutions to Spur Gear Dynamics Introduction Problem Formulation Semi-Analytical Solutions to the SDOF Spur Gear Dynamic Formulation Direct Application of Multi-Term Harmonic Balance (MHBM) ix

11 4.3.2 Semi-Analytical Solutions Based on One-Term HBM Iterative MHBM Algorithm Analysis of Sub-Harmonic Response and Dynamic Instability Semi-Analytical Solutions to 6DOF Spur Gear Dynamic Formulation Conclusion References for Chapter Chapter 5 Effect of Sliding Friction on the Vibro-Acoustics of Spur Gear System Introduction Source Sub-System Model Structural Path with Friction Contribution Bearing and Housing Models Experimental Studies and Validation of Structural Model Comparison of Structural Paths in LOA and OLOA Directions Prediction of Noise Radiation and Contribution of Friction Prediction using Rayleigh Integral Technique Prediction using Substitute Source Method Prediction vs. Noise Measurements Conclusion References for Chapter Chapter 6 Inclusion of Sliding Friction in Helical Gear Dynamics Introduction Problem Formulation Mesh Forces and Moments with Sliding Friction Shaft and Bearing Models Twelve DOF Helical Gear Pair Model Role of Sliding Friction Illustrated by an Example Conclusion x

12 References for Chapter Chapter 7 Analysis of Helical Gear Dynamics using Floquet Theory Introduction Linear Time-Varying Formulation Analytical Solutions by Floquet Theory Response to Initial Conditions Forced Periodic Response Conclusion References for Chapter Chapter 8 Conclusion Summary Contributions Future Work References for Chapter Bibliography xi

13 LIST OF TABLES Table Page 2.1 Parameters of Example I: NASA-ART spur gear pair (non-unity ratio) Parameters of Example II-A and II-B: NASA spur gear pair (unity ratio). Gear pair with the perfect involute profile is designated as II-A case and the one with tip relief is designated as II-B case Averaged coefficient of friction predicted over a range of operating conditions for Example II by using Benedict and Kelly s empirical equation [2.14] Parameters of the example case: NASA spur gear pair with unity ratio (with long tip relief) Comparison of measured natural frequencies and finite element predictions Relationship between Contact Zones and Contact Regions for the NASA-ART helical gear pair xii

14 LIST OF FIGURES Figure Page 1.1 (a) Schematic of the spur gear contact, where LOA and OLOA represent the lineof-action direction and off line-of-action direction, respectively. (b) Directions of the sliding velocity (V), normal mesh load and friction force in spur gears Comparison of periodic mesh stiffness functions for a spur gear pair MDOF helical gear dynamic model (proposed in Chapter 6) and its contact mechanics with sliding friction Block diagram for the vibro-acoustics of a simplified geared system with two excitations at the gear mesh (as proposed in Chapter 5) Snap shot of contact pattern (at t = 0) in the spur gear pair of Example I Tooth mesh stiffness functions of Example I calculated by using the FE/CM code (in the static mode). (a) Individual and combined stiffness functions. (b) Comparison of the combined stiffness functions Schematic of the bearing-shaft model Normal and friction forces of analytical (MDOF) spur gear system model Validation of the analytical (MDOF) model by using the FE/CM code (in the dynamic mode). Here, results for Example I are given in terms of δ () t and its spectral contents ( f ) with t c = 2.4 ms and f m = Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = Validation of the analytical (MDOF) model by using the FE/CM code (in the dynamic mode). Here, results for Example I are given in terms of FpBx () t and its spectral contents FpBx ( f ) with t c = 2.4 ms and f m = Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = xiii

15 2.7 Validation of the analytical (MDOF) model by using the FE/CM code (in the dynamic mode). Here, results for Example I are given in terms of FpBy ( t ) and its spectral contents FpBy ( f ) with t c = 2.4 ms and f m = Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = Effect of µ on δ () t based on the linear time-varying SDOF model for Example I at T p = 2000 lb-in. Here, t c = 1 s Coefficient of friction µ as a function of the roll angle for Example II, as predicted by using Benedict & Kelley s empirical equation [2.14]. Here, oil temperature is 104 deg F and T = 500 lb-in Mesh harmonics of the static transmission error (STE) calculated by using the FE/CM code (in the static mode) for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B) Tooth stiffness functions of a single mesh tooth pair for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B) Combined tooth stiffness functions for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B) Dynamic loads predicted for Example II at 500 lb-in, 4875 RPM and 140 F with t c = 0.44 ms: (a) Normal loads of gear pair with perfect involute profile (II-A); (b) normal loads of gear par with tip relief (II-B); (c) friction forces of gear pair with perfect involute profile (II-A); (d) friction forces of gear pair with tip relief (II-B) Dynamic shaft displacements predicted for Example II at 500 lb-in, 4875 RPM and 140 F with t c = 0.44 ms and f m = 2275 Hz: (a) xp ( t ) ; (b) X p ( f ) ; (c) yp ( t ) and (d) Yp ( f ) Dynamic transmission error predicted for Example II at 500 lb-in, 4875 RPM and 140 F with t c = 0.44 ms and f m = 2275 Hz: (a) δ ( t) ; (b) ( f ) Sensors inside the NASA gearbox (for Example II-B) Mesh harmonic amplitudes of X p as a function of the mean torque at 140 F. All values are normalized with respect to the amplitude of Y p at the first mesh harmonic xiv

16 2.18 Mesh harmonic amplitudes of y p as a function of the mean torque at 140 F for Example II-B. All values are normalized with respect to the amplitude of y p at the first mesh harmonic Predicted dynamic transmission errors (DTE) for Example II over a range of torque at 140 F: (a) gear pair with perfect involute profile (II-A); (b) with tip relief (II-B). All values are normalized with respect to the amplitude ofδ (II-A) at the first mesh harmonic with 100 lb-in Mesh harmonic amplitudes of x p as a function of temperature at 500 lb-in for Example II-B. All values are normalized with respect to the amplitude of y p at the first mesh harmonic Mesh harmonic amplitudes of y p as a function of temperature at 500 lb-in for Example II-B. All values are normalized with respect to the amplitude of y p at the first mesh harmonic (a) Snap shot of contact pattern (at t = 0) in the spur gear pair; (b) MDOF spur gear pair system; here kt ( ) is in the LOA direction (a) Comparison of Model II [3.7] and Model III [3.8] given T p = 22.6 N-m (200 lb-in) and Ω p = 1000 RPM. (b) Averaged magnitude of the coefficient of friction predicted as a function of speed using the composite Model V with T p = 22.6 N- m (200 lb-in). Here, t c is one mesh cycle Comparison of normalized friction models. Note that curve between 0 t/ t c < 1 is for pair # 1; and the curve between 1 t/ t c < 2 is for pair # Combined normal load and friction force time histories as predicted using alternate friction models given T p = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM Predicted LOA and OLOA displacements using alternate friction models given T = 56.5 N-m (500 lb-in) and Ω = 4875 RPM p p 3.6 Predicted dynamic transmission error (DTE) using alternate friction models given T = 56.5 N-m (500 lb-in) and Ω = 4875 RPM p p xv

17 3.7 Validation of the normal load and sliding friction force predictions: (a) at T p = 79.1 N-m (700 lb-in) and Ω p = 800 RPM; (b) at T p = 79.1 N-m (700 lb-in) and Ω p = 4000 RPM Realistic mesh stiffness functions of the spur gear pair example (with tip relief) given T p = 550 lb-in. (b) Periodic frictional functions (a) Normal (mesh) and friction forces of 6DOF analytical spur gear system model. (b) Snap shot of contact pattern (at t = 0) for the example spur gear pair Semi-analytical vs. numerical solutions for the SDOF model, expressed by Eq. (4.3), given T p = 550 lb-in, Ω p = 500 RPM, µ = (a) Time domain responses; (b) mesh harmonics in frequency domain Semi-analytical vs. numerical solutions for the SDOF model as a function of pinion speed with µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = Normalized determinant of the sub-harmonic matrix K u as a function of ωn / Ω NS with µ = 0.04: (a) T p = 100 lb-in.; (b) T p = 550 lb-in (a) 6DOF spur gear pair model and its subset of unity gear pair (3DOF model) used to study the natural frequency distribution; (b) Natural frequencies Ω N as a function of the stiffness ratio K / k B m 4.7 Semi-analytical vs. numerical solutions for the 6DOF model as a function of Ω p with µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = Semi-analytical vs. numerical solutions of the LOA displacement x p for the 6DOF model as a function of Ω p with K B /k m = 100, µ = 0.04 (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = Semi-analytical vs. numerical solutions of the OLOA displacement y p for the 6DOF model as a function of Ω p with K B /k m = 100, µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = xvi

18 4.10 Semi-analytical vs. numerical solutions of the LOA displacement x p for the 6DOF model as a function of Ω p with K B /k m = 0.37, µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = Semi-analytical vs. numerical solutions of the OLOA displacement y p for the 6DOF model as a function of Ω p with K B /k m = 0.37, µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = Block diagram for the vibro-acoustics of a simplified geared system with two excitations at the gear mesh Bearing forces predicted under varying T p given Ω p = 4875 RPM and 140 F. (a): LOA bearing force; (b) OLOA bearing force. Key: m is the mesh frequency index (a) Schematic of NASA gearbox; (b) Finite element model of NASA gearbox with embedded bearing stiffness matrices Comparison of the gearbox mode shape at 2962 Hz: (a) modal experiment result [5.12]; (b) finite element prediction (a) Experiment used to measure structural transfer functions; (b) comparison of transfer function magnitudes from gear mesh to the sensor on top plate Magnitudes of the combined transfer mobilities in two directions calculated at the sensor location on the top plate Comparison of normal surface velocity magnitudes and substitute source strength vectors under T p = 500 lb-in and Ω p = 4875 RPM. (a) Line 1: interpolated surface velocity on top plate; (b) Line 2: simplified 2D gearbox model with 15 substitute source points; (c) Line 3: substitute source strengths in complex plane for 2D gearbox. Column 1: mesh frequency index m = 1; Column 2: m = 2; Column 3: m = Normalized sound pressure (ref 1.0 Pa) predicted at the microphone 6 inch above the top plate under varying torque T p given Ω p = 4875 RPM and 140 F Overall sound pressure levels (ref: 2e-5 Pa) and their contributions predicted at the microphone 6 in above the top plate under Ω p = 4875 RPM and 140 F. (a) T p = 500 lb-in (optimal load for minimum transmission error); (b) T p = 800 lb-in. 149 xvii

19 6.1 Schematic of the helical gear pair system Tooth stiffness function calculated using Eq. (6.1) based on the FE/CM code [6.11] Contact zones at the beginning of a mesh cycle. (a) In the helical gear pair; (b) contact zones within contact plane. Key: PP is the pitch line; AA is the face width W; AD is the length of contact zone Z Predicted tooth stiffness functions Schematic of the bearing-shaft model. Here, the shaft and bearing stiffness elements are assumed to be in series to each other. Only pure rotational or translational stiffness elements are shown. Coupling stiffness terms K x θ, K y y θ are x not shown Time and frequency domain responses of translational pinion displacements uxp, uyp, u zp at T p = 2000 lb-in and Ω p = 1000 RPM. All displacements are normalized with respect to µinch (1 µm) Time and frequency domain responses of pinion bearing forces F SB, xp, F SB, yp and F SB, zp at T p = 2000 lb-in and Ω p = 1000 RPM. All forces are normalized with respect to 1 lb Time and frequency domain responses of composite displacements δ x, δy, δ z and velocity δ z at T p = 2000 lb-in and Ω p = 1000 RPM. All motions are normalized with respect to 39.37µinch (1µm) or 39.37µinch (1µm/s) Schematic of the helical gear pair system Contact zones at the beginning of a mesh cycle within the contact plane. Key: PP is the pitch line; AA is the face width W; AD is the length of contact zone Z Individual effective stiffness K e,i (t) along the contact zone, where T mesh is one mesh cycle Piece-wise effective stiffness function defined in six regions within one mesh cycle with µ = xviii

20 7.5 (a) Effective stiffness and (b) homogeneous responses predictions within two mesh cycles given x 0 = in., v 0 = 20 in./s at Ω p = 1000 RPM Predictions of damped homogeneous responses within two mesh cycles given x 0 = in., v 0 = 20 in./s, µ = 0.2 at Ω p = 1000 RPM Predictions of (undamped) forced periodic responses within two mesh cycles given x 0 = in., v 0 = 20 in./s, T p = 2000 lb-in, µ = 0.2 and Ω p = 1000 RPM Steady state forced periodic responses given x 0 = in., v 0 = 20 in./s, T p = 2000 lb-in., µ = 0.1 and Ω p = 1000 RPM: (a) DTE vs. time; (b) DTE spectra Predicted mesh harmonics of (undamped) forced periodic responses as a function of µ given x 0 = in., v 0 = 20 in./s, T p = 2000 lb-in and Ω p = 1000 RPM: (a) DTE; (b) slope of DTE Mapping of eigenvalue κ (absolute value) maxima as a function of the ratio of time-varying mesh frequency f mesh (t) to the system natural frequency f n xix

21 LIST OF SYMBOLS List of Symbols for Chapter 1 O P t V x y z ε pinion/gear center location pitch point time (s) contact point speed (in./s) line-of-action direction off line-of-action direction axial direction unloaded static transmission error (µin.) Subscripts 1 pinion 2 gear Abbreviations ART DOF DTE EHL LOA MDOF OLOA SDOF STE Advanced Rotorcraft Transmission degree-of-freedom dynamic transmission error elasto-hydrodynamic lubrication line-of-action multi-degree-of-freedom off line-of-action single degree-of-freedom static transmission error List of Symbols for Chapter 2 a, b shaft distance (in) C R surface roughness constant c viscous damping (lb-s/in) E Young s modulus (psi) F force (lb) f frequency (Hz) I area moment of inertia (in 4 ) i, j indices of gear tooth xx

22 J polar moment of inertia (lb-s 2 -in) K stiffness matrix (lb/in) K stiffness (lb/in) k tooth mesh stiffness (lb/in) L geometric length (in) N normal contact force (lb) n mesh index R tooth surface roughness (in) r radius (in) T torque (lb-in) t time (s) V e entraining velocity (in/s) V s sliding velocity (in/s) W n normal load per unit length of face width (lb/in) X moment arm (in) X Γ load sharing factor x motion variable along LOA axis (in) Y frequency spectrum of motion along OLOA axis (in) y motion variable along OLOA axis (in) frequency spectrum of dynamic transmission error (in) δ dynamic transmission error (in) ε static transmission error (in) γ nominal roll angle (rad) λ base pitch (in) µ coefficient of friction υ 0 dynamic viscosity (lb-s/in 2 ) θ vibratory angular displacement (rad) σ contact ratio Ω angular speed (rad/s) ζ viscous damping ratio Subscripts 0, 1 n indices of meshing teeth/modes avg average B bearing b base c (mesh) cycle e effective f friction g gear i index of gear tooth m mesh P pitch point p pinion xxi

23 S x y z shaft LOA direction OLOA direction axial direction Superscripts. first derivative with respect to time.. second derivative with respect to time Abbreviations ART DOF DTE FE/CM HPSTC LPSTC LTV LOA MDOF OLOA SDOF Operators ceil floor mod sgn Advanced Rotorcraft Transmission degree-of-freedom dynamic transmission error finite element/contact mechanics highest point of single tooth contact lowest point of single tooth contact linear time-varying line-of-action multi-degree-of-freedom off line-of-action single degree-of-freedom ceiling function floor function modulus function sign function List of Symbols for Chapter 3 b b H c E F G H J K k L m N empirical coefficient semi-width of Hertzian contact band viscous damping (lb-s/in) Young s modulus (GPa) force (lb) dimensionless material parameter dimensionless central film thickness polar moment of inertia (lb-s 2 -in) stiffness (lb/in) tooth mesh stiffness (lb/in) geometric length (in) mass (lb s 2 /in) normal contact force (lb) xxii

24 P h maximum Hertzian pressure (GPa) r radius (in) S surface roughness (µm) SR slide-to-roll ratio T torque (lb-in) t time (s) U speed parameter V, v velocity (m/s) W load parameter w n normal load per unit length of face width (N/mm) X moment arm (in) x motion variable along LOA axis (in) y motion variable along OLOA axis (in) Z face width (mm) α pressure angle (rad) η M dynamic viscosity (N-s/mm 2 ) Λ film parameter λ base pitch (in) µ coefficient of friction ν Poisson s ratio θ vibratory angular displacement (rad) ρ profile radii of curvature (mm) Φ regularizing factor σ contact ratio Ω angular speed (rad/s) ζ viscous damping ratio Subscripts 0, 1 n indices of meshing teeth avg average B bearing b base C Coulomb friction c (mesh) cycle comp composite e entraining component f friction g gear i index of gear tooth p pinion r rolling component S smoothened friction model s sliding component X Xu and Kahraman model xxiii

25 x y LOA direction OLOA direction Superscripts. first derivative with respect to time.. second derivative with respect to time effective value Abbreviations DOF DTE EHL HPSTC LPSTC LTV LOA MDOF OLOA Operators floor mod sgn degree-of-freedom dynamic transmission error elasto-hydrodynamic lubrication highest point of single tooth contact lowest point of single tooth contact linear time-varying line-of-action multi-degree-of-freedom off line-of-action floor function modulus function sign function List of Symbols for Chapter 4 A, B harmonic balance coefficients C damping parameter (lb-s/in) c viscous damping (lb-s/in) D Fourier differentiation matrix E gear constant F harmonic balance matrix F force (lb) f frictional function i, j indices J polar moment of inertia (lb-s 2 -in) K harmonic balance matrix K stiffness (lb/in) k tooth mesh stiffness (lb/in) L geometric length (in) M (friction) torque (lb-in) or mass (lb s 2 /in) m mass (lb s 2 /in) N normal contact force (lb) or harmonic order xxiv

26 n mesh index R base radius (in) r radius (in) S index T (normalized) period t time (s) X moment arm (in) Fourier coefficient vector δ dynamic transmission error (in) ε static transmission error (in) ϑ angle (rad) λ base pitch (in) µ coefficient of friction θ vibratory angular displacement (rad) σ contact ratio τ dimensionless time υ sub-harmonic index Ω angular speed (rad/s) ω mesh frequency (rad) ζ viscous damping ratio Subscripts 0, 1 n indices of meshing teeth B bearing b base e effective f friction h super harmonic matrix g gear i index of gear tooth k stiffness coefficient P pitch point p pinion u sub-harmonic matrix x LOA direction y OLOA direction δ dynamic transmission error coefficient Superscripts. first derivative with respect to time.. second derivative with respect to time first derivative with respect to dimensionless time second derivative with respect to dimensionless time ^ differential operator xxv

27 nominal value ~ iterative harmonic balance parameter + pseudo-inverse Abbreviations DFT DOF DTE FFT LOA LTV MDOF MHBM OLOA SDOF discrete Fourier transform degree-of-freedom dynamic transmission error fast Fourier transform line-of-action linear time-varying multi-degree-of-freedom multi-term harmonic balance method off line-of-action single degree-of-freedom Operators floor floor function mod modulus function sgn sign function matrix determinant List of Symbols for Chapter 5 e error F force (lb) f frequency (Hz) H transfer function H v Hankel function I identity matrix i, j indices of gear tooth J polar moment of inertia (lb-s 2 -in) L geometric length (in) K stiffness (lb/in) k tooth mesh stiffness (lb/in) k(ω) wave number m mass (lb s 2 /in) N normal contact force (lb) n natural (frequency) P sound pressure (Pa) Q source strength (Pa-in 2 ) xxvi

28 r radius (in) S surface area (in 2 ) T torque (lb-in) t time (s) V velocity (in/s) w weighting function X moment arm (in) x motion variable along LOA axis (in) Y mobility (in/s/lb) y motion variable along OLOA axis (in) α angle (rad) λ base pitch (in) µ coefficient of friction θ vibratory angular displacement (rad) ρ air density (lb s 2 /in 4 ) σ contact ratio Ω angular speed (rad/s) ω angular frequency (rad) Ξ velocity error matrix ζ viscous damping ratio Subscripts B b e f g i m P p S x y bearing base effective parameter friction gear index of gear tooth mean component path pinion shaft or source LOA direction OLOA direction Superscripts. first derivative with respect to time.. second derivative with respect to time ~ complex value vector -1 matrix inverse * complex conjugate Abbreviations xxvii

29 DOF LOA MIMO OLOA RMS STE degree-of-freedom line-of-action multi-input multi-output off line-of-action mean square root static transmission error Operators floor floor function mod modulus function sgn sign function absolute value List of Symbols for Chapter 6 E Young s modulus (psi) e unit vector along axis F force (lb) K tooth mesh stiffness (lb/in) k tooth mesh stiffness density (lb/in 2 ) I area moment of inertia (in 4 ) J polar moment of inertia (lb-s 2 -in) L length of contact line (in) l variable along contact line (in) M moment (lb-in) m mass (lb s 2 /in) N normal contact force (lb) N mesh index r radius (in) T torque (lb-in) T mesh mesh period (s) t time (s) u translational motion (in) W face width (in) v velocity of contact point (in/s) x LOA coordinate of contact point (in) z axial coordinate of contact point (in) β helical angle deformation of contact point (in) µ coefficient of friction λ base pitch (in) δ dynamic transmission error (in) φ pressure angle (deg) σ contact ratio xxviii

30 Θ θ Ω ζ (static) angular deflection (rad) vibratory angular displacement (rad) angular speed (rad/s) viscous damping ratio Subscripts 0, 1 n indices of meshing teeth A (shaft) cross section area (in 2 ) b base c contact point g gear h (coordinate) upper limit i index of gear tooth l (coordinate) lower limit P pitch point p pinion S shaft s sliding component V viscous component x LOA direction y OLOA direction z axial direction Superscripts. first derivative with respect to time.. second derivative with respect to time mean component <-1> matrix inverse T matrix transverse Abbreviations ART DOF FE/CM LOA LTV MDOF OLOA Advanced Rotorcraft Transmission degree-of-freedom finite element/contact mechanics line-of-action linear time-varying multi-degree-of-freedom off line-of-action Operators cross product ceil ceiling function floor floor function xxix

31 mod sgn modulus function sign function List of Symbols for Chapter 7 C viscous damping coefficient (lb-s/in) e unit vector along axis F force (lb) f frequency (Hz) G state matrix H transition matrix J polar moment of inertia (lb-s 2 -in) K tooth mesh stiffness (lb/in) k tooth mesh stiffness (lb/in) L length of contact line (in) M moment (lb-in) m mass (lb s 2 /in) r radius (in) T torque (lb-in) t time (s) v velocity (in/s) W face width (in) X state space response x LOA coordinate of contact point (in) Z contact zone z axial coordinate of contact point (in) β helical angle δ dynamic transmission error (in) ε static transmission error (in) Φ state transition matrix φ pressure angle (deg) γ basis solution κ eigenvalue λ base pitch (in) µ coefficient of friction Π Wronskian matrix θ vibratory angular displacement (rad) σ contact ratio τ integration variable Ω angular speed (rad/s) ζ viscous damping ratio Subscripts xxx

32 0, 1 n indices of meshing teeth c contact point e effective parameter g gear m mesh (frequency) p pinion x LOA direction y OLOA direction z axial direction Superscripts. first derivative with respect to time.. second derivative with respect to time time average Abbreviations DOF DTE FE/CM LOA LTV OLOA SDOF degree-of-freedom dynamic transmission error finite element/contact mechanics line-of-action linear time-varying off line-of-action single-degree-of-freedom Operators ceil ceiling function floor floor function LommelS1 Lommel function mod modulus function sgn sign function absolute value List of Symbols for Chapter 8 µ coefficient of friction Abbreviations DOF DTE LTV LOA MDOF OLOA degree-of-freedom dynamic transmission error linear time-varying line-of-action multi-degree-of-freedom off line-of-action xxxi

33 SDOF single-degree-of-freedom xxxii

34 CHAPTER 1 INTRODUCTION 1.1 Motivation Spur and helical gears are widely used in vehicles and mechanical devices to transmit large torques while maintaining a constant input-to-output speed ratio. One remaining challenge for modern gear engineering is the reduction of gear noise in ground and air vehicles such as heavy duty trucks and helicopters. Typically, steady state gear (whine) noise is generated by several sources [ ]. Virtually all of the prior researchers [ ] have assumed that the main source is static transmission error (STE), which is defined as the derivation from the ideal (kinematic) tooth profile induced by manufacturing errors and elastic deformations. Accordingly, design engineers tend to reduce STE, via improved manufacturing processes and tooth modifications [1.7]. Yet, at higher torque loads, noise levels are still relatively high though STE might be somewhat minimal (say at the design loads). In other cases, the trend in sound pressure levels does not necessarily match the STE vs. torque curves [1.2]. Typical examples include experimental data on the Advanced Rotorcraft Transmission (ART) gears tested by 1

35 NASA Glenn and OSU [ ]. These suggest that additional vibro-acoustic sources must be considered. The relative speed between V 2 and V 1 of two meshing gear teeth (with centers at O 1 and O 2 ), as depicted in Figure 1.1(a), changes direction at the pitch point P during each contact event, thus providing additional periodic excitations normal to the direction of contact, as shown in Figure 1.1(b). Certain unique characteristics of the gear tooth sliding make it a potentially dominant factor, despite the somewhat lower magnitudes of friction force. First, due to the reversal in the direction during meshing action, friction is associated with a large oscillatory component, which causes both higher magnitudes as well as higher bandwidth in dynamic responses. Furthermore, friction is more significant at higher torque and lower speeds. In reality, frictional source mechanism is associated with surface roughness, lubrication regime properties, time-varying friction forces/torques and mesh interface dynamics. These lead to interesting gear dynamic phenomena, such as super-harmonic response, unstable regimes, sub-harmonic resonance and angular modulation [ ]. Clearly, the diverse effects of friction can only be analyzed by adopting an intra-disciplinary approach, wherein the principles of meshing kinematics, contact and tribological characteristics, dynamics and noise propagation mechanisms are integrated into a cohesive model. 2

36 (a) Sliding Velocity V 2 V 1 Normal Load Friction Force (b) Figure 1.1 (a) Schematic of the spur gear contact, where LOA and OLOA represent the line-of-action direction and off line-of-action direction, respectively. (b) Directions of the sliding velocity (V), normal mesh load and friction force in spur gears. 3

37 Historically, the friction between gear teeth and its cyclic nature have been either ignored or incorporated as an equivalent viscous damping term [ ]. Such an approach is clearly inadequate since viscous damping is essentially a passive characteristic and it cannot act as the external excitation to the governing system. Neither does it consider the dynamic effects in the off-line-of-action (OLOA) direction. Hence, there is a definite need for new or improved models that could predict the dynamic and vibro-acoustic responses of a geared system and clarify the role of sliding friction. This is the salient focus of this study. 1.2 Literature Review In a series of recent articles, Vaishya and Singh [ ] have provided an extensive review of prior work. They developed a spur gear pair model with sliding friction and rectangular mesh stiffness by assuming that load is equally shared among all the teeth in contact, as shown in Figure 1.2. They also solved the SDOF system equations in terms of the dynamic transmission error (DTE) by using the Floquet theory and the harmonic balance method [ ]. While the assumption of equal load sharing yields simplified expressions and analytically tractable solutions, it may not lead to a realistic model (as shown in Figure 1.2 and then Chapter 2). Houser et al. [ ] experimentally demonstrated that the friction forces play a pivotal role in determining the load transmitted to the bearings and housing in the OLOA direction; this effect is more pronounced at higher torque and lower speed conditions. 4

38 Tooth stiffness (lb/in) Figure 1.2 Comparison of periodic mesh stiffness functions for a spur gear pair. Key:, realistic load sharing (proposed in Chapter 2);, equal load sharing assumed by Vaishya and Singh [1.13]. Velex and Cahouet [1.18] described an iterative procedure to evaluate the effects of sliding friction, tooth shape deviations and time-varying mesh stiffness in spur and helical gears and compared simulated bearing forces with measurements. They reported significant oscillatory bearing forces at lower speeds that are induced by the reversal of friction excitation with alternating tooth sliding direction. In a subsequent study, Velex and Sainsot [1.19] analytically found that the Coulomb friction should be viewed as a non-negligible excitation source to error-less spur and helical gear pairs, especially for translational vibrations and in the case of high contact ratio gears. However, their work was confined to a study of excitations and the effects of tooth modifications were not 5

39 considered. Lundvall et al. [1.20] considered profile modifications and manufacturing errors in a multi-degree-of-freedom (MDOF) spur gear model and examined the effect of sliding friction on the angular dynamic motions. By utilizing a numerical method, they reported that the profile modification has less influence on the dynamic transmission error when frictional effects are included. However, incorporation of the time-varying sliding friction and the realistic mesh stiffness functions into an analytical (MDOF) formulation and their dynamic interactions remain unsolved. In all of the work mentioned above and related literature [ ], the sliding friction phenomenon has been typically formulated by assuming the Coulomb formulation with a constant coefficient of friction for modeling convenience. This is partially related to the difficulty associated with the measurement of friction force in a gear mesh. In reality, tribological conditions change continuously due to varying mesh properties, dynamic fluctuations and lubricant film thickness as the gears roll through a full cycle [ ]. Thus, coefficient of friction varies instantaneously with the spatial position of each tooth and the direction of friction force changes at the pitch point. Alternate tribological theories, such as elasto-hydrodynamic lubrication (EHL), boundary lubrication or mixed regime, have been employed to explain the sliding friction under varying operating conditions [ ]. For instance, Benedict and Kelley [1.21] proposed an empirical dynamic friction coefficient under mixed lubrication regime based on measurements on a roller test machine. Xu et al. [ ] recently proposed yet another friction formula that is obtained by using a non-newtonian, thermal EHL formulation. Duan and Singh [1.27] developed a smoothened Coulomb model for dry friction in torsional dampers; it could be applied to gears to obtain a smooth transition at 6

40 the pitch point. Hamrock and Dawson [1.28] suggested an empirical equation to predict the minimum film thickness for two disks in line contact. They calculated the film parameter, which could lead to a composite, mixed lubrication model for gears. Rebbechi et al. [1.8] have successfully used root strains to compute friction force under dynamic conditions. Recently, Vaishya and Houser [1.9] have shown that quasi-static measurement of friction force is possible by using the technique of digital filtering to eliminate the dynamic effects. However, no comprehensive work could be found which critically evaluates the existing lubrication theories in the framework of an actual gear mesh. Also, no prior work has incorporated the time-varying coefficient of friction into MDOF gear dynamics or examined its effect. Sliding friction at gear teeth also manifests as a noise source, as contended by the dynamic tests by Houser et al. [1.16]. Borner and Houser [1.17] predicted the dynamic forces due to friction and qualitatively discussed the radiated sound from the housing. Most studies on gearbox system dynamics [1.2] have relied on a combination of detailed finite element, boundary element and semi-analytical methods. Van Roosmalen [1.29] formulated a gearbox model including analytical formulations for the vibration at the gears due to tooth deflections and the vibration transfer through the bearings. Lim et al. [ ] developed a lumped parameter model with a rigid casing and a finite element model with a flexible casing for a simple geared system. However, finite and boundary element methods often require extensive computational time. Over the last four decades, some simplified lumped parameter models have been developed though few have incorporated the torsional and translational motions in both the line-of-action (LOA) and OLOA directions. Steyer [1.34] examined a single mesh geared system with 6 DOF. By 7

41 assuming the housing mass is much larger than the gears and shafts, an impedance mismatch was created with a rigid boundary condition at the bearing location. Thus, the internal geared system was modeled separately and analytical expressions were presented for a unity gear pair in terms of the resulting force transmissibility curve. Kartik [1.35] developed a frequency-response based model to predict noise radiation from gearbox housings with a multi-mesh gear set. His work showed that the bearing and mesh stiffness significantly affects the sound pressure in the high frequency range while the casing stiffness controls the response in the range below 4 khz. However, the transfer function relating the bearing forces to the equivalent force at the housing panel was based on limited experiments. Overall, the above mentioned system models fall short of providing a complete vibro-acoustic model. 1.3 Problem Formulation Key Research Issues Governing equations for gear dynamics should lead to a class of damped inhomogeneous periodic differential equations [ ] with multiple interacting coefficients [ ]. Although similar equations may also be found in a variety of disciplines such as communication networks [1.36] and electrical circuits [1.39], the gear friction problems, however, significantly differ from existing models such as the classical Hill s equations [1.36] in several ways. First, unlike classic friction problems in most mechanical systems, the direction of gear friction is normal rather than in the direction of 8

42 nominal motions. Second, the frictional forces and moments emerge on both sides of the governing equations as either excitations or periodically-varying parameters. Also, the periodic damping should capture not only the kinematic effects but also the energy dissipation due to sliding friction. Third, the periodic mesh stiffness is not confined to a rectangular wave assumed by Manish and Singh [ ], or a simple sinusoid as in the Mathieu s equation [1.36]. Instead, they should describe realistic, yet continuous, profiles of Figure 1.2 resulting from a detailed finite element/contact mechanics analysis [1.40]. Lastly, the stiffness and viscous damping terms incorporate combined (but phase correlated) contributions from all (yet changing) tooth pairs in contact. Historically, such periodic differential equations are seldom investigated and limited prior research efforts, as reported in the literature review [ ], are based largely on numerical integration and the Fast Fourier Transform algorithm. Consequently, there is a clear need for closed form analytical (say by using the Floquet theory) and semi-analytical (say by using the multi-term harmonic balance method) solutions to the dynamic responses of spur and helical gear pairs under the influence of sliding friction. Recently, Velex and Ajmi [1.41] implemented a harmonic analysis to approximate the dynamic factors in helical gears (based on tooth loads and quasi-static transmission errors). Their work, however, does not describe the multi-dimensional system dynamics or include the frictional effect, which may lead to multiplicative terms as described earlier. The parametric friction force excitation may have an influence on the stability of the homogenous system. Further, for a satisfactory understanding of dynamic behaviors of gears, a higher number of degrees-of-freedom are required for analysis, such as the MDOF helical gear 9

43 model of Figure 1.3. This is essential for additional phenomena like friction force, torsional-flexural coupling, shaft wobble and axial shuttling, which are yet to be fully understood. Also, to represent practical geared systems, a generalized model is required that incorporates different gear design configurations, lubrication conditions and meshing parameters. Existing solution methodology [ ] has to be improved to compute the dynamic response of the entire gearbox, for a combined excitation of transmission error, sliding friction, mean torque and other sources. Subsequently, the relative contribution of various parameters and the resulting noise characteristics need to be understood. This requires an improved source-path-receiver model for the entire gearbox system that incorporates competing noise sources. ε () t Figure 1.3 MDOF helical gear dynamic model (proposed in Chapter 6) and its contact mechanics with sliding friction. 10

44 1.3.2 Scope, Assumptions and Objectives Chief goal of this research is to improve the earlier work by Vaishya and Singh [ ] by developing improved mathematical models and proposing new analytical solutions that will enhance our understanding of the influence of friction on gear dynamics and vibro-acoustic behavior. Many dynamic phenomena that emerge due to interactions between parametric variations (time-varying mesh stiffness and viscous damping) and sliding friction will be predicted, along with a better understanding of the relative contributions of transmission error versus sliding friction noise to the gear whine noise. The specific objectives of this study are therefore as follows: Extend Vaishya and Singh s work [ ] by developing improved MDOF dynamic system for a spur gear pair that incorporates realistic time-varying mesh stiffness functions, accurate representations of sliding friction and load sharing between meshing tooth pairs. (Chapter 2) Comparatively evaluate alternate sliding friction models [ ] and predict the interfacial friction forces and motions in the OLOA direction. Also, validate dynamic system models and analytical solutions by comparing predictions to numerical solutions, the benchmark finite element/contact mechanics code as well as measurements. (Chapters 2 and 3) Propose a semi-analytical algorithm based on both single- and multi-term harmonic balance methods to quickly construct frequency responses of multidimensional spur gear dynamics with sliding friction. This should provide new insights into the dynamic interactions between parametric excitations. (Chapter 4) 11

45 Propose a refined source-path-receiver model that characterizes the structural paths in two directions and develop analytical tools to efficiently predict the whine noise radiated from gearbox panels and quantify the contribution of sliding friction to the overall whine noise. Analytical predictions of the structural transfer function and noise radiation will be compared with measurements. (Chapter 5) Propose a new three-dimensional formulation for helical gears to characterize the dynamics associated with the contact plane including the reversal at the pitch line due to sliding friction. A 12 DOF model will be developed which includes the rotational and translational motions along the LOA, OLOA and axial directions as well as the bearing/shaft compliances. (Chapter 6) Develop improved closed form solutions for the linear time-varying helical gear system in terms of the dynamic transmission error under the effect of sliding friction by using the Floquet theory. (Chapter 7) Scope and assumptions include the following: For the internal spur and helical gear pair sub-systems, the pinion and gear are modeled as rigid disks. The elastic deformations of the shaft and bearings are modeled using lumped elements which are connected to a rigid casing. Also, vibratory angular motions are small in comparison to the mean motion, and the mean load is assumed to be high such that the dynamic load is not sufficient to cause tooth separations [1.42]. If these assumptions are not made, the system model would have implicit non-linearities. Consequently, the position of the line of contact and relative sliding velocity depend only on the nominal angular motions; this leads to a linear time-varying system formulation. Note that different mesh stiffness schemes are assumed for spur and helical gears: For the spur gear analysis, the realistic 12

46 and continuous mesh stiffness is considered based on an accurate finite element/contact mechanics analysis code [1.40]; Thus, the time-varying stiffness is indeed an effective function which may also include the effect of profile modifications. For the helical gear analysis, however, only those gears with perfect involute profiles are considered and the mesh stiffness per unit length along the contact line (or stiffness density) is assumed to be constant [1.19]. This is equivalent to the equal load sharing assumption by Vishya and Singh [ ]. Such limitation may be further examined in future work. For the structure-borne whine noise model of the gearbox system, a source-pathreceiver model of Figure 1.4 is used. All the assumptions as mentioned above are embedded in the modeling of the internal gear pair sub-system. The unloaded static transmission error and sliding friction are considered as the two main excitations to the system; these are assumed to be most dominant in the LOA and OLOA directions, respectively. Hence, only corresponding structural paths in these two directions are considered by neglecting the moment transfer in the bearing matrices. Also, by assuming the housing mass is much larger than the gears and shafts, an impedance mismatch is created with a rigid boundary condition at the bearing location. Thus, the internal geared system could be modeled separately and its resulting force response provides force excitations to the structural paths. Finally, for the NASA gearbox used as the case study, the box plate is assumed to be the main radiator due to its relatively high mobility as well as the way the gearbox was assembled. Finally, it is worthwhile to mention that all chapters of this thesis are written in a self-contained manner in terms of formulation, literature review, methods and results. 13

47 Transmission error Sliding friction SOURCE 6 DOF linear-timevarying spur gear pair model + shafts LOA bearing forces OLOA bearing forces Coupling at bearings RECEIVER PATH Sound pressure Radiation model Housing velocity Housing structure model Figure 1.4 Block diagram for the vibro-acoustics of a simplified geared system with two excitations at the gear mesh (as proposed in Chapter 5). References for Chapter 1 [1.1] Ozguven, H. N., and Houser, D. R., 1988, Mathematical Models Used in Gear Dynamics - a Review, Journal of Sound and Vibration, 121, pp [1.2] Lim, T. C., and Singh, R., 1989, A Review of Gear Housing Dynamics and Acoustic Literature, NASA-Technical Memorandum, 89-C-009. [1.3] Oswald, F. B., Seybert, A. F., Wu, T. W., and Atherton, W., 1992, Comparison of Analysis and Experiment for Gearbox Noise, Proceedings of the International Power Transmission and Gearing Conference, Phoenix, pp [1.4] Baud, S., and Velex, P., 2002, Static and Dynamic Tooth Loading in Spur and Helical Geared Systems-Experiments and Model Validation, American Society of Mechanical Engineers, 124, pp [1.5] Comparin R. J., and Singh, R., 1990, An Analytical Study of Automotive Neutral Gear Rattle, ASME Journal of Mechanical Design, 112, pp [1.6] Mark, W. D., 1978, Analysis of the Vibratory Excitation of Gear Systems: Basic Theory, Journal of Acoustical Society of America, 63(5), pp

48 [1.7] Munro, R. G., 1990, Optimum Profile Relief and Transmission Error in Spur Gears, Proceedings of IMechE, Cambridge, England, 9-11 Apr., pp [1.8] Rebbechi, B. and Oswald, F. B., 1991, Dynamic Measurements of Gear Tooth Friction and Load, NASA-Technical Memorandum, [1.9] Vaishya, M., and Houser, D. R., 1999, Modeling and Measurement of Sliding Friction for Gear Analysis, American Gear Manufacturer Association Technical Paper, 99FTMS1, pp [1.10] Schachinger, T., 2004, The Effects of Isolated Transmission Error, Force Shuttling, and Frictional Excitations on Gear Noise and Vibration, MS Thesis, The Ohio State University. [1.11] Blankenship, G. W., and Singh, R., 1995, A New Gear Mesh Interface Dynamic Model to Predict Multi-Dimensional Force Coupling and Excitation, Mechanism and Machine Theory Journal, 30(1), pp [1.12] Padmanabhan, C., and Singh, R., 1995, Analysis of Periodically Excited Non- Linear Systems by a Parametric Continuation Technique, Journal of Sound and Vibration, 184(1), pp [1.13] Vaishya, M., and Singh, R., 2001, Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory, Journal of Sound and Vibration, 243(3), pp [1.14] Vaishya, M., and Singh, R., 2001, Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics, Journal of Sound and Vibration, 248(4), pp [1.15] Vaishya, M., and Singh, R., 2003, Strategies for Modeling Friction in Gear Dynamics, ASME Journal of Mechanical Design, 125, pp [1.16] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, Vibro-Acoustic Effects of Friction in Gears: An Experimental Investigation, SAE Paper # [1.17] Borner, J., and Houser, D. R., 1996, Friction and Bending Moments as Gear Noise Excitations, SAE Paper # [1.18] Velex, P., and Cahouet, V., 2000, Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics, ASME Journal of Mechanical Design, 122(4), pp

49 [1.19] Velex, P., and Sainsot. P, 2002, An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears, Mechanism and Machine Theory, 37, pp [1.20] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, A Flexible Multi-body Approach for Frictional Contact in Spur Gears, Journal of Sound and Vibration, 278(3), pp [1.21] Benedict, G. H., and Kelley B. W., 1961, Instantaneous Coefficients of Gear Tooth Friction, Transactions of the American Society of Lubrication Engineers, 4, pp [1.22] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, Prediction of Mechanical Efficiency of Parallel-Axis Gear Pairs, ASME Journal of Mechanical Design, 129 (1), pp [1.23] Xu, H., 2005, Development of a Generalized Mechanical Efficiency Prediction Methodology, PhD dissertation, The Ohio State University. [1.24] Seireg, A. A., 1998, Friction and Lubrication in Mechanical Design, Marcel Dekker, Inc., New York. [1.25] Baranov, V. M., Kudryavtsev, E. M., and Sarychev, G. A., 1997, Modeling of the Parameters of Acoustic Emission under Sliding Friction of Solids, Wear, 202, pp [1.26] Drozdov, Y. N., and Gavrikov, Y. A., 1968, Friction and Scoring under the Conditions of Simultaneous Rolling and Sliding of Bodies, Wear, 11, pp [1.27] Duan, C., and Singh, R., 2005, Super-Harmonics in a Torsional System with Dry Friction Path Subject to Harmonic Excitation under a Mean Torque, Journal of Sound and Vibration, 285(2005), pp [1.28] Hamrock, B. J., and Dowson, D., 1977, Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part III Fully Flooded Results, Journal of Lubrication Technology, 99(2), pp [1.29] Van Roosmalen, A., 1994, Design Tools for Low Noise Gear Transmissions, PhD Dissertation, Eindhoven University of Technology. [1.30] Lim, T. C., and Singh, R., 1990, Vibration Transmission Through Rolling Element Bearings. Part I: Bearing Stiffness Formulation, Journal of Sound and Vibration, 139(2), pp

50 [1.31] Lim, T. C., and Singh, R., 1990, Vibration Transmission Through Rolling Element Bearings. Part II: System Studies, Journal of Sound and Vibration, 139(2), pp [1.32] Lim, T. C., and Singh R., 1991, Statistical Energy Analysis of a Gearbox with Emphasis on the Bearing Path, Noise Control Engineering Journal, 37(2), pp [1.33] Lim, T. C., and Singh, R., 1991, Vibration Transmission Through Rolling Element Bearings. Part III: Geared Rotor System Studies, Journal of Sound and Vibration, 151(1), pp [1.34] Steyer, G., 1987, Influence of Gear Train Dynamics on Gear Noise, NOISE- CON 87 proceedings, pp [1.35] Kartik, V., 2003, Analytical Prediction of Load Distribution and Transmission Error for Multiple-Mesh Gear-Trains and Dynamic Studies in Gear Noise and Vibration, MS Thesis, The Ohio State University. [1.36] Richards, J. A., 1983, Analysis of Periodically Time-varying Systems, New York, Springer. [1.37] Jordan, D. W., and Smith, P., 2004, Nonlinear Ordinary Differential Equations, 3 rd Edition, Oxford University Press. [1.38] Thomsen, J. J., 2003, Vibrations and Stability, 2 nd Edition, Springer. [1.39] Kenneth S. K., Jacob K. W. and Alberto S-V, 1990, Steady-State Methods for Simulating Analog and Microwave Circuits, Kluwer Academic Publishers, Boston. [1.40] External2D (CALYX software), 2003, Helical3D User s Manual, ANSOL Inc., Hilliard, OH. [1.41] Velex, P., and Ajmi, M., 2007, Dynamic Tooth Loads and Quasi-Static Transmission Errors in Helical Gears Approximate Dynamic Factor Formulae, Mechanism and Machine Theory Journal, 42(11), pp [1.42] Blankenship, G. W., and Kahraman, A., 1995, Steady State Forced Response of a Mechanical Oscillator with Combined Parametric Excitation and Clearance Type Nonlinearity, Journal of Sound and Vibration, 185(5), pp

51 CHAPTER 2 SPUR GEAR DYNAMICS WITH SLIDING FRICTION AND REALISTIC MESH STIFFNESS 2.1 Introduction In a series of recent articles, Vaishya and Singh [ ] developed a spur gear pair model with periodic tooth stiffness variations and sliding friction based on the assumption that load is equally shared among all the teeth in contact. Using the simplified rectangular pulse shaped variation in mesh stiffness, they solved the single-degree-offreedom (SDOF) system equations in terms of the dynamic transmission error (DTE) using the Floquet theory and the harmonic balance method [ ]. While the assumption of equal load sharing yields simplified expressions and analytically tractable solutions, it may not lead to a realistic model. This chapter aims to overcome this deficiency by employing realistic time-varying tooth stiffness functions and the sliding friction over a range of operational conditions. New linear time-varying (LTV) formulation will be extended to include multi-degree-of-freedom (MDOF) system dynamics for a spur gear pair. 18

52 Vaishya and Singh [ ] have already provided an extensive review of prior work. In addition, Houser et al. [2.4] experimentally demonstrated that the friction forces play a pivotal role in determining the load transmitted to the bearings and housing in the off-line-of-action (OLOA) direction; this effect is more pronounced at higher torque and lower speed conditions. Velex and Cahouet [2.5] described an iterative procedure to evaluate the effects of sliding friction, tooth shape deviations and time-varying mesh stiffness in spur and helical gears and compared simulated bearing forces with measurements. They reported significant oscillatory bearing forces at lower speeds that are induced by the reversal of friction excitation with alternating tooth sliding direction. In a subsequent study, Velex and Sainsot [2.6] analytically found that the Coulomb friction should be viewed as a non-negligible excitation source to error-less spur and helical gear pairs, especially for translational vibrations and in the case of high contact ratio gears. However, their work was confined to a study of excitations and the effects of tooth modifications were not considered. Lundvall et al. [2.7] considered profile modifications and manufacturing errors in a MDOF spur gear model and examined the effect of sliding friction on the angular dynamic motions. By utilizing a numerical method, they reported that the profile modification has less influence on the dynamic transmission error when frictional effects are included. Nevertheless, two key questions remain unresolved: How to concurrently incorporate the time-varying sliding friction and the realistic mesh stiffness functions into an analytical (MDOF) formulation? How to quantify dynamic interactions between sliding friction and mesh stiffness terms especially when tip relief is provided to the gears? This chapter will address these issues. 19

53 2.2 Problem Formulation Objectives and Assumptions Chief objective of this chapter is to propose a new method of incorporating the sliding friction and realistic time-varying stiffness into an analytical MDOF spur gear model and to evaluate their interactions. Key assumptions are: (i) pinion and gear are modeled as rigid disks; (ii) shaft-bearings stiffness in the line-of-action (LOA) and OLOA directions are modeled as lumped elements which are connected to a rigid casing; (iii) vibratory angular motions are small in comparison to the mean motion; and (iv) Coulomb friction is assumed with a constant coefficient of friction µ. If assumption (iii) is not made, the system model would have implicit non-linearities. Consequently, the position of the line of contact and relative sliding velocity depend only on the nominal angular motions. An accurate finite element/contact mechanics (FE/CM) analysis code [2.8] will be employed, in the static mode, to compute the mesh stiffness at every time instant under a range of loading conditions. Here, the time-varying stiffness is calculated as an effective function which may also include the effect of profile modifications. The realistic mesh stiffness is then incorporated into the LTV spur gear model with the contributions of sliding friction. The MDOF formulation should describe both the LOA and OLOA dynamics; a simplified SDOF model will also be derived that describes the vibratory motion in the torsional direction. Proposed methods will be illustrated via two spur gear examples (designated as I and II) whose parameters are listed in Table 2.1 and Table 2.2. The MDOF model of Example I will be validated by using the FE/CM code 20

54 [2.8] in the dynamic mode. Issues related to tip relief will be examined in Example II in the presence of sliding friction. Finally, experimental results of Example II will be used to further validate our method. Parameter/property Pinion Gear Number of teeth Diametral pitch, in Pressure angle, deg Outside diameter, in Root diameter, in Face width, in Tooth thickness, in Gear mass, lb s 2 in E E-02 Polar moment of inertia, lb s 2 in 8.48E E-02 Bearing stiffness (LOA and OLOA), lb/in 20E6 Center distance, in 3.5 Profile contact ratio 1.43 Elastic modulus, psi 30E6 Density, lb s -2 in E-04 Poisson s ratio 0.3 Table 2.1 Parameters of Example I: NASA-ART spur gear pair (non-unity ratio) 21

55 Parameter/property Pinion/Gear Number of teeth 28 Diametral pitch, in -1 8 Pressure angle, deg 20 Outside diameter, in Root diameter, in Face width, in 0.25 Tooth thickness, in Roll angle where the tip modification starts (for II-B), deg 24.5 Straight tip modification (for II-B), in 7E-04 Center distance, in 3.5 Profile contact ratio 1.63 Elastic modulus, psi Density, lb s -2 in -4 30E6 7.30E-04 Poisson s ratio 0.3 Range of temperatures, F 104, 122, 140, 158, 176 Range of input torques, lb in 500, 600, 700, 800, 900 Table 2.2 Parameters of Example II-A and II-B: NASA spur gear pair (unity ratio). Gear pair with the perfect involute profile is designated as II-A case and the one with tip relief is designated as II-B case 22

56 2.2.2 Timing of Key Meshing Events Analytical formulations for a spur gear pair are derived via Example I (NASA- ART spur gear pair) with parameters of Table 2.1. For a generic spur gear pair with noninteger contact ratio σ, n = ceil(σ ) meshing tooth pairs need to be considered, where the ceil function rounds the σ element to the nearest integer towards a higher value. Consequently, two meshing tooth pairs need to be modeled for Example I (σ = 1.43). First, transitions in key meshing events within a mesh cycle need to be determined from the undeformed gear geometry for the construction of the stiffness function. Figure 2.1 is a snapshot for Example I at the beginning of the mesh cycle (t = 0). At that time, pair #1 (defined as the tooth pair rolling along line AC) just comes into mesh at point A and pair #0 (defined as the tooth pair rolling along line CD) is in contact at point C, which is the highest point of single tooth contact (HPSTC). As the gears roll, when pair #1 approaches the lowest point of single tooth contact (LPSTC) of point B at t = t B, pair #0 leaves contact. At t = t P, pair #1 passes through the pitch point P, and the relative sliding velocity of the pinion with respect to the gear is reversed, resulting in a reversal of the friction force. This should provide an impulse excitation to the system. Finally, pair #1 goes through point C at t = t c, completing one mesh cycle (t c ). These key events are defined below, where length L AC is equal to one base pitch λ. Ω p is the nominal pinion speed, r bp is the base radius of the pinion, t c λ =, Ω r p bp t t b c L λ AB =, t t p AP =. (2.1) c L λ 23

57 Ω g Ω p Figure 2.1 Snap shot of contact pattern (at t = 0) in the spur gear pair of Example I Calculation of Realistic Time-Varying Tooth Stiffness Functions The realistic time-varying stiffness functions are calculated using a FE/CM code, External2D [2.8]. An input torque T p is applied to the pinion rotating at mean braking torque T g on the gear and its angular velocity Ω p, and the Ω g obey the basic gear kinematics. Superposed on the nominal motions are oscillatory components denoted as θ p and θg for the pinion and gear, respectively. The normal contact forces N 0 (t), N 1 (t) and pinion deflection θ () t are then computed by performing a static analysis using p FE/CM software [2.8]. The stiffness function of the i th meshing tooth pair for a generic 24

58 spur gear pair is given by Eq. (2.2), where the floor function rounds the contact ratio σ to the nearest integer towards a lower value, i.e. floor( σ ) = 1 for Example I. Ni () t ki ( t) =, i= 0, 1,..., n= floor( σ ). (2.2) r θ () t bp p The stiffness function k(t) for a single tooth pair rolling through the entire meshing process is obtained by following the contact tooth pair for n = ceil(σ ) number of mesh cycles. Due to the periodicity of the system, expanded stiffness function ki ( t) of the i th meshing tooth pair is calculated at any time instant t as: [ ] k ( t) = k ( n i) t + mod( t, t ), i= 0, 1,..., n= floor( σ ). (2.3) i c c Here, mod is the modulus function defined as: mod( xy, ) = x y floor( x/ y), if y 0. (2.4) For Example I, calculated k 0 () t, k () 1 t functions and their combined stiffness are shown in Figure 2.2(a). Note that k () t and 0 k () 1 t are, in fact, different portions of kt () as 25 described in Eq. (2.3). Figure 2.2(b) compares the continuous kt () of the realistic load sharing model against the rectangular pulse shaped discontinuous kt () based on the equal load sharing formulation proposed earlier by Vaishya and Singh [ ].

59 Figure 2.2 Tooth mesh stiffness functions of Example I calculated by using the FE/CM code (in the static mode). (a) Individual and combined stiffness functions. Key:, total stiffness; stiffness of pair #0;, stiffness of pair #1. (b) Comparison of the combined stiffness functions. Key:, realistic load sharing;, equal load sharing as assumed by Vaishya and Singh [ ]. 26

60 2.3 Analytical Multi-Degree-of-Freedom Dynamic Model Shaft and Bearing Stiffness Models Next, we develop a generic spur gear pair model with 6 DOFs including rotational motions ( θ p and θ g ), LOA translations ( xp and x g ) and OLOA translations ( y p and y g ). The governing equations are derived in the subsequent sections. First, a simplified shaft model, as shown in Figure 2.3, is developed based on the Euler s beam theory [2.9]. Corresponding to the 6 DOFs mentioned above, only the diagonal term in the shaft stiffness matrix needs to be determined as follows, where E is the Young s modulus, I = π r 4 s /4 is the area moment of inertia for the shaft, and a and b are the distances from pinion/gear to the bearings. a+ b K 3 ( ) 2 Sx = KSy = EI a b + ab 3 3 ab, K 0 S θ =. (2.5) z The rolling element bearings are modeled using the bearing stiffness matrix K Bm formulation (of dimension 6) as proposed by Lim and Singh [2.10, 2.11]. Assume that each shaft is supported by two identical axially pre-loaded ball bearings with a mean axial displacement; the mean driving load T m generates a mean radial force F xm in the LOA direction and a moment M ym around the OLOA direction. The time-varying friction force and torque are not included in the mean loads. 27

61 K Bx /2 a K By /2 x b K Bx /2 θ z K By /2 y z Figure 2.3 Schematic of the bearing-shaft model. Corresponding to the 6 DOFs considered in our spur gear model, only two significant coefficients, K Bxx and K Byy, are considered for K Bm [2.10, 2.11]. The combined bearing-shaft stiffness ( K and K By in the LOA and OLOA directions) are Bx derived by assuming that the bearing and shaft stiffness elements act in parallel Dynamic Mesh and Friction Forces Figure 2.4 shows the mean torque and internal reaction forces acting on the pinion for Example I. For the sake of clarity, forces on the gear are not shown, which are equal in magnitude but opposite in direction to the pinion forces. Based on the Coulomb 28

62 friction law, the magnitude of friction force ( F ) is proportional to the nominal tooth f load (N) as Ff = µ N where µ is constant. The direction of F f is determined by the calculation of nominal relative sliding velocity, which results in the LTV system formulation. Denote X () t as the moment arm on the pinion for the friction force acting on the i th meshing tooth pair pi X ( t) = L + ( n i) λ + mod( Ω r t, λ), i = 0, 1,..., n= floor( σ). (2.6) pi XA p bp The corresponding moment arm for the friction force on the gear is X ( t) = L + iλ mod( Ω r t, λ), i = 0, 1,..., n= floor( σ). (2.7) gi YC g bg Assume time-varying mesh (viscous) damping coefficient and relate it to ki ( t ) by a time-invariant damping ratio m 2 2 ζ as follows, where Je = JpJg / ( J prbg + Jgrbp) c ( t) = 2 ζ k ( t) J, i= 0, 1,..., n= floor( σ ). (2.8) i mi i e The normal forces acting on the pinion are Npi () t = Ngi () t = ki () t rbpθp () t rbgθg () t ε p () t + xp () t xg () t + ci() t r bpθ p() t r bgθg() t ε p() t + x p() t x g() t, i = 0, 1,..., n= floor( σ). (2.9) 29

63 Here ε ( t ) is the profile error component of the static transmission error (STE), and x p(t) P and x g (t) denote the translational bearing displacements of pinion and gear, respectively. For a generic spur gear pair whose j th meshing pair passes through the pitch point within the mesh cycle, the friction forces in the i th meshing pair are derived as follows µ Npi ( t), i = 0, 1,..., j 1, Fpfi () t = µ Npi()sgn t mod( Ω prbp t, λ) + ( n i) λ L AP, i = j, µ Npi ( t), i = j, j + 1,..., n = floor( σ ), µ Ngi ( t), i = 0, 1,..., j 1, Fgfi ( t) = µ Ngi ( t)sgn mod( Ω grbgt, λ) + ( n i) λ LAP, i = j, µ Ngi ( t), i = j, j + 1,..., n = floor( σ ). (2.10a) (2.10b) Consequently, the friction forces for Example I of Figure 2.4 are given as: F () t = µ N () t, (2.10c) pf 0 p0 Fpf 1() t = µ Np 1()sgn t mod( Ωprbp t, λ) L AP, (2.10d) F () t = µ N () t, (2.10e) gf 0 g 0 Fgf1() t = µ Ng1()sgn t mod( Ωgrbgt, λ) L AP. (2.10f) 30

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

65 2.3.3 MDOF Model The governing equations for the torsional DOFs are n= floor( σ) n= floor( σ) J θ () t = T + X () t F () t r N () t, (2.11) p p p pi pfi bp pi i= 0 i= 0 n= floor( σ) n= floor( σ) J gθ g () t = Tg + Xgi () t Fgfi () t + rbg Ngi () t i= 0 i= 0. (2.12) The governing equations of the translational DOFs in the LOA direction are n= floor( σ ) mx () t+ 2 ζ K mx () t+ K x() t+ N () t = 0, (2.13) p p pbx pbx p p pbx g pi i= 0 n= floor( σ ) mx () t + 2 ζ K mx () t + K x() t+ N () t = 0. (2.14) g g gbx gbx g g gbx g gi i= 0 Here, K pbx and K gbx are the effective shaft-bearing stiffness in the LOA direction, and ζ pbx and ζ gbx are their damping ratios. Similarly, the governing equations of the translational DOFs in the OLOA direction are n= floor( σ ) my () t+ 2 ζ K my () t+ K y() t F () t= 0, (2.15) p p pb y pby p p pby p pfi i= 0 32 n= floor( σ ) my () t+ 2 ζ K my () t+ K y() t F () t = 0. (2.16) g g gb y gby g g gby g gfi i= 0

66 The composite DTE, which is the relative dynamic displacement of pinion and gear along the LOA direction, is defined as δ() t = r θ () t r θ () t + x () t x () t. (2.17) bp p bg g p g Finally, the dynamic bearing forces are as: F () t = K x () t 2 ζ K m x () t, (2.18a) pbx pbx p pbx pbx p p F () t = K y () t 2 ζ K m y () t, (2.18b) pby pby p pbb pby p p F () t = K x () t 2 ζ K m x () t, (2.18c) gbx gbx g gbb gbx g g F () t = K y () t 2 ζ K m y () t. (2.18d) gby gby g g Bb gby g g 2.4 Analytical SDOF Torsional Model When only the torsional DOFs of the spur gear pair are of interest, a simplified but equivalent SDOF model can be derived by assuming that the shaft-bearings stiffness is much higher than the mesh stiffness. After eliminating θ p (t) and θ g (t) in terms of the DTE, δ() t = r θ () t r θ () t, the governing SDOF model is obtained for a generic spur bp p bg g gear pair whose j th meshing pair passes through the pitch point within the mesh cycle: 33

67 µ µ n= floor( σ ) i= 0 n= floor( σ ) i= 0 n= floor( σ ) J δ( t) + c ( t) δ( t) + k ( t) δ( t) + e i i i= 0 sgn mod( Ω prbpt, λ) + ( n j) λ L AP X pj () t Jgrbp + Xgj () t J pr bg = ci() t δ() t + ki() t δ() t 2 2 J prbg+ Jgrbp. (2.19) n= floor( σ ) Te + c () () () () 2 2 i t ε p t + ki t ε p t + Jr + Jr p bg g bp i= 0 sgn mod( Ω prbpt, λ) + ( n j) λ L AP X pj () t Jgrbp + Xgj () t J pr bg ci() t ε p() t + ki() t ε p() t 2 2 J prbg+ Jgrbp Here the effective polar moment of inertia J e is consistent with that defined in Eq. (2.8) and the effective torque is T e = T p J g r bp + T g J p r bg. The dynamic response δ () t is controlled by three excitations: (i) time-varying T e, (ii) ε p () t and its derivative ε p () t and (iii) sliding friction. For Example I, the governing Eq. (2.19) could be simplified as [ ] [ ] J δ( t) + c ( t) + c ( t) δ( t) + k ( t) + k ( t) δ( t) + e X p1() t Jgrbp + Xg1() t Jpr bg µ c1() t δ() t + k1() t δ() t sgn mod(, ) 2 2 Ωprbpt λ L AP + Jr p bg Jr + g bp µ ( X () t J r + X () t J r ) p0 g bp g 0 p bg c0 () t δ() t + k0() t δ() t 2 2 Jr p bg+ Jr g bp T = + [ c1( t) + c0( t) ] ε p( t) + [ k1( t) + k0( t) ] ε p( t) + Jr e 2 2 p bg+ Jr g bp X p1() t Jgrbp + Xg1() t Jpr bg µ c1() t εp() t + k1() t εp() t 2 2 sgn mod( Ωprt bp, λ) L AP + Jr p bg+ Jr g bp 34 ( X p0() t Jgrbp + Xg0() t Jprbg) µ c ( t) ε ( t) + k ( t) ε ( t) + 0 p 0 p 2 2 Jr p bg Jr g bp (2.20)

68 2.5 Effect of Sliding Friction in Example I Validation of Example I Model using the FE/CM Code The governing equations of either SDOF or MDOF system models are numerically integrated by using a 4 th -5 th order Runge-Kutta algorithm with fixed time step. The ε () t and ε () t components are neglected, i.e. no manufacturing errors other p p than specified profile modifications are considered. Concurrently, the dynamic responses are independently calculated by running the FE/CM code [2.8] using the Newmark method. Predicted and computed results are compared with good correlations in terms of the DTE, and LOA and OLOA forces, as shown in Figure 2.5 to Figure 2.7. Note that time domain comparisons include both transient and steady state responses but the frequency domain results report only the steady state responses. Figure 2.5 shows that the sliding friction introduces additional DTE oscillations when the contact teeth pass through the pitch point. Figure 2.6 illustrates that the sliding friction enhances the dynamic bearing forces in the LOA direction, especially at the second mesh harmonic. This is because the moments associated with Fpfi() t and Fgfi () t are coupled with the moments of N pi (t) and N gi (t). 35

69 (t) (in) (f) (in) (t) (in) (f) (in) Figure 2.5 Validation of the analytical (MDOF) model by using the FE/CM code (in the dynamic mode). Here, results for Example I are given in terms of δ () t and its spectral contents ( f ) with t c = 2.4 ms and f m = Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = 0.2. Key:, Analytical (MDOF) model;, FE/CM code (in t domain); o, FE/CM code (in f domain). 36

70 x 10 3 x (a) Normalized time t / t (b) c x 10 3 x Mesh order n (c) Normalized time t / t c (d) Mesh order n Figure 2.6 Validation of the analytical (MDOF) model by using the FE/CM code (in the dynamic mode). Here, results for Example I are given in terms of FpBx () t and its spectral contents FpBx ( f ) with t c = 2.4 ms and f m = Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = 0.2. Key:, Analytical (MDOF) model;, FE/CM code (in t domain); o, FE/CM code (in f domain). 37

71 Further, the normal loads mainly excite the vibration in the LOA direction, as illustrated by Eqs. (2.9), (2.11) and (2.13). The scales of the bearing forces of Figure 2.7(a-b) for µ = 0 case are the same as those of Figure 2.7(c-d) for the sake of comparison. The bearing forces predicted by the MDOF model for µ = 0 case approach zero (within the numerical error range). This is consistent with the mathematical description of Eqs. ( ). Larger deviations at this point are observed in Figure 2.7(a-b) for the FE/CM analysis. Figure 2.7shows that the OLOA dynamics are more significantly influenced by the sliding friction when compared with the LOA results of Figure 2.6. In order to accurately predict the higher mesh harmonics, refined time steps (say more than 100 increments per mesh cycle) are needed. Consequently, the FE/CM analysis tends to generate an extremely large data file that demands significant computing time and postprocessing work. Meanwhile, the lumped model allows much finer time resolution while being computationally more efficient (by at least two orders of magnitude when compared with the FE/CM). Hence, the lumped model could be effectively used to conduct parametric design studies. 38

72 x x F pby (t) (lb) 0-5 F pby (f) (lb) (a) Normalized time t / t c (b) Mesh order n x x (t) (lb) F pby 0-5 F pby (f) (lb) (c) Normalized time t / t c (d) Mesh order n Figure 2.7 Validation of the analytical (MDOF) model by using the FE/CM code (in the dynamic mode). Here, results for Example I are given in terms of FpBy () t and its spectral contents FpBy ( f ) with t c = 2.4 ms and f m = Hz. Sub-figures (a-b) are for µ = 0 and (c-d) are for µ = 0.2. Key:, Analytical (MDOF) model;, FE/CM code (in t domain); o, FE/CM code (in f domain). 39

73 2.5.2 Effect of Sliding Friction Figure 2.8 shows the calculated DTE without any friction is almost identical to the STE at a very low speed ( Ω = 2.4 rpm). However, the sliding friction changes the p shape of the DTE curve. During the time interval t [0, t P ], the friction torque on the pinion opposes the normal load torque as shown in Figure 2.4, resulting in a higher value of the normal load that is needed to maintain the static equilibrium. Also, friction increases the peak-peak value of the DTE as compared with the STE. For the remainder of the mesh cycle t [ t, t ], friction torque acts in the same direction as the normal load P c torque. Thus a small value of normal load is sufficient to maintain the static equilibrium. Detailed parametric studies show that the amplitude of second mesh harmonic increases with the effect of sliding friction. 40

74 Figure 2.8 Effect of µ on δ () t based on the linear time-varying SDOF model for Example I at T p = 2000 lb-in. Here, t c = 1 s. Key:, µ = 0;, µ =

75 2.5.3 MDOF System Resonances For Example I, the nominal bearing stiffness KpBx = KpBy = KgBx = KgBy = lb/in are much higher than the averaged mesh stiffness k m. The couplings between the rotational and translational DOFs in the LOA direction are examined by using a simplified 3 DOF model as suggested by Kahraman and Singh [2.12]. Note that the DTE is defined here as δ = rbpθxp rbgθxg, and the undamped equations of motion are km km k m e δ m δ 0 0 mp 0 x p km ( km KpBx) k + + m xp = 0. (2.21) 0 0 m g x g k x g 0 m km ( km + KgBx) 2 2 Here, the effective mass is defined as me JpJg / ( rgj p rpjg) = +. The eigensolutions of Eq. (2.21) yield three natural frequencies: Two coupled transverse-torsional modes ( f 1 and f 3 ) and one purely transverse mode ( f 2 ); numerical values are: f = 5,130 Hz, 1 f 2 = 8,473 Hz and f 3 = 11,780 Hz. Predictions of Eq. (2.21) match well with the numerical simulations using the formulations of section 2.3 (though these results are not shown here). A comparative study verifies that one natural frequency of the MDOF model shifts away from that of the SDOF model (6,716 Hz) due to the torsional-translational coupling effects. In the OLOA direction, simulation shows that only one resonance is present at 1 f = K / m = 9,748 Hz, which is dictated by the bearing-shaft stiffness. pby 2π pby p 42

76 2.6 Effect of Sliding Friction in Example II Next, the proposed model is applied to Example II with the parameters of Table 2.2. The chief goal is to examine the effects of tip relief and sliding friction. Further, analogous experiments were conducted at the NASA Glenn Research Center Gear Noise Rig [2.13]. Comparisons with measurements will be given in section Empirical Coefficient of Friction The coefficient of friction varies as the gears travel through mesh, due to constantly changing lubrication conditions between the contact teeth. An empirical equation for the prediction of the dynamic friction variable, µ, under mixed lubrication has been suggested by Benedict and Kelley [2.14] based on a curve-fit of friction measurements on a roller test machine. Rebbechi et al. [2.15] verified this formulation by measuring the dynamic friction forces on the teeth of a spur gear pair. Their measurements seem to be in good agreement with the Benedict and Kelley equation except at the meshing positions close to the pitch point. This empirical equation, when modified to account for the average gear tooth surface roughness ( R avg ), is X ( γ ) W Γ n µγ ( ) = CRavg log10 2, νovs( γ) Ve ( γ) C Ravg 44.5 = 44.5 R avg. (2.22a,b) 43

77 where C Ravg is the surface roughness constant, W n is the normal load per unit length of face width, and υ o is the dynamic viscosity of the lubricant. Here Vs ( γ ) is the sliding velocity, defined as the difference in the tangential velocities of the pinion and gear, and V ( ) e γ is the entraining velocity, defined as the addition of the tangential velocities, for roll angle γ along the LOA. Further, R avg in our case was measured with a profilometer using a standard method [2.13]. Lastly, X Γ ( γ ) is the load sharing factor as a function of roll angle, and it was assumed based on the ideal profile of smooth meshing gears. Figure 2.9 shows µ as a function of roll angle calculated using Eq. (2.22). Since µ was assumed to be a constant earlier, an averaged value is found by taking an average over the roll angles between 19.8 and 21.8 degrees. Table 2.3 lists the µ values that were computed at each mean torque and oil temperature for Example II (with R avg = µ m). Temperature ( F) Torque (lb-in) Table 2.3 Averaged coefficient of friction µ predicted over a range of operating conditions for Example II by using Benedict and Kelly s empirical equation [2.14] 44

78 Figure 2.9 Coefficient of friction µ as a function of the roll angle for Example II, as predicted by using Benedict & Kelley s empirical equation [2.14]. Here, oil temperature is 104 deg F and T = 500 lb-in. Key P: Pitch point at deg Effect of Tip Relief on STE and k(t) The STE is calculated as a function of mean torque for both the perfect involute gear pair (designated as II-A) and then one with tip relief (designated as II-B) using FE/CM code. Figure 2.10 compares the amplitudes of STE spectra at mesh harmonics for both cases. (In this and following figures, predictions are shown as continuous lines for the sake of clarity though they are calculated only at discrete torque points.) 45

79 2.5 x Transmission error (in) (a) Torque (lb-in) Transmission error (in) Figure 2.10 Mesh harmonics of the static transmission error (STE) calculated by using the FE/CM code (in the static mode) for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B). Key:, n = 1;, n = 2;, n = 3. 46

80 The first two mesh harmonics are most significantly affected by the tip relief and they are minimal at the optimal mean torque around 500 lb-in. For both the II-A and II- B cases, typical kt () functions of a single meshing tooth over two complete mesh cycles are calculated using Eq. (2.2) for various mean torques, as shown in Figure 2.1. Note that kt () is defined as the effective stiffness since it incorporates the effect of profile modification such as the linear tip relief (II-B). Observe that although the maximum stiffness remains the same, application of the tip relief significantly changes the stiffness profile. For the perfect involute profile (II-A), steep slopes are observed in the vicinities near the single or two teeth contact regimes, and a smooth transition is observed in between these steep regimes. Also, kt () is found to be insensitive to a variation in the mean torque. However, with tip relief, an almost constant slope is found throughout the transition profile between single and two teeth contact regimes. Moreover, a smaller profile contact ratio (around 1.1 at 100 lb-in) is observed for the tip relief case when compared with around 1.6 (at all loads) for the perfect involute pair. The realistic kt () function is then incorporated into the lumped MDOF dynamic model. Figure 2.12 shows the combined kt () with contributions of both meshing tooth pairs over two mesh cycles for Example II. Observe that the profile of case II-A is insensitive to a variation in the mean torque, but the profile of case II-B shows a minimum around 500 lb-in. Frequency domain analysis reveals that the first two mesh harmonics are most significantly affected by the linear tip modification. Overall, it is evident that significant changes take place in the STE, tooth load distribution and mesh stiffness function due to the profile modification (tip relief), which may be explained by an avoidance of the corner contact at an optimized mean torque. 47

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

82 Figure 2.12 Combined tooth stiffness functions for Example II: (a) gear pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B). Key:, 100 lb-in;, 500 lb-in;, 900 lb-in. 49

83 2.6.3 Phase Relationship between Normal Load and Friction Force Excitations Using the 6DOF spur gear model with parameters consistent with the experimental conditions, dynamic studies are conducted for Example II. First, a mean torque of 500 lb-in is used corresponding to the optimal case with minimal STE. Equations ( ) show that the normal loads Ni and friction forces Ffi excite the LOA and OLOA dynamics, respectively. The force profile of a single tooth pair undergoing the entire meshing process is obtained by tooth pairs #0 and #1 for two continuous meshing cycles as shown in Figure 2.13(a-b) and (c-d) for II-A and II-B cases respectively. Observe that the peak-to-peak magnitude of combined pinion normal load N pi is minimized for the tip relief gear due to reduced STE at 500 lb-in. However, the combine pinion friction force Ffpi with tip relief has a higher peak-to-peak magnitude when compared with the perfect involute gear. This implies that the tip relief amplifies F fi in the OLOA direction while minimizing Ni in the LOA direction. Such contradictory effects are examined next using the phase relationship between F fpi. N pi and 50

84 Figure 2.13 Dynamic loads predicted for Example II at 500 lb-in, 4875 RPM and 140 F with t c = 0.44 ms: (a) Normal loads of gear pair with perfect involute profile (II-A); (b) normal loads of gear par with tip relief (II-B); (c) friction forces of gear pair with perfect involute profile (II-A); (d) friction forces of gear pair with tip relief (II-B). Key:, combined;, tooth pair #0;, tooth pair #1. 51

85 At points A, B, C and D, corner contacts are observed for N pi of the perfect involute gear, corresponding to the time instants when meshing tooth pairs come into or out of contact. These introduce discontinuous points in the slope of the N pi profile. Note that N p1 and N p2 between A and B (or C and D) are in phase with each other, which should amplify the peak-to-peak variation of Npi. For the F fpi profile of Figure 2.13(c), an abrupt change in the direction is observed at the pitch point P in addition to the corner contacts. Unlike N, the profiles of F 1 and F 2 of Figure 2.13(c) between A pi and B (or C and D) are out of phase with each other. This should minimize the peak-topeak variation of Ffpi fp. When tip relief is applied in Figure 2.13(b), corner contacts of N pi are reduced and smoother transitions are observed at points A, B, C and D. Unlike the perfect involute gear, N p1 and N p2 are now out of phase with each other between A and B (or C and D), which reduces the peak-to-peak variation of fp N pi. However, the profiles of F fp1 and F fp2 of Figure 2.13(d) are in phase with each other in the same region, which amplifies the variation of Ffpi. The out of phase relationship between N pi and F fpi explains why the tip relief (designed to minimize the STE) tends to increase the friction force excitations. This relationship is mathematically embedded in Eq. (2.10) and graphically illustrated in Figure 2.13, where N p1 and N p2 are in phase while F fp1 and F fp2 are out of phase. Consequently, a compromise would be needed to simultaneously address the dynamic responses in both the LOA and OLOA directions. 52

86 2.6.4 Prediction of the Dynamic Responses Dynamic responses including xp () t, yp () t, Fpbx() t, Fpby () t and DTE δ () t are predicted by numerically integrating the governing equations. Predictions from both perfect and tip relief gears are compared to examine the effect of profile modification in the presence of sliding friction. Figure 2.14 shows that the normalized x () t at 500 lb-in is much smaller (over 90% reduction) when the tip relief is applied. This is because that the STE is the most dominant excitation in the LOA direction and it is minimized at 500 lb-in when the tip relief is applied. An alternate explanation is that the peak-to-peak variation of to increased N pi is minimized with the tip relief as shown in Figure In the OLOA direction, more significant oscillations are observed for y () t due Ffpi N pi are larger than that of Ffpi excitations with tip relief. Despite that the vibratory components of, predicted yp () t is actually higher than xp () t. This shows the necessity of including sliding friction when other excitations such as the STE are minimized. Note that a phase difference is present in simulated y () t before and after the tip relief is applied. Predicted pinion bearing forces are not shown here since they depict the same features as the displacement responses of Figure Figure 2.15 shows the DTE predictions, as defined by Eq. (2.17), with and without the tip relief. Similarity between Figure 2.14(a-b) and Figure 2.15(a-b) suggests that the relative LOA displacement plays a dominant role in the DTE responses. However, this conclusion is somewhat case specific as the DTE results depend on the mesh stiffness, bearing stiffness, and gear geometry. 53 p p p

87 Figure 2.14 Dynamic shaft displacements predicted for Example II at 500 lb-in, 4875 RPM and 140 F with t c = 0.44 ms and f m = 2275 Hz: (a) xp () t ; (b) X p ( f ); (c) yp () t and (d) Yp ( f ). Key:, gear pair with perfect involute profile in t domain (II-A); o, with perfect involute profile in f domain (II-A);, with tip relief (II-B). 54

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

89 2.7 Experimental Validation of Example II Models Experiments corresponding to Example II-B were conducted at the NASA Glenn Research Center (Gear Noise Rig) to validate the MDOF spur gear pair model and to establish the relative influence of friction force excitation on the system. Figure 2.16 shows the inside of the gearbox, where a bracket was built to hold two shaft displacement probes one inch away from the center of the gear in the LOA and OLOA directions [2.13]. The probes face a steel collar that was machined to fit around the output shaft with minimal eccentricity. Accelerometers were mounted on the bracket, so the motion of the displacement probes could be subtracted from the measurements, if necessary. A thermocouple was installed inside the gearbox to measure the temperature of the oil flinging off the gears as they enter into mesh. The thermocouple position was chosen to be consistent with Benedict and Kelley s [2.14] experiment. A common shaft speed of 4875 rpm is used in all tests so that the first five harmonics of the gear mesh frequency (2275, 4550, 6825, 9100, and Hz) do not excite system resonances. Data of shaft displacement in the LOA and OLOA directions are collected from the proximity sensors under oil inlet temperatures over the range of temperatures (104, 122, 140, 158, and 176 F). At each temperature the torque is varied from 500 to 900 lb-in increments of 100 lbin. 56

90 Thermocouple OLOA LOA Proximity Probes Bracket Accelerometer Figure 2.16 Sensors inside the NASA gearbox (for Example II-B). Parametric studies are conducted to examine the dynamic responses under varying operational conditions of temperature and nominal torque. Benedict and Kelly s [2.14] friction model is used to calculate the empirical µ as given in Table 2.3 and the realistic kt () calculated using FE/CM under varying torques are incorporated into the dynamic model. Since the precise parameters of the experimental system are not known [2.13], both simulated and measured data are normalized with respect to the amplitude of their first mesh harmonic of the OLOA displacement (which is then designed as 100%). This 57

91 facilitates the comparison of trends and allows simulations and measurements to be viewed in the same graphs from 0 to 100%. Figure 2.17 compares the first five mesh harmonics of the LOA displacement as a function of mean torque. (In this and other figures, predictions are shown as continuous lines for the sake of clarity though they are calculated only at discrete points like the measurements.) It was observed that overall simulation trend matches well with the experiment. Magnitudes of the first two mesh harmonics are most dominant and they have minimum values around the optimized load due to the linear tip relief. Figure 2.17 also shows predicted first two harmonics for the prefect involute gear (II-A). Compared with the tip relief gear, they increase monotonically with the mean torque and have much higher values than the tip relief gear around the optimal torque. Figure 2.18 compares the first five mesh harmonics of the OLOA displacement, on a normalized basis, as a function of mean torque. The overall simulation trend again matches well with the experiment. However, unlike the LOA responses, the first harmonic of OLOA displacement grows monotonically with an increase in the mean torque. This is because the friction forces increase almost proportionally with normal loads as predicted by the Coulomb law, but the frictional contribution of each meshing tooth pair tends to be in phase with each other for the tip relief gear (II-B). Thus it should amplify the combined friction force excitation in the OLOA direction. Consequently it is not reducing the OLOA direction responses induced by the sliding friction, even though the profile modification can be efficiently used to minimize gear vibrations in the LOA direction. 58

92 Figure 2.17 Mesh harmonic amplitudes of X p as a function of the mean torque at 140 F. All values are normalized with respect to the amplitude ofy p at the first mesh harmonic. Key:, n = 1 (prediction of II-B);, n = 2 (prediction of II-B);, n = 3 (prediction of II-B);, n = 1 (prediction of II-A);, n = 2 (prediction of II-A);, n = 1 (measurement of II-B);, n = 2 (measurement of II-B); O, n = 3 (measurement of II-B). 59

93 Figure 2.18 Mesh harmonic amplitudes of Y p as a function of the mean torque at 140 F for Example II-B. All values are normalized with respect to the amplitude ofy p at the first mesh harmonic. Key:, n = 1 (prediction of II-B);, n = 2 (prediction of II-B);, n = 3 (prediction of II-B);, n = 1 (measurement of II-B);, n = 2 (measurement of II-B); O, n = 3 (measurement of II-B). 60

94 Figure 2.19 compares first five mesh harmonics of the normalized DTE for II-A and II-B cases over a range mean torques. Observe that the DTE spectral trends are very similar to the STE spectral trends of Figure For example, the harmonic amplitudes of the perfect involute gear grow monotonically with mean torque while the harmonic amplitudes of the tip relief gear have minimum values around the optimal torque. Also, the DTE spectra show a dominant second harmonic, whose magnitude is comparable to that at the first harmonic. In some cases for the tip relief gear the second harmonic becomes the most dominant component especially when the mean torque is lower than 350 lb-in. Finally, Figure 2.20 compares the first five mesh harmonics of the normalized LOA displacement as a function of operational temperature. The changes in temperature are converted into variation in µ of Table 2.3. Compared with the OLOA motions, both predictions and measurements in the LOA direction give almost identical results at all temperatures. Figure 2.21 shows the first five mesh harmonics of the OLOA displacement with a change in temperature. The first harmonic varies quite significantly even though the changes in µ are relatively small. Consequently, the OLOA dynamics tends to be much more sensitive to a variation in µ as compared with the LOA motions. Measured data of Figure 2.21 show some variations due to the experimental errors [2.13]. 61

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

96 Figure 2.20 Mesh harmonic amplitudes of X p as a function of temperature at 500 lb-in for Example II-B. All values are normalized with respect to the amplitude ofy p at the first mesh harmonic. Key:, n = 1 (prediction of II-B);, n = 2 (prediction of II-B);, n = 3 (prediction of II-B);, n = 1 (measurement of II-B);, n = 2 (measurement of II-B); O, n = 3 (measurement of II-B). 63

97 Temperature (deg F) Figure 2.21 Mesh harmonic amplitudes of Y p as a function of temperature at 500 lb-in for Example II-B. All values are normalized with respect to the amplitude ofy p at the first mesh harmonic. Key:, n = 1 (prediction of II-B);, n = 2 (prediction of II-B);, n = 3 (prediction of II-B);, n = 1 (measurement of II-B);, n = 2 (measurement of II-B); O, n = 3 (measurement of II-B). 64

98 2.8 Conclusion Chief contribution of this study is the development of a new multi-degree of freedom, linear time-varying model. This formulation overcomes the deficiency of Vaishya and Singh s work [ ] by employing realistic tooth stiffness functions and the sliding friction over a range of operational conditions. Refinements include: (1) an accurate representation of tooth contact and spatial variation in tooth mesh stiffness based on a FE/CM code in the static mode; (2) Coulomb friction model for sliding resistance with empirical coefficient of friction as a function of operation conditions; (3) a better representation of the coupling between the LOA and OLOA directions including torsional and translational degrees of freedom. Numerical solutions of the MDOF model yield the dynamic transmission error and vibratory motions in the LOA and OLOA directions. The new model has been successfully validated first by using the FE/CM code while running in the dynamic mode and then by analogous experiments. Since the lumped model is more computationally efficient when compared with the FE/CM analysis, it could be quickly used to study the effect of a large number of parameters. One of the main effects of sliding friction is the enhancement of the DTE magnitude at the second gear mesh harmonic. A key question whether the sliding friction is indeed the source of the OLOA motions and forces is then answered by our model. The bearing forces in the LOA direction are influenced by the normal tooth loads, but the sliding frictional forces primarily excite the OLOA motions. Finally, effect of the profile modification on the dynamic transmission error has been analytically examined under the influence of frictional effects. For instance, the tip relief introduces an amplification in the OLOA motions and forces due to an out of phase relationship between the normal 65

99 load and friction forces. This knowledge should be of significant utility to the designers. Future modeling work should examine the effects of other profile modifications and find the conditions for minimal dynamic responses when both STE and friction excitations are simultaneously present. Also, the model could be further refined by incorporating alternate friction formulations. References for Chapter 2 [2.1] Vaishya, M., and Singh, R., 2001, Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory, Journal of Sound and Vibration, 243(3), pp [2.2] Vaishya, M., and Singh, R., 2001, Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics, Journal of Sound and Vibration, 248(4), pp [2.3] Vaishya, M., and Singh, R., 2003, Strategies for Modeling Friction in Gear Dynamics, ASME Journal of Mechanical Design, 125, pp [2.4] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, Vibro-Acoustic Effects of Friction in Gears: An Experimental Investigation, SAE Paper # [2.5] Velex, P., and Cahouet, V., 2000, Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics, ASME Journal of Mechanical Design, 122(4), pp [2.6] Velex, P., and Sainsot. P, 2002, An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears, Mechanism and Machine Theory, 37, pp [2.7] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, A Flexible Multi-body Approach for Frictional Contact in Spur Gears, Journal of Sound and Vibration, 278(3), pp [2.8] External2D (CALYX software), 2003, Helical3D User s Manual, ANSOL Inc., Hilliard, OH. 66

100 [2.9] Vinayak, H., Singh, R., and Padmanabhan, C., 1995, Linear Dynamic Analysis of Multi-Mesh Transmissions Containing External, Rigid Gears, Journal of Sound and Vibration, 185(1), pp [2.10] Lim, T. C., and Singh, R., 1990, Vibration Transmission Through Rolling Element Bearings. Part I: Bearing Stiffness Formulation, Journal of Sound and Vibration, 139(2), pp [2.11] Lim, T. C., and Singh, R., 1990, Vibration Transmission Through Rolling Element Bearings. Part II: System Studies, Journal of Sound and Vibration, 139(2), pp [2.12] Kahraman, A., and Singh, R., 1991, Error Associated with a Reduced Order Linear Model of Spur Gear Pair, Journal of Sound and Vibration, 149(3), pp [2.13] Singh, R., 2005, Dynamic Analysis of Sliding Friction in Rotorcraft Geared Systems, Technical Report submitted to the Army Research Office, grant number DAAD [2.14] Benedict, G. H., and Kelley B. W., 1961, Instantaneous Coefficients of Gear Tooth Friction, Transactions of the American Society of Lubrication Engineers, 4, pp [2.15] Rebbechi, B., Oswald, F. B., and Townsend, D. P., 1996, Measurement of Gear Tooth Dynamic Friction, ASME Power Transmission and Gearing Conference proceedings, DE-Vol. 88, pp

101 CHAPTER 3 PREDICTION OF DYNAMIC FRICTION FORCES USING ALTERNATE FORMULATIONS 3.1 Introduction Gear dynamic researchers [ ] have typically modeled sliding friction phenomenon by assuming Coulomb formulation with a constant coefficient (µ) of friction (it is designated as Model I in this chapter). In reality, tribological conditions change continuously due to varying mesh properties and lubricant film thickness as the gears roll through a full cycle [ ]. Thus, µ varies instantaneously with the spatial position of each tooth and the direction of friction force changes at the pitch point. Alternate tribological theories, such as elasto-hydrodynamic lubrication (EHL), boundary lubrication or mixed regime, have been employed to explain the interfacial friction in gears [ ]. For instance, Benedict and Kelley [3.7] proposed an empirical dynamic friction coefficient (designated as Model II) under mixed lubrication regime based on measurements on a roller test machine. Xu et al. [3.8, 3.9] recently proposed yet another friction formula (designated as Model III) that is obtained by using a non-newtonian, thermal EHL formulation. Duan and Singh [3.11] developed a smoothened Coulomb 68

102 model for dry friction in torsional dampers; it could be applied to gears to obtain a smooth transition at the pitch point and we designate this as Model IV. Hamrock and Dawson [3.10] suggested an empirical equation to predict the minimum film thickness for two disks in line contact. They calculated the film parameter Λ, which could lead to a composite, mixed lubrication model for gears (designated as Model V). Overall, no prior work has incorporated either the time-varying µ () t or Models II to V, into multi-degreeof-freedom (MDOF) gear dynamics. To overcome this void in the literature, specific objectives of this chapter are established as follows: 1. Propose an improved MDOF spur gear pair model with time-varying coefficient of friction, µ () t, given realistic mesh stiffness profiles of Chapter 2; 2. Comparatively evaluate alternate sliding friction models and predict the interfacial friction forces and motions in the off-line-of-action (OLOA) direction; and 3. Validate one particular model (III) by comparing predictions to the benchmark gear friction force measurements made by Rebbechi et al. [3.12]. 3.2 MDOF Spur Gear Model Transitions in key meshing events within a mesh cycle are determined from the undeformed gear geometry. Figure 3.1(a) is a snapshot for the example gear set (with a contact ratio σ of about 1.6) at the beginning (t = 0) of the mesh cycle (t c ). At that time, pair # 1 (defined as the tooth pair rolling along line AC) just comes into mesh at point A and pair # 0 (defined as the tooth pair rolling along line CD) is in contact at point C, which is the highest point of single tooth contact (HPSTC). When pair # 1 approaches the 69

103 lowest point of single tooth contact (LPSTC) at point B, pair # 0 leaves contact. Further, when pair #1 passes through the pitch point P, the relative sliding velocity of the pinion with respect to the gear is reversed, resulting in a reversal of the friction force. Beyond point C, pair # 1 will be re-defined as pair # 0 and the incoming meshing tooth pair at point A will be re-defined as pair # 1, resulting in a linear-time-varying (LTV) formulation. The spur gear system model is shown in Figure 3.1(b) and key assumptions for the dynamic analysis include the following: (i) pinion and gear are rigid disks; (ii) shaft-bearings stiffness elements in the line-of-action (LOA) and OLOA directions are modeled as lumped springs which are connected to a rigid casing; (iii) vibratory angular motions are small in comparison to the kinematic motion. Overall, we obtain a LTV system formulation, as explained in Chapter 2 with a constant µ. Refinements to the MDOF model of Figure 3.1(b) with time-varying sliding friction µ () t are proposed as follows. The governing equations for the torsional motions θ () t and θ () t are as follows: p g n= floor( σ) n= floor( σ) J θ () t = T + X () t F () t r N () t, (3.1) p p p pi pfi bp pi i= 0 i= 0 n= floor( σ) n= floor( σ) J θ () t = T + X () t F () t + r N () t. (3.2) g g g gi gfi bg gi i= 0 i= 0 70

104 (a) T g, Ω g J g, m g K gby kt () K gbx K pby J p, m p K pbx Ω p, T p y θ x (b) Figure 3.1 (a) Snap shot of contact pattern (at t = 0) in the spur gear pair; (b) MDOF spur gear pair system; here () kt is in the LOA direction. 71

105 Here, the floor function rounds off the contact ratio σ to the nearest integer (towards a lower value); J p and J g are the polar moments of inertia for the pinion and gear; Tp and T g are the external and braking torques; N pi () t and N gi () t are the normal loads defined as follows: Npi () t = Ngi () t = ki () t rbp θp () t rbg θg () t + xp () t xg () t + ci() t rbp θ p() t rbg θg() t + x p() t x g() t, i = 0, 1,..., n= floor( σ). (3.3) where ki ( t ) and ci ( t ) are the time-varying realistic mesh stiffness and viscous damping profiles; r bp and r bg are the base radii of the pinion and gear; xp () t and xg () t denote the translational displacements (in the LOA direction) at the bearings. The sliding (interfacial) friction forces Fpfi() t and Fgfi () t of the i th meshing pair are derived as follows; note that five alternate µ () t models will be described later. F () t = µ () t N (), t F () t = µ () t N (), t i = 0,..., n. (3.4a,b) pfi pi gfi gi The frictional moment arms X pi () t and Xgi() t acting on the i th tooth pair are: X ( t) = L + ( n i) λ+ mod( Ω r t, λ), i = 0,..., n, (3.5a) pi XA p bp X ( t) = L + iλ mod( Ω r t, λ), i = 0,..., n. (3.5b) gi YC g bg 72

106 where mod is the modulus function defined as: mod( xy, ) = x y floor( x/ y), if y 0 ; sgn is the sign function; Ω p and is the base pitch. Refer to Figure 3.1(a) for length L AP Ωg are the nominal operational speeds (in rad/s); and λ translational motions xp () t and xg () t in the LOA direction are:. The governing equations for the n= floor( σ ) mx () t+ 2 ζ K m x () t+ K x() t+ N () t = 0, (3.6) p p pbx pbx p p pbx p pi i= 0 n= floor( σ ) mx () t + 2 ζ K m x () t + K x() t+ N () t = 0. (3.7) g g gbx gbx g g gbx g gi i= 0 Here, m p and m g are the masses of the pinion and gear; K pbx and K gbx are the effective shaft-bearing stiffness values in the LOA direction, and ζ pbx and ζ gbx are their damping ratios. Likewise, the governing equations for the translational motions y () t and y () t in the OLOA direction are written as: g p n= floor( σ ) my () t+ 2 ζ K m y () t+ K y() t F () t= 0, (3.8) p p pb y pby p p pby p pfi i= 0 n= floor( σ ) my () t+ 2 ζ K m y () t+ K y() t F () t = 0. (3.9) g g gb y gby g g gby g gfi i= 0 73

107 3.3 Spur Gear Model with Alternate Sliding Friction Models Following a similar modeling strategy of Chapter 2, we obtain a LTV system formulation. Refinements to the MDOF model with time-varying sliding friction µ () t are proposed as follows. The sliding (interfacial) friction forces Fpfi() t and Fgfi () t of the i th meshing pair are F () t = µ () t N (), t F () t = µ () t N (), t i = 0,..., n. (3.10a,b) pfi pi gfi gi Five alternate µ () t models are described as follows: Model I: Coulomb Model The Coulomb friction model with time-varying (periodic) coefficient of friction µ Ci ( t) for the i th meshing tooth pair is derived as follows, where µ avg is the magnitude of the time-average. µ Ci ( t) = µ avg sgn mod( Ω prbpt, λ) + ( n i) λ L AP, i= 0,..., n. (3.11) 74

108 3.3.2 Model II: Benedict and Kelley Model The instantaneous profile radii of curvature (mm) ρ () t of i th meshing tooth are: ρ ( t) = L + ( n i) λ+ mod( Ω r t, λ), i = 0,..., n. (3.12a) pi XA p bp ρ ( t) = L ρ ( t), i = 0,..., n. (3.12b) gi XY pi The rolling (tangential) velocities vr ( t )(m/s) of i th meshing tooth pair are: v rpi Ω pρ pi() t () t =, 1000 Ωgρgi() t vrgi( t) =, i= 0,..., n. (3.13a,b) 1000 tooth pair are: The sliding velocity vs ( t ) and the entraining velocity ve ( t ) (m/s) of i th meshing v () t = v () t v () t, v ( t) = v ( t) + v ( t), i = 0,..., n. (3.14a,b) si rpi rgi ei rpi rgi The unit normal load (N/mm) is wn Tp /( Z rwpcosα ) =, where α is the pressure angle, Z is the face width (mm), T p is torque (N-mm) and r wp is the operating pitch radius of pinion (mm). Our µ () t prediction for the i th meshing tooth pair is based on the Benedict and Kelley model [3.7], though it is modified to incorporate a reversal in the direction of friction force at the pitch point. Here, Savg 0.5( Sap Sag ) 75 = + is the averaged

109 surface roughness ( µ m ), and η M is the dynamic viscosity of the oil entering the gear contact w n µ Bi ( t) = log10 sgn mod(, ) ( ) 2 Ω prbpt λ + n i λ L AP 1.13 Savg ηm vsi ( t) vei ( t), i = 0,..., n. (3.15) Model III: Formulation Suggested by Xu et al. The composite relative radius of curvature ρ ( t) (mm) of i th meshing tooth pair is: r ρ pi () t ρgi () t ρri () t =, i = 0,..., n (3.16) ρ () t + ρ () t pi gi The effective modulus of elasticity (GPa) of mating surfaces is ν p 1 ν g E = 2/ +, where E and ν are the Young s modulus and Poisson s ratio, Ep Eg respectively. The maximum Hertzian pressure (GPa) for the i th meshing tooth pair is: we n Phi () t =, i = 0,..., n. (3.17) 2000 πρ ( t) ri 76

110 Define the dimensionless slide-to-roll ratio SR() t and oil entraining velocity V () e t (m/s) of i th meshing tooth pair as: 2 vsi ( t) SRi () t = v () t, ei vei () t Vei () t =, i = 0,..., n. (3.18a,b) 2 The empirical sliding friction expression (for the i th meshing tooth pair), as proposed by Xu et al. based on non-newtonian, thermal EHL theory [ ], is modified in our work to incorporate a reversal in the direction of the friction force at the pitch point: f ( SRi(), t Phi(), t ηm, Savg) b b 2 3 b6 b7 b8 µ Xi () t = e Phi SRi () t Vei () t ηm Ri ()sgn t mod( Ω prbpt, λ) + ( n i) λ L AP, SR t P t 10 ( i hi ηm avg ) i hi ηm i() hi()log ( ηm ) S avg f SR(), t P (), t, S = b + b SR() t P ()log t ( ) + b e + b e, i = 0,..., n. (3.19a,b) Xu [3.9] suggested the following empirical coefficients (in consistent units) for the above formula: b 1 = , b 2 = , b 3 = , b 4 = , b 5 = , b = , 6 b 7 = , b 8 = , and b 9 =

111 3.3.4 Model IV: Smoothened Coulomb Model Xu [3.9] conducted a series of friction measurements on a ball-on-disk test machine and measured the µ () t values as a function of SR; these results resemble the smoothening function reported by Duan and Singh [3.11] near the pitch point (SR = 0) especially at very low speeds (boundary lubrication conditions). By denoting the periodic displacement of i th meshing tooth pair as x ( t) = mod( Ω r t, λ) + ( n i) λ L, a i p bp AP smoothening function could be used in place of the discontinuous Coulomb friction of Chapter 2. The arc-tangent type function is proposed as follows though one could also use other functions [3.11]: 2µ avg 2µ avgσ µ Si ( t) = arctan [ Φ xi ( t) ] + xi ( t), i = 0,..., n (3.20) 2 2 π π 1 +Φ xi ( t) In the above, the regularizing factor Φ is adjusted to suit the need of smoothening requirement. A higher value of Φ corresponds to a steeper slope at the pitch point. In our work, Φ = 50 is used for a comparative study Model V: Composite Friction Model Alternate theories (Models I to IV) seem to be applicable over specific operational conditions. This necessitates a judicious selection of an appropriate lubrication regime as indicated by the film parameter, Λ, that is defined as the ratio of minimum lubrication 78

112 2 2 film thickness and composite surface roughness R = R + R measured with a comp rms, g rms, p filter cutoff wave length L x, where R rms is the rms gear-tooth surface roughness [3.13]. The film parameter for rotorcraft gears usually lies between 1 and 10. In the mixed lubrication regime the films are sufficiently thin to yield partial asperity contact, while in the EHL regime the lubrication film completely separates the gear surfaces. Accordingly, a composite friction model is proposed as follows: µ C ( t) simplified Coulomb model, computationally efficient (Model I) µ B ( t) 1< Λ<4, mixed lubrication, (Model II) µ () t = (3.21) µ X ( t) 4 Λ<10, EHL lubrication, (Model III) µ S( t) low Ωp, high Tp, Λ< 1, boundary lubrication (Model IV) Application of Model II, III or IV would, of course, depend on the operational and tribological conditions though Model I could be easily utilized for computationally efficient dynamic simulations. Note that the magnitude µ avg of Model I or IV should be determined separately. For instance, the averaged coefficient based on Model II was used in Chapter 2. Also, the critical Λ value between different lubrication regimes must be carefully chosen. The film thickness calculation employs the following equation developed by Hamrock and Dowson [3.10, 3.13], based on a large number of numerical solutions that predict the minimum film thickness for two disks in line contact. Here, G is the dimensionless material parameter, W is the load parameter, U is the speed parameter, H is the dimensionless central film thickness, and b H is the semi-width of Hertzian contact band: 79

113 3 Hci () t ρ r1() t 10 LX Λ i () t =, i = 0,..., n (3.22a) R 2 b ( t) comp Hi b 8 w ρ ( t) =, n r1 H1() t π Er G = kη E, (3.22b-c) s M r H ci G U () t ( t) = 3.06 (), (3.22d) i 0.10 Wi t η v () t U () t 10 i W t M ei 6 =, i () 2 Erρri( t) = wn E ρ () t. (3.22e.f) r ri 3.4 Comparison of Sliding Friction Models Figure 3.2(a) compares the magnitudes of µ () t as predicted by Model II and III for the spur gear set of Chapter 2 given T p = 22.6 N-m (200 lb-in) and Ω p = 1000 RPM. The LTV formulations for meshing tooth pairs # 0 and 1 result in periodic profiles for both models. Two major differences between these two models are: (1) The averaged magnitude from Model II is much higher compared with that of Model III since friction under mixed lubrication is generally higher than under EHL lubrication; and (2) while Model III predicts nearly zero friction near the pitch point, Model II predicts the largest µ value due to the entraining velocity term in the denominator. 80

114 µ Normalized time t/t c (a) µ(t) Ω p (RPM) (b) Figure 3.2 (a) Comparison of Model II [3.7] and Model III [3.8] given T p = 22.6 N-m (200 lb-in) and Ω p = 1000 RPM. Key:, pair # 1 with Model II;, pair # 0 with Model II;, pair # 1 with Model III;, pair # 0 with Model III; (b) Averaged magnitude of the coefficient of friction predicted as a function of speed using the composite Model V with T p = 22.6 N-m (200 lb-in). Here, t c is one mesh cycle. 81

115 As explained by Xu [3.9], three different regions could be roughly defined on a µ versus SR curve. When the sliding velocity is zero, there is no sliding friction, and only rolling friction (though very small) exists. Thus, the µ value should be almost zero at the pitch point. When the SR is increased from zero, µ first increases linearly with small values of SR. This region is defined as the linear or isothermal region. When the SR is increased slightly further, µ reaches a maximum value and then decreases as the SR value is increased beyond that point. This region is referred to as non-linear or non-newtonian region. As the SR is increased further, the friction decreases in an almost linear fashion; this is called as the thermal region. Model II seems to be valid only in the thermal region [3.8, 3.9]. Figure 3.2(b) shows the averaged magnitude of µ avg predicted as a function of Ω p using the composite formulation (Model V) with T p = 22.6 N-m (200 lb-in). An abrupt change in magnitude is found around 2500 RPM corresponding to a transition from the EHL to a mixed lubrication regime. Similar results could be obtained by plotting the composite µ () t as a function of T p. Though our composite model could be used to predict µ () t over a large range of lubrication conditions, care must be exercised since the calculation of Λ itself is based on an empirical equation [3.10]. Figure 3.3 compares four friction models on a normalized basis. The curves between 0 t/ t c < 1 are defined for pair # 1 and those between 1 t/ t c < 2 are defined for pair # 0. Discontinuities exist near the pitch point for Models I and II, and these might serve as artificial excitations to the OLOA dynamics. On the other hand, smooth transitions are observed for Models III and IV corresponding to the EHL lubrication condition. 82

116 1.5 1 Normalized µ(t) Normalized time t/t c Figure 3.3 Comparison of normalized friction models. Key:, Model I (Coulomb friction with discontinuity);, Model II [3.7];, Model III [3.8];, Model IV (smoothened Coulomb friction). Note that curve between 0 t/ t c < 1 is for pair # 1; and the curve between 1 t/ t c < 2 is for pair # 0. Figure 3.4 compares the combined normal loads and friction force time histories as predicted by four friction models given T p = 56.5 N-m (500 lb-in) and 83 Ω p = 4875 RPM. Note that while Figure 3.3 illustrates µ () t for each meshing tooth pair the friction forces of Figure 3.4 include the contributions from both (all) meshing tooth pairs. Though alternate friction formulations dictate the dynamic friction force profiles, they have negligible effect on the normal loads.

117 1400 N p (N) F fp (N) Normalized time t/t c Figure 3.4 Combined normal load and friction force time histories as predicted using alternate friction models given T p = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Key:, Model I;, Model II;, Model III;, Model IV. 84

118 3.5 Validation and Conclusion Figure 3.5 compares the predicted LOA and OLOA displacements with alternate friction models given T p = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Note that the differences between predicted motions are not significant though friction formulations and friction force excitations differ. This implies that one could still employ the simplified Coulomb formulation (Model I) in place of more realistic time-varying friction models (Models II to IV). Similar trend is observed in Figure 3.6 for the dynamic transmission errors (DTE), defined as δ() t = r θ () t r θ () t + x () t x () t. The most bp p bg g significant variation induced by friction formulation is at the second harmonic, which matches the results reported by Lundvall et al. [3.6]. Finally, predicted normal load and friction force time histories (with Model III) are validated using the benchmark friction measurements made by Rebbechi et al. [3.12]. Results are shown in Figure 3.7. Based on the comparison, µ is found to be about since it was not given in the experimental study. Here, we have made the periodic LTV definitions of meshing tooth pairs # 0 and 1 to be consistent with those of measurements, where meshing tooth pairs A and B are labeled in a continuous manner. Predictions match well with measurements at both low ( Ω = 800 RPM) and high ( Ω = 4000 RPM) speeds. Ongoing research focuses on the development of semi-analytical solutions given a specific µ () t model and an examination of the interactions between tooth modifications and sliding friction. p p g p 85

119 x p (µm) x p (µm) y p (µm) y p (µm) Normalized time t/t c Mesh order n Figure 3.5 Predicted LOA and OLOA displacements using alternate friction models given T = 56.5 N-m (500 lb-in) and Ω = 4875 RPM. Key: in time domain:, Model I; p p, Model II;, Model III;, Model IV; in frequency (mesh order n) domain:, Model I;, Model II;, Model III; +, Model IV. 86

120 DTE (µm) DTE (µm) Normalized time t/t c Mesh order n Figure 3.6 Predicted dynamic transmission error (DTE) using alternate friction models given T p = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Key: in time domain:, Model I;, Model II;, Model III;, Model IV; in frequency (mesh order n) domain:, Model I;, Model II;, Model III; +, Model IV. 87

121 N p (N) F fp (N) Normalized time t/t c (a) 2000 N p (N) F fp (N) Normalized time t/t c (b) Figure 3.7 Validation of the normal load and sliding friction force predictions: (a) at T p = 79.1 N-m (700 lb-in) and Ω p = 800 RPM; (b) at T p = 79.1 N-m (700 lb-in) and 88 Ω p = 4000 RPM. Key:, prediction of tooth pair A with Model III;, prediction of tooth pair B with Model III; X, measurement of tooth pair A [3.12];, measurement of tooth pair B [3.12].

122 References for Chapter 3 [3.1] Vaishya, M., and Singh, R., 2001, Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory, Journal of Sound and Vibration, 243(3), pp [3.2] Vaishya, M., and Singh, R., 2001, Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics, Journal of Sound and Vibration, 248(4), pp [3.3] Vaishya, M., and Singh, R., 2003, Strategies for Modeling Friction in Gear Dynamics, ASME Journal of Mechanical Design, 125, pp [3.4] Velex, P., and Cahouet, V., 2000, Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics, ASME Journal of Mechanical Design, 122(4), pp [3.5] Velex, P., and Sainsot. P, 2002, An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears, Mechanism and Machine Theory, 37, pp [3.6] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, A Flexible Multi-body Approach for Frictional Contact in Spur Gears, Journal of Sound and Vibration, 278(3), pp [3.7] Benedict, G. H., and Kelley B. W., 1961, Instantaneous Coefficients of Gear Tooth Friction, Transactions of the American Society of Lubrication Engineers, 4, pp [3.8] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, Prediction of Mechanical Efficiency of Parallel-Axis Gear Pairs, ASME Journal of Mechanical Design, 129 (1), pp [3.9] Xu, H., 2005, Development of a Generalized Mechanical Efficiency Prediction Methodology, PhD dissertation, The Ohio State University. [3.10] Hamrock, B. J., and Dowson, D., 1977, Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part III Fully Flooded Results, Journal of Lubrication Technology, 99(2), pp [3.11] Duan, C., and Singh, R., 2005, Super-Harmonics in a Torsional System with Dry Friction Path Subject to Harmonic Excitation under a Mean Torque, Journal of Sound and Vibration, 285(2005), pp [3.12] Rebbechi, B., Oswald, F. B., and Townsend, D. P., 1996, Measurement of Gear Tooth Dynamic Friction, ASME Power Transmission and Gearing Conference proceedings, DE-Vol. 88, pp

123 [3.13] AGMA Information Sheet 925-A03, 2003, Effect of Lubrication on Gear Surface Distress. 90

124 CHAPTER 4 CONSTRUCTION OF SEMI-ANALYTICAL SOLUTIONS TO SPUR GEAR DYNAMICS 4.1 Introduction Periodic differential equations [ ] are usually needed to describe the gear dynamics [ ] since significant variations in mesh stiffness k(t) and damping c(t) are observed, within the fundamental period t c (one mesh cycle). Additionally, dynamic friction force F f (t) and torque M f (t) also undergo periodic variations, with the same period t c, due to changes in normal mesh loads and coefficient of friction µ, as well as a reversal in the direction of F f (t) at the pitch point [ ], especially in spur and helical gears. For the sake of illustration, typical k i (t) profiles and frictional functions f i (t) for the i th meshing pair in spur gears are shown in Figure 4.1; derivations of f i (t) will be explained later along with particulars of the example case. The fundamental nature of the linear time-varying (LTV) system is illustrated in Figure 4.2(a); the system model is described in Chapter 2. 91

125 k i (t) (lb/in) (a) f(t) (b) Figure 4.1(a) Realistic mesh stiffness functions of the spur gear pair example (with tip relief) given T p = 550 lb-in. Key:, k () t ;, 0 k () 1 t. (b) Periodic frictional functions. Key:, f () t ;, f () t ;, f () t

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

127 The governing single degree-of-freedom (SDOF) equation in terms of dynamic transmission error (DTE) δ() t = r θ () t r θ () t is given below, where subscripts p and bp p bg g g correspond to the pinion and gear, respectively; θ is the vibratory component of the rotation; and r b is the base radius. S S J eδ() t + J b ci() t δ() t + ki() t δ () t + µ sgn mod( Ω prbpt, λ) + ( S j) λ L AP i= 0 i= 0 (4.1a) ci() t δ() t + ki() t δ() t X pi() t Jgrbp + Xgi() t Jpr bg = Te + T() t, J = J J, e p g 2 2 Jb = Jr g bp + Jr p bg, Te = TpJgrbp + TgJprbg, (4.1b-d) X ( t) = L + ( S i) λ + mod( Ω r t, λ), i = 0,..., S = floor( σ), (4.1e) pi XA p bp X ( t) = L + iλ mod( Ω r t, λ), i = 0,..., S = floor( σ). (4.1f) gi YC g bg Further, J is the moment of inertia; T and Ω are the nominal torque and rotation speed; and λ is the base pitch. Tooth pairs #1 and #0 are defined as the pairs rolling along line AC and CD in Figure 4.2(b), respectively. The j th tooth pair passes though the pitch point P during the meshing event, and the reversal at P is characterized by the sign function sgn with a constant coefficient of Coulomb friction µ [4.4]. The modulus function (mod(x, y) = x y floor(x/y), if y 0) is used to describe the periodic friction force F f (t) and the moment arm X(t). The floor function rounds off the contact ratio σ to the nearest integers towards a lower value, i.e. S = 1 for the example case. Finally, L corresponds to the geometric length in Figure 4.2(b). 94

128 The chief goal of this chapter is to find semi-analytical solutions to Eq. (4.1) type periodic systems which significantly differ from the classical Hill s equation [4.1] in several ways. First, the periodic ki ( t ) is not confined to a rectangular wave assumed by Manish and Singh [ ], or a simple sinusoid as in the Mathieu s equation [4.1]. Instead, Eq. (4.1) should describe realistic, yet continuous, profiles of Figure 4.1(a) resulting from a detailed finite element/contact mechanics analysis [4.7]. Hence, multiple harmonics of ki ( t ) should be considered. Second, the periodic viscous ci ( t ) term should dissipate vibratory energy due to the sliding friction besides its kinematic effect. Third, S the δi() tci() t and i= 1 S δi() tki() t terms of Eq. (4.1) incorporate combined (but phase i= 1 correlated) contributions from all (yet changing) tooth pairs in contact. Consequently, the relative phase between neighboring tooth pairs should play an important role in the resulting response δ () t. Fourth, multiplicative effects between k ( t ), c ( t ), X ( t ) and δ () t should result in higher mesh harmonics, which poses difficulty in constructing i closed-form solutions. Lastly, Tt () of Eq. (4.1) represents the time-varying component of the forcing function due to unloaded (manufacturing) static transmission error ε () t. This indicates that the frictional forces and moments reside on both sides of Eq. (4.1) as either periodically-varying parameters or external excitations, thus posing further mathematical complications. i i i 95

129 4.2 Problem Formulation Sliding friction has been found as a non-negligible excitation source in spur and helical gear dynamics by Houser et al. [4.8], Velex and Cahouet [4.9], Velex and Sainsot [4.10], and Lundvall et al. [4.11]. Earlier, Vaishya and Singh [ ] developed a SDOF spur gear model with rectangular k(t) and sliding friction profiles; they solved the δ(t) response by using the Floquet theory [4.4] and multi-term harmonic balance method (MHBM) [4.5]. Their work was recently refined and extended to helical gears in our work (refer to Chapters 5 and 6) where closed-form solutions of δ(t) for a SDOF system are derived under frictional excitations. While the equal load sharing assumption [ ] yields simplified expressions and analytically tractable solutions, they do not describe realistic conditions. This particular deficiency has been partially overcame in Chapters 2 and 3 where we proposed a multi-degree-of-freedom (MDOF) model with realistic k(t) and sliding friction functions. However, we utilized numerical integration and fast Fourier transform (FFT) analysis methods in Chapters 2 and 3 that are often computationally sensitive. Hence, a semi-analytical algorithm based on MHBM is highly desirable for quick constructing frequency responses without any loss of generality. Recently, Velex and Ajmi [4.12] implemented a similar harmonic analysis to approximate the dynamic factors in helical gears (based on tooth loads and quasi-static transmission errors). Their work, however, does not describe the multi-dimensional system dynamics or include the frictional effect, which may lead to multiplicative terms as described earlier. 96

130 The prime objective of this chapter is thus to extend the above mentioned publications [4.4]. In particular, we intend to develop semi-analytical harmonic balance solutions to the 6DOF spur gear model of Chapter 2 with realistic k(t) and sliding friction functions. The example case used for this study is the unity ratio NASA spur gear (with tip relief); refer to Table 2.2 of Chapter 2 for its parameters. Key assumptions include: (i) the pinion and gear are rigid disks; (ii) vibratory motions are small in comparison to the nominal motion; this would lead to a linear time-varying model; (iii) Coulomb friction is assumed with a constant µ though sign is reversed at the pitch point; (iv) when the torsional component is dominant over the translational component of δ(t) for the 6DOF model of Chapter 2, the harmonic solutions of the SDOF system could be extended to predict translational responses in the line-of-action (LOA) and off-line-of-action (OLOA) directions. Note that semi-analytical method analyzes the 6DOF system as a 5DOF model as it calculates the δ(t) and not absolute angular displacements θ p (t) and θ p (t); all 6 motion terms are determined in the numerical method. 4.3 Semi-Analytical Solutions to the SDOF Spur Gear Dynamic Formulation Consider the example case with only the mean load T e, i.e. T(t) = 0 including ε () t = 0. Equation (4.1) can be rewritten over the mesh cycle 0 t tc as follows: 97

131 [ ] m eδ() t + c0() t δ() t + k0() t δ() t 1+ E1+ E3t L, (4.2a) AP + c1( t) δ( t) + k1( t) δ( t) 1 + ( E2 + E3t) sgn( t ) = Fe Ω prbp ( ) E1 = µ LXA + λ Jgrbp + LYC Jpr bg / Jb, (4.2b) ( ) E2 = µ LXAJgrbp + LYC + λ Jpr bg / Jb, (4.2c) ( ) E = Ω r J r J r J, m = J / J, F = T / J. (4.2d-f) 3 µ p bp g bp p bg / b e e b e e b Next express Eq. (4.2) in terms of the dimensionless time τ = t/ t c δ ( τ) = dδ / dτ = t δ( t) andδ ( τ) = d δ / d τ = t δ( t) : c c, such that + [ ][ ] [ ( τδ ) ( τ) ( τδτ ) ( )][ 1 ( τ) ( τ) ] meδ ( τ) tc c ( τ) δ ( τ) tck ( τ) δ( τ) 1 E tce f ( τ), (4.3a) + t c + t k + E f + E f = t F 0 2 c 1 c c e f ( τ ) mod( τ,1) =, f ( τ ) sgn [ mod( τ,1) τ ] sgn [ f ( τ) τ ] = =, (4.3b-c) 1 P 0 [ ] f ( τ ) = mod( τ,1)sgn mod( τ,1) τ = f ( τ) f ( τ), τ = L / λ. (4.3d-e) 2 P 0 1 P AP P Each periodic function, ki ( τ ), ci ( τ ), fi ( τ ) or δ ( τ ) now has a period of T = 1; Figure 4.1(b) shows the typical f ( τ ), f ( τ ) and 0 1 f ( ) 2 τ functions, which describe the periodic moment arm and sliding friction excitations for the example case. 98

132 4.3.1 Direct Application of Multi-Term Harmonic Balance (MHBM) Define the Fourier series expansions of the periodic ki( τ ) and c i (t) in Eq. (4.3) up to N mesh harmonics as follows, where ω = 2π n (in rad/s) and n is the mesh order. n N k ( τ ) = A + A cos( ωτ) + B sin( ωτ) i ki0 kin n kin n n= 1 n= 1 N. (4.4a) ki0 1 A = k ( τ ) dτ, 0 i 1 A = 2 k cos( ω τ) dτ, kin 0 in n 1 B = 2 k sin( ω τ) dτ. (4.4b-d) kin 0 in n N c ( τ ) = 2 ζ k ( τ) I = A + A cos( ωτ) + B sin( ωτ) i i e ci0 cin n cin n n= 1 n= 1 N. (4.5a) di0 1 A = c ( τ ) dτ, 0 i 1 A = 2 c cos( ω τ) dτ, cin 0 in n 1 B = 2 c sin( ω τ) dτ. (4.5b-d) cin 0 in n The f i (τ) functions could be expanded explicitly as shown below: N 1 1 f0( τ ) = sin( ωτ n ), (4.6a) 2 nπ 1 n= 1 ( ωτ N ) ( ωτ n P) 4sin 4 cos 1 f ( τ ) = 1 2τ cos( ω τ) + sin( ω τ) N n P P n n n= 1 ωn n= 1 ωn, (4.6b) N 1 sin( ) cos( ) cos( ) ωτ n P ωτ n P ωτ n P ωτ n + n= 1 f2( τ) = τp + 2 N. (4.6c) 2 ωn ωnτpcos ( ωnτp) sin ( ωnτp) 0.5ωn sin( ωnτ) n= 1 99

133 Finally, assume that the periodic dynamic response δ ( τ ) is of the following form: N δ ( τ) = A + A cos( ω τ) + B sin( ω τ) δ0 δn n δn n n= 1 n= 1 N. (4.7) Substitute Fourier series expansions of Eqs. ( ) into Eq. (4.3) and balance the mean and harmonic coefficients of sin( ωnτ ) and cos( ωnτ ). This converts the linear periodic differential equation into easily solvable linear algebraic equations (as expressed below) where K h is a square matrix of dimension (2N+1) consisting of known coefficients of ki ( τ ), ci ( τ ) and fi ( τ ). By calculating the inverse of coefficients of δ ( τ ) could be computed at any gear mesh harmonic (n). K h, the 2N+1 Fourier Aδ 0 Fe A 1 0 δ B δ1 0 K h =. (4.8) A δ N 0 B δ N Semi-Analytical Solutions Based on One-Term HBM Next, we construct one-term HBM [4.13] solutions to conceptually illustrate the method. Set the harmonic order N = 1 (only the fundamental mesh, in addition to the mean term) in Eqs. ( ) and balance the harmonic terms in Eq. (4.3). This leads to a 100

134 K h matrix of dimension 3. Three of its typical coefficients are given as follows and the rest could be found in a similar manner: K h11 1 ta c k11af 21E3 + Bk11Bf11E2 ( tb c k01e3 / π ) + 2Ak00 + 2Ak10 + Ak11 Af11E2 = 2 2Ak00E1 tcak00 E3 2Ak10 Af10E2 2tcAk10 Af 20E3 tcbk11bf 21E K = A + A + A E + A A E + t A A E + A A E h21 k01 k k10 f11 2 c k10 f 21 3 k11 f10 2 ( ) + ta / 2 c k11af 20E3 + ta c k01e3 (4.9a) (4.9b) K = B A E + B + t B A E + A B E + t A B E + B E + B h31 k11 f10 2 k11 c k11 f 20 3 k10 f11 2 c k10 f 21 3 k01 1 k01 t t B E A E + π 2 c c k01 3 k 00 3 (4.9c) The Fourier series coefficients of δ ( τ ) are then obtained by inverting K. Figure 4.3 shows that one-term HBM solution predicts the overall tendency (mean and first harmonic) fairly well when compared with numerical simulations at T p = 550 lb-in and h Ω p = 500 RPM. This confirms that the one-term HBM (and likewise the MHBM) approach coverts the periodic differential Eq. (4.3) with multiple interacting coefficients into simpler algebraic calculations that are computationally more efficient than numerical integrations and subsequent FFT analyses. Thus, the semi-analytical solution provides an effective design tool. Also, most coefficients of Eq. (4.9) show side-band effects that are introduced by k(t) (or c(t)) and the f i (t) functions. 101

135 10.4 x (in) (a) t/t c (in) (b) Figure 4.3 Semi-analytical vs. numerical solutions for the SDOF model, expressed by Eq. (4.3), given T p = 550 lb-in, Ω p = 500 RPM, µ = (a) Time domain responses; (b) mesh harmonics in frequency domain. Key:,, numerical simulations;,, semi-analytical solutions using one-term HBM;,, semi-analytical solutions using 5-term HBM. 102

136 4.3.3 Iterative MHBM Algorithm When N 5, we can utilize a symbolic software [4.14] to balance multiple harmonic terms and calculate element of K h. However, the computational cost involved with each K h increases by N 3 due to the triple multiplication of periodic coefficients in Eq. (4.3). Consequently, for higher N (say >5), a direct computation of K h becomes inefficient and thus inadvisable. Instead, we apply a matrix-based iterative MHBM algorithm [4.15, 4.16]. First, define variables Ω and ϑ: π ( υt ), ϑ =Ω t [ 0, 2π ), ϑ mod ( υϑ/ 2 π, 1 ) Ω= 2 / c =. (4.10a-c) where υ is the sub-harmonic index; also define the differential operator ^ as: xˆ dx dϑ 1 = =Ω x, x xˆ = Ω. (4.11a,b) Equation (4.3) is then converted into the following form: ( ) ( ) ( ) ( ) ( ) 2 Ω m ˆ ˆ eδ ϑ + c0 ϑ δ ϑ k0 ϑ δ ϑ 1 E1 E4ϑ Ω + + +, (4.12) + Ω c + k + E + E L = F ( ϑ) ˆ δ ( ϑ) ( ϑ) δ ( ϑ) 1 ( ϑ) sgn ( λϑ / AP 1) e Or, express it more compactly as: 103

137 ~ Ω 2 ~ m ˆ δ =, (4.13a) e ( ϑ) + ΩC( ϑ) ˆ δ ( ϑ) + K( ϑ) δ ( ϑ) Fe ( ϑ) = 0( ϑ)( ϑ) + 1( ϑ) 1+ ( 2 + 4ϑ) sgn ( λϑ / 1) C c E E c E E L AP, (4.13b) ( ϑ) = 0( ϑ)( ϑ) + 1( ϑ) 1+ ( 2 + 4ϑ) sgn ( λϑ / 1) K k E E k E E L AP. (4.13c) For the MHBM, a discrete Fourier transform (DFT) matrix is formed as follows, where ϑ = i2π M and M 2 N + 1: i / F ( ϑ1) ( ϑ1) ( Nϑ1) ( Nϑ1) ( ϑ ) ( ϑ ) ( Nϑ ) ( Nϑ ) 1 sin cos sin cos 1 sin cos sin cos 1 sin cos sin cos = ( ϑ ) ( ϑ ) ( Nϑ ) ( Nϑ ) M M M M, (4.14a) δ ( 1) ( ) δ ϑ δ ϑ δ ( ϑm ) F 2 =, ˆ δ ( 1) ( ) ˆ δ ϑ ˆ δ ϑ ˆ δ ( ϑ M ) FD 2 =, ˆ δ ˆ δ ( ϑ 1) ˆ δ ( ϑ ) ˆ δ ( ϑm ) 2 2 = FD (4.14b-d) Here, is a vector of 2 N + 1 Fourier coefficients; and the Fourier differentiation matrix is given as: 104

138 D =, N N 0 D = N N (4.14e,f) Applying the DFT to the equation of motion yields the following MHBM equations where + F is the Moore-Penrose or pseudo-inverse of the DFT matrix: m e D F CF F KF F, (4.15a) Ω +Ω + = ({ ( ϑ1) ( ϑ2) ( ϑm )}) K diag K K K, (4.15b) ({ ( ϑ1) ( ϑ2) ( ϑm )}) C diag C C C. (4.15c) Figure 4.3 shows that the five-term HBM solutions compare well with numerical simulations. Likewise, an increase in N captures higher frequency components around the 10 th mesh harmonic, as observed in the numerical simulations. The semi-analytical solutions are efficiently used in Figure 4.4 for parametric studies of δ(t) at the gear mesh harmonics over a range of computational cost. Ω p ; and, these calculations are indeed achieved with reduced 105

139 Ω p Ω p Ω p Ω p Figure 4.4 Semi-analytical vs. numerical solutions for the SDOF model as a function of pinion speed with µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key:, numerical simulations;, semi-analytical solutions using 5-term HBM. 106

140 4.4 Analysis of Sub-Harmonic Response and Dynamic Instability Vaishya and Singh [4.5] examined the parametric instability of a spur gear pair (with equal load sharing) via the sub-harmonic analysis. A similar approach is implemented along with following improvements. First, since realistic rather than rectangular k(t) profile is examined, system stability could now be evaluated as a function of T p with contributions from profile modifications. Second, the sub-harmonic matrix was constructed earlier [4.5] as an external forcing function without the frictional effects. In the proposed work, sliding friction is characterized as a parametric excitation and then its effect on instability is examined. Re-expand δ ( τ ) in terms of the first sub-harmonic term at 0.5ω 0 and higher sub-harmonics such as1.5ω 0 : δu ( τ) = Aδ0.5 cos( ω0.5τ) + Bδ0.5 sin( ω0.5τ) + Aδ1.5 cos( ω1.5τ) + Bδ1.5 sin( ω1.5τ) (4.16) Substitute δu ( τ ) into Eq. (4.3) and expand coefficients at sub-harmonic frequencies. By balancing harmonics at 0.5ω 0 and 1.5ω 0, the following equation is obtained Aδ B δ K u =. (4.17) 4 4 A δ1.5 0 Bδ

141 For a non-trivial (unstable) solution to the above homogeneous equation, matrix K u must be singular. Such points correspond to period-doubling instability [4.5, 4.17] and are found by computing the determinant K u as a function of the ratio of mesh frequency ω n and the natural frequency Ω NS of the corresponding linear time-invariant SDOF system, which could be estimated from I e and time-averaged mesh stiffness. Figure 4.5 shows the normalized K u for the example case given T p = 100 lb-in (light load) and T p = 550 lb-in (design load); each is calculated for four damping ratios ζ. The zero-crossing points of the K u curve suggest the onset of instability [4.5]. Note that the unstable zone depends on the value of T p, which also influences k(t) and the effective contact ratio. For instance, larger period-doubling unstable zone is observed under a light load condition in Figure 4.5(a), as compared with the design loading condition in Figure 4.5(b). Further, instability could be effectively controlled by an increase in ζ. For example, when T p = 550 lb-in, 2% value is sufficient to stabilize the system under period-doubling condition; however, about 8% is needed at T p = 100 lb-in to achieve stability under the same condition. Also, a variation in µ seems to have negligible influence on K u ; however, in reality the energy dissipated by the sliding friction is usually embedded in an equivalent ζ. Thus an increase in µ should enhance the stability. 108

142 (a) K u (b) Figure 4.5 Normalized determinant of the sub-harmonic matrix K u as a function of ωn / Ω NS with µ = 0.04: (a) T p = 100 lb-in.; (b) T p = 550 lb-in. Key:, ζ = 0;, ζ = 0.01;,ζ = 0.05;, ζ =

143 4.5 Semi-Analytical Solutions to 6DOF Spur Gear Dynamic Formulation When the excitation (mesh) frequencies do not coincide with any natural frequency, the semi-analytical solution, δ ( τ) = r θ ( τ) r θ ( τ), of the SDOF model bp p bg g could approximate the DTE, δ ( τ) = r θ ( τ) r θ ( τ) + x ( τ) x ( τ), for a 6DOF spur bp p bg g p g gear system of Chapter 2, where x(τ) and y(τ) are the bearing displacements in the LOA and OLOA directions respectively. First, the distribution of natural frequencies is examined by using a 6DOF linear time-invariant model of Figure 4.6(a). Since the OLOA motions y p (τ) and y g (τ) are decoupled from other DOFs, we will focus on the coupling (in terms of DTE) between transverse and torsional motions in the LOA direction. This leads to a simplified 3DOF system model [4.18] in terms of x p (τ), x g (τ) and δ () t. Define the following parameters for the example case: Mass of pinion (gear) mp = mg = M ; moment of inertia J p = J g = J ; basic radius r bp = r bg = R ; the equivalent mass 2 e = /(2 ) ; m J R time-averaged mesh stiffness k m (T p =550 lb-in.); shaft-bearing stiffness K Bp = KBg = KB. The natural frequencies of the 3DOF system are found as follows [4.18]. Ω = 2 N1, N3 [ ] [ ] 2 k M + (2 k + K ) m ± k M + (2 k + K ) m 4Mm k K m m B e m m B e e m B 2Mm e, (4.18a) K M 2 B Ω N 2 =. (4.18b) 110

144 θ g J, M K B k m K B K B J, M x K B θ p y θ (a) (b) Figure 4.6 (a) 6DOF spur gear pair model and its subset of unity gear pair (3DOF model) used to study the natural frequency distribution; (b) Natural frequencies Ω N as a function of the stiffness ratio K / k. Key:, Ω SN of SDOF system (torsional only, in terms B m of DTE);, Ω N1 of 3DOF system;, Ω N2 of 3DOF system;, Ω N3 of 3DOF system. 111

145 Mode 1 ( Ω N1 ) and Mode 3 ( Ω N 3 transverse-torsional modes, respectively; and Mode 2 ( Ω N2 ) correspond to the first and second coupled ) is the purely transverse mode. Also, the natural frequency of the corresponding SDOF torsional system can be estimated as Ω NS = km / me. Figure 4.6(b) compares the natural frequencies ( Ω N ) of the 3DOF and SDOF models as a function of the stiffness ratio K / k. For easy comparison with the excitation (mesh) frequencyω, which is determined by the nominal pinion RPM, n all natural frequencies are converted in Figure 4.6(b) from rad/s into RPM units. Observe B m that Ω asymptotically approaches Ω 3 (or Ω 1 ) when K / k < 1 (or K / k > 10 ). NS N N B m B m Moreover, ω does not excite any resonance in Zone I ( ω <<Ω 1 ) and Zone III n n N ( n N3 ω >> Ω ). Additionally, non-resonant Zone II could be found for both soft B m N n NS B m NS n N ( K / k < 1, Ω 2 << ω <<Ω ) and stiff ( K / k > 10, Ω << ω << Ω 2 ) shaftbearing cases. In these non-resonant zones, the semi-analytical solution δ(t) of the SDOF model could be extended to the 6DOF system. Figure 4.7 compares the 5-term HBM prediction of δ(t) for the SDOF system with numerical simulation for a 6DOF model for two limiting value of K / k. When K / k = 100, prediction matches well with B m B numerical simulation. However, when K / k = 0.37 (i.e. nominal case of Chapter 2), B good correlation is observed only away from system resonances Ω in Zones I, II and III. m m N 112

146 Ω p Ω p Ω p Ω p Figure 4.7 Semi-analytical vs. numerical solutions for the 6DOF model as a function of Ω p with µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key:, predictions using five-term HBM;, numerical simulations with nominal K B (K B /k m = 0.37);, numerical simulations with stiff K B (K B /k m = 100). 113

147 The dynamic normal (mesh) loads Ni ( τ ) of the pinion and gear are equal in magnitude but opposite in direction. To account for interactions between ki ( τ ) and δ ( τ ) as well as between ci ( τ ) and δ ( τ ), Fourier series is expanded to find the Ni( τ ) terms up to 2N mesh harmonics. 2N 2N N ( τ ) = k ( τδτ ) ( ) + c ( τδτ ) ( ) = A + A cos( ωτ) + B sin( ωτ), (4.19a) N00 N0n n N0n n n= 1 n= 1 2N 2N N ( τ ) = k ( τδτ ) ( ) + c ( τδτ ) ( ) = A + A cos( ωτ) + B sin( ωτ). (4.19b) N10 N1n n N1n n n= 1 n= 1 Similarly, the dynamic friction forces Ffi ( τ ) are expanded in Eq. (4.20). Since tooth pair #1 is associated with a periodic change of friction force at the pitch point, F ( ) f 1 τ is expanded up to 3N mesh harmonics due to a multiplication of k 1 ( τ ), δ ( τ ) and f ( τ ). 1 2N 2N, (4.20a) F ( τ ) = µ N ( τ) = A + A cos( ωτ) + B sin( ωτ) f 0 0 F00 F0n n F0n n n= 1 n= 1 3N 3N. (4.20b) F ( τ ) = µ N ( τ) f ( τ) = A + A cos( ωτ) + B sin( ω τ) f1 1 1 F10 F1n n F1n n n= 1 n= 1 In the LOA (or x) direction, the transfer function (at frequency ω a) with N p (τ) as input and xp ( τ ) as output is found by using the corresponding linear time-invariant model as follows, where K pbx and ζ pbx are the shaft-bearing stiffness and damping terms. 114

148 X t =. (4.21a) N m + K t + j t K m 2 p c ( ω) 2 2 p ( ω p pbx c ) 2ω cζ pbx pbx p Magnitude M pxn and phase α pxn at the n th mesh order, ωn = 2πnτ (rad/s), are M px ( ω ) = n t 2 c ( ω m + K t ) + 4ω t ζ K m n p pbx c n c pbx pbx p, (4.21b) α ( ω ) = tan px n 1 2ω t ζ K m n c pbx pbx p ω m K t 2 2 n p pbx c. (4.21c) Thus the pinion bearing displacement xp ( τ ) in the LOA (or x) direction is expanded as n= 1 2N xp( τ) = M px(0)( AN00 + AN10) + M px( ωn)( AN0 n + AN1n)cos ωτ n + αpx( ωn) n= 1. (4.22) 2N + M px ( ωn)( BN 0n + BN1 n)sin ωτ n + αpx( ωn) In the OLOA (or y) direction, the magnitude and phase of the transfer function from friction force F f (τ) to pinion displacement y p ( τ ) could be found at ω n as M py ( ω ) = n t 2 c ( ω m + K t ) + 4ω t ζ K m n p pby c n c pby pby p, (4.23a) α ( ω ) = tan py n 1 2ω t ζ K m n c pby pby p ω m K t 2 2 n p pby c 115. (4.23b)

149 Thus the pinion displacement y p ( τ ) in the OLOA (or y) direction is expanded as: y ( τ ) = M (0) A + M (0) A p py F00 py F10 2N 2N + M ( ω ) A cos ωτ + α ( ω ) + M ( ω ) B sin ωτ + α ( ω ). (4.24) py n F 0n n py n py n F 0n n py n n= 1 n= 1 3N 3N + M ( ω ) A cos ωτ + α ( ω ) + M ( ω ) B sin ω τ + α ( ω ) py n F1n n py n py n F1n n py n n= 1 n= 1 Equations (4.22, 4.24) confirm that multiplications between periodic coefficients ki ( τ ) (or ci ( τ ) ), δi ( τ ) (or c i ( τ ) ) and fi ( τ ) lead to higher mesh harmonic components which are commonly observed in spur gears [4.8]. 4.6 Conclusion function of Figure 4.8 compares the semi-analytical xp ( τ ) with numerical prediction as a Ω p with K B /k m = 100. Good correlations are observed up to 16,000 RPM (in Zone 1 with high K B /k m ) including the LOA shaft-bearing resonances at n = 3 and n = 4. Likewise, the semi-analytical y p in the OLOA direction compares well with numerical simulations in Figure 4.9, where the shaft-bearing resonances are also observed at n = 3 and n =

150 Ω p Ω p Ω p Ω p Figure 4.8 Semi-analytical vs. numerical solutions of the LOA displacement x p for the 6DOF model as a function of Ω p with K B /k m = 100, µ = 0.04 (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key:, numerical simulations;, predictions using five-term HBM. 117

151 Ω p Ω p Ω p Ω p Figure 4.9 Semi-analytical vs. numerical solutions of the OLOA displacement y p for the 6DOF model as a function of Ω p with K B /k m = 100, µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key:, numerical simulations;, predictions using five-term HBM. 118

152 Although good correlations are usually expected for stiff shaft-bearings, it is worthwhile to examine the case where K B is comparable or less than k m. By using the nominal parameters of Chapter 2 where K B /k m = 0.37, semi-analytical predictions are compared with numerical simulations in Figure 4.10 and Figure 4.11 for LOA and OLOA responses, respectively. Observe that semi-analytical x p matches numerical simulations only from system resonances Ω N, as explained earlier in Figure 4.6. Nonetheless, good correlation is observed in the OLOA direction over the operating speed range in Figure 4.11 since the bearing resonance dictates the y p dynamics. Overall, this chapter has successfully developed semi-analytical solutions to periodic differential equations with time-varying parameters of spur gears including realistic mesh stiffness and sliding friction. Proposed one-term and multi-term HBM predictions compare well with numerical simulations; the computational efficiency is achieved by converting the periodic differential equations into easily solvable algebraic equations, while providing more insight into the dynamic behavior. Both super-and subharmonic analyses are successfully conducted to examine the higher mesh harmonics due to multiplicative coefficients and the system stability, respectively. Finally, semianalytical solutions are developed for a 6DOF system model for the predictions of (normal) mesh loads, friction forces and bearing displacements in the LOA and OLOA directions, under non-resonant conditions. Methods of this work could be extended to multi-mesh spur gear dynamics. 119

153 1.2 x 10-4 Ω N1 Ω N 2 Ω NS (a) 2 x 10-5 (b) 0.8 Z 1 Z 2 Z Ω N x x Ω (RPM) 10-5 p (c) x10 3 x 10-5 Ω (RPM) p 0.8 (d) x x10 3 (RPM) (RPM) Ω p Ω p Figure 4.10 Semi-analytical vs. numerical solutions of the LOA displacement x p for the 6DOF model as a function of Ω p with K B /k m = 0.37, µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key:, numerical simulations;, predictions using five-term HBM. 120

154 Ω p Ω p Ω p Ω p Figure 4.11 Semi-analytical vs. numerical solutions of the OLOA displacement y p for the 6DOF model as a function of Ω p with K B /k m = 0.37, µ = (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key:, numerical simulations;, predictions using five-term HBM. 121

155 References for Chapter 4 [4.1] Richards, J. A., 1983, Analysis of Periodically Time-varying Systems, New York, Springer. [4.2] Jordan, D. W., and Smith, P., 2004, Nonlinear Ordinary Differential Equations, 3 rd Edition, Oxford University Press. [4.3] Thomsen, J. J., 2003, Vibrations and Stability, 2 nd Edition, Springer. [4.4] Vaishya, M., and Singh, R., 2001, Analysis of Periodically Varying Gear Mesh Systems with Coulomb Friction Using Floquet Theory, Journal of Sound and Vibration, 243(3), pp [4.5] Vaishya, M., and Singh, R., 2001, Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics, Journal of Sound and Vibration, 248(4), pp [4.6] Vaishya, M., and Singh, R., 2003, Strategies for Modeling Friction in Gear Dynamics, ASME Journal of Mechanical Design, 125, pp [4.7] External2D (CALYX software), 2003, Helical3D User s Manual, ANSOL Inc., Hilliard, OH. [4.8] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, Vibro-Acoustic Effects of Friction in Gears: An Experimental Investigation, SAE Paper # [4.9] Velex, P., and Cahouet, V., 2000, Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics, ASME Journal of Mechanical Design, 122(4), pp [4.10] Velex, P., and Sainsot. P, 2002, An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears, Mechanism and Machine Theory, 37, pp [4.11] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, A Flexible Multi-body Approach for Frictional Contact in Spur Gears, Journal of Sound and Vibration, 278(3), pp [4.12] Velex, P., and Ajmi, M., 2007, Dynamic Tooth Loads and Quasi-Static Transmission Errors in Helical Gears Approximate Dynamic Factor Formulae, Mechanism and Machine Theory Journal, 42(11), pp [4.13] Kim, T. C., Rook, T. E., and Singh, R., 2005, Super- and Sub-Harmonic Response Calculations for A Torsional System with Clearance Non-Linearity using Harmonic Balance Method, Journal of Sound and Vibration, 281(3-5), pp

156 [4.14] Maple 10 (symbolic software), 2005, Waterloo Maple Inc., Waterloo, Ontario. [4.15] Padmanabhan, C., Barlow, R. C., Rook, T.E., and Singh, R., 1995, Computational Issues Associated with Gear Rattle Analysis, ASME Journal of Mechanical Design, 117, pp [4.16] Duan, C., and Singh, R., 2005, Super-Harmonics in a Torsional System with Dry Friction Path Subject to Harmonic Excitation under a Mean Torque, Journal of Sound and Vibration, 285(2005), pp [4.17] Den Hartog, J. P., 1956, Mechanical Vibrations, New York, Dover Publications. [4.18] Kahraman, A., and Singh, R., 1991, Error Associated with A Reduced Order Linear Model of Spur Gear Pair, Journal of Sound and Vibration, 149(3), pp [4.19] Singh, R., 2005, Dynamic Analysis of Sliding Friction in Rotorcraft Geared Systems, Technical Report submitted to the Army Research Office, grant number DAAD

157 CHAPTER 5 EFFECT OF SLIDING FRICTION ON THE VIBRO-ACOUSTICS OF SPUR GEAR SYSTEM 5.1 Introduction Gears are known to be one of the major vibro-acoustic sources in many practical systems including ground and air vehicles such as heavy-duty trucks and helicopters. Typically, steady state gear (whine) noise is generated by several sources and the reduction of gear noise is often challenging for most products. Virtually all prior researchers [ ] have assumed the main exciter to be the static transmission error (STE) that is defined as the derivation from the ideal tooth profile induced by manufacturing errors and elastic deformations. However, high precision gears are still unacceptably noisy in practice. When the transmission error has been minimized (say via modifying the tooth profile), the sliding friction remains as a potential contributor to gear noise and vibration. Further, most prior research on gear friction [ ] has been confined to the dynamic analysis of the gear pair source sub-system and no attempt has been made to examine the friction related structural path and noise radiation issues. To 124

158 fill in this void, the main objectives of this chapter are thus to: First, propose a refined source-path-receiver model that characterizes the structural paths in two directions and, second, propose analytical tools to efficiently predict the whine noise and quantify the contribution of sliding friction to the overall whine noise. The system model is depicted in Figure 5.1. The source sub-system includes the spur gear pair and shafts inside the gearbox; these are characterized by a 6 degree-of-freedom (DOF) linear-time-varying model of Chapter 2. The transmission error dominated bearing forces in the line-of-action (LOA) direction and friction dictated bearing forces in the off line-of-action (OLOA) direction are coupled and transmitted to the housing structure. Radiated sound pressure p(ω) from gearbox panels (at gear mesh frequencies) are then received by microphone(s). Analytical predictions of the structural transfer function and noise radiation will be compared with measurements. Transmission error Sliding friction SOURCE 6 DOF linear-timevarying spur gear pair model + shafts LOA bearing forces OLOA bearing forces Coupling at bearings RECEIVER PATH Sound pressure Radiation model Housing velocity Housing structure model Figure 5.1 Block diagram for the vibro-acoustics of a simplified geared system with two excitations at the gear mesh. 125

159 5.2 Source Sub-System Model The source sub-system is described by the 6DOF, linear time-varying spur gear pair model of Chapter 2 that incorporates the sliding friction and realistic mesh stiffness, which is calculated by an accurate finite element/contact mechanical code [5.6]. Rigid bearing is assumed as boundary conditions due to the impedance mismatch at the shaft/bearing interface. Overall, the system formulations are summarized as following. The governing equations for the torsional motions θ p (t) and θ g (t) of pinion and gear are: n J θ () t = T + X () t F () t r N () t (5.1) p p p pi pfi bp pi i= 0 i= 0 n n J θ () t = T + X () t F () t + r N () t (5.2) g g g gi gfi bg gi i= 0 i= 0 n where n = floor(σ) in which the floor function rounds off the contact ratio σ to the nearest integer (towards a lower value); J p and J g are the polar moments of inertia of the pinion and gear; T p and T g are the external and braking torques; r bp and r bg are base radii of the pinion and gear; and, N pi (t) and N gi (t) are the normal loads defined as follows: N pi () t = Ngi () t = ki () t rbpθp () t rbgθg () t + xp () t xg () t + ci( t) r bpθp( t) r bgθg( t) + x p( t) x g( t) (5.3) 126

160 where k i (t) and c i (t) are the realistic mesh stiffness and viscous damping profiles; x p (t) and x g (t) denote the LOA displacements of pinion and gear centers. The sliding friction forces F pfi (t) and F gfi (t) as well as their moment arms X pi (t) and X gi (t) of the i th meshing pair are derived as: F () t = µ () t N () t, F () t = µ () t N () t (5.4a,b) pfi i pi gfi i gi X ( t) = L + ( n i) λ + mod( Ω r t, λ), (5.5a) pi XA p bp X ( t) = L + iλ mod( Ω r t, λ) (5.5b) gi YC g bg where the sliding friction is formulated by µ ( t) = µ 0 sgn mod( Ω r t, λ) + ( n i) λ L i p bp AP is the base pitch; sgn is the sign function; the modulus function mod(x, y) = x ; λ y floor(x/y), if y 0; Ω p and Ω g are the nominal speeds (in rad/s); and, L AP, L XA and L YC are geometric length constants of Chapter 2. The governing equations for x p (t) and x g (t) motions in the LOA direction are: mx () t+ 2 ζ K mx () t+ K x() t+ N () t = 0 (5.6) p p psx psx p p psx p pi i= 0 n mx () t + 2 ζ K mx () t+ K x() t + N () t = 0 (5.7) g g gsx gsx g g gsx g gi i= 0 n Here, m p and m g are the masses of the pinion and gear; K psx and K gsx are the effective shaft stiffness values in the LOA direction, and ζ psx and ζ gsx are the damping ratios. Likewise, the translational motions y p (t) and y g (t) in the OLOA direction are governed by: 127

161 m y () t + 2 ζ K m y () t + K y () t F () t = 0 (5.8) p p ps y psy p p psy p pfi i= 0 n my () t+ 2 ζ K my () t+ K y() t F () t = 0 (5.9) g g gs y gsy g g gsy g gfi i= 0 n Both LOA and OLOA bearing forces are predicted for the example case (unityratio NASA spur gear pair whose parameters are listed in Table 5.1) and compared in Figure 5.2 at the first three gear mesh frequencies as a function of pinion torque T p. Observe that the friction dominated OLOA dynamic responses are less sensitive to a variation in T p. Parameter/property Pinion/Gear Parameter/property Pinion/Gear Number of teeth 28 Face width, in 0.25 Diametral pitch, in -1 8 Tooth thickness, in Pressure angle, º 20 Center distance, in 3.5 Outside diameter, in Elastic modulus, psi Root diameter, in Shaft stiffness, lb/in Table 5.1 Parameters of the example case: NASA spur gear pair with unity ratio (with long tip relief) 128

162 F pbx (lb) m = 1 m = 2 m = Torque (lb-in) (a) F pby (lb) Torque (lb-in) (b) Figure 5.2 Bearing forces predicted under varying T p given Ω p = 4875 RPM and 140 F. (a): LOA bearing force; (b) OLOA bearing force. Key: m is the mesh frequency index. 129

163 5.3 Structural Path with Friction Contribution Bearing and Housing Models Predicted bearing forces by the source sub-system provide excitations to the multi-input multi-output (MIMO) structural paths for the gearbox of Figure 5.3(a). Force excitations are coupled at each bearing via a 6 by 6 stiffness matrix [K] Bm which is calculated by using the algorithm proposed by Lim and Singh [5.7]. Nominal shaft loads and bearing preloads are assumed to ensure a time-invariant [K] Bm. In order to focus on the transmission error path and frictional path in the LOA and OLOA directions respectively, [K] Bm is further simplified into a 2 by 2 matrix by neglecting the moment transfer [5.8] and assuming that no axial force is generated by the spur gear sub-system. Calculated nominal bearing stiffness [5.7] are K Bx = K By = lb/in at mean operating conditions; these are much larger than the shaft stiffness of lb/in. This is consistent with the impedance mismatch assumption made at the shaft/bearing interface. 130

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

165 The implementation of [K] Bm into the finite element gearbox model of Figure 5.3(b) is given special attention [ ]. At high mesh frequencies (say up to 5 khz), the dimensions of the bearings are comparable to the plate flexural wavelength. Hence the holes may significantly alter the plate dynamics and such effects must be modeled [5.10]. A rigid (with Young s modulus 100 times higher than the casing steel) and massless (with density 1% of the casing steel) beam element is used to model the interface from shaft to bearing. Its length is chosen to be very short to avoid the introduction of any beam resonances in the frequency range of interest. The shaft beam element is connected to the central bearing node though orthogonal foundation stiffness (K Bx and K By ) in the LOA and OLOA directions, respectively. The central node is then connected to the circumferential bearing nodes by 12 rigid and mass-less beams (one at each rolling element s angular position) which form a star configuration, such that the displacement of the plate around the bearing hole are equal to the housing node at the center Experimental Studies and Validation of Structural Model The finite element model of Figure 5.3 is created by using I-DEAS for the NASA gearbox with bearing holes, embedded stiffness matrices [K] Bm, stiffening plates as well as clamped boundary conditions at four rigid mounts. Although the gear pair and shafts are not included, it has been shown [5.7, 5.9] that an "empty" gearbox tends to describe the dynamics of the entire gearbox system. Table 5.2 confirms that the natural frequencies predicted by the finite element model correlate well with measurements 132

166 reported by Oswald et al. [5.11] despite minor modifications made to the gearbox. Mode shape predictions also match well with modal tests, and Figure 5.4 gives a typical comparison of structural mode at the 8 th natural frequency (f n = 2962 Hz). Method/mode index Measurements [5.11] (Hz) Finite element predictions (Hz) Table 5.2 Comparison of measured natural frequencies and finite element predictions 133

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

168 In order to validate the structural paths, several transfer functions were measured for the NASA gearbox by assuming that the quasi-static system response is similar to the response under non-resonant rotating conditions. The gearbox was modified to allow controlled excitations to be applied to the gear-mesh and measured. Brackets were welded to the bedplate of the gear-rig to mount shakers in the LOA and OLOA directions outside the gearbox, as shown in Figure 5.5(a). Stinger rods were connected from the shakers through two small holes in the gearbox and attached to a collar on the input shaft. Two mini accelerometers were fastened to a block behind the loaded gear tooth to measure the LOA and OLOA mesh accelerations. Band-limited random noise signals were then used as excitation signals and tests were done with only one shaker activated at a time with a 600 lb-in static preload. Dynamic responses were measured to generate vibro-acoustic transfer functions. Sensor # 1 of Figure 5.5(a) is a tri-axial accelerometer mounted on the output shaft bearing cap to measure the LOA, OLOA, and axial vibrations. Sensors #2 and #3 are unidirectional accelerometers mounted on the top and back plates, respectively. The transfer function of the combined source-path sub-systems is predicted as following: Y H S P( ω) = H S( ω) H P( ω) = H S( ω) Y plate bearing ( ω) ( ω) (5.10) where H ( ω) is the motion transmissibility from mesh excitation to translational bearing S responses (in LOA or OLOA directions) by using a 6DOF linear time-invariant spur gear 135

169 model [5.13]. Note that such a lumped model is insufficient to capture the bending and flexural modes of the gear flanks and shafts. Here, Y ( ω) and Y ( ω ) are the transfer and driving point mobilities for the (top) plate and the bearing; these are derived from the finite element gearbox model by using the modal expansion method with 1% structural damping for all modes. Figure 5.5(b) shows that the predicted motion transmissibility from gear mesh to the top plate correlates reasonably well with measurement given the complexity of the geared system. The highest frequency is chosen such that the shortest wave-length is 4 times larger than the mesh dimension on the top plate. Recall that the interactions between the shaft and bearings/casing were neglected in our model by the impedance mismatch assumption. Consequently, a 10 db empirical (but uniformly applied) weighting function w is used to tune the magnitude of transfer mobility prediction in Figure 5.5(b) for a better comparison. Further work is needed to explain this shift. plate bearing 136

170 40 (a) 20 H top plate (db) Frequency (Hz) (b) Figure 5.5 (a) Experiment used to measure structural transfer functions; (b) comparison of transfer function magnitudes from gear mesh to the sensor on top plate. Key:, measurements;, predictions,. 137

171 5.3.3 Comparison of Structural Paths in LOA and OLOA Directions First, assume that (i) the bearing forces predicted by the lumped source model [5.6] are in phase at either bearing end for both the pinion and gear shafts; and (ii) the bearing forces of pinion and gear are same in magnitude but opposite in directions due to the symmetry of unity ratio gear pair. Second, the overall structural paths are derived for the transmission error controlled LOA (or x) path and the friction dominated OLOA (or y) path in terms of combined effective transfer mobilities Y, ( ω) and Y, ( ω) : ex ey Y ( ω) = w Y ( ω) w Y ( ω) (5.11a) ex, pxn,, pxn,, gxn,, gxn,, n n Y ( ω) = w Y ( ω) w Y ( ω) (5.11b) ey, pyn,, pyn,, gyn,, gyn,, n n where w is the empirical weighting function (10 db, as discussed in the previous section); and the subscript n is the index of the two ends of pinion/gear shafts. Figure 5.6 compares the magnitudes of Y, ( ω) and Y, ( ω) at the sensor location on the top plate. Different ex ey peaks are observed in the LOA and OLOA paths spectra. This implies that at certain frequencies (e.g. 650 and 1700 Hz), the OLOA path (and thus the frictional effects) could be dominant over the LOA path (and thus the transmission error effects) given comparable force excitation levels. The proposed method thus provides a design tool to quantify and evaluate the relative contribution of structural path due to sliding friction. The top plate velocity distribution V ( ω) could then be predicted by using Eq. (5.12), top where F,,( ω) and F,,( ω) are the pinion bearing forces predicted by the lumped pbx pby 138

172 source model in the LOA and OLOA directions. Figure 5.7(a) shows the surface interpolated velocity distributions on the top plate at three mesh harmonics (m =1, 2, 3) given T p = 500 lb-in and Ω p = 4875 RPM. 1 1 V top ( ω) = F p, B, x( ω) Y e, x( ω) + F p, B, y ( ω) Y e, y ( ω) 2 2 (5.12) Figure 5.6 Magnitudes of the combined transfer mobilities in two directions calculated at the sensor location on the top plate. Key:, mobility of the OLOA path;, mobility of the LOA path. 139

173 5.4 Prediction of Noise Radiation and Contribution of Friction Prediction using Rayleigh Integral Technique Since the rectangular top plate is the main radiator [5.14] of the gearbox due to its relatively high mobility, Rayleigh integral [5.15] is used to approximate the sound pressure radiation by assuming that the top plate is included in an infinite rigid baffle and each elementary plate surface is an equivalent point source in the rigid wall. The sound pressure amplitude is given as follows where ρ is the air density, Q ( ω) = V ( ω) S is the source strength of i th equivalent source with area r i is the distance of i th source to the receiving point. i i i Si, k( ω) is the wave number and jωρ Q ( ω) = e 2π r i jk ( ω ) r ( ω) i (5.13) P i i Compared with conventional boundary element analysis, Rayleigh integral approximates sound pressure in a fraction of the required computation time [5.16]. Hence, it is most suitable for parametric design studies. Although some researchers [5.16] have pointed out that Rayleigh integral may give large errors for sound pressure prediction if applied to strongly directional, three dimensional (3D) fields, such errors are not significant here due to the flat (rather than curved) top plate and favorable surroundings (such as rigid side plates and anechoic chamber). 140

174 5.4.2 Prediction using Substitute Source Method As an alternative to Rayleigh integral, a newly developed algorithm based on the substitute source approach [5.17] is used to compute radiated or diffracted sound field. It is conducted by removing the gearbox and introducing acoustic sources within the liberated space which yield the desired boundary conditions at the box surface (Neumann problem). Solutions are obtained in terms of the locations and/or the strengths of the substitute sources by minimizing the error function between original and estimated particle velocity normal to the interface surface [5.17]. Since the surface velocity distributions of gearbox are essentially symmetric along the center lines due to geometric symmetry, velocity distributions along the border lines of EFGH plane in Figure 5.3(b) are chosen to simplify the 3D gearbox into a 2D radiation model for simpler data representation as well as faster computation. Zero (negligible) velocity distribution is assumed along lines EF, FG and HE since the microphone (receiver) is positioned above the center of major radiator, i.e. the top plate. A 2D line source uniformly pulsating with unit-length volume velocity Q is chosen as the substitute source. Its radiation field is the same in any plane perpendicular to the source line. Amplitudes of the sound pressure and radial velocity of such source are given by the following, where (2) H v is the Hankel function of second kind and order v. k( ) c (2) P ω ρ ( ω) = Q ( ω) H0 [ k( ω) r] 4 (5.14a) k( ) (2) V ω r ( ω) = j QH 1 [ k( ω) r] 4 (5.14b) 141

175 A greedy search algorithm [5.17] is used to search for optimal substitute sources: First, a large number of candidate source positions within the vibrating body are defined, e.g. at the vertices of a square grid. Second, a single position is first found which allows the point source to produce the smallest deviation between the original and estimated normal velocity of surface vibration. The estimation is then subtracted from the original velocity to get a velocity residual. Third, among the rest of candidate points, a new position is found which makes the second source acting at it, maximally reduce the velocity residual of the first step. Once found, the source strengths of both sources are adjusted for a best fit of the original surface velocity and a new residual velocity. Each subsequent step defines a new optimum source position among the ones not already used. The curve fitting of source strengths is done by minimizing the mean square root (RMS) value of the velocity error. The vector of complex source strength Q is related (as shown below) to the vector V n of complex normal surface velocity at control points via the source-velocity transfer matrix T where r ij = r i r j and α ij is the angle between vector r r and the outer normal to the surface. i 1 ( ω) ( ω) ( ω), (5.15a) Q = T V n k( ) (2) T ω ij ( ω) = j H1 [ k( ω) rij ]cos( αij ) 4 (5.15b) To minimize the impact of an ill-conditioned matrix, the number of control points is kept well above that of independent source points. Minimization of the RMS error 142

176 using pseudo-inverse yields the following, where the asterisk signifies the conjugate transpose: 1 * * ( ω) ( ω) ( ω) ( ω) ( ω) (5.16) Q = T T T V n The difference between synthesized and original surface normal velocities is: V ( ω) =Ξ( ω) V n ( ω), (5.17a) 1 * * Ξ ( ω) = T( ω) T( ω) T( ω) T( ω) I( ω) (5.17b) where I ( ω) is the identity matrix. The matrix Ξ ( ω) appears as a velocity error matrix. The RMS velocity error is normalized by dividing with the RMS value of original velocity as: e RMS E ( ω) = = V V V * * ( ) ( ) ( ) ( ) RMS ( ω) Vn ω Ξ ω Ξ ω Vn ω * nrms, ( ω) n( ω) n( ω) (5. 18) 143

177 5.4.3 Prediction vs. Noise Measurements Figure 5.7(a) shows predictions of surface interpolated velocity distribution on the top plate at the first three mesh harmonics under T p = 500 lb-in and Ω p = 4875 RPM. Note that predictions at high frequencies (e.g. mesh index m = 3) are less reliable due to the limitation of element dimensions as compared the wave length. The symmetry of surface velocity distribution leads to the simplification into a 2D gearbox model of Figure 5.7(b). To ensure necessary accuracy for the acoustic radiation, the selected central lines of the 2D plane should capture the dominant structural modes of Figure 5.7(a). Also, the structural wavelength along the central line should be higher than the acoustic wavelength of interest to ensure the validity of the 2D approach. The source points of Figure 5.7(b) are chosen from a mesh grid of candidate points not too close to the boundary to prevent forming steep gradients of surface pressure. Observe that only 15 substitute sources tend to predict well the surface distribution of velocity magnitude at the gear mesh harmonics. Figure 5.7(c) illustrates the predicted source strengths of the substitute sources in the complex plane for evaluation of the acoustic source properties. A single dominant substitute source is observed at the first mesh harmonic (monopole-like acoustic source); however, several dominant substitute sources are present and these are more equally distributed in the complex plane at the higher harmonics (multi-pole acoustic source). 144

178 (a) (b) e e-5 1e e e-5 5e e e-5 5e-6 30 (c) (c) Figure 5.7 Comparison of normal surface velocity magnitudes and substitute source strength vectors under T p = 500 lb-in and Ω p = 4875 RPM. (a) Line 1: interpolated surface velocity on top plate; (b) Line 2: simplified 2D gearbox model with 15 substitute source points; Key:, original surface velocity magnitude;, surface velocity magnitude by substitute sources;, locations of substitute sources. (c) Line 3: substitute source strengths in complex plane for 2D gearbox. Column 1: mesh frequency index m = 1; Column 2: m = 2; Column 3: m =