Vibration and Sound Radiation Analysis of Vehicle Powertrain

Vibration and Sound Radiation Analysis of Vehicle

Powertrain Systems with Right-Angle Geared Drive

A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mechanical Engineering, Department of Mechanical and Materials Engineering College of Engineering & Applied Science University of Cincinnati

March 2017 By

Yawen Wang

M.S., Mechanical Engineering, University of Cincinnati, Cincinnati, USA, 2013 B.S., Mechanical Engineering, Chongqing University, Chongqing, P. R. China, 2010

Committee chair: Dr. Teik C. Lim Members: Dr. Jay Kim Dr. Manish Kumar Dr. Michael J. Alexander-Ramos

1 ABSTRACT

Hypoid and bevel gears are widely used in the rear axle systems for transmitting torque at right angle. They are often subjected to harmful dynamic responses which cause gear whine noise and structural fatigue problems. In the past, researchers have focused on gear noise reduction through reducing the transmission error, which is considered as the primary excitation of the geared system. Effort on the dynamic modeling of the gear-shaft-bearing-housing system is still limited. Also, the noise generation mechanism through the vibration propagation in the geared system is not quite clear. Therefore, the primary goal of this thesis is to develop a system-level model to evaluate the vibratory and acoustic response of hypoid and bevel geared systems, with an emphasis on the application of practical vehicle powertrain system. The proposed modeling approach can be employed to assist engineers in quiet driveline system design and gear whine troubleshooting.

Firstly, a series of comparative studies on hypoid geared rotor system dynamics applying different mesh formulations are performed. The purpose is to compare various hypoid gear mesh models based on pitch-cone method, unloaded and loaded tooth contact analysis. Consequently, some guidelines are given for choosing the most suitable mesh representation. Secondly, an integrated approach is proposed for the vibro-acoustic analysis of axle systems with right-angle geared drive. The approach consists of tooth contact analysis, lumped parameter gear dynamic model, finite element model and boundary element model. Then, a calculation method for tapered roller bearing stiffness matrix is introduced, which is based on the Finite Element/Contact

Mechanics model of axle system with right-angle geared system. The effect of rigid bearing support and flexible bearing support is studied by comparing the flexible axle system model with rigid supported gear pair model. Other important system dynamic factors are also investigated,

ii such as the gear-shaft interaction, rotordynamics effect and housing flexibility. Finally, some conclusions and recommendations for future studies are given.

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ACKNOWLEDGEMENTS

“Trust in the Lord with all your heart, and do not lean on your own understanding. In all your ways acknowledge him, and he will make straight your paths.” Proverbs 3:5-6

First and above all, I would like to thank God for giving me strength and wisdom to write this dissertation.

I would like to express my sincere gratitude to my advisor Dr. Teik C. Lim for his guidance and support in the preparation of this dissertation and throughout my graduate study. I am very thankful to Dr. Jay Kim, Dr. Manish Kumar and Dr. Michael J. Alexander-Ramos for serving as my committee members and giving me valuable instructions. My sincere appreciation also goes to

Dr. Sandeep Vijayakar at Advanced Numerical Solutions for his help in developing the gear contact model.

I am grateful for the financial support provided throughout my graduate work by the

Hypoid and Bevel Gear Mesh and Dynamic Modeling Consortium, the Vibro-Acoustic and Sound

Quality Research Laboratory, the Department of Mechanical Engineering and the Graduate school.

I would also like to express my gratitude to all my colleagues at the Vibro-Acoustic and Sound

Quality Research Laboratory in University of Cincinnati. I am greatly indebted to Dr. Junyi Yang, who gave me many academic suggestions and shared with me his insights. My special thanks also go to Dr. Mingfeng Li and Dr. Guohua Sun, for their support and review of this thesis.

Finally, I would like to thank my parents Rongfu Wang and Qimei Zhang for their unconditional love and support.

TABLE OF CONTENTS

ABSTRACT ...... ii

ACKNOWLEDGEMENTS ...... v

TABLE OF CONTENTS ...... vi

LIST OF PUBLICATIONS ...... xi

LIST OF TABLES ...... xiii

LIST OF FIGURES ...... xiv

LIST OF ABBREVIATIONS ...... xx

LIST OF SYMBOLS ...... xxi

Chapter 1 Introduction ...... 1

1.1 Motivation ...... 1

1.2 Literature Review ...... 2

1.3 Scope and Objectives ...... 5

1.4 Organization ...... 6

Chapter 2 Comparative Analysis of the Hypoid Geared Rotor System Dynamics Applying

Dissimilar Tooth Meshing Formulations ...... 9

2.1 Introduction ...... 9

2.2 Gear Mesh Models ...... 10

2.2.1 Pitch Cone-based mesh model ...... 10

2.2.2 Unloaded tooth contact analysis based mesh model ...... 12

2.2.3 Loaded tooth contact analysis based mesh model ...... 13

2.2.4 Mesh stiffness calculation in pitch cone and TCA based mesh models ...... 16

2.3 Hypoid Geared Rotor System Dynamics ...... 18

2.3.1 Light Load Case ...... 19

2.3.2 Heavy Load Case ...... 20

2.3.3 Multi-point Mesh Case...... 21

2.4 Conclusion ...... 23

Chapter 3 Vibration and Sound Radiation Analysis of Vehicle Axle Systems Using an Integrated

Approach ...... 25

3.1 Introduction ...... 25

3.2 Gear Mesh and Dynamic Model ...... 26

3.2.1 Gear Mesh Model ...... 26

3.2.2 Gear Dynamics Model ...... 29

3.3 Vibration and Sound Radiation Analysis ...... 32

3.3.1 Housing Vibration Analysis ...... 32

3.3.2 Sound Radiation Analysis ...... 33

3.4 Conclusions ...... 35

Chapter 4 Tapered Roller Bearing Contact Analysis and Stiffness Calculation ...... 36

4.1 Introduction ...... 36

4.2 Finite Element/Contact Mechanics Model ...... 37

4.2.1 Gear Mesh Model ...... 37

4.2.2 Bearing Model ...... 38

4.3 Bearing Stiffness Calculation ...... 39

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4.4 Numerical Simulation ...... 40

4.4.1 Comparison against Analytical Method ...... 43

4.4.2 Bearing Stiffness ...... 45

4.4.3 Contact Pattern ...... 47

4.4.4 Transmission Error ...... 49

4.5 Conclusion ...... 51

Chapter 5 Effect of Component Flexibility on Axle System Dynamics ...... 52

5.1 Introduction ...... 52

5.2 Contact Analysis ...... 54

5.2.1 Gear Pair Model ...... 54

5.2.2 Axle System Model ...... 55

5.3 Dynamic Analysis ...... 57

5.4 Simulation Results ...... 59

5.4.1 Contact Results ...... 59

5.4.2 Modal Characteristics ...... 60

5.4.3 Dynamic Results ...... 61

5.5 Conclusions ...... 64

Chapter 6 Effect of Gear-Shaft Coupling Dynamics and Gyroscopic Moments ...... 65

6.1 Introduction ...... 65

6.2 Gear Mesh and Dynamic Formulation ...... 68

6.2.1 Gear Mesh Model ...... 68

6.2.2 Propeller Shaft Model ...... 70

6.2.3 Geared Rotor System Model ...... 72

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6.2.4 Shaft-bearing Stiffness Model ...... 75

6.2.5 Coupled Driveline System Model ...... 77

6.3 Sound Radiation Analysis ...... 79

6.4 Results and Discussion ...... 81

6.4.1 Modal Analysis ...... 82

6.4.2 Parametric Studies ...... 84

6.4.3 Design Enhancement ...... 95

6.5 Conclusions ...... 99

Chapter 7 Effect of Housing Flexibility...... 101

7.1 Introduction ...... 101

7.2 Simulation Flowchart ...... 102

7.3 Multi-Body Gearbox Dynamic Analysis ...... 104

7.3.1 Reduced Housing Model ...... 104

7.3.2 Coupled Multi-body Dynamic Model ...... 106

7.4 Vibro-Acoustic Analysis ...... 108

7.4.1 Housing Forced Vibration Analysis ...... 108

7.4.2 Boundary Element Model ...... 109

7.5 Results and Discussion ...... 109

7.6 Conclusions ...... 113

Chapter 8 Summary ...... 115

8.1 Conclusions ...... 115

8.2 Proposed Future Studies ...... 115

BIBLIOGRAPHY ...... 117

LIST OF PUBLICATIONS

Y. Wang, G. Qiao, X. Li, T. Lim, "Effect of Component Flexibility on Axle System Dynamics". SAE Technical Paper, Grand Rapids, Michigan, 2017.

Y. Wang, J. Yang, D. Guo, T. Lim, "Vibration and Sound Radiation Analysis of the Final Drive Assembly Considering the Gear-Shaft Coupling Dynamics". Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering and Science, 230(7-8), pp. 1258-1275, 2016.

Y. Wang, X. Li, T. C. Lim, "Modeling of Axle System with Emphasis on Tapered Roller Bearing Contact Analysis and Stiffness Calculation," Proceedings of the International Conference on Power Transmissions, Chongqing, China, 2016.

Y. Wang, J. Yang, X. Li, G. Sun, T. Lim, "Interaction of Gear-Shaft Dynamics Considering Gyroscopic Effect of Compliant Driveline System," SAE International Journal of Passenger Cars-Mechanical Systems, 8(2), pp. 742-747, 2015.

Y. Wang, T. C. Lim, J. Yang, "Multi-Point Mesh Modeling and Nonlinear Multi-Body Dynamics of Hypoid Geared System," SAE International Journal of Passenger Cars-Mechanical Systems, 6(2), pp. 1127-1132, 2013.

Y. Wang, D. Guo, S. Gopalakrishana, T. C. Lim, "Vibration and Sound Radiation Analysis of Vehicle Axle Systems Using an Integrated Approach," Proceedings of the National Conference on Noise Control Engineering, San Francisco, California, 2015.

Y. Wang, J. Yang, D. Guo, G. Sun, T. C. Lim, "A System Approach for Vibro-Acoustic Analysis of Right-Angle Gearbox," Proceedings of the National Conference on Noise Control Engineering, Ft. Lauderdale, Florida, 2014.

J. Yang, Y. Wang, D. Guo, T. C. Lim, "Comparative Analysis of the Hypoid Geared Rotor System Dynamics Applying Dissimilar Tooth Meshing Formulations," International Gear Conference, Lyon Villeurbanne, France, 2014.

Y. Wang, T. C. Lim, J. Yang, "Torque Load Effects on Mesh and Dynamic Characteristics of Hypoid Geared System," ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Portland, Oregon, DETC2013- 13594, 2013.

Y. Wang, " Torque Load Effect on Multi-Point Mesh and Dynamics of Right-angle Geared Drives," MS Thesis, University of Cincinnati, USA, 2013.

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LIST OF TABLES

Table 2.1 Coefficients and pressure angles...... 12

Table 3.1 Design parameters for the hyoid geared rotor system...... 28

Table 4.1 Hypoid gear design parameters...... 42

Table 4.2 Design parameters of pinion bearings...... 42

Table 4.3 Design parameters of gear bearings...... 43

Table 4.4 Diagonal terms of bearing stiffness matrices predicted by current model and Lim/Singh’s

model...... 44

Table 6.1 Gear Design Parameters and system parameters...... 81

Table 6.2 Geometry dimensions and material property of the propeller shaft...... 82

Table 7.1 Design parameters for the hyoid geared rotor system...... 109

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LIST OF FIGURES

Figure 2.1 Coordinate system definition for pitch cone-based mesh model [5]...... 11

Figure 2.2 Coordinate system definition for unloaded TCA-based mesh model [5]...... 13

Figure 2.3 Illustration of loaded tooth contact analysis results [15]...... 14

Figure 2.4 Schematic of mesh models: (a) Single point mesh model; (b) multi-point mesh model.

...... 16

Figure 2.5 Mesh stiffness for a sample gear set...... 16

Figure 2.6 (a) Cantilever beam; (b) simplification of pinion and gear...... 17

Figure 2.7 Dynamic mesh force (DMF) for (a) a face-hobbed gear and (b) a face-milled gear. 19

Figure 2.8 Multi-harmonic DMF for (a) a face-hobbed and (b) a face-milled gear...... 20

Figure 2.9 DMF for (a) a face-hobbed and (b) a face-milled gear...... 21

Figure 2.10 Multi-harmonic DMF for (a) a face-hobbed gear and (b) a face-milled gear...... 21

Figure 2.11 (a) Mesh stiffness (b) DMF for a light load case...... 22

Figure 2.12 (a) Mesh stiffness (b) DMF for a heavy load case...... 23

Figure 3.1 Simulation flowchart for axle whine ...... 28

Figure 3.2 Illustrations of gear tooth contact analysis...... 28

Figure 3.3 A 14-DOF lumped parameter model of hypoid geared rotor system ...... 30

Figure 3.4 The dynamic bearing force at actual bearing locations ...... 32

Figure 3.5 Axle housing model for forced vibration analysis...... 33

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Figure 3.6 Frequency response of housing surface accerleration (pinion nose) ...... 33

Figure 3.7 Radiated sound power level of the axle housing...... 34

Figure 3.8 Far field sound pressure level in a spherical field at 6 times the mesh frequency

(3000Hz)...... 35

Figure 4.1 Axle system model...... 38

Figure 4.2 Pinion head bearing model...... 39

Figure 4.3 Dimensional parameters of tapered roller bearing...... 41

Figure 4.4 Clearance between rolling elements and raceway...... 41

Figure 4.5 Time-varying bearing stiffness...... 45

Figure 4.6 Time-varying bearing stiffness...... 47

Figure 4.7 Von Mises stress distribution on the roller surface...... 48

Figure 4.8 Maximum Von Mises stress on the roller surface...... 49

Figure 4.9 Gear transmission error in one mesh cycle...... 50

Figure 4.10 Gear TE harmonics...... 50

Figure 5.1 Illustrations of: a) multi-point coupling of hypoid gear pair; b) contact cells on

engaging tooth surface...... 54

Figure 5.2 Axle system model...... 56

Figure 5.3 A lumped parameter model of hypoid geared rotor system...... 58

Figure 5.4 The TE harmonics...... 60

Figure 5.5 Modal frequencies and mode shapes of hypoid geared rotor system with simply

supported assumption...... 60

Figure 5.6 Modal frequencies and mode shapes of hypoid geared rotor system with flexible

bearings...... 61

Figure 5.7 Dynamic transmission error simply-supported bearing, flexible bearing . 62

Figure 5.8 Dynamic mesh force: simply-supported bearing, flexible bearing...... 62

Figure 5.9 Dynamic pinion bearing forces: simply supported (horizontal direction),

simply supported (axial direction), simply supported (vertical direction), flexible

bearings (horizontal direction), flexible bearings (axial direction), flexible

bearings (vertical direction)...... 63

Figure 5.10 Dynamic gear bearing forces: simply supported (horizontal direction), simply

supported (axial direction), simply supported (vertical direction), flexible

bearings (horizontal direction), flexible bearings (axial direction), flexible

bearings (vertical direction)...... 63

Figure 6.1 Illustrations of: a) multi-point coupling of hypoid gear pair; b) contact cells on

engaging tooth surface...... 69

Figure 6.2 Propeller shaft model a) beam with two different cross sections; b) cross section

dimensions...... 72

Figure 6.3 Lumped parameter model of hypoid geared rotor system...... 72

Figure 6.4 Illustrations of a) shaft-bearing lumped model; b) shaft-bearing layout...... 76

Figure 6.5 Schematic of propeller shaft and pinion coupling...... 77

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Figure 6.6 Axle housing structure a) finite element model b) boundary element model...... 80

Figure 6.7 Modal frequencies and mode shapes of the geared system...... 83

Figure 6.8 Modal strain energy distribution of the geared system: 1 – pinion translational

compliance; 2 – gear translational compliance; 3 – pinion torsional compliance; 4 – gear

torsional compliance; 5 – pinion bending compliance; 6 – gear bending compliance; 7 –

mesh compliance...... 83

Figure 6.9 Dynamic mesh responses: a) dynamic transmission error spectrum; b) dynamic mesh

force spectrum: baseline, only gear gyroscopic effect, only shaft

flexibility, with both gyroscopic effect and shaft flexibility...... 86

Figure 6.10 Dynamic bearing forces at pinion bearings: a) pinion bearing 1 in horizontal direction;

b) pinion bearing 1 in axial direction; c) pinion bearing 1 in vertical direction; d) pinion

bearing 2 in horizontal direction; e) pinion bearing 2 in axial direction; f) pinion bearing

2 in vertical direction. baseline, only gear gyroscopic effect, only

shaft flexibility, with both gyroscopic effect and shaft flexibility...... 87

Figure 6.11 Dynamic bearing forces at gear bearings: a) gear bearing 3 in horizontal direction;

b) gear bearing 3 in axial direction; c) gear bearing 3 in vertical direction; d) gear bearing

4 in horizontal direction; e) gear bearing 4 in axial direction; f) gear bearing 4 in vertical

direction. baseline, only gear gyroscopic effect, only shaft flexibility,

with both gyroscopic effect and shaft flexibility...... 88

Figure 6.12 Dynamic mesh responses: a) dynamic transmission error spectrum; b) dynamic mesh

force spectrum. baseline, half pinion mass moment of inertia, double

pinion mass moment of inertia...... 90

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Figure 6.13 Dynamic bearing forces: a) pinion bearing 1; b) pinion bearing 2; c) gear bearing 3;

d) gear bearing 4. baseline, half pinion mass moment of inertia,

double pinion mass moment of inertia...... 91

Figure 6.14 Dynamic mesh responses: a) dynamic transmission error spectrum; b) dynamic mesh

force spectrum. baseline, reduced shaft stiffness, increased shaft

stiffness...... 92

Figure 6.15 Dynamic bearing forces: a) pinion bearing 1; b) pinion bearing 2; c) gear bearing 3;

d) gear bearing 4. baseline, half pinion mass moment of inertia,

double pinion mass moment of inertia...... 93

Figure 6.16 Dynamic mesh responses: a) dynamic transmission error spectrum; b) dynamic mesh

force spectrum. baseline, reduced pinion bearing stiffness,

increased pinion bearing stiffness...... 94

Figure 6.17 Dynamic bearing forces: a) pinion bearing 1; b) pinion bearing 2; c) gear bearing 3;

d) gear bearing 4. baseline, reduced pinion bearing stiffness,

increased pinion bearing stiffness...... 95

Figure 6.18 Dynamic bearing forces: a) pinion bearing 1 for optimized case 1; b) pinion bearing

2 for optimized case 1; c) pinion bearing 1 for optimized case 2; d) pinion bearing 2 for

optimized case 2. baseline, optimized case...... 97

Figure 6.19 Predicted sound power level of gearbox radiated noise. baseline, case

1, case 2...... 98

Figure 6.20 Sound pressure distribution comparison a) baseline at 660Hz; b) case 1 at 660Hz; c)

baseline at 1020Hz; d)case 2 at 1020Hz...... 99

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Figure 7.1 Gearbox radiated noise prediction flowchart...... 103

Figure 7.2 Housing schematic and retained nodes for CMS reduction...... 105

Figure 7.3 Multi-body dynamic model of hypoid geared rotor system...... 107

Figure 7.4 Time-varying mesh parameters in one mesh cycle: (a) transmission error; (b) mesh

stiffness...... 110

Figure 7.5 Dynamic mesh responses: (a) dynamic transmission error; (b) dynamic mesh force

(solid line , rigid housing; dotted line , flexible housing)...... 110

Figure 7.6 Dynamic bearing force on pinion bearing: (a) in horizontal direction; (b) in axial

direction; (c) in vertical direction (solid line , rigid housing; dotted line ,

flexible housing)...... 112

Figure 7.7 Dynamic bearing force on gear bearing: (a) in horizontal direction; (b) in axial

direction; (c) in vertical direction (solid line , rigid housing; dotted line ,

flexible housing)...... 112

Figure 7.8 Predicted sound power level: (solid line , rigid housing; dotted line ,

flexible housing)...... 113

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LIST OF ABBREVIATIONS

BE boundary element

BEM boundary element method

CMS component mode synthesis

DBF dynamic bearing force

DMF dynamic mesh force

DOF degree-of-freedom

DTE dynamic transmission error

EHL elastohydrodynamic lubrication

FE finite element

FEM finite element method

FMM fast multi-pole method

FRF frequency response function

LH left-hand

LTCA loaded tooth contact analysis

NVH noise, vibration and harshness

RH right-hand

TCA tooth contact analysis

TE transmission error

4WD four-wheel-drive

LIST OF SYMBOLS

bc gear backlash

[C] system damping matrix cm mesh damping coefficient eL translational loaded transmission error fi normal contact force magnitude at contact cell i

Ftotal, Fn static total normal contact force, i.e. static normal mesh force

Fm dynamic normal mesh force h directional cosine and rotation radius vector

I moment of inertia km mesh stiffness

L line-of-action directional cosine vector

M total moment due to contact

M, [M] mass and inertia matrix ni (nix, niy, niz) surface normal vector at contact cell i n directional cosine of line-of-action

{q} general coordinate vector

R effective mesh point position vector ri (rix, riy, riz) position vector at contact cell i

T external torque t time x, y, z translational coordinates

ε0 translational unloaded transmission error

xxi

δd dynamic transmission error

θ angular coordinate

λ directional rotation radius

Subscripts

E, D Engine/driver g gear i contact cell i

L Load/driven component l label for pinion (l = p) and gear (l = g) p pinion x, y, z x, y and z coordinate directions

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Chapter 1 Introduction

1.1 Motivation

Hypoid and spiral bevel gear drives are widely used in automotive and aerospace applications because of their ability to transfer power through non-intersecting and non-parallel shafts. However, vibration induced by the gear mesh often generates annoying noise, especially gear tonal whines at the gear mesh frequencies. The vibration induced by the gear mesh also affects the mechanical gear mesh efficiency, and contributes to the fatigue failure of driveline components, particularly the gears and bearings. Therefore, in order to design a more silent, efficient and durable driveline, a systematic study on the dynamics of the hypoid/bevel geared rotor system is needed.

Compared to parallel axis gears, such as spur or helical gears, the studies on right-angle geared rotor system dynamics are less intensive, and this is partially due to the more complicated gear geometry and time and spatial-varying gear mesh characteristics.

The demand for quite driveline system is driven by regulations and customers’ expectation.

The vibration of axle system not only radiates noise to the surrounding environment through axle casing structure, but also affects vehicle interior sound quality. It is a source of intense annoyance even when it is not the loudest noise component, because the way it is perceived. Gear noise is a collection of pure tones which the human ear can detect even when they are 10dB lower than the overall noise level.

Since axle gear whine is a system-level NVH problem, it is necessary to evaluate the system responses and vibration propagation through the gear-shaft-bearing structure. Most of the previous studies neglect the contribution of component flexibility and system dynamics in the analysis of gear noise radiation. However, some experimental works have shown more complicated dynamic

1 behaviors as opposed to the prediction from simplified gear dynamic model and considerable discrepancies between experimental and simulation results.

The goal of this study is to provide an efficient system approach to analyze the vibration and sound radiation of right-angle gear transmission system with an emphasis on the application of vehicle axle system. The system responses, such as bearing reaction forces, housing surface responses and radiated noise are analyzed and crucial system design parameters are investigated.

1.2 Literature Review

Vibration and noise are always among major concerns for gear applications. There are many published studies on the parallel axis geared rotor system dynamics, but the dynamics of right-angle geared rotor system is studied less extensively due to its complicated tooth geometry and mesh mechanism. Right-angle geared rotor system is very common in automotive rear axle application and has received increasing attention in the past few decades. Although many experimental work and numerical simulations have been carried out in the past to investigate the physics of axle whine noise generation [1-4], it is still not fully understood how component flexibility and system dynamics could affect radiated noise from the driveline system. In order to predict the noise radiation more precisely and use the approach to tackle axle whine noise problem during the design process, an accurate and realistic driveline system model is needed.

There are few publications talking about the experimental correlations for hypoid/bevel geared rotor dynamics, because of the lack of analytical capability and the difficulties in experimental measurement. Remmers [1] used a lumped mass-spring model for a rear axle driveline and predicted the pinion resonance spectrum. Hirasaka et al. [3] conducted an experiment study to estimate dynamic mesh force, and they found that the dynamic mesh force is very sensitive

2 to the torsional vibration characteristics of the driveline system and the stiffness of propeller shaft.

Chen [5] measured the sound pressure and vibration response under drive and coast conditions for a hypoid gear set used in a truck application. In their simulation results and test results, reasonably good correlation between dynamic mesh force and overall noise spectrum is observed. The analysis also shows that the vertical motion of the pinion bearing location matches well with the radiated noise level. Peng [6] presented a dynamic analysis on a spiral bevel gear set for off- highway vehicle application and compare the simulated dynamic mesh force and housing response with test results. The results also show reasonably good agreement and thus validate the analytical model.

There are generally two approaches to control the vibration and noise issues of the hypoid and bevel geared rotor systems. The first approach is to reduce and modify the shape of the kinematic transmission error, since the kinematic transmission error is known as major excitation.

In the case of hypoid or bevel gears, various tooth surface generation models have been proposed to represent the gear tooth surfaces mathematically from the machine setting and cutter specifications. The tooth contact analysis is then applied along with geometry generation to predict load distribution and loaded transmission error. The optimization of transmission error and contact pattern can also be found in the references.

The other approach to control the gear noise and vibration problems is to tune the sensitivity of the vibration response to the transmission error and study the system dynamics of geared rotor system. Most early studies on hypoid and bevel gear dynamics are limited to experimental or simple analytical models of the gear mesh. The dynamic characteristics of the meshing hypoid gear pair due to gear transmission error, namely the dynamic transmission error and mesh force have been studied by many researchers – some are noted below. Chen and Lim [7,

8] proposed a generalized 3-dimensional dynamic model based on idealized gear geometry and quasi-static loaded tooth contact analysis. Wang and Lim {Wang, 2007 #68} studied the interaction between time-varying mesh characteristic and backlash nonlinearity, and observed some interesting nonlinear behaviors such as jump phenomenon, sub-harmonic and chaotic responses. They have also found that mean load and mesh damping are two key factors for tooth impact condition. Besides the dynamic mesh forces, friction of meshing teeth pair also plays an important role in gear dynamics and NVH (noise, vibration and harshness) responses as an energy sink. Peng [6] proposed a sliding friction model for hypoid and spiral bevel gears based on a quasi- static loaded tooth contact analysis and the synthesis of the effective lumped friction coefficients.

The sliding friction was found to have significant influence on the magnitude of dynamic response, especially for spiral bevel gears under medium and heavy load due to the inherent friction direction reversal behavior. Feng, et.al. [9] enhanced the friction model by applying the mixed elastohydrodynamic lubrication (EHL) formulation. The friction also affects the transmission efficiency, which is another major concern for vehicle differential hypoid gears. In practice, the differential gearbox efficiency and NVH refinement possess contradictory desired attributes [10].

Mohammadpour, et.al. [10, 11] proposed a combined multi-body dynamics and lubricated contact mechanics model of vehicular differential hypoid gear pairs to obtain a balanced approach for transmission efficiency and NVH performance. In their study, realistic torque load was applied considering the road data, aerodynamic effects and engine torque variation. The results showed that gear tooth pair separation and friction loss have significant influence on axle whine generation, which is also confirmed previously by Koronias, et.al. [4].

The dynamic mesh force can be transmitted through shaft-bearing structure to cause the vibration of the whole structure that radiates noise. Even with minimized transmission error and

4 proper lubrication, the system can still be sensitive to the excitation force. Moreover, minimizing the transmission error normally requires more involved efforts and cost. Hence, in some cases, it is more reasonable to tune the system dynamics for noise reduction purpose. Many researchers investigated the effect of system dynamics on gear vibration. For example, Hua, et.al. [12] proposed a shaft-bearing model for computing the effective support stiffness of spiral bevel geared rotor system and compared the dynamic responses of different shaft-bearing configurations. Yang, et.al. [13, 14] studied the effect of time-varying bearing stiffness and housing flexibility on the hypoid geared rotor system responses. Wang, et.al. [15] studied the effect of torque load on the mesh and dynamic characteristics of hypoid geared system. Mohammadpour, et.al. [11] investigated the effect of system damping and bearing stiffness on the dynamic transmission error and found that lower system damping and higher bearing rigidity can deteriorate the tooth separation condition and worsen NVH performance.

From the previous studies, one can see that hypoid/bevel gear pair dynamics is studied extensively but less attention is given to the sensitivity of structural responses to different system vibration modes, even though the experimental work indicates there is a significant influence. In addition, no direct link was shown to reveal the relationship between the gear whine noise and system design parameters, which are addressed in this study.

1.3 Scope and Objectives

The focus of this dissertation is to develop a more realistic and effective system model for vibration and sound radiation analysis of right-angle geared drive. An integrated approach is proposed, which utilizes the 3-dimensional loaded tooth contact analysis, gear lumped parameter

5 geared rotor model, finite element and boundary element housing model. The following tasks are expected to be accomplished:

a) Propose an integrated approach to effectively analyze and troubleshoot the NVH

performance of right-angle geared drive and assist driveline system design engineers;

b) Develop a geared rotor system dynamic model incorporating the contribution of system

dynamic effects in the rear axle system, such as propeller shaft, tapered roller bearings,

housing, etc.

c) Study the effect of component flexibility on the modal properties and dynamic

responses of the geared rotor system.

d) Study the sensitivity of the system responses to the design parameters and operation

conditions.

e) Simulate the NVH performance enhancement of the driveline system due to design

changes, structural modifications.

1.4 Organization

Chapter 1 presents the introduction, literature review, scope and objectives for the thesis research. This chapter discusses previous work in gear dynamic research, with an emphasis on hypoid and bevel gear dynamics. Then the objectives of this thesis are further given, which is to propose a more realistic and accurate axle system model and study the effect of system dynamics on gear mesh and structural responses.

Chapter 2 discusses a study comparing the effects of dissimilar tooth meshing formulations is performed. The results provide a guideline for applying the most suitable mesh representations for specific applications.

Chapter 3 presents an integrated approach for analyzing the vibration and noise radiation of vehicle axle system efficiently. Load tooth contact analysis (LTCA) is performed and multi- body non-linear dynamic model and time-varying bearing forces at actual bearing locations are calculated and used to excite the finite element (FE) model of an axle housing. The surface velocities from the housing forced vibration analysis are taken into the boundary element (BE) model and sound pressure levels around the axle housing and sound power level are calculated.

Chapter 4 presents the rear axle system modeling with an emphasis on bearing dynamics.

The tapered roller bearing stiffness is calculated based on the contact mechanics/finite element model and numerical method. The proposed method is then compared against the traditional analytical method. A series of parametric studies are also performed to investigate the effects of bearing preloads on the transmission error, bearing stiffness, and bearing contact pattern. The proposed approach demonstrates the capability to calculate fully populated bearing stiffness matrix and time-dependent contact characteristics between rollers and raceways, and thus can be employed in the axle system design and further dynamic analysis.

Chapter 5 presents the comparison analysis of hypoid geared rotor system with different assumption on the component flexibility, aiming at investigating the effect of the elasticity of the shafts, bearings and housing. The load distribution results and gear transmission errors are calculated and compared between the hypoid gear pair model and axle system model with shaft- bearing-housing flexibility. The modal characteristics and dynamic responses by assuming simply- support bearing and flexible bearing have also been compared.

Chapter 6 presents an analysis for driveline system composing a coupled propeller shaft model and a lumped model of hypoid geared rotor system using the component mode synthesis approach. In the proposed formulation, the gyroscopic effect of both the gear and propeller shaft

7 is considered. The influence of pinion bending moment of inertia, propeller shaft stiffness and bearing stiffness on the system dynamic responses are investigated, and the dynamic parameters are tuned to reduce the radiated noise due to the torsional and bending modes of the coupled driveline system.

Chapter 7 presents a system approach for vibro-acoustic analysis of right-angle gearbox which consists of four stages: tooth contact analysis, gear mesh and dynamic analysis, housing forced vibration analysis and vibro-acoustic analysis. The effect of housing flexibility on the vibratory and acoustic responses of the gearbox is incorporated by using the component mode synthesis (CMS) method.

Chapter 8 gives the conclusions produced from this dissertation and some recommendation for future work.

Chapter 2 Comparative Analysis of the Hypoid Geared Rotor System Dynamics Applying Dissimilar Tooth Meshing Formulations

2.1 Introduction

The fully dynamic simulation of the hypoid geared rotor system dynamics is almost

computationally prohibited due to the complicated tooth geometry. To save the computational

costs, the gear mesh model has been used to represent the dynamic coupling between engaging

gears [7, 16-18], and the gears are taken as rigid bodies [5, 7, 15-19]. The gear mesh model is

comprised of several mesh parameters including the mesh point, line of action, mesh stiffness, and

transmission error (TE). These mesh parameters can be obtained through simply using the gear

design parameters (Pitch Cone-based mesh model) [5], or extracting the unloaded/loaded tooth

contact (TCA/LTCA-based mesh model) analysis results [5, 7, 20-22].

The accuracy of the mesh parameters is critical to get a precise prediction of the dynamic

responses. Any variations in the mesh stiffness or directional rotation radii, which are calculated

from the mesh point and line of action, could result in changes of both the amplitude and peak

frequency of the dynamics responses [19]. The kinematic TE, the excitation to the geared rotor

systems, was also reported to play a significant role in the dynamic responses [23]. Besides, it was

found that the input torque had a great effect on the mesh parameters and the dynamic responses

too [15, 23]. In this study, the dynamic responses of a same hypoid geared rotor system predicted

using the Pitch Cone, TCA and LTCA based gear mesh models for both the light and heavy

operating load cases are compared. Based on the comparison, guidance about how to choose a

suitable mesh model for different applications is given.

2.2 Gear Mesh Models

2.2.1 Pitch Cone-based mesh model

The basic assumption of the pitch cone-based mesh model is that the pitch point and its normal vector are effective estimations of the mean mesh point and the line of action of engaging hypoid gear pairs. The formulation of calculating the pitch point and its normal vector using the gear blank design parameters was derived by Chen [5], with the help of the Gleason’s “trial and error” technique [24] and the formulation published by Chaing [25]. This mesh model is very computational efficient and can be used even before a detailed gear design is available, due to the fact that it uses only the gear blank design parameters. For the same reason, by using this mesh model, the time-varying and load-dependent characteristics of the gear mesh process cannot be captured, and the kinematic TE and mesh stiffness cannot be obtained directly either.

The formulations for calculating the mean mesh point and the line of action, in the coordinate system Xf − Yf − Zf shown in Figure 2.1, are given in this study without derivation. The coordinate system Xf − Yf − Zf is fixed to the reference frame, which supports the geared rotor system. The Zf axis is in the direction that has the shortest distance between the gear rotational axes, and the Yf axis coincident with the pinion rotational axis. The position vector 푹of the mesh point can be expressed as:

푹 = 푥풊 + 푦풋 + 푧풌 (2.1) where

2 2 푥 = 푅퐺푡푎푛훾퐺 − 퐸푠푖푛훾퐺/√푐표푠 훾푃 − 푠푖푛 훾퐺 (2.2a)

푦 = 푅퐺푠푖푛훾푃/푐표푠훾퐺 (2.2b)

2 2 푧 = 퐸 − 푅퐺√푐표푠 훾푃 − 푠푖푛 훾퐺/푐표푠훾퐺 (2.2c)

In the above equations, 퐸 is the pinion and gear the offset, 푅퐺 represents the gear pitch cone radius, and 푝and 퐺 are the pitch angles of the pinion and gear respectively.

Figure 2.1 Coordinate system definition for pitch cone-based mesh model [5].

The normal vector can be expresses as:

풏 = 푛푥풊 + 푛푦풋 + 푛푧풌, (2.3) where

푛푥 = 푐표푠휃푃푐표푠훾푃푠푖푛휙 − 휅1푠푖푛휃푃푐표푠훽푃푐표푠휙 + 휅2푠푖푛훾푃푐표푠휃푃푠푖푛훽푃푐표푠휙 (2.4a)

푛푦 = −푠푖푛훾푃푠푖푛휙 + 휅2푐표푠훾푃푠푖푛훽푃푐표푠휙 (2.4b)

푛푧 = 푠푖푛휃푃푐표푠훾푃푠푖푛휙 + 휅1푐표푠휃푝푐표푠훽푃푐표푠휙 + 휅2푠푖푛휃푃푠푖푛훾푃푠푖푛훽푃푐표푠휙. (2.4c)

In the above equations, the symbols 훽푃and 훽퐺represent the pinion and gear spiral angel

−1 respectively, the parameter 휃푃 is defined as 푐표푠 (−푠푖푛훾퐺/푐표푠훾푃), and 휙represents the pressure angle. The coefficient 1 and 2 are used to distinguish specific geometrical configuration and operating condition of the hypoid gear pair. The pinion could have either right-hand (RH) or left- hand (LH) spiral teeth, and the hypoid gear set could be working under either forward or reverse conditions. For different cases, the coefficients and pressure angles are given in Table 2.1, where

11 the symbols 푐 and 푣 represent the pressure angles of the concave and convex sides of the pinion respectively.

Table 2.1 Coefficients and pressure angles.

LH pinion with RH gear

Forward 휅1 = 1, 휅2 = 1, 휙 = 휙푐

Reverse 휅1 = 1, 휅2 = −1, 휙 = 휙푣

RH pinion with LH gear

Forward 휅1 = 1, 휅2 = −1, 휙 = 휙푣

Reverse 휅1 = −1, 휅2 = 1, 휙 = 휙푐

2.2.2 Unloaded tooth contact analysis based mesh model

Hypoid gear tooth surface is the basis of the unloaded TCA-based mesh model. The hypoid gear tooth surfaces can be generated by simulating the dynamics of the gear manufacture process

(please refers to the book by Litvin [26] for more information). The calculated pinion and gear tooth surfaces are usually put into two different rotating coordinate systems determined by the relative position of the pinion and gear in the gear set assembly. Several coordinate systems are given in Fig.2 for illustration purpose, where the 푆푤1(푥푤1, 푦푤1, 푧푤1) and 푆푤2(푥푤2, 푦푤2, 푧푤2) are the pinion and gear rotational reference systems. The 푧푤1 and 푧푤2 axes are defined to be the pinion and gear rotational axles, 퐸 represents the offset of the pinion and gear rotational axles, and

′ ′ 휑1 and 휑2 represent the roll angles of pinion and gear respectively. In order to apply the surface continuous tangency condition, the rotating pinion and gear tooth surfaces are transferred into the same fixed reference Sf(xf, yf, zf) shown in Figure 2.2, where the symbols ΔH and ΔQ are the axial shifts of the pinion and gear in assembly.

Figure 2.2 Coordinate system definition for unloaded TCA-based mesh model [5].

′ The surface continuous tangency condition is applied for each given pinion roll angle 휑1, to compute the contact point and its normal vector. Therefore, the resulted mesh point and line of

′ action are functions of the pinion roll angle. The gear roll angle 휑2can be also calculated for each

′ given pinion roll angle휑1, and the angular transmission error can be calculated as:

′ ′ ′ ′∗ 푁1 ′ ′∗ ∆φ2(휑1) = (휑2 − 휑1 ) − ⁄ (휑1 − 휑2 ), (2.5) 푁2

′∗ ′∗ where 휑1 and 휑2 are the initial roll angles for the pinion and gear at a mean point, and

푁1and푁2 represent the number of teeth of pinion and gear respectively. Similar as the pitch cone- based mesh model the mesh stiffness cannot be obtained directly either.

2.2.3 Loaded tooth contact analysis based mesh model

After getting the tooth surface, the loaded tooth contact analysis can be performed by using various approaches, such as finite element method [20], the ease-off topology and the shell theory

[21], and a combined surface integral and finite element method [22]. The mesh parameters can then be condensed from the results of a loaded tooth contact analysis [14]. A simplified illustration of the loaded tooth contact analysis results is shown in Figure 2.3, where the grids on the gear tooth surface represent contacting cells.

n fi

Figure 2.3 Illustration of loaded tooth contact analysis results [15].

For contact cell 푖 in the mesh coordinate system 푆(푥, 푦, 푧)shown in Figure 2.3, the position vector is given as 푟푖(푟푖푥, 푟푖푦, 푟푖푧), the normal vector is represented by 푛푖(푛푖푥, 푛푖푦, 푛푖푧), and the contact force is represented by 푓푖. A vector summation process is performed to obtain the total mesh force 퐹(퐹푥, 퐹푦, 퐹푧) given by

푁 푧 2 퐹푗 = ∑푖=1 푛푖푗 푓푖, 퐹 = √∑푗=푥 퐹푗 , (푗 = 푥, 푦, 푧), (2.6)

and then the line-of-action 푁(푛푥, 푛푦, 푛푧) can be calculated as:

푛푗 = 퐹푗/퐹, (푗 = 푥, 푦, 푧). (2.7)

The mesh point 푅(푥, 푦, 푧)is calculated by applying a method that is commonly used to compute the mass center of gravity of a group of rigid bodies. It can be obtained by solving the following algebraic equations

푥 = (푀푧 + 퐹푥푦)/퐹푦, (2.8a)

푦 = (푀푥 + 퐹푥푧)/퐹푧, (2.8b)

푥 = (푀푦 + 퐹푧푥)/퐹푥, (2.8c)

14 where 푀푗, (푗 = 푥, 푦, 푧) is the moment about the axis 푗 caused by the combined loads at all the contact cells. The expression for the moment can be explicitly given by

푁 {푀푥, 푀푦, 푀푧} = ∑푖=1 푓푖 ∙ (푟푖×푛푖). (2.9)

The loaded angular transmission error can be calculated in the similar way as the unloaded angular transmission error was calculated using Equation (2.5). The loaded and unloaded translational transmission errors 푒 and 푒0 are the projections of respective angular transmission error along the line of action of the net mesh force. Finally, the mesh stiffness is defined by

푘푚 = 퐹/(푒 − 푒0). (2.10)

Applying the above condense procedure to the whole engaging gear pair leads to the single point mesh model [14] shown in Figure 2.4(a), where a single set of mesh parameters is established.

The condense process can also be applied to each contacting gear tooth pair, which leads to the multi-point mesh model [18] shown in Figure 2.4(b). For the second mesh model, a set of mesh parameters is needed to be established for each contacting gear tooth pair. If applying the single point mesh model to the dynamic simulation, a net dynamic mesh force (DMF) for the analyzed gear pair will be obtained, which is enough as a concern of the vibration and noise issues. If one is also interested in getting the dynamic mesh forces for each contacting gear tooth pair, the multi- point mesh model will be a reasonable choice.

(a) (b) a)

Figure 2.4 Schematic of mesh models: (a) Single point mesh model; (b) multi-point mesh model.

2.2.4 Mesh stiffness calculation in pitch cone and TCA based mesh models

As mentioned previously, the mesh stiffness cannot be calculated directly for the pitch cone-based and unloaded TCA-based mesh models. However, the mesh stiffness is one of the critical mesh parameters, which affects the dynamics of the hypoid geared rotor systems to a great extent. In this section, an indirect approach, using both the mesh stiffness condensed from the loaded tooth contact analysis results (using Equation (2.10)) and the beam theory, is introduced.

The mesh stiffness 푘푠1 as a function of the input torque for a sample hypoid gear set is given in

Figure 2.5, which was pre-calculated from the loaded tooth contact analysis-based mesh model, and it is taken as a known for the estimation of the mesh stiffness of another gear pair.

10 10

′ (푇1, 퐾푚1)

10 Ks1 (N/m)

8 10 0 0.5 1 1.5 2 2.5 4 Torque (N-m) x 10

Figure 2.5 Mesh stiffness for a sample gear set.

A cantilever beam, whose cross-section has the width of 푐 and the height of 푑, is shown in

Figure 2.6(a). Assuming a point force 퐹 is applied at a distance of 푎 from the fixed end, and the

3 ( ) 4퐹푎 ⁄ transverse deformation will be 푦 푎 = 퐸푐푑3, where the 퐸 represents the Young’s modulus.

Therefore, the stiffness at a distance of 푎 from the fixed end is:

퐹 3 푘 = = 퐸푐푑 ⁄ . ( 2.11) 푦(푎) 4푎3

(a) (b)

Figure 2.6 (a) Cantilever beam; (b) simplification of pinion and gear.

The pinion and gear can be simplified as two contact cantilever beams as shown in Figure

2.6(b). Similarly, a rough estimation of the mesh stiffness can be obtained as:

3 3 푐푝푑푝 푐푔푑푔 3 ∙ 3 푘푝푘푔 퐸 푎푝 푎푔 푘푚 = = ( 3 3 ), (2.12) 푘푝+푘푔 4 푐푝푑푝 푐푔푑푔 3 + 3 푎푝 푎푔

where the subscript 푝 and 푔 represent the pinion and gear respectively. From Equation

(2.12), it can be seen that the gear mesh stiffness is proportional to the quantity in the brackets.

Therefore, for a given gear pair, the estimation of the mesh stiffness can be obtained as:

퐵푠2 푘푠2 = 푘푠1 , (2.13) 퐵푠1

17 where 푘 and 퐵 are the mesh stiffness at any input torque and the quantities included in the brackets of Equation (2.12). The subscripts 푠1 and 푠2 represent the parameters are for the gear Set-1 (with known mesh stiffness as a function of input torque) and the gear Set-2 (whose mesh stiffness is needed to be calculated) respectively. To take into account the input load effect, Equation (2.13) can be further improved as:

퐾푚1 퐵푠2 푘푠2 = ′ 푘푠1 . (2.14) 퐾푚1 퐵푠1

′ In the above equation, 퐾푚1is the pre-calculated mesh stiffness at an input torque of 푇1 for the gear Set-1 shown in Figure 2.5. The 퐾푚1represents the mesh stiffness at the same torque 푇1 for the gear Set-2, and it should be known for applying Equation (2.14), which means that at least one mesh stiffness for the gear Set-2 need to be known in order to include the input torque effect on the mesh stiffness.

2.3 Hypoid Geared Rotor System Dynamics

The linear time-varying 14 degrees of freedom dynamic models [14] of two practical hypoid geared rotor systems (face-hobbed and face-milled gears) were analyzed using a research code called HGSim [27]. The predicted dynamic mesh forces using the different mesh models were analyzed and compared for both the face-hobbed and face-milled gears. The unloaded TCA- based mesh model is not available yet for the face-hobbed gears in HGSim [27], and it is not included in this study. The TE calculated from the LTCA mesh model is also used for all the cases adopting the Pitch-Cone based mesh model, and this is because the TE cannot be obtained from the gear blank design parameters.

2.3.1 Light Load Case

The dynamic mesh forces of the face-hobbed gears under a light load condition are compared in Figure 2.7(a). Small discrepancy between the curves, predicted using the pitch cone based and the loaded TCA based mesh models (indicated by Calyx), can be observed, especially at the high frequencies. For the face-milled case, the comparison of the predicted dynamic mesh forces using different mesh models are given in Figure 2.7(b), where small discrepancies at high frequencies can also be observed. However, for the face-milled case, the dynamic mesh forces predicted using the pitch cone based and unloaded TCA based mesh models are almost the same.

This is mainly because the same mesh stiffness calculation approach, explained in the Section 2.4, was applied for these two mesh models. While, a different mesh stiffness calculation procedure was adopted for the loaded TCA mesh model, which is believed to be the most accurate.

2 10

10 Magnitude Magnitude (N)

Magnitude Magnitude (N) TCA PitchCone Calyx Calyx 0 PitchCone 10 1000 2000 3000 4000 1000 2000 3000 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Figure 2.7 Dynamic mesh force (DMF) for (a) a face-hobbed gear and (b) a face-milled gear.

For the cases using the loaded TCA mesh model, dynamic mesh forces due to different gear mesh harmonic excitations can also be obtained. The dynamic mesh forces due to different harmonics of the TE for the face-hobbed case are given in Figure 2.8(a), and the ones for the face- milled case are given in Figure 2.8(b). For both the two cases, the dynamic mesh forces due to the first harmonic of the TE dominate the total dynamic mesh forces at most of the frequency points.

4 10 2 10

2 10

0 10 0 Mesh Order Num1 Mesh Order Num1 10 Mesh Order Num2

Mesh Force (N) Mesh Force Mesh Order Num2 Mesh Force (N) Mesh Force Mesh Order Num3 Mesh Order Num3 Total Response Total Response -2 -2 10 10 1000 2000 3000 4000 1000 2000 3000 4000 Pinion Shaft Speed (RPM) Pinion Shaft Speed (RPM) (a) (b)

Figure 2.8 Multi-harmonic DMF for (a) a face-hobbed and (b) a face-milled gear.

2.3.2 Heavy Load Case

For the face-hobbed gears with a heavy input torque, the comparison of the predicted dynamic mesh forces using different mesh models is given in Figure 2.9(a). The dynamic mesh forces of the face-milled case are compared in Figure 2.9(b). For both cases, there is nearly no noticeable discrepancy in the predicted dynamic mesh forces. This is because that the mesh stiffness condensed from the loaded TCA results and the one calculated using Equation (2.14), has very small discrepancy for large input torque cases. Actually, in Equation (2.14), linear proportional relation between mesh stiffness and different loads is assumed. It is easily to observe from Figure 2.5 that, the mesh stiffness is almost linear to the input torque in the large input torque range. While, in the small input torque range, the mesh stiffness increases rapidly and nonlinearly as the input torque increases.

3 4 10 10

2 10 2 10

1 TCA Magnitude Magnitude (N) 10 Magnitude (N) PitchCone Calyx Calyx 0 10 PitchCone 1000 2000 3000 4000 1000 2000 3000 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Figure 2.9 DMF for (a) a face-hobbed and (b) a face-milled gear.

Similarly, the dynamic mesh forces due to different harmonics of the TE for the face- hobbed case are given in Figure 2.10(a), and the ones for the face-milled case are given in Figure

2.10(b). For both the two cases, the dynamic mesh forces due to the first harmonic of the TE dominate the total dynamic mesh forces at most of the frequency points.

4 10

2 10 2 10

0 10 0

Mesh Order Num1 10 Mesh Order Num1 Mesh Force (N) Mesh Force Mesh Order Num2 (N) Mesh Force Mesh Order Num2 Mesh Order Num3 Mesh Order Num3 -2 Total Response Total Response 10 -2 10 1000 2000 3000 4000 (b)1000 2000 3000 4000 Pinion Shaft Speed (RPM) Pinion Shaft Speed (RPM) (a) (b)

Figure 2.10 Multi-harmonic DMF for (a) a face-hobbed gear and (b) a face-milled gear.

2.3.3 Multi-point Mesh Case

The multi-point mesh model for an example gear set was condensed from the loaded TCA analysis results for both the light load and heavy load cases. For the light load case, the maximum number of gear tooth pairs in contact is two, which is indicated by the two curves (the pinion roll

21 angle dependent mesh stiffness) shown in Figure 2.11(a). The dynamic mesh forces for the example gear set under light load condition are given in Figure 2.11(b), where two dynamic mesh forces are obtained as expected for different gear tooth pairs in contact. It is easily observed that the dynamic mesh forces for the two gear tooth pairs in contact are about the same, this is mainly because their mesh stiffness curves are almost symmetric about their intersection point.

4 3 x 10 10 15 Mean mesh force = 364.39 N

2 10 10

Mesh Point 1 1

5 Mesh Point 2 10 Km N/mm( ) Mesh Force (N) Mesh Force Mesh point 1 Mesh point 2 0 0 10 0 5 10 15 20 25 0 1000 2000 3000 4000 Pinion roll angle (degree) Frequency (Hz) (a) (b)

Figure 2.11 (a) Mesh stiffness (b) DMF for a light load case.

For the heavy load case, the multi-point mesh stiffness is given in Figure 2.12(a), and the maximum number of gear tooth pairs in contact is four. The mesh stiffness of the mesh point 1 and

4 are almost symmetric about their intersection point, and the same symmetric feature is observed for the mesh stiffness of the mesh point 2 and 3 expect for their relative high magnitude. The dynamic mesh forces of all the engaging gear tooth pairs are shown in Figure 2.12(b). As expected, the dynamic mesh forces of the tooth pairs, of which the magnitudes of the mesh stiffness are about the same, are very close, and larger dynamic mesh forces can be seen for the tooth pair with large mesh stiffness.

5 x 10 5 3 10 Mean mesh force = 18190.41 N 4 10 2 Mesh point 1 Mesh point 2 3 10 Mesh point 3 Mesh Point 1 Mesh point 4 Mesh Point 2 1 Mesh Point 3

Km N/mm( ) 2

Mesh Point 4 (N) Mesh Force 10

1 0 10 0 5 10 15 20 25 0 1000 2000 3000 4000 Pinion roll angle (degree) Frequency (Hz) (a) (b)

Figure 2.12 (a) Mesh stiffness (b) DMF for a heavy load case.

2.4 Conclusion

The pitch cone mesh option is the simplest form of a single-point coupling gear mesh model, which works for both face-milled and face-hobbed gears. This mesh option only requires the gear blank design parameters, and it can be used before the gear tooth surface is designed. If the user wants to obtain an approximate gear mesh model quickly, this option is a good choice. It is also most suitable for large scale geared rotor system model, for which the accurate transmission error and mesh stiffness are not required. The unloaded TCA mesh option is a single-point coupling mesh model based on idealized tooth contact analysis, where the gear tooth surface geometry must be known. It is slightly complicated than pitch cone mesh model, and is only works for face-milled gears for now. The loaded TCA-based mesh option is a single/multi-point coupling mesh model based on quasi-static loaded tooth contact analysis results, which works for both face-milled and face-hobbed gears. Both the gear tooth surface geometry and the elasticity of the material are taken into account when generating the mesh model. This mesh option is the most complicated, and it is the most exact too. The obtained pinion angle dependent mesh parameters make the dynamic simulation using the multi-harmonic TE excitations possible. Using this mesh option, the dynamic mesh forces of each gear tooth pair in contact can also be obtained, in addition to the net dynamic

23 mesh force for the engaging gear pair. For an accurate prediction of the hypoid geared rotor system dynamics, the loaded TCA mesh option is strongly recommended.

Chapter 3 Vibration and Sound Radiation Analysis of Vehicle Axle Systems Using an Integrated Approach

3.1 Introduction

The vehicle interior sound quality has long been one of the key concerns for automotive industry [1, 2, 4, 28]. The demand for reduced vehicle interior noise is driven by regulations and consumer expectations. The axle gear whine noise is one of the major noise sources in a vehicle cabin, especially for four-wheel-drive (4WD) vehicles. The excitation of the axle gear whine noise is believed to be the gear transmission error (TE) by many researchers. The dynamic mesh force, which is generated by the transmission error and time-varying mesh stiffness, is propagated through the gear-shaft-bearing structures, and finally radiate noise from housing surface. The control of axle gear whine noise requires a full system level modelling and analysis. Thus, the main goal of this study is to provide an end-to-end solution to understand the noise and vibration characteristics of the vehicle axle system and to assist in trouble-shooting and improve axle whine noise.

A number of investigations have been made in the past to find out the cause of axle whine and identify the relationship between axle whine with the excitation force of the meshing gear pair.

Yoon, et.al. [29] measured transmission error of a hypoid gear pair and found that the hypoid gear whine noise can be minimized when the input torque is in the target range of the torque load for optimal transmission error. Koronias, et.al. [4] investigated on axle whine vibration and noise by a combined experimental and numerical approach. They compared the vibration modes of the drivetrain with experimental results and established a causal relationship between axle whine and the flexural mode response of system components. Most of the research papers on the practical

25 work of axle whine reduction adopted tooth profile modification and structure modification as the means to achieve sound quality [30-32].

Minimizing the TE excitation is always an effective way to reduce the overall NVH performance of the axle system. Many researchers have proposed different methods to reduce gear noise by optimizing gear microgeometry and tooth profile modification [33, 34]. However, the axle whine problem can still remain if the system is too sensitive to the excitation at gear mesh.

Thus, the vibration characteristics of the main component on the structure transfer path, such as shafts, bearings, and housing, should be considered in the system model for maximum NVH improvements.

In this chapter, an integrated approach is presented to analyze the axle noise generation.

The analysis includes detailed contact modeling between gear pairs, the multi-body dynamic modeling of the hypoid geared rotor system and the structural and acoustic responses of the axle housing. The simulation flowchart is shown in Fig. 1. The rest of this chapter is structured as follows: Section 2 presents gear mesh and dynamic model, and the dynamic bearing forces are shown. Section 3 presents the vibro-acoustic analysis of the axle housing, and the housing surface response, sound pressure levels and sound power level are shown. Finally, a conclusion is given in Section 4.

3.2 Gear Mesh and Dynamic Model

3.2.1 Gear Mesh Model

This section introduces the simulation flowchart. Figure 3.1 shows the inputs and the simulation steps to predict the noise radiation of the gearbox. The modeling details about reduced housing model and simulation steps will be discussed later in the following section.

The gear tooth contact analysis (TCA) is performed by a 3D quasi-static loaded tooth contact analysis program, in which the semi-analytical theory combined with finite element method is used to solve for the contact forces. The design parameters of the hypoid geared rotor system is shown in Table 3.1, and the illustration of a hypoid gear pair in contact is shown in Fig.

2. Because the load distribution results cannot be directly used in the dynamic model, it is necessary to synthesize the mesh parameter in one mesh cycle from the load result of each contact cell.

Bearing Bearing Operation Gear Dynamic

Configuration Geometry Condition Geometry Parameters

Loaded Tooth Contact Analysis,

Shaft-bearing Assembly Stiffness Nonlinear 14-DOF Gear Dynamic Model

Transient Dynamic Bearing Loads

Axle Housing Vibration Analysis using FEA

Axle Housing Surface Vibration Responses

Radiated Noise of Axle Housing using BEM

Radiated Sound Pressure Levels, Sound Power Level, etc.

Figure 3.1 Simulation flowchart for axle whine

Table 3.1 Design parameters for the hyoid geared rotor system.

Parameters Pinion Gear

Number of Teeth 10 43

Spiral Angle [rad] 0.803 0.591 Pitch Angle [rad] 0.295 1.269 Pitch Radius [m] 0.048 0.168 Face Width [kg/m3] 0.0522 0.0478 Type [m] Left hand Right hand Loaded Side Concave Convex Offset [m] 0.0318 Gear Backlash [mm] 0.1 Torque Load [Nm] 320 Rotational Speed [r/min] 3000

(a) finite element mesh of a hypoid gear pair (b) contact pattern of pinion tooth

Figure 3.2 Illustrations of gear tooth contact analysis

The transmission error, mesh stiffness, mesh point, line-of-action are synthesized from load distribution results obtained through loaded tooth contact analysis, and served as the link between tooth contact process and dynamic model. At this stage, one can reduce the transmission error and find a favorable shape of the transmission error by tooth profile modification, machine settings and gear geometry optimization, etc. The bearing dynamic responses at each bearing location can be calculated through multi-body gear dynamic analysis and served as the excitation forces to the housing finite element (FE) model. At this stage, the control strategy is to tune the system dynamic parameters to make the system less sensitive to the excitation TE. Housing FRFs are then calculated in the forced vibration analysis and used as the input excitation for the boundary element model, and finally output sound power and sound pressure. At this stage, housing structure can be optimized to reduce gearbox noise. It is also noted that the housing flexibility can be considered in the multi-body dynamic analysis by using CMS technique to condense the full housing FE model. The dynamic and acoustic results with and without considering the housing flexibility will be compared and shown in the results, and its effect will be discussed.

3.2.2 Gear Dynamics Model

A 14-DOF lumped parameter model of hypoid geared rotor system is shown in Figure 3.3.

The equations of motion in matrix form can be written as:

[푀]{푥̈} + [퐶]{푥̇} + [퐾]{푥} + [퐺]{푥̇} + [퐺푎]{푥} = {퐹} (3.1) where

푇 {푥1} = {휃퐷, 푥푝, 푦푝, 푧푝, 휃푝푥, 휃푝푦, 휃푝푧, 푥푔, 푦푔, 푧푔, 휃푔푥, 휃푔푦, 휃푔푧, 휃퐿} , (3.2)

{푀1} = 푑푖푎푔{퐼퐷, 푀푝, 푀푝, 푀푝, 퐼푝푥, 퐼푝푦, 퐼푝푧, 푀푔, 푀푔, 푀푔, 퐼푔푥, 퐼푔푦, 퐼푔푧, 퐼퐿}. (3.3)

Figure 3.3 A 14-DOF lumped parameter model of hypoid geared rotor system

The stiffness matrix [퐾] is the lumped shaft-bearing assembly support stiffness. The damping matrix [퐶] is assumed to be viscous type and derived from damping ratio. The gyroscopic matrix [퐺] and [퐺푎] are associated with the absolute rolling velocities and accelerations of pinion and gear. The force vector can be written as

{퐹1} = {푇퐷, ℎ푝퐹푚, −ℎ푔퐹푚, −푇퐿 }. (3.4)

where ℎ푝,푔 is the directional transformation vector and the dynamic mesh force 퐹푚 can be expressed as

̇ 푘푚(훿푑 − 휀0 − 푏푐) + 푐푚(훿푑 − 휀0̇ ) 푖푓 훿푑 − 휀0 ≥ 푏푐 퐹푚 = {0 푖푓 − 푏푐 < 훿푑 − 휀0 < 푏푐 (3.5) ̇ 푘푚(훿푑 − 휀0 + 푏푐) + 푐푚(훿푑 − 휀0̇ ) 푖푓 훿푑 − 휀0 ≤ 푏푐

The dynamic transmission error is given by

푇 푇 훿푑 = ℎ푝{푥푝, 푦푝, 푧푝, 휃푝푥, 휃푝푦, 휃푝푧} − ℎ푔{푥푔, 푦푔, 푧푔, 휃푔푥, 휃푔푦, 휃푔푧} (3.6)

The time-varying mesh parameters 푘푚, 푐푚 and 휀0 are obtained from load distribution results of gear teeth.

Finally, the bearing force are calculated based on the displacements at pinion and gear support and the shaft-bearing assembly stiffness. The shaft-bearing assembly stiffness are calculated through a series of load equilibrium equations and the detailed formulations of can be found in Ref [6]. Figure 3.4 shows the dynamic pinion and gear bearing responses in time domain and frequency domain.

40 20 0

Bearing Force (kN) Force Bearing 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 Time (s)

(a) Time history of dynamic pinion bearing force 4000

2000

Amplitude (N) 0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) (b) Frequency spectrum of dynamic pinion bearing force

5 4 3

Bearing Force (kN) Force Bearing 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 Time (s)

2000

1000

Amplitude (N) 0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) (d) Frequency spectrum of dynamic gear bearing force Figure 3.4 The dynamic bearing force at actual bearing locations

3.3 Vibration and Sound Radiation Analysis

3.3.1 Housing Vibration Analysis

The finite element model of the differential housing used for the forced vibration analysis is shown in Figure 3.5. The interface connected to the rest of the vehicle is fixed. The bearing loads calculated from the lumped parameter model are added to the center of the bearing circles and rigid couplings are assumed between the center point and bearing outer race sleeves. Then, structure responses are computed and the frequency response of the acceleration of a node pinion nose location is shown in Figure 3.6. The major frequency components in this case occur at 5 times and 6 times the mesh frequency (500Hz). This is due to the natural frequencies of the housing are closely distributed in this frequency range and the mode shapes are associated with the deformation of the housing surface at pinion nose.

=1m

(a) 3-dimensional model (b) finite element mesh

Figure 3.5 Axle housing model for forced vibration analysis.

15 6x

) 2 10

Mesh frequency

5 Acceleration(m/s

0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Figure 3.6 Frequency response of housing surface accerleration (pinion nose)

3.3.2 Sound Radiation Analysis

The frequency responses of the housing surface velocities are taken into a BE model for the calculation of sound pressure levels around the axle housing and radiated sound power level.

The sound power level result is shown in Fig. 7. The major frequency components in the sound power spectrum are 4 time, 5 times and 6 times the mesh frequency between 2k to 3k Hz. As a result, the natural modes of the hypoid geared rotor system and axle housing in this frequency

33 range need be checked to further analyze the vibration transfer path for noise reduction. The sound pressure level contour plot over a spherical field mesh at 6 times the mesh frequency is shown in Fig. 8.

90 4x 5x 6x

50 Sound Power Level (dBA) 40

1000 2000 3000 4000 Frequency (Hz) Figure 3.7 Radiated sound power level of the axle housing.

Figure 3.8 Far field sound pressure level in a spherical field at 6 times the mesh frequency (3000Hz).

3.4 Conclusions

This chapter presents a series of analysis procedures for vehicle axle noise prediction and identification. The primary source of axle whine noise is believed to be the hypoid gear transmission error, but the sensitivity of the axle system to the gear transmission error and the radiation efficiency of the housing are also important for achieving vehicle interior sound quality.

The proposed approach combines the gear tooth contact analysis, lumped parameter gear rotor system model, FE and BE model of the axle housing, with an emphasis on high computational efficiency and sufficient accuracy. The methodology is supposed to assist engineers in product development and troubleshooting axle gear whine. As a next step, the structural vibration and acoustic responses should be calculated under various operation conditions to study the effect of rotational speed and torque load on the responses. Then, design optimization should be suggested and compared with baseline results.

Chapter 4 Tapered Roller Bearing Contact Analysis and Stiffness Calculation

4.1 Introduction

Bearing stiffness is an important contributor to the transmission of vibration from the rotating shaft to the housing. The investigation on the transmission of vibrations through bearings started decades ago, and many bearing models have been proposed. The early models describe bearings as ideal boundary conditions for the shafts, or as purely translational stiffness elements.

Later, more precise bearing models have been proposed based on the work of Lim and Singh [35], which includes all possible rigid-body degrees of freedom of a bearing system. Liew and Lim [36] ex-tended the aforementioned rolling element bearing stiffness formulation to include the time- variation effect of raceway rotation. Sheng et al. [37] studied the effect of speed on bearing stiffness and the dynamics of ball bearing-support rotor system.

To obtain such bearing stiffness, the bearing load vector need to be a known input for above bearing analytical models. This limits the application to statically determined system in which the static equilibrium equations are sufficient to determine the support reaction forces. However, the bearing load vector is usually a function of bearing stiffness, shaft stiffness and housing stiffness.

This paradox is explained by Razpotnik et al. [38]. To solve the bearing stiffness for a statically undetermined system, there are usually two methods. The first method is to make an initial guess for bearing load vector and use an iterative process to find the equilibrium between the two quantities by applying the analytical model. The second method is to use finite element model which considers the precise bearing geometry and the contact between the rolling elements and the raceways. Recently, a finite element/contact mechanics (FE/CM) model of ball bearing is proposed by Guo and Parker [39] for bearing stiffness calculation. The FE/CM model was

36 compared against the existing analytical models in the literature as well as the experimental results.

The same method is also implemented in a parallel axis gearbox to predict the vibration response and sound radiation of the gearbox [40].

Thus far, the studies on bearing models have been focusing on the bearing itself or parallel axis rotor-bearing system. Very few studies addressed the tapered roller bearing modeling and its effect on the right-angled geared system. The bearing stiffness matrices are usually calculated based on analytical model, and the effect of system component flexibility is neglected [12, 41, 42].

Mohammadpour et al. [43] considered the integrated lubricated tapered roller bearing and gear contacts in a tribo-dynamic analysis. Some re-searchers studied the effect of bearing preload on the contact load distribution and proposed optimal de-signs for tapered roller bearing for automotive application [44, 45].

The goal of this study is therefore to investigate the tapered roller bearing contact condition and stiffness determination in the context of axle system modeling. Bearing preload is also considered and its effect on the bearing stiffness, contact load distribution and gear transmission error is studied.

4.2 Finite Element/Contact Mechanics Model

4.2.1 Gear Mesh Model

Figure 4.1 illustrates the vehicle rear axle system considered. The model includes pinion shaft, ring gear with the carrier, four tapered roller bearings and axle housing. For simplification purpose the bevel gears in the differential are neglected.

37 displacements in the near-field contact area and far-field elastic deformation [22]. The algorithm is implemented in a computational program [47]. The output of the FE/CM model is the gear transmission error, tooth load distribution, bearing deflections and loads. Based on these output results, the mesh parameters and bearing stiffness can be calculated.

Figure 4.1 Axle system model.

4.2.2 Bearing Model

Figure 4.2 shows the 3-D pinion hear bearing model. The bearing model includes the important design details which are absent in most of the previous analytic models, such as the roller and raceway crowning, clearance, roller length, bearing width. The discrepancy due to neglecting these details has been shown in previous study [39].

The deflections of the bearings calculated from quasi-static analysis of the axle system are used for the determination of bearing stiffness. Small perturbation is added on the bearing deflection vector for each direction at a time. The forces due to the perturbed deflection vector are calculated. The bearing stiffness matrix is calculated based on the forces and displacements and the calculation method will be introduced later in the next section.

Figure 4.2 Pinion head bearing model.

4.3 Bearing Stiffness Calculation

The stiffness matrix of rolling element bearings is 푘 푘 푘 푘 푘 0 푥푥 푥푦 푧푧 푥휃푥 푥휃푦 푘 푘 푘 푘 푘 0 푦푥 푦푦 푦휃푥 푦휃푥 푦휃푦 푘 푘 푘 푘 푘 0 퐾 = 푧푥 푧푦 푧푧 푧휃푥 푧휃푦 (4.1) 푘 푘 푘 푘 푘 0 휃푥푥 휃푥푦 휃푥푧 휃푥휃푥 휃푥휃푦

푘휃푦푥 푘휃푦푦 푘휃푦푧 푘휃푦휃푥 푘휃푦휃푦 0 [ 0 0 0 0 0 0] where 푥 and 푦 denote the bearing in-plane axes and z denotes the axial direction. The symbols

휃푥, 휃푦 are the out-of-plane angular deflections about the 푥 and 푦 axes, respectively.

k k k k k 0 xx xy zz xθx xθy k k k k k 0 yx yy yθx yθx yθy k k k k k 0 K = zx zy zz zθx zθy (4.2) k k k k k 0 θxx θxy θxz θxθx θxθy

kθyx kθyy kθyz kθyθx kθyθy 0 [ 0 0 0 0 0 0]

The stiffness matrix is calculated numerically through second order finite difference method by

k k k k k 0 xx xy zz xθx xθy k k k k k 0 yx yy yθx yθx yθy 퐅(퐪 +훿퐪)−퐅(퐪 −훿퐪) k k k k k 0 0 0 K = zx zy zz zθx zθy 2훿퐪 k k k k k 0 θxx θxy θxz θxθx θxθy

kθyx kθyy kθyz kθyθx kθyθy 0 [ 0 0 0 0 0 0]

(4.3)

where 퐪0 is the calculated bearing deflection vector at given load 퐅0 about which the stiffness

matrix is desired and 훿퐪 = {훿푞푥, 훿푞푦, 훿푞푧, 훿푞휃푥, 훿푞휃푦} is the specified small disturbance vector at

퐪0.

4.4 Numerical Simulation

In this section, a practical design of axle system with hypoid gear pair and tapered roller bearings is taken as an example for numerical simulation. The dimensional parameters and clearance of tapered roller bearing are shown in Figure 4.3 and Figure 4.4, respectively. The hypoid design parameters are shown in Table 4.1. The design parameters of pinion bearings and gear bearings are shown in Table 4.2 and Table 4.3, respectively. The quasi-static analysis is per-formed for one mesh cycle and the number of total time steps is 21. Five cases with no axial clearance and

0.02mm, -0.02mm, -0.05mm and -0.1mm axial clearance are compared to investigate the influence of preload on the roller-race contact stress and bearing stiffness.

Figure 4.3 Dimensional parameters of tapered roller bearing.

Figure 4.4 Clearance between rolling elements and raceway.

Table 4.1 Hypoid gear design parameters.

Parameters Pinion Gear

Number of Teeth 10 43

Spiral Angle [rad] 0.803 0.591

Pitch Angle [rad] 0.295 1.269

Pitch Radius [m] 0.048 0.168

Face Width [m] 0.0522 0.0478

Type Left hand Right hand

Loaded Side Concave Convex

Offset [m] 0.0318

Torque Load [Nm] 320

Table 4.2 Design parameters of pinion bearings

Parameters Pinion Pinion head tail

Number of rollers 16 24

Roller length [mm] 19 17.1

Roller large end diameter 16.67 7.84 [mm]

Cup angle [degree] 15 12.86

Bearing width [mm] 22 18.2

Outer raceway diameter 110 80 [mm]

Inner raceway diameter 50 50 [mm]

Cup inner diameter [mm] 90.36 67.9

Distance of thrust center 27 17 [mm]

Table 4.3 Design parameters of gear bearings.

Parameters Wheel bearings

Number of rollers 28

Roller length 23.3

Roller large end diameter 13.23

Cup angle [m] 23.31

Bearing width [mm] 26.3

Outer raceway diameter 145 [mm]

Inner raceway diameter 95 [mm]

Cup inner diameter [mm] 124.93

Distance of thrust center 33.11 [mm]

4.4.1 Comparison against Analytical Method

The proposed bearing stiffness determination method is compared with a program based on Herzian contact theory [35]. The differences of the diagonal terms of calculated bearing stiffness matrix are highlighted in Table 4.4. In general, the stiffness terms predicted by current

FE/CM model are lower than those predicted by analytical model. This is due to the different assumptions between these two methods. While Herzian contact theory assumes unrealistically thick and wide bearing races, the combined sur-face integral and finite element approach does not re-quire assumptions about bearing dimensions. Also, it assumes the bearing inner ring and outer ring are rigid for the analytical model. Nevertheless, the FE/CM model includes the elasticity of the bearing inner ring and shaft.

Figure 4.5 shows the time-varying bearing stiffness calculated by the FE/CM model in two ball pass period. The time-varying characteristics of the bearing stiffness is due to the fact that the number of rollers change periodically in one ball pass period. It can excite the gearbox vibration as the primary structure-borne path. It can also be observed that the bearing stiffness is larger when there are more number of rollers in contact, and smaller when there are less number of rollers in contact. In this case, it is found that there are 5 rollers in contact at the valley and 6 rollers in contact at the peak of the curve.

Table 4.4 Diagonal terms of bearing stiffness matrices predicted by current model and Lim/Singh’s model.

푘푥푥 푘푦푦 푘푧푧 푘휃푥휃푥 푘휃푦휃푦

(N/mm) (N/mm) (N/mm) (Nmm/rad)

Current 1.36×106 1.36×106 1.95×105 4.21×107 4.96×107

Lim/Singh 2.17×106 2.01×106 3.00×105 9.71×107 1.11×108

Figure 4.5 Time-varying bearing stiffness.

4.4.2 Bearing Stiffness

The bearing preload can have significant influence on the bearing stiffness. The effect of bearing preloads, represented by the negative internal clearance, are studied in this section. Figure

4.6 shows the influence of bearing preloads on the diagonal terms of the bearing matrix. When the preload is present, the bearing stiffness generally increase nonlinearly as expected, due to the enlarged contact area.

(a) Bearing radial stiffness

(b) Bearing axial stiffness

Figure 4.6 Time-varying bearing stiffness.

4.4.3 Contact Pattern

The preload will also affect the contact load distribution between rolling elements and bearing races. Since the tapered roller bearings are designed to handle the combination of radial load and axial load, the contact stress will concentrate on the roller end. The Von Mises stress distribution on the roller and maximum stress are shown in Figure 4.7 and Figure 4.8, respectively.

The maximum stress is reduced when proper preload is applied, due to the load sharing effect.

However, under excessive preload, such as the -0.1mm case in this study, the trend tends to reverse.

Therefore, to avoid bearing failure and extend the fatigue life, a proper bearing preload should be applied.

(a) 0.02 mm axial clearance

(b) -0.1 mm axial clearance

Figure 4.7 Von Mises stress distribution on the roller surface.

Figure 4.8 Maximum Von Mises stress on the roller surface.

4.4.4 Transmission Error

The bearing preload will affect the supporting stiffness of pinion and influence the gear transmission error (TE) as a result. It can be shown in Figure 4.9 that the absolute value of transmission error is reduced for the case with negative axial clearance, due to the stiffening effect when the bearing preload is present. The bearing preload also reduces the first order harmonic of transmission error, as shown in Figure 4.10, but the effect is not very significant. The peak-to-peak value of transmission error is a major excitation source for the vibration of geared system and the reduction indicates improvement in the dynamic performance of axle system when proper bearing preload is present.

0.02mm

0mm

-0.02mm

-0.05mm

-0.1mm

Figure 4.9 Gear transmission error in one mesh cycle.

Figure 4.10 Gear TE harmonics.

(z = 0mm: no preload; z = -0.01mm: with preload).

4.5 Conclusion

In this study, a finite element/contact mechanics mod-el of an axle system with tapered roller bearings are proposed based on realistic axle design parameters. The bearing stiffness under operating bearing load, including the bearing preload, is calculated applying the proposed model and stiffness determination method. The results are compared with the diagonal terms of bearing stiffness matrix predicted by analytical meth-od. The effect of bearing dimensional preload on the bearing stiffness, roller-race contact stress distribution and gear transmission error is investigated.

The simulation results show that proper bearing preload is needed for the purpose of system rigidity and bearing fatigue life. In the next chapter, the effect of bearing stiffness on dynamic responses of the axle system will be investigated.

Chapter 5 Effect of Component Flexibility on Axle System Dynamics

5.1 Introduction

The vehicle interior sound quality has long been one of the major concerns for automotive

industry. The axle gear whine is one of the major noise sources in a vehicle cabin, especially for

four-wheel-drive (4WD) vehicles. The excitation of the axle gear whine is believed to be the gear

transmission error (TE) by many researchers. The vibratory energy of the gear pair caused by

transmission error is transmitted structurally through shaft-bearing-housing assembly and radiates

off from exterior housing. Thus, the understanding of the source of excitation and the primary

transmission path is the key to solve gearbox noise problem. Most of the earlier research work

neglect the flexibility of other components, such as shaft, bearing, etc., or oversimplify the gear

contact process and bearing support stiffness. Thus, the goal of this study is to provide an in-depth

understanding of the role of flexible components played in the gear tooth deflection and dynamic

responses.

A number of investigations have been made in the past to find out the cause of axle whine

and identify the relationship between axle whine with the excitation force of the meshing gear pair.

Yoon, et.al. [29] measured transmission error of a hypoid gear pair and found that the hypoid gear

whine noise can be minimized when the input torque is in the target range of the torque load for

optimal transmission error. Koronias, et.al. [4] investigated on axle whine vibration and noise by

a combined experimental and numerical approach. They compared the vibration modes of the

drivetrain with experimental results and established a causal relationship between axle whine and

the flexural mode response of system components. Most of the research papers on the practical

work of axle whine reduction adopted tooth profile modification and structure modification as the

means to achieve sound quality [30-32].

As the lightweight materials and design become a popular trend in the development of vehicles, the influence of component flexibility becomes increasingly important. The vibration characteristics of the main component on the structure transfer path, such as shafts, bearings, and housing, should be considered in the system model for maximum NVH improvements. Yang [48] and Wang [42] studied the influence of propeller shaft bending modes and analyze the gear-shaft interaction by using component mode synthesis (CMS). The same method was also applied to investigate the influence of housing flexibility on hypoid gear dynamic responses [49]. Ognjanović and Kostić [50] used a combined approach of theoretical, numerical and experimental analyses to explain the process of spreading disturbance power through the elastic housing and found that the gear unit housing has a dominant role in the transformation of disturbance power and modulation of the sound emitted to the surroundings. Vanhollebeke [51] investigated the structure-borne and airborne noise radiated from a wind turbine gearbox by using multi-body gearbox model and an acoustic FE model. They found that the flexibility of the housing will make a difference in both mode shapers and eigenfrequencies and thus is not negligible for evaluating the noise, vibration and harshness (NVH) responses of wind turbine gearboxes.

In this chapter, the influence of elastic shaft-bearing-housing structure on the transmission error is firstly investigated by comparing the gear pair and axle system model. The axle system model includes detailed contact modeling between gear pairs, the roller bearings and housing. The bearing stiffness matrix can be calculated using finite difference method for the same model. Then, the dynamic responses are simulated and compared between the model with simply-supported bearing assumption and the model with flexible bearings.

5.2 Contact Analysis

The hypoid gear mesh model, including mesh point, line-of-action, mesh stiffness and kinematic TE can be derived from the load distribution results calculated by a 3-dimensional quasi- static loaded tooth contact analysis program for hypoid and spiral bevel gears [47]. This program combines the semi-analytical theory with finite element method [22], which can solve the gear tooth contact problem very efficiently.

5.2.1 Gear Pair Model

A hypoid gear pair with multiple contact interfaces is shown in Figure 1(a), and Figure 1(b) shows contact cells on each tooth pair. The position vector of the contact cell in the mesh 4 coordinate system is 푟푖(푟푖푥, 푟푖푦, 푟푖푧), the contact force is 푓푖, and the normal vector is 푛푖(푛푖푥, 푛푖푦, 푛푖푧).

ni n

z ( fi f

a) ri r x y

(a) (b)

Configuration: CONFIG.DAT M esh file 1: pinion.msh M esh file 2: gear.msh ( Advanced Numerical Solutions. Figure 5.1 Illustrations of: a) multi-point coupling of hypoid gear pair; b) contact cells on b) engaging tooth surface.

The total contact force is calculated by summing the contact forces on each contact cell

(assuming there are N contact cells)

푁 푧 2 퐹푗 = ∑푖=1 푛푖푗 푓푖, 퐹 = √∑푗=푥 퐹푗 , (푗 = 푥, 푦, 푧) (5.1)

The line-of-action vector 퐿푚(푛푥, 푛푦, 푛푧) can be obtained from 54

퐹 푛 = 푗 , (푗 = 푥, 푦, 푧) (5.2) 푗 퐹

The total contact moment is given by

푁 푀푥 = ∑푖=1 푓푖 [푛푖푧푟푖푦 − 푛푖푦푟푖푧] (5.3a)

푁 푀푦 = ∑푖=1 푓푖 [푛푖푥푟푖푧 − 푛푖푧푟푖푥] (5.3b)

푁 푀푧 = ∑푖=1 푓푖 [푛푖푦푟푖푥 − 푛푖푥푟푖푦] (5.3c)

The mesh point 푅푚(푥푚, 푦푚, 푧푚) can be obtained from

푁 ∑푖=1 푟푖푦푓푖 푦푚 = 푁 (5.4a) ∑푖=1 푓푖

푥푚 = (푀푧 + 퐹푥푦푚)/퐹푦 (5.4b)

푧푚 = (푀푦 + 퐹푧푥푚)/퐹푥 (5.4c)

The translational loaded and unloaded transmission errors 푒퐿 and 휀0 are the projections of corresponding angular transmission error along the line of action. Finally, the mesh stiffness is defined by

푘푚 = 퐹/(푒퐿 − 휀0) (5.5)

The above representations are then combined with the geared rotor system dynamic model to enable TE excitation to be applied reasonably well and also to compute meaningful gear mesh response.

5.2.2 Axle System Model

Figure 2 illustrates the vehicle rear axle system considered. The model includes pinion shaft, ring gear with the carrier, four tapered roller bearings and axle housing. For simplification purpose the bevel gears in the differential are neglected.

The FE/CM model takes account of the macro- and micro-geometry of the gear tooth profile, bearing rollers and raceway. Its efficiency and accuracy has been proven in the previous studies [40, 46]. The output of the FE/CM model is the gear transmission error, tooth load distribution, bearing deflections and loads. Based on these output results, the mesh parameters and bearing stiffness can be calculated.

Figure 5.2 Axle system model.

The bearing model includes the important design details which are absent in most of the previous analytic models, such as the roller and raceway crowning, clearance, roller length, bearing width. The discrepancy due to neglecting these details has been shown in previous study.

The deflections of the bearings calculated from quasi-static analysis of the axle system are used for the determination of bearing stiffness [39, 52]. Small perturbation is added on the bearing deflection vector for each direction at a time. The forces due to the perturbed deflection vector are calculated. The bearing stiffness matrix is calculated based on the forces and displacements

k k k k k 0 xx xy zz xθx xθy k k k k k 0 yx yy yθx yθx yθy k k k k k 0 K = zx zy zz zθx zθy (5.6) k k k k k 0 θxx θxy θxz θxθx θxθy

kθyx kθyy kθyz kθyθx kθyθy 0 [ 0 0 0 0 0 0]

where 푥 and 푦 denote the bearing in-plane axes and z denotes the axial direction. The symbols 휃푥, 휃푦 are the out-of-plane angular deflections about the 푥 and 푦 axes, respectively. The stiffness matrix is calculated numerically through second order finite difference method by

퐅(퐪 +훿퐪)−퐅(퐪 −훿퐪) 0 0 (5.7) 2훿퐪

where 퐪0 is the calculated bearing deflection vector at given load 퐅0 about which the stiffness

matrix is desired and 훿퐪 = {훿푞푥, 훿푞푦, 훿푞푧, 훿푞휃푥, 훿푞휃푦} is the specified small disturbance vector at

퐪0.

5.3 Dynamic Analysis

The 14-DOF lumped parameter model of the hypoid geared rotor system is similar to the model used in Tao’s study [6]. The equations of motion of the geared rotor system in matrix form can be written as

[푀1]{푥̈1} + [퐶1]{푥̇1} + [퐾1]{푥1} + Ω[퐺1]{푥̇1} = {퐹1} (5.8)

Where 푇 {푥1} = {휃퐷, 푥푝, 푦푝, 푧푝, 휃푝푥, 휃푝푦, 휃푝푧, 푥푔, 푦푔, 푧푔, 휃푔푥, 휃푔푦, 휃푔푧, 휃퐿} (5.9)

{푀1} = 푑푖푎푔{퐼퐷, 푀푝, 푀푝, 푀푝, 퐼푝푥, 퐼푝푦, 퐼푝푧, 푀푔, 푀푔, 푀푔, 퐼푔푥, 퐼푔푦, 퐼푔푧, 퐼퐿} (5.10)

The stiffness matrix [퐾1] is the lumped shaft-bearing assembly support stiffness. The damping matrix [퐶1] is assumed to be viscous type and derived from damping ratio. The gyroscopic matrix [퐺1] can be expressed as given below

−퐼푝푦 퐼푝푦 ↑(5,7)

↑(7,5) {퐺1} = (5.11) −퐼푔푦/푅 퐼푔푦/푅 ↑(11,13) ↑(13,11) [ ]

Figure 5.3 A lumped parameter model of hypoid geared rotor system.

The force vector can be written as

{퐹1} = {푇퐷, ℎ푝퐹푚, −ℎ푔퐹푚, −푇퐿 } (5.12)

The dynamic mesh force 퐹푚 and directional transformation vectors ℎ푙 are defined as

̇ 퐹푚 = 푘푚(훿푑 − 휀0) + 푐푚(훿푑 − 휀0̇ ) (5.13)

ℎ푙 = {푛푙푥, 푛푙푦, 푛푙푧, 휆푙푥, 휆푙푦, 휆푙푧 }, 푙 = 푝, 푔 (5.14)

where 휀0 is unloaded transmission error and rotational radius are defined as

휆푙푥 = 퐿푙(푋푙×푅푙), 휆푙푦 = 퐿푙(푌푙×푅푙), 휆푙푧 = 퐿푙(푍푙×푅푙) (5.15)

where 퐿푙 and 푅푙 is the line of action and mesh point in pinion coordinate system (푙 = 푝) or gear coordinate system (푙 = 푔) , and 푋푙, 푌푙, 푍푙 are the unit vectors in the pinion or gear coordinate system. The dynamic transmission error is given by

푇 푇 훿푑 = ℎ푝{푥푝, 푦푝, 푧푝, 휃푝푥, 휃푝푦, 휃푝푧} − ℎ푔 {푥푔, 푦푔, 푧푔, 휃푔푥, 휃푔푦, 휃푔푧} (5.16) which accounts for the system dynamic response in addition to the kinematic effect of transmission error.

5.4 Simulation Results

In this section, a practical design of axle system with hypoid gear pair is taken as an example for numerical simulation. The results are compared between the gear pair model and axle system model to show the effect of component flexibility on the mesh and dynamic characteristics of the geared rotor system.

5.4.1 Contact Results

The effect of component flexibility on the transmission error (TE) excitation is shown in

Fig. 4. It can be seen that all the harmonic orders have increased, when the elasticity of the system is considered. For the further linear dynamic analysis, the first order of TE harmonics will be used as the excitation to the system.

Figure 5.4 The TE harmonics.

5.4.2 Modal Characteristics

The modal frequencies and mode shapes for different support assumptions are compared in Figure 5 and Figure 6. It is observed that not only the modal frequencies have been shifted by applying actual bearing stiffness matrices, but the mode shapes have also changed, due to the change of support stiffness.

Figure 5.5 Modal frequencies and mode shapes of hypoid geared rotor system with simply supported assumption.

Figure 5.6 Modal frequencies and mode shapes of hypoid geared rotor system with flexible bearings.

5.4.3 Dynamic Results

The dynamic mesh responses, including the dynamic transmission error and dynamic mesh force are calculated separately for simply supported case and flexible bearing case, and the comparison results are presented in Figure 7 and Figure 8. The shifts of peak frequencies can be observed. The magnitudes of the peaks have also changed for different assumptions on support stiffness.

The dynamic bearing forces are also largely affected by the support stiffness. Figure 9 and

Figure 10 show the dynamic bearing forces for pinion bearing and gear bearing, respectively. It could be observed that both the magnitudes and peak frequencies have changed for bearing forces in all the three directions, due to the change of modal frequencies and mode shapes of the geared rotor system.

Figure 5.7 Dynamic transmission error simply-supported bearing, flexible bearing

Figure 5.8 Dynamic mesh force: simply-supported bearing, flexible bearing.

Figure 5.9 Dynamic pinion bearing forces: simply supported (horizontal direction), simply supported (axial direction), simply supported (vertical direction), flexible bearings (horizontal direction), flexible bearings (axial direction), flexible bearings (vertical direction).

Figure 5.10 Dynamic gear bearing forces: simply supported (horizontal direction), simply supported (axial direction), simply supported (vertical direction), flexible bearings (horizontal direction), flexible bearings (axial direction), flexible bearings (vertical direction).

5.5 Conclusions

In this study, a system level model of axle system with hypoid gear pair is developed and compared with a hypoid gear pair model. The effect of the elasticity of the shafts, bearings and housing on TE as well as the contribution of flexible bearings on the dynamic responses are investigated by comparing the results between the system model and simplified model. The results reveal that the peak-to-peak transmission error as well as the system responses are sensitive to the system elasticity and shaft-bearing flexibility. Thus the component flexibility should not be ignored if one desires to predict the system vibration responses accurately and improve the system design in an effort to maximize the overall NVH behavior.

Chapter 6 Effect of Gear-Shaft Coupling Dynamics and Gyroscopic Moments

6.1 Introduction

Automotive rear axle gear noise problem has received attention in the past few decades.

Although many experimental work and numerical simulations have been carried out in the past to

investigate the physics of axle whine noise generation [1-4], it is still not fully understood how

component flexibility and system dynamics could affect radiated noise from the driveline system.

In order to predict the noise radiation more precisely and use the approach to tackle axle whine

noise problem during the design process, an accurate and realistic driveline system model is

needed.

It is now well established that gear transmission error is the primary source of gear whine

noise. The dynamic characteristics of the meshing hypoid gear pair due to gear transmission error,

namely the dynamic transmission error and mesh force have been studied by many researchers –

some are noted below. Chen and Lim [7, 8] proposed a generalized 3-dimensional dynamic model

based on idealized gear geometry and quasi-static loaded tooth contact analysis. Wang and Lim

[53] studied the interaction between time-varying mesh characteristic and backlash nonlinearity,

and observed some interesting nonlinear behaviors such as jump phenomenon, sub-harmonic and

chaotic responses. They have also found that mean load and mesh damping are two key factors for

tooth impact condition. Besides the dynamic mesh forces, friction of meshing teeth pair also plays

an important role in gear dynamics and NVH (noise, vibration and harshness) responses as an

energy sink. Peng [6] proposed a sliding friction model for hypoid and spiral bevel gears based on

a quasi-static loaded tooth contact analysis and the synthesis of the effective lumped friction

coefficients. The sliding friction was found to have significant influence on the magnitude of

65 dynamic response, especially for spiral bevel gears under medium and heavy load due to the inherent friction direction reversal behavior. Feng, et.al. [9] enhanced the friction model by applying the mixed elastohydrodynamic lubrication (EHL) formulation. The friction also affects the transmission efficiency, which is another major concern for vehicle differential hypoid gears.

In practice, the differential gearbox efficiency and NVH refinement possess contradictory desired attributes [10]. Mohammadpour, et.al. [10, 11] proposed a combined multi-body dynamics and lubricated contact mechanics model of vehicular differential hypoid gear pairs to obtain a balanced approach for transmission efficiency and NVH performance. In their study, realistic torque load was applied considering the road data, aerodynamic effects and engine torque variation. The results showed that gear tooth pair separation and friction loss have significant influence on axle whine generation, which is also confirmed previously by Koronias, et.al. [4].

The dynamic mesh force can be transmitted through shaft-bearing structure to cause the vibration of the whole structure that radiates noise. Even with minimized transmission error and proper lubrication, the system can still be sensitive to the excitation force. Moreover, minimizing the transmission error normally requires more involved efforts and cost. Hence, in some cases, it is more reasonable to tune the system dynamics for noise reduction purpose. Many researchers investigated the effect of system dynamics on gear vibration. For example, Hua, et.al. [12] proposed a shaft-bearing model for computing the effective support stiffness of spiral bevel geared rotor system and compared the dynamic responses of different shaft-bearing configurations. Yang, et.al. [13, 14] studied the effect of time-varying bearing stiffness and housing flexibility on the hypoid geared rotor system responses. Wang, et.al. [15] studied the effect of torque load on the mesh and dynamic characteristics of hypoid geared system. Mohammadpour, et.al. [11] investigated the effect of system damping and bearing stiffness on the dynamic transmission error

66 and found that lower system damping and higher bearing rigidity can deteriorate the tooth separation condition and worsen NVH performance.

Most previous studies neglect the gyroscopic moments and shaft dynamics to reduce the model complexity and computational cost. However, the gyroscopic effect can cause misaligned contact of gear tooth pairs and may alter the gear dynamic behavior. It has been shown the importance of gyroscopic effect and coupled lateral-torsional vibration modes on parallel axis geared rotor systems [54-58] and angled rotor system [59]. However, the aforementioned studies have been limited to gear free vibration analysis. The gear gyroscopic effect on hypoid gear dynamics was studied by Peng and Lim [60], and it was found that the bending and torsional moments of inertia of the gear bodies are important factors to determine the importance of gyroscopic effect. Kartik and Houser [61] investigated the shaft dynamic effects on gear vibration and noise excitation. In fact, many researchers have studied the geometric variation and damping treatment of the shaft for the purpose of vibration and noise reduction [62-64] because of the potential of noise reduction through improved design of the shaft. The effect of shaft bending flexibility on the hypoid gear vibration has been studied by Yang and Lim [48]. In their study, the propeller shaft is modeled applying the Euler-Bernoulli beam elements and the gyroscopic moments of the shaft and gear bodies are neglected. Since the shaft dynamics, including the rotordynamic and gyroscopic effects of the gear bodies, can affect the lateral motions of the gears, it is necessary to understand the coupling between the gyroscopic moments and shaft vibrations.

No prior study has been performed that takes both of them into consideration in the driveline system model. Thus, one of the goals of the current work is to address this gap in the literature. In addition, previous studies mainly focus on gear pair dynamics and give less attention to the sensitivity of structural responses to different vibration modes, which is addressed in this study.

This chapter is an extended version of work published in [65], with extended vibro-acoustic modeling and structural responses. The sensitivity of the structural responses to the various system parameters and design changes is also investigated in this study. Firstly, a lumped parameter model of the propeller shaft is developed with Timoshenko beam elements that includes the effects of rotary inertia, shear deformation and gyroscopic moments. The propeller shaft model is then coupled with a linear hypoid gear pair representation using the component mode synthesis (CMS) approach. The gyroscopic effect of the gear bodies is also considered. A series of dynamic analyses applying the proposed model with gyroscopic moments of the gear bodies only, propeller shaft coupling effect only, and both effects are performed to determine their severity and impact on gear dynamics and ultimately radiated noise response. In addition, the effects of the key parameters, such as gear inertia, and shaft and bearing stiffness are also studied to shed light on the feasibility of vibration and noise reductions through dynamic tuning at a system level. Two example cases for design enhancement are examined. Finally, the conclusions derived from this study are presented and suggestions for future investigations are discussed.

6.2 Gear Mesh and Dynamic Formulation

6.2.1 Gear Mesh Model

The hypoid gear mesh model, including mesh point, line-of-action, mesh stiffness and kinematic transmission error (TE) can be derived from the load distribution results calculated by a 3-dimensional quasi-static loaded tooth contact analysis program for hypoid and spiral bevel gears [66]. This program combines the semi-analytical theory with finite element method [22], which can solve the gear tooth contact problem very efficiently.

ni n

z fi f

ri r x y

(a) (b)

Configuration: CONFIG.DAT M esh file 1: pinion.msh M esh file 2: gear.msh Advanced Numerical Solutions. Figure 6.1 Illustrations of: a) multi-point coupling of hypoid gear pair; b) contact cells on engaging tooth surface.

A hypoid gear pair with multiple contact interfaces is shown in Figure 6.1(a), and Figure

6.1(b) shows contact cells on each tooth pair. The position vector of the contact cell in the mesh coordinate system is 푟푖(푟푖푥, 푟푖푦, 푟푖푧), the contact force is 푓푖, and the normal vector is 푛푖(푛푖푥, 푛푖푦, 푛푖푧).

The total contact force is calculated by summing the contact forces on each contact cell (assuming there are N contact cells)

푁 푧 2 퐹푗 = ∑푖=1 푛푖푗 푓푖, 퐹 = √∑푗=푥 퐹푗 , (푗 = 푥, 푦, 푧) (6.1)

The line-of-action vector 퐿푚(푛푥, 푛푦, 푛푧) can be obtained from

퐹 푛 = 푗 , (푗 = 푥, 푦, 푧) (6.2) 푗 퐹

The total contact moment is given by

푁 푀푥 = ∑푖=1 푓푖 [푛푖푧푟푖푦 − 푛푖푦푟푖푧] (6.3a)

푁 푀푦 = ∑푖=1 푓푖 [푛푖푥푟푖푧 − 푛푖푧푟푖푥] (4.3b)

푁 푀푧 = ∑푖=1 푓푖 [푛푖푦푟푖푥 − 푛푖푥푟푖푦] (4.3c)

The mesh point 푅푚(푥푚, 푦푚, 푧푚) can be obtained from

푁 ∑푖=1 푟푖푦푓푖 푦푚 = 푁 (6.4a) ∑푖=1 푓푖

푥푚 = (푀푧 + 퐹푥푦푚)/퐹푦 (4.4b)

푧푚 = (푀푦 + 퐹푧푥푚)/퐹푥 (4.4c)

The translational loaded and unloaded transmission errors 푒퐿 and 휀0 are the projections of corresponding angular transmission error along the line of action.

Finally, the mesh stiffness is defined by

푘푚 = 퐹/(푒퐿 − 휀0) (6.5)

The above representations are then combined with the geared rotor system dynamic model to enable TE excitation to be applied reasonably well and also to compute meaningful gear mesh response.

6.2.2 Propeller Shaft Model

The propeller shaft is modeled as two hollow beams with two different cross sections as shown in Figure 6.2. Only the bending flexibility of the shaft is considered in the coupling with geared rotor system. The torsional flexibility is considered separately as a spring between engine inertia and pinion inertia of moments. The finite element (FE) model of the propeller shaft is built by applying Timoshenko beam finite rotating shaft element, in which the rotary inertia, shear deformation and gyroscopic moment of the beam are considered. The details of the element matrices can be found in Reference [67].

Since the full shaft finite element (FE) model contains thousands of nodes, it is difficult to directly apply it to geared rotor system model. Thus, a fixed-interface CMS method, which is also known as Craig-Bampton method, is adopted here to condense the shaft model [68]. The size of the mass and stiffness matrices of reduced shaft model is determined by the degrees of freedom of

70 retained nodes and truncated set of normalized modes, which is usually much smaller than the size of original matrices. In this case, the retained nodes are at the ends of the propeller shaft, which interface with the geared rotor system representation. The equations of motion for the shaft model can be written as

[푀퐶퐶] [0] {푢̈ 퐶} [퐾퐶퐶] [0] {푢퐶} {퐹퐶} [ ] { } + [ ] { } = { } (6.6) [0] [푀퐼퐼] {푢̈ 퐼} [0] [퐾퐼퐼] {푢퐼} {퐹퐼} where the coupling coordinates and interior coordinates are noted by superscripts C and I, respectively. The physical coordinates {푢} can be represented by transformation matrix and modal coordinate {푥}:

{푢퐶} {푥퐶} { } = [Φ] { } (6.7) {푢퐼} {푥퐼}

The transformation matrix that maps the physical coordinates of the housing into the modal coordinates can be obtained from

[퐼] [0] [Φ] = [ ] (6.8) [Φ퐶] [Φ푁] where [Φ퐶] is matrix of constraint modes and [Φ푁] is matrix of normal modes. The constraint modes [Φ퐶] can be computed by

[Φ퐶] = −[퐾퐼퐼]−1[퐾퐼퐶] (6.9)

Finally, the condensed mass and stiffness matrices of the housing can be obtained by

푇 푀푠 = Φ 푀Φ (6.10a)

푇 퐾푠 = Φ 퐾Φ (4.10b)

푇 퐺푠 = Φ 퐺Φ (4.10c)

Similarly, the load vector can be obtained using 71

푇 퐹푠 = Φ 푓 (4.10d) which is employed in the shafting model.

(a) (b)

Figure 6.2 Propeller shaft model a) beam with two different cross sections; b) cross section dimensions.

6.2.3 Geared Rotor System Model

The 14-DOF (degrees of freedom) lumped parameter model of the hypoid geared rotor system is shown in Figure 6.3, which is similar to the model used in Peng’s study [6].

Figure 6.3 Lumped parameter model of hypoid geared rotor system.

The equations of motion of the geared rotor system in matrix form can be written as

[푀1]{푥̈1} + [퐶1]{푥̇1} + [퐾1]{푥1} + Ω[퐺1]{푥̇1} = {퐹1} (6.11)

where the coordinate vector and mass matrix are given by

푇 {푥1} = {휃퐷, 푥푝, 푦푝, 푧푝, 휃푝푥, 휃푝푦, 휃푝푧, 푥푔, 푦푔, 푧푔, 휃푔푥, 휃푔푦, 휃푔푧, 휃퐿} (6.12)

[푀1] = 푑푖푎푔[퐼퐷, 푀푝, 푀푝, 푀푝, 퐼푝푥, 퐼푝푦, 퐼푝푧, 푀푔, 푀푔, 푀푔, 퐼푔푥, 퐼푔푦, 퐼푔푧, 퐼퐿] (6.13)

The stiffness matrix [퐾1] includes the gear mesh stiffness and lumped shaft-bearing assembly support stiffness, which will be detailed in the next section. The damping matrix [퐶1] is assumed to be viscous type and represents the combined effect of structural damping present in the system except for mesh damping. The damping ratio is nominally assumed to be 0.02 according to the literature for most practical transmissions [8, 11]. The symbol Ω is the mean shaft speed. The gyroscopic matrix [퐺1] can be expressed as given below:

−퐼푝푦 퐼푝푦 ↑(5,7)

↑(7,5) [퐺1] = (6.14) −퐼푔푦/푅 퐼푔푦/푅 ↑(11,13) ↑(13,11) [ ]

where 푅 is the gear ratio. The force vector can be written as

{퐹1} = {푇퐷, 풉풑퐹푚, −풉품퐹푚, −푇퐿 } (6.15)

The mean input torque 푇퐷 and output torque 푇퐿 are assumed to be constant. Assuming there is no transmission loss, they should satisfy the following relation,

푇퐿 = 푅 ∗ 푇퐷 (6.16)

The directional transformation vectors 풉풑 and 풉품 are defined as

푇 풉풍 = {푛푙푥, 푛푙푦, 푛푙푧, 휆푙푥, 휆푙푦, 휆푙푧 } , 푙 = 푝, 푔 (6.17)

The rotational radius 휆푙푥, 휆푙푦, 휆푙푧 are defined as

휆푙푥 = 푳풍풎 ∙ (푿풍×푹풍풎), 휆푙푦 = 푳풍풎 ∙ (풀풍×푹풍풎), 휆푙푧 = 푳풍풎 ∙ (풁풍×푹풍풎) (6.18)

푇 where 푳풍풎 = {푛푙푥, 푛푙푦, 푛푙푧} is the normal line-of-action directional cosine vector and 푹풍풎 =

푇 {푥푙푚, 푦푙푚, 푧푙푚} is the mesh point in pinion coordinate system (푙 = 푝) or gear coordinate system

(푙 = 푔) , and 푋푙, 푌푙, 푍푙 are the unit vectors in the pinion or gear coordinate system. The dynamic mesh force 퐹푚 is defined as

̇ 퐹푚 = 푘푚(훿푑 − 휀0) + 푐푚(훿푑 − 휀0̇ ) ( 6.19)

where 푘푚 is the mesh stiffness obtained from quasi-static contact analysis and 푐푚 is the mesh damping calculated from this empirical relation,

푐푚 = 2휉√푘푚푚푒 (6.20)

where 휉 is the mesh damping ratio and 푚푒 is the equivalent mass given as

2 2 휆푝푦 휆푔푦 푚푒 = 1/ ( + ) (6.21) 퐼푝푦 퐼푔푦

The dynamic transmission error is given by

푻 푻 훿푑 = 풉풑{푥푝, 푦푝, 푧푝, 휃푝푥, 휃푝푦, 휃푝푧} − 풉품{푥푔, 푦푔, 푧푔, 휃푔푥, 휃푔푦, 휃푔푧} (6.22)

74 that accounts for the system dynamic response in addition to the kinematic effect of transmission error.

6.2.4 Shaft-bearing Stiffness Model

The effective lumped parameter shaft-bearing support stiffness matrix is synthesized from the shaft-bearing stiffness model. A static beam FE model is applied for the calculation of effective shaft-bearing lumped stiffness matrix, and the static stiffness of the shaft and bearings under mean mesh load are used. The pinion overhung and gear straddled mounting is assumed as shown in

Figure 6.4. Each bearing is represented as a stiffness matrix that is applied as a spring type element between the node at the actual bearing center location and a node for the rigid housing. The bearing stiffness matrix is computed based on a set of nonlinear bearing load-displacement equations. It is assumed that the operating load variation does not have significant influence on the bearing stiffness.

(a)

(b)

Figure 6.4 Illustrations of a) shaft-bearing lumped model; b) shaft-bearing layout.

The load-displacement relation at the effective lumped support point can be derived by

[∆푆퐵]5×5 −1 [퐹푆퐵]5×5 { } = [퐾퐹퐸]푛×푛 { } (6.23) [∆표푡ℎ푒푟] 5×푛 0 5×푛 where [∆푆퐵] is the displacement in the five degrees of freedom (DOF) of the node at the lumped stiffness point, [∆표푡ℎ푒푟] is the other DOF displacement, [퐾퐹퐸] is the assembly stiffness matrix of the beam finite element mode, [퐹푆퐵] is a matrix formed by five sets of forcing vectors acting on the reference node. The effective lumped stiffness matrix can be determined by

−1 [퐾푆퐵]5×5 = [퐹푆퐵][∆푆퐵] (6.24)

The full lumped stiffness matrix in Eq. (15) can be assembled by

[퐾1] = 퐷푖푎푔[∙∙∙ [퐾푆퐵] ∙∙∙] + 퐷푖푎푔[∙∙∙ [퐾푚] ∙∙∙] + 퐷푖푎푔 [[퐾푡푝] ∙∙∙ [퐾푡푔]] (6.25)

where [퐾푚] is the gear mesh coupling stiffness matrix, [퐾푡푝] is the coupling stiffness matrix of the torsional spring that connects the pinon and the engine, and [퐾푡푔] is the the coupling stiffness matrix of the torsional spring that connects the gear and the load.

6.2.5 Coupled Driveline System Model

Two transverse springs and damping elements along x and z-axis are assumed to represent the universal joint that connects the propeller shaft to the pinion input shaft. Extra moments are generated by the distance L between pinion shaft end and its mass center, as shown in Figure 6.5.

Figure 6.5 Schematic of propeller shaft and pinion coupling.

Therefore, the coupling transverse forces can be calculated as

퐹푥 = 퐾푐표푢푝(푥푠 − 푥푝 − 퐿 ∗ 휃푧) (6.26a)

퐹푧 = 퐾푐표푢푝(푧푠 − 푧푝 − 퐿 ∗ 휃푥) (4.26b)

The moments induced by the transverse forces can be calculated as

푀푥 = 퐹푧퐿, 푀푧 = 퐹푥퐿 (6.27)

These forces and moments are used in the subsequent derivation of the equations of motion of the full geared-rotor system.

The 14 equations of motion for the geared rotor system can be written as:

퐼퐸휃̈퐷 + 푐푝푦푟(휃̇퐷 − 휃푝푦̇ ) + 푘푝푦푟(휗퐷 − 휗푝푦) = 푇퐷 (6.28)

푚 푥̈ + 푐 푥̇ + 푘 푥 = −푛 퐹 − 퐹 (6.29) 푝 푝 푝푥푡 푝 푝푥푡 푝 푝푥 푚 푥

푚 푦̈ + 푐 푦̇ + 푘 푦 = −푛 퐹 (6.30) 푝 푝 푝푦푡 푝 푝푦푡 푝 푝푦 푚

푚 푧̈ + 푐 푧̇ + 푘 푧 = −푛 퐹 − 퐹 (6.31) 푝 푝 푝푧푡 푝 푝푧푡 푝 푝푧 푚 푧

퐼 휃̈ + 푐 휃̇ + 푘 휗 − 퐼 휃̇ 휃̇ = −휆 퐹 + 푀 (6.32) 푝푥 푝푥 푝푥푟 푝푥 푝푥푟 푝푥 푝푦 푝푧 푝푦 푝푥 푚 푥

퐼 휃̈ + 푐 (휃̇ − 휃̇ ) + 푘 (휗 − 휗 ) = −휆 퐹 (6.33) 푝푦 푝푦 푝푦푟 푝푦 퐸 푝푦푟 푝푦 퐸 푝푦 푚

퐼 휃̈ + 푐 휃̇ + 푘 휗 + 퐼 휃̇ 휃̇ = −휆 퐹 + 푀 (6.34) 푝푧 푝푧 푝푧푟 푝푧 푝푧푟 푝푧 푝푦 푝푥 푝푦 푝푧 푚 푧

푚 푥̈ + 푐 푥̇ + 푘 푥 = 푛 퐹 (6.35) 푔 푔 푔푥푡 푔 푔푥푡 푔 푔푥 푚

푚 푦̈ + 푐 푦̇ + 푘 푦 = 푛 퐹 (6.36) 푔 푔 푔푦푡 푔 푔푦푡 푔 푔푦 푚

푚 푧̈ + 푐 푧̇ + 푘 푧 = 푛 퐹 (6.37) 푔 푔 푔푧푡 푔 푔푧푡 푔 푔푧 푚

퐼 휃̈ + 푐 휃̇ + 푘 휗 − 퐼 휃̇ 휃̇ = 휆 퐹 (6.38) 푔푥 푔푥 푔푥푟 푔푥 푔푥푟 푔푥 푔푦 푔푧 푔푦 푔푥 푚

퐼 휃̈ + 푐 (휃̇ − 휃̇ ) + 푘 (휗 − 휗 ) = 휆 퐹 (6.39) 푔푦 푔푦 푔푦푟 푔푦 퐿 푔푦푟 푔푦 퐿 푔푦 푚

퐼 휃̈ + 푐 휃̇ + 푘 휗 + 퐼 휃̇ 휃̇ = 휆 퐹 (6.40) 푔푧 푔푧 푔푧푟 푔푧 푔푧푟 푔푧 푔푦 푔푥 푔푦 푔푧 푚

̈ ̇ ̇ 퐼퐿휃퐿 + 푐푔푦푟(휃퐿 − 휃푝푦) + 푘푔푦푟(휗퐿 − 휗푔푦) = 푇퐿 (6.41) which includes the coupling forces and moments.

Finally, the hypoid geared rotor system and propeller shaft model are coupled by incorporating the mass, stiffness and damping matrices. The resultant equations of motion of the complete assembly in the matrix form can be expressed as

푥̈ 1,푖푛푡 [[푀1,푖푛푡][푀1,푐표푢푝]] [0] 푥̈ [ ] 1,푐표푢푝 + 푥̈푠,푖푛푡 [0] [[푀푠,푖푛푡][푀푠,푐표푢푝]] {푥̈푠,푐표푢푝}

푥̇1,푖푛푡 [[퐶 ][퐶 ]] [[0] − [퐶 ]] 1,푖푛푡 1,푐표푢푝 1,푐표푢푝 푥̇1,푐표푢푝 [ 푇 ] + 푥̇푠,푖푛푡 [[0] − [퐶푠,푐표푢푝]] [[퐶ℎ,푖푛푡] [퐶푠,푐표푢푝] + [퐶1,푐표푢푝]] {푥̇푠,푐표푢푝}

푥 푥̇ 1,푖푛푡 1,푖푛푡 [[퐾1,푖푛푡][퐾1,푐표푢푝]] [[0] − [퐾1,푐표푢푝]] 푥1,푐표푢푝 [퐺1] [0] 푥̇1,푐표푢푝 [ 푇 ] { 푥 } + Ω [ ] = 푠,푖푛푡 [0] [퐺s] 푥̇푠,푖푛푡 [[0] − [퐾푠,푐표푢푝]] [[퐾푠,푖푛푡] [퐾푠,푐표푢푝] + [퐾1,푐표푢푝]] 푥 푠,푐표푢푝 {푥̇푠,푐표푢푝}

퐹 1,푖푛푡 퐹 1,푐표푢푝 (6.42) 퐹푠,푖푛푡 {퐹푠,푐표푢푝 } where the subscript ‘int’ represents interior coordinates, and subscript ‘coup’ represents the coupling coordinates.

6.3 Sound Radiation Analysis

The forced vibration and acoustic analysis of the final drive gearbox housing are presented in this section. Finite element (FE) model is used for the forced vibration analysis. The FE model of a gearbox housing for a typical vehicle axle application is shown in Figure 6.6(a). The bearing loads calculated from previous dynamic analyses are applied at the center point of the bearing circles and rigid couplings are assumed between the center point and bearing outer race sleeves.

The frequency response functions (FRFs) are calculated for the frequency up to 2 kHz.

(a) (b) Figure 6.6 Axle housing structure a) finite element model b) boundary element model.

The housing FRFs will be used as input to a boundary element (BE) model to further evaluate the sound radiation from the gearbox housing. The boundary element model of the gearbox housing is shown in Figure 6.6(b). Fast multi-pole BE method is applied to calculate the gearbox sound radiation, which accelerates the computation with a realistic system modeling [40,

69, 70]. To obtain accurate estimate of sound power, 20 numerical microphones are mounted on a sphere surface enveloping the gearbox according to the international standard ISO 3745. The sound power is computed by integrating the sound pressure at these locations in the sphere. The sound power level is calculated as

푠 퐿푤 = 퐿푤 + 10푙표푔10 ( ) + 푐1 + 푐1 ( 6.43) 푠0

퐵 313.14 0.5 푐1 = −10푙표푔10 [ ( ) ] (6.44) 퐵0 273.15+휃

퐵 296.15 0.5 푐2 = −15푙표푔10 [ ( ) ] (6.45) 퐵0 273.15+휃

2 where 푠 is the area of the measurement surface and 푠0 = 1푚 ; the symbol 퐵 is the barometric air pressure during measurements, in Pascals; the symbol 퐵0 is the reference barometric pressure; the

80 symbol 휃 is the air temperature during measurement; and 퐿푝 is the weighed surface sound pressure level over the measurement surface in decibels:

1 퐿 = 10푙표푔 {( [∑ 100.1(퐿푝,푖+푊)])} (6.46) 푝 10 푁 푖

푝푖 퐿푝,푖 = 20푙표푔20( ) (6.47) 푝0 where 푁 is the number of microphone positions and 푊 is the weighting function applied by the

−5 filter at the frequency of analysis. Also, 푝푖 is the root mean square pressure in Pa and 푝0 = 2×10 .

6.4 Results and Discussion

A typical hypoid geared rotor assembly for vehicle axle application is adopted here for case study. The gear design parameters are shown in Table 6.1. The propeller shaft design parameter are shown in Table 6.2. Modal analysis is performed to calculate the modes of the original 14-

DOF geared rotor system model. Applying the coupled driveline system model described by Eq.

(6.30), a linear time-invariant dynamic analysis is performed. Then, the effects of gyroscopic moments and gear-shaft coupling, and the effects of system design parameters, such as pinion moment of inertia, propeller shaft properties and pinion bearing compliance, on gear dynamics and sound radiation are examined.

Table 6.1 Gear Design Parameters and system parameters.

Gear parameters Pinion Gear Number of Teeth 10 43 Spiral Angle [rad] 0.803 0.591 Pitch Angle [rad] 0.295 1.269 Pitch Radius [m] 0.048 0.168 Face Width [kg/m3] 0.0522 0.0478 Type [m] Left hand Right hand Loaded Side Concave Convex System parameters Driver Load Torsional moment of inertia (kg-m2) 2.613 5.16 Pinion Gear

Mass (kg) 24.71 122.67 Torsional moment of inertia (kg-m2) 0.0585 1.91 Bending moment of inertia (kg-m2) 0.496 2.07 Shaft torsional stiffness (Nm/rad) 1.2E4 7.4E4 Shaft-bearing bending stiffness (Nm/rad) 3.1E7 9.8E7 Axial support stiffness (N/m) 1E9 2.3E9 Lateral support stiffness (N/m) 8.8E9 1.3E10 Gear Backlash [mm] 0.1 Mesh damping ratio 0.12 Support structure damping ratio 0.02 Torque Load [Nm] 350 Rotational Speed [r/min] 3000

Table 6.2 Geometry dimensions and material property of the propeller shaft.

Desity Young’s Parameters l (m) r (m) t (m) (kg/m^2) Module (pa)

Section 1 0.4 0.025 0.0016 7850 2.1e11

6.4.1 Modal Analysis

In this case, the bending modes of the condensed propeller shaft selected to be coupled with geared system covers the frequency range up to 4 kHz. The geared rotor system modal analysis is also performed by solving the eigenvalues and eigenvectors of the original 14-DOF lumped parameter dynamic model. The natural frequencies and mode shapes of the baseline case are shown in Figure 6.7, and the modal strain energy distribution is shown in Figure 6.8.

0.6 0.4 2 4 1

0.4 0.2 1 2 0.5 0.2 0 0 0 0 0 -0.2

-0.2 -0.4 -1 -2 -0.5 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 (0.0 Hz) (16.1 Hz) (47.1 Hz) (338.4 Hz) (663.9 Hz) 1 0.5 0.1 0.4 0.3

0.5 0 0 0.2 0.2

0 -0.5 -0.1 0 0.1

-0.5 -1 -0.2 -0.2 0

-1 -1.5 -0.3 -0.4 -0.1 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 Mode 6 Mode 7 Mode 8 Mode 9 Mode 10 (668.3 Hz) (681.2 Hz) (857.1 Hz) (868.7 Hz) (1027.8 Hz) 0.5 0.6 0.4 0.6

0.4 0.2 0.4

0 0.2 0 0.2

0 -0.2 0

-0.5 -0.2 -0.4 -0.2 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 1 2 3 4 5 6 7 8 91011121314 Mode 11 Mode 12 Mode 13 Mode 14 (1327.2 Hz) (1330.3 Hz) (3159.5 Hz) (3173.7 Hz)

Figure 6.7 Modal frequencies and mode shapes of the geared system.

1 1 1 1 1 1 1

0.8 0.8 0.8 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2 0.2 0.2

0 0 0 0 0 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 (0.0 Hz) (16.1 Hz) (47.1 Hz) (338.4 Hz) (663.9 Hz) (668.3 Hz) (681.2 Hz) 1 1 1 1 1 1 1

0.8 0.8 0.8 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2 0.2 0.2

0 0 0 0 0 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Mode 8 Mode 9 Mode 10 Mode 11 Mode 12 Mode 13 Mode 14 (857.1 Hz) (868.7 Hz) (1027.8 Hz) (1327.2 Hz) (1330.3 Hz) (3159.5 Hz) (3173.7 Hz) Figure 6.8 Modal strain energy distribution of the geared system: 1 – pinion translational compliance; 2 – gear translational compliance; 3 – pinion torsional compliance; 4 – gear torsional compliance; 5 – pinion bending compliance; 6 – gear bending compliance; 7 – mesh compliance.

6.4.2 Parametric Studies

6.4.2.1 Effect of Gyroscopic Moments and Shaft Dynamics

The effect of gear gyroscopic moments, propeller shaft flexibility and their interactions are studied numerically for the baseline design. Dynamic models with only gear gyroscopic effect, with only shaft flexibility and with both gear gyroscopic effect and shaft flexibility are compared, and the corresponding dynamic mesh results are shown in Figure 6.9. Dynamic transmission error and mesh force spectrum are the main mesh responses and they are directly linked to gear whine problem. In Figure 6.9(a), it is shown that the gyroscopic effect of the gear bodies has little influence on the dynamic transmission error, and it’s almost overlapped with baseline response.

This is because the gear gyroscopic effect only becomes significant when the rotation speed is large and the ratio of gear bending moment of inertia over the torsional moment of inertia is small.

Shaft bending flexibility is observed to have limited influence on the dynamic transmission error spectrum in the frequency range of pinion bending modes. When both gear gyroscopic effect and shaft flexibility are considered, however, the influence is more significant and additional structural modes are observed in the frequency range between 600Hz and 1000Hz, where the pinion bending modes and 3rd propeller shaft bending mode lie in. Same observation is obtained for the dynamic mesh force comparison in Figure 6.9(b). Neither gear gyroscopic effect nor shaft bending flexibility alone has significant influence on the response, whereas their interaction could cause discrepancy in the peaks around pinion bending resonance frequencies. The influenced frequency range depends on the mode shapes and the overall mesh response is still dominant by out-of-phase gear torsional modes.

The lumped support reaction forces can be calculated by the displacements of gear bodies and lumped support stiffness matrices. Then the dynamic bearing forces at actual bearing location

84 can be calculated through a load equilibrium analysis of shaft-bearing model. The dynamic bearing forces can excite the gearbox housing and cause structure-borne noise. The bearing forces are also compared with only gear gyroscopic effect, with only shaft flexibility and with both gear gyroscopic effect and shaft flexibility. Figure 6.10 shows the bearing forces at pinion bearing location in horizontal, axial and vertical directions. For horizontal and vertical bearing responses, it is observed that the resonance peak corresponding to the pinion bending mode around 670Hz splits into two resonance peaks when both gear gyroscopic effect and shaft flexibility are considered. The splitting of the resonance is typical of rotating systems that have a non-negligible gyroscopic effect [71]. Figure 6.11 shows the bearing forces at gear bearing location in horizontal, axial and vertical directions. Similarly, for horizontal and vertical responses the resonance peak corresponding to the gear bending mode around 870Hz splits into two resonance peaks when both gear gyroscopic effect and shaft flexibility are considered. The discrepancy is not significant when only gear gyroscopic effect alone or only shaft flexibility alone is considered. In addition, for both pinion and gear bearing responses, the axial direction responses are not affected much by the gyroscopic effect and shaft flexibility, this is probably due the fact that gyroscopic effect and shaft flexibility mainly interact with the pinion and gear bending coordinates.

-5 10

DTE (m) DTE -6 10

0 500 1000 1500 2000 Frequency (Hz) (a)

3 10

10 DMF (N) DMF

1 10

0 500 1000 1500 2000 Frequency (Hz) (b) Figure 6.9 Dynamic mesh responses: a) dynamic transmission error spectrum; b) dynamic mesh force spectrum: baseline, only gear gyroscopic effect, only shaft flexibility, with both gyroscopic effect and shaft flexibility.

4 4 10 10

3 10 2 10

2 10