Nonlinear Dynamics of Driveline Systems with Hypoid Gear Pair
A dissertation submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
in the Program of School of Dynamic Systems
of the College of Engineering and Applied Science
April 2012
by
Junyi Yang
M. S. Southeast University, Nanjing, P. R. China, 2007
B. S. Southeast University, Nanjing, P. R. China, 2004
Academic Committee Chair: Dr. Teik C. Lim
Members: Dr. Dong Qian
Dr. David F. Thompson
Dr. Murali M. Sundaram
ABSTRACT
This dissertation research focuses on evaluating the nonlinear dynamics of driveline systems employed in motor vehicles with emphasis on characterizing the excitations and response of right-angle, precision hypoid-type geared rotor structure. The main work and contribution of this dissertation is divided into three sections. Firstly, the development of an asymmetric and nonlinear gear mesh coupling model will be discussed. Secondly, the enhancement of the multi-term harmonic balance method (HBM) is presented. Thirdly and as the final topic, the development of new dynamic models capable of evaluating the dynamic coupling characteristics between the gear mesh and other driveline structures will be addressed.
A new asymmetric and nonlinear mesh model will be proposed that considers backlash, and the fact that the tooth surfaces of the convex and concave sides are different. The proposed mesh model will then be fed into a dynamic model of the right-angle gear pair to formulate the dimensionless equation of motion of the dynamic model. The multi-term HBM will be enhanced to simulate the right-angle gear dynamics by solving the resultant dimensionless equation of motion. The accuracy of the enhanced HBM solution will be verified by comparison of its results to the more computationally intensive direct numerical integration calculations. The stability of both the primary and sub-harmonic solutions predicted by applying multi-term HBM will be analyzed using the Floquent Theory. In addition, the stability analysis of the multi-term HBM solutions will be proposed as an approximate approach for locating the existence of sub- harmonic and chaotic motions.
In this dissertation research, a new methodology to evaluate the dynamic interaction between the nonlinear hypoid gear mesh mechanism and the time-varying characteristics of the rolling element bearings will also be developed. The time-varying mesh parameters will be
II
obtained by synthesizing a 3-dimensional loaded tooth contact analysis (TCA) results. The time- varying stiffness matrix approach will be used to represent the dynamic characteristics of the rolling element bearings.
An overall nonlinear dynamic model of the hypoid gear box considering elastic housing
structure will be developed as well. A lumped parameter model of the flexible housing will be
extracted form an appropriate set of frequency response functions through modal parameter identification method. In order to obtain the rotational coordinates, a rigid body interpolation of
the translational responses at the bearing locations on the housing structure will be applied. The
reduced model will be then coupled with the hypoid gear-shaft-bearing assembly model by
applying a proposed dynamic coupling procedure.
Finally, a hypoid geared rotor system model considering the propeller shaft flexibility
will be established. The propeller shaft bending flexibility will be modeled as lumped parameter
model through using the component mode synthesis (CMS). The torsional flexibility of propeller
shaft will be simplified as a torsional spring connecting the inertia of moment of engine and
pinion. Physically, the pinion input shaft is driven by the propeller shaft through a universal
joint, which will be modeled as a flexible simple supported boundary condition as well as
fluctuating rotation speed and torque excitation.
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IV
ACKNOWLEDGEMENTS
I would like to thank Prof. Teik C. Lim, who is serving as my academic advisor and the chair of my academic committee, for his great instructions and support throughout my graduate study. I would also like to thank Dr. Dong Qian, Dr. David F. Thompson and Dr. Murali M.
Sundaram for serving as my doctoral academic committee members.
I wish to thank all my colleagues at the Vibro-Acoustic and Sound Quality Research
Laboratory in University of Cincinnati for their friendship. I would like to express my special gratitude to Dr. Tao Peng for his valuable academic suggestions.
Finally, I would like to thank my parents and my wife Si Chen for their support and patience during my graduate study.
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TABLE OF CONTENTS
ABSTRACT II
ACKNOWLEDGEMENTS ...... V
TABLE OF CONTENTS ...... VI
LIST OF TABLES ...... X
LIST OF FIGURES ...... XI
LIST OF SYMBOLS ...... XVIII
Chapter 1. Introduction ...... 1
1.1 Literature Review ...... 1
1.2 Scope and Objectives ...... 6
1.3 Organization ...... 8
Chapter 2. A Review of Loaded Tooth Contact Analysis Approaches ...... 12
2.1 Introduction ...... 12
2.2 Gear Body Flexible Deflection ...... 14
2.3 Gear Tooth Flexible Deflection ...... 17
2.3.1 Cantilever Beam ...... 17
2.3.2 Cantilever Plate ...... 18
2.3.3 Shell ...... 21
2.3.4 FEM Method ...... 23
2.3.5 Finite Strip Method...... 26
2.3.6 Finite Prism Method ...... 27
2.3.7 Combined Surface Integral and Finite Element Method ...... 27
2.4 Conclusion ...... 29 VI
Chapter 3. An Enhanced Multi-term Harmonic Balance Solution for Non-linear Period-one
Dynamic Motions in Right-angle Gear Pairs ...... 31
3.1 Introduction ...... 31
3.2 Mathematical Model ...... 34
3.3 Period-one Dynamics ...... 42
3.4 Parametric Studies ...... 46
3.4.1 Numerical Validation ...... 47
3.4.2 Numerical Analysis ...... 50
3.5 Conclusion ...... 58
Chapter 4. An Enhanced Multi-term Harmonic Balance Solution for Non-linear Period-
Dynamic Motions in Right-angle Gear Pairs ...... 59
4.1 Introduction ...... 59
4.2 Period- Sub-harmonic Response ...... 62
4.3 Results and Discussion ...... 65
4.3.1 Comparison of HBM and numerical integration results ...... 66
4.3.2 Effect of Static Load ...... 75
4.3.3 Effect of Static Transmission Error Excitation ...... 80
4.3.4 Sub-harmonic and Chaotic Motions ...... 83
4.4 Conclusion ...... 84
Chapter 5. Dynamics of Coupled Nonlinear Hypoid Gear Mesh and Time-varying Bearing
Stiffness Systems ...... 86
5.1 Introduction ...... 86
5.2 Nonlinear Geared Rotor System Model ...... 88
VII
5.2.1 Basic Assumptions ...... 88
5.2.2 Mesh Model ...... 89
5.2.3 Shaft-bearing Assembly Model ...... 92
5.2.4 Time-varying Bearing Stiffness ...... 93
5.2.5 Formulation ...... 95
5.3 Case Study ...... 97
5.3.1 Mesh and Bearing Interaction ...... 99
5.3.2 Backlash and Bearing Interaction ...... 100
5.4 Summary ...... 104
Chapter 6. Nonlinear Dynamic Simulation of Hypoid Gearbox with Elastic Housing ...... 105
6.1 Introduction ...... 105
6.2 Geared Rotor System Dynamic Model ...... 107
6.3 Lumped Parameter Model of Housing ...... 109
6.4 Coupled Dynamic Model ...... 115
6.5 Case Study ...... 116
6.5.1 Methodology Validation ...... 118
6.5.2 Parametric Study ...... 120
6.6 Conclusion ...... 125
Chapter 7. Propeller Shaft Bending and Effect on Gear Dynamics ...... 126
7.1 Introduction ...... 126
7.2 Mathematical Model ...... 128
7.2.1 Simplified Propeller Shaft Model ...... 128
7.2.2 Geared Rotor System Model ...... 130
VIII
7.2.3 Coupled Boundary Condition ...... 133
7.3 Numerical Results and Discussion ...... 134
7.4 Conclusion ...... 141
Chapter 8. Conclusions and Proposed Future Studies ...... 143
8.1 Conclusions ...... 143
8.2 Proposed Future Studies ...... 145
BIBLIOGRAPHY ...... 147
IX
LIST OF TABLES
Table 3.1, Dimensionless dynamic parameters for a typical automotive hypoid gear pair ...... 47
Table 3.2, Physical parameters of a real application hypoid gear pair ...... 48
Table 4.1, Dimensionless dynamic parameters for a light load automotive hypoid gear pair ...... 66
Table 5.1, Design parameters: (a) Gear data, (b) System data, (c) Data of bearing on pinion shaft,
(d) Data of bearing on gear shaft ...... 97
Table 6.1, Design parameter: (a) Gear data, (b) System data, (c) Data of bearing on pinion shaft,
(d) Data of bearing on gear shaft ...... 117
Table 6.2, Stiffness and damping of the analytical model ...... 119
Table 7.1, Geometry dimension of propeller shaft ...... 134
Table 7.2, Design parameter: (a) Gear data, (b) System data, (c) Data of bearing on pinion shaft,
(d) Data of bearing on gear shaft ...... 134
X
LIST OF FIGURES
Fig. 2.1, Linear normal stress distribution (Weber, 1949) ...... 14
Fig. 2.2, Schematic of test rig (O’Donnell, 1960) ...... 15
Fig. 2.3, Finite element model of gear tooth (Stegemiller and Houser, 1993) ...... 17
Fig. 2.4, Cantilever beam model of gear tooth (Weber, 1949) ...... 18
Fig. 2.5, Cantilever plate model of gear tooth (MacGregor, 1936) ...... 18
Fig. 2.6, Moment imaging: (a) plate of infinite length, (b) plate of finite length (Wellauer et al.,
1960) ...... 19
Fig. 2.7, Tapered cantilever plate model of gear tooth (Yau, et al., 1994) ...... 20
Fig. 2.8, Annual sector plate model of gear tooth (Vaidyanathan, et al., 1994) ...... 21
Fig. 2.9, Thick cylinder circular shell model of gear tooth (Vaidyanathan, et al. 1993) ...... 22
Fig. 2.10, Discretization of instant contact line (Wilcox, 1981) ...... 24
Fig. 2.11, Finite mesh of gear tooth (Gosselin, et al., 1995)...... 26
Fig. 2.12, Finite strip model of gear tooth (Gagnon, et al., 1996) ...... 26
Fig. 2.13, Finite prism element (Vijayakar and etc., 1987) ...... 28
Fig. 2.14, Two-dimensional formulation of finite strip element (Vijayakar and etc., 1989) ...... 28
Fig. 3.1, (a) Two degrees of freedom torsional vibration model of a hypoid gear pair, (b) Pinion
and gear coordinate systems ...... 35
Fig. 3.2, Asymmetric nature of hypoid gear mesh stiffness: (a) Drive side (b) Coast side ...... 38
Fig. 3.3, Comparison of multi-term HBM and numerical integration: (a) RMS of the dynamic
displacement. (b) Mean value of the dynamic displacement. Stable steady-state
solution by multi-term HBM; Unstable steady-state solution by multi-terms HBM; ○
Solutions by numerical integration ...... 49
XI
Fig. 3.4, Effect of the directional rotation radii variation on the dynamic response: (a) RMS of
the dynamic displacement. (b) Primary resonance of the dynamic displacement.
Stable steady-state solution by multi-term HBM; Unstable steady-state solution by
multi-term HBM ...... 52
Fig. 3.5, Effect of directional rotation radii asymmetric nonlinearity on the dynamic response: (a)
RMS of the dynamic displacement. (b) Primary resonance of the dynamic displacement.
Stable steady-state solution by multi-term HBM; Unstable steady-state
solution by multi-term HBM ...... 53
Fig. 3.6, Effect of the mesh stiffness variation on the dynamic response: (a) RMS of the dynamic
displacement. (b) Primary resonance of the dynamic displacement. Stable steady-
state solution by multi-term HBM; Unstable steady-state solution by multi-term
HBM ...... 55
Fig. 3.7, Effect of the mesh stiffness asymmetric nonlinearity on the dynamic response: (a)
RMS of the dynamic displacement. (b) Primary resonance of the dynamic displacement.
Stable steady-state solution by multi-term HBM; Unstable steady-state
solution by multi-term HBM ...... 57
Fig. 4.1, RMS of dynamic transmission error. Stable steady solution by multi-terms
HBM; Unstable steady solution by multi-terms HBM; ○ Solutions by Numerical
Integration ...... 67
Fig. 4.2, Bifurcation diagram of baseline case (frequency sweep up) ...... 68
Fig. 4.3, a) Period 1 HBM solutions Stable steady solution; Unstable steady
solution; Δ Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d)
Trajectory of NI result ...... 69
XII
Fig. 4.4, a) Period 2 HBM solutions Stable steady solution; Unstable steady
solution; Δ Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d)
Trajectory of NI result ...... 71
Fig. 4.5, a) Period 3 HBM solutions Stable steady solution; Unstable steady
solution; Δ Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d)
Trajectory of NI result ...... 72
Fig. 4.6, a) Period 2 HBM solutions Stable steady solution; Unstable steady
solution; Δ Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d)
Trajectory of NI result ...... 73
Fig. 4.7, a) Period 3 HBM solutions Stable steady solution; Unstable steady
solution; Δ Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d)
Trajectory of NI result ...... 74 ~ ~ Fig. 4.8, a) Period 2 HBM solutions (Tp 0. 3 ); b) Period 2 HBM solutions (Tp 0. 5 ); c)
~ ~ Bifurcation diagram (Tp 0. 3 ); d) Bifurcation diagram (Tp 0. 5 ); Stable steady
solution; Unstable steady solution ...... 77
Fig. 4.9, a) Trajectory on phase plane (~ 1.33 ); b) Poincare plot (~ 1.33 ); c) Trajectory on
phase plane (~ 1.45 ); d) Poincare plot (~ 1.45 ); e) Trajectory on phase plane
(~ 1.51 ); f) Poincare plot (~ 1.51 );g) Trajectory on phase plane (~ 1.69 ); h) Poincare
plot (~ 1.69 ); i) Trajectory on phase plane (~ 1.95 ); j) Poincare plot (~ 1.95 ) ...... 79 ~ ~ Fig. 4.10, a) Period 2 HBM solutions (e3 0. 225 ); b) Period 2 HBM solutions (e3 0. 115 );
~ ~ c) Bifurcation diagram (e3 0. 225 ); d) Bifurcation diagram (e3 0. 115 ); Stable
steady solution; Unstable steady solution ...... 81
XIII
Fig. 4.11, a) Trajectory on phase plane (~ 1.1); b) Poincare plot (~ 1.1); c) Trajectory on
phase plane (~ 1.38 ); d) Poincare plot (~ 1.38 ); e) Trajectory on phase plane
(~ 1.91 ); f) Poincare plot (~ 1.91 ) ...... 83
Fig. 4.12, a) Existence spaces of sub-harmonic motions; b) Existence spaces of chaotic motions;
Ο start point; + end point ...... 84
Fig. 4.13, a) Existence spaces of sub-harmonic motions; b) Existence spaces of chaotic motions;
Ο start point; + end point ...... 84
Fig. 5.1, Schematic of a 14-DOF nonlinear dynamic model of hypoid geared rotor system...... 89
Fig. 5.2, Loaded tooth contact analysis model: (a) Gear pair geometry, and (b) Contact cells on
engaging tooth surface...... 90
Fig. 5.3, Shaft bearing assembly: (a) Physical structure, (b) Beam finite element representation.
...... 92
Fig. 5.4, Rolling element bearing kinematics and the corresponding coordinate systems...... 93
Fig. 5.5, Shaft-bearing assembly: (a) Pinion, (b) Gear...... 97
Fig. 5.6, Time-varying bearing stiffness ...... 98
Fig. 5.7, Time-varying meshes stiffness ...... 99
Fig. 5.8, Dynamic mesh force comparison: time-invariant bearing stiffness, time-
varying bearing stiffness...... 100
Fig. 5.9, Comparison of bearing axial reaction force: time-invariant bearing stiffness;
time-varying bearing stiffness...... 100
Fig. 5.10, Dynamic mesh force: frequency sweep up; frequency sweep down.
...... 101
XIV
Fig. 5.11, Dynamic mesh force comparison: time-invariant bearing stiffness;
time-varying bearing stiffness ...... 102
Fig. 5. 12, Comparison of bearing radial reaction force: time-invariant bearing stiffness;
time-varying bearing stiffness ...... 102
Fig. 5.13, Dynamic mesh force comparison: with backlash; without backlash. 103
Fig. 5. 14, Comparison of bearing reaction moment: with backlash; without
backlash...... 103
Fig. 6.1, Dynamic Model of hypoid geared rotor assembly ...... 107
Fig. 6.2, a) Housing continuous structure; b) Excitation and response points ...... 110
Fig. 6.3, Frequency response function of housing structure ...... 111
Fig. 6.4, Poles estimation from FRFs ...... 112
Fig. 6.5, Plot of Enhanced frequency response function; and Single-DOF
frequency response function fit ...... 113
Fig. 6.6, Illustration of 2-D rigid body interpolation ...... 113
Fig. 6.7, Shaft-bearing assembly: (a) Pinion, (b) Gear ...... 117
Fig. 6.8, Lumped parameter model representing a housing structure ...... 118
Fig. 6.9, Comparison of dynamic mesh force: Full coupling; Partial coupling;
No coupling...... 120
Fig. 6.10, Acceleration of mass block 1: Full coupling; Partial coupling ...... 120
Fig. 6.11, Comparison of dynamic mesh force: 10-modes coupling; 5-modes
coupling; No coupling...... 121
Fig. 6.12, Comparison of surface acceleration: 10-modes coupling; 5-modes
coupling ...... 122
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Fig. 6.13, Comparison of dynamic mesh force: No external excitation; Excitation
exerted on point 1; Excitation exerted on point 2...... 123
Fig. 6.14, FFT of dynamic mesh force with external excitation on point 1...... 124
Fig. 6.15, FFT of dynamic mesh force with external excitation on point 2...... 124
Fig. 6.16, Comparison of surface acceleration: No external excitation; Excitation
exerted on point 1; Excitation exerted on point 2 ...... 125
Fig. 7.1, Simplified propeller shaft (a) beam with different cross-section, (b) Cross-section
dimension definition ...... 128
Fig. 7.2, (a) Static system with unit forces applied at interface, (b) Dynamic system with fixed
interface boundary condition ...... 130
Fig. 7.3, Lumped parameter model of hypoid geared rotor system ...... 130
Fig. 7.4, Schematic of propeller shaft and pinion coupling ...... 133
Fig. 7.5, Shaft-bearing assembly: (a) Pinion, (b) Gear ...... 134
Fig. 7.6, Comparison of dynamic mesh force: Rigid Propeller Shaft; Propeller
Shaft Bending (L=0.1); Propeller Shaft Bending (L=0.15) ...... 136
Fig. 7.7, Modal shape of the linear model ...... 136
Fig. 7.8, Comparison of reaction forces on pinion bearing: Rigid Propeller Shaft;
Propeller Shaft Bending (L=0.1) (a) radial force, (b) axial force, (c) moment ...... 138
Fig. 7.9, Comparison of reaction forces on gear bearing: Rigid Propeller Shaft;
Propeller Shaft Bending (L=0.1) (a) radial force, (b) axial force, (c) moment ...... 139
Fig. 7.10, Effect of coupling stiffness: (a) comparison of dynamic mesh force, (b) affected
resonance Baseline Coupling Stiffness ( ); Low Coupling
Stiffness ( ); High Coupling Stiffness ( )...... 140
XVI
Fig. 7.11, Effect of coupling damping: (a) comparison of dynamic mesh force, (b) affected
resonance Baseline Coupling Damping ( ); High Coupling
Damping ( ); High Coupling Damping ( ) ...... 140
Fig. 7.12, Comparison of dynamic mesh force: Baseline Modal Damping Ratio (
. ); Low Modal Damping Ratio ( . ; High Modal Damping
Ratio ( . )...... 141
Fig. 8.1, Potential test configuration ...... 146
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LIST OF SYMBOLS
[A ], [A ], [B ], 0 1 0 unknown coefficient matrices of FRF matrix bc gear backlash cc , cd , coast/drive side mesh damping coefficient c , ccoup , mod propeller coupling/modal damping ratio [C] system damping matrix e, e0 loaded/unloaded static transmission error f ( e) non-linear displacement F equivalent gear contact force vector
{Fbxm , Fbym , Fbzm , M xm , M ym } bearing force vector h arc-length h p , hg , directional rotation radius vector of pinion and gear [H()], [eH()] FRF matrix/ enhanced FRF matrix
I p , I g pinion and gear inertias jl unit vector along shaft axis J Jacobian matrix kc , kd , coast/drive side mesh stiffness kcoup propeller shaft coupling stiffness
[k p ], [k g ] pinion/gear supporting stiffness matrices ke , kl , torsional stiffness of input/output shafts [K] system stiffness matrix L pinion end to mass center distance
me , m p , mg equivalent, pinion and gear mass [M ] mass matrix nlm line of action unit vector q modal displacement rlm mesh point r mean value ratio r , r b d roller and bearing inner raceway curvature radius
XVIII
sl pinion/gear coordinate system t time
T p ,Tg load on pinion and gear x – e x, y, z perturbation/displacement of x, y and z y perturbation state vector
0 pressure angle dynamic transmission error
{ xm , ym , zm , xm , ym } bearing displacement vector
lm directional rotation radius frequency damping ratio rotational displacement
[], [m ] modal shape/modified modal shape
z mean shaft speed
s t roller orbit position
Subscripts c coast side coup coupling d drive side da damping ext external int internal k label for stiffness l pinion (l p ) and gear (l g ) m drive (m=d) and coast (m=c) sides n natural mode pr propeller shaft x, y, z translational coordinates
XIX
1 mean value 2,3... alternative value
Superscripts • derivative w.r.t. time ′ derivative w.r.t. dimensionless time → vector quantities ~ dimensionless quantities
XX
Chapter 1. Introduction
Hypoid and spiral bevel gear drives are widely used in the axle and transmission of many
types of vehicles and machinery, including automotive, off-road vehicle, helicopter and other
gear reducers. This is a result of their ability to transfer power through non-intersecting and non-
parallel shafts. However, vibration often generates annoying noise, affects the mechanical gear
mesh efficiency, and reduces the fatigue life of driveline components, particularly the gears and bearings. The vibration is actually the structural dynamic response excited by the kinematic
transmission error and the time-varying gear mesh characteristics. Therefore, optimizing the
kinematic transmission error as well as systematic research on the dynamics of the gear pair and
the geared rotor system are required in order to design a more silent, efficient and durable
transmission. Based on a review of published literature, few publications have been made on the
topic of right-angle geared rotor system dynamics. Most of the previous studies are limited to
inefficient numerical or analytical evaluation of the vibration, with insufficient consideration of the coupling effect between the gear mesh and other driveline components. Thus, the goal of this dissertation is to evaluate the nonlinear dynamics of the right-angle gear pair efficiently through enhancing the multi-term HBM method, to characterize the coupling between the nonlinear gear mesh and the dynamic characteristics of other driveline components. Particular attention is paid to the time-varying dynamic attributes of the rolling element bearings, the flexibility of the housing structure and the elasticity of the rotating propeller shaft.
1.1 Literature Review
From a survey of public domain literature, there are generally two approaches to control
the vibration of the hypoid/bevel geared rotor systems. The first one is to design a favorable
shape of the kinematic transmission error and to reduce its magnitude (Litvin, F. L. and Zhang,
1
Y., 1991; Lewicki et al., 1994, 2007), considering that the kinematic transmission error is widely
accepted as the root cause of geared rotor system vibration and noise issues. The other one is to
tune the sensitivity of the vibration response to the kinematic transmission error excitation in the frequency ranges of concern (Remmers, 1971; Abe and Hagiwara, 1990; Hirasaka, Sugita and
Asai, 1991; Cheng, 2000; Peng, 2011). Besides vibration and noise control, there are other potential research topics such as gear tooth wear, mechanism of lapping process, gear tooth contact and bending fatigue, gear tooth health monitoring, gear tooth lubrication, and efficiency
which are not reviewed in this dissertation.
The kinematic transmission error can be obtained through TCA, since the gear tooth
geometry is the basis of the TCA. Currently, there are two cutting processes of manufacturing
hypoid and spiral bevel gears: face-hobbing and face-milling. The method of simulating different
cutting processes to generate the gear tooth surface was developed by many researchers, such as
for face-hobbing gears (Litvin et al. 1991; Fan 2006; Vimercati 2007) and for face-milling gears
(Litvin et al., 1995; Fan, 2007). The manufactured tooth surface may deviate from the theoretical
tooth geometry due to the machine dynamics, temperature, heat treatment and lapping. The gear
tooth topology of the manufactured gear tooth can be measured by the Coordinate Measuring
Machine (CMM). Many publications focused on minimizing the gear tooth deviation by
adjusting machine tool settings (Litvin et al., 1993; Lin, Tsay and Fong, 1998, 2001; Shih and
Fong, 2007, 2008; Fan, 2010). In some publications (Gosselin et al., 1998; Artoni et al., 2008),
useful algorithms were developed to determine the theoretical machine settings from the
measured discrete tooth coordinates. In other cases (Gosselin, 1991; Kin, 1992; Zhang, 1994),
the CMM was first applied to measure the deviations of the manufactured gear tooth geometry
2
from the theoretical one at several predetermined discrete points, and then the real tooth surface
was represented by the theoretical tooth surface plus the interpolated deviations.
Without considering the elasticity of the gear tooth, the unloaded TCA was performed
based on the obtained gear tooth geometry to predict the kinematic transmission error and
contact bearing by many researchers (Zhang, Litvin et al., 1994; Litvin, F. L., Chen, N. X. and
Chen, J.-S., (1995); Vogel et al., 2002; Fan, 2006, 2007; Shih et al., 2007). The accuracy of the
unloaded TCA was demonstrated by correlating the predicted and measured kinematic TE and
the contact pattern (Fuentes et al., 2002; Kawasaki and Tsuji, 2010).
In order to reduce the sensitivity of the kinematic TE and the contact bearing to
misalignment, the gear tooth flank is often modified by properly adjusting machine settings.
Typical modifications include profile changes, longitudinal adjustment, and flank torsion
modification. The unloaded TCA together with the Local Synthesis method were applied to
modify the gear tooth flank in order to obtain a parabolic shape kinematic TE and a desired
oriented contact eclipse by Litvin and his colleagues (Litvin and Zhang, 1991; Litvin, 1995,
2004; Litvin et al., 1996). This approach was developed based on the traditional cradle style
hypoid and spiral bevel gear generator. The development of the computer numerical control
(CNC) hypoid generating machine makes it possible to cut hypoid and spiral bevel gears in free
form. With the help of the CNC, a fourth-order motion curve to reduce the running noise of the
hypoid and spiral bevel gear set was implemented (Stadtfeld, 2000; Stadtfeld and Gaiser, 2000;
Wang and Fong, 2006). The transformation of the machine settings between traditional cradle
style gear generator and the modern CNC one was outlined by Goldrich (1989). Some
researchers also tried to modify the tooth flank by using an ease-off based method (Shih and
Fong, 2007; Artoni, Kolivand and Kahraman, 2010; Shih, 2010). Besides adjusting the machine
3
settings, applying different cutter profiles was also found to be a feasible way to generate
favorable tooth flank (Gosselin, Masseth, and Liang, 2003; Litvin, Fuentes, and Hayasaka,
2006).
Considering the elasticity of the gear tooth, the loaded TCA can be performed to obtain
the loaded TE, the contact pattern as well as the contact and bending stresses. Several different
loaded TCA approaches were developed by many researchers, such as the FEM-based method
(Bibel et al., 1995; Vilmos, 2000; Fuentes et al., 2002; Litvin, Fuentes and Hayasaka, 2006), the
simplified shear and bending flexibility representation based formulation (Gosselin et al., 2000),
the semi-analytical and semi-FEM approach (Vijayakar, 1991; Piazza and Vimercati, 2007), and
the ease-off topography and shell theory based procedure (Kolivand and Kahramna, 2009, 2010).
The testing and simulation results were correlated to show the accuracy of some of the loaded
TCA methods (Handschuh and Bibel, 1999; Gosselin et al, 2000; Litvin et al., 2006).
Some interesting experimental studies indicated that the vibration and noise of an axle is affected by the dynamics of the geared rotor system. The existence of a pinion resonance was predicted as well as experimentally proven by Remmers (1971). In that study, the existence of the pinion resonance was predicted through the use of a single degree of freedom (DOF) mass spring model with infinite mesh stiffness. Abe and Hagiwara (1990) showed experimentally that
axle gear noise can be reduced by adding an inertia disk at either side flange of the final drive or
decreasing the pinion shaft diameter and stiffness. Hirasaka, Sugita and Asai (1991) conducted
experimental studies of the body and driveline sensitivity to the unit transmission error of an axle
hypoid gear pair for gear noise, and found that the estimated dynamic mesh force of an engaging
gear pair was affected significantly by the torsional vibration characteristics of the driveline
system. Considering the effects of all the driveline components, Brecher, Gorgels, Hesse and
4
Hellmann (2011) proposed a new methodology to measure the driveline level dynamic
transmission error, and correlated the dynamic transmission error with the driveline noise. Yoon,
Choi, Yang and Oh (2011) developed a tri-axial transmission error measurement procedure to
identify the gear whine noise for an automotive rear axle with assembly errors.
A mathematical framework for analyzing both linear and nonlinear pure vibration characteristics of hypoid and spiral bevel geared rotor systems was developed (Lim and Cheng,
1999; Cheng, 2000). A single-point mesh model comprised of mesh stiffness, mesh point, and line of action, was formulated to represent the dynamic coupling between the engaging gear pair.
Due to the curvilinear feature of gear tooth surface and kinematics, the dynamic coupling between engaging gear pairs is time-varying. Backlash was also modeled to represent the clearance of the working gear pair. Wang (2002) further developed a multi-point mesh model.
Instead of using a single mesh stiffness coupling for the whole gear pair, a multi-point mesh representation is aimed at constructing an individual mesh stiffness coupling for each tooth pair
that is in contact. As a result, the dynamic mesh force at each individual tooth pair can be
obtained. Jiang (2002) derived a 2 DOF NLTV dynamic model for the hypoid gear pair
considering only torsional coordinates and investigated the dynamic response through the
application of both numerical integration and the HBM. Wang (2007) further examined
nonlinearities of the hypoid gear pair considering the asymmetric mesh stiffness, backlash and
time-varying parametric excitation. More recently, a coupled multi-body dynamics and vibration
model was developed by Peng (2010). In that formulation, the gear perturbation responses are
analyzed simultaneously with the rigid body rotation of the gear pair. This requires the mesh
model be defined as a function of gear angular position rather than strictly time. Thus, the
dynamic load of the teeth in contact for practical load and operating cases can be simulated. The
5
validation of the methodology was partially demonstrated by correlating the calculated dynamic
mesh force and measured noise of a relatively noisy truck carrier component (Cheng, 2000), and by correlating the calculated and indirectly measured dynamic mesh force of an off-road vehicle differential box (Peng, 2010).
1.2 Scope and Objectives
From the above literature review, relatively few publications studying the dynamics of
hypoid and spiral bevel geared rotor systems can be found, compared with the extensive research
about the optimization of gear tooth flank for vibration and noise concern. Most of the previous
studies are limited to inefficient numerical or analytical evaluation of the vibration as well as
insufficient consideration of coupling effect between nonlinear gear mesh and the dynamic
characteristics of other driveline components.
The main objectives of this dissertation are to evaluate the nonlinear dynamics of the
right-angle gear pair efficiently through enhancing the multi-term HBM method, to evaluate the
coupling between the nonlinear gear mesh and the dynamic characteristics of other driveline
components, such as the time-varying dynamic attributes of rolling element bearings, the
flexibility of the housing structure and the elasticity of the rotating propeller shaft. A more
accurate gear mesh model is developed by taking into account that the gear tooth geometry of
concave and convex sides is different. Considering the backlash and asymmetric mesh
nonlinearity, a complete nonlinear dynamic model for right-angle gear pairs is proposed. In the
proposed formulation, the mesh stiffness directional rotation radius and mesh damping are
assumed to be time-varying and asymmetric. The multi-term HBM coupled with a modified
discrete Fourier Transform (DFT) process and the numerical continuation method are applied to
solve the dimensionless equation of motion for dynamic displacement. A throughout study,
6
including primary and sub-harmonic motions, of a practical applied hypoid gear pair is performed to fill the gap that no right-angle gear pair dynamics is evaluated systematically using
analytical method. A time-varying bearing stiffness matrix representation is coupled with the nonlinear gear mesh model to study the dynamic interaction between non-linear hypoid gear mesh and time-varying bearing model, which fills the gap that there is no prior publication directly dealing with this issue. An overall nonlinear dynamic model of gearbox is developed with a lumped parameter representation of housing extracted from an appropriate set of FRFs.
This new model significantly reduces the computational costs as compared with traditional FEM, while a good accuracy of the prediction is maintained. A lumped parameter model of propeller shaft is obtained using CMS, and it is further coupled with geared rotor system to evaluate the dynamics of the driveline system. Parametric studies indicate that the increasing of propeller shaft damping can reduce the dynamic mesh force between engaging gear pair, which provides a potential approach to attenuate driveline vibration and noise.
a) Review loaded tooth contact analysis approaches. A mesh model to represent the dynamic coupling between engaging gear pair will be extracted from the results of loaded tooth contact analysis. So the accuracy of the loaded tooth contact analysis will be critical. Based on the review, a combined semi analytical and three-dimension finite element method is found to be the most suitable approach for this dissertation.
b) Build a new mesh model of the right-angle gear pair characterized with backlash nonlinearity as well as asymmetric and time-varying mesh coupling. Formulate the dimensionless equation of motion of the right-angle gear pair dynamic model. Enhance and formulate the multi-term HBM to make it easier for right-angle gear dynamics applications.
Study period-one, sub-harmonic and chaotic motions of a hypoid gear pair numerically and
7 analytically. Determine the frequency range where the sub-harmonic and chaotic motions may exist through the analysis of the stability of the multi-term HBM solutions.
c) Develop a methodology to evaluate the coupling between nonlinear hypoid gear mesh and time-varying rolling element bearing dynamic characteristics. The changing orbital position of the rolling elements is considered by adopting a time-varying stiffness matrix to represent the dynamic characteristics of the rolling element bearings. Interactions among time- varying mesh parameters, backlash nonlinearity as well as the time-varying bearing stiffness matrix are systematically studied.
d) Establish an overall nonlinear dynamic model of a hypoid gear box considering the elastic housing structure. A lumped parameter model of the housing is extracted from an appropriate set of frequency response functions, and is then coupled with a hypoid geared rotor system through a dynamic coupling procedure. The effect of the housing flexibility on the dynamic mesh force is examined. Also the effect of the external excitation on the dynamic mesh force and housing surface acceleration is evaluated.
e) Develop a new multi-DOF dynamic model of a hypoid geared rotor system considering the propeller shaft bending effect. The bending elasticity of propeller shaft is modeled as a lumped parameter model through using CMS. The effect of propeller shaft bending damping on gear dynamics is discussed, and this provides a potential new design methodology to reduce the right-angle gear box vibration and noise.
1.3 Organization
In this dissertation, a mesh model will be applied to represent the dynamic coupling between engaging gear pair. The mesh parameters will be extracted from the results of loaded tooth contact analysis. So the accuracy of the loaded tooth contact analysis is critical.
8
In Chapter 2, different loaded tooth contact analysis approaches are reviewed. The deformation of gear under load consists of contact deformation, gear tooth deformation and additional deflection induced by gear body flexibility. Different methods were applied to predict the different types of deformation. Based on the review, a combined semi analytical and three- dimension finite element method is found to be the most suitable approach for this dissertation.
The author plans to present the work contained in this chapter on a technical conference.
The difference between the actual output gear position and the theoretical one is defined as the transmission error of gear pairs, which is known as the source of vibration and noise issues of geared rotor systems. The transmission error will directly excite the engaging gear pair to vibrate. The vibration will then be transferred through gear shafts to the supporting bearing systems. Through bearings, the vibration will further be transferred to the gearbox housing structure, and vibrating housing surface will emit noise to the environment. Meanwhile, the vibration will also be transferred to the propeller shaft and then to the vehicle body through universal joint and propeller shaft mounts. The remaining chapters are arranged in the sequence of the vibration transfer path.
In Chapter 3, the dimensionless representation of a proposed dynamic model with backlash nonlinearity as well as asymmetric and time-varying mesh coupling for the right-angle gear pair is formulated. The traditional multi-term HBM is enhanced and applied to solve the dimensionless equation for period-one solutions. Parametric studies are performed, and particular attention is paid to the effect of the asymmetric mesh coupling on the dynamic response. This chapter is extracted from the author’s work in the paper published in Nonlinear Dynamics (Yang,
Peng and Lim, 2012).
9
In Chapter 4, the dimensionless equation of motion formulated in the last chapter is
further solved for the period-n motions through use of the enhanced multi-term HBM. A
numerical method is also adopted to validate the multi-term HBM and study the chaotic motions.
Stability of the multi-term HBM solutions is analyzed by applying the Floquent theory, and it is found that stability analysis is able to determine the frequency range where the sub-harmonic and
chaotic motions may exist. This chapter is extracted from the author’s work in a paper which is ready to be submitted to Nonlinear Dynamics.
In Chapter 5, a new methodology to evaluate the coupling between a nonlinear gear mesh and the time-varying dynamic characteristics of rolling element bearings is developed. The dynamic mesh force and the dynamic bearing reaction load are evaluated and compared by using different gear mesh and bearing representation combinations. This chapter is extracted from the author’s work presented at the SAE 2011 Conference (Yang and Lim, 2011), and it was published in SAE International Journal of Passenger Cars– Mechanical Systems (Yang and Lim,
2011).
In Chapter 6, an overall dynamic model of a hypoid gear box considering the housing flexibility is established. A lumped parameter model of the housing is extracted from an appropriate set of frequency response functions, and is further combined with the dynamic model of hypoid gear-shaft-bearing system through a dynamic coupling procedure. The effect of the
housing elasticity and the external excitations on the dynamic response is evaluated through parametric studies. This chapter is extracted from the author’s work presented at the ASME 2011
Conference (Yang and Lim, 2011).
In Chapter 7, a new multi-DOF dynamic model of a hypoid geared rotor system
considering the propeller shaft bending effect is established. The effect of propeller shaft bending
10 flexibility as well as bending damping on geared rotor system dynamics is discussed, and this provides a potential methodology to reduce the right-angle gear box vibration and noise by doing damping treatment on propeller shaft. The author plans to present the work contained in this chapter on a technical conference.
11
Chapter 2. A Review of Loaded Tooth Contact Analysis Approaches
2.1 Introduction
In this dissertation, the mesh model is applied to represent the dynamic coupling between
engaging gear pair. The mesh parameters of the mesh model are extracted from the results of
loaded tooth contact analysis. So the accuracy of the loaded tooth contact analysis is critical, and
various loaded tooth contact analysis approaches are reviewed in this chapter. The purposes of
different approaches could be summarized as calculating the contact and bending stresses with
the concern of the gear tooth fatigue, and predicting the transmission error and contact pattern
with the concern of noise and vibration issues of geared rotor systems. The deflections of gear
under static load were found to be comprised of contact deformation, gear tooth flexible
deformation including shear, bending and axial compression, as well as the additional
deformation induced by gear body flexibility. The contact deformation was calculated based on
the classical Hertzian theory and its variants, which were reviewed and compared by Cornell
(1981) and Gosselin, et al. (1994).
Three different approaches to calculate the contact deformation were reviewed and evaluated through using two different cases by Cornell (1981), including 1) an approximate
Hertzian and compression method from Hamilton Standard, 2) a semi-empirical method
developed by Palmgren, and 3) a closed form method developed by Weber. The comparison of
the predicted local deformations indicated that the third method was the most accurate one, while
the first approach did not include the nonlinearity properly, and the second approach was width
dependent.
The analytical contact stress and deformation formulations for both spur and spiral bevel
gear tooth were reviewed and compared with finite element method (FEM) by Gosselin, et al.
12
(1994). The analytical methods were variants of the basic Hertzian theory, including the
formulations for spur gears developed by Palmgren, Richardson, Roark, Johnson, Terauchi and
Weber, and the formulations for the spiral bevel gears developed by Roark, Brewe and Hamrock.
The gear tooth was meshed with H8 or H20 element with axially, radially and thickness-wise
defined number of elements. The fine meshed area was obtained by intersecting the pinion and
gear tooth surface to a depth equals to the contact deformation predicted by Hertzian theory for
two contact spheres and cylinders. The potential contact surface was defined as a slide-line or
slide-surface. The difference between the contact point and its corresponding point in the gear
tooth middle plane before and after the load application was taken as the FE-based contact
deformation. Comparison indicated that the contact stresses were of comparable levels between
analytical and FEA approaches, while the contact deformations differed 20% to 150%.
Coy and Chao (1981) derived an empirical formulation to help select the grid size in
finite element analysis of spur gear to account for Hertz deformation. The deflection of a semi-
cylinder under point load was calculated from a plane strain finite element model, and compared
with classical Hertzian solution for deflection. The element aspect ratio and the ratio of element
size to the contact width were considered as variables, and a linear relation between these two ratios were derived from the analysis. The resultant grid size selection formulation was applied to calculate the deflection of a spur gear, and validated by comparing the obtained spur gear deformation results with two Hertzian theory based solutions. Besides, the results from Cornell’s model (1981) indicated that the Hertz deflection was a very important component, approximately
25 percent of the total tooth deflection, which emphasized the importance of the proper grid size.
For the calculation of the gear tooth flexible deformation, different approaches were adopted, including cantilever beam and plate model based methods, semi-empirical formulation,
13 numerical methods as well as semi-analytical and three-dimensional finite element based method. In order to calculate the gear body elasticity induced deflection, various empirical equations were extracted either from actual measurement results or from numerical simulation solutions, and analytical formulations based on certain assumptions was also developed. In this chapter, different methods used to calculate gear tooth and body flexible deflections will be reviewed.
2.2 Gear Body Flexible Deflection
The spur/helical gear body was modeled by Weber (1949) as a semi-plane with a linear bending stresses distribution assumption at the interface of the tooth and the supporting semi- plane as shown in Fig. 2.1. The resultant bending stress as well as the uniform distributed shearing and axial stresses, due to a point load exerted along the surface normal direction, were obtained through the equilibrium condition with a rigid gear tooth assumption. The rotation of the loaded surface of the semi-plane was obtained by balancing the stress energy and work done by external force.
y
x
Fig. 2.1, Linear normal stress distribution (Weber, 1949)
The formulations developed by Weber, Muskhelishvili and Vogt were compared by
O’Donnell (1960), and it was found that both bending and shear stresses would cause the rotation of the loaded surface of the support. A symmetric test setup as shown in Fig. 2.2 was also designed by O’Donnell (1960) to measure the cantilever beam deflections with flexible support.
14
The reference cantilever beam deflection was subtracted from the testing beam deformation to
remove the normal compression effect of the center section. The cantilever beam deflection with
rigid support was calculated using classic bending deflection theory, and was subtracted from the
measured deflection to get the additional deflection induced by the elastic support. The
experiment results indicated that cubic bending and linear shear stress distributions would result
in more accurate predictions. Empirical formulations for calculating additional deflection of a
cantilever beam due to elastic support were developed considering the more realistic stress
distributions for both plane strain and plane stress cases.
Load
Reference beam
Testing beam
Fig. 2.2, Schematic of test rig (O’Donnell, 1960)
The spur/helical gear body was considered as an elastic circular ring by Sainsot and
Velex (2004). A new analytical bidimensional formula allowing fast and accurate calculation of
gear body-induced tooth deflection due to point load on gear tooth was developed, assuming the
linear bending and uniform shear stress distributions on the root cycle. The more realistic cubic
distribution of normal stress and parabolic distribution of shear stress at the root cycle were shown to have very limited improvement of the additional deflection due the elastic gear body.
The bending of a cantilever plate, which is infinitely long with a finite width and a constant thickness, supported from an elastic half space was studied by Small (1961). Both the transverse displacement and rotational flexibility of this half space were included. It was
15
assumed that the normal stress distributed linearly and the shear stress was constant over the
plate thickness at the supporting half space. The average rotation and displacement of the half
space were defined in the case that the work done by the bending moment and shear force equals
the work done by distributed bending and shear stresses. The explicit integral form of moment,
shear force, rotation and displacement of half space were derived through using the classical
Boussinesq results and the inversion theorem of Fourier integrals. The obtained general forces
and displacements were applied as the boundary conditions for the cantilever plate to calculate
the additional deflection due to the flexibility of the elastic half space. Experiment was also performed, and a good correlation between the predicted and measured results was obtained.
The base flexibility of spur gear teeth of different face width was studied by Stegemiller and Houser (1993) through applying the three-dimension finite element analyses. The rigid base boundary condition was represented by constraining the nodes on the support interface, while the flexible base boundary condition was defined by constraining the nodes at a distance of twice the tooth base thickness from support interface as shown in Fig. 2.3. The base translation was obtained by applying the load at zero moment arms. The base rotation was obtained by subtracting the fixed-base tooth deflection and base translational from the flexible-base tooth deflection. Point loads were exerted at the middle of the tooth height and tooth tip at different planes along the tooth width direction indicated by arrows in Fig. 2.3. The base rotation and translation were studied for different load cases, and found to be functions of both tooth height and width. Moment image method developed by Wellauer and Seireg (1960) was applied to consider the tooth end effect, and the results generated from the middle face loading were considered as the infinite plate solutions. Exponential functions were derived to represent both
16 the base translation and rotation by fitting the solutions predicted by using moment image method. The effect of tooth thickness was also considered by properly adjusting the exponent.
Fig. 2.3, Finite element model of gear tooth (Stegemiller and Houser, 1993)
2.3 Gear Tooth Flexible Deflection
2.3.1 Cantilever Beam
The spur/helical gear tooth was modeled by Weber (1949) as a cantilever beam of continually variable cross-section as shown in Fig. 2.4. The tooth deformation under the point load along the tooth surface normal direction was derived by equating the strain energy to the work of the external force, with rigid gear body assumption. The profile was extended to the gear root circle without considering the fillets. The stain energy was comprised of the energy due to bending stress, shear stress and normal stress. Normal force was found to be very small, about
1/20 of the shear force, and therefore it could be neglected.
17
Fig. 2.4, Cantilever beam model of gear tooth (Weber, 1949) The cantilever beam model of Weber (1949) was extended by Cornell (1981) to take into account the tooth fillet. Similarly, the bending and shear deflections for spur gear teeth were obtained through using elementary knowledge of strength of material.
2.3.2 Cantilever Plate
The spur gear tooth was treated by MacGregor (1935) as a thin and infinitely long cantilever plate of rectangular profile. The elasticity theory was applied to obtain an exact solution for the deflection along the free edge caused by a point load applied perpendicularly to the plate on the free edge as shown in Fig. 2.5. Due to the mathematical complexity, only the deflections at several critical points were obtained, and were correlated with measured data.
Fig. 2.5, Cantilever plate model of gear tooth (MacGregor, 1936)
18
The semi-empirical solutions of the moment at fixed edge of the cantilever plate of finite
length under transversely applied load at any location on its surface was given by Wellauer and
Seireg (1960), based on the cantilever plate theory and a proposed moment image method. The
method could be applied to spur, helical, bevel or worm gears. For an infinitely long cantilever
plate, the bending moment distribution at the fixed edge due to any transversely applied load is
known as shown in Fig. 2.6 (a). Cantilever plate of finite length can be considered as it is cut
from an infinitely long one. The bending moment distribution of the cantilever plate of finite
length was assumed to be the sum of the original moment distribution of the remaining portion and the mirror image of the moment distribution of the cut portion as illustrated in Fig 2.6 (b).
Due to the linear property of the cantilever plate problem, the superposition was applied to obtain the moment distribution as a result of distributed loads. The semi-empirical solutions were found to generally match with previous experimental data for thin plate. Experiment was also performed to obtain the moment distribution from measured root strain data for thick plate and
rack tooth, assuming that the moment is proportional to the strain. The effect of different loads,
both uniform and non-uniform line loads with the load lien parallel with or inclined from the free
edge, on the moment distribution was studied both semi-empirically and experimentally.
(a) (b)
Fig. 2.6, Moment imaging: (a) plate of infinite length, (b) plate of finite length (Wellauer et al., 1960)
19
The spur gear tooth was modeled by Yau, et al. (1994) as the tapered cantilever plate as
shown in Fig. 2.7. The shear and bending deflections were obtained through using a Rayleigh-
Ritz approach. The free-free and clamp-free beam modal shapes were applied as the admissible
functions to represent the gear tooth bending and shear deformations. The gear tooth deflections
were solved analytically, and were compared with thin plate model based theoretical, measured
and finite element results. The comparison indicated that thin plate model based solutions were
smaller for stubby gear teeth, considering that it did not consider the shear deflection. A simple
test stand is also designed to measure the gear tooth deflections. Both the measured data in
previous publications and the one obtained from the newly designed test stand were slightly
bigger than the solutions predicted by Rayleigh-Ritz approach, due to the base flexibility. The
Rayleigh-Ritz solutions were shown to be in good agreement with finite element results.
Fig. 2.7, Tapered cantilever plate model of gear tooth (Yau, et al., 1994)
The straight bevel gear teeth was modeled by Vaidyanathan, et al. (1994) as annual sector
Mindlin plate with linear varied thickness in both radial and angular direction as shown in Fig.
2.8. The similar Rayleigh-Ritz approach used by Yau, et al. (1994) was applied to calculate the bending and shear deflections, while the polynomials were chosen as the admissible functions.
Clamped and flexibly supported boundary conditions along a radial edge were considered by using the polynomials with different boundary conditions. For elastically supported case, the
20
energy due to the elastic support was added to the gear tooth potential energy. A general
procedure to determine the load distribution along the line of contact and transmission error for straight bevel gear was also developed. A spur gear tooth was as sector plate with a constant base
thickness and a negligible height taper along face wide. The deflection of the spur gear tooth was
evaluated, and the accuracy of the method was validated through comparing with the results
published by Yau, et al. (1994). The validation of the sector plate model and Rayleigh-Ritz
method on predicting bevel gear tooth deflection was also validated by comparing with the finite
element results. (Compatibility and equilibrium conditions)
Fig. 2.8, Annual sector plate model of gear tooth (Vaidyanathan, et al., 1994)
2.3.3 Shell
The face-milled FORMATE spiral bevel gear tooth was modeled as a thick circular
cylinder shell with linearly varied thickness and height by Vaidyanathan, et al. (1993) as shown
in Fig. 2.9. The shear deformation was considered using different shear deformation theories,
assuming parabolic and constant shear strain along the tooth thickness direction respectively.
The gear tooth deformations and stresses due to point loads were calculated using a Rayleigh-
Ritz based approach, taking the polynomials as the admissible functions. Only the rigid
supporting was considered. The predicted gear tooth deformations and stresses using the assumption of constant shear strain distribution along tooth thickness were shown to have
21
comparable accuracy to the finite element solutions. The shear deformation model assuming
parabolic shear strain distribution along the tooth thickness direction predicted higher stresses,
and its computational costs were also higher.
Fig. 2.9, Thick cylinder circular shell model of gear tooth (Vaidyanathan, et al. 1993)
Kolivand and Kahraman, (2009) developed a load distribution model applicable for both
the face-milled and face-hobbed hypoid gears. The unloaded tooth contact analysis was
performed using a proposed method based on the ease-off, which was defined as the deviation of
the real gear surface from the conjugate of its real mating gear, to get the potential contact curves. The tooth compliance matrix of the gear tooth with rigid support was extracted from a shell representation of gear tooth using Rayleigh-Ritz approach, which is very similar as what
Vaidyanathan, et al. (1993) did. The local contact deformation and the base rotation effects on the total tooth compliance were also included by using the closed-form formula by Weber (1949)
and an approximate interpolation method similar to the one developed for helical gears by
Stegmiller and Houser (1993) respectively. The compatibility and equilibrium conditions were
applied for the contact gear pair to obtain the load distribution.
22
2.3.4 FEM Method
Wang and Howard (2005) performed the quasi-static tooth contact analysis of high contact ratio spur gear with and without tooth profile modification using 2D finite element.
Adaptive mesh for 4-node plain stress elements was applied to obtain an accurate Hertzian deformation, and the 2-node line contact elements were used to represent the friction contact pairs. The gear tooth tips were modified to a fillet with a radius of 0.2mm to eliminate the singularity caused by tooth tip contact. All the nodes on the driving gear hub were constraint from the radial movement and coupled with a master node in rotation about the hub center in order to obtain a unique solution, while the nodes on the driven gear hub were completely constraint for each driving gear roll angle. The predicted transmission error, mesh stiffness and contact stresses were thus the functions of the driving gear roll angle. The effect of different tooth profile modifications on the contact stresses and load distribution was evaluated.
Lu, et al. (1998) built a one tooth FEM model in three-dimension for the double circular- are helical pinion and gear separately to analyze the load sharing and FEM stress. The parts on the two sides of and below the loaded tooth sufficiently far from the fillet and wheel respectively were chosen for the fixed boundary. The contact zone was assumed to be ellipse, and the contact pressure was considered as a semi-ellipsoid, based on Hertzian assumption. For simplicity, an averaged pressure was applied for each element. The load condition, which was the same as contact pressure, for the FEM model was obtained through the equilibrium condition of a single gear tooth due to the total input torque. The displacement of the contact center was obtained for each contacting teeth pair by running the FE calculation multiple times. The shared torque for contacting each teeth pair was calculated based on the position errors caused by surface mismatch and the calculated elastic deformations. The corresponding stress for each tooth was obtained through solving for the FE solutions.
23
Wilcox (1981) applied a combined flexibility matrix and finite element method to
analyze stresses in bevel and hypoid gear teeth. The governing equations were comprised of the
flexibility equation and constrained equation. The flexibility equations related the load
distribution to the flexibility matrices of pinion and gear teeth, surface mismatch, and rigid
rotation induced displacements. The constraint equation related applied torque to load
distribution. In calculation of the flexibility matrix, the instant contact line was discretized into a
serial of nodes as shown in Fig. 2.10. Unit load was applied to each node individually, and the resultant displacement vectors obtained from finite element solutions were arranged properly to generate the flexibility matrix. The misalignments were included into the flexibility matrix by assuming they were linearly related to the applied torque.
Fig. 2.10, Discretization of instant contact line (Wilcox, 1981)
Bibel and etc. (1995) analyzed the contact stresses of spiral bevel gears based on a multi
tooth finite element model with gap element between each contact node pair. The detailed gear tooth geometry was applied in this analysis, and obtained by simulating the kinematics of
manufacturing process developed by Litvin and Fuentes (2004). For any node on the pinion tooth
surface, a corresponding contact node on gear tooth surface was defined as the intersection of the
normal vector of the point on pinion and the gear tooth surface. The obtained initial FEA stresses
24
seemed to compare favorably with predicted Hertzian contact stresses. However, the big contact stress gradient indicated that finite element refinement was needed.
Vilmos (2000) derived the equations for the calculation of tooth deflection and fillet stresses by summarizing results obtained through performing a big number of FEM computer runs, using regression and interpolation functions. The curved 20-node elements were used to
discretize the pinion and gear teeth, while concentrated loads were applied to pinion and gear
tooth separately to approximate contact pressure. The FEM model with five teeth was found to
give accurate stress and deflection predictions, as compared with one tooth and three teeth
models.
Gosselin, et al. (1995) developed a general approach to do loaded tooth contact analysis
of spiral bevel and hypoid gears. The initial tooth separation due to profile mismatch was
obtained from tooth geometry. The Hertzian theory was applied to consider contact deformation,
while the bending and shear deformations were calculated using FEM. One tooth of both pinion
and gear on a solid rim were meshed using HEX-20 element separately. Unit vector loads were
applied in the normal direction to a chosen set of nodes on both pinion and gear tooth surfaces,
and the resultant deflections matrix was actually the compliance matrix. The stiffness of the
pinion and gear was considered as connection in serial. A relative coarse mesh was applied as
shown in Fig. 2.11, and the load and deflection other than the node position were obtained by
interpolation. The compatibility condition, that the blank rotation caused by the displacement of any tooth must be the same, and the equilibrium condition, that the sum of torque contribution of each meshing tooth pair must balance the total input torque, were formulated and solved for deflections and stresses.
25
Fig. 2.11, Finite mesh of gear tooth (Gosselin, et al., 1995)
2.3.5 Finite Strip Method
Gagnon, et al. (1996) considered the tooth of spur, helical and straight bevel gears as cantilever thick plate of varying thickness, modeled by Finite Strip Method based on Mindlin’s theory. The rectangular Finite Strip discretization, as shown in Fig. 2.12, was applied for spur gear tooth, while the circular and radial discretization was adopted for straight bevel and helical gear tooth respectively. The displacement function along the transverse direction was assumed to be linear, while it was assumed to be a cubic-spline serial along the longitudinal direction.
Fig. 2.12, Finite strip model of gear tooth (Gagnon, et al., 1996)
26
2.3.6 Finite Prism Method
Guingand, et al. (2004) applied the finite prism method (FPM) and the Boussinesq theory
to perform quasi-static tooth contact analysis of helical gears. The transverse section of the prism
was modeled by using the isoparametric 8-node element, while the solution of the spatial
differential equation of a vibrating beam with proper boundary conditions was used along the
longitude direction. Initial contact points were obtained from unloaded tooth contact analysis. A set of so-called bulk influence coefficients separately for pinion and gear teeth was obtained by considering the tooth elastic deflection using FPM and the contact deformation using Boussinesq theory, and was used to calculate load sharing. Once the load sharing was known, the FPM was applied to calculate the 3D stresses at the root of the teeth. The calculated stresses of the driven gear were compared with the measured results to demonstrate the accuracy of the FPM method.
2.3.7 Combined Surface Integral and Finite Element Method
Vijayakar and etc. (1987, 1988, 1989, 1991 and 1991) developed a combined surface integral and finite element solution for a three-dimensional contact problem. The basic assumption for this combined method was that the surface integral for the elastic half space model could predict relative displacement of points near the contact zone accurately, while the finite element model could predict deformation well beyond a certain distance away from the contact zone. The quasi-prismatic type of element using Chebyshev, as shown in Fig2.13, was developed by Vijayakar and etc. (1987). In order to obtain an accurate load distribution, the error in the tooth geometry representation had to be much smaller than the deformation due to the conformal natural of the gear tooth contact. A special 20 nodes two-dimensional formulation using Hermit cubic shape function was applied by Vijayakar and etc. (1989) to allow C continuous representation of the gear tooth surface within the prescribed tolerance as shown in
27
Fig. 2.14, for the elements sharing part of the contact surface. The compatibility condition for the primary contact point and its adjacent area within a prescribed separation, and the equilibrium condition for the contact gear pair were applied to derive the final equations. The Simplex type of algorithm was applied to solve the obtained equations for load distribution by Vijayakar and etc. (1988). Detailed gear body and even the shaft bearing structures could be included in tooth contact analysis, due to FEM was applied for the structures beyond a certain distance away from the contact zone.
Fig. 2.13, Finite prism element (Vijayakar and etc., 1987)
Fig. 2.14, Two-dimensional formulation of finite strip element (Vijayakar and etc., 1989)
28
2.4 Conclusion
The deflections of gear under static load are comprised of contact deformation, gear tooth
flexible deformation including shear, bending and axial compression, as well as the additional
deformation induced by gear body flexibility. There is a considerable difference between the
contact deformations predicted by FEM and analytical approaches including Hertzian theory and
its variants, even though the predicted contact stresses are in the comparable level. Furthermore,
the contact deformation results of FEM are sensitive to the different mesh.
Compared with semi-plane, semi-space and circular elastic ring models of gear body, the
FEM is more accurate for calculating the extra gear deformation induced by gear body
flexibility. This is because that more detailed gear geometry and more realistic boundary
conditions can be defined through using FEM. Besides, for FEM there is no linear, parabolic or
cubic stress distribution simplification on the gear root. The efficiency and computational cost of
FEM is acceptable as well, since a relative coarse mesh is good enough to get accurate extra
deformations induced by gear body flexibility.
The tapered cantilever plate model of spur/helical gear tooth, annular sector plate model
of straight bevel gear tooth and thick cylinder circular shell model of spiral bevel/hypoid gear
tooth can represent gear tooth geometrically more realistic, as compared with the simple
cantilever beam and thin plate models of gear tooth. The FEM approach can better describe gear
tooth geometry by using very fine mesh. FPM is capable of representing gear tooth, especially
hypoid gear tooth, accurately with much less elements as compared with the traditional FEM.
Thus, a combined surface integral and FEM method is believed to be the most suitable
approach for establishing mesh model of hypoid gear pair. It is because that, in the combined method, the gear tooth is modeled using FPM element, and the gear body and other supporting
29 structure are modeled using traditional FEM elements. One of the other most important reasons is that the contact deformation is calculated through using a particular procedure which combines the surface integral and FEM.
30
Chapter 3. An Enhanced Multi-term Harmonic Balance Solution for Non-
linear Period-one Dynamic Motions in Right-angle Gear Pairs
3.1 Introduction
In the past two decades, the non-linear frequency response spectra of parallel axis gears
have been extensively studied. In those studies (Comparin and Singh, 1989; Kahraman and
Singh, 1990, 1991; Blankenship and Kahraman, 1995; Kahraman and Blankenship, 1996;
Theodossiades and Natsiavas, 2000; Ma and Kahraman, 2005, 2006; Al-shyyab and Kahraman,
2005, 2007), particular attention was given to the formulation of the non-linear gear dynamic
models and the development of analytical, semi-analytical and numerical solution methods.
Using different mesh stiffness models, including piecewise linear time-invariant (PLTI)
(Comparin and Singh, 1989; Kahraman and Singh, 1990), piecewise linear time-varying (PLTV)
(Kahraman and Singh, 1991; Blankenship and Kahraman, 1995; Kahraman and Blankenship,
1996; Theodossiades and Natsiavas, 2000) and piecewise nonlinear time-varying (PNTV) (Ma and Kahraman, 2005, 2006) ones, various non-linear representations of the parallel axis gear pair were constructed and the corresponding one degree-of-freedom (DOF) dimensionless equation of motions were formulated. The method was also extended to develop the multiple DOFs non- linear models of multi-mesh gear trains (Al-shyyab and Kahraman, 2005), geared rotor-bearing systems (Kahraman and Singh, 1991) and planetary gear sets (Al-shyyab and Kahraman, 2007).
Using various solution techniques, including analog simulation (Comparin and Singh, 1989), digital simulation (Comparin and Singh, 1989; Kahraman and Singh, 1990), numerical integration (Theodossiades and Natsiavas, 2000; Ma and Kahraman, 2005, 2006; Al-shyyab and
Kahraman, 2005), harmonic balance method (HBM) (Comparin and Singh, 1989; Kahraman and
Singh, 1990), multi-scale method (Kahraman and Singh, 1990, 1991; Theodossiades and
31
Natsiavas, 2000) and multi-term HBM (Kahraman and Singh, 1991; Blankenship and Kahraman,
1995; Kahraman and Blankenship, 1996; Ma and Kahraman, 2005, 2006; Al-shyyab and
Kahraman, 2005, 2007), the dimensionless equations of motion were solved for the gear dynamic response. Amongst the various solutions analyzed, the multi-term HBM coupled with the discrete Fourier Transform was shown to be capable of studying a wide range of non-linear dynamic models of parallel axis gears (Kahraman and Singh, 1991; Ma and Kahraman, 2005,
2006; Al-shyyab and Kahraman, 2005, 2007), and the effectiveness was demonstrated by comparison to experimental results (Blankenship and Kahraman, 1995; Kahraman and
Blankenship, 1996) as well as direct numerical integration calculations (Kahraman and Singh,
1991; Ma and Kahraman, 2005, 2006; Al-shyyab and Kahraman, 2005, 2007). Also, the computational cost of the multi-term HBM is much lower as compared to other solution schemes, and it appears all the harmonic solutions, including stable and unstable dynamics, can be predicted. Research studies on geared rotor dynamics have also been performed extensively in
Europe, for example by Velex and his colleagues (Maatar and Velex, 1996; Ajim and Velex,
2005) in France. The history of the mathematical models used in gear dynamics up until 1987 are well documented in the review paper (Ozguven and Houser, 1988) that presented five overlapping categories used to classify the nature of models proposed.
In spite of the vast number of studies directed at parallel axis gears, relatively few open publications on the topic of non-linear dynamics of right-angle gears such as hypoid and spiral bevel types can be found. This is in part due to the complex mesh mechanism in this class of gears. Unlike the spur gear, the lines of action in these right-angle gears are not constant during the meshing process. This complexity is mainly due to the curvature feature of the hypoid gear tooth geometry and kinematics. For the same reason, the mesh coupling between the pinion and
32
gear are not symmetric unlike parallel axis gears. In earlier studies, dynamic studies of right
angle gears are limited to system level trial-and-error testing and relatively simple mathematical
models. A vibration model having two degrees of freedom for a bevel gear pair was adopted
(Kiyono, Fujii and Suzuki, 1981), in which the line-of-action vector was simulated by a sine
curve. In study (Abe and Hagiwara, 1990), it was shown experimentally that axle gear noise can
be reduced in some cases by modifying the vibration mode with an inertia disk mounted on the
final drive. Most recently, in a set of studies by Cheng and Lim (Lim and Cheng, 1999; Cheng
and Lim, 1991,2001; Cheng, 2000), an analytical modeling framework for analyzing the
vibration characteristics of hypoid and bevel geared rotor systems were developed. Using that framework, a non-linear model with PLTV mesh stiffness, time-varying (TV) directional rotation radius and backlash nonlinearity was formulated and solved using the direct numerical integration (Cheng, 2000; Jiang, 2002; Wang, 2002; Wang, Lim and Li, 2007). Also, a non-
linear model considering PLTV mesh stiffness and backlash nonlinearity was proposed and
analyzed using HBM by Jiang (2002). Although the effect of tooth mesh stiffness asymmetric
nonlinearity on the dynamic response was analyzed using direct numerical integration (Wang
and Lim, 2009), there is no known study that has addressed the directional rotation radius
asymmetric nonlinearity, which is one of the motivations of the present analysis.
In this study, considering both backlash and asymmetric mesh nonlinearity, a complete
non-linear dynamic model for right-angle gear pairs is proposed. In the proposed formulation,
the mesh stiffness, directional rotation radius and mesh damping are assumed to be time-varying
and asymmetric. The multi-term HBM coupled with a modified discrete Fourier Transform
(DFT) process and the numerical continuation method (Allgower and Georg, 1990; Kim, Rook
and Singh, 2005) are applied to solve the dimensionless equation of motion for the gear dynamic
33
displacement response. The symmetric or asymmetric non-linear restoring force, damping force
and external excitation forces are first calculated at each time point within one mesh period.
Second, the DFT process is performed to obtain the coefficients of the fundamental frequency and its harmonics. Third, the dynamic response is derived from the solution of the harmonic balance equations. The results of the proposed enhanced multi-term HBM is validated by comparison to the more computationally intensive, direct numerical integration calculations.
Finally, the effects of key parameters including the variation and asymmetry in the mesh stiffness and directional rotation radius on the gear dynamic responses is studied systematically through a series of parametric studies.
3.2 Mathematical Model
The two degree-of-freedom torsional vibration model and the coordinate systems of a
right-angle gear pair are shown in Fig. 3.1. The shaft and bearing are considered as rigid, and
pinion and gear are modeled as rigid bodies. The mesh coupling between the pinion and gear is
represented using a set of mesh stiffness and damping elements acting along a line of action
dictated by the directional rotation radius. These mesh parameters are all considered as time-
varying and asymmetric.
34
Gear
Ig Zp Zg θg
Gear axis Yg b
Cm Km O e g Xg pinion Gear axis Op θp Pinion axis Pinion axis Ip Xp Yp
(a) (b)
Fig. 3.1, (a) Two degrees of freedom torsional vibration model of a hypoid gear pair, (b) Pinion and gear
coordinate systems
The equation of motion can be derived as,
(3.1a) I pp p ( )c( )( e) p ( )k( ) f ( e) Tp ,
(3.1b) I gg g ( )c( )( e) g ( )k( ) f ( e) Tg ,
where I p and I g are the mass moments of inertial of pinion and gear, Tp and Tg are the torque applied on pinion and gear, and c( ) and k( ) are the asymmetric time-varying mesh damping
and stiffness coefficients given by,
cd , 0, c( ) (3.2) cc , 0,
k d , 0, k ( ) (3.3) kc , 0,
35
A (3.4) kd kd1 (kd (2a) cos(at) kd (2a1) sin(at)), a1
B (3.5) kc kc1 (kc(2b) cos(bt) kc(2b1) sin(bt)), b1
H (3.6) cd cd1 (cd (2h) cos(ht) cd (2h1) sin(ht)), h1
J (3.7) cc cc1 (cc(2 j) cos( jt) cc(2 j1) sin( jt)), j1
where cm and km (m d,c for drive and coast side, respectively) are the mesh damping and
stiffness for different tooth sides in contact. Typical drive and coast side mesh stiffness, obtained
by synthesizing from the loaded tooth contact analysis results (Cheng, 2000), are shown in Fig.
3.2. The plot clearly illustrates the asymmetric nature of the mesh stiffness for the drive and
coast side cases. Similarly, other mesh parameters such as mesh damping, mesh points and line-
of-action are all asymmetric. For brevity, they are not plotted here. Also, in Eq. 3.1, p ( ) and
g ( ) are the asymmetric time-varying directional rotation radii of the pinion and gear. They
can be expressed as,
, 0, pd (3.8) p ( ) pc , 0,
, 0, gd (3.9) g ( ) gc , 0,
L (3.10) pd pd1 ( pd (2l) cos(lt) pd (2l1) sin(lt)), l1
36
M (3.11) pc pc1 ( pc(2m) cos(mt) pc(2m1) sin(mt)), m1
U (3.12) gd gd1 (gd (2u) cos(ut) gd (2u1) sin(ut)), u1
V (3.13) gc gc1 (gc(2v) cos(vt) gc(2v1) sin(vt)), v1
where pm and gm are the pinion and gear directional rotation radii for drive (m=d) and coast
(m=c) sides of the teeth in contact. The directional rotation radii are explicitly defined as,
(3.14) lm nlm ( jl rlm ),
where nlm and rlm are the line of action and mesh point respectively in the coordinate system sl
for different tooth sides in contact (l p,q for pinion and gear, m d,c for drive side and coast
side), and jl is the unit vector along the pinion or gear rotating axis in the coordinate system sl (
l p,q for pinion and gear). Additionally, in Eq.3.1, the non-linear displacement function
f ( e) is given by
e b, e b, f ( e) 0, b e b, (3.15) e b, e b,
where 2b is the total gear backlash.
37
5.3
10 ) 5.2 10
5.2 N/mm
( 10
5.1 10 5.1 10
Mesh Stiffness 0 10 20 30 40 0 10 20 30 40 Pinion rolling angle
(a) (b) Fig. 3.2, Asymmetric nature of hypoid gear mesh stiffness: (a) Drive side (b) Coast side
The mesh stiffness, mesh damping, directional rotation radii and non-linear displacement
are all a function of the dynamic transmission error that is defined as,
(3.16) p ( ) p g ( ) g .
The kinematic transmission error e under no or very light load is the source of internal
excitation in the system. Its harmonic expression can be shown to be:
Y (3.17) e (e(2 y) cos(yt) e(2 y1) sin(yt)). y1
The mean value of the kinematic transmission error is set to zero, since it will not affect the
dynamic responses. This can be seen clearly from Eq. 3.18a, where only the second order derivative of the kinematic transmission error with respect to time is present on the right hand side as an excitation.
The above two degrees-of–freedom, semi-definite vibration system expressed by Eq. 3.1 can be simplified into a single degree-of-freedom, definite vibration model, since there is no constraint on the rotational coordinates of both the pinion and gear. Let x e , Equations
(3.1a-3.1b) can be combined into the following single equation of motion,
38
( )T ( )T m ( )x c( )x k( ) f (x) m ( ) p p g p e , (3.18a) e e I p I p
m , 0, ed (3.18b) me ( ) mec , 0,
2 2 (3.18c) med 1/( pd / I p gd / I g ),
2 2 (3.18d) mec 1/( pc / I p gc / I g ),
x b, x b f (x) 0, b x b. (3.18e) x b, x b
Also, during the simplification process, lm and lm are assumed to be zero. It is a reasonable assumption since the mesh point and line of action are typically continuous without abrupt change even though they are inherently time-varying.
2 2 For simplicity, consider med1 1/( pd1 / I p gd1 / I g ), andn kd1 / med1 . Then, to derive the dimensionless equation of motion, the following transformations are applied:
~x x / b, (3.19)
~ (3.20) t nt,
~ (3.21) /n ,
A ~ ~ ~~ ~ ~~ (3.22) kd kd / kd1 1 (kd (2a) cos(at ) kd (2a1) sin(at )), a1
B ~ ~ ~~ ~ ~~ (3.23) kc kc / kc1 1 (kc(2b) cos(bt ) kc(2b1) sin(bt )), b1
39
H ~ ~ ~~ ~ ~~ (3.24) cd cd / cd1 1 (cd (2h) cos(ht ) cd (2h1) sin(ht )), h1
J ~ ~ ~~ ~ ~~ (3.25) cc cc / cc1 1 (cc(2 j) cos( jt ) cc(2 j1) sin( jt )), j1
L ~ ~ ~~ ~ ~~ (3.26) pd pd / pd1 1 ( pd (2l) cos(lt ) pd (2l1) sin(lt )), l1
M ~ ~ ~~ ~ ~~ (3.27) pc pc / pc1 1 ( pc(2m) cos(mt ) pc(2m1) sin(mt )), m1
U ~ ~ ~~ ~ ~~ (3.28) gd gd / gd1 1 (gd (2u) cos(ut ) gd (2u1) sin(ut )), u1
V ~ ~ ~~ ~ ~~ (3.29) gc gc / gc1 1 (gc(2v) cos(vt ) gc(2v1) sin(vt )), v1
Y ~ ~ ~~ ~ ~~ (3.30) e e / b (e(2 y) cos(yt ) e(2 y1) / bsin(yt )). y1
Accordingly, the simplified dimensionless equation of motion can be represented as,
g(~x) ~ ~ ~ ~ ~x 2c(~x)g(~x)~x k(~x) f (~x) T (~x) T (~x) e~, (3.31a) 1 p p g g where the symbols in the above equation are given by,
2 c pd1 d1 , (3.31b) 2I pn
2 I gd1 p (3.31c) 2 , pd1I g
40
T ~ pd1 p (3.31d) Tp , bn I p
~ ~ (3.31e) Tg Tp ,
c~ ~x 1 ~ d (3.31f) c(x) ~ ~ , rda cc x 1
c c1 (3.31g) rda , cd1
~ k , ~x 1, ~ d (3.31h) k(x) ~ ~ rk kc , x 1,
k c1 (3.31i) rk , kd1
~x 1, ~x 1, ~ ~ f (x) 0, 1 x 1, (3.31j) ~ ~ x 1 x 1,
~2 ~2 ~ , x 0, g(~x) pd gd (3.31k) 2 ~2 2 ~2 ~ rp pc rg gc , x 0,
~ , ~x 1, ~ ~ pd (3.31l) p (x) ~ ~ rp pc , x 1,
~ , ~x 1, ~ ~ gd (3.31m) g (x) ~ ~ rg gc , x 1,
pc1 (3.31n) rp , pd1
41
gc1 (3.31o) rg . gd1
Next, the period-one solution of Eq. 3.31a is discussed.
3.3 Period-one Dynamics
The multi-term harmonic balance method coupled with DFT, which has been
successfully applied by References (Kahraman and Singh, 1991; Blankenship and Kahraman,
1995; Kahraman and Blankenship, 1996; Ma and Kahraman, 2005, 2006; Al-shyyab and
Kahraman, 2005, 2007) to analyze spur gear non-liner dynamics, is adopted in this study to solve the dimensionless equation of motion for ~x . In those previous studies, the DFT is applied to
calculate the Fourier coefficients of the non-linear displacement. Here in this analysis, the DFT is
extended to calculate the Fourier coefficients of the damping, non-linear restoring and external excitation forces.
To start the derivation, the steady-state solution is assumed to be of the form,
R ~ ~ ~ ~~ ~ ~~ (3.32) x(t) x1 (x2r cos(rt ) x2r1 sin(rt )), r1
which can be differentiated to yield,
R ~ ~~ ~~ ~~ ~~ (3.33) x (t) (rx2r sin(rt ) rx2r1 cos(rt )). r1
Then, the time series of the damping, non-linear restoring and external excitation forces can be
obtained by sampling even number of n points within one fundamental mesh period. Here, n must be larger than 2 R where R is the highest harmonics of the solution of interest. Using this process, the time series of the damping force can be shown to be:
42
~ ~ ~ ~ ~ ~ ~ (3.34) Fda (ti ) 2c(x(ti ))g(x(ti ))x(ti ), i 0,1,2...n 1.
Similarly, the time series of the non-linear restoring force is,
~ ~ ~ g(x(ti )) ~ ~ F (t ) k(~x(t )) f (~x(t )), i 0,1,2...n 1; (3.35) k i 1 i i
and the time series of the external excitation forces are
~ ~ ~ ~ ~ (3.36) Fp (ti ) Tp p (x(ti )), i 0,1,2...n 1;
~ ~ ~ ~ ~ (3.37) Fg (ti ) Tg g (x(ti )), i 0,1,2...n 1.
In order to use the multi-term HBM, all of these force terms must be represented by
Fourier series as follows:
R ~~ ~~ (3.38) Fda Fda1 Fda(2r) cos(rt ) Fda(2r1) sin(rt ), r1
R ~~ ~~ (3.39) Fk Fk1 Fk(2r) cos(rt ) Fk(2r1) sin(rt ), r1
R ~~ ~~ (3.40) Fp Fp1 Fp(2r) cos(rt ) Fp(2r1) sin(rt ), r1
R ~~ ~~ (3.41) Fq Fq1 Fq(2r) cos(rt ) Fq(2r1) sin(rt ), r1
where the coefficients of each series can be calculated using the discrete Fourier Transform.
They are explicitly given by,
n1 n1 1 2 (3.42a-b) Fda1 Fda (ti ), Fda(2r) Fda (ti )cos(2rti ), n i0 n i0
43
n1 2 (3.42c) Fda(2r1) Fda (ti )sin(2rti ), r 0,1,2...R , n i0
n1 n1 1 2 (3.43a-b) Fk1 Fk (ti ), Fk(2r) Fk (ti )cos(2rti ), n i0 n i0
n1 2 (3.43c) Fk(2r1) Fk (ti )sin(2rti ), r 0,1,2...R , n i0
n1 n1 1 2 (3.44a-b) Fp1 Fp (ti ), Fp(2r) Fp (ti )cos(2rti ), n i0 n i0
n1 2 (3.44c) Fp(2r1) Fp (ti )sin(2rti ), r 0,1,2...R , n i0
n1 n1 1 2 (3.45a-b) Fg1 Fg (ti ), Fg(2r) Fg (ti )cos(2rti ), n i0 n i0
n1 2 (3.45c) Fg(2r1) Fg (ti )sin(2rti ), r 0,1,2...R . n i0
Next, the algebraic sums of the coefficients of the terms of the like frequencies are forced to be equal. Doing so will yield,
(3.46a) S1 Fda1 Fk1 Fp1 Fq1,
~ 2 ~ ~ 2 ~ (3.46b) S2r (r) x2r Fda(2r) Fk(2r) Fp(2r) Fq(2r) (r) e2r ,
~ 2 ~ ~ 2 ~ (3.46c) S2r1 (r) x2r1 Fda(2r1) Fk(2r1) Fp(2r1) Fq(2r1) (r) e2r1.
Finally, the Newton-Raphson method is applied to solve the vector equations S 0 for
~ ~ ~ ~ T the solution vector x [x1 x2 ... x2R1 ] . The problem statement can be expressed as follows:
~x (m1) ~x (m) [J 1 ](m) S (m) , (3.47)
44
where the ( m 1 )-th iteration of solution vector ~x (m1) is calculated using the previous iteration
values of ~x (m) , S (m) and the Jacobian matrix J (m) . The process is repeated until the steady-state
~ ~ ~ ~ T solution x [x1 x2 ... x2R1 ] converges to within a predefined error limit for that excitation
frequency. If the response at the next frequency point is needed, a control parameter is then set to
the next value to restart the iteration process again. Also, it may be noted that corresponding to
one excitation frequency, multiple solutions may exist for the non-linear gear dynamic system. If
the frequency is chosen as the control parameter, which is normally done, the calculations may
only yield parts of the whole solution set unless one properly switches the frequency sweep
directions as also seen in the earlier spur gear dynamic analysis (Blankenship and Kahraman,
1995; Kahraman and Blankenship, 1996; Ma and Kahraman, 2005, 2006). Fortunately, there are
other choices for control parameter such as the components of the solution vector (Al-shyyab and
Kahraman, 2005) and the arc-length (Allgower and Georg, 1990). In this study, the arc-length is
used as the control parameter since it is believed to be more effective and easy to formulate. The
arc-length and tangential vector of the solution curve are used to predict the initial value of the
next solution point given by,
~ ~ (3.48) xn1 xn Tn h,
~ ~ where xn1 represents initial value for the (n+1)-th point iteration, xn represents the solution of
the n -th point, Tn is the tangential vector corresponding to the solution at the n -th point, and h is the arc-length increment.
The Floquet theory (Nayfeh and Mook, 1979) is applied to perform the stability analysis of the steady-state solution obtained above. Consider the perturbation equation,
45
~ ~ ~ ~ g(x(t )) ~ ~ ~ ~x 2c(~x(t ))g(~x(t ))~x k (~x(t )) f (~x(t ))~x 0, (3.49) 1 where ~x is the small perturbation of the periodic solution ~x(t) ~x(t T) . By defining y [~x ~x ]T , the perturbation equation can be rewritten as
~ ~ ~ y(t ) H(t )y(t ) , (3.50a)
0 1 ~ ~ ~ H(t ) g(x(t )) ~ ~ ~ ~ ~ ~ ~ ~ ~ (3.50b) k (x(t )) f (x(t )) 2c(x(t ))g(x(t )) 1 .
The above differential equation is then solved for y(T) assuming the initial condition of
y(0) I 2 where I 2 is the square identity matrix of dimension two. Then, the local stability of the steady-state solutions can be determined by analyzing the eigenvalues of y(T) . The solution is considered stable when the modulus of the eigenvalues is less than unity; otherwise the solution is unstable.
3.4 Parametric Studies
The light load case studied in Reference (Wang and Lim, 2007) is taken as the baseline set, and only the period-one motions are of interest. The dimensionless right-angle hypoid gear parameters for the numerical calculations are listed in Table 3.1. To simplify the parametric analysis, only the first harmonic of the mesh stiffness, directional rotation radii and transmission error are considered. Also, the torque transmitted through the hypoid gear pair and the mesh damping coefficients are taken as constants.
46
Table 3.1, Dimensionless dynamic parameters for a typical automotive hypoid gear pair
Parameter symbols Numerical value ~ Tp 0.1 0.03 0.75 ~ ~ cd ,cc 1 ~ ~ kd 2 ,kc2 -0.0405 ~ ~ kd 3 ,kc3 0.0294
rp ,rq ,rk ,rda 1 ~ ~ ~ ~ pd 2 , pc2 ,gd 2 ,gc2 0.01 ~ ~ ~ ~ pd 3 , pc3 ,gd 3 ,gc3 0 ~ e2 0 ~ e3 -0.5
First, the predictions of the multi-term HBM method are compared to the results of the
less efficient, direct numerical integration. Then, the results of the parametric studies are
examined to understand the effects of key parameters on the gear dynamic response.
3.4.1 Numerical Validation
For verification purpose, the nonlinear dynamic response of the baseline right-angle
hypoid gear case is obtained using both the multi-term HBM and the Runge-Kutta integration routine with variable step (Wang and Lim, 2007). Their predicted dynamic responses are compared in Fig. 3.3, and can be seen to match with each other very well. Here, Figure 3.3(a)
shows the comparison of the root mean square (RMS) values of the displacement response, while the corresponding mean values are plotted in Fig. 3.3 (b). The dimensionless results can be easily transformed back to the physical coordinate. For example, for a practical application, whose
physical parameters are listed on Table 3.2, the dimensionless parameters are calculated as
2 2 I p 0.03kg m , I g 0.08kg m , pd1 0.026m , gd1 0.1119m , as well as
47
8 kd1 1.210 N / m , and the backlash is assumed to be b 20m , then the natural angular
frequency can be obtained asn 4893rad / s . Thus, in Fig. 3.3(a), the physical frequency range is from about 154 to 1947 Hz, and the displacement range is from 0.2 to 200 m.
Table 3.2, Physical parameters of a real application hypoid gear pair
Parameter symbols Numerical value Pinion/Gear Tooth Number 10/43 Pinion/Gear Pitch Radius 0.048/0.1681 (m) Offset 0.03175 (m) Input Torque 0.5 (N*m)
48
1 10
0 10 lacement lacement p
-1 Dis 10
-2 10 0 0.5 1 1.5 2 2.5 Frequency (a)
0.2 10 lacement p
Dis 0.1 10
0 0.5 1 1.5 2 2.5 Frequency
(b)
Fig. 3.3, Comparison of multi-term HBM and numerical integration: (a) RMS of the dynamic displacement. (b) Mean value of the dynamic displacement. Stable steady-state solution by multi-
term HBM; Unstable steady-state solution by multi-terms HBM; ○ Solutions by numerical
integration
49
The right-angle gear example selected for the numerical analysis is a hypoid type, as noted earlier. From past studies (Cheng, 2000; Wang and Lim, 2007, 2009), it is known that hypoid gears can manifest both single-sided and double-sided tooth impacts. This behavior is also seen here in the simulation results of both multi-term HBM and numerical integration. A closer examination of Fig. 3.3(a) reveals that, in the low frequency range, the hypoid gear pair behaves quite linearly with no evidence of tooth impact response. As the dimensionless frequency increases to ~ 0.8064 , the single-sided tooth impact begins to emerge. At that
point, the dynamic displacement response curve veers left towards the lower frequency range
unlike a typical linear system that would continue to climb in both response and frequency. This
type of characteristic indicates the effect of backlash nonlinearity causing single-sided tooth
impact, which is similar to a softening spring case. The response curve continues along that same
trajectory as the dimensionless frequency decreases to ~ 0.7116 from ~ 0.8064 until double-sided tooth impact begins to show up. At this point, the effect of backlash nonlinearity manifests itself as a hardening spring case because of the additional impact with the preceding tooth. When this happens, the trajectory of the response takes a sharp turn towards the right with increasing frequency again. The dynamic displacement response reaches the peak amplitude at
~ 0.8439 . After that point, the hypoid gear pair response changes back to only single-sided
tooth impact and subsequently becoming linear again as the frequency increases further.
3.4.2 Numerical Analysis
In this section, a series of numerical simulations are performed. The purpose is to
examine the effects of the magnitude variation and asymmetric nonlinearity of the mesh stiffness
and directional rotation radii on the dynamic displacement response.
50
Effect of directional rotation radii variation
In this analysis, only the effect of the directional rotation radii variation is analyzed. No
asymmetry is included by setting rp rq 1. The variation is represented by five different values
~ ~ ~ ~ of the second Fourier coefficient given by pd 2 pc2 gd 2 gc2 0.01,0.1,0.15,0.18,0.25 , which are related to the coefficients of the first order harmonic of the directional rotation radii.
The numerical results presented in Fig. 3.4(a) shows that the increase in second Fourier coefficient or first order harmonic of the directional rotation radii seems to amplify the peak ~ ~ ~ ~ values of the second and third super-harmonic responses. When pd 2 pc2 gd 2 gc2 0. 01, no obvious peak around ~ 0.5 and ~ 0.33 can be found. As the second Fourier coefficient
increases to 0.1, the second and third super-harmonic peaks begin to appear. The peak values of the second and third super-harmonics continue to amplify as the second Fourier coefficient are further increased.
Similar effect of the directional rotation radii variation on the primary resonance can be observed as shown in the enlarged plot around the primary resonance in Fig. 3.4(b). As the second Fourier coefficient of the directional rotation radii for both pinion and gear are increased, the peak value and the corresponding resonance frequency of the primary resonance seems to rise appreciably.
51
1 10
0 10 0.25
0.18 lacement p -1
Dis 10 0.15 0.1
0.01 -2 10 0 0.5 1 1.5 2 2.5 Frequency (a)
0.25 0.4 10 0.18
0.3 10 lacement
p 0.15 0.1 Dis
0.2 0.01 10
0.7 0.75 0.8 0.85 0.9 Frequency (b)
Fig. 3.4, Effect of the directional rotation radii variation on the dynamic response: (a) RMS of the dynamic displacement. (b) Primary resonance of the dynamic displacement. Stable steady-state
solution by multi-term HBM; Unstable steady-state solution by multi-term HBM
52
1 10
0 10 lacement p -1
Dis 10
-2 10 0 0.5 1 1.5 2 2.5 Frequency (a)
0.85 0.95 1.0 1.05
1.15 lacement p Dis
Frequency
(b)
Fig. 3.5, Effect of directional rotation radii asymmetric nonlinearity on the dynamic response: (a) RMS of
the dynamic displacement. (b) Primary resonance of the dynamic displacement. Stable steady-
state solution by multi-term HBM; Unstable steady-state solution by multi-term HBM
53
Effect of directional rotation radii asymmetric nonlinearity
To examine the effect of asymmetry in the directional rotation radii on the dynamic displacement responses, five different coast to drive sides mean value ratios given by
rp rq 0.85,0.95,1.00,1.05,1.15 are assumed. The second Fourier coefficient is set to the fixed
~ ~ ~ ~ value of pd 2 pc2 gd 2 gc2 0. 01.
The analysis results shown in Fig. 3.5 indicate that the asymmetric nonlinearity of directional rotation radii only affects the dynamic displacement response in the double-sided tooth impact range. The peak frequency of the primary resonance increases as the ratio of the directional rotation radius mean value of coast side over the drive side one increases as shown in the enlarged plot Fig. 3.5(b).
Effect of mesh stiffness variation
In this analysis, similar to the directional rotation radii study discussed above, the mesh
stiffness is assumed symmetric withrk 1, in order to examine the effect of mesh stiffness variation. Five different groups of second and third Fourier coefficients of the mesh stiffness ~ ~ ~ ~ values given by (kd 2 kc2 ,kd 3 kc3 ) (0.0405,0.0294), (0.081,0.0588), (0.162,0.1176),
(0.243,0.1764) , (0.324,0.2352) are specified.
As shown in Fig. 3.6, when the second and third Fourier coefficients of the mesh stiffness increase, the peak values of the super-harmonic resonances are amplified and both the resonance frequency and peak values of the primary resonance are amplified as well. This trend is similar to the directional rotation radius results.
54
1 10
0 10 (-0.324, 0.2352)
(-0.243, 0.1764) lacement p (-0.162, 0.1167)
Dis -1 10 (-0.081, 0.0588)
(-0.0405, 0.0294)
-2 10 0 0.5 1 1.5 2 2.5 Frequency (a)
(-0.324, 0.2352)
0.7 10 (-0.243, 0.1764)
(-0.162, 0.1167) 0.5 10 (-0.081, 0.0588) lacement p
0.3 Dis 10 (-0.0405, 0.0294)
0.1 10
0.6 0.7 0.8 0.9 1 1.1 Frequency (b)
Fig. 3.6, Effect of the mesh stiffness variation on the dynamic response: (a) RMS of the dynamic displacement. (b) Primary resonance of the dynamic displacement. Stable steady-state solution by
multi-term HBM; Unstable steady-state solution by multi-term HBM
55
Effect of mesh stiffness asymmetric nonlinearity
Five different ratio values given by rk 0.25,0.50, 1.00, 2.00, 4. 00 of mesh stiffness mean value of coast side over the drive side one are selected to check their effects on the dynamic displacement responses. The second and third Fourier coefficients of the mesh stiffness ~ ~ ~ ~ are set to the fixed set of (kd 2 kc2 ,kd 3 kc3 ) (0.0405,0.0294).
In Fig. 3.7, the mesh stiffness asymmetry nonlinearity can be seen to only affect the dynamic displacement response in the double side impact range. This is again similar to the directional rotation radius study. The frequency of the primary resonance increases as the ratio of the mesh stiffness mean value of coast side over the drive side one increases. The effect appears
like a hardening spring, as seen in the enlarged plot Fig. 3.7(b). When rk 0. 25 and rk 0. 5 , all the double-sided tooth impact steady-state solutions remain stable, which means that the mesh stiffness asymmetric nonlinearity seems to affect the stability of the steady-state solution of the double-sided tooth impact. This is because that double-sided impact behaves similar to a hardening spring, as seen from earlier results. As the ratio of mean value of coast side over the drive side one decreases, the effect of hardening spring becomes weaker, and the solution
trajectory veers lesser to the right. As a result, for the case rk 0. 25 and rk 0. 5 , when gear pair undergoes the transition from single-sided tooth impact to double-sided tooth impact with the frequency sweeping down, there is no jump phenomenon seen in these cases at all. For the limiting case when the ratio of mean value of coast side and the drive side one equals zero, there is no double-sided tooth impact, and the solution trajectory does not veer to the right.
56
1 10
0 10 lacement lacement p
-1 Dis 10
-2 10 0 0.5 1 1.5 2 2.5 Frequency (a)
4.0 2.0
0.5 1.0 10 0.5 0.25
0.3 lacement 10 p Dis
0.1 10
0.7 0.8 0.9 1 1.1 Frequency (b)
Fig. 3.7, Effect of the mesh stiffness asymmetric nonlinearity on the dynamic response: (a) RMS of the dynamic displacement. (b) Primary resonance of the dynamic displacement. Stable steady-
state solution by multi-term HBM; Unstable steady-state solution by multi-term HBM
57
3.5 Conclusion
A nonlinear time-varying dynamic model for right-angle gear pair, considering both the
backlash and mesh coupling asymmetric nonlinearity, is formulated. The resultant equation of
motion is solved for the period-one dynamics using the multi-term harmonic balance method
(HBM) coupled with a modified discrete Fourier Transform procedure and numerical continuation method. The accuracy of the HBM solution is demonstrated by comparison to the computationally more intensive direct numerical integration solutions. The Floquet theory is applied to determine the stability of the steady-state harmonic balance solutions. A light load case using the example of a right-angle hypoid gear set is chosen for the parametric studies. The effects of the variation of mesh stiffness and directional rotation radii and their asymmetric nonlinearity on the dynamic displacement are analyzed. The increase in the variation of directional rotation radii and mesh stiffness can be seen to amplify the magnitude of the super- harmonic responses, and also raise both the magnitude and resonance frequency of the primary resonances. On the other hand, the directional rotation radius and mesh stiffness asymmetric nonlinearity seems to only affect the double-sided tooth impact response. The increase of the ratio of directional rotation radius mean value of coast side over drive side one seems to amplify both the magnitude and resonance frequency of the primary resonance. The increase of the ratio of mesh stiffness mean value of coast side over drive side more substantially amplifies both the
magnitude and peak frequency of the primary resonance. Furthermore the existence of mesh
stiffness asymmetry nonlinearity influence the stability of the double-sided tooth impact steady-
state solution and the transition between single and double-sided tooth impacts, and thus affect
the existence of the jump phenomenon.
58
Chapter 4. An Enhanced Multi-term Harmonic Balance Solution for Non-
linear Period- Dynamic Motions in Right-angle Gear Pairs
4.1 Introduction
Sub-harmonics and chaotic motions of piecewise linear systems were observed and
analyzed through using numerical methods in previous open publications, including Shaw and
Holmes (1983), Thompson and Elvey (1984), as well as Mahfouz and Badradhan (1990) For
systems with clearance type nonlinearity (Kahraman1 1992; Mahfouz and Badrakhan, 1990) or
systems with asymmetric nonlinearity (Natsiavas and Gonzalez, 1982) were also found to
experience sub-harmonics and chaotic motions. Geared rotor systems seem to be an even more
complicated type of mechanical system. Since it suffers from piecewise linear or nonlinear time- varying mesh parameters especially the mesh stiffness, backlash nonlinearity, as well as the asymmetric mesh parameters when double-side impact happens. Kahraman and Blankenship
(1996) investigated the sub-harmonics of parallel axis gear pair considering time-varying mesh stiffness and backlash nonlinearity. Ma and Kahraman (2005) further evaluated the sub- harmonics of parallel axis gear pair with piecewise nonlinear mesh stiffness. The sub-harmonics of a multi-mesh gear train were also studied by Al-shyyab and Kahraman (2005). In these studies, the typical multi-harmonic balance method was applied to solve for the gear dynamic motions. Wang and Lim (2007) formulated the equation of motion for hypoid gear pair system considering both backlash nonlinearity and time-varying mesh parameters including mesh points, line of action, directional rotational radii and mesh stiffness, and studied its sub-harmonics and chaotic motions through using numerical integration methods. Asymmetric mesh stiffness effect on the nonlinear gear dynamics was also investigated by Wang and Lim in their publication
59
(2009). Other approaches such as analog simulation, digital simulation, harmonic balance method, and multi-scale method were also applied to study the nonlinear gear motions in the past two decades. A more detailed literature review can be found in the paper by Yang, Peng and Lim
(2012).
In a previous paper of the author Yang, Peng and Lim (2012), the equation of motion of a right-angle geared system consisting of pinion and gear is given in dimensionless form: ~ ~ ~ g(x) ~ ~ ~ ~ ~x (t ) 2c(~x)g(~x)~x (t ) k(~x) f (~x) T (~x) T (~x) e~, (4.1) 1 p p g g where ~x is the relative dimensionless displacement between contacting teeth of gear and pinion. ~ In Eq. (1), ~x and ~x are its first and second derivative with respect to the dimensionless time t ; ~ ~ is the system parameter; Tp and Tg are the dimensionless forms of applied torque on pinion and gear respectively. The applied torque may vary with the change of pinion angular position, ~ ~ in this paper these torques are assumed to be constant for simplicity, and the relationTg Tp holds. The function c(~x)is the time-varying and asymmetric damping ratio, where and c(~x) are its mean value and normalized form respectively. The normalized damping ratioc(~x) is described as the fundamental dimensionless frequency~ component and its harmonics as follows:
H ~ ~ ~~ ~ ~~ ~ cd 1 [cd (2h) cos(ht ) cd (2h1) sin(ht )]; x 1, ~ h1 c(x) J (4.2) ~ ~ ~~ ~ ~~ ~ rda cc rda {1 [cc(2 j) cos( jt ) cc(2 j1) sin( jt )]}; x 1, j1
g(~x) where r da is the asymmetric index of damping ratio. The symbol is a function of directional radii of pinion and gear,
60
~2 ~2 ~ , x 0, g(~x) pd gd 2 ~2 2 ~2 ~ (4.3) rp pc rg gc , x 0,
where rp and rg are asymmetric index of pinion and gear directional radii respectively. The
~ ~ dimensionless directional radius of ponion and gear and are time-varying and asymmetric, p p
L ~ ~ ~~ ~ ~~ ~ pd 1 [ pd (2l) cos(lt ) pd (2l1) sin(lt )]; x 1, ~ ~ l1 p (x) M (4.4) ~ ~ ~~ ~ ~~ ~ rp pc rp {1 [ pc(2m) cos(mt ) pc(2m1) sin(mt )]}; x 1, m1
U ~ ~ ~~ ~ ~~ ~ gd 1 [gd (2u) cos(ut ) gd (2u1) sin(ut )]; x 1 ~ ~ u1 g (x) V . (4.5) ~ ~ ~~ ~ ~~ ~ rg gc rg {1 [gc(2v) cos(vt ) gc(2v1) sin(vt )]}; x 1 v1
The time-varying and asymmetric mesh stiffness is represented by k(~x) ,
A ~ ~ ~~ ~ ~~ ~ kd 1 [kd (2a) cos(at ) kd (2a1) sin(at )]; x 1 ~ a1 k(x) B . (4.6) ~ ~ ~~ ~ ~~ ~ rk kc rk {1 [kc(2b) cos(bt ) kc(2b1) sin(bt )]}; x 1 b1
where r is the asymmetric index of mesh stiffness. The nonlinear displacement function is k symbolized as f (~x) , which describes the clearance type nonlinearity,
~x 1, ~x 1 ~ ~ f (x) 0, 1 x 1. (4.7) ~ ~ x 1 x 1
The e~ represents the second derivative of dimensionless unloaded transmission error, the transmission error is the internal excitation of the geared system, and they are given by:
Y ~ ~ ~~ ~ ~~ e (e(2 y) cos(yt ) e(2 y1) sin(yt )), (4.8) y1
61
Y ~ ~ 2 ~ ~~ ~ ~~ e {(y) [e(2 y) cos(yt ) e(2 y1) sin(yt )]}. (4.9) y1
In this formulation, the mesh frequency, directional radii of pinion and gear, and damping ration are all asymmetric. Hence, the model contains asymmetric nonlinearity in addition to clearance nonlinearity. Furthermore, the model is also time-varying, and the system is subject to both parametric and external excitations. The period-one dynamic motions of the system were obtained by solving the dimensionless equation of gear motion using an enhanced multi-term harmonic balance method (HBM) with a modified discrete Fourier Transform process and numerical continuation method in another paper by the author.
This paper will focus on both the quantitative and qualitative analysis on the period- dynamic motions derived from the model described above using the same solution technique.
Bifurcation plots obtained using the numerical integration show that the system undergoes chaotic motions, which cannot be obtained by using the multi-term HBM. However, the stability analysis of the dynamic motions can reveal clues of the existence of bifurcation and chaos.
Similar procedure was used in a paper written by Benedettini, Rega and Salvatori (1992) to establish the existence conditions of sub-harmonic and chaotic motions. Parametric studies will also be performed to evaluate the effect of important parameters, such as drive load and kinematic TE, on the gear pair nonlinear dynamics.
4.2 Period- Sub-harmonic Response
First, let~ , where is the subharmonic index. The steady state solution is assumed to be a function of the fundamental frequency and its harmonics given by,
R ~ ~ ~ ~ ~ ~ ~ x(t ) x1 (x2r cos(rt ) x2r1 sin(rt )). (4.10) r1
62
Taking the derivatives yield,
R ~ ~ ~~ ~ ~~ ~ x (t ) (rx2r sin(rt ) rx2r1 cos(rt )), (4.11) r1
R ~ ~ 2 ~ ~ ~ ~ x (t ) {(r) [x2r cos(rt ) x2r1 sin(rt )]}. (4.12) r1
The transmission error and its second derivative are given by,
Y ~ ~ ~ ~ ~ e (e(2 y ) cos(yt ) e(2 y 1) sin(yt )), (4.13) y1
Y ~ 2 ~ ~ ~ ~ e {(y) [e(2 y ) cos(yt ) e(2 y 1) sin(yt )]}. (4.14) y1
Then the time series of the damping force, the non-linear restoring force and the external excitation forces can be gained by sampling n number of even points for one fundamental period ofT 2 / . It may be noted that the number of the time points n within one fundamental period, which must be larger than 2 R , where R is the highest harmonics of interest. Using this process, the time series of the damping force can be shown to be: ~ ~ ~ ~ ~ ~ ~ Fda (ti ) 2c(x(ti ))g(x(ti ))x(ti ), i 0,1,2...n 1. (4.15)
Similarly, the time series of the non-linear restoring force is, ~ ~ ~ g(x(ti )) ~ ~ F (t ) k(~x(t )) f (~x(t )), i 0,1,2...n 1; (4.16) k i 1 i i and the time series of the external excitation forces are
~ ~ ~ ~ ~ Fp (ti ) Tp p (x(ti )), i 0,1,2...n 1; (4.17)
~ ~ ~ ~ ~ Fg (ti ) Tg g (x(ti )), i 0,1,2...n 1. (4.18)
In order to use the multi-term HBM, all these force terms must be represented in the form of
Fourier series as shown,
63
R ~ ~ Fda Fda1 Fda(2r) cos(rt ) Fda(2r1) sin(rt ), (4.19) r1
R ~ ~ Fk Fk1 Fk (2r) cos(rt ) Fk(2r1) sin(rt ), (4.20) r1
R ~ ~ Fp Fp1 Fp(2r) cos(rt ) Fp(2r1) sin(rt ), (4.21) r1
R ~ ~ Fq Fq1 Fq(2r) cos(rt ) Fq(2r1) sin(rt ), (4.22) r1 where the coefficients of each Fourier series can be calculated using the discrete Fourier
Transform,
1 n1 2 n1 Fda1 Fda (ti ), Fda(2r) Fda (ti )cos(2rti ), (4.23a-b) n i0 n i0
2 n1 Fda(2r1) Fda (ti )sin(2rti ), i 0,1,2...n 1, (4.23c) n i0
1 n1 2 n1 Fk1 Fk (ti ), Fk(2r) Fk (ti )cos(2rti ), (4.24a-b) n i0 n i0
2 n1 Fk(2r1) Fk (ti )sin(2rti ), i 0,1,2...n 1, (4.24c) n i0
1 n1 2 n1 Fp1 Fp (ti ), Fp(2r) Fp (ti )cos(2rti ), (4.25a-b) n i0 n i0
2 n1 Fp(2r1) Fp (ti )sin(2rti ), i 0,1,2...n 1, (4.25c) n i0
1 n1 2 n1 Fg1 Fg (ti ), Fg(2r) Fg (ti )cos(2rti ), (4.26a-b) n i0 n i0
2 n1 Fg(2r1) Fg (ti )sin(2rti ), i 0,1,2...n 1. (4.26c) n i0
64
where Flm represents the Fourier coefficient of damping l daand restoring forces
l k , as well as the external forces exerted on pinionl p and gearl q respectively, where the subscript m 1,2...2R 1 represents the order of harmonics. Next, the algebraic sums of the coefficients of the terms with same frequencies and triangle function are forced to be equal.
S1 Fda1 Fk1 Fp1 Fq1, (4.27a)
2 ~ 2 ~ S 2r (r) x2r Fda(2r) Fk(2r) Fp(2r) Fq(2r) (r) e2r, (4.27b)
2 ~ 2 ~ S2r1 (r) x2r1 Fda(2r1) Fk(2r1) Fp(2r1) Fq(2r1) (r) e2r1. (4.27c)
where Si i 1,2...2R 1 represents the sum of the coefficients of the terms of different frequencies.
Finally, the Newton-Raphson method is applied to solve the vector equations S 0 for
~ ~ ~ ~ T the solution vector x [x1 x2 ... x2R1 ] which is given by
~x (m1) ~x (m) [J 1 ](m) S (m) , (4.28) where the m 1 th iteration of solution vector ~x (m1) is calculated using the previous iteration value of ~x (m) , S (m) and the Jacobian matrix J (m1) . The process is repeated until the steady
~ ~ ~ ~ T solution x [x1 x2 ... x2R1 ] converges within a predefined error limit. Once convergence is satisfied for the current iteration, a control parameter is set to the next value to restart the calculation process for the next iteration. In order to obtain multi-solutions corresponding to one excitation frequency, the response curve is described as the parametric function of its arc-length
(Yang, Peng and Lim, 2012). The Floquet theory procedure (Yang, Peng and Lim, 2012) is applied to determine the stability of the obtained solutions.
4.3 Results and Discussion 65
4.3.1 Comparison of HBM and numerical integration results
Light load case of a hypoid gear pair analyzed previously (Yang, Peng and Lim) is studied here. The dimensionless parameter values of this hypoid gear pair are listed in Table 4.1.
Only the variation component of the dynamic transmission error is plotted and analyzed in this paper, as it is the cause of the vibration and noise problem in geared rotor systems.
Table 4.1, Dimensionless dynamic parameters for a light load automotive hypoid gear pair
Parameter symbols Numerical value ~ Tp 0.1 0.03 0.75 ~ ~ cd ,cc 1 ~ ~ kd 2 ,kc2 -0.0405 ~ ~ kd 3 ,kc3 0.0294
rp ,rq ,rk ,rda 1 ~ ~ ~ ~ pd 2 , pc2 ,gd 2 ,gc2 0.01 ~ ~ ~ ~ pd 3 , pc3 ,gd 3 ,gc3 0 ~ e2 0 ~ e3 -0.5
Dynamic transmission errors predicted by both HBM and NI are compared quantitatively in Fig. 4.1. The comparison indicates that results obtained using HBM as well as NI match each other very well. The primary, period-2 and period-3 motions are clearly disconnected and easy to discern when HBM is applied. However, an extra step of data processing is required in order to distinguish the primary and sub-harmonic motions when NI is applied. Bifurcation diagram obtained from NI for the frequency sweep up case is plotted in Fig. 4.2. The result indicates that besides primary and sub-harmonic motions this hypoid gear pair also undergoes chaotic motions, which are indicated by unlimited discrete points for a single excitation frequency in the bifurcation plots. The HBM inherently has difficulties in predicting the level of chaotic motions.
66
However, in this case, the motions predicted by HBM and NI still match each other pretty well in the frequency range where the system undergoes chaotic motions. This interesting phenomenon will be demonstrated and explained in the following paragraphs.
Five excitation frequency points are chosen to demonstrate the primary, period 2, period
3, chaotic and period 6 motions, and the results are plotted in Fig.s 4.3-4.7. RMS of displacement
Frequency
Fig. 4.1, RMS of dynamic transmission error. Stable steady solution by multi-terms HBM;
Unstable steady solution by multi-terms HBM; ○ Solutions by Numerical Integration
67
Displacement Displacement
Frequency
Fig. 4.2, Bifurcation diagram of baseline case (frequency sweep up)
The primary motion predicted using HBM in the frequency range of interest, which is up to 2.5 times of the peak frequency, is plotted in Fig. 4.3(a). The FFT, Poincare section and phase plane trajectory of the motion, obtained by using NI for the case when the fundamental frequency of the internal excitation equals 0.45, are plotted in Fig. 4.3(b)-4.3(d). The Poincare map containing one discrete point indicates that it is a primary motion. The corresponding solution obtained using HBM, marked in Fig. 4.3(a) with a triangle symbol, is stable. The situation that the phase plane trajectory of the motion does not cross any of the -1 and 1 displacement lines indicates that it is a no impact motion (that is gear teeth remain in contact at all times), since the backlash is normalized into 1. Similarly, it is a single-side or double-side tooth impact motion when the phase plane trajectory of the motion crosses one or both of the 1
and -1 displacement lines. The response excited by the internal excitation with a fundamental
frequency of 0.45 is a no impact motion, as indicated by Fig. 4.3(d).
68
Displacement
Frequency a) b) Velocity Velocity
Displacement c) d) Fig. 4.3, a) Period 1 HBM solutions Stable steady solution; Unstable steady solution; Δ
Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d) Trajectory of NI result
Similarly, the period-2 and period-3 dynamic motions, as predicting using HBM, are plotted in Fig. 4.4(a) and Fig. 4.5(a). The FFT, Poincare section and phase plane trajectory of the period 2 and period 3 motions are plotted in Fig. 4.4(b)-4.4(d) and Fig. 4.5(a)-4.5(d). The period-
2 and period-3 responses for the NI cases are excited by the internal excitation with the fundamental frequencies of 0.95 and 1.75 respectively. The corresponding HBM solutions are stable, which are marked in Fig. 4.4(a) and Fig. 4.5(a). The period-2 and period-3 responses are
69 double-side impact motions as indicated by Fig. 4.4(d) and Fig. 4.5(d). Thus, the conclusion can be drawn that stable HBM solutions is capable of predicting both the level and the types of the hypoid gear dynamic motions.
Two peaks of the period-2 dynamic motions can be observed in Fig. 4.4(a), while only one peak of the period-3 dynamic motions can be seen in Fig. 4.5 (a). It is clear that the dynamic motions in the vicinity of the small peak of the period-2 dynamic solutions are unstable, and the hypoid gear pair undergoes chaotic motions as indicated in Fig. 4.2. Below, more parametric studies will be performed to analyze the complex motions and jump phenomenon around the small peak of the period-2 dynamic motions.
70
lacement p Dis
Frequency a) b) Velocity
Displacement c) d) Fig. 4.4, a) Period 2 HBM solutions Stable steady solution; Unstable steady solution; Δ
Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d) Trajectory of NI result
71
lacement p Dis
Frequency a) b) Velocity
Displacement c) d) Fig. 4.5, a) Period 3 HBM solutions Stable steady solution; Unstable steady solution; Δ
Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d) Trajectory of NI result
The response excited by the internal excitation with a fundamental frequency of 1.18 is computed using NI. The corresponding HBM solution is unstable as indicated by Fig. 4.6(a). It is easy to observe from Fig. 4.6(b)-4.6(d) that the hypoid gear pair undergoes chaotic motions. The chaotic response is also a double-side tooth impact type of behavior as evident from Fig. 4.6 (d).
Chaotic motion manifests itself as a broadband frequency response as shown in Fig. 4.6 (b).
However, for the case studied here, it is important to notice that most of the energy of the chaotic
72 motion is concentrated on two spikes as shown in Fig 4.6(b) with the frequencies that equal to the fundamental frequency or a half of it. This is probably the reason why the amplitude of the dynamic transmission error predicted by HBM and NI are very close; even though the solution obtained using HBM is unstable. Displacement
Frequency a) b) Velocity Velocity
Displacement c) d) Fig. 4.6, a) Period 2 HBM solutions Stable steady solution; Unstable steady solution; Δ
Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d) Trajectory of NI result
73
lacement p Dis
Frequency a) b) Velocity Velocity
Displacement c) d) Fig. 4.7, a) Period 3 HBM solutions Stable steady solution; Unstable steady solution; Δ
Compared HBM solution; b)FFT of NI result; c) Poincare plot of NI result; d) Trajectory of NI result
The period-3 response predicted by HBM is unstable as shown in Fig. 4.7(a), when the fundamental frequency of the internal excitation equals 1.61. The FFT, Poincare section and phase plane trajectory of the corresponding response obtained using NI are plotted in Fig. 4.7(b)-
4.7(d), and these plots implies that this is a period-6 motion. The vibratory energy of this period-
6 dynamic motion is concentrated on one spike as shown I Fig. 4.7(b) with a frequency of one third of the fundamental frequency, as shown by Fig. 4.7 (b). This explains that the amplitudes of
74 the dynamic transmission error predicted by the HBM and NI are very close to each other, in spite of the face that the HBM solutions are unstable. Further increasing the harmonic terms as introduced in section 2 to 6, the stable solution of period-6 motions definitely can be predicted.
For simplification, they are not plotted in this paper. From the study of unstable HBM solutions, another conclusion can be drawn that the unstable solution predicted by the HBM implies more complex motions, even chaotic motions.
4.3.2 Effect of Static Load
The period-2 solutions predicted by using HBM for the cases with the drivec load of 0.3 and 0.5 are plotted in Fig. 4.8(a) and 4.8(b). The frequency sweep up bifurcation diagrams of corresponding cases obtained using NI are plotted in Fig. 4.8(c) and 4.8(d). Observing the HBM solution plots, as the drive load is increased from 0.1 (baseline case shown in Fig. 4.1) to 0.3, the small peak of the period-2 solution is changed from unstable to stable, and the small peak is eliminated by further decreasing the static TE to 0.115.
More complex jump response is observed for the case that has two stable peaks as shown in Fig. 4.8(a), as compare to the cases which only have one stable peak as shown in Fig. 4.8(b).
For the case with the static load of 0.3, as shown in Fig. 4.8(a) and 4.8(c), the gear motion first jumps from a period 1 motion to a single-side impact period 2 motion, as shown in Fig. 4.9(a) and 4.9(b), with the jump frequency of 1.22. The gear motion then jumps to a double-side impact motion, as shown in Fig. 4.9(c) and 4.9(d), at the switch frequency of 1.42. With the further increase of the frequency to 1.48, the gear motion jumps to an unstable branch and undergoes a period doubling procedure to chaos as shown in Fig. 4.9(e)-4.9(h). After the chaotic motions, it jumps to a single-side impact period 2 motion, as shown in Fig. 4.9(i) and 4.9(j), with a jump frequency of 1.88. Finally it jumps back to a period 1 motion at the swift frequency of 2
75 approximately. As a comparison, as shown in Fig. 4.8(b) and 4.8(d), the gear motion of the 0.5 drive load case experiences a relatively simple jump process. It jumps from a primary motion to a double-side impact period-2 motion, then jumps to a single-sided tooth impact motion, finally jumps back to a period-1 motion. The phase plane trajectory and the Poincare map of the drive load case with the static load of 0.5 are similar as what are plotted in Fig. 4.9, and are not shown here for the simplification purpose.
As the static load is increased, the hypoid gear pair seems to experience less complex motions. For the baseline case with the static load of 0.1, gear motion undergoes primary, period-
2, period-3, period-6 and several period doubling process to chaotic motions. As a comparison, for the case with a 0.3 static load, it undergoes primary, period-2 and a period doubling process to chaotic motions; and for the case where the static load has a value of 0.5, it only undergoes period 1 and 2 motions.
76
lacement a) b) p Dis
c) d) Frequency
~ ~ Fig. 4.8, a) Period 2 HBM solutions (Tp 0.3 ); b) Period 2 HBM solutions (Tp 0.5 ); c) Bifurcation
~ ~ diagram (Tp 0.3 ); d) Bifurcation diagram (Tp 0.5 ); Stable steady solution; Unstable
steady solution
77
a) b) Velocity Velocity
c) d)
e) f)
78
g) h) Velocity Velocity
i) j) Displacement
Fig. 4.9, a) Trajectory on phase plane (~ 1.33 ); b) Poincare plot (~ 1.33 ); c) Trajectory on phase plane (~ 1.45 ); d) Poincare plot (~ 1.45 ); e) Trajectory on phase plane (~ 1.51 ); f) Poincare plot
(~ 1.51 );g) Trajectory on phase plane (~ 1.69 ); h) Poincare plot (~ 1.69 ); i) Trajectory on
phase plane (~ 1.95 ); j) Poincare plot (~ 1.95 )
79
4.3.3 Effect of Static Transmission Error Excitation
The period-2 solutions predicted using HBM for the cases with the static transmission error of 0.225 and 0.115 are plotted in Fig. 4.10(a) and 4.10(b). The frequency sweep up bifurcation diagrams of the corresponding cases obtained using NI are plotted in Fig. 4.10(c) and
4.10(d). Similarly, as the static TE is decreased from 0.5 (baseline case shown in Fig. 4.1) to
0.225, the small peak of period 2 HBM solution is changed from unstable to stable, and the small peak is eliminated by further decreasing the static TE to 0.115.
As the static TE increases, the gear motion seems to experience less complex jump process as well as less complex motions. For the case with a static TE of 0.225, as shown in Fig.
4.10(a) and 4.10(c), as the frequency is increased, the gear motion first jumps from the primary motion to a single-side impact period-2 motion, as shown in Fig. 4.11(a) and 4.11(b), at the jump frequency of 0.99. The gear motion then jumps to a chaotic motion rapidly, as shown in Fig.
4.11(c) and 4.11(d), with a switch frequency of 1.21. After the chaotic motion, it jumps to another single-side impact period-2 motion, as shown in Fig. 4.11(e) and 4.11(f), at a jump frequency of 1.84. Finally, it jumps back to period 1 motions at the frequency of 1.94. As a comparison, the gear motion of the case with a 0.115 static TE experiences relatively simple jump process, as shown in Fig. 4.10(b) and 4.10(d). As the frequency is increased, the gear dynamic motion first jumps from the period-1 dynamic motion to a double-sided tooth impact behavior, and then it jumps back to the primary motions. The phase plane trajectory and the
Poincare map of the case with a 0.115 static TE are not shown for simplification. For baseline case with a TE equals 0.5, the gear motion undergoes primary, period-2, period-3, period-6 and several period doubling processes to chaotic motions; for the case with a static TE of 0.225, it undergoes primary, period-2 and chaotic motions; and for the case with a static TE of 0.115, it only experiences period 1 and 2 motions. 80
a) b)
c) d) Frequency
~ ~ Fig. 4.10, a) Period 2 HBM solutions ( e3 0.225 ); b) Period 2 HBM solutions ( e3 0.115 ); c)
~ ~ Bifurcation diagram ( e3 0.225 ); d) Bifurcation diagram ( e3 0.115 ); Stable steady
solution; Unstable steady solution
81
a) b) Velocity Velocity
c) d)
e) Displacement f)
82
Fig. 4.11, a) Trajectory on phase plane (~ 1.1); b) Poincare plot (~ 1.1); c) Trajectory on phase plane (~ 1.38 ); d) Poincare plot (~ 1.38 ); e) Trajectory on phase plane (~ 1.91 ); f) Poincare plot
(~ 1.91 )
4.3.4 Sub-harmonic and Chaotic Motions
From the results of the previous sections, it can be concluded that the unstable HBM solution implies more complex motions. This clue is applied to locate the sub-harmonic as well as the chaotic motions approximately. The existence spaces of the sub-harmonic motions, indicated by the unstable primary solutions, as a function of drive load and TE are plotted in Fig.
4.12 (a) and 4.13(a) respectively. Similarly, the unstable period-2 and period-3 responses are used to locate the chaotic motions as a function of static load and static TE are plotted in Fig.
4.12 (b) and 4.13 (b) respectively. In those predicted sub-harmonic or chaotic motions frequency ranges, the primary or period- solutions may coexist, and the initial condition determines the gear motions.
The frequency range of probably existing of both sub-harmonic motions is reduced as the static load is increased or the TE is decreased, as indicated by Fig. 4.12 (a) and 4.13(a). The same trend for the chaotic motions can be seen from Fig. 4.12(b) and 4.13(b). In this case, the existence space of chaotic motions seems to have more than one frequency range for higher TE or lighter load condition. In the first one or two frequency range as shown in Fig. 4.12 (b), the gear motions seem to be double-side impact chaotic motions similar like Fig. 4.6, while in the last frequency range the gear motions seem to be single-side impact chaotic motions similar like
Fig. 4.10 (g-h) and 4.12 (c-d) .
83
Static load Static load
Frequency a) b) Fig. 4.12, a) Existence spaces of sub-harmonic motions; b) Existence spaces of chaotic motions; Ο start
point; + end point Static TE
Frequency a) b) Fig. 4.13, a) Existence spaces of sub-harmonic motions; b) Existence spaces of chaotic motions; Ο start
point; + end point
4.4 Conclusion
A nonlinear, time-varying dynamic model of right-angle gear pair systems is adopted to study the sub-harmonics and chaotic motions. This pure torsional gear pair model is
84 characterized with the time-varying excitations, as well as the clearance and asymmetric nonlinearities. The sub-harmonics and chaotic motions are studied using by solving the dimensionless equation of gear motion using the multi-term harmonic balance method with a modified discrete Fourier Transform process and numerical continuation method. The HBM steady-state solutions including period-1, period-2 and period-3 motions are shown to be in excellent agreement with the numerical integration results. Parametric studies reveal that the decrease in drive load or the increase of kinematic transmission error can result in more complex gear dynamic motions, especially for the period-2 motions. The frequency ranges for potential existing chaotic motions are predicted approximately using the stability analysis of obtained
HBM solutions for the studied case.
85
Chapter 5. Dynamics of Coupled Nonlinear Hypoid Gear Mesh and Time-
varying Bearing Stiffness Systems
5.1 Introduction
The dynamic mesh force of engaging gear teeth, excited by transmission error, is believed to be one of the major sources of vibration and noise coming from the geared transmission systems. Vibration of gear pair can be transmitted through shaft and bearing system into the gearbox housing and subsequently to the rest of the vehicle structures. The structure- borne noise is then radiated from the vibrating housing and vehicle body structures. To be able to control this response, it is imperative that we understand the underlying physics controlling the generation of the gear excitation. In geared rotor systems, gears and rolling element bearings are precision elements and are some of the most complex components inside the gearbox. Both of these machine elements are inherently nonlinear. However, their effects on the generation of gear excitation are still not well defined. Therefore, gaining an accurate representation and understanding of the dynamic interactions between the gears and bearing systems are essential in having the capability to predict the severity of gear vibration and noise produced.
The general stiffness matrix approach of representing the dynamic characteristics of rolling element bearing was formulated analytically by Lim and Singh (1990). Their theory was further implemented in general mechanical rotor and linear spur geared rotor systems (Lim and
Singh, 1990, 1991, 1992). For high frequency applications, the statistical energy analysis method was combined with the stiffness matrix representation to study the vibration energy transmitted through rolling element bearing (Lim and Singh, 1992). Later on, the formulation was refined by considering the effect of load distribution along the roller length (Lim and Singh, 1994). Based on those work (Lim and Singh, 1990, 1990, 1991, 1992, 1994), Liew and Lim (2005) developed 86 the time-varying stiffness formulation considering the raceway rotation, and implemented the new theory into a linear parallel geared rotor system. An alternative approach was used by
Lahmar and Velex (2003) to study the interaction between rolling element bearing and linearly behaving helical gear dynamics. In their study, time-varying property of rolling element bearing was formulated by recalculating the bearing stiffness matrix at each time step of the numerical integration process. Interaction of bearing radial clearance and spur gear pair backlash clearance was studied by using multiple degrees of freedom nonlinear dynamic model with multiple clearances (Comparin and Singh, 1990; Kahraman and Singh, 1991; Padmanabhan and Singh,
1992). However, no prior publications that directly deal with the issue of dynamic interaction between non-linear hypoid gear mesh and time-varying bearing model was found in the open literature. The attempt to fill this gap is one of the primary objectives of the present investigation.
An analytical framework for analyzing both linear and nonlinear pure vibration characteristics of hypoid and bevel geared rotor system was developed by Cheng and Lim (1998,
1999, 2000, 2000, 2001). The single-point mesh model was formulated to represent the coupling between engaging gear pair. Due to the curvilinear feature of gear tooth and kinematics, the coupling between engaging gear pair is time-varying. Backlash was also modeled to represent the clearance of the working gear pair. Later, Wang and Lim (2002) further developed a multi- point mesh model. Instead of using a single mesh stiffness coupling for the whole gear pair, a multi-point mesh representation is aimed at constructing an individual mesh stiffness coupling for each tooth pair that is in contact. As a result, the dynamic mesh force at each individual tooth pair can be obtained. More recently, a coupled multi-body dynamics and vibration model is developed by Peng (2010). In that formulation, the gear perturbation responses are analyzed simultaneously with the rigid body rotation of the gear pair. This requires the mesh model be
87 defined as a function of gear angular position rather than strictly time. Thus, the dynamic load of teeth in contact for practical load and operating cases can be simulated. However, the major drawback of both the multi-point mesh theory and coupled multi-body dynamics and vibration model is the high computational cost due to the significantly more complex mathematical expressions. In fact, the designs of these complex models make them more suitable for a wide variety of problems including fatigue, duty cycle and transient issues in addition to simply noise and vibration response. In this paper, since the main focus is on the vibration and noise response, the single-point mesh representation and pure vibration assumption are sufficient.
In this investigation, the dynamic interaction between nonlinear hypoid gear mesh and time-varying bearing stiffness is studied. Both time-varying mesh parameters and backlash nonlinearity of hypoid gear are also considered. The dynamic characteristics of the rolling element bearing are represented by a set of time-varying stiffness matrices. Numerical result seems to show that dynamic mesh force is insensitive to time-varying bearing stiffness effect.
On the other hand, the dynamic bearing load is found to be affected significantly by the temporal variation in its stiffness, as expected. The jump response frequency and amplitude of both the dynamic mesh force and dynamic bearing load are affected only slightly by the time-varying bearing stiffness property. On the other hand, both the dynamic mesh force and bearing loads are found to be sensitive to backlash.
5.2 Nonlinear Geared Rotor System Model
5.2.1 Basic Assumptions
A generic hypoid geared rotor system consisting of a coupling gear pair, shafts, bearings, engine and load is modeled with a nonlinear, 14 degrees of freedom (DOF) lumped parameter representation as shown in Fig. 5.1.
88
Fig. 5.1, Schematic of a 14-DOF nonlinear dynamic model of hypoid geared rotor system.
Pinion and gear are considered as rigid bodies, and the engine and load are each represented by a rotational coordinate. The flexibility of both pinion and gear shaft bearing assemblies is condensed into a set of stiffness matrices to support the pinion and gear bodies.
The coupling between the engaging gear pair is represented using single-point mesh model. The proposed mesh model consists of mesh point, line-of-action, and mesh stiffness, backlash and transmission error. The model used in this paper is similar to the one proposed by Peng (2010) except for the following feature. Due to the time-varying property of rolling element bearing, in this analysis, the support stiffness matrices are time-varying. This differs from the constant coefficient model previously employed by Peng (2010).
5.2.2 Mesh Model
The gear mesh model can be obtained by several ways, including pitch cone design theory, unloaded tooth contact analysis and loaded tooth contact analysis (Cheng, 2000; Wang,
2002; Peng, 2010). The first two approaches use idealized tooth surface geometry only. The third one uses both detailed tooth surface geometry and the effect of material elasticity. In this study,
89 the mesh model is condensed from the results of a loaded tooth contact analysis performed using a formulation that combines a semi-analytical theory with a 3-dimensional finite element (FE) approach (Vijayakar, 1991).
The geometry of hypoid gear pair is shown in Fig. 2(a), and Fig. 2(b) shows the result of
contact analysis qualitatively. For contact cell i in the mesh coordinate system Sm , its position
vector is represented by ri (rix , riy , riz ) , contact force is represented by fi , and ni (nix ,niy ,niz ) stands for its normal vector.
ni z fi
ri x y
(a) (b) Fig. 5.2, Loaded tooth contact analysis model: (a) Gear pair geometry, and (b) Contact cells on engaging
tooth surface.
A vector summation process is performed to obtain the total mesh force F given by
z N 2 Fj nij f i , F Fj , ( j x, y, z), (5.1) i1 jx
while the line-of-action Nnx ,n y ,nz can be obtained from
n j F j / F, ( j x, y, z). (5.2)
90
The mesh point Rxm ,ym , zm is computed 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
xm (M z Fx ym ) / Fy , (5.3a)
ym (M x Fx zm ) / Fz , (5.3b)
z m (M y Fz xm ) / Fx , (5.3c)
where M j ,( j x, y, z) is the moment about the axis j caused by the combined loads at all the contact cells. The expression for the moment can be explicitly given by
N {M x , M y , M z } f i (ri ni ). (5.4) i1 The angular transmission error is a measure of the deviation of the relative rigid body rotational motions of the gear pair from the ideal position as dictated by the gear ratio. The
loaded and unloaded translational transmission errors e and e0 are the projections of respective angular transmission error along the line of action of the net mesh force. Finally, the mesh stiffness is define by
k F /(e e ). (5.5) m 0 The mesh model will be utilized in the dynamic model to represent the mesh coupling between pinion and gear. Note that the mesh formulation established above is in the mesh model
coordinate system denoted by Sm . Therefore, it should be transformed into pinion or gear coordinate systems when assimilated into the dynamic model.
91
5.2.3 Shaft-bearing Assembly Model
Support stiffness matrices for the pinion [k p ] and gear [k g ] can be obtained computationally, analytically or experimentally. In this study, the FE approach utilizing beam elements is applied to extract the flexibility of shat-bearing assemblies. This proposed approach is applied to model the pinion/gear overhung position as shown in Fig. 5.3. Figure 5.3(a) is the physical structure and Fig. 5.3(b) is the corresponding FE representation.
Reference point
Bearing 1 Bearing 2
(a) (b)
Fig. 5.3, Shaft bearing assembly: (a) Physical structure, (b) Beam finite element representation.
In the shaft-bearing assembly, each rolling element bearing is represented by a set of time-varying stiffness matrices. Pinion/gear shaft is modeled as beam elements, and each node has 6-DOF. The relation between the applied load and deflection at the reference point will give the support stiffness matrix. In this investigation, six load cases are designed. In each load case,
one column of a 6 6 identity matrix is taken as the excitation force vector{Fi },i 1,2,... 6 exerted on the reference point. For each load case, the finite element equations are solved for the
displacement vector of reference point represented by{d i },i 1,2,... 6 . The support stiffness matrix can be calculated using
1 [kl ] [{d1},{d 2 },{d 3},{d 4 },{d 5 },{d 6 }] , l p, g. (5.6)
92
This modeling process for the mean support stiffness matrices combined with the geared rotor system dynamic model was previously validated by experimental results Peng (2010).
However, in this paper, the resultant support stiffness matrices are time-varying due to the time- varying property of rolling element bearings.
5.2.4 Time-varying Bearing Stiffness
The kinematics and coordinate system of a rolling element bearing is shown as Fig. 5.4.
The same nomenclature employed by Lim and Singh (1990, 1990, 1991, 1992, 1994), and Liew and Lim (2005) are adopted here. Raceways and rolling elements are assumed to be rigid except at the localized contact area. The Hertzian contact theory is used to calculate the normal contact forces.
y
ym M bym Fbym ym
Fbxm xm x
s (t) Mbxm xm
Fbzm zm
z
Fig. 5.4, Rolling element bearing kinematics and the corresponding coordinate systems.
The bearing system is assumed to be able to rotate freely about the z-axis, and
{Fbxm , Fbym , Fbzm , M xm , M ym } is the net load vector on the rolling element bearing. The vector
93
{ xm , ym , zm , xm , ym } is used to represent the displacements of the bearing system. The analytical relation between the load and displacement vector was previously formulated as a function of geometrical parameters by Lim and Singh (1990, 1990, 1991, 1992, 1994). The
elements of the bearing stiffness matrix kb are defined as the partial derivatives of load with respect to displacement. For the diagonal element for the x translational coordinate, this partial derivative is given by
Fbxm k xx , (5.7) xm and for the coupling element between x and y translational coordinates, it is given by
Fbxm k xy . (5.8) ym
The load magnitude and vector, and the rolling element orbital position affect the bearing stiffness matrix. In this paper, the load variation is assumed negligible, and only the effect of the orbital position of rolling element is considered. Furthermore, pure rolling motion between raceways and rolling element is assumed (Liew and Lim, 2005). Given these assumptions, the orbital position of the rolling element can be expressed as,
1 r 2 (s 1) b s t 1 cos( 0 ) z t , s 1,2,...Z, (5.9) 2 r d Z
where rb stands for the radius of rolling element, rb is the inner raceway curvature center radius
for ball bearing or pitch radius for roller type bearing, 0 is the unloaded contact angle, z is the mean shaft speed, s is the number index of rolling element, and Z stands for the total number of rolling elements. Therefore, for a given shaft speed and basic parameters of the rolling element bearing, the stiffness matrix can be expressed as the function of time.
94
5.2.5 Formulation
The ordinary differential equations of motion of the system can be written as
[M ]{x} [C]{x} [K]{x)} {F} (5.10)
T {x}{ E , x p , y p , z p , px , py , pz ,...xg , yg , zg , gx , gy , gz , L } (5.11)
[M] diag[I E , M p , M p , M p , I px , I py , I pz ,...M g , M g , M g , I gx , I gy , I gz , I L ] (5.12)
where E and L stand for the torsional coordinates of engine and load, and (xl , yl , zl , lx , ly , lz ) stand for the translational and rotational coordinates of the pinion (l p) or gear (l g) .
Correspondingly, I E and I L are the inertia masses of engine and load, and (M l , I lx , I ly , I lz ) are the masses and inertia masses of the pinion (l p) or gear (l g) . Note that the coordinates of
pinion and gear are represented in pinion coordinated system (S p ) and gear coordinate system
(S g ) as shown in Fig. 5.1.
The stiffness matrix [K ] consists of the contributions from condensed shaft and bearing support stiffness matrices of both pinion and gear, and it is defined as follows:
k E K K p (5.13) K g k L
where the two constants k E and k L stand for the torsional stiffnesses of the input and output shafts. The value of system damping matrix C depends on the type of damping model employed; for example, modal damping, system damping and component damping models.
The force vector {F} in Eq. 5.10 includes the external excitation and internal forces. The external excitation force can be torque fluctuation of engine, and the internal excitation force is a time-varying spatial vector caused by transmission error. In order to compute the excitation force
95 vector, the rotational radius for both pinion p px , py , pz and gear g gx ,gy ,gz need to be defined first:
lx N l (X l Rl ), l p, g (5.14a)
ly N l (Yl Rl ) , l p, g (5.14b)
lz Nl (Zl Rl ) , l p, g (5.14c)
where N l (nlx ,nly ,nlz ) and Rl stand for line of action and mesh point in pinion coordinate
system (l p) or gear coordinate system (l p) . Similarly, X l ,Yl , Z l are the unit vectors in the pinion or gear coordinate system. Then, the excitation force vector {F} in Eq. 5.1 can be obtained from:
{F} {TE , hp Fm ,hg Fm ,TL } {Fext } (5.15)
where TE and TL are the input and output torques, {Fext } represents the external excitation force
vector, and hl is defined as
hl {nlx ,nly ,nlz ,lx ,ly ,lz }, l p, g (5.16)
T Also, hl represents the transpose of hl , and Fm stands for the mesh force at any given point in time, which includes both spring and mesh damping forces as follows:
Fm km f e0 cm e0 , (5.17)
where cm is the mesh damping ratio. The dynamic transmission error is defined as:
T T hp {x p , y p , z p , px , py , pz } hg {xg , yg , zg , gx , gy , gz }. (5.18)
The nonlinear function f ( e) is applied to model the clearance nonlinearity inherent in the hypoid gear system, and is defined as:
96
e0 b, e0 b, f ( e) 0, b e0 b, (5.19) e0 b, e0 b, where b is the backlash in the hypoid gear pair.
5.3 Case Study
A typical industrially applied hypoid geared rotor system is studied here. The gear and bearing data is listed in Table 5.1. The configuration of pinion and gear shaft bearing assembly is shown in Fig. 5.5. The X direction translation stiffness for bearing 1 is given in Fig. 5.6.
Reference point Reference point
14mm 130mm 19mm 180mm
Bearing 1 Bearing 2 Bearing 3 Bearing 4 (a) (b)
Fig. 5.5, Shaft-bearing assembly: (a) Pinion, (b) Gear.
Table 5.1, Design parameters: (a) Gear data, (b) System data, (c) Data of bearing on pinion shaft, (d) Data of bearing on gear shaft (a) Gear data Pinion Gear Number of Teeth 14 45 Spiral Angle (rad) 0.48 0.48 Pitch Angle (rad) 0.3 1.27 Face Width (m) 0.06 0.038
(b) System data Engine Load Torsional moment (kg) 1.3 2.7 Input Output Torsional stiffness 143400 62000 Pinion Gear Mass (kg) 22 68.4 Torsional moment (kg*m) 0.038 0.78 Bending moment (kg*m) 0.34 0.61
97
(c) Data of bearing on pinion shaft Bearing #1 Bearing #2 Number of Rollers 17 21 Pitch Radius (m) 0.055 0.049 Contact Angle (rad) 0.237 0.32 Axial load (N) 2500 2500 Length of Roller (m) 0.04 0.023
(d) Data of bearing on gear shaft Bearing #3 Bearing #4 Number of Rollers 20 26 Pitch Radius (m) 0.061 0.054 Contact Angle (rad) 0.262 0.288 Axial load (N) 20000 20000 Length of Roller 0.027 0.02
2550
2540
2530
2520
2510 BearingX-X 1 stiffness (kN/mm) 0 10 20 30 Bearing angular position(degree) Fig. 5.6, Time-varying bearing stiffness
Interaction between the time-varying mesh parameters and time-varying bearing stiffness is studied using a heavy load case without jump phenomenon. The light load case with time- invariant mesh parameters is applied to study the interaction between backlash nonlinearity and time-varying bearing stiffness.
98
5.3.1 Mesh and Bearing Interaction
In this case, only the interaction between time-varying bearing stiffness and time-varying mesh model is studied. The effect of backlash is not considered. In order to do so, a heavy load case of hypoid geared rotor system is specified since previous studies (Cheng, 2000; Peng, 2010) show that teeth of pinion and gear do not lose contact under most heavy load cases. Then, the time-varying mesh model for the heavy load case is derived through the method discussed in the previous section, and the time-varying mesh stiffness is shown as Fig. 5.7. Both time-invariant bearing and time-varying bearing stiffness matrices are implemented here and the dynamic results including dynamic mesh force and dynamic bearing loads are calculated and compared.
60
59
58
57
56 Mesh stiffness (kN/mm) stiffness Mesh 55 0 10 20 30 Pinion roll angle (degree) Fig. 5.7, Time-varying meshes stiffness
Dynamic mesh forces, calculated using different bearing formulation, are compared as shown in Fig. 5.8. There is no jump phenomena seen on the dynamic mesh force plot, and thus the backlash nonlinearity does not appear to affect the dynamic simulation results, as expected.
As indicated by the comparison, the dynamic mesh force seems to be insensitive to the time- varying bearing stiffness. On the other hand, the bearing axial reaction force seem to be sensitive to the bearing stiffness, especially for the bearing with lower nominal axial load, as shown by
99
Fig. 5.9. Bearing radial reaction force and reaction moment have the trend for this case, and they are shown here. The results show the same trend reported by Lahmar and Velex (2003).
4 10
2 10 Dynamic mesh force (N) 0 10 0 200 400 600 800 Mesh frequency (Hz) Fig. 5.8, Dynamic mesh force comparison: time-invariant bearing stiffness,
time-varying bearing stiffness.
300 600
200 400
100 200
0 0 Bearing axial 2 reaction force(N) 0 200 400 600 800 Bearing axial 4 reaction force(N) 0 200 400 600 800 Mesh frequency(Hz) Mesh frequency(Hz) Fig. 5.9, Comparison of bearing axial reaction force: time-invariant bearing
stiffness; time-varying bearing stiffness.
5.3.2 Backlash and Bearing Interaction
In this case, only the interaction between time-varying bearing stiffness and gear backlash nonlinearity is studied. The effect of time-varying mesh model is not considered. Here, a time-invariant mesh model of a light load case is adopted to represent the mesh coupling
100 between pinion and gear. Three different combinations of bearing formulation and backlash nonlinearity are studied and compared to reveal the nature of the interaction between them: (1) time-invariant bearing stiffness with backlash nonlinearity; (2) time-varying bearing stiffness with backlash nonlinearity; and (3) time-varying bearing stiffness without backlash nonlinearity.
The dynamic mesh forces of case 2 for frequency sweep up and sweep down are shown in Fig.
5.10. And it indicates that jump phenomenon exists for this case.
The frequency sweeping up results of the first two combination cases are compared as shown by Fig. 5.11 and 5.12. The dynamic mesh force is insensitive to the time-varying formulation of the bearing system as indicated by Fig. 5.11, and this is similar to the conclusion gained in the previous section. However, the bearing radial reaction force is not sensitive to bearing characteristics as indicated in Fig. 5.12. The comparison of dynamic results also indicated that the jump frequencies and the magnitude of dynamic results show slight differences. The dynamic bearing axial reaction force, reaction moment and frequency sweep down case show the same trend for this case, and they are not shown in this paper.
3 10
2 10
1 10 Dynamic mesh force (N) force mesh Dynamic 0 10 0 200 400 600 800 Mesh frequency (Hz)
Fig. 5.10, Dynamic mesh force: frequency sweep up; frequency sweep down.
101
3 10
2 10
1 10 Dynamic mesh force (N) 0 10 0 200 400 600 800 Mesh frequency (Hz)
Fig. 5.11, Dynamic mesh force comparison: time-invariant bearing stiffness; time-
varying bearing stiffness
150 150
100 100
50 50
0 0 Bearing 2 radial reaction force(N) 0 200 400 600 800 Bearing 4 radial reaction force(N) 0 200 400 600 800 Mesh frequency(Hz) Mesh frequency(Hz)
Fig. 5. 12, Comparison of bearing radial reaction force: time-invariant bearing stiffness;
time-varying bearing stiffness
The frequency sweep up dynamic response results of the last two combination cases are compared as shown by Fig. 5.13 and 5.14. Both the dynamic mesh forces and dynamic bearing reaction load are affected significantly by backlash nonlinearity.
102
4 10
2 10 Dynamic mesh force (N) 0 10 0 200 400 600 800 Mesh frequency (Hz)
Fig. 5.13, Dynamic mesh force comparison: with backlash; without backlash.
4 5
4 3
3 2 2
1 1
0 0 0 200 400 600 800 0 200 400 600 800 Bearing 2 net reaction moment(N*m) Bearing 4 net reaction moment(N*m) Mesh frequency(Hz) Mesh frequency(Hz)
Fig. 5. 14, Comparison of bearing reaction moment: with backlash; without backlash.
Also, the dynamic mesh force and dynamic bearing loads seem to be reduced by the backlash nonlinearity effect. Previous study by Blankenship and Kahraman (1995) shows that backlash nonlinearity can stabilize the parametric excitation cause by time-varying mesh model.
Similar conclusion that the backlash nonlinearity is able to stabilize the parametric resonance caused by time-varying bearing representation can be drawn here. The bearing reaction force and frequency sweep down case also possesses the same trend for this set of condition, and they are not shown in this paper for brevity sake.
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5.4 Summary
A new approach to study the dynamic interaction of nonlinear hypoid gear mesh and time-varying bearing stiffness is proposed. Both backlash nonlinearity and time-varying mesh parameters, such as mesh stiffness, mesh point and line-of-action, are included in the nonlinear hypoid gear mesh model. The time-varying bearing stiffness due to the changing orbital position of rolling elements is modeled. Also, a practical hypoid geared rotor system is studied to reveal the characteristics of the complex interactions.
The dynamic simulation results of the interaction between the time-varying mesh parameters and time-varying bearing stiffness for a heavy load case without jump phenomenon reveal that the dynamic mesh force is relatively insensitive to the time-varying bearing stiffness.
On the other hand, the dynamic bearing loads are found to be quite sensitive to the time-varying bearing property, especially for the bearing installations with lower nominal axial load.
The analysis of the dynamic interaction between the time-varying bearing stiffness and gear backlash nonlinearity for a light load case with time-invariant mesh parameters yields the following findings for the three different combinations of bearing model and backlash nonlinearity studied: (1) time-invariant bearing stiffness with backlash nonlinearity; (2) time- varying bearing stiffness with backlash nonlinearity; and (3) time-varying bearing stiffness without backlash nonlinearity. The comparison of the first two cases shows that both dynamic mesh force and dynamic bearing loads are insensitive to the time-varying bearing stiffness for the case of light load condition with time-invariant mesh parameters. The comparison of the last two cases shows that backlash nonlinearity can stabilize the parametric resonance caused by time-varying bearing stiffness for the case of light load condition with time-invariant mesh parameters and time-varying bearing stiffness.
104
Chapter 6. Nonlinear Dynamic Simulation of Hypoid Gearbox with Elastic
Housing
6.1 Introduction
The overall dynamic model of hypoid gearbox with flexible housing representation is critical to predict accurately the vibration and acoustic noise response level. The root cause of the vibration and noise issues of gearbox is believed to be the kinematic transmission error of the gear pair. Dynamic mesh force of gear teeth in engagement is generated by the transmission error excitation, which in turn causes gear structure vibration. The vibration is transmitted into the housing through the shaft-bearing structure. Ultimately, the vibrating housing structure radiates noise to the surrounding space.
The basic framework for studying hypoid geared rotor system dynamics is proposed by
Cheng and Lim (Lim and Cheng, 1999; Cheng, 2000), and has been further developed by Wang
(2002), Wang (2007) and Peng (2010). Both pinion and gear are assumed to be rigid bodies, and the coupling between them is represented by the gear mesh model. The mesh model comprises of several key parameters, including mesh stiffness, mesh damping, line of action and mesh point that are all essentially time-varying. In order to accommodate assembly errors, backlash between gear teeth is included in the model as clearance type nonlinear characteristic. The stiffness matrix representation of the bearing dynamic characteristic applied here is the same one developed by
Lim and Singh (1990). The stiffness matrix formulation can be expressed analytically by basic rolling element bearing geometry parameters. This model has been applied successfully to predict the vibration transmitted through rolling element bearings in geared rotor systems (Lim and Singh, 1991). Relatively very few efforts emphasize the modeling of overall dynamic model
105 of hypoid gearbox with flexible housing representation. In one study, the modal model of housing is constructed and coupled with the internal geared assembly for modal analysis
(Hasselbring, 1999); however, the coupling between bearing and housing is relatively simple.
Later on, the same model is used to predict the dynamic responses of axle assembly by Mennem
(Mennem, 2004). To fill the present gap in the literature, one primary objective of this study is to formulate a nonlinear dynamic model of a hypoid gearbox.
Due to the nonlinearity of the hypoid gear meshing behavior, general frequency response function-based substructure technique (Lim and Li, 2000; Pan, Lim et al., 2003; Wang and Lim,
2005) is not applied, since it is only good for linear system. Instead of using the frequency response function directly, the lumped parameter model of the housing structure, extracted from an appropriate set of frequency response functions using modal parameter identification and modal synthesis method (Hurty, Collins, and Hart, 1971; Kuang and Tsuei, 1985; Allemang,
Brown et al., 1994; Allemang and Brown, 1998;), is used to represent the flexibility of the gearbox housing. A dynamic condensation procedure is applied to couple the hypoid geared rotor assembly with the lumped parameter model of the housing structure, since the proposed lumped parameter model of the housing is usually not a definite system. In order to couple the bearing and housing rotational coordinates, rigid body interpolation is also applied to obtain the housing rotational degrees of freedom at the bearing location.
To demonstrate the salient feature of the proposed approach, an analytical model of the housing is first analyzed. Subsequently, a practical application is considered to analyze the effect of housing flexibility on the dynamic mesh force, and the effect of external excitation, which is exerted on the housing, on the dynamic mesh force and housing surface vibration. The
106 parametric analysis reveals the sensitivity of dynamic mesh force to housing flexibility, and the significance of the effect of external force exerted on the housing on gear dynamic responses.
6.2 Geared Rotor System Dynamic Model
The hypoid geared rotor assembly, comprised of load, engine and a pair of gears in mesh supported by shafts and bearings, is modeled as a lumped parameter model with 14 degrees of freedom (DOF) as shown in Fig. 6.1. Pinion and gear are assumed to be rigid bodies with 6-DOF respectively, and only the torsional coordinates of the engine and load is considered. Prior studies (Cheng, 2000; Peng, 2010) using the comparison of simulation and testing results have shown the validity of the lumped parameter.
Fig. 6.1, Dynamic Model of hypoid geared rotor assembly
Time-varying and nonlinear mesh model is applied to represent the complex mesh coupling between the pinion and gear as shown in Fig. 6.1. Mesh parameters including mesh
stiffness km , mesh damping cm , mesh point Rxm ,ym , z m and line of action L nx , n y ,nz are essentially time-varying, due to the complex curvature of the hypoid gear tooth surface.
Unloaded kinematic transmission error 0 is modeled as the internal displacement excitation.
107
Clearance type nonlinearity is applied to represent the effect of backlashbc . All these parameters except backlash are obtained from the results of loaded tooth contact analysis, which is performed using a formulation that combines a semi-analytical theory with a 3-dimensional finite element (FE) approach (Vijayakar, 1991), through a spatial vector summation (Cheng, 2000;
Peng 2010).
The support stiffness matrix is applied to represent the flexible effect of the shaft-bearing assembly supporting either the pinion or gear. It is obtained by a finite element based condensation procedure (Peng, 2010). This condensation procedure is applied to the linear time- invariant part of the hypoid geared rotor assembly. Shafts are modeled using beam elements with the ability to account for non-uniform cross sectional areas, rolling element bearings are represented by a 5 5 stiffness matrix with rotational coordinates except the free rotation about the bearing-shaft axis (Lim and Singh, 1990).
Finally, the 14 equations of motion of hypoid geared rotor system are:
[M 1 ]{x1}[C1 ]{x1}[K1 ]{x1} {F1}, (6.1) where
T {x1}{ E , x p , y p , z p , px , py , pz , xg , yg , zg , gx , gy , gz , L } , (6.2)
[M1 ] diag[I E , M p ,M p , M p , I px , I py , I pz , M g , M g , M g , I gx , I gy , I gz , I L ]. (6.3)
The stiffness matrix is
k E K p K1 , (6.4) K g k L
108 where[K p ]and[K g ] are shaft-bearing assembly support stiffness matrices of pinion and gear
respectively, K E and K L are torsional stiffness of input and output shafts. The excitation force vector is
{F1} {TE , h p Fm ,hg Fm ,TL }{Fext }, (6.5)
where mesh force Fm between coupling gear pair and hl are defined as
Fm km f 0 cm 0 , (6.6)
hl {nlx , nly , nlz ,lx ,ly ,lz }, l p, g . (6.7)
The dynamic transmission error and nonlinear function f ( e)are defined as
T T hp {x p , y p , z p , px , py , pz } hg {xg , yg , z g , gx , gy , gz }., (6.8)
e0 bc , e0 bc , f ( e) 0, bc e0 bc , , (6.9) e0 bc , e0 bc ,
The rotational radius for both pinion p px , py , pz and gear g gx ,gy ,gz are defined as
lx Ll (X l Rl ), l p, g , (6.10)
ly Ll (Yl Rl ) , l p, g , (6.11)
lz Ll (Z l Rl ) , l p, g , (6.12)
where Ll (nlx , nly , nlz ) and Rl stand for line of action and mesh point in pinion coordinate system
(l p) or gear coordinate system (l g). Similarly, X l ,Yl , Z l are the unit vectors in the pinion or gear coordinate system.
6.3 Lumped Parameter Model of Housing
109
It is clear that the simplest way to obtain a lumped parameter model of the continuous structure is to discretize it using finite elements. However, the resultant size of the mass and stiffness matrices may be prohibitively large. In fact, in most cases, after these mass and stiffness matrices are coupled with the nonlinear time-varying hypoid geared rotor assembly, the computational requirement is too large to be practical. Alternatively, the modal model of the housing extracted from an appropriate set of frequency response functions (FRFs) is coupled with the internal geared rotor assembly. The housing structure considered in this study is shown
in Fig. 6.2(a).
Constrained Point 1 Pinion Bearing Point 2
Point 3 Gear Bearing
Fig. 6.2, a) Housing continuous structure; b) Excitation and response points
There are 14 response points shown in Fig. 6.2(b) on the housing surface and bearing locations; each possesses 3 orthogonal translation coordinates. Also, 3 of these 14 points are excited externally. The first order formulation, as described in Eq. (6.13-6.15), of the Unified
Matrix Polynomial Approach (Allemang, Brown et al., 1994; Allemang and Brown, 1998) is then applied to extract the modal parameters from the FRFs[H()]as shown in Fig. 6.3.
From the equation,
1 ( j) [A1][H()][A0 ][H()] [B0 ][I]. (6.13)
110 either [A1 ] or [Ao ] is assumed to be the identity matrix. This results in two equations whose solutions bound the “true” solution given by
-6 10
-8 10
-10 10
Displacement (m) Displacement -12 10
-14 10 200 400 600 800 1000 Frequency (Hz)
Fig. 6.3, Frequency response function of housing structure
[H ()] ( j)[H ()] ( j)[H ()] [A0 ] [B0 ] 2 (6.14) [I] ( j) [H ()] ( j)[H ()]
( j)[H ()] 2 ( j) [H ()] [H ()] [A1 ] [B0 ] 1 (6.15) [I] ( j) [H ()] ( j)[H ()]
The eigenvalues and eigenvectors of [Ao ]or the inverse of the matrix [A1 ] are essentially the roots and modal shapes[]. Fig. 6.4 shows the poles of the housing structure calculated using the method described above.
111
5.2
5.1
5
4.9
4.8
4.7 Damping (%)
4.6
4.5
4.4 -1000 -500 0 500 1000 Frequency (Hz) Fig. 6.4, Poles estimation from FRFs
The enhanced FRFs (eFRFs) are then generated by multiplying the system FRFs by the mode shape of the mode to be enhanced then summing all of the system FRFs into one eFRF.
This process is done per spectral line and can be visualized as “weighting” the FRFs that have more participation in a particular mode of interest more heavily than response points that have less participation, nodal points for example. Once the system FRFs are condensed down to the
SDOF representations for each of the mode shapes, an analytical SDOF equation can be fit to the eFRF, using a best fit solution. In this case, the Least Square scheme is used. The formulation of the least square problem is given as follows:
j[eH ()] i [eH ()] Si 0 (6.16)
i [eH ()] Si j[eH ()] 0 (6.17)
where i is thei th root, and S i is the residue of i th mode. Here, S i is also the modal scaling factor for the corresponding mode. eFRF and its SDOF fit of the sixth mode of the housing structure are shown as an example in Fig 6.5.
112
-5 10 Displacement (m) Displacement
-6 10
0 200 400 600 800 1000 1200 Frequency (Hz)
Fig. 6.5, Plot of Enhanced frequency response function; and Single-DOF
frequency response function fit
The housing is coupled with the hypoid geared rotor assembly through the rolling element bearings. To obtain the rotational coordinates of housing at bearing locations, a rigid body interpolation procedure is applied. This process is more easily described using a 2-D example with 3 DOFs, as shown in Fig. 6.6. Suppose a 2-D cross-section of a beam is assumed to move rigidly and the 3 DOFs of some point p are known. The displacement at some other point on the cross-section, suppose pointi , can be calculated using Eq.s (18) and (19) given as follows:
Fig. 6.6, Illustration of 2-D rigid body interpolation
113
X i X p Yi sin z X i (1 cosz ) (6.18)
Yi Y p Yi sin z Yi (1 cosz ) (6.19)
where(X i ,Yi ) is the distance of pointi from point p , and represents displacement. If the small
angle assumption is applied to z , the above equation can be written in a matrix format
X p X i 1 0 Yi Y p . (6.20) Y 0 1 X i i p
This process can be generalized to the 3 dimensional case in which it is necessary to calculate 6 DOFs of the rigid body. An example of the 3 dimensional rigid body equation is shown below:
X 1 0 0 0 Z Y i i i X Y 0 1 0 Z 0 X p i i i Y p Z i 0 0 1 Yi X i 0 Z p X 1 0 0 0 Z Y . (6.21) j j j Y 0 1 0 Z 0 X X , p j j j Z 0 0 1 Y X o Y , p j j j Z , p ... ......
With more measured points, the least square can be performed to obtain the six rigid body coordinates of the housing at the bearing location. The modal shape [] can then be
modified to be [m ] by substituting coordinates of measured points with the rigid body coordinates.
The [A] and [B] matrices consisting of mass, stiffness and damping matrices of the housing can be obtained as:
114
T [0] [M 2 ] [A] [m ] * diag (1/ S1...1/ S i ...1/ S n ) *[m ] (6.22) [M 2 ] [C 2 ]
T [M 2 ] [0] [B] [m ] * diag (1 / S1...i / Si ...n / S n ) *[m ] (6.23) [0] [K 2 ]
6.4 Coupled Dynamic Model
A direct coupling process is performed to couple a hypoid geared rotor assembly model with a housing lumped parameter model as follows:
x1,ext [[M 1,ext ] [M 1,coup ]] [0] x1,coup [0] [[M ] [M ]] x 2,ext 2,coup 2,ext x2,coup
x1,ext [C1,ext ] [C1,coup ] [0] [C1,coup ] x1,coup T (6.24) [0] [C ] [C ] [C ] [C ] x 1,coup 2,ext 2,coup 1,coup 2,ext x2,coup
x1,ext F1,ext [K1,ext ] [K1,coup ] [0] [K1,coup ] x1,coup F1,coup T [0] [K ] [K ] [K ] [K ] x F 1,coup 2,ext 2,coup 1,coup 2,ext 2,ext x2,coup F2,coup where the subscript coup represents coupled coordinates between hypoid geared rotor assembly model and housing lumped parameter model, andext represents uncoupled coordinates. The coupled coordinates include all the coordinates of pinion and gear except the rotational ones around the pinion and gear axis.
Because the modes of interest may be less than the measured degrees of freedom, the lumped parameter model of the housing can be a non-definite model, and convergence problem may be encountered during numerical simulation. Dynamic condensation is applied to solve this problem. Assume a transform matrix[] as:
115
x1 [I] [0] x1 x1 [] . (6.25) x 2 [0] [ m ] q q
Pre-multiplying the equation by the transpose of[] , it is condensed to a definite dynamic model,
x1,ext [[M 1,ext ] [M 1,coup ]] [0] x1,coup [0] [M r ] q
x1,ext [C1,ext ] [C1,coup ] [0] [C1,coup ][m ] T T T x1,coup [ ] [0] [C ] [C ] [ ] [[0] [C ]][ ] m 1,coup r m 1,coup m (6.26) q
F1,ext x1,ext [K1,ext ] [K1,coup ] [0] [K1,coup ][m ] T F1,coup T T T x1,coup [] [ ] [0] [K ] [K ] [ ] [[0] [K ]][ ] F m 1,coup r m 1,coup m q 2,ext F2,coup
where q is the modal coordinates of the housing structure, and [M r ],[Cr ]and[K r ]are modal mass, damping and stiffness matrices, respectively for the housing. The final displacement vector is defined in semi-physical and modal coordinates.
6.5 Case Study
A typical industrially applied hypoid geared rotor assembly is studied here. The gear and bearing data is listed in Table 6.1. The configuration of pinion and gear shaft bearing assemblies is shown in Fig. 6.7.
116
Reference point
30mm 100mm Bearing 1 Bearing 2
(a) Reference point
15mm 50mm Bearing 3 Bearing4
(b)
Fig. 6.7, Shaft-bearing assembly: (a) Pinion, (b) Gear
Table 6.1, Design parameter: (a) Gear data, (b) System data, (c) Data of bearing on pinion shaft, (d) Data of bearing on gear shaft
(a) Gear data Pinion Gear Number of Teeth 14 45 Spiral Angle ( rad ) 0.48 0.48 Pitch Angle ( rad ) 0.3 1.3 Face Width ( m ) 0.06 0.04
(b) System data Engine Load Torsional moment ( kg * m 2 ) 2 5 Input Output Torsional stiffness ( N / m ) 143400 72000 Pinion Gear Mass ( kg ) 24 120 Torsional moment ( kg * m 2 ) 0.06 2 Bending moment ( kg * m 2 ) 0.5 2
(c) Data of bearing on pinion shaft Bearing #1 Bearing #2 Number of Rollers 17 21 Pitch Radius ( m ) 0.055 0.049 Contact Angle ( rad ) 0.237 0.32 Axial load ( N ) 2500 2500 Length of Roller ( m ) 0.04 0.023
117
(d) Data of bearing on gear shaft Bearing #3 Bearing #4 Number of Rollers 20 26 Pitch Radius ( m ) 0.061 0.054 Contact Angle ( rad ) 0.26 0.29 Axial load ( N ) 20000 20000 Length of Roller ( m ) 0.027 0.02
6.5.1 Methodology Validation
In order to validate the above procedure, an analytical model with 10 degrees of freedom, as shown in Fig. 6.8, is coupled with the dynamic model of the hypoid geared rotor assembly.
This coupling is performed in two ways. One uses a full coupling with all ten modes; and the other uses a partial coupling with the first five modes. Equation (6.24) is used to fully couple the analytical model and the geared rotor assembly dynamic model; concurrently, Equation (6.26) is
used for the partial coupling in which[m ] is the mode shape matrix of the first five modes.
Fig. 6.8, Lumped parameter model representing a housing structure
The first 5 coordinates are coupled with the pinion coordinated except rotation around the pinion axis. Similarly, the last 5 coordinates are coupled with the gear coordinates except
118 rotation around the gear axis. The specific parameters of this analytic model are given in Table 2, with viscous damping applied. Each of the mass blocks is assumed to be 1 kg.
Table 6.2, Stiffness and damping of the analytical model
No. K (N/m) C (N*s/m) 1 2250000 150 2 9000000 300 3 20250000 450 4 36000000 600 5 56250000 750 6 81000000 900 7 110250000 1050 8 144000000 1200 9 182250000 1350 10 225000000 1500
Fig. 6.9 and 6.10 show the comparison of dynamic mesh force and acceleration of the first mass block obtained by assuming full, partial and no coupling between the geared rotor assembly and the analytic model. There is no acceleration for the “no coupling” case as indicated by Fig. 6.10. The results show that the analytical model affects the dynamic mesh force significantly, and the accuracy of the partial coupling case is comparable to the full coupling case. Therefore, only the partially coupling will be applied for nonlinear simulation of the hypoid geared rotor assembly with a continuous elastic housing structure, as described in the following section.
Equations (6.6-6.8) indicate that the dynamic mesh force is a function of all the 14 degrees of freedom of a geared rotor assembly. Thus, the dynamic mesh force is not only a function of resonance frequencies but also a function of modal shapes. This explains that the peak value of the dynamic mesh force seems to be shifted to higher frequency for this case when flexible housing is coupled to the geared rotor assembly.
119
2500
2000
1500
1000
Dynamic Mesh Force(N) 500
0 0 100 200 300 400 500 Mesh Frequency (Hz) Fig. 6.9, Comparison of dynamic mesh force: Full coupling; Partial coupling;
No coupling.
0.25
0.2
0.15
0.1 Acceleration
0.05
0 0 100 200 300 400 500 Mesh Frequency (Hz)
Fig. 6.10, Acceleration of mass block 1: Full coupling; Partial coupling
6.5.2 Parametric Study
A practical housing structure is coupled with the hypoid geared rotor assembly in the following study. Applying the rigid body interpolation to obtain the rotational coordinates, the
120 partially coupled procedure is applied. Two partial coupling cases are studied, one to couple the first 5 modes of the housing with a geared rotor system, and the other to couple the first 10 modes. Coupling with the first 10 modes of housing is believed to be representative when the mesh frequency is less than 500 Hz, since the natural frequency of the tenth mode is around 1000
Hz, as shown in Fig. 6.4.
2500
2000
1500
1000
Dynamic Mesh Force(N) 500
0 0 100 200 300 400 500 Mesh Frequency (Hz)
Fig. 6.11, Comparison of dynamic mesh force: 10-modes coupling; 5-modes coupling;
No coupling.
121
1 10
0 10
-1 10
-2 10 Housing Surface Acceleration
-3 10 0 100 200 300 400 500 Mesh Frequency (Hz)
Fig. 6.12, Comparison of surface acceleration: 10-modes coupling; 5-modes coupling
Figures 6.11 and 6.12 show the comparison of dynamic mesh force and housing surface acceleration, respectively, using 5-mode, 10-modes and no coupling cases. No housing surface acceleration data is available for the “no coupling” case, and the Y-direction acceleration of
Point 1, defined previously in Fig. 6.2(b), is plotted in Fig. 6.12. The results indicate that the 5- modes coupling assumption is accurate enough for this case to predict dynamic mesh force and housing acceleration when the mesh frequency is less than 500 Hz. Based on the curve shown, the dynamic mesh force appears to be sensitive to the housing elastic characteristics.
Comparisons of housing surface accelerations at other locations and directions also support this conclusion. For simplification purposes, these results are not included as an additional proof, and the data will not be shown in this study.
External excitation on the housing surface then is applied simultaneously with the internal TE excitation to study the effect of external excitation on the dynamic mesh force and housing surface acceleration. Two cases with the external excitation exerted on different locations, Points 1 and 2 as shown in Fig. 6.2(b), on the housing are studied. The external
122 excitation has a magnitude of 2000 N and a frequency of 360 Hz. As indicated by Fig. 6.13, the dynamic mesh force is more sensitive to the external excitation exerted on Point 2. This is similar to the linear cases, where the sensitivity of response to excitation is determined by the mode shape. A FFT of dynamic mesh force for a single mesh frequency with a value of 300 Hz for the two cases is plotted in Fig. 6.14 and 6.15. As shown, the magnitudes of dynamic mesh force at 360 Hz for the two cases are different, while the magnitudes of the dynamic mesh forces at mesh frequency and harmonics show very little difference.
2500
2000
1500
1000
Dynamic Mesh Force (N) 500
0 0 100 200 300 400 500 Mesh Frequency (Hz)
Fig. 6.13, Comparison of dynamic mesh force: No external excitation; Excitation
exerted on point 1; Excitation exerted on point 2.
123
1000
800 X: 360 Y: 673.8
600
400
Dynamic Force Mesh (N) 200
0 0 200 400 600 800 1000
Frequency (Hz) Fig. 6.14, FFT of dynamic mesh force with external excitation on point 1
1000
800
600
400
X: 360 Y: 156.5
Dynamic Force Mesh (N) 200
0 0 200 400 600 800 1000
Frequency (Hz) Fig. 6.15, FFT of dynamic mesh force with external excitation on point 2
Housing surface accelerations in the Y-direction of Point 1, for different cases are plotted and compared in Fig. 6.16. Based on mix results, the housing surface acceleration is sensitive to the external excitations, and the effect is location dependent.
124
1 10
0 10
-1 10
-2 10 Housing Surface Acceleration
-3 10 0 100 200 300 400 500 Mesh Frequency (Hz)
Fig. 6.16, Comparison of surface acceleration: No external excitation; Excitation
exerted on point 1; Excitation exerted on point 2
6.6 Conclusion
A methodology to predict the nonlinear dynamic response of hypoid gearboxes with an elastic housing structure is proposed. The dynamic substructure strategy is applied to assemble the hypoid gear pair, shaft-bearing assembly and elastic housing models. An analytical housing model with 10 degrees of freedom is first applied to demonstrate the accuracy of the proposed formulation by comparing the dynamic responses obtained under full and partial coupling assumptions. Then a practical housing structure is examined by coupling with the hypoid geared rotor assembly to study the effect of housing elasticity on the dynamic responses. The parametric analysis reveals the sensitivity of dynamic mesh force to housing flexibility. Finally, the effect of external force exerted on the housing on gear dynamic responses can be significant and also appears to be location dependent.
125
Chapter 7. Propeller Shaft Bending and Effect on Gear Dynamics
7.1 Introduction
The propeller shaft is found to be very important when dealing with the automotive and off-highway vehicle driveline system noise, vibration and harshness (NVH). It is because that the propeller shaft has its bending and torsional resonance in the frequency ranges of multiple sources, particularly the axle gear mesh excitation frequency range.
The basic framework of studying the dynamics of the driveline system with hypoid gear pair was built by Cheng and Lim (1998, 1999, 2000, and 2001), and was further enhanced by
Wang (2002), Wang (2007), Peng (2010), and Yang, et al. (2011). Both pinion and gear were assumed to be rigid bodies, and only the torsional degrees of freedom of the engine and load were considered. The dynamic coupling between the gear pair in contact was represented as an efficient and comprehensive mesh model comprised of several time-varying mesh parameters including mesh stiffness, mesh damping, line of action and mesh point. The clearance type nonlinearity was also included to take into account the effect of gear teeth backlash. However, a simple torsional spring representation of the propeller shaft was adopted to consider its torsional flexibility which was applied to couple the inertia of moments of pinion and the engine. One of the primary purposes of this paper is to model and evaluate the effect of the bending flexibility of the propeller shaft on the nonlinear driveline dynamics with hypoid gear pair. It was found that the damping of the propeller shaft could reduce the driveline vibration from an experimental study by Sun (2011). Considering that it is easier to do damping treatment for the propeller shaft as compared to do so inside the rear axle, one of the other purposes of this study is to numerically evaluate the effect of propeller shaft bending damping on the dynamics of driveline systems with hypoid gear pair.
126
The clonk (300-5000Hz) was analyzed by Theodossiades, et al. (2004, 2005) using a multi-body dynamic model of the three-piece driveline and the rear axle gear pair of a light truck in ADAMS. The flexibility of the lightly damped thin-walled drive shafts with high modal density was included by using the finite element technique in Nastran and component mode synthesis method (CMS). Universal joints were simplified as constraints for those rigid bodies.
The obtained vibration of the thin wall drive shafts was applied to predict the noise radiated into the environment. Some other researchers, Mazzei, et al. (1999) and DeSmidt, et al. (2002, 2004), considered the universal joint as a nonlinear component, and studied the dynamics of rotating shaft systems driven through universal joint. It was observed that the driven shaft would experience rotational speed, bending moments and torque fluctuation while the torque and rotating speed of the driving shaft was constant.
In this study, a lumped parameter model of the propeller shaft is modeled through using the component mode synthesis. The frequency range of interested is below 1000 Hertz, and thus the first several bending resonances are sufficient to represent the bending flexibility of the propeller shaft. A nonlinear rigid body model is established for the hypoid geared rotor system following the framework built by Cheng and Lim (1998, 1999, 2000 and 2001). The pinion input shaft is driven by the propeller shaft through a universal joint, which could be modeled as a flexible simple supported boundary condition as well as fluctuating rotation speed and torque excitation. Those excitations will not be included in this study. Since the effects of the input rotation speed and torque fluctuations on the dynamics of the hypoid geared rotor system were studied by Tao (2011). The methodology then is applied to evaluate the effect of the propeller shaft bending flexibility and damping on the dynamics of a practical applied off-highway rear axle system. The numerical studies seem to indicate that the effect of the propeller shaft bending
127 elasticity and damping on dynamic responses is mode shape dependent, and that the increasing of propeller shaft bending damping is able to reduce the dynamic mesh force.
7.2 Mathematical Model
Both geared rotor system and the bending flexibility of the propeller shaft are represented as lumped parameter models. The torsional flexibility of the propeller shaft is simplified as a torsional spring and damping elements to couple the inertia of moments of engine and pinion.
The universal joint could be modeled as a flexible simple supported boundary condition as well as fluctuating rotation speed and torque excitation. However, fluctuations of speed and torque are not included in this study. Since those excitations were modeled and evaluated by Tao (2011).
7.2.1 Simplified Propeller Shaft Model
It is reasonable to simplify the propeller shaft as beam with different cross sections as shown in Fig. 7.1, considering only the bending flexibility of the shaft will be included. The torsional flexibility is modeled separately as a torsional spring between engine inertia and pinion inertia of moments.
y r
l t x
(a) (b)
Fig. 7.1, Simplified propeller shaft (a) beam with different cross-section, (b) Cross-section
dimension definition
128
Component mode synthesis method is capable of mapping the physical structure except a pre-chosen interface into the mode coordinate system. The interface of the propeller shaft is obviously the free end at right which is physically connected with pinion input shaft. According to CMS, the dynamic displacements of constrain-free propeller shaft are the sum of the static displacements caused by unit forces exerted at the interface coordinates shown in Fig. 7.2 (a) and the dynamic displacements of the propeller shaft fixed at the interface as shown in Fig. 7.2 (b).
The CMS mapping matrix Ψ can be calculated through using the following formulation
x x I0x x Ψ , (7.1) x K K x Φq K K Φ q q where q and q indicate the internal and coupling coordinates. The symbol Φ and η represent mode shapes and the corresponding modal coordinate vector of the constraint system shown in Fig. 7.2 (b). The symbols K i, j 1, 2 represent the stiffness matrices of the static system shown in Fig. 7.1 (a). From the static equilibrium equation of motion shown in Eq. 7.2
, (7.2) 0 it is easy to obtain the displacements at internal coordinates as a function of the displacements at the interface shown in Eq. 7.3
. (7.3)
For a point on beam, there are two general displacements, transverse displacement and rotation of the cross-section. So, unit forces f exerted at the interface is a 2*2 identical matrix.
129
Unit forces Constraint interface
(a) (b)
Fig. 7.2, (a) Static system with unit forces applied at interface, (b) Dynamic system with fixed interface
boundary condition
The mass, stiffness and damping matrices of the system after applying the CSM can be obtained by the following formulations
(7.4 a)
K Ψ kΨ (7.4 b)
C Ψ cΨ (7.4 c)
Similarly, the new load vector can be calculated using F Ψ f (7.4 d)
7.2.2 Geared Rotor System Model
The hypoid geared rotor system, comprised of pinion, gear, engine, load and the supporting shaft-bearing assemblies, is represented as lumped parameter model as shown in Fig. 7.3.
Fig. 7.3, Lumped parameter model of hypoid geared rotor system
130
Both the pinion and gear are modeled as rigid bodies, and only the inertia of engine and load are included. The mass matrix and the displacement vector of the geared system are obtained as
[M 1 ] diag[I E , M p , M p , M p , I px , I py , I pz , M g , M g , M g , I gx , I gy , I gz , I L ] , (7.5)
{x }{ , x , y , z , , , , x , y , z , , , , }T 1 E p p p px py pz g g g gx gy gz L . (7.6)
The supporting stiffness matrices for pinion k and gear k are extracted from a finite element model of the supporting shaft-bearing assembly similarly as illustrated in Chapter 5.
However, in this chapter, the mean stiffness matrices are applied to represent the rolling element bearings. The torsional flexibility of propeller and output shafts is simplified as a torsional spring connecting driver and pinion, as well as load and gear. The stiffness matrix of the geared system is obtained as
k E K p K1 . (7.7) K g k L
The dynamic mesh coupling between the pinion and gear is represented by a mesh model, which is essentially nonlinear and time-varying due to the complex curvature of the hypoid gear
tooth surface. Mesh parameters include mesh stiffness km , mesh damping cm , mesh point
Rxm ,ym , zm , line of action L nx ,ny ,nz , backlashbc , as well as the loaded and unloaded
kinematic transmission error and 0 . All these parameters except backlash are obtained from the results of loaded tooth contact analysis, which is performed using a formulation that combines a semi-analytical theory with a 3-dimensional finite element (FE) approach (Vijayakar, 1991), through a spatial vector summation (Cheng, 2000; Peng 2010). The contact force along the line of action applied at mesh point is calculated
131
Fm km f 0 cm 0 (7.8) where f ( e) represent the nonlinear displacement function caused by backlash
0 bc , 0 bc f ( 0 ) 0, bc 0 bc . (7.9) 0 b c , 0 bc
The excitation vector of the geared system is obtained by projecting the contact force into pinion and gear coordinate systems
{F1} {TE , hp Fm ,hg Fm ,TL }, (7.10)
where the projection vector hl are defined as
hl {nlx , nly , nlz ,lx ,ly ,lz }, l p, g . (7.11)
The dynamic transmission error can be obtained as the difference of the pinion and gear displacements along the line of action
T T hp {x p , y p , z p , px , py , pz } hg {xg , yg , z g , gx , gy , gz }. (7.12)
The rotational radius for both pinion p px , py , pz and gear g gx ,gy ,gz are defined as
lx Ll (X l Rl ), l p, g , (7.13a)
ly Ll (Yl Rl ) , l p, g , (7.13b)
lz Ll (Z l Rl ) , l p, g , (7.13c)
where Ll (nlx ,nly ,nlz ) and Rl stand for line of action and mesh point in pinion coordinate system
(l p) or gear coordinate system (l g) . Similarly, X l ,Yl , Z l are the unit vectors in the pinion or gear coordinate system.
Finally, the equation of motion of the geared system is obtained
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[M 1 ]{x1}[C1 ]{x1}[K1 ]{x1} {F1}. (7.14)
7.2.3 Coupled Boundary Condition
Physically, the propeller shaft is connected with the pinion input shaft through universal joint. There is no bending moment transferred directly from propeller shaft to pinion input shaft.
Two transverse springs and damping elements along both x and z axis are assumed to represent the joint. Due to a distance L between the free end of the pinion shaft assembly and its mass center as shown in Fig. 7.4, extra moments applied to the pinion can also be generated by the transverse forces.
Mass
Fig. 7.4, Schematic of propeller shaft and pinion coupling
In pinion coordinate system, the pinion rotates along the y-axis, while the propeller shaft is assembled along the negative y-axis as shown in Fig. 7.4. In the same coordinate system, the coupling transverse forces between propeller shaft and pinion can be calculated
F K x x L∗θ (7.15a)
F K z z L∗θ . (7.15b)
The moments induced by the transverse forces to the pinion can be computed
(7.16a)
. (7.16b)
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7.3 Numerical Results and Discussion
The studied propeller shaft dimension, illustrated in Fig. 7.1, is listed in Table 7.1. For this set of data, the fifth bending natural frequency is 3455 Hz, which is good enough for studying the propeller shaft bending flexibility effect on gear dynamics under 1000 Hz.
Table 7.1, Geometry dimension of propeller shaft
Session 1 0.4 0.025 0.0016 Session 2 0.8 0.03095 0.0016
A typical industrially applied hypoid geared rotor assembly is studied here. The gear and bearing data is listed in Table 7.2. The configuration of both pinion and gear shaft bearing assemblies is shown in Fig. 7.5.
Reference point
30mm 100mm Bearing 1 Bearing 2
(a) Reference point
15mm 50mm Bearing 3 Bearing4
(b)
Fig. 7.5, Shaft-bearing assembly: (a) Pinion, (b) Gear
Table 7.2, Design parameter: (a) Gear data, (b) System data, (c) Data of bearing on pinion shaft, (d) Data of bearing on gear shaft
(a) Gear data Pinion Gear Number of Teeth 14 45 Spiral Angle ( rad ) 0.48 0.48 Pitch Angle ( rad ) 0.3 1.3 Face Width ( m ) 0.06 0.04
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(b) System data Engine Load Torsional moment ( kg * m 2 ) 2 5 Input Output Torsional stiffness ( N / m ) 143400 72000 Pinion Gear Mass ( kg ) 24 120 Torsional moment ( kg * m 2 ) 0.06 2 Bending moment ( kg * m 2 ) 0.5 2
(c) Data of bearing on pinion shaft Bearing #1 Bearing #2 Number of Rollers 17 21 Pitch Radius ( m ) 0.055 0.049 Contact Angle ( rad ) 0.237 0.32 Axial load ( N ) 2500 2500 Length of Roller ( m ) 0.04 0.023
(d) Data of bearing on gear shaft Bearing #3 Bearing #4 Number of Rollers 20 26 Pitch Radius ( m ) 0.061 0.054 Contact Angle ( rad ) 0.26 0.29 Axial load ( N ) 20000 20000 Length of Roller ( m ) 0.027 0.02
The comparison of dynamic mesh forces of the cases with rigid and flexible propeller shaft bending assumptions is shown in Fig. 7.6. It is easily to observe that propeller shaft bending flexibility only affects the dynamic mesh force around 625 Hz. The linear modal analysis of the geared rotor system shows that the pinion rotations are the dominant degrees of freedom of the
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2000
1500
1000
500 Dynamic Mesh Force (N)
0 0 200 400 600 800 1000 Mesh Frequency (Hz) Fig. 7.6, Comparison of dynamic mesh force: Rigid Propeller Shaft; Propeller Shaft
Bending (L=0.1); Propeller Shaft Bending (L=0.15) mode shape of the resonance around 625 Hz as shown in Fig. 7.7. Further increasing the distance between the end and the center of pinion shaft assembly L from 0.1 m to 0.15 m, the change of dynamic mesh force around 625 Hz becomes more evident as expected. This is because that the moments induced by transverse forces to the pinion are increased with the increase of L.
Explanation of x-axis component 1 2 3 4 5 6 7 8 9 10 11 12 13 14 2
1.5
1
0.5
0
-0.5
-1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Fig. 7.7, Modal shape of the linear model
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The bearing reaction forces are also found to be affected by the propeller shaft bending elasticity. The radial force and the moment seem to be affected more significant as compared with the axial force as shown in Fig. 7.8. Comparing Fig 7.8 and 7.9, it is observed that reaction forces on pinion bearing are more sensitive to the propeller shaft bending flexibility.
1400
1200
1000
800
600
400 Bearing Radial Force (N) 200
0 0 200 400 600 800 1000 Mesh Frequency (Hz) (a)
250
200
150
100
Bearing Axial Force (N) Axial Force Bearing 50
0 0 200 400 600 800 1000 Mesh Frequency (Hz) (b)
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25
20
15
10 Bearing Moment (Nm) 5
0 0 200 400 600 800 1000 Mesh Frequency (Hz) (c) Fig. 7.8, Comparison of reaction forces on pinion bearing: Rigid Propeller Shaft;
Propeller Shaft Bending (L=0.1) (a) radial force, (b) axial force, (c) moment
3500
3000
2500
2000
1500
1000 Bearing Radial Force (N) 500
0 0 200 400 600 800 1000 Mesh Frequency (Hz) (a)
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1400
1200
1000
800
600
400 Bearing Axial Force (N) Axial Force Bearing 200
0 0 200 400 600 800 1000 Mesh Frequency (Hz) (b)
100
80
60
40 Bearing Moment(Nm) 20
0 0 200 400 600 800 1000 Mesh Frequency (Hz) (c) Fig. 7.9, Comparison of reaction forces on gear bearing: Rigid Propeller Shaft; Propeller
Shaft Bending (L=0.1) (a) radial force, (b) axial force, (c) moment
The empirical values of the coupling stiffness and damping are chosen in the previous studies. So it is important to study the sensitivity of the dynamic mesh force to the coupling stiffness and damping. The dynamic mesh force seems to be affected slightly by the variation of the coupling stiffness and damping for this case as shown in Fig. 7.10 and 7.11, as compared
139 with the results shown in Fig. 7.6. This is probably because that the effect of propeller shaft flexibility of dynamic mesh force is mode shape dependent.
2000 1100
1500 1000
1000 900
800 500 Dynamic Mesh Force (N) Dynamic Mesh Force (N) 700
0 0 200 400 600 800 1000 600 650 700 750 Mesh Frequency (Hz) Mesh Frequency (Hz) (a) (b) Fig. 7.10, Effect of coupling stiffness: (a) comparison of dynamic mesh force, (b) affected
resonance Baseline Coupling Stiffness ( ); Low Coupling Stiffness
( ); High Coupling Stiffness ( ).
2000 1050
1000 1500 950
900 1000 850
800 500 Dynamic Mesh Force (N) Dynamic Mesh Force (N) Force Mesh Dynamic 750
700 0 0 200 400 600 800 1000 600 650 700 750 Mesh Frequency (Hz) Mesh Frequency (Hz)
(a) (b) Fig. 7.11, Effect of coupling damping: (a) comparison of dynamic mesh force, (b) affected
resonance Baseline Coupling Damping ( ); High Coupling Damping
( ); High Coupling Damping ( )
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The modal damping model is applied to represent the damping of propeller shaft in this study. Simulation results of different propeller shaft damping ratios are compared to study the effect of the bending damping on the gear dynamics. The results shown in Fig. 7.12 indicate that increasing propeller shaft bending damping can reduce the dynamic mesh force around 625 Hz for this particular case studied. It is relative simple to do damping treatment on the propeller shaft as compared with to do damping treatment inside the gear box. Performing damping treatment on propeller shaft is believed to be a potential approach to reduce the vibration and noise excited by gear mesh.
2000
1500
1000
500 Dynamic Mesh Force (N) Force Mesh Dynamic
0 0 200 400 600 800 1000 Mesh Frequency (Hz)
Fig. 7.12, Comparison of dynamic mesh force: Baseline Modal Damping Ratio (
. ); Low Modal Damping Ratio ( . ; High Modal Damping Ratio
( . ).
7.4 Conclusion
A new multi-DOF model of hypoid geared rotor system considering propeller shaft flexibility is developed. The propeller shaft bending flexibility is modeled by a lumped
141 parameter through using component model synthesis, and its torsional elasticity is simplified as a torsional spring connecting inertia of moments of engine and pinion. The universal joint could be modeled as a flexible simple supported boundary condition as well as fluctuating rotation speed and torque excitation. However, fluctuations of speed and torque are not discussed, considering their effects on the gear dynamics were studied by Tao (2011). Numerical studies of a practical applied off-highway vehicle rear axle seem to indicate that the effect of propeller shaft bending elasticity on the dynamic responses, including dynamic mesh force and bearing reaction forces, is mode shape dependent. The radial reaction forces and moments of pinion supporting bearings are affected considerably by the propeller shaft bending elasticity, while the axial reaction forces are slightly affected. Meanwhile the reaction forces of gear supporting bearing are insensitive to propeller shaft bending elasticity. The parametric studies also indicate that the increasing of modal damping ratios of propeller shaft can reduce the dynamic mesh force. Considering that it is relative simple to do damping treatment on the propeller shaft as compared with to do damping treatment inside the gear box, increasing the damping of propeller shaft is a more realistic approach to reduce the vibration and noise excited by gear mesh.
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Chapter 8. Conclusions and Proposed Future Studies
8.1 Conclusions
This dissertation research primarily evaluates the nonlinear dynamics of the right-angle gear pair analytically, and to develop new methodologies for studying the complex coupling phenomenon between the nonlinear gear mesh and other driveline components, such as the rolling element bearing, the housing and the propeller shaft. The main work and achievements are listed as follows:
a) Analytical, numerical, and experimental loaded tooth contact analysis approaches are reviewed. The deformation of gear under load consists of contact deformation, gear tooth deformation and additional deflection induced by gear body flexibility. The variants of Hertzian theory, FEM as well as a combined surface integral and 3-D FEM approach were applied to calculate the contact deformation. Cantilever beam, plate and shell models, FEM, FSM and FPM were used to compute tooth elastic deformation. Semi-plane model, experimental approaches, elastic ring and FEM were adopted to predict extra deformation caused by gear body flexibility.
Based on the review, a combined semi analytical and three-dimension finite element method is chosen to determine the values of mesh parameters.
b) A mesh coupling model of the right-angle gear pair with backlash nonlinearity as well as asymmetric and time-varying mesh coupling is proposed, considering the convex and concave side tooth geometry is different. The new mesh model allows more accurate representation of the coupling between gear pair, especially when double-side impact occurs, without increasing the complexity of the problem. The dimensionless equation of motion of the dynamic model of a right-angle gear pair using the new mesh model is formulated. The conversional multi-term
HBM for parallel gear pair analysis is enhanced by applying DFT to all forces instead of only the
143 spring force, and is formulated to accommodate the right-angle gear pair dynamics applications.
The enhanced multi-term HBM reduces the computational time from hours to seconds without sacrificing the accuracy as compared with the numerical integration method. The asymmetric directional rotation radius and mesh stiffness are found to affect both the peak value and peak frequency of the dynamic displacement.
c) The enhanced HBM is further formulated to study the sub-harmonic motions of the right-angle gear pair. Complicated phenomenon, such as sub-harmonics motions, chaotic motions, bifurcation, single-side impact, double-side impact and jump, are also observed and discussed numerically and analytically. It seems that the complexity of the responses can be reduced by either increasing the input torque or decreasing the TE, and unstable HBM solutions always indicate that more complex solutions exist. The frequency ranges, where the sub- harmonic and chaotic motions may exist, are predicted by the unstable period-one and sub- harmonic solutions.
d) The coupling between nonlinear hypoid gear mesh and time-varying rolling element bearing dynamic characteristics is evaluated using a new proposed methodology. Time-varying stiffness matrix representation is applied to describe the dynamic characteristics of the rolling elements bearing, considering the changing of the orbital position of the rolling elements. A practical example case is analyzed to study the nonlinear dynamic coupling of time-varying mesh parameters and the time-varying characteristics of the bearing, and the interaction between backlash nonlinearity and time-varying bearing representation. The dynamic mesh force is found to be insensitive to different bearing representation, while the bearing reaction force is considerably affected. It is also revealed that the backlash nonlinearity can stabilize the parametric resonance excited by time-varying bearing stiffness.
144
e) A lumped parameter model of the housing is extracted form an appropriate set of frequency response functions, which can be obtained either by simulation or measurement.
Considering the flexibility of the housing, an overall nonlinear dynamic model of hypoid gear box is established applying a proposed dynamic coupling procedure, which results in the displacement vector in semi-physical and modal coordinates. The effect of the elasticity of housing on the dynamic mesh force is evaluated for a practical rear axle sytem. The numerical results indicate that the dynamic mesh force is sensitive to the housing elasticity, and the effect of external excitation on the dynamic mesh force as well as the housing surface acceleration depends on the location of the external excitation.
f) A new multi-DOF dynamic model of a hypoid geared rotor system considering the propeller shaft bending effect is established. The flexibility of propeller shaft is modeled as lumped parameter model using CMS. The effect of propeller shaft bending flexibility and damping on the geared rotor system dynamics are found to be mode shape dependent. It is also observed that increasing the damping ratio of propeller shaft can reduce the dynamic mesh force, which indicates that application of damping treatment on propeller shaft would be a potential approach to reduce the right-angle gear box vibration and noise.
8.2 Proposed Future Studies
a) Feed the multi-point gear mesh model to the coupled multi-body dynamics and nonlinear vibration model to evaluate the dynamic mesh force for each teeth pair in contact.
b) Combine the rotating flexible propeller shaft model with coupled multi-body dynamics and nonlinear vibration model to study the effect of gear teeth impact on the propeller shaft surface acceleration.
145
c) Synthesize the hydrodynamic effect of lubricant with the elastic contact of gear teeth pair to predict gear rattle considering under lubrication condition.
d) Explore the feasibility of coupled contact-dynamic problem of hypoid and/or spiral bevel geared rotor system.
e) Formulate multi-term HBM for multiple degrees of freedom representation of hypoid geared rotor system.
Dynamometer
Flexible Couplings Housing Bearings Motor Propeller shaft
Gear Pinion
Fig. 8.1, Potential test configuration
f) Correlate testing and simulation results predicted by using the proposed nonlinear dynamic model of driveline systems with hypoid gear pair. The potential test configuration is shown in Fig. 8.1, which is similar as what was adopted by Peng (2010). In Peng’s dissertation, the dynamic mesh force was obtained by multiplying the pinion shaft torque by the rotational directional radius which is obtained from 3D loaded tooth contact analysis. The strain gauge was put on the shaft close to the pinion to measure the strain for the purpose of calculating pinion shaft torque. Similarly, the bearing reaction force can be calculated from the outputs of the strain gauge put close to the bearing location. Housing/propeller shaft can be included or excluded during the test in order to represent rigid or flexible housing/propeller shaft.
146
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