DYNAMIC MODELING OF DOUBLE-HELICAL PLANETARY GEAR SETS
DISSERTATION DRAFT
Presented in Partial Fulfillment of the Requirements for The Degree of Doctor of Philosophy in the Graduate School of The Ohio State University
By
Sondkar Prashant, M.Tech.
Graduate Program in Mechanical Engineering
The Ohio State University 2012
Dissertation Committee:
Dr. Ahmet Kahraman, Advisor
Professor Daniel Mendelsohn
Professor Manoj Srinivasan
Professor Junmin Wang
© Copyright by
Sondkar Prashant
2012
ABSTRACT
This dissertation aims at investigating the dynamic response of double-helical
planetary gear sets theoretically. A three-dimensional discrete dynamic model of a
double-helical planetary gear set is proposed, including all gear mesh, bearing and
support structure compliances. The model is presented in three levels of complexity: (i) a
linear time-invariant (LTI) model, (ii) a LTI model with gyroscopic effects included, and
(iii) a nonlinear time-varying (NTV) model with parametrically time-varying gear mesh
stiffnesses and nonlinear tooth separation effects included.
As the first step, a generic linear (no tooth separations), time-invariant (constant
gear mesh stiffnesses) dynamic model is formulated to analyze any N-planet double-
helical planetary gear system. The model includes any planet phasing conditions dictated
by the number of planets, number of gear teeth and planet position angles as well as any
phase shifts due to the designed stagger between the right and left sides of the gear set.
The forced response due to gear mesh transmission errors excitations is computed by
using the modal summation technique with the natural modes found from the
corresponding Eigen value problem for the undamped system. The strain energies of the
mode shapes are computed to identify the modes excitable by these excitations.
Parametric studies are presented to demonstrate sizable influences of planet phasing,
ii
stagger conditions, gear and carrier support conditions as well as the number of planets
on the steady-state forced response.
In the second modeling step, the LTI model is modified to include a class of
gyroscopic effects due to vibratory skew of spinning gears for the case of a stationary
carrier. The complex Eigen solutions are examined to quantify the influence of rotational
speed of the gear set through gyroscopic effects on the natural modes. A complex modal
summation formulation is used to compute the forced response with gyroscopic effects.
Results indicate that the influence of gyroscopic moments on natural frequencies is
modest within typical speed ranges, with only a sub-set of modes exhibiting dominant
tilting motions impacted by the gyroscopic effects. Effect of gyroscopic moments on
forced response curves is found to be limited to slight changes in amplitudes and
frequencies of certain resonance peaks.
As the final step, mesh stiffness fluctuations due to change in number of tooth pairs
are introduced as internal parametric excitation along with the transmission error
excitations at the same phasing relations. Tooth separation functions are also applied to
obtain a set of NTV equations of motion, which are solved by using direct numerical
integration. Differences observed between the forced response curves for time-varying
and time-invariant systems are characterized by additional resonance peaks and overall
increases in response amplitudes while no signs of nonlinear behavior are noted.
iii
DEDICATION
Dedicated to
My dear family
iv
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to Prof. Ahmet Kahraman for the opportunity, guidance, and support throughout my research at The Ohio State University.
I am grateful to Prof. Daniel Mendensohn, Prof. Junmin Wang, and Prof. Manoj
Srinivasan for serving on my dissertation committee.
Special thanks to Mr. Jonny Harianto for providing software support. I am thankful to Mr. Sam Shon and Dr. David Talbot for their technical expertise and willingness to share it. I would also like to extend my thanks to all my lab mates, including, but not limited to Devin Hilty and Mohammad Hotait for their friendship and support throughout my work at Gear Lab. I am thankful to Pratt & Whitney for sponsoring this research activity.
I want to sincerely thank my dear family for their continuous support and encouragement without which this work would not have been possible.
v
VITA
June 2004 ...... Bachelor of Engineering Pune University, India
June 2006 ...... Master of Technology Indian Institute of Technology (IIT), Madras, India
2005-2006 ...... DAAD Fellowship Technical University of Darmstadt, Germany
2006-2008 ...... Engineer (Engine Dynamics) General Electric, India
2008-2012 ...... Graduate Research Associate The Ohio State University, OH
FIELDS OF STUDY
Major Field: Mechanical Engineering
Focus on Gear Dynamics.
vi
TABLE OF CONTENTS
ABSTRACT ...... ii
DEDICATION ...... iv
ACKNOWLEDGMENTS ...... v
VITA ...... vi
LIST OF TABLES ...... xi
LIST OF FIGURES ...... xii
NOMENCLATURE ...... xv
CHAPTERS:
1. Introduction ...... 1
1.1 Background and Motivation ...... 1
1.2 Literature Review ...... 6
1.3 Scope and Objectives ...... 11
1.4 Dissertation Outline ...... 13
References for Chapter 1 ...... 15
vii
2. A Linear Time-invariant Dynamic Model of a Double-Helical Planetary Gear Set ...... 20 2.1 Introduction ...... 20
2.2 Discrete Model and its Assumptions ...... 21
2.2.1 A Sun-Planet i Pair Formulation ...... 23
2.2.2 A Ring-Planet i Pair Formulation ...... 29
2.2.3 A Carrier-Planet i Pair Formulation ...... 34
2.2.4 Coupling of the Left and Right Sides ...... 38
2.2.5 The Overall System Equations ...... 41
2.2.6 Excitations ...... 43
2.3 Solution Methodology ...... 48
2.4 An Example Simulation ...... 51
2.4.1 Influence of Right-to-left Stagger ...... 54
2.4.2 Influence of Planet Phasing Conditions ...... 65
2.4.3 Influence of Number of Planets ...... 68
2.4.4 Influence of Radially Floating Sun Gear ...... 71
2.5 Mode Identification using Modal Strain Energy ...... 73
2.6 Summary ...... 80
References for Chapter 2 ...... 81
3. Influence of Gyroscopic Effects on Dynamic Behavior of Double-Helical Planetary Gear Sets ...... 83 3.1 Introduction ...... 83
3.2 Incorporation of Gyroscopic Moments in the Dynamic Model ...... 84
3.2.1 A Sun-Planet i Pair with Gyroscopic Effects ...... 85
viii
3.2.2 A Ring-Planet i Pair with Gyroscopic Effects ...... 87
3.2.3 A Carrier-Planet i Pair with Gyroscopic Effects ...... 88
3.2.4 Coupling Elements with Gyroscopic Effects ...... 89
3.2.5 The Overall System Equations with Gyroscopic Effects ...... 90
3.3 Solution Methodology ...... 91
3.4 An Example Simulation ...... 94
3.5 Summary ...... 103
References for Chapter 3 ...... 105
4. Investigation of Time-Varying Gear Mesh Stiffness Effects on Dynamics of Double-Helical Planetary Gear Sets ...... 106 4.1 Introduction ...... 106
4.2 A Nonlinear Time-Varying Dynamic Model ...... 109
4.2.1 Definition of Time-Varying Gear Mesh Stiffnesses ...... 109
4.2.1 Equations of Motions for NTV Model ...... 112
4.3 Solution of the NTV System Equations ...... 115
4.4 Numerical Results ...... 117
4.4.1 Verification and Analysis of Time-domain Solutions ...... 118
4.4.2 Example System Analyses ...... 119
4.5 Summary ...... 131
References for Chapter 4 ...... 132
5. Conclusions and Future Recommendations ...... 135
5.1 Summary ...... 135
5.2 Conclusions and Contributions ...... 137 ix
5.3 Recommendations for Future Work ...... 140
BIBLIOGRAPHY ...... 141
Appendix A Beam Element Matrices ...... 148
Appendix B Overall System Matrices ...... 153
Appendix C Elements of Forcing Vector ...... 158
x
LIST OF TABLES
Table Page
2.1 Basic design parameters of the example gear system ...... 52
2.2 Harmonic amplitudes and phase angles of the transmission error excitations of the example gear set of Table 2.1 [2.6] ...... 53
2.3 Predicted natural frequencies and mode types of the example gear set ...... 55
2.4 Strain Energy distribution for the modes of the example gear set (excited modes are shown in italic characters) ...... 78
3.1 Strain energy distribution for the modes exhibiting change in natural frequencies due to gyroscopic effects ...... 96
4.1 Harmonic amplitudes and phase angles of the transmission error and mesh stiffness excitations of the example gear set of Table 2.1 [4.19] for different torque levels ...... 121
xi
LIST OF FIGURES
Figure Page
1.1 Main components of planetary gear set ...... 2
1.2 An example double helical planetary gear set [1.2] ...... 4
2.1 Dynamic model of a double-helical planetary gear system ...... 24
2.2 Dynamic model of sun-planet i pair ...... 25
2.3 Dynamic model of ring-planet i pair ...... 30
2.4 Dynamic model of carrier-planet i pair ...... 35
2.5 (a) Geometry of a double helical external gear, and (b) three-piece model of the double helical gear ...... 40
2.6 Illustration of the right-to-left stagger conditions in a double helical gear pair; (a) 0 and (b) ...... 46 stg stg 2.7 Maximum dynamic mesh force amplitudes at the left side (a) s-pi and (b) r-pi meshes ...... 56
2.8 Maximum dynamic mesh force amplitudes at the right side (a) s-pi and (b) r-pi meshes ...... 57
2.9 Mode shapes representative of Eq. (2.32) at (a) 2 1262 Hz and (b) 38 11047 Hz ...... 60
2.10 Mode shape representative of Eq. (2.33) at (a) 14 2994 Hz and (b) 15 3166 Hz ...... 61
2.11 Dynamic factors at the left side (a) s-pi and (b) r-pi meshes ...... 63
2.12 Dynamic factors at the right side (a) s-pi and (b) r-pi meshes ...... 64
xii
2.13 Maximum dynamic mesh force amplitudes at the left/right side s-pi meshes for (a) 0 and (b) for different stg stg planet phasing conditions ...... 69
2.14 Maximum dynamic mesh force amplitudes at the left/right side (a) s-pi and (b) r-pi meshes for different number of planet gears ...... 70
2.15 Effect of radially floating sun gear on dynamic mesh force amplitudes for (a) stg 0 and (b) stg ...... 72
2.16 Normalized modal strain energy components of modes at (a) 8398 Hz (b) 9451Hz ...... 79
3.1 Variation of certain natural frequencies with the rotational speed due to gyroscopic effects (C0 ) ...... 95
3.2 Strain energy distribution for support spring and planet bearings for modes at (a) 7 2076 Hz and (b) 14 2994 Hz. (Tr - Translational, Ti - Tilting, Ax - Axial) ...... 98
3.3 Maximum percent change in natural frequencies with an input speed range of 0 to 20,000 rpm as a function of (a) sun gear support stiffness in tilting direction and (b) ring gear support stiffness in tilting direction ...... 99
3.4 Maximum percent change in natural frequencies with an input speed range of 0 to 20,000 rpm as a function of polar mass moment of inertia of (a) the sun gear and (b) the ring gear ...... 100
3.5 Maximum dynamic mesh force amplitudes at the left side (a) s-pi and (b) r-pi meshes with and without gyroscopic effect ...... 102
3.6 Maximum dynamic mesh force amplitudes at the left side (a) s-pi and (b) r-pi meshes with and without gyroscopic effect for radially floating sun ( stg ) ...... 104
4.1 Comparison of modal summation and direct numerical integration solutions for the LTI case. Maximum dynamic mesh force amplitudes on the left side (a) s-pi and (b) r-pi meshes ...... 120
4.2 Root-mean-square. values of dynamic mesh forces at the left side (a) s-pi and (b) r-pi meshes for different excitation models ( stg 0 ) ...... 123
xiii
4.3 Steady-state dynamic mesh force at an s-pi mesh on the left side obtained by using Model II. (a) Time history at m 8852 Hz and (b) the corresponding frequency spectrum ...... 124
4.4 Steady-state dynamic mesh force at an s-pi mesh on the left side obtained by using Model II. (a) Time history at m 5523Hz and (b) the corresponding frequency spectrum ...... 125
4.5 Maximum dynamic mesh force amplitudes on the left side of (a) s-pi and (b) r-pi meshes with different excitation models ( stg 0 ) ...... 126
4.6 Maximum dynamic mesh force amplitudes at the left side (a) s-pi and (b) r-pi meshes for different excitation models ( stg ) ...... 128
4.7 Maximum dynamic mesh force amplitudes on the left side (a) s-pi and (b) r-pi meshes of an in-phase system for different gear mesh models ( stg 0 ) ...... 129
4.8 Maximum dynamic mesh force amplitudes on the left side of (a) s-pi and (b) r-pi mesh at various input torque levels. Model II was used here ...... 130
A.1 An Euler beam element ...... 152
xiv
NOMENCLATURE
Symbol Definition
C Damping matrix
D Diameter e Transmission error eˆ Amplitude of l-th harmonics of transmission error
F Force acting on planet i due to coupling between carrier and planet i
Fspi Dynamic mesh force for sun-planet i gear mesh
Frpi Dynamic mesh force for ring-planet i gear mesh
F Force vector
FW Face width
FWg Length of gap between two sides of double helical gears
G Gyroscopic matrix h Unite step function representing tooth separation
H Angular momentum vector
I Diametral mass moment of inertia
J Polar mass moment of inertia
xv k Planet bearing stiffness ksp Mean value of sun-planet i gear mesh stiffness krp Mean value of ring-planet i gear mesh stiffness kx Support bearing stiffness in x direction ( src,, ) ky Support bearing stiffness in y direction ( src,, ) kz Support bearing stiffness in z direction ( src,, ) kx Support bearing stiffness in x direction ( src,, ) ky Support bearing stiffness in y direction ( src,, ) kˆ Amplitude of l-th harmonics of mesh stiffness
KD Dynamic factor
L Modal matrix of left eigen vector m Mass
M Moment acting on planet i due to coupling between carrier and planet i
M Mass matrix
N Number of Planets p Relative mesh displacement pˆ Modal relative mesh displacement q Forced vibration response vector
Q Mode shape r Base radii of gear ( sr,, pi) xvi rc Planet pin hole radius r State vector
R Modal matrix of right eigen vector
T Torque
Tmesh Mesh period
U Strain energy x Forced vibration response in x direction xˆ Modal displacement in x direction y Forced vibration response in y direction yˆ Modal displacement in y direction z Forced vibration response in z direction zˆ Modal displacement in z direction
Z Number of teeth on gear
pi Position angle for planet i
Helix angle
spi phase angle between the s-pi mesh on the left side and the reference s-p1 mesh
rpi phase difference between the r-pi mesh and the r-p1 mesh on the left side
stg phase difference between the left and right side due to intentional nominal stagger of the teeth
rs phase difference between the reference s-p1 mesh and the r-p1 mesh on left side
xvii
Stiffness multiplier for proportional damping
Damping factor
Mass multiplier for proportional damping
x Forced vibration response in x direction
y Forced vibration response in y direction
z Forced vibration response in z direction
ˆ x Modal displacement in x direction
ˆ x Modal displacement in y direction
ˆ z Modal displacement in z direction
Dynamic compliance
Phase angle of l-th harmonics of transmission error excitation
Phase difference
Phase angle of l-th harmonics of mesh stiffness
Transvers pressure angle
Angle made by plane of action with vertical y axis
Natural frequency
m Mesh frequency
Velocity vector
Rotational speed
xviii
Subscripts b Bore c Carrier i Index for number of planets l Harmonics number m Mean value mesh Gear mesh n Number of beam element r Ring gear pb Planet bearing pi Planet i gear s Sun gear sup Support bearing x x direction y y direction z z direction
x x direction
y y direction
z z direction
Number of degrees of freedom
xix
Superscript
L Left side of double helical gear
R Right side of double helical gear
xx
CHAPTER 1
Introduction
1.1 Background and Motivation
Planetary gear sets (also known as epicyclic gear sets) are commonly used in many automotive, industrial, aerospace and wind turbine industries. The typical applications include jet propulsion systems, rotorcraft transmissions, passenger vehicle automatic transmission, wind turbine gearboxes and other industrial gearboxes. Figure 1.1 shows the components of a simple planetary gear set consisting of an N number of planet gears
(typically N [3,7] ) that are in mesh with an external sun gear (s) and an internal ring
gear (r). The planet gears are supported by a rigid structure called carrier (c) through various types of planet bearings (needle bearings, rolling element bearings, and in some aerospace applications, journal bearings).
One of the main advantages of planetary gear sets is that the input power is split
into a number of parallel power transmission paths, providing higher power density
(power to weight ratio) values. This lowers the forces carried by individual planet meshes to allow smaller tooth modules with higher gear contact ratios, resulting in quieter gear 1
carrier carrier planet planet
planet pin
rolling element planet pin bearing ring
Figure 1.1: Main components of planetary gear set
2
set designs [1.1]. Different speed ratios can be obtained from the same gear set by
changing input, output and stationary (reaction) members. This capability is the basis for
their extensive use in automotive automatic transmissions and transfer cases. In addition,
the axi-symmetric (coaxial) configuration of planetary gear sets eliminates the most of
the radial loads on bearings [1.1]. Furthermore, planetary gear sets have the ability to
self-center themselves radially to compensate for various manufacturing errors such as
carrier pin-hole position errors and gear eccentricities.
Planetary gear sets consist of either spur, (single) helical or double-helical gears.
Spur planetary gear sets can be commonly found in heavy machinery and off-highway
gearboxes and transmissions, while the helical planetary gear sets are the norm for all
automotive applications as in automatic transmissions and transfer cases. When helical
gears are used in a planetary gear set, axial thrust forces are created on the sun and ring
gears that must be balanced by bearings or the thrust forces on the adjacent planetary gear sets. Double-helical planetary gear sets, as the one shown in Figure 1.2 [1.2] for a jet engine turbofan application, do not have this problem as the static axial gear mesh forces acting on the right and left sides cancel each other. In addition, they have higher load carrying capacity and better noise characteristics. For these reasons, they have been used for jet engine and rotorcraft gearbox applications in spite the cost and manufacturing challenges associated with their production.
Planetary gear sets have several unique issues/features associated with their functional behavior. As each planet branch is a split power transmission path, an equal planet-to-planet load sharing is often not possible especially when certain manufacturing 3
Figure 1.2: An example double helical planetary gear set [1.2].
4
tolerances of the gear and the carrier are not designed to be very tight and all of the central members (sun, ring or carrier) are supported rigidly in the radial direction [1.3, 1.
4]. Reduced gear rim thicknesses not only reduce the mass but also shown to reduce the influence of internal gear and carrier errors [1.5]. It is also reported that a flexible internal gear improves the equal load sharing amongst the planets. The structural modal properties of planetary gears with equally spaced planets possess choices of certain phasing conditions such that cancellation or neutralization of dynamic gear mesh forces can be achieved through design strategies to reduce vibration and noise levels. Proper planet mesh phasing can eliminate many troublesome vibration modes [1.6]. Planetary gear sets also exhibit unique vibration frequency spectra with amplitude or frequency modulated sideband components associated either with the carrier rotation or with the eccentricities of gear components [1.7-1.9]. These vibration signatures were exploited for fault diagnosis purposes like detection of cracks in the carrier [1.10].
Dynamic analysis of a double-helical planetary gear set, as any other gear system, is required for two main reasons. One reason is associated with the resultant noise outcome. High-frequency dynamic forces created at the gear meshes are transmitted to gearbox housing and the surrounding support structures to generate structure-borne noise.
Therefore, reduction of vibrations necessitates a better understanding of the dynamic behavior. The other main reason stems from the durability and reliability requirements of the planetary gear set. Dynamic gear mesh and bearing forces that alternate about the static forces transmitted cause dynamic stresses and dynamic factors to impact the fatigue lives of the gears and bearings adversely. Hence, dynamic analysis of a double helical
5
planetary gear set becomes a critical step in the design process, which aims at developing
quiet and reliable transmissions.
In this research, a theoretical will be performed to understand the dynamic behavior
of double helical planetary gear sets. Towards this aim, an analytical modeling
framework will be formulated and executed, which will serve as a fundamental tool for
design and further research. The proposed models will be used to investigate the free and
forced vibration response of the double helical planetary gear sets. Gyroscopic effects,
time varying mesh stiffness and gear mesh excitations will all be incorporated in the
model.
1.2 Literature Survey
Numerous analytical models on planetary gear dynamics consisting of spur or
helical gears can be found in the literature while research on double-helical gears is quiet
limited. In their review paper, Yang and Dai [1.11] presented an overview of various
planetary gear dynamics models found in literature, most of which are focused on
analytical work comprising of discrete-parameter models with gears bodies assumed to
be rigid and tooth flexibilities are modeled as a springs. These models vary in terms of
degrees of freedom included (purely torsional, two-dimensional, or three-dimensional)
and type of formulation employed (linear time-invariant (LTI) models with constant gear
mesh stiffnesses and no backlash, linear time-varying models (LTV) with fluctuating
mesh stiffnesses and no gear backlash, or nonlinear time-varying models (NTV) with
both fluctuating mesh stiffnesses and backlash included). 6
Types of analyses performed in these studies also vary as some focused on
prediction of natural frequencies and mode shapes, others emphasized the excitation
mechanism and cancellation/minimization of these excitations by properly phasing the
planets while a number of studies predicted forced vibration response and dynamic tooth
load. Botman [1.12] carried out a study to investigate the modes of planetary gear set
with a non-rotating or rotating carrier. Kahraman [1.13] developed a torsional model of
planetary gear set and established closed-form expressions for natural frequencies and
mode shapes of planetary system consisting of any number of planets. A dynamic model
for planetary gear set was presented by Saada and Velex [1.14] to study the influence of
gear mesh stiffnesses and support stiffnesses on natural frequencies of the system.
Kahraman [1.1] was first to investigate the free torsional vibration characteristics of a
compound planetary gear set. Effort was made to classify modes as rigid body mode,
asymmetric planet modes and axi-symmetric overall modes. Inalpolat and Kahraman
[1.15] developed a model to study free and forced vibration characteristics of any
planetary gear train formed by N stages of different types (single-planet, double-planet, or
complex-compound).
Studies on reducing the dynamic mesh forces by proper planet phasing to cancel or
neutralize the mesh excitations goes back to Seager [1.6] who established conditions for
neutralization of harmonic components of excitations of the central members of a
planetary gear set formed by spur gears, which was achieved by suitable choice of number of gear teeth. Toda and Botman [1.16] showed that the vibration excitations can be reduced significantly with relative indexing of planets. They focused their effort on
7
establishing how indexing of planets can minimize the effect of tooth spacing errors in
the planets. Kahraman and Blankenship [1.17] provided the first generalized phasing
formulation, defining the relationships amongst all of the gear meshes of an N-planet,
helical planetary gear set. This formulation showed that there is no specific phasing
condition that can neutralize the excitations in all directions, but it is possible to find a
phasing condition that can yield a desirable response for certain applications, operating
within a given speed range. Parker and Lin [1.18] provided the analytical description of
mesh phasing relationships between sun-planet and ring-planet meshes. Platt and
Leopold [1.19] carried out experiments to study the effect of planet mesh phasing on
noise levels. Different phasing conditions including in-phase and sequentially phased
planets were investigated for their respective noise levels.
Third group of studies focused on prediction of the forced vibration response and
dynamic gear mesh and bearing forces. Cunliffe et al [1.20] studied the variation of
dynamic tooth load with planet pin stiffness for single stage planetary gear set consisting
of spur gears. Hidaka and Terauchi [1.21] conducted experiments to study the dynamic
load sharing among planet meshes and compared the results with analytical model [1.22].
Influence of run-out errors on the motion of floating sun gear was investigated by Hidaka
et al [1.23]. August and Kasuba [1.24] developed a torsional model of single stage
planetary gear system with shafts, input and output units. Effect of a floating, partially
floating and fixed sun gear on dynamic mesh forces was investigated by incorporating
additional transverse degrees of freedom for the sun gear. Motion of sun gear was found
to be influenced by mesh stiffness variation as well. Kahraman [1.25] presented a two-
8
dimensional NTV model of a planetary gear set. This study is focused on investigating
the influence of design, manufacturing and assembly variations on dynamic planet-to-
planet load sharing characteristics.
The first generalized three-dimensional model for a helical planetary gear set was
proposed by Kahraman [1.26]. This LTI model included the planet phasing formulation
of Kahraman and Blankenship [1.17] as well as all six degrees of freedom for each gear
and the carrier. A purpose of this model was to simulate the dynamic loads at individual
gear meshes in relation to particular meshing conditions to identify the best possible
phasing configurations. Velex and Flamand [1.27] extended their earlier modeling effort
[1.14] to include the contact formulation in the dynamic model of a planetary gear
system. A numerical method was introduced to define the instantaneous positions of
contact and corresponding mesh parametric excitations. Time integration method was
employed to solve the system of equations. A generalized torsional NTV model was
presented by Al-Shyyab and Kahraman [1.28] by including gear backlash type clearance
nonlinearities and time-varying stifffnesses. The multi-term Harmonic Balance Method
was used to solve nonlinear equations of motion analytically. Chaari et al [1.29]
investigated the effect of manufacturing errors on the dynamic behavior of planetary gears. Influence of eccentricity of gears and profile errors on the frequency response of system was studied. Parametric instability of planetary gear system due to variable mesh stiffness was analyzed by Hbaieb et al [1.30]. A rectangular waveform was used to represent the mesh stiffness variations. A perturbation technique was employed to solve
9
the system of equations. Stability boundaries were identified, which included primary,
secondary and combination instabilities.
In recent years, some researchers have used deformable gear body dynamic models.
Parker et al [1.31] employed contact model developed by Vijayakar [1.32] to build deformable body dynamic model of planetary gear system and studied the dynamic behavior of the system over a range of speeds. Effects of gear rim thickness parameters
on the gear stresses were analyzed by Kahraman et al [1.5]. Yuksel and Kahraman [1.33]
combined a surface wear model and deformable-body dynamic model to study the
influence of tooth wear on dynamic mesh forces. A hybrid three dimensional lumped
parameter and finite element model was developed by Abousleiman et al [1.34] to
investigate the effect of flexible ring gear on the dynamic response. The same
investigators published another study [1.35] to look at the effects of geometrical errors
and centrifugal forces on the dynamic response.
Literature on double helical gears is quiet sparse and limited to a single gear pair
arrangement, not multi-mesh systems such as planetary gear sets. Thomas [1.36]
developed an analytical model for investigating load distribution and transmission error
of a double helical gear pair under quasi-static conditions. The model was later used by
Clapper and Houser [1.37] to investigate the root stresses of double helical gears and
comparison was made with experiments performed. Zhang et al [1.38] carried out a noise
optimization of double helical parallel shaft gearbox by developing a three-dimensional
FE model. Noise reduction was achieved by varying the thickness of internal bearing
supporting panels and external walls of gearbox. Wang et al [1.39] presented study about 10
tooth modification of double helical gear pair to reduce the transmission error.
Experiments were performed on optimized tooth geometry to study its transmission error characteristics. Jauregui and Gonzalez [1.40] developed single degree-of-freedom model to study axial vibrations of double helical gear pair due to manufacturing errors. Quasi- static and dynamic analyses of a double helical gear pair was carried out by Ajmi and
Velex [1.41]. A 12-DOF model of a left side of gear pair was combined with another 12-
DOF model of the companion right side pair using Euler beam elements. The effect of floating pinion and staggering of teeth on the quasi-static and dynamic behavior of a gear pair was investigated. Anderson et al [1.42] conducted experiments on double helical planetary gear set to measure efficiency, vibration amplitudes and stress levels.
1.3 Scope and Objectives
The literature review presented above reveals that the dynamic behavior of double- helical planetary gear sets has not been studied. Modeling of double-helical planetary gear sets involve additional complications over (single) helical planetary gear sets in terms of left-to-right side load sharing and left-to-right gear mesh phasing relations. It is evident that a methodology to analyze double helical gears does not exist and no published model can be found in the literature describing dynamic characteristics of double-helical planetary gear systems. In fact, double helical systems attracted very little attention even in terms of their single gear pair dynamic behavior. Accordingly, overall objective of this dissertation is to develop analytical models to study essential dynamic characteristics of a double helical planetary gear system.
11
The models will be based on a discrete representation of the planetary gear set. All
gear bodies will be assumed to be rigid disks with gear tooth compliance represented by
spring and damper elements. The same discrete treatment will be applied to bearings as
well. As the damping mechanisms of such systems are the least known, modal and proportional damping formulations will be used. The dissertation objective stated above
will be achieved in three steps of increasing complexity:
(i) Development of a Linear Time Invariant (LTI) model: A new LTI model of a
double helical planetary gear set will be developed to predict the free vibration
characteristics and forced vibration response under simplified damping
conditions. This generalized three-dimensional model will include any
number of equally or unequally positioned planets, as well as any planet
phasing and support conditions. Torsional, transverse, axial and rotational
(tilting) motions of gears and the planet carrier will be included in this three-
dimensional model. A detailed parametric study will be performed on this
model to investigate the influence of different parameters on the dynamic
response.
(ii) Incorporation of Gyroscopic Effects: The LTI model will be expanded to
include high speed effects such as gyroscopic moments to study their
influence on the dynamic behavior of the system. Sensitivity of Eigen values
and the forced response to rotational speed and gyroscopic effects will be
investigated.
12
(iii) Development of a Nonlinear Time Varying (NTV) Model: In order to quantify
the nonlinear and time varying effects, periodically varying mesh stiffnesses
due to variable number of teeth in contact and tooth separation nonlinearity
will be incorporated in the earlier LTI model. Furthermore, a piecewise
clearance function will be incorporated, to model any tooth separations.
Numerical time integration scheme will be adopted to solve the system of
nonlinear equations.
The purpose of this research is to gain insight into the dynamic behavior of double helical planetary system with simplified model without resorting to deformable-body modeling of gear sets, due to computational limitations. Support structures and bearings will modeled in a simple manner as done in most of previous gear dynamic models.
1.4 Dissertation Outline
As stated in the previous section, the modeling effort will be carried out in three steps. Each of these steps will be described in an individual chapter. Chapter 2 is focused on development of the LTI model of a double helical planetary gear set. The modeling assumptions along with details of the modeling methodology will be presented.
The equations of motion will be derived and solved to predict natural modes and the forced response of the gear set within a given range of operating speed. This model will serve as the foundation for the other models. A parametric study will be carried out to investigate the influence of different design parameters and variations such as right-to-
13
left gear tooth stagger, support stiffnesses, planet position angles, planet mesh phasing
conditions and number of planets on the dynamic behavior of the system.
In Chapter 3, the LTI formulation of Chapter 2 will be modified to include gyroscopic moments. Gyroscopic effects are often hypothesized to be important in high- speed gear applications in aerospace industry with little work to substantiate it.
Inclusion of gyroscopic effects will result in an asymmetric damping matrix. The governing complex Eigen Value problem will be solved to quantify the combined effect of speed and gyroscopic moments on natural modes of the planetary gear set. The modal summation technique will be used to determine the forced response. Direct comparisons between the cases when the gyroscopic effect included and ignored will be presented to determine conditions when gyroscopic models must be included in the model.
Chapter 4 provides a further expansion of earlier models by including periodic time variation of gear mesh stiffnesses as well as contact loss induced by the backlash present at the gear meshes. As inclusion of these effects makes the stiffness matrix a periodically time-varying one, subject to a nonlinear backlash (clearance) constraint, the frequency- domain (modal summation) solutions of the previous chapters are not applicable to this
NTV system. Gyroscopic effects will also be included in this NTV model. The system equations will be put into the state-space form and solved by the direct numerical integration method. The results of the NTV model will be compared to the corresponding LTI models with and without gyroscopic effects to determine whether time-varying gear mesh stiffness effects and gear backlash should be included in modeling of double helical planetary gear sets. 14
Finally, Chapter 5 summarizes entire work, and lists major conclusions and the
contributions of the proposed research. Recommendations for future research on this
topic to improve the modeling effort are also included in this chapter.
References for Chapter 1:
[1.1] Kahraman, A., 2001, “Free Torsional Vibration Characteristics of Compound Planetary Gear Sets,” Mechanism and Machine Theory, 36, pp. 953-971.
[1.2] Mraz, S., 2009, “Gearing up for Geared Turbofan,” Machine Design by Engineers for Engineers. (http://machinedesign.com/article/gearing-up-for-geared-turbofans- 0202)
[1.3] Ligata, H. and Kahraman, A., 2008, “An Experimental Study of the Influence of Manufacturing Errors on the Planetary Gear Stresses and Planet Load Sharing,” ASME Journal of Mechanical Design, 130, 041701-1 – 041701-9.
[1.4] Bodas, A. and Kahraman, A., 2004, “Influence of Carrier and Gear Manufacturing Errors on the Static Load Sharing behavior of Planetary Gear Sets,” JSME International Journal, 47, pp. 908-915.
[1.5] Kahraman, A., Kharazi, A., and Umrani, M., 2003, “A Deformable Body Dynamic Analysis of Planetary Gears with Thin Rims,” Journal of Sound and Vibration, 262, pp. 752-768.
[1.6] Seager, D., 1975, “Conditions for the Neutralization of Excitation by the Teeth in Epicyclic Gearing,” Journal Mechanical Engineering Science, 17(5), pp. 293- 298.
[1.7] McFadden, P. and Smith, J., 1985, “An Explanation for the Asymmetry of the Modulation Sidebands about the Tooth Meshing Frequency in Epicyclic Gear Vibration,” Proceedings of Institution of Mechanical Engineers, 199(C1), pp. 65- 70.
15
[1.8] Inalpolat, M. and Kahraman, A., 2009, “A Theoretical and Experimental Investigation of Modulation Sidebands of Planetary Gear Sets,” Journal of Sound and Vibration, 323, pp. 677-696.
[1.9] Inalpolat, M. and Kahraman, A., 2010, “A Dynamic Model to predict Modulation Sidebands of a Planetary Gear Set having Manufacturing Errors,” Journal of Sound and Vibration, 329, pp. 371-393.
[1.10] Blunt, D. and Keller, J., 2006, “Detection of Fatigue Crack in a UH-60A Planet Gear Carrier using Vibration Analysis,” Mechanical Systems and Signal Processing, 20, pp. 2095-2111.
[1.11] Yang, J. and Dai, L., 2008, “Survey of Dynamics of Planetary Gear Trains,” International Journal of Materials and Structural Integrity, 1, pp. 302-322.
[1.12] Botman, M. 1976, “Epicyclic Gear Vibrations,” Journal of Engineering for Industry, 97, pp. 811-815.
[1.13] Kahraman, A., 1994, “Natural Modes of Planetary Gear Trains,” Journal of Sound and Vibration, 173(1), pp. 125-130.
[1.14] Saada, A. and Velex, P., 1995, “An Extended Model for the Analysis of the Dynamic Behavior of Planetary Trains, ” ASME Journal of Mechanical Design, 117, pp. 241-247.
[1.15] Inalpolat, M. and Kahraman, A., 2008, “Dynamic Modeling of Planetary Gears of Automatic Transmissions,” Proceedings of Institution of Mechanical Engineers Part K: Journal of Multi-body Dynamics, 222, pp. 229-242.
[1.16] Toda, A. and Botman, M., 1979, “Planet Indexing in Planetary Gears for Minimum Vibrations,” ASME paper, 79-DET-73.
[1.17] Kahraman, A. and Blankenship, G., 1994, “Planet Mesh Phasing in Epicyclic Gear Sets,” Proceedings of ASME Power Transmission and Gearing Conference, San Diego.
16
[1.18] Parker, R. and Lin, J., 2004, “Mesh Phasing Relationships in Planetary and Epicyclic Gears,” ASME Journal of Mechanical Design, 126, pp. 365-370.
[1.19] Platt, R. and Leopold, R., 1996, “A Study on Helical Gear Planetary Phasing Effects on Transmission Noise,” VDI Berichte, 1230, pp.793-807.
[1.20] Cunliffe, F., Smith, J., and Welbourn, D., 1974, “Dynamic Tooth Loads in Epicyclic Gears,” ASME Journal of Engineering for Industry, 95, pp. 578-584.
[1.21] Hidaka, T. and Terauchi, Y., 1976, “Dynamic Behavior Planetary Gear, 1st Report: Load Distribution in Planetary Gear,” Bulletin of JSME, 19, pp. 690-698.
[1.22] Hidaka, T., Terauchi, Y., and Fujii, M., 1980, “Analysis of Dynamic Tooth Load on Planetary Gear,” Bulletin of JSME, 23, pp. 315-323.
[1.23] Hidaka, T., Terauchi, Y., and Dohi, K., 1979, “On the Relation between the Run- Out Errors and the Motion of the Center of Sun Gear in Stoeckicht,” Bulletin of JSME, 22(167), pp. 748-754.
[1.24] August, R. and Kasuba, R., 1986, “Torsional Vibration and Dynamic Loads in a Basic Planetary Gear System,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 108, pp. 348-353.
[1.25] Kahraman, A., 1994, “Load Sharing Characteristics of Planetary Transmission,” Mechanism and Machine Theory, 29(8), pp. 1151-1165.
[1.26] Kahraman, A., 1994, “Planetary Gear Train Dynamics,” ASME Journal of Mechanical Design, 116, pp. 713-720.
[1.27] Velex, P. and Flamand, L., 1996, “Dynamic Response of Planetary Trains to Mesh Parametric Excitations,” ASME Journal of Mechanical Design, 118, pp. 7- 14.
[1.28] Al-Shyyab, A. and Kahraman, A., 2007, “A Nonlinear Dynamic Model for Planetary Gear Sets,” Proceedings of the Institution of Mechanical Engineers Part K: Journal of Multi-Body Dynamics, 221, pp. 567-576.
17
[1.29] Chaari, F., Fakhfakh, T., Hbaieb, R., Louati, J., and Hadder, M., 2006, “Influence of Manufacturing Errors on the Dynamic Behavior of Planetary Gears,” International Journal of Advance manufacturing Technology, 27, pp. 738-746.
[1.30] Hbaieb, R., Chaari, F., Fakhfakh, T., Hadder, M., 2006, “Dynamic Stability of a Planetary Gear Train under the Influence of Variable Meshing Stiffnesses,” Proceedings of IMechE. Part D: Journal of Automobile Engineering, 229(D12), I711-I725.
[1.31] Parker, R., Agashe, V., and Vijayakar, S., 2000, “Dynamic Response of a Planetary Gear System using a Finite Element/Contact Mechanics Model,” ASME Journal of Mechanical Design, 122, pp. 304-310.
[1.32] Vijayakar, S., 1991, “A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem,” International Journal for Numerical Methods in Engineering, 31, pp. 525-545.
[1.33] Yuksel, C. and Kahraman, A., 2004, “Dynamic Tooth Loads of Planetary Gear Sets having Tooth Profile Wear,” Mechanism and Machine Theory, 39, pp. 695- 715.
[1.34] Abousleiman, V., and Velex, P., 2006, “A Hybrid 3D Finite Element/Lumped Parameter Model for Quasi-Static and Dynamic Analyses of Planetary/Epicyclic Gear Sets,” Mechanism and Machine Theory, 41, pp. 725-748.
[1.35] Abousleiman, V., Velex, P., and Becquerelle, S., 2007, “Modeling of Spur and Helical gear Planetary Drives with Flexible Ring Gears Planet Carriers,” ASME Journal of Mechanical Design, 129, pp. 95-106.
[1.36] Thomas, J., 1991, “A Procedure for Predicting the Load Distribution and Transmission Error Characteristics of Double Helical Gears,” MS Thesis, The Ohio State University.
[1.37] Clapper, M. and Houser, D., 1993, “Prediction of Fully Reversed Stresses at the Base of the Root in Spur and Double Helical Gears in a Split Torque Helicopter Transmission,” Proceedings of American Helicopter Society Rotor Wing Specialists Meeting, Williamsburg, VA.
18
[1.38] Zhang, T., Kohler, H., and Lack, G., 1994, “Noise Optimization of a Double Helical Parallel shaft Gearbox,” International Gearing Conference, UK, pp. 93- 98.
[1.39] Wang, C., Fang, Z., and Jia, H., 2010, “Investigation of Design Modification for Double Helical Gears Reducing Vibration and Noise,” Journal of Marine Science and Applications, 9, pp. 81-86.
[1.40] Jauregui, J. and Gonzalez, O., 1999, “Modeling Axial Vibrations in Herringbone Gears,” Proceedings of ASME Design Engineering Technical Conference, Nevada, DETC99/VIB-8109.
[1.41] Ajmi, M. and Velex, P., 2001, “A Model for Simulating the Quasi-Static and Dynamic Behavior of Double Helical Gears,” The JSME International Conference on Motion and Power Transmission, MPT-2001, pp. 132-137.
[1.42] Anderson, N., Nightingale, L., and Wagner, A., 1989, “Design and Test of Turbofan Gear System,” Journal of Propulsion, 5(1), pp. 95-102.
19
CHAPTER 2
A Linear Time-invariant Dynamic Model of a Double-Helical Planetary Gear Set
2.1 Introduction
In this chapter, a three-dimensional discrete Linear Time-invariant (LTI) model of a
double-helical planetary gear set will be proposed. The modeling methodology and
governing assumptions will be stated. Equations of motion will be derived and solved
using Modal Summation technique in order to predict the steady-state response. The
proposed model will be used to investigate both free and forced vibration characteristics
of an example double-helical planetary gear set within ranges of several key design
parameters. The model will be formulated in a general and modular form such that any
number of equally and unequally positioned planets, any planet-to-planet phasing
conditions as well as any typical support conditions can be simulated effectively.
Furthermore, the proposed model formulation will form the basis for further studies in
Chapters 3 and 4 on gyroscopic and time-varying effects.
20
2.2 Discrete Model and its Assumptions
A three-dimensional discrete model of an N-planet double-helical planetary gear set
is proposed. The formulation adapts 6(N 3) degree-of-freedom (DOF) model of
Kahraman [2.1] to represent the right and left sides of the double-helical planetary gear
independently. It then uses a method proposed by Ajmi and Velex [2.2] to connect the
right and left sides to each other.
The following assumptions are made in the model formulation:
(i) The bodies representing the sun, planet and ring gears and the carrier are all
assumed to be rigid.
(ii) Flexibilities of gear mesh are represented by linear springs acting on the plane
of action normal to gear tooth surfaces (inclined by helix angle ).
(iii) Time-varying component of mesh stiffness due to fluctuation of number of
tooth pairs in contact are neglected in line with the findings of several helical
gear dynamics studies [2.3, 2.4]. Validity of this assumption will be
investigated later in Chapter 4 through a time-varying formulation.
(iv) Gear teeth at the mesh interfaces are assumed to maintain contact all the time
(i.e. tooth separations do not occur). The time-varying formulation of Chapter
4 will also include a clearance type separation function to explore whether
such nonlinear effects are important.
21
(v) The model in this chapter does not include any gyroscopic effects while they
will be investigated in detail in Chapter 3.
(vi) Planets are assumed to be identical to each other such that each planet-sun and
planet-ring mesh can be assumed to have the same geometric properties and
contact characteristics.
(vii) Frictional forces arising from tooth sliding are considered to be negligible in
accordance with the off-line-of-action helical gear vibration measurements of
Kang and Kahraman [2.5].
(viii) Left and right sides of double-helical gears are assumed identical in geometry
except the hands of the teeth are opposite. Right-to-left stagger angles are
assumed to be exactly the same for each corresponding mesh. Any stagger
deviations due to manufacturing will be ignored.
(ix) A class of potential manufacturing errors associated with the gears and the
carrier will be neglected. Such errors including gear run-out and tooth
indexing errors, planet tooth thickness errors, carrier eccentricity, planet pin
hole position errors and ring gear roundness error would impact the dynamic
response in two ways. First of all, each of the run-out and eccentricity errors
would constitute low frequency excitations to be included in the dynamic
model. Secondly, many of these errors prevent an equal load sharing amongst
the planets such that the gear mesh frequency excitation and mesh stiffness of
22
each planet mesh would differ. As the intended applications will be high-
precision aerospace gearing, these errors are of secondary importance.
(x) Damping of the system is represented by either a constant modal damping or a
proportional damping matrix.
Figure 2.1 shows the overall dynamic model of an entire double-helical planetary gear set
with only one planet shown for simplicity purposes. Under the assumptions listed above,
the model formulations will be done first for three basic sub-systems: (i) a sun-planet i
pair (left or right side), (ii) a ring-planet i pair (left or right side), and (iii) a carrier-planet
i pair (left or right side). A beam formulation will then be introduced to combine left and
right sides of each double-helical gear. A general assembly process will then be
employed to obtain the overall mass and stiffness matrices including support bearing
conditions.
2.2.1 A Sun-Planet i Pair Formulation
Figure 2.2 illustrate a dynamic model of an external helical gear pair, which
represents one side (either left or right side) of the sun gear (subscript s) meshing with the
same side of planet-i (subscript pi) located at arbitrary position angle pi . As shown in
Figure 2.2, the plane of action of the gear pair makes an angle with the vertical y spi
axis. This angle can be defined in terms of the transverse pressure angle of the sun- sp
planet pair and as pi
23
y ring gear
planet i
x
sun gear left side
right side z
Figure 2.1: Dynamic model of a double-helical planetary gear system.
24
ypi
ypi spi planet i
ys xpi xpi ys etspi ()
ksp zpi pi
zpi x s xs
zs
zs
Sun gear
Figure 2.2: Dynamic model of sun-planet i pair.
25
sp pi, T s : Counterclockwise, spi (2.1) sp pi, T s : Clockwise,
where Ts is external torque acting on the sun gear. The undamped equations of motion for this s-pi pair are derived using the helical gear pair formulations of Kahraman [2.1].
These equations for sun gear degrees of freedom are
mys s() t k sp cos cos spi p spi () t 0, (2.2a)
mx() t k cossin p () t 0, (2.2b) s s sp spi spi
mz() t k sin p () t 0, (2.2c) s s sp spi
Itkr() sin cos pt () 0, (2.2d) sys sps spispi
Itkr() sin sin pt () 0, (2.2e) sxs sps spispi
JtkrptTN() cos () /(2 ). (2.2f) szs sps spi s
The corresponding equations of motion for the degrees of freedom of planet i are
myppi () t k sp coscos spispi p () t 0, (2.3a)
mx() t k cossin p () t 0, (2.3b) ppi sp spispi
mz() t k sin p () t 0, (2.3c) ppi sp spi
Itkr() sin cos pt () 0, (2.3d) p ypi sp p spi spi
Itkr() sin sin pt () 0, (2.3e) p xpi sp p spi spi
Jtkrpt() cos () 0. (2.3f) p zpi sp p spi 26
In these equations, m , I and J are mass, the diametral mass moment of inertia and
the polar mass moment of inertia of one side (left or right), and r is base circle radius of
gear ( spi, ). ksp is the average value of the gear mesh stiffness for the sun-planet
pair. p ()t represents the relative mesh displacement of the s-pi mesh in the direction spi
normal to tooth surface such that
ptspi() ( yy s pi )cos spi ( xx s pi )sin spi r s zs r p zpi cos + (rrsys p ypi )cos spi ( rr s xs p xpi )sin spi zz s pi sin et spi ( ) (2.4)
where etspi () is the loaded static transmission error excitation at the s-pi mesh and is the helix angle. Equations (2.2) and (2.3) are written in matrix form as
11 12 M0q ()ttt KK q () ff () ss spispissmsi ksp . (2.5a) 0M q ()ttt22 q () f () ppisym. Kspi pi spi where
yts () ytpi () xs ()t xpi ()t zts () ztpi () qs ()t , q pi ()t , (2.5b,c) ys ()t ypi ()t ()t ()t xs xpi zs ()t zpi ()t
Ms DiagmmmI ssssss I J, (2.5d) 27
M ppppppp Diag m m m I I J , (2.5e)
cc22 csc 2 ccs rccsrcscs 2 rcc 2 ss s 22 2 2 s c s cs rcsss cs rs cs rsc s 222 11 srcsrssrcsss s (2.5f) K , spi 22 2 2 2 2 rcsrcssrccsss s 22 2 2 Sym. rss s s r s c s 22 rcs
c22 c c s c 2 c cs rc 2 cs rc s cs rc c 2 pp p csc222 s c scs rcscs rs 2 cs rsc 2 pp p 222 ccs scs s rcspp rss rcs p K12 , spi 22222 rcss c s rc s c s rc sspspsp s rr c s rr c s s rr c c s 22222 rcscssssspspsp rs cs rss rrcss rrs s rrscs 22 2 rcsssspspsp c rs c rc s rr c c s rr s c s rr c (2.5g)
c22 c c s c 2 c cs rc 2 cs rc s cs rc c 2 pp p s22 c s cs rc s cs rs 2 cs rs c 2 pp p 222 srcsrssrcspp p K22 , spi 22 2 2 2 2 rpp c s r css r p ccs 22 2 2 Sym. rpp s s r s c s 22 rcp (2.5h)
28
cc cc cs cs s s fsi ()tket sp spi () , fspi ()tket sp spi ()rsc , (2.5i,j) rss c p rs s rssp s rc s rcp
0 0 0 fsm . (2.5k) 0 0 TNs /2
In the above equations (2.5f-k), c cosspi , s sinspi , c cos , and ssin .
2.2.2 A Ring-Planet i Pair Formulation
Next consider the same planet i meshing with the ring gear (subscript r) on one side of the double-helical gear set as shown in Figure 2.3. With pi being at the same angular position pi , the plane of action of this internal gear mesh makes an angle rpi with the vertical y axis, which is defined in terms of the transverse pressure angle rp of the ring-planet pair and pi as
rp pi, T s : Counterclockwise, rpi (2.6) rp pi, T s : Clockwise
29
y pi
ring gear pi planet i rpi
yr xpi xpi
yr x etrpi () r
pi zpi krp xr zpi
zr zr
Figure 2.3: Dynamic model of a ring-planet i pair.
30
The undamped equations of motion for the r-pi pair shown in Figure 2.3 are derived for the motions of the ring as
myr r() t k rp coscos rpi p rpi () t 0, (2.7a)
mxrr () t k rp cossin rpirpi p () t 0, (2.7b)
mz() t k sin p () t 0, (2.7c) r r rp rpi
Itkrryr() rpr sin cos rpirpi pt () 0, (2.7d)
Itkr() sin sin pt () 0, (2.7e) rxr rpr rpirpi
JtkrptTNrzr() rpr cos rpi () r /(2 ). (2.7f)
Here m , I and J are mass, the diametral mass moment of inertia and the polar mass r r r moment of inertia of one side of the ring gear, and r is base circle radius of the ring r gear. The corresponding motions of planet i are governed by
myppi () t k rp cos cos rpirpi p () t 0, (2.8a)
mx() t k cossin p () t 0, (2.8b) ppi rp rpirpi
mzppi () t k rp sin p rpi () t 0, (2.8c)
Itkrp ypi() rp p sin cos rpi pt rpi () 0, (2.8d)
Itkrp xpi() rp p sin sin rpi pt rpi () 0, (2.8e)
Jtkrptp zpi() rp p cos rpi () 0. (2.8f)
31
In Eq. (2.7) and (2.8), krp is the average gear mesh stiffness of the ring-planet pair, and p ()t is the relative displacement of the r-pi mesh along the plane of action normal to rpi the tooth surfaces, defined as
ptrpi() ( yy r pi )cos rpi ( x pi x r )sin rpi r r zr r p zpi cos + (rrp ypi r yr )cos rpi ( rr r xr p xpi )sin rpi zzet r pi sin rpi ( ) (2.9)
with etrpi () being the loaded static transmission error excitation at the r-pi mesh. After substituting Eq. (2.9) in, equations (2.7) and (2.8) can be put into the matrix form as
11 12 M0q ()ttt KK q () ff () rr rpirpirrmri krp . (2.10a) 0M q ()ttt22 q () f () ppisym. Krpi pi rpi
where q pi ()t is defined in Eq. (2.5c),
ytr () xr ()t ztr () qr ()t , (2.10b) yr ()t ()t xr zr ()t
Mr Diag m rrrrrr m m I I J , (2.10c)
32
c22 c csc 2 ccs rc 2 cs rcscs rcc 2 rr r 22 2 2 s c s cs rcsrrr cs rs cs rsc 222 11 srcsrssrcsrr r K , rpi 22 2 2 2 2 rcrr s rcss rccs r 22 2 2 Sym. rrr s s r s c s 22 rcr (2.10d)
c22 c c s c 2 c cs rc 2 cs rc s cs rc c 2 pp p csc222 s c scs rcscs rs 2 cs rsc 2 pp p 222 c cs s cs s rcppp s rs s rcs K12 , rpi 22222 rcrr c s rc s c s rc rrprprp s rr c s rr c s s rr c c s 22222 rcscsrr rs cs rss rrprprp rrcss rrs s rrscs 22 2 rcrr c rs c rc rrprprp s rr c c s rr s c s rr c (2.10e)
cc22 csc 2 ccs rccsrcscs 2 rcc 2 pp p s22 c s cs rc s cs rs 2 cs rs c 2 ppp 222 srcsrssrcs pp p K22 , rpi 22 2 2 2 2 rpp c s r css r p ccs 22 2 2 Sym. rpp s s r s c s 22 rcp (2.10f)
33
cc cc cs cs s s fri ()tket rp rpi () , frpi ()tket rp rpi ()rsc . (2.10g,h) rsr c p rs s rssp r rc r rcp
0 0 0 frm . (2.10i) 0 0 TNr /2
In equations (2.10d-i), ccos , s sin . c cos rpi and s sinrpi .
2.2.3 A Carrier-Planet i Pair Formulation
Figure 2.4 shows a one side of a carrier-planet i pair, with planet i positioned at the same angle pi and attached to the carrier via a bearing. Rotation center of the planet i represented by the z axis is at a distance r from the rotational axis z of the carrier. pi c c
The planet bearing that is modeled as a diagonal stiffness matrix
Kbp Diag k x k y k z k x k y 0 couples planet pi to the carrier c along a circle of radius rc . Bearing forces and moments acting on planet pi due to any arbitrary motions of pi and c of this particular side are defined as
34
carrier
planet i
Figure 2.4: Dynamic model of carrier-planet i pair.
35
Ftyycpiczcpi() k [( y y ) r cos ], (2.11a)
Ftx () kxc [( x x piczcpi ) r sin ], (2.11b)
Ftzzcpicycpicxcpi()[()cossin], k z z r r (2.11c)
Mt() k ( ), (2.11d) y y yc ypi
Mtx () kxxcxpi ( ). (2.11e)
Here, pi can rotate in the z direction with no resistance such that Mtz () 0. With these bearing forces and moments defined, the equations of motion of the c-pi pair are written as
mycc () t F y () t 0, (2.12a)
mxcc () t F x () t 0, (2.12b)
mzcc () t F z () t 0, (2.12c)
ItMtrcyc() y () c cos piz Ft () 0, (2.12d)
ItMtrcxc() x () c sin piz Ft () 0, (2.12e)
Jtrczc() c sin pix Ftr () c cos piy FtTN () c /(2 ), (2.12f)
myppi () t F y () t 0, (2.13a)
mx() t F () t 0, (2.13b) ppi x
mz() t F () t 0, (2.13c) ppi z
ItMt() () 0, (2.13d) pypi y 36
ItMt() () 0. (2.13e) pxpi x
In matrix form, Eq. (2.12) and (2.13) reduce to
M0q()tt KK11 12 q () cccpicpic fcm . (2.14a) 0Mppi q ()tt22 q pi () 0 sym. Kcpi
where q pi ()t is defined in Eq. (2.5c),
ytc () xc ()t ztc () qc ()t , (2.14b) yc ()t ()t xc zc ()t
Mccccccc Diag m m m I I J , (2.14c)
kkrcyyc00 0 0 kkrs00 0 xxc kkrckrszzczc 0 11 Kcpi kkrckrcs22 2 0 , (2.14d) yzc zc 22 Sym.0 kxzc k r s 22 22 krsxc krc yc
37
ky 00000 0k 0000 x 00k 000 K12 z , (2.14e) cpi 00krc k 00 zc y 0000krszc k x krcyc krs xc 0000
ky 00000 k 0000 x k 000 K22 z . (2.14f) cpi k 00 y Sym.0 kx 0
0 0 0 fcm . (2.14g) 0 0 TNc /2
In equations (2.14d-f), ccos and s sin . pi pi
2.2.4 Coupling of the Left and Right Sides
The sub-system models shown in Figures 2.2 to 2.4 consist of only one side of the double-helical gears. In an actual double-helical system, left and right sides of a gears and the carrier are either one-piece (for planet and sun gears) or connected rigidly (for the
38 ring gear and the carrier). As done earlier by Ajmi and Velex [2.2], left and right sides of gear (s, r, pi) are connected by using Euler type finite beam elements.
In Figure 2.5, a double-helical gear is divided into three pieces: the left side gear segment at an inner diameter of Db , the right side gear segment at the same inner diameter, and a connecting structure of outside diameter Dg . The connecting structure spans from gear face mid-point of the right side to the gear face mid-point of the left side.
Here, partitioning is done such that the total mass of the one-piece gear equals the sum of individual masses of the left and right sides and the connecting structure. The same is true for the inertias as well. The ring gear and carrier connecting structures are also handled the same way by using a two-element finite element model, thus representing each gear by three nodes. The node in middle can be used to connect the gear to the support structure through a stiffness matrix that represents a bearing or spline support.
With this the stiffness and mass sub-matrices for the connecting structures are given as
KK11 12 0 ee11 KKKK22 11 12 , (2.15a) eeee12 2 sym. K22 e2
MM11 12 0 ee11 MMMM22 11 12 (2.15b) eeee12 2 sym. M22 e2 where sr,, pic , and subscript e represents beam elements for each component. 39
(a)
(b)
left side right side gear gear segment segment
node L node M node R
beam beam element 1 element 2
Figure 2.5: (a) Geometry of a double helical external gear, and (b) three-piece model of the double helical gear.
40
The sub-matrices in Eq. (2.15) are defined for the n-th ( n[1, 2] ) beam element as
KK11 12 MM11 12 en en en en Ken , Men , (2.15c,d) sym K22 sym M22 en en
The individual elements of Ken and Men are given in Appendix A. These sub- matrices corresponds to the displacement sub-vector
()q L qqeM () . (2.16) ()q R where subscripts L, R and M indicate left side, right side and middle nodes of each components respectively. In case where the connecting structure for a gear (sr,, pi),
()q L and ()q R take place of gear displacement vectors in Eq. (2.5), (2.10) and (2.14).
2.2.5 The Overall System Equations
These sub system matrices defined by Eq. (2.5), (2.10) and (2.14) are assembled systematically along with right-to-left coupling matrices defined by Eq. (2.15) to obtain the overall equations of motion of a double-helical planetary gear set consisting of N planets (a total of NNdof 18( 3) degrees of freedom) as
Mq()tt +Cq () +Kq () tt F (). (2.17)
41 where q()t is the overall displacement vector, M is the mass matrix, F()t is the force vector and K is overall stiffness matrix ( KK mesh K b ) consisting of gear stiffness matrix Kmesh and support stiffness matrix Kb . M and Kmesh are given in the
Appendix B. Vector q()t includes all of the 18(N 3) degrees of freedom
qse()t qre()t qce ()t q()t , (2.18) q ()t pe1 q pNe ()t
Forcing vector F()t can be represented as assembly of sub-vectors as
Fs ()t Fr ()t 0 F()t , (2.19a) F ()t p1 FpN ()t where
N N ffsmsiL () ffrm () ri L i1 i1 F0s ()t , F0r ()t , (2.19b,c) N N ff () ff () smsiR rm ri R i1 i1
42
(fspi f rpi) L F0pi ()t . (2.19d) (fspi f rpi) R
A support stiffness matrix Kb is defined in the form
Kbbsbrbc Diag 0K 00K 00K 0 0, (2.20a)
and incorporated in Eq. (2.17) to include the matrices for the sun ( Kbs ), ring ( Kbr ), and carrier ( Kbc ) supports. Here, Kbs , Kbr and Kbc are matrices of dimension 6 that are applied to the middle nodes of the respective connecting structures. They are defined as
Kbyxzyx Diag k k k k k 0, (2.20b) with src,,. Finally, C in Eq. (2.17) is the proportional damping matrix that is given as
CK+M . (2.21) where and are proportionality constants.
2.2.6 Excitations
In Eq. (2.19a), individual forcing vectors forming F()t include transmission error excitations that are given as a part of relative mesh displacements in Eq. (2.4) and (2.9).
There are a total of 4N of such periodic excitations (one for each individual gear mesh)
43 for an N-planet gear set. These excitations have the same fundamental frequency that is equal to the gear mesh (tooth passing) frequency m . Transmission error excitations at the s-pi and r-pi meshes can be computed by using a gear load distribution model [2.6] as well as the average gear mesh stiffnesses ksp and krp . Each of the periodic excitations et() along the meshes of the right and left sides of the gear set have the same spi waveforms at the same harmonic amplitudes since the planets are assumed to be identical. Yet they possess a unique phasing relationship defined by N, planet spacing angles pi (p1 0 and iN[2, ]), and the numbers of teeth Zs on the sun gear. The same is true for the r-pi excitations et() where N, (iN[1, ] ), and the numbers of rpi pi teeth Zr on the ring gear define the relative phasing.
Without loss of generality, the s-p1 mesh on the left side of the gear set is chosen here as the reference mesh. The transmission error excitation at this reference mesh is given in Fourier series form as
L ()L etsp1 () eˆspl cos( lt m spl ), (2.22a) l1 where eˆ and are the amplitude and phase angle of the l-th harmonic of this spl spl excitation as predicted by the gear load distribution analysis [2.6]. The superscript L in parenthesis indicates a left side mesh. The transmission error functions on the other s-pi meshes ( iN[2, ]) on the left side can be defined relative to the reference s-p1 mesh as
44
L ()L etspi () eˆspl cos( lt m spl l spi ), i [2, N ]. (2.22b) l1
Here is the phase angle between the s-pi mesh on the left side and the reference s-p1 spi mesh [2.1, 2.7]
Zspi , for CW planet rotation, spi (2.22c) Zspi, for CCW planet rotation.
where Zs is the number of teeth on the sun gear.
An angle is defined as the phase difference between the left and right side due stg to intentional nominal stagger of the teeth. For the case when right and left side teeth are aligned perfectly (i.e. they are mirror images of each other) as illustrated in Figure 2.6(a),
0 . Meanwhile, for a 50% stagger condition is shown in Figure 2.6(b), stg stg where the tip of a tooth on the left side aligns with a tooth root on the right side. In practice, the stagger condition is a design parameter whose impact on the dynamic response is yet to be described. With the stagger phase angle defined, the excitation on the s-pi mesh of the right side of the gear set is defined as
L ()R etspi () eˆspl cos( lt m spl l spi l stg ), i [1, N ]. (2.22d) l1
Meanwhile, the transmission error excitation at the r-p1 mesh on the left side is defined in relation to the reference s-p1 mesh of the left side as
45
(a)
Left Right
(b)
a1
a1
Left Right
Figure 2.6: Illustration of the right-to-left stagger conditions in a double helical gear pair; (a) and (b) .
46
L ()L etrp1 () eˆrpl cos( lt m rpl l rs ), (2.23a) l1 where rs is phase difference between the reference s-p1 mesh and the r-p1 mesh, both on the left side as defined in Ref. [2.7]. eˆ and are the amplitude and phase angle rpl rpl of the l-th harmonic of the excitation, again predicted by using a gear load distribution model [2.6]. Similarly, excitations on any other r-pi meshes on the left side are given as
L ()L etrpi () eˆrpl cos( lt m rpl l rpi l rs ), i [2, N ], (2.23b) l1 where is the phase difference between the r-pi mesh and the r-p1 mesh on the left rpi side, which is given by
Zrpi, for CW planet rotation, rpi (2.23c) Zrpi , for CCW planet rotation.
Here Z denotes number of teeth on ring gear. With the same stagger defined r stg between the left and right sides, excitations of the ring-planet meshes of the right side of the gear set are defined as
L ()R etrpi () eˆrpl cos( lt m rpl l rpi l rs l stg ), i [1, N ]. (2.23d) l1
47
In above equations, the gear mesh frequency m defined above can be determined from kinematic relationships as a function of rotational speeds of the sun and ring gears s and r as [2.8]
ZZsr s/ ( Z s Z r ), fixed ring gear, msrrsrZZ / ( Z Z ), fixed sun gear, (2.24) Zss , fixed carrier.
2.3 Solution Methodology
By setting F0()t and C0 , Eq. (2.17) is reduced to
Mq()tt + ( Kmesh K b ) q () 0 . (2.25)
2 The corresponding Eigen Value problem KQ MQ (withKKmesh K b ) of this undamped free system is solved to find the undamped natural frequencies and the
[1,N ] corresponding mode shapes Q ( dof ).
The steady state response of the LTI system due to the transmission error excitations defined in Section 2.2.6 is obtained using the Modal Summation Technique.
All of these excitations at meshes are, in general, out-of-phase of each other.
Furthermore, each individual harmonic term l of any excitation also has a different phase angle. As such, the response of the gear set to each harmonic term of each gear mesh transmission error excitation must be determined individually using the Modal
48
Summation Technique. These individual responses must then be summed according to the superposition principle to compute the total steady state response.
The Modal Summation Technique makes the use of the expansion theorem in conjunction with the superposition principle to determine the response from and normalized Q . First, the forcing vector given in Eq. (2.19a) is expressed as a sum of
4N vectors (each representing the excitation at one gear mesh) as
4N FF()tt k (). (2.26) k 1 where subscript k denotes a particular gear mesh. Explicit expressions for these forcing vectors Fk ()t are given in Appendix C.
Response to an individual forcing vector Fk ()t is obtained by modal summation as
L Ndof qFkkplmplmpl()tk ( jelt )ˆ cos( ), (2.27a) l11 where sr, (s if k is representing an external mesh and r if the gear mesh is an internal one), j 1 and represents appropriate phasing terms defined by Eq.
(2.22) or (2.23). Fk is the vector of amplitudes of Fk ()t . lm()j is the complex dynamic compliance matrix given by
49
QQ ()j , (2.27b) lm 222 ()(2)ljl mm where Q is the -th normalized mode shape.
Modal or proportional type damping can be employed in this formulation. For the case of modal damping, . When the user assumes a form of damping that is proportional to the mass and stiffness matrices according to Eq. (2.21), the proportional damping ratio of the -th mode is defined as
2 . (2.28) 2 where and are the proportional damping constants. The overall displacement vector is given as sum of steady-state responses to each of individual excitations:
4N qq()tt k (). (2.29) k 1
With q()t known, relative gear mesh displacements are computed according to Eq.
(2.4) and (2.9), from which dynamic mesh forces at each of the 4N gear meshes can be obtained as
Ftkpt() (), sr , (2.30) pi p pi
50 where Ftpi () and ppi ()t correspond to the same right or left side gear mesh. Likewise the components of the support bearing forces are computed by using user defined support stiffness matrices and the corresponding gear (or carrier) displacements. For practical design purposes, dimensionless dynamic factors KD are defined as the ratio of the maximum gear mesh force to the static gear mesh force FTss (2 rN s ). Dynamic factor for a given mesh is given as
Max F() t pi pi KD 1. (2.31) Fs
2.4 An Example Simulation
A double helical planetary gear set consisting of four equally spaced planets (
N 4 , 0, , and 3 ) with a stationary (non-rotating) carrier is used here as an pi 2 2 example gear set. Table 2.1 lists its basic gear design parameters of this example gear set. Total number of degrees of freedom for the model is NNdof 18( 3) 126 .
Individual contact analyses of one sun-planet and one ring-planet mesh were carried out by using a gear pair load distributions model [2.6] under specified loading conditions to determine the average mesh stiffness values ( ksp and krp ), and the harmonic components of the transmission error excitations (amplitudes eˆspl and eˆrpl , and phase angles and in equations (2.22) and (2.23)). Only the first three harmonics of spl rpl
51
Table 2.1: Basic design parameters of the example gear system
Sun Planet Ring Carrier
Number of Teeth 47 39 125 -- Normal module (mm) 1.81 1.81 -- Helix angle (o) 21.5 21.5 -- Normal Pressure angle (o) 22.5 22.5 -- Base radius (mm) 41.8 34.7 111.2 -- Mass (kg) 2.4 0.73 6.8 14.4 Mesh Stiffness (N/µm) 564 531 --
kkyx, (N/µm) 100 100 1000 --
kkyx , (1e6 Nm/rad) 5 5 10 -- D (mm) g 75 250 64 -- DD/ (mm) -- bo 30 263.8 42 (s) 1.35e-6 (s1 ) 50
kp (N/µm) 564.75 -- 531.15 --
52
Table 2.2: Harmonic amplitudes and phase angles of the transmission error excitations of the example gear set of Table 2.1. [2.6]
Harmonics, eˆ ( µm) eˆ (µm) (rad) (rad) l spl rpl spl rpl 1 0.405 0.483 -0.769 -0.709 2 0.088 0.135 -1.299 -1.543 3 0.021 0.005 0.604 0.927
53 the excitations were considered (i.e.l [1, 3] ) as harmonic terms with l 3 have negligibly small amplitudes. Connecting structures between left and right sides have the dimensions listed in Table 2.1 according to Figure 2.5. Table 2.1 also lists the support stiffness values for the sun and ring gears and the carrier as well as the planet bearing stiffness values used in these simulations.
Eigen value solution is carried out for the system defined in Table 2.1 to predict the undamped natural frequencies as listed in Table 2.3. The corresponding modes are classified as in-phase, sequentially phased and counter-phased, adapting the planetary gear set mode classification of Refs. [2.1, 2.9]. Such classifications that are based on predominantly two-dimensional motions along the transverse plane (x-y plane) of the gears are not fully descriptive here since most of the natural modes exhibit three- dimensional motions with dominant axial (z) and rotational ( x and y ) motions. It is noted in Table 2.3 that there are numerous natural modes within the operating frequency range of the gear set while it is not obvious which modes would be excited by the excitations defined in Section 2.2.
2.4.1 Influence of Right-to-left Stagger
Figures 2.7 shows variation of the maximum dynamic mesh force amplitudes
()FMaxFt () and ()FMaxFt () for sun-planet and ring-planet spi L spi L rpi L rpi L meshes of the left side of double helical gears, respectively, with the gear mesh frequency
54
Table 2.3: Predicted natural frequencies and mode types of the example gear set
Mode index Natural Frequency (kHz) Mode Type
Q
1 0.987 In phase 2, 3 1.262 Sequentially phased 4 1.902 In phase 5,6 1.965 Sequentially phased 7,8 2.076 Sequentially phased 9 2.506 In phase 10 2.518 In phased 11,12 2.522 Sequentially phased 13,14 2.994 Sequentially phased 15 3.166 Sequentially phased 29 8.398 Counter phased 32,33 8.852 Sequentially phased 34 9.037 In phase 35,36 9.451 Sequentially phased 38 11.047 Counter phased
55
2000 (a) - - - - 1600
1200
800 (N)
400
0 3000 (b)
2400
1800
(N) 1200
600
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 2.7: Maximum dynamic mesh force amplitudes at the left side (a) s- pi and (b) r-pi meshes.
56
2000 (a)
1600 - - - -
1200
(N) 800
400
0 3000 (b) 2400
1800
(N) 1200
600
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 2.8: Maximum dynamic mesh force amplitudes at the right side (a) s-pi and (b) r-pi meshes.
57
m (mssrrZZ for the case of fixed carrier where s and r are the rotational speeds of the sun and ring gears in rad/s, respectively). Figure 2.8 presents the same for the right side maximum dynamic mesh force amplitudes
()FMaxFt () and()FMaxFt () . In both figures, results for three spi R spi R rpi R rpi R right-to-left gear teeth stagger phase angles of 0, and are compared. Several stg 2 observations can be made from Figures 2.7 and 2.8:
Maximum dynamic sun-planet mesh force amplitudes at all planet meshes at a
given (left or right) side are equal (i.e. ()()()()FFsp1234 L sp L FF sp L sp L )
and ( ()()()()FFspR1234 spR FF spR spR). The same is true for the ring-
planet mesh forces as well.
Each of the resonance peaks is associated with a particular mode excited by a
certain harmonic amplitude l of the excitations. The frequencies of these
modes as well as the harmonics exciting these modes are specified as labels of
each resonance peak.
Maximum dynamic mesh forces vary considerably with stg considered. Not
only amplitudes of response but also the frequencies of resonance peaks
change with stg . Different values of stagger are seen to excite different types
of modes. For stg 0, excited modes exhibit motion where both left and
right sides of double helical gears move together as one piece as in the case of
58
modes at natural frequencies of 1262 and 11047 Hz. These modes are
illustrated in Figure 2.9. Modes 1262 Hz is classified in Table 2.3 as
sequentially phased mode while 11047 which is excited by 2nd harmonic of
excitation is Counter Phased mode. If the modal displacements for
sequentially phased mode 1262 is applied to relative gear mesh
displacement expressions of equations (2.4) and (2.9) with transmission error
terms discarded, one would arrive at templates for this mode in the form
LR LR sp1 a11 a rp1 b11 b sp2 a22 a rp2 b22 b . (2.32) sp3 a11 a rp3 b11 b sp4 a22 a rp4 b22 b
It is noted in these modes that the right and left side gear meshes move in
unison, and hence get excited when stg 0. On the other hand the
sequentially phased modes at 2994 and 3166 Hz exhibit equal but
opposite motions on the right and left sides (as shown in Figure 2.10)
represented by the relative gear mesh displacement templates
LR LR sp1 a11 a rp1 b11 b sp2 a22 a rp2 b22 b (2.33) sp3 a11 a rp3 b11 b sp4 a22 a rp4 b22 b
59
Figure 2.9: Mode shapes representative of Eq. (2.32) at (a) Hz and (b) Hz.
60
Figure 2.10: Mode shape representative of Eq. (2.33) at (a) Hz
and (b) Hz.
61
As a result, these modes are excited when stg . Any stagger angles other
than 0 and excite both types of motions at lower levels, as illustrated in
response curves for . stg 2
For the cases when stg 0 or maximum gear mesh forces at the left and
right side meshes are equal, i.e. ()()FFspi L spi R and ()()FFrpi L rpi R
while ()()FF and ()()FF for any other value. spi L spi R rpi L rpi R stg
The second harmonic components of the excitations for stg excite the
modes that follow the template of Eq. (2.32) with left and right sides moving
8398 in unison. For instance, the resonance peak at 1 Hz is a m 2 2
result of this.
Predicted dynamic factor curves corresponding to Figures 2.7 and 2.8 are shown in
Figures 2.11 and 2.12, respectively. Dynamic factors show maximum dynamic loads are
spi higher than static load carried by the system. In Figure 2.11 dynamic factor KD of 1.28 for the mesh frequency of m 8852 Hz indicates dynamic load is 28% higher than the static load experienced by the corresponding mesh which must be taken into account while designing the gears at that speed.
62
1.3 (a - - - -
1.2
1.1
1 1.5
(b 1.4
1.3
1.2
1.1
1 0 2000 4000 6000 8000 10000
(Hz)
Figure 2.11: Dynamic factors at the left side (a) s-pi and (b) r-pi meshes.
63
1.3 (a - - - -
1.2
1.1
1 1.5
(b 1.4
1.3
1.2
1.1
1 0 2000 4000 6000 8000 10000 (Hz)
Figure 2.12: Dynamic factors at the right side (a) s-pi and (b) r-pi meshes.
64
2.4.2 Influence of Planet Phasing Conditions
Two earlier studies by Kahraman [2.1] and Kahraman and Blankenship [2.10] provided the framework which established a well-structured relationship between the mode shapes excited and the phasing of the excitations as formulated in Section 2.2.
These studies classified the natural modes of a four-planet helical (or spur) planetary gear set (with equally spaced planets) in three groups:
In-phase (IP) modes: In these axisymmetric modes, all planets move in an
identical manner relative to the central member of the gear set (sun, ring and
carrier). These modes were shown to be excited by in phase harmonic terms
of the excitations.
Sequentially phased (SP) modes: In these modes, each planet moves in a
certain way relative to the central members such that none of the central
members exhibit any motions. In other words, these modes are limited to
planet motions. Kahraman [2.1] and Kahraman and Blankenship [2.10]
showed that these modes are excited by the harmonics of the excitation that
are sequentially phased.
Counter phased (CP) modes: As a special case of sequentially phased modes
unique to four-planet gear sets, diametrically opposed planets move the same
way relative to the central members while the motions of two adjacent planets
are 180-degrees out-of-phase. The central members do not move in these
65
modes. The same studies also indicated that these modes are excited by
counter-phased harmonics of the excitations.
These above rules were only valid for gear sets consisting of a single flank (spur or single helical). Therefore, they might not be fully applicable to the double-helical arrangement in hand. In order to investigate the influence of such planet phasing conditions, three variations of the example gear set are considered
A Sequentially Phased Gear Set: The example gear set specified in Table 2.1
has number of teeth values of Z 47, Z 125 and Z 39. As s r p
ZNs 47 4 11.75 and ZNr 125 4 31.25, the following phase angles
apply in Eq. (2.2):
For l 1: lZ 0, , , 3 for i 1, 2, 3, 4. spi 22 lZ 0, , 0, i 1, 2, 3, 4. For l 2 : spi for
For l 3 : lZ 0,3 , , for i 1, 2, 3, 4. spi 22
For l 4 : lZspi 0, 0, 0, 0 for i 1, 2, 3, 4.
A very similar phasing relationship applies to the excitations of the ring gear
meshes as well. These excitation phasing conditions indicate that the first and
third harmonics of the excitations of this gear set should excite sequentially
phased modes while the second and fourth harmonic should excite counter-
phased and in-phase modes, respectively.
66
An In-phase Gear Set: A gear set with Z 48, Z 124 and Z 39 has s r p
ZNs 48 4 12 and ZNr 124 4 31 with lZspi0, 0, 0, 0 for
i 1, 2, 3, 4 regardless of the value of l. The same is true for the ring gear meshes as well. This suggests that all harmonics of the excitation should
excite the in-phase modes only.
Z 46, Z 126 A Counter-phased Gear Set: A third variation with s r and
Z p 39 has ZNs 46 4 11.5 and ZNr 126 4 31.5 with the
resultant phase angles:
For l 1, 3 : lZspi 0, , 0, for i 1, 2, 3, 4.
l 2, 4 lZ 0, 0, 0, 0 i 1, 2, 3, 4. For : spi for
These excitation phasing conditions indicate that the first and third harmonics
of the excitations of this gear set should excite counter-phased modes while
the second and fourth harmonic should excite in-phase modes.
In Figure 2.13(a), the maximum s-pi gear mesh forces of these three gear sets are compared for the stagger condition of stg 0 while Figure 2.13(b) shows the same for
stg . For stg 0 , the SP gear set is seen to excite SP modes while the IP gear set exciting IP modes and the CP gear set being concerned with the CP modes. These SP, IP and CP modes follow the following templates for relative mesh displacements, respectively:
67
LR LR LR
sp1 a11 a sp1 a11 a sp1 a11 a sp2 a22 a sp2 a11 a sp2 a11 a (2.34a-c) sp3 a11 a sp3 a11 a sp3 a11 a sp4 a22 a sp4 a11 a sp4 a11 a
It is noted here that right and left side motions are in unison in these excited modes regardless of the planet phasing conditions since stg 0 .
For a 50% stagger ( stg ) in Figure 2.13(b), the phasing between the right and left sides of the double helical gear set become evident with the types of modes compared in Figure 2.13(a). The SP, IP and CP modes excited by the corresponding gear sets have the following respective relative mesh displacement templates:
LR LR LR
sp1 a11 a sp1 a11 a sp1 a11 a sp2 a22 a sp2 a11 a sp2 a11 a . (2.35a-c) sp3 a11 a sp3 a11 a sp3 a11 a sp4 a22 a sp4 a11 a sp4 a11 a
All three of templates above, of these types of modes point to motions that are 180- degrees out-of-phase between the right and left sides of the double-helical gear set.
These are the modes that require a 50% stagger in order for them to be energized.
2.4.3 Influence of Number of Planets
In order to study the effect of number of planet gears N of the dynamic response of the gear set, certain basic design parameters of the example gear set are modified. Three 68
3600 (a) - - - -
2700
1800 (N)
900
0 2000
(b)
1500
1000
(N)
500
0
(Hz)
Figure 2.13: Maximum dynamic mesh force amplitudes at the left/right side s- pi meshes for (a) and (b) for different planet
phasing conditions.
69
1600
(a) - - - -
1200
800 (N)
400
0 1600
(b)
1200
800 (N)
400
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 2.14: Maximum dynamic mesh force amplitudes at the left/right side (a) s-pi and (b) r-pi meshes for different number of planet gears.
70 to five planet variations of the same example gear set are devised all having Zs 53,
Zr 127 and Z p 37 . This number of teeth combination allows the three, four and five-planet gear sets to have equally spaced planets as well as all three gear sets having sequential phasing conditions. Figure 2.14 compares the dynamic responses of these three gear set with stg 0. Considerable changes in dynamic response can be observed under same loading and damping conditions for different N. While the three-planet gear set exhibits the largest resonance peaks, there are no clear trends on which gear set would be better dynamically regardless of the operating speed ranges and torques.
2.4.4 Influence of Radially Floating Sun Gear
In planetary gear sets, the sun gear is often allowed to float radially (i.e. not supported or piloted radially by a bearing support) in order to allow the gear set to “self- center” itself to compensate for certain types of carrier and gear manufacturing errors as explained by and Ligata and Kahraman [2.11] and Bodas and Kahraman [2.12]. In Figure
2.15, the influence of floating the sun gear radially on the forced response curves is demonstrated for both stg 0 and on the baseline design of Table 2.1. In case of a floating sun, diagonal terms of the sun gear support stiffness matrix K in Eq. (2.20b) bs are taken to be kk1(10)6 N/m, kk5(10)4 Nm/rad while the piloted ys xs ys xs gear set uses the values given in Table 2.1. Modes of the form given by Eq. (2.33) appear to be most sensitive to the values of the sun gear radial support stiffnesses. As a result,
71
2000 Piloted Sun (a) - - - - Floating sun 1500
1000 (N)
500
0 2000 (b)
1500
(N) 1000
500
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 2.15: Effect of radially floating sun gear on dynamic mesh force amplitudes for (a) and (b) .
72 the response of the gear set with the stagger condition of shown in Figure stg
2.15(b) exhibits significant changes at lower frequency ranges (say m 5000 Hz) when the sun gear is radially floating. However very little influence of sun gear support is evident in Figure 2.15(a) for stg 0 .
2.5 Mode Identification using Modal Strain Energy
The Eigen value analysis of the undamped free system yields natural frequencies
Q and corresponding mode shapes ( [1,Ndof ] ) as described in Section 2.3. The forced response curves shown in Figures 2.7, 2-8, 2.13-2.15 indicate that only a few of these modes are excited by the transmission error excitations applied at the gear meshes.
Unless a forced response computation is performed, the question of which modes should be expected to be excited is left unanswered. In an attempt to answer this question, modal strain energies of each mode will be formulated and computed in this section.
The same strain energy computation should also identify the most heavily loaded component and corresponding degrees of freedom within a given mode shape.
Modal strain energy U of the -th vibration mode can be expressed as
1 U QKQT . (2.36a) 2
73 where K is overall stiffness matrix as given in Eq. (2.17). As a modal quantity, the value of U has no meaning since the mode shapes are given in terms of modal displacements that are relative quantities as
qˆ se ˆ qre qˆ ce Q , (2.36b) qˆ pe1 ˆ q pNe
where qˆe ( srcpi,,, .) can be expressed as
()qˆ L qqˆˆeM () . (2.36c) ()qˆ R
Using these modal vectors, total modal strain energy can be grouped in three categories
[2.13] such that
UU sup U mesh U pb. (2.37)
Here U represents the total modal strain energy in support bearings for sun and ring sup
U gears and the carrier, mesh represents the total strain energy in all gear meshes and
U denotes the total strain energy associated with the planet bearings. pb
Strain energy in support springs U is defined as sup 74
UUUUsup s r c , (2.38a)
where Us , Ur and Uc are the strain energies in support springs of the sun, ring and
carrier, respectively. These strain energies components are defined as
11 Ukykxkzkk()qKqˆˆT () ˆˆ222ˆ ˆˆ 2 2 , (2.38b) 22Mb M y x z yy xx where K represents the support stiffness matrix for sun, ring and carrier ( src,, ) b
applied to middle nodes of the respective connecting structures as given by Eq. (2.20b)
and
yˆ xˆ zˆ ()qˆ . (2.38c) M ˆ y ˆ x ˆ z
is the modal displacement vector of the corresponding middle node. This strain energy
can be further broken down into its translational, axial and tilting components by
considering appropriate degrees of freedom.
U Strain energy in gear meshes mesh can be divided into sun-planet and ring-planet
meshes as follows
75
N UUUUU UUUU . mesh sp rpLR sp rp spi rpi L spi rpi R i1 (2.39) where U and U are the strain energies in sun-planet i and ring-planet i meshes, spi rpi respectively, and subscripts R and L denote right and left sides. These strain energies can be expressed in terms of modal relative mesh displacements pˆ for sun gear meshes spi and pˆ for ring gear meshes. They are defined from Eq. (2.4) and (2.9) as rpi
ˆˆˆ ˆˆ ˆˆ pyyspi()cos()sin s pi spi xx s pi spi rr s zs p zpi cos (2.40a) ˆˆ ˆˆ ˆˆ + (rrs ys p ypi )cos spi ( rr s xs p xpi )sin spi zz s pi sin
ˆˆˆ ˆˆ ˆˆ pyyrpi()cos()sin r pi rpi xx pi r rpi rr r zr p zpi cos (2.40b) ˆˆ ˆˆ ˆˆ + (rrp ypi r yr )cos rpi ( rr r xr p xpi )sin rpi zz r pi sin
With these, U and U are given as spi rpi
1 2 Ukp ˆ , (2.41a) spi2 sp spi
1 2 Ukp ˆ . (2.41b) rpi2 rp rpi
Finally the modal strain energy in planet bearings involves the relative modal displacements between the carrier and planets on left and right sides such that
76
1 N 22 ˆˆˆˆ ˆˆ Ukyyrkxxrpb y c pi c zccos pi x c pi c zc sin pi 2 i1 2 ˆˆ ˆˆ kzz c z pi r c yccos pi r c xc sin pi (2.42) ˆˆ22 ˆˆ kky yc ypi x xc xpi . which can be further separated into its translational, axial and tilting components.
As stated earlier, the absolute value of U in Eq. (2.36a) has no meaning since the modal displacements in Eq. (2.36b) are relative quantities. For this reason, the value of each strain energy component can be normalized such that U 1.
Strain energy distribution for different modes of the example gear set is given in the
Table 2.4. The modes excited by the transmission error are presented in italic characters in this table. Most of the excited modes in the higher frequency region (say 3000
Hz) exhibit higher strain energy content in gear meshes as compared to strain energies in
8398 support structures and planet bearings. The excited modes at frequencies 29 and
35,36 9451Hz shows normalized Umesh values of 0.89 and 0.93, respectively. This indicates that most of the energy is represented by the relative gear mesh displacements in these modes, making them the most likely modes excitable by the transmission error excitations. Further break down of strain energies at these modes is shown in Figure 2.16.
8398 The mode at 29 Hz exhibits nearly same level of energy in the sun-planet and ring-planet meshes, while the mode at 35,36 9451 Hz shows more strain energy in the ring-planet meshes. 77
Table 2.4: Strain Energy distribution for the modes of the example gear set (excited modes are shown in italic characters)
Mode Index Natural Frequency (kHz) Modal Strain Energy
Umesh Usup U pb 1 0.987 0.07 0.00 0.93 2, 3 1.262 0.02 0.62 0.36 4 1.902 0.93 0.00 0.07 5,6 1.965 0.00 0.83 0.17 7,8 2.076 0.18 0.82 0.00 9 2.506 0.00 0.00 1.00 10 2.518 0.00 0.00 1.00 11,12 2.522 0.00 0.01 0.99 13,14 2.994 0.35 0.63 0.02 15 3.166 0.98 0.00 0.02 29 8.398 0.89 0.00 0.11 32,33 8.852 0.91 0.00 0.09 34 9.037 0.89 0.00 0.10 35,36 9.451 0.93 0.00 0.06 38 11.047 0.99 0.00 0.01
78
0.6
0.4
0.2 Normalized strain energy strain Normalized
0 0.6
0.4
0.2 Normalized strain energy Normalized
0
Figure 2.16: Normalized modal strain energy components of modes at (a) Hz (b) Hz.
79
2.6 Summary
In this chapter, a linear, time-invariant model of a double-helical planetary gear set was developed which allows the analysis of a gear set with any number of planets, any planet phasing and spacing configurations and any support conditions. The model included all rigid body degrees of freedom of gears and the carrier in a three-dimensional manner. Planets were allowed to be positioned equally or unequally spaced around the sun gear. The model captures the phasing relationships between the planet meshes as well as the right-to-left phase differences associated with the staggering the teeth of gears.
Free and forced vibration analyses of the model were carried out by solving the governing Eigen Value problem and applying the modal summation technique with proportional/modal damping.
An example gear set analyses point to various unique dynamic behaviors of double- helical planetary gear sets. Numerous modes with dominant tilting motions are predicted to indicate that a three-dimensional formulation is must for double-helical planetary gear sets. The results of parametric studies also show that right-to-left stagger conditions and planet phasing conditions are equally critical in defining what modes are excited and what resonance peaks are formed in the forced response of the gear set. Change in number of planet gears in the gear set was also shown to change the dynamic response considerably. Additionally, the influence of the sun support conditions was shown to be more pronounced for the case of 50% stagger. At the end, the modal strain energies associated the gear meshes, planet bearings and support structures of the central members
80 is formulated to demonstrate that most of the modes excited by the transmission error excitations are those with high gear mesh modal strain energies.
References for Chapter 2:
[2.1] Kahraman, A., 1994, “Planetary Gear Train Dynamics,” ASME Journal of Mechanical Design, 116, pp. 713-720.
[2.2] Ajmi, M. and Velex, P., 2001, “A Model for Simulating the Quasi-Static and Dynamic Behavior of Double Helical Gears,” The JSME International Conference on Motion and Power Transmission, MPT-2001, pp. 132-137.
[2.3] Seager, D., 1975, “Conditions for the Neutralization of Excitation by the Teeth in Epicyclic Gearing,” Journal Mechanical Engineering Science, 17(5), pp. 293- 298.
[2.4] Kubur, M., Kahraman, A., Zini, D., and Kienzle, K., 2004, “Dynamic Analysis of a Multi-Shaft Helical Gear Transmission by Finite Elements: Model and Experiment,” ASME Journal of Vibration and Acoustics, 126, pp. 398-406.
[2.5] Kang, M. and Kahraman, A., 2012, “Measurement of Vibratory Motions of Gears Supported by Compliant Shafts,” Mechanical Systems and Signal Processing, 29, pp. 391-403.
[2.6] LDP Gear Load Distribution Program, 2011, Gear and Power Transmission Research Laboratory, The Ohio State University, USA.
[2.7] Parker, R. and Lin, J., 2004, “Mesh Phasing Relationships in Planetary and Epicyclic Gears,” ASME Journal of Mechanical Design, 126, pp. 365-370.
[2.8] Kahraman, A., 1994, “Load Sharing Characteristics of Planetary Transmission,” Mechanism and Machine Theory, 29(8), pp. 1151-1165.
[2.9] Platt, R. and Leopold, R., 1996, “A Study on Helical Gear Planetary Phasing Effects on Transmission Noise,” VDI Berichte, 1230, pp.793-807. 81
[2.10] Kahraman, A. and Blankenship, G.W., 1994, “Planet Mesh Phasing in Epicyclic Gear Sets,” International Gearing Conference, Newcastle upon Tyne, pp. 99-104.
[2.11] Ligata, H., Kahraman, A., and Singh, A., 2009, “Closed-form Planet Load Sharing Formulae for Planetary Gear Sets using Translational Analogy,” Journal of Mechanical Design, 131, 021007-1 to 021007-7.
[2.12] Bodas, A. and Kahraman, A., 2004, “Influence of Carrier and Gear Manufacturing Errors on the Static Load Sharing Behavior of Planetary Gear Sets,” JSME International Journal, Series C, 47(3), 908-915.
[2.13] Lin, J., 2000, “Analytical Investigation of Planetary Gear Dynamics,” PhD Thesis, The Ohio State University.
82
CHAPTER 3
Influence of Gyroscopic Effects on Dynamic Behavior of Double-Helical Planetary
Gear Sets
3.1 Introduction
In this chapter, the three-dimensional discrete Linear Time-invariant (LTI) model proposed in Chapter 2 is expanded to include certain classes of gyroscopic effects to study their potential impact on the dynamic behavior of the double-helical planetary gear sets. Two types of gyroscopic effects might exist in a planetary gear set:
(i) gyroscopic moment due to resistance of a spinning body to its change in plane
of rotation (as per principle of conservation of angular momentum), and
(ii) gyroscopic effect due to a rotating carrier, which introduces additional
centripetal and Coriolis acceleration components.
In most geared turbo fan application, carrier is stationary (non-rotating), often due to its weight, mounting complexities as well as lubrication system implementation issues.
With geared turbofan applications in focus, gyroscopic effects due to rotating carrier are 83 not considered in this chapter. Gyroscopic moments due to skew or tilting are included in the linear time-invariant equations of motion. A complex Eigen value solver is employed to determine the natural modes. A complex modal summation formulation is then used to predict the forced response with gyroscopic effects included. Influences of several key design parameters and operating conditions on the dynamic response of the system due to additional gyroscopic moments are studied at the end.
3.2 Incorporation of Gyroscopic Moments in the Dynamic Model
Principle of conservation of angular momentum indicates that a spinning body resists any force tending to change its plane of rotation. Resistance to this change depends on mass moment of inertia of the spinning body and its angular rotational velocity. Accordingly, any tilting/rocking motion of a gears or carrier is resisted by a gyroscopic couple (moment). A generalized formulation for this gyroscopic moment is derived here for any spinning body ( sr,, pi), rotating at constant angular velocity of k where k is the unit vector along the z axis. The angular momentum of the spinning body with no tilting motion (rotating in its plane of rotation) is
HkJ , (3.1)
The velocity vector for the same body due to tilting and spinning motions can be expressed as
xyi+ j z k, (3.2)
84 where i, j and k are unit vectors along x, y and z axis respectively and x , y and
z are vibratory velocities in x, y and z direction respectively. As per principle of conservation of angular momentum, rate of change of angular momentum due to tilting motion leads to additional moment M given by
MH H, (3.3a)
Substituting Eq. (3.1) and (3.2) in Eq. (3.3a) gives expression for additional moment acting on body as
MijJJyx. (3.3b)
The components of this additional moment in x and y direction (i and j components respectively) can be incorporated in linear time invariant equations of motion as derived in chapter 2. Using Eq. (3.3) the equations of motion for each of the sub systems as described in section 2.1 are modified as follows
3.2.1 A Sun-Planet i Pair with Gyroscopic Effects
Equations (2.2) and (2.3) derived earlier for a sun-planet pair shown in Figure 2.2 is modified to account for gyroscopic moments, defined by Eq. (3.3). Due to the vibratory displacements xs ()t and ys ()t of the sun gear, gyroscopic moments are created to modify the sun gear equations of motion in and direction (Eq. (2.2d,e) as xs ys
85
ItJsys() s sxs () tkr sps sin cos spispi pt () 0, (3.4a)
ItJsxs() s sys () tkr sps sin sin spispi pt () 0. (3.4b) while other sun gear equations of motion 2.2(a-c,f) remain the same.
Similarly, Eq. (2.3a-c,f) for planet motions remain unchanged and Eq. (2.3d,e) are modified to account for gyroscopic moments as
ItJtkrpypi() p pxpi () spp sin cos spispi pt () 0, (3.5a)
ItJp xpi() p p ypi () tkr sp p sin sin spi pt spi () 0. (3.5b)
Eq. (2.2a-c,f), (3.4), Eq. (2.3a-c,f) and (3.5) are written in matrix form as
M0qssss ()tt G0q () 0Mppi q()tt 0G ppi q ()
11 12 KKspi spi q s()tt ff sm si () k . (3.6a) sp 22 sym. K qfpi()tt spi () spi
Here velocity sub-vectors are given as
ys ()t y pi ()t xs ()t x pi ()t zts () ztpi () q s ()t , q pi ()t , (3.6b,c) ys ()t ypi ()t ()t ()t xs xpi zs ()t zpi ()t and the sub-matrices of the gyroscopic matrix are found as 86
000000 000000 000000 Gss , (3.6d) 0000Js 0 000J 00 s 000000
000000 000000 000000 G . (3.6e) pp0000J 0 p 000J p 00 000000
where psspZ Z is the rotational speed of the planet (with a non-rotating carrier).
It is evident from Eq. (3.6) that gyroscopic effects introduce an additional skew- symmetric component to the overall damping matrix that was defined in Chapter 2 by considering a proportional damping mechanism.
3.2.2 A Ring-Planet i Pair with Gyroscopic Effects
In a similar manner, equations of motion for the ring-planet i pair shown in Figure
2.3 are obtained by modifying Eq. (2.7) and (2.8) to account for gyroscopic moments.
The resultant matrix equation for the ring-planet i pair with gyroscopic effects included is as follows
87
M0qrrrr ()tt G0q () 0Mppi q()tt 0G ppi q () KK11 12 q()tt ff () rpi rpi r rm ri krp . (3.7a) sym. K22 qfpi()tt rpi () rpi where
yr ()t xr ()t ztr () q r ()t , (3.7b) yr ()t ()t xr zr ()t
000000 000000 000000 Grr . (3.7c) 0000Jr 0 000J 00 r 000000
The rest of the matrices in Eq. (3.7a) are as defined in Eq. (2.10). In Eq. (3.7c), r is the rotational speed of the ring gear that is defined as Z Z for a stationary carrier. rrss
3.2.3 A Carrier-Planet i Pair with Gyroscopic Effects
The gyroscopic moments associated with a carrier-planet i pair shown in Figure 2.4 is obtained through modifications to Eq. (2.12) and (2.13). Here the gyroscopic effects
88 caused by the planet motion must be accounted for while no gyroscopic moment acts on the stationary carrier (c 0 ). Matrix form of the equations of motion of this pair is as follows
11 12 M0qcc ()tt 00q ccpicpic () KK q () t0 .(3.8a) 0Mppi q()tt 0G ppi q ()22 q pi () t 0 sym. Kcpi where
yc ()t xc ()t ztc () q c ()t . (3.8b) yc ()t ()t xc zc ()t
The remaining matrices in Eq. (3.7a) are as in Eq. (2.14).
3.2.4 Coupling Elements with Gyroscopic Effects
With gyroscopic moments for one side of gears are accounted for, gyroscopic matrices for connecting beam elements are defined next. As described in Section 2.2.4 and illustrated in Figure 2.5, each double-helical gear is divided into three pieces with left and right sides connected using Euler type finite beam elements. The gyroscopic matrix for each connecting structure is given as
89
GG011 12 ee11 GGGG22 11 12 , (3.9a) eeee12 2 skew sym. G22 e2 where sr,, pi and subscript e denotes beam elements for each component. The sub- matrices in Eq. (3.9a) are defined for the n-th ( n[1, 2] ) beam element as
11 12 GGen en G . (3.9b) en 22 skew sym. Gen
The individual elements of Gen are given in Appendix A.
3.2.5 The Overall System Equations with Gyroscopic Effects
The sub-system gyroscopic matrices for gears and carrier as defined by Eq. (3.6) to
(3.8) along with gyroscopic matrices for connecting structures given in Eq. (3.9) are assembled systematically to form equations of motion of the entire double-helical planetary gear set as
Mq()tt + G+C q () + Kmesh K b q () tt F (). (3.10)
All of the system matrices here are defined in Eq. (2.17) and Appendix B.
90
3.3 Solution Methodology
A free vibration analysis of a given system is carried out for undamped case (C0
) to study the influence of gyroscopic moments on the natural frequencies of the system.
The response of the linear, time-invariant system to the transmission error excitations is again obtained by using the Modal Summation Technique in complex domain [3.1]. For forced vibration response a proportional damping CK+M is considered here together with the gyroscopic matrix G . For this purpose, Eq. (3.10) is put in the state- space form
rAr+BF()ttt () (), (3.11a) with the state vector r()t , and matrices A and B defined as
q()t r()t , (3.11b) q()t
0I 0 A , B= . (3.11c,d) 11 1 MK M[G+C] M
Here I is the identity matrix of dimension Ndof . The free vibration solution of Eq.
(3.11a) can be carried out by setting F0()t which reduces Eq. (3.11a) to
rAr()tt (). (3.12)
The corresponding algebraic and adjoint Eigen value problems for a system are given, respectively, as 91
AR R , (3.13a)
LATT L. (3.13b) where a diagonal matrix of Eigen values and R and L are the modal matrices of so- called right and left Eigenvectors, respectively. The modal matrices R and L are biorthogonal such that LRT I and LART .
As explained earlier a free vibration analysis is carried out first for an undamped system (C0 ). The Eigen values here come in complex conjugate pairs. The -th Eigen value ()j or its complex conjugate ()j ) obtained from Eq. (3.13) is used to determine the -th natural frequency as
()imag ( j ) imag ( j ). (3.14)
The Eigen values and natural frequencies obtained are speed dependent.
Similar to LTI system without gyroscopic effect, a proportional damping matrix is considered for calculating the response of the system to transmission error excitations.
The forcing vector is separated to its components based on their phasing using same methodology presented in Section 2.3,
4N FF()tt k (), (3.15) k 1
92 with each of the 4N terms represent the excitation caused by a gear mesh. Expansion theorem is used in conjunction with the above biorthogonality relations to find the response of the given system to individual forcing function Fk ()t [3.1]
L 2Ndof ˆ rkp()tk lmplmpl ( j ) eˆ cos( lt ), (3.16a) l11 where sr, , j 1 and represents appropriate phasing terms defined by Eq.
ˆ ()j (2.22) or (2.23). The term lm is given by
T ˆ LBF k lm()j R . (3.16b) jm
Here Fk is the vector of amplitudes of Fk ()t , and R and L are the -th pair of
Eigenvectors. Linearity of the model allows superposition principle to be applied to find the total steady-state response as
4N rr()tt k (). (3.17) k 1
The state vector r()t consists of displacement vector q()t and the velocity vector q()t as given by Eq. (3.11b). With q()t known, relative gear mesh displacements are computed according to Eq. (2.4) and (2.9), from which dynamic mesh forces at each of the 4N gear meshes can be obtained. The dynamic factors as defined by Eq. (2.31) are valid here as well.
93
3.4 An Example Simulation
The example double-helical planetary gear set used in Chapter 2 with parameters listed in Table 2.1 will be used as well. The gyroscopic effects formulated above are included in the analysis of this four-planet gear set ( N 4 ) with a non-rotating carrier.
Complex Eigen value solution was carrier out to predict the undamped natural frequencies . Influence of different gear parameters on the natural frequencies of the system was studied. Complex modal summation was carried out to quantify the influence of gyroscopic terms on the forced vibration response of the same system with the same transmission error excitations as in Chapter 2 with additional proportional damping.
As the gyroscopic matrix G is speed dependent (i.e. GG () ), modal behavior becomes speed dependent as well. Rotational speed of input member (sun gear) is varied from 0 to 020,000s rpm ( 0 s 2094 rad/s) and Eigen value problem was solved at various s values within this range and influence of rotational speed on was quantified. Figure 3.1 shows variation of with s . Only some of the natural frequencies within the frequency range of interest are seen to vary with s while others remain the same regardless of s . These affected natural frequencies are listed in Table
3.1 together with the strain energy components associated with the corresponding mode shapes. The modes with lower frequencies ( 7 2076 and 14 2994 Hz.) exhibit
94
10000
p2 9500
(Hz) p2
9000 p1
p1
8500 4000
3500
(Hz) 3000
2500
2000 0 5000 10000 15000 20000
(rpm)
Figure 3.1: Variation of certain natural frequencies with the rotational speed due to gyroscopic effects ( ).
95
Table 3.1: Strain energy distribution for the modes exhibiting change in natural frequencies due to gyroscopic effects.
Mode Index Natural Frequency (kHz) Modal Strain Energy
U U U mesh sup pb 7, 8 2.076 0.18 0.82 0.00 13, 14 2.994 0.35 0.63 0.02 17, 18 3.757 0.67 0.32 0.02 32, 33 8.852 0.91 0.00 0.09 35, 36 9.451 0.93 0.00 0.06
96 more strain energy in the bearings (support or planet bearings) with most of it caused by the tilting action as shown in Figure 3.2.
All the affected modes are double modes (two modes at the same natural frequency) when gyroscopic effects are not included. When gyroscopic effects are included, they split and diverge from each other with increasing s as observed in Figure 3.1. For one mode, the natural frequency increases due to gyroscopic stiffening while the frequency of the companion mode reduces due to gyroscopic softening. The maximum change in natural frequency in Figure 3.1 is observed for the mode at 17,18 3757 Hz. This frequency value predicted without gyroscopic effects changes to 17 3675 and
18 3887 Hz at s 20,000 rpm, representing a modest 3.4% change.
Maximum percent change in natural frequencies within the same speed range was used as a metric to quantify the influence of support stiffness on the gyroscopic moments.
Figure 3.3(a) shows that the maximum percentage change in natural frequencies due to gyroscopic effects with the sun gear bearing stiffnesses in tilting/rocking directions. The changes in the natural frequencies is the most significant at the lowest values of k and ys kxs . This is rather predictable since reducing stiffnesses in rocking direction result in larger amplitudes of ys and xs (and hence, ys and xs ). Similar effect is observed as a result of the changes in the ring gear bearing stiffnesses k and k in yr xr tilting/rocking directions as well, as shown in Figure 3.3(b).
97
0.8 (a)
0.6
0.4
0.2
0 0.8
(b)
0.6
0.4
0.2
0
Figure 3.2: Strain energy distribution for support spring and planet bearings for modes at (a) Hz and (b) Hz. (Tr - Translational, Ti - Tilting, Ax - Axial).
98
20 (a)
15
Max change 10 in [%]
5
0 012345 (Nm/µrad)
20 (b)
15
Max change 10 in [%]
5
0 012345 (Nm/µrad)
Figure 3.3: Maximum percent change in natural frequencies with an input speed range of 0 to 20,000 rpm as a function of (a) sun gear support stiffness in tilting direction and (b) ring gear support stiffness in tilting direction.
99
40 (a)
30
Max change 20 in [%]
10
0 0.010 0.015 0.020 0.025 0.030 2 (kg-m )
40 (b)
30
Max change 20 in [%]
10
0 0.10 0.15 0.20 0.25 0.30 2 (kg-m )
Figure 3.4: Maximum percent change in natural frequencies with an input speed range of 0 to 20,000 rpm as a function of polar mass moment of inertia of (a) the sun gear and (b) the ring gear.
100
Next, the influence of the polar mass moments of inertia on gyroscopic effects is investigated. Two cases are considered, one with a floating sun gear configuration (
6 4 kkys xs 10 N/m, kkys xs 5(10) Nm/rad) and other with floating ring gear (
7 5 kkyr xr 10 N/m, kkyr xr 10 Nm/rad). Polar mass moments of inertia of the sun and ring gears are varied in Figure 3.4(a) and (b) for the above cases to observe the maximum percentage change in natural frequencies. This figure shows that, as polar mass moments of inertia of gears increase, maximum percent change in natural frequencies with speed increases as well.
Gyroscopic effects can be expected to influence the forced frequency response curves of the gear set in two ways. One is the resonant frequencies assuming the modes listed in Table 3.1 are excited by the transmission error excitations. The other is the changes done to the damping matrix with the addition of G, which should influence the amplitudes of some of the resonance peaks. Dynamic gear mesh force amplitude curves with gyroscopic effects included are compared in Figure 3.5 to those of the LTI system without gyroscopic effects for a case of no right-to-left stagger ( stg 0). Somewhat lower proportional damping values with coefficients (5(10)7 s, 10 s-1) are used here as compared those in Chapter 2. It is seen in Figure 3.5 that there is little difference between the curves with and without gyroscopic effects. Some minor differences are observed at resonance peaks associated with modes at frequencies 32,33 8852 and
35,36 9451Hz. From Figure 3.1, 32,33 8852 Hz are split to 32 8790 Hz and
101
5000
without gyroscopic effect (a) - - - - with gyroscopic effect 4000
3000
2000
1000
0 7000
6000 (b)
5000
4000
3000
2000
1000
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 3.5: Maximum dynamic mesh force amplitudes at the left side (a) s- pi and (b) r-pi meshes with and without gyroscopic effect.
102
32 8901Hz at m 8850 Hz (corresponding to s 11,300 rpm) when the gyroscopic effects are included. The above variation in natural frequency is marked by line p1-p1 in Figure 3.1. This change in natural frequency is reflected in Figure 3.5 with a shift in resonance frequency from 8852 Hz to 8790 Hz. Similarly shift in resonance peak at 9451 Hz can be explained by the change in the natural frequency marked by line p2-p2 in Figure 3.1.
The same analysis is repeated for the case of radially floating sun gear with a 50% stagger ( stg ). Figure 3.6 shows dynamic mesh forces for sun-planet and ring-planet meshes on left side, with and without gyroscopic effects included. It can be seen that resonance peak associated with 14 891 Hz shows considerable change in amplitude, accompanied by a slight change in resonant frequency. Including gyroscopic effects at this frequency ( m 891 Hz or s 1,137 rpm), splits one pair of these four modes to
1 881 Hz and 4 902 Hz while the other two modes remain unchanged at
2,3 891 Hz. The rest of the excited modes do not exhibit any change in natural frequency due to gyroscopic effect. As a result, the impact of the gyroscopic effects in the force response of the example gear set can be deemed secondary.
3.5 Summary
In this chapter, a certain class of gyroscopic effects is included in the linear time- invariant model of Chapter 2. Gyroscopic moments due to the resistance of a spinning gear to its change in plane of rotation are incorporated in the model. Free and forced 103
8000 without gyroscopic effect (a) - - - - with gyroscopic effect
6000
4000
2000
0 10000
(b) 8000
6000
4000
2000
0 0 200040006000800010000 (Hz)
Figure 3.6: Maximum dynamic mesh force amplitudes at the left side (a) s- pi and (b) r-pi meshes with and without gyroscopic effect for radially floating sun ( ).
104 vibration analyses are carried out for the resulting set of equations by solving complex
Eigen value problem and applying modal summation technique to the state-space form of the system equations. The example simulations indicate that only a portion of the natural modes, those exhibiting significant tilting motions, are impacted by gyroscopic moments.
Softer tilting bearing support stiffness conditions and larger polar mass moments of inertia were shown to increase the impact of gyroscopic effects on these natural frequencies. Inclusion of the gyroscopic effects were shown to influence the forced frequency response of the example gear set very little, as certain resonant peak amplitudes and frequencies are varied slightly.
References for Chapter 3:
[3.1] Meirovitch, L., 2001, Fundamentals of Vibration, McGraw-Hill Higher Education, NY.
105
CHAPTER 4
Investigation of Time-Varying Gear Mesh Stiffness Effects on Dynamics of
Double-Helical Planetary Gear Sets
4.1 Introduction
Models presented in Chapters 2 and 3 employed two related assumptions in modeling of the gear meshes. First was the assumption of constant mesh stiffness values.
It is well-established through accurate gear contact load distribution models that the overall stiffness of a gear mesh varies with roll angle (or time) in a periodic manner. The most apparent reason for such parametric fluctuation in the stiffness of a gear mesh is the fact that number of tooth pairs in contact fluctuates between two integers. A spur gear pair with a typical profile contact ratio value has one tooth pair of two tooth pairs in contact at a given time. With the stiffness of a tooth pair is relatively insensitive to the rotational position, the overall gear mesh stiffness is either formed by single tooth pair stiffness (in the zone of single-tooth contact) or two in-parallel tooth pair springs. In line with this, time dependency of a mesh stiffness of a spur gear pair was experimentally demonstrated to impact the steady-state response amplitudes of a spur gear significantly
106
[4.1]. It was also shown experimentally by the same investigators that mesh stiffness fluctuations are also responsible for other unique behavior such as parametric resonances
[4.2]. With such a experimental evidence in place, various modeling studies (e.g. refs.
[4.3-4.7]) were shown to correlate with spur gear pair measurements closely. However such time-varying stiffness effects were discounted by experimental studies on helical gear pairs including Kubur et al [4.8] showed very good match between the predictions from a three-dimensional Linear Time-Invariant (LTI) model of a helical gear system to their own helical gear pair measurements. Various three-dimensional helical gear models adapted this LTI approach (e.g. [4.9-4.11]). This was deemed reasonable since the loads in helical gear meshes are carried by a larger number of tooth pairs and contact lines are diagonal to the tooth surfaces, the overall stiffness fluctuations of helical meshes can be significantly smaller than their spur counterparts.
The second assumption used in Chapters 2 and 3 was that the gears forming the pair maintain their contact all the time, i.e. no tooth separations are allowed to take place. As any gear pair must have certain amount of backlash (clearance) for other design reasons such as lubrication and assembly, one would expect that the teeth should loose contact as soon as the dynamic gear mesh force amplitudes exceed the mean gear mesh force representing the transmitted torque. Experiments of Munro [4.12], Kubo [4.13] as well as more recent measurements by Kahraman and Blankenship [4.1-4.4, 4.14] all indicate that such tooth separations are common for spur gear pairs especially near the primary and sub-harmonic resonance peaks where dynamic gear mesh force amplitudes become large. As such, softening type forced response curves with discontinuities were reported
107 in these experiments as well as numerous Nonlinear Time-Varying (NTV) modeling studies published within the last two decades. Ma and Kahraman [4.15, 4.16] described these interactions between the parametric and nonlinear effect on a single-degree-of- freedom oscillator theoretically. Such nonlinear behavior is not evident for helical gear pairs, as the limited published studies [4.8, 4.17] indicate that the mesh stiffness fluctuations and the transmission error amplitudes are not large enough to trigger nonlinear behavior.
Such simplifying assumptions that reduce the gear dynamics model to LTI ones can be justified to a certain extent in view of published helical gear experiments, real motivation in using them often been the conveniences afforded by linear systems. The
Eigen value solutions leading to the forced response using frequency domain analysis makes larger-scale LTI system very desirable for computational purposes. The solution of the NTV model of a double-helical planetary gear set must rely on time-domain numerical integration techniques for their solutions. This chapter proposes an NTV model of a double-helical planetary gear set with periodic gear mesh stiffness conditions and tooth separations to investigate whether these effects play a tangible role on the forced response. For this, the LTI model of Chapter 3 will be modified in the next section to include mesh stiffness functions with phasing relations similar to static transmission error functions as well as subjecting them to a piecewise-linear backlash function. The
NTV model results will be compared at the end to the LTI results to assess the significance of these effects for double-helical planetary gear sets.
108
4.2 A Nonlinear Time-Varying Dynamic Model
For computational concerns, it has been a common practice to reduce the degrees of freedom of the gear dynamics model when time-varying and non-linear effects are included. The NTV models of planetary gear sets are often torsional [4.7] or two- dimensional [4.18]. Such a reduction to the LTI model of Chapters 2 and 3 is not prudent here as the motions predicted were truly three-dimensional as evident from the components of the gear mesh strain energies as well as the excited mode shapes. With this, the task in hand is to modify the 18(N 3) degree-of-freedom model of the double- helical planetary gear set (N is the number of planet branches) from Chapter 2 with the gyroscopic effects from Chapter 3 to include the time-varying gear mesh stiffnesses and tooth separations.
4.2.1 Definition of Time-Varying Gear Mesh Stiffnesses
The excitations for the LTI model consisted of the motion transmission errors applied in the form of periodic displacement functions. For this, a gear load distribution model [4.19] was used to predict the transmission error excitation at an individual external (sun-planet) and internal (ring-planet) mesh. In Section 2.2.6, these excitations were written in Fourier series form and modified to impose planet mesh phasing conditions determined by the planet position angles ( pi with p1 0 ), number of planets (N), the number of teeth on the sun and ring gears ( Zs and Zr ), and the stagger
109 between the right and left sides of the gear set ( stg ). The same methodology will be applied here to the gear mesh stiffness functions as well.
With the first sun-planet mesh on the left side as the reference mesh as in Section
2.2.6, the mesh stiffness of this mesh is predicted using the load distribution model [4.19] and written in Fourier series form as
L ()L ˆ ktksp1 ()spsplmspl k cos( lt ). (4.1a) l1 where kˆ and are the amplitude and phase angle of the l-th harmonic component, spl spl
()L and ksp is the mean value. Here, ktsp1 () is simultaneous with the transmission error excitation in the same mesh, given in Chapter 2 (Eq. (2.22a) as
L ()L etsp1 () eˆspl cos( lt m spl ), (4.1b) l1 where eˆ and are the amplitude and phase angle of the l-th harmonic. In other spl spl
()L ()L words, phase angles spl and spl are such that ktsp1 () and etsp1 () at any time t represent the same exact gear mesh position.
The gear mesh stiffness at other s-pi meshes (iN[2, ]) on the left side are defined relative to the reference s-p1 mesh as
110
L ()L ˆ ktkspi ()sp k spl cos( lt m spl l spi ), i [2, N ], (4.2a) l1 where
Zspi , for CW planet rotation, spi (4.2b) Zspi, for CCW planet rotation.
With representing intentional nominal stagger of the teeth between the right stg
and left sides, stiffness functions on the right side sun-planet meshes are written as
L ()R ˆ ktkspi ()sp k spl cos( lt m spl l spi l stg ), i [1, N ]. (4.3) l1
The gear mesh stiffness at the r-p1 mesh on the left side is defined in relation to the reference s-p1 mesh of the left side as
L ()L ˆ ktkrp1 ()rp k rpl cos( lt m rpl l rs ), (4.4) l1 where is phase difference between the reference s-p1 mesh and the r-p1. Here, kˆ rs rpl and are the amplitude and phase angle of the l-th harmonic of the mesh stiffness and rpl
krp is mean value of gear mesh stiffness for ring planet mesh. Similarly, mesh stiffness
of other r-pi meshes on the left side are given as
L ()L ˆ ktkrpi ()rp k rpl cos( lt m rpl l rpi l rs ), i [2, N ], (4.5a) l1 111 where
Zrpi, for CW planet rotation, rpi (4.5b) Zrpi , for CCW planet rotation.
Finally, with the same stagger stg , gear mesh stiffness functions for ring-planet meshes on the right side of the gear set are defined as
L ()R ˆ ktkrpi ()rp k rpl cos( lt m rpl l rpi l rs l stg ), i [1, N ]. (4.6) l1
In above equations, the gear mesh frequency m can be obtained using same kinematic relationship as defined in Eq. (2.24).
4.2.2. Equations of Motions for NTV Model
Equations of motion of the Linear Time Invariant (LTI) model with gyroscopic effects are modified here to take into account time-varying mesh stiffness and tooth separation nonlinearities.
With the same set of parameters, equations of motion for sun-planet i pair as defined in Eq. (2.2a-c, f and 3.4a,b) and Eq. (2.3a-c, f and 3.5a,b) are modified to include time varying mesh stiffness and tooth separation nonlinearities. The new set of equations for sun gear motion are given as
mys s() t h spi k spi ()cos t cos spi p spi () t 0, (4.7a)
112
mxss () t h spispi k ()cossin t spispi p () t 0, (4.7b)
mzss () t h spispi k ()sin t p spi () t 0, (4.7c)
ItJsys() s sxs () thktr spispi () s sin cos spispi pt () 0, (4.7d)
ItJsxs() s sys () thktr spispi () s sin sin spispi pt () 0, (4.7e)
Jszs()thktr spispi () s cos ptT spi () s /(2 N ). (4.7f)
The corresponding equations of motion for planet i are also modified as
myp pi() t h spi k spi ()cos t cos spi p spi () t 0, (4.8a)
mxppi () t h spispi k ()cos t sin spispi p () t 0, (4.8b)
mzp pi() t h spi k spi ()sin t p spi () t 0, (4.8c)
ItJthktrp ypi() p p xpi () spi spi () p sin cos spi pt spi () 0, (4.8d)
ItJp xpi() p p ypi () thktr spi spi () p sin sin spi pt spi () 0, (4.8e)
Jthktrptp zpi() spi spi () p cos spi () 0. (4.8f)
The gear mesh stiffness term ktspi () carries superscript L or R based on whether left of right side of double-helical gear set is considered. The term h in these equations is the spi unit step function representing tooth separation as
1, ptspi ( ) 0, hspi (4.9) 0, ptspi ( ) 0.
113
A negative/zero relative mesh displacement ptspi () 0 represents a tooth separation condition, resulting in a zero gear mesh spring force. The back collisions of the teeth are not included in this formulation based on the experimental evidence provided in Ref.
[4.1-4.4]. Equations of motion, Eq. (4.7) and (4.8) can be written in matrix form
M0qssss ()tt G0q () 0Mppi q()tt 0G ppi q ()
11 12 KKspi spi q s()tht f sm spi f si () hk() t . (4.10) spi spi 22 qf()tht () sym. Kspi pi spi spi
In similar manner, equations of motion given by Eq. (3.7a) for a ring planet i sub- system are modified to obtain
M0qrrrr ()tt G0q () 0Mppi q()tt 0G ppi q ()
11 12 KKrpi rpi q r()tht f rm rpi f ri () hk() t , (4.11a) rpi rpi 22 qf()tht () sym. Krpi pi rpi rpi where
1, ptrpi ( ) 0, hrpi (4.11b) 0, ptrpi ( ) 0.
The formulation for a carrier-planet i sub-system given in Eq. (3.8a) is not influenced by the time-varying and nonlinear effects, which is repeated here for completeness purposes: 114
M0q()tt 00q () KK11 12 q () t cc ccpicpic fcm . 0Mppi q()tt 0G ppi q ()22 q pi () t0 sym. Kcpi (4.12)
As it is done for the LTI formulation, these equations of motion are applied to an entire double-helical planetary gear set configurations to obtain the overall equations of motion including time-varying gear mesh stiffnesses, tooth separation nonlinearities and gyroscopic effects as
Mq()tt + [ C () +G ] q () t + [ Kmesh ( q ,)+ t K b ] q () t F m F (). t (4.13a)
Here it is noted that both the stiffness matrix and damping matrix are time-varying since a proportional damping is used in the form
CK+M()tt () . (4.13b)
4.3 Solution of the NTV System Equations
A direct numerical time integration scheme is employed to solve the equations governing the NTV system. As most of the numerical time integration algorithms are written for a first order system of equations, the second order ordinary differential equations of Eq. (4.13a) are first converted to first order differential equations by writing them in state-space form as
rAr+BF+F()ttt () () [m ()], t (4.14a)
115 with the state vector r and matrices A and B defined as
q()t r()t , (4.14b) q()t
0I 0 A()t , B= . (4.14c,d) 11 1 MKq(,)tt M[G+C] () M
Here K(,)q tt Kmesh (,)+q K b .
This system of 36(N 3) first order NTV ordinary differential equations is solved using Gear’s BDF (Backward Differentiation Formulae) method [4.20]. Due to large scale of the problem (Eigen values for the corresponding LTI system ranges from 0 to
1012 ), large differences in stiffness properties of the system and large number of degrees of freedom, equations of motion behave stiff, requiring a stiff solver as the one used here. The numerical integration must be carried out for extended periods to pass the transients such that the steady-state response can be obtained. As in any dynamic system, larger damping levels and more appropriate selection of initial conditions r(0) should shorten the transient region of the response. It is also expected that the transients last longer in the vicinity of the resonance peaks than off resonance regions. While not much can be done about damping and proximity to the resonance peaks, the initial conditions at a given gear mesh frequency (speed) increment were chosen as the final steady-state solution from the previous mesh frequency increment to minimize the simulations required to avoid transient solutions.
116
The state vector r()t consists of displacement vector q()t and the velocity vector q()t . With q()t known, relative gear mesh displacements are computed according to Eq.
(2.4) and (2.9), from which dynamic mesh forces at each of the 4N gear meshes can be obtained. The dynamic factors as defined by Eq. (2.31) are valid here as well.
4.4 Numerical Results
The NTV model formulation employs two separate excitations at a gear mesh j, a static transmission error excitation et() and a parametric gear mesh stiffness excitation j kt(). The static transmission error et() has a load-depended component et() that j j jd represents the tooth deflections and a kinematic component etjk () that represents the intentional tooth modifications and manufacturing errors, both of which cause the tooth surfaces to deviate from a perfect involute. With this, etjjdjk() e () t e () t.
In Chapters 2 and 3 where ktj () kjm constant, both components of the etj () were included by considering the loaded static transmission error (LSTE) as etj (). This was proposed by Ozguven and Houser [4.21] as a good approximation for spur gears as well. Focusing on helical gear mesh modeling, Blankenship and Singh [4.22] suggested that using LSTE along with a time-varying gear mesh stiffness might not be correct as the same tooth deflections contribute to both et() and kt(). Per Tamminana et al [4.23] j j four versions of gear mesh models for spur gear pair: (i) ktj () and etjjdjk() e () t e () t
, (ii) ktj () and etjjk() e () t, and (iii) ktj () and etj () 0. Through comparison to 117
NTV spur gear responses and predictions of a 2D deformable-body dynamics model,
Tamminana et al [4.23] recommended that second option with et() e () tshould be jjk used for spur gear pairs having tooth modifications while the third option with etj () 0 is the best in case of unmodified (perfect involute) gears.
No such extensive study on the gear mesh modeling exists for helical gears for two reasons. One is that only a very small fraction of published gear dynamics data is available for helical gears. In addition, there is no 3D deformable-body model available to simulate helical gears under dynamic conditions. For this reason, the NTV model proposed in Section 4.2 will be exercised with four variations of excitation parameters:
Model I: LTI system with ktj () kjm constant and etjjdjk() e () t e () t ,
Model II: NTV system with ktj () and etjjdjk() e () t e () t,
Model III: NTV system with ktj () and etjjk() e () t, and
Model IV: NTV system with ktj () and etj () 0.
4.4.1 Verification and Analysis of Time-domain Solutions
The numerical time integration scheme employed to solve Eq. (4.14a) is first verified by comparing its steady-state solutions for the limiting LTI case to those from the frequency-domain modal summation solution from Chapter 2. Figure 4.1 compares the maximum dynamic gear mesh force amplitudes predicted for the example system defined in Table 2.1 for the same proportional damping values of (1.35(10)6 s, 118
50 s-1) with sun torque of 2000 Nm. Here numerical integration solutions were carried out for 500 gear mesh periods (tTmax 500 mesh whereTmesh2 m ) to pass the transient region such that steady-state response can be obtained for this comparison.
As shown in Figure 4.1(a) and (b), respectively, both dynamic sun-planet and ring-planet mesh forces predicted by the two methods match perfectly for the limiting LTI case indicating that the state-space form of the equations of motion as well as the numerical integration scheme works accurately.
4.4.2 Example System Analyses
In this section, steady-state responses obtained by all three NTV gear mesh models specified in Section 4.4 will be compared to each other as well as that of the corresponding LTI system. With the transmission error and mesh stiffness excitation parameters defined in Table 4.1 for sun torque of 2000 Nm (Ts 2000 Nm), Figure 4.2 provides such a comparison for the same system used in Figure 4.1. Here the amplitude of the response is the root-mean-square values of the dynamic gear mesh forces and is defined as
L ˆ 2 ()Fpi rms Fsr pil , ,. (4.15) l1
ˆ where Fpil is the l-th harmonic of the steady-state gear mesh force. It is seen in this figure that the r.m.s. dynamic amplitudes follow the same trends exciting similar set of
119
2000
(a) - - - -
1600
1200
(N) 800
400
0 3000
(b)
2400
1800 (N)
1200
600
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 4.1: Comparison of modal summation and direct numerical integration solutions for the LTI case. Maximum dynamic mesh force amplitudes on the left side (a) s-pi and (b) r-pi meshes.
120
Table 4.1: Harmonic amplitudes and phase angles of the transmission error and mesh stiffness excitations of the example gear set of Table 2.1 [4.19] for different torque levels.
Ts 2000 Nm
eˆ kˆ eˆ kˆ l spl spl rpl rpl spl spl rpl rpl ( µm) ( N/µm) (µm) (N/µm) (rad) (rad) (rad) (rad) 1 0.405 36.55 0.483 36.41 -0.769 2.429 -0.709 2.501 2 0.088 5.88 0.135 7.89 -1.299 1.981 -1.543 1.674 3 0.021 1.09 0.005 0.01 0.604 -2.291 0.927 -1.899
Ts 1000 Nm
eˆ kˆ eˆ kˆ l spl spl rpl rpl spl spl rpl rpl ( µm) ( N/µm) (µm) (N/µm) (rad) (rad) (rad) (rad) 1 0.068 32.08 0.09 31.74 -0.75 2.49 -0.75 2.56 2 0.025 5.87 0.04 7.62 -1.19 2.09 -1.19 1.79 3 0.014 1.15 0.01 0.26 0.760 -2.06 0.76 -1.76
Ts 667 Nm
eˆ kˆ eˆ kˆ l spl spl rpl rpl spl spl rpl rpl ( µm) ( N/µm) (µm) (N/µm) (rad) (rad) (rad) (rad) 1 0.048 24.73 0.032 24.44 2.36 2.46 2.59 2.58 2 0.005 7.01 0.008 8.46 -0.52 2.00 -1.48 1.85 3 0.011 1.22 0.011 0.25 0.85 -2.05 0.76 -1.66
121 modes at same resonance frequencies while the amplitudes of certain resonance peaks differ with the gear mesh model used. In Figure 4.2, the peak at
11(8398) 4199Hz is excited for mesh models I and II suggesting that m 22 transmission error component of the excitations excite this mode. Resonance peak at
8852 m Hz are excited somewhat equally regardless of the gear mesh model used, while the peak at 11(11047) 5523 Hz is excited in case of models II, m 22
III and IV indicating that mesh stiffness is responsible for exciting this mode. Meanwhile the peak at m 9451Hz is seen for Models II, III and IV indicating again that mesh stiffness variation is responsible for exciting this mode.
Some of the time histories of the steady state response are presented next to demonstrate their shape as well as harmonic content. In relation to Figure 4.2, Figure
4.3(a) shows dynamic gear mesh force time history for the left side s-p1 mesh at
8852 m Hz, which represents a resonance peak in Figure 4.2. As evident from the
Fast Fourier spectrum of the same data shown in Figure 4.3(b), the response at
m 5523 Hz is dominated by the second gear mesh harmonic, indicating that this mode is excited by the second harmonic terms of excitations
The root-mean-square representation of the steady-state response used in Figure 4.2 was not used in earlier chapters. Instead, maximum dynamic response amplitude defined in Chapter 2 was used. Using the same maximum dynamic mesh force amplitudes, Figure
4.2 is reproduced in Figure 4.5. Comparison of these two figures indicate that the
122
2500
(a)
2000
1500
(N) 1000
500
0 3000
(b) 2400
1800
(N) 1200
600
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 4.2: Root-mean-square values of dynamic mesh forces at the left side (a) s-pi and (b) r-pi meshes for different excitation models ().
123
9000 (a)
8000
7000
(N) 6000
5000
4000 0 1 2 3 4 Mesh cycles
2500 (b)
2000
1500
(N) 1000
500
0 0 1 2 3 4 5 Mesh order
Figure 4.3: Steady-state dynamic mesh force at an s-pi mesh on the left side obtained by using Model II. (a) Time history at Hz and (b) the corresponding frequency spectrum.
124
7500 (a)
7000
6500
(N) 6000
5500
5000 0 1 2 3 4 Mesh cycles 1000 (b)
800
600
(N) 400
200
0 0 1 2 3 4 5 Mesh order
Figure 4.4: Steady-state dynamic mesh force at an s-pi mesh on the left side obtained by using Model II. (a) Time history at Hz and (b) the corresponding frequency spectrum.
125
2500 (a)
2000
1500
(N) 1000
500
0 3000
(b) 2400
1800
(N) 1200
600
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 4.5: Maximum dynamic mesh force amplitudes at the left side (a) s-pi and (b) r-pi meshes with different excitation models ( ).
126 response curves are very similar for both cases such that r.m.s. or maximum gear mesh force amplitudes can be used as the metric for the forced response plots.
Figure 4.6 shows maximum amplitudes of dynamic mesh forces for same gear set with 50% stagger ( stg ) between the right and left sides of the gear set. All for gear mesh models exhibit resonance peaks at 2999Hz, with models II to IV causing higher resonance peaks as shown in Figure 4.6(b).
Next, an In-phase gear set analyzed in Chapter 2 is revisited here to examine the influence of different gear mesh models on the system response. Figure 4.7 shows the maximum dynamic mesh forces for this in-phase system for all four types of gear mesh models. As observed earlier, resonance frequencies remain same with amplitude of response varying considerably across models I to IV.
As the final analysis, influence of torque levels on the dynamic mesh forces is shown in Figure 4.8 using Model II as an example. For the example system of Table 2.1, the excitation parameters change with torque as shown in Table 4.1 for three sun gear torque values of Ts 2000 , 1000 and 667 Nm. It can be seen from Figure 4.8 that the shapes of the resonance curves remain same but amplitudes increase with an increase in mean torque levels. This is partly because the excitation parameters vary with torque.
More importantly, the increase in mean torque increases the influence of time-varying stiffness excitations, which is in agreement with References [4.1, 4.2, 4.24].
127
10000 (a)
8000
6000
(N) 4000
2000
0 8000
(b)
6000
4000 (N)
2000
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 4.6: Maximum dynamic mesh force amplitudes at the left side (a) s-pi and (b) r-pi meshes for different excitation models ( ).
128
5000
(a)
4000
3000
(N) 2000
1000
0 5000 (b)
4000
3000
(N) 2000
1000
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 4.7: Maximum dynamic mesh force amplitudes at the left side (a) s-pi and (b) r-pi meshes of an in-phase system for different gear mesh models ( ).
129
2500 (a)
2000
1500
(N) 1000
500
0 2500
(b) 2000
1500
(N) 1000
500
0 0 2000 4000 6000 8000 10000 (Hz)
Figure 4.8: Maximum dynamic mesh force amplitudes at the left side of (a) s-pi and (b) r-pi mesh at various input torque levels. Model II was used here.
130
4.5 Summary
This chapter is focused on potential effects of time-varying gear mesh stiffness and nonlinearities on the steady-state response of a double-helical planetary gear set. The model preserves generic nature of the existing model such that any number of planets, any phasing condition, spacing configurations and any support conditions can still be analyzed. Direct numerical integration scheme was employed to solve the resulting set of nonlinear time varying equations of motion in time domain. Based on past studies on discrete dynamic modeling of helical gear pairs, four gear mesh model variations (the first one being the LTI system of Chapter 2) were defined and the model was exercised with all mesh models. Sizable differences were observed in forced response curves with inclusion of time-varying stiffnesses, characterized by new resonance peaks and overall increased response amplitudes. In the absence of detailed experimental data, it was not possible to discriminate for or against any of these gear mesh models. However their closeness to the LTI solution suggests that the LTI system approximation might be sufficient for practical design purposes, which is in agreement with the conclusions of
Kubur et al [4.8] for regular helical gears.
While the variations of the model used in this chapter all implemented tooth separation functions defined by Eq. (4.9) and (4.11b) to allow contact loss in cases the relative gear mesh displacements pspi ()t and prpi ()t reach zero. None of the simulations presented however show any signs of nonlinear behavior. There are no jump discontinuities and peaks exhibit symmetric, linear behavior. This confirms the
131 conclusions of Kubur et al [4.8] for regular helical systems. In view of these simulations, the model can be reduced to a linear time-varying one by setting hhspi rpi 1 in Eq.
(4.9) and (4.11b).
References for Chapter 4:
[4.1] Kahraman, A. and Blankenship, G. W., 1999, “Effect of Involute Contact Ratio on Spur Gear Dynamics,” ASME Journal of Mechanical Design, 121, pp. 112- 118.
[4.2] Kahraman, A. and Blankenship, G. W., 1997, “Experiments on Nonlinear Dynamic Behavior of an Oscillator with Clearance and Time-varying Parameters,” ASME Journal of Applied Mechanics, 64, pp. 217-226.
[4.3] Kahraman, A. and Blankenship, G. W., 1996, “Interactions between Commensurate Parametric and Forcing Excitations in a Systems with Clearance,” Journal of Sound and Vibration, 194, pp. 317-336.
[4.4] Blankenship, G. W. and Kahraman, A., 1995, “Steady State Forced Response of a Mechanical Oscillator with Combined Parametric Excitation and Clearance Type Nonlinearity,” Journal of Sound and Vibration, 185(5), pp. 743-765.
[4.5] Al-Shyyab, A. and Kahraman, A., 2005, “Non-Linear Dynamic Analysis of a Multi-Mesh Gear Train using Multi-term Harmonic Balance Method: Period-one Motions,” Journal of Sound and Vibration, 284, pp. 151-172.
[4.6] Al-Shyyab, A. and Kahraman, A., 2005, “Non-Linear Dynamic Analysis of a Multi-Mesh Gear Train using Multi-term Harmonic Balance Method: Subharmonic motions,” Journal of Sound and Vibration, 279, pp. 417-451.
[4.7] Al-shyyab, A. and Kahraman, A., 2007, “A Non-linear Dynamic Model for Planetary Gear Sets,” Proc. ImechE, Part K: Journal of Multi-Body Dynamics, 221, pp. 567-576.
132
[4.8] Kubur, M., Kahraman, A., Zini, D., and Kienzle, K., 2004, “Dynamic Analysis of a Multi-shaft Helical Gear Transmission by Finite Elements: Model and Experiment,” ASME Journal of Vibrations and Acoustics, 126, pp. 398-406.
[4.9] Kahraman, A., 1994, “Dynamic Analysis of a Multi-mesh Helical Gear Train,” ASME Journal of Mechanical Design, 116, pp. 706-712.
[4.10] Kahraman, A., 1994, “Planetary Gear Train Dynamics,” ASME Journal of Mechanical Design, 116, pp. 713-720.
[4.11] Kahraman, A., 1993, “Effect of Axial Vibrations on the Dynamics of a Helical Gear Pair,” ASME Journal of Vibration and Acoustics, 115, pp. 33-39.
[4.12] Munro, R. G., 1962, “The Dynamic Behaviors of Spur Gears,” PhD Dissertation, Cambridge University, UK.
[4.13] Kubo, A., Yamada, K., Aida, T., and Sato, S., 1972, “Research on Ultra High Speed Gear Devices (reports 1-3),” Transaction of Japan Society of Mechanical Engineer, 38, pp. 2692-2715.
[4.14] Kahraman, A. and Blankenship, G. W., 1999, “Effect of Involute Tip Relief on Dynamic Response of Spur Gear Pairs,” ASME Journal of Mechanical Design, 121, pp. 313-315.
[4.15] Ma, Q. and Kahraman, A., 2006, “Sub harmonic Motions of a Mechanical Oscillator with Periodically Time-varying, Piecewise Non-linear Stiffness,” Journal of Sound and Vibration, 294(3), pp. 924-636.
[4.16] Ma, Q. and Kahraman, A., 2005, “Period-one motions of a Mechanical Oscillator with Periodically time-varying, Piecewise Non-linear Stiffness,” Journal of Sound and Vibration, 284, pp. 893-914.
[4.17] Umezawa, K., Ajima, T., and Houjoph, H., 1986, “Vibration of Three Axis Geared System,” Transaction of Japan Society of Mechanical Engineer, 29, pp. 950-957.
133
[4.18] Kahraman, A., 1994, “Load Sharing Characteristics of Planetary Transmissions,” Mechanisms and Machine Theory, 29, pp. 1151-1165.
[4.19] LDP Gear Load Distribution Program, 2011, Gear and Power Transmission Research Laboratory, The Ohio State University, USA.
[4.20] Gear, C. W., 1971, “The Simultaneous Numerical Solution of Differential- Algebraic Equations,” IEEE Trans. Circuit Theory, TC-(18), pp. 89-95.
[4.21] Ozguven, H., and Houser, D., 1988, “Dynamic Analysis of High Speed Gears by Using Loaded Static Transmission Error,” Journal of Sound and Vibration, 125(1), pp. 71-83.
[4.22] Blankenship, G. W., and Singh, R., 1995, “A New Gear Mesh Interface Dynamic Model to Predict Multi-Dimensional Force Coupling and Excitation,” Mechanism and Machine Theory, 30(1), pp. 43-57.
[4.23] Tamminnana, V. K, Kahraman A. and Vijayakar, S., 2007, “On the Relationship between the Dynamic Factors and Dynamic Transmission Error of Spur Gear Pairs,” ASME Journal of Mechanical Design, 129, pp. 75-84.
[4.24] Al-Shyyab, A., 2003, “Nonlinear Analysis of Multi Mesh Gear Train Using Multi-Term Harmonic Balance Method,” PhD Dissertation, The University of Toledo, USA.
134
CHAPTER 5
Conclusions
5.1 Summary
This research is focused on investigating the dynamic response of double-helical planetary sets through a development of an analytical model. The proposed model used a three-dimensional discrete-parameter formulation with all gear mesh, bearing and support structure compliances included. The model was presented in three levels of complexity, starting with linear time-invariant (LTI) model, then the same LTI formulation with gyroscopic effects included, and finally a nonlinear time-varying (NTV) version with parametrically time-varying gear mesh stiffnesses and nonlinear tooth separation effects included.
As the first step, a generic three-dimensional linear (no tooth separations), time- invariant (constant gear mesh stiffnesses) model was proposed to simulate dynamic behavior of any N-planet double-helical planetary gear system. The model allowed any planet phasing conditions dictated by the number of planets, number of gear teeth and planet position angles. In addition, gear mesh phasing conditions associated with any 135 stagger angle between the right and left sides of the gear set were also accounted for in the model. As excited by the loaded gear mesh motion transmission errors applied at the gear meshes as displacement excitations, forced response of the gear set was computed by using the modal summation technique, with the natural modes found from the corresponding real Eigen value solution for the undamped system. As the least known set of parameters, damping at the gear mesh and bearing interfaces were represented by a proportional damping formulation. Strain energies of the mode shapes were computed to identify the modes excitable by the transmission error excitations. Parametric studies on an example gear set showed significant influences of planet phasing, stagger conditions, gear and carrier support conditions as well as the number of planets on the steady-state forced response.
High-speed double-helical planetary gear set applications requires an investigation of the influence of gyroscopic effects on the system response. In the second step, LTI model was modified to include a class of gyroscopic effects due to vibratory skew of spinning gears, with the carrier being stationary. The corresponding complex Eigen value solutions were examined to quantify the influence of rotational speed of the gear set through gyroscopic effects on the natural frequencies and the mode shapes. A complex modal summation formulation was employed to compute the forced response with gyroscopic effects. Influence of gyroscopic moments on natural frequencies was found to be modest within typical aerospace speed ranges, with only a sub-set of modes exhibiting dominant tilting motions impacted by the gyroscopic effect. In the forced
136 response curves, effect of gyroscopic moments was limited to modest changes in amplitudes and frequencies of certain resonance peaks.
As the final step in the model development, mesh stiffness fluctuations due to change in number of tooth pairs in contact were included in the second version of the model. The stiffness fluctuations serve as internal parametric excitation, applied at the gear meshes along with the transmission error excitations with the same mesh-to-mesh phasing relations. Tooth separation nonlinearities were also included here to arrive at a set of NTV equations of motion in state-space form, which were solved by using direct numerical integration. Based on past studies on discrete dynamic modeling of helical gear pairs, three variations of transmission error excitations (zero, unloaded and loaded) were considered with the time-varying gear mesh stiffnesses. Sizable differences were noted between the forced response curves for time-varying and time-invariant systems characterized by additional resonance peaks and overall increases in response amplitudes while no signs of nonlinear behavior were evident.
5.2 Conclusions and Contributions
This research provided a mathematical tool to predict dynamic behavior of a double-helical planetary gear system. Based on the results of the simulations of example systems using this tool, a number of conclusions can be made in regards to the dynamic behavior of double-helical planetary gear sets:
137
Natural modes exhibit shapes that can allow a classification based on the
planet gear displacements relative to the central members. Certain modes can
be characterized as axi-sysmmetric with same planet motions relative to sun
gear while the others are not.
Numerous natural modes are predicted to have significant tilting and/or axial
motions, indicating that a three-dimensional model must be used in simulating
double-helical planetary gear sets.
The modes excited by the excitation mechanisms devised, are shown to have
phasing conditions that match the imposed phasing of the excitations. The
modes with large gear mesh modal strain energies are the ones excited by the
gear mesh excitations.
Both the right-to-left stagger angle and the planet-to-planet phasing conditions
are shown to change the forced response significantly, in the process exciting
different sets of natural modes. This indicates that an accurate dynamic model
of a double-helical planetary gear set must use the actual excitation phasing
conditions defined by the design parameters such as stagger angle, number of
planets, numbers of gear teeth and planet position angles.
Only few of the natural modes exhibiting significant tilting motions are
impacted by gyroscopic moments while most of these modes are not the ones
excited by the gear mesh excitations. As such, influence of gyroscopic
138
moments on the forced vibration response was found to be secondary for the
systems with non-rotating carriers.
Different forms of the transmission error excitations used in the time-varying
model showed differences in the resultant force response curves, highlighting
the need for experimental data to determine the accurate form of transmission
errors used in time-varying models.
In agreement with the previous experimental observations from single-helical
gear sets, the double-helical planetary gear sets are predicted to exhibit linear
behavior. Accordingly, backlash-induced nonlinearities can be neglected in
modeling double-helical gear systems.
This proposed model represents the first generalized analytical tool to study the dynamic characteristics of double-helical planetary gear system. Given its generality, the model is applicable to systems with any number of planets, any support condition and any phasing configurations. Model can be useful to study the influence of key design parameters like stagger, phasing condition, number of planets on the dynamic response of the system within ranges of speed and torque of operation. While found to be not vitally influential for the example systems analyzed, the model provides the means for investigating gyroscopic effects as well as time-varying effects and nonlinearities of any double-helical planetary system, filling the void in these aspects of aerospace gearing as well.
139
5.3 Recommendations for Future Work
Assumptions made in development of this discrete analytical model can be removed to investigate their influence on the dynamic characteristics of the double helical planetary gear system. Some of the recommendations which can enhance the work presented in this study are
The model can be complemented by incorporating different types of
manufacturing and assembly errors (gear run-out and tooth indexing errors,
planet tooth thickness errors, carrier eccentricity, planet pin hole position
errors and ring gear roundness error) to study their influence on the dynamic
characteristics of the gear system.
Flexibility of ring gear can also be taken into account by modeling hybrid
finite element and discrete model by using techniques like model
condensation or employing complete deformable body model.
Gyroscopic effect due to rotating carrier can also be included in the
formulation by using rotating frame of reference.
Analytical model presented in this study must be validated through
comparisons to experiments. For this, a new experimental set-up must be
developed to execute tightly-controlled high-speed planetary gear experiments
to generate data suitable for the validation of the model.
140
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APPENDIX A
Beam Element Matrices
The formulations for the beam elements used to connect left and right sides of a gear sets are defined here. Considering an Euler beam element en ( srcp,,,1,, pN, n=1,2) having 6 degrees of freedom at each of its two nodes as shown in Figure A1, the displacement vector of the element is given as
y x z y x z left qen . (A1) y x z y x z right
Element mass matrix Men in Eq. (2.15b) is given as sum of three matrices such that
MMMMen e123 e e , (A2)
148
156 0156 00140Sym . 2 02204 2 220004 m 000000 Me1 , 420 54 0 0 0 13 0 156 054013000156 007000000140 22 01303 0 0 02204 22 1300030220004 000000000000
36 036 000Sym . 2 03 04 2 30004 Ime 000000 Me2 , 30 36 0 0 0 3 0 36 0360300036 000000000 22 030 000304 22 3000 030004 000000000000
0 00 000Sym . 0000 00000 J 000001 M me . e3 0000000 3 00000000 000000000 0000000000 00000000000 000001/2000001
149
Here, m is the mass of the beam element per unit length, Ime and Jme are the diametral and polar mass moments of inertia of the element per unit length, and is the length of the element. The stiffness matrix of the same element en in Eq. (2.15a) is defined as
KKKKen e123 e e , (A3) where
12 012 000Sym . 2 06 04 2 60004 EIae 000000 Ke1 , 3 12 0 0 0 6 0 12 0120600012 000000000 22 0 6 02 0 0 0 6 04 22 60002060004 000000000000
0 00 001Sym . 0000 00000 EA K 000000 e2 0000000 , 00000000 00 1000001 0000000000 00000000000 000000000000
150
0 00 000Sym . 0000 00000 GJ K ae 000001 e3 0000000 . 00000000 000000000 0000000000 00000000000 00000 1000001
Here, Iae and Jae are diametral and polar area moments of inertia, A is the cross- sectional area of the element, and E and G are the Young’s modulus and the shear modulus of elasticity, respectively. Similarly, gyroscopic matrix for the same element
en used in Eq. (3.9a) is defined as
0 36 0 000Skew Sym . 3000 2 03040l mIae 000000 . (A4) Gen 15A 03603000l 3600030360l 000000000 2 3000ll 03000 22 030ll 0003040 000000000000
151
y
x y y x y z z x x z z
Area A
Figure A.1: An Euler beam element.
152
APPENDIX B
Overall System Matrices
Overall gear mesh matrix described in Eq. (2.17) is given as
KKKKmesh m123 m m (B1)
where
N 11 11 12 KKse11 () spi L K se 0 0 0 i1 21 22 11 12 KKKKse1122 se se se 0 0 N 0KKK021 22 () 11 0 se22 se spi R i1 N 11 11 12 000KKKre11 () rpi L re i1 21 22 11 000 KKKre112 re re Km1 12 000 0Kre2 000 00 000 00 000 00 ()K00K021 () 21 sp11 L rp L 000 00 21 00K( sp1)R 00
153
0000 0000 0000 0000 12 K000re2 N 22 11 KKre2 () rpi R 0 0 0 i1 N 11 11 12 0KKKce11 () cpi L ce 0 Km2 i1 21 22 11 12 0KKKKce1122 ce ce ce N 21 22 11 00KKKce22 ce () cpi R i1 21 0K()cp1 L 00 0000 ()K00K21 () 21 rp11 R cp R
()K0012 sp1 L 000 12 00K()sp1 R 12 ()K00rp1 L 000 12 00K()rp1 R 12 Km3 ()K00cp1 L 000 12 00K()cp1 R 11 22 22 22 12 KK+KKp11 e() sp 1 rp 1 cp 1 L K p 11 e 0 21 22 11 12 KKKKpe11 pe 11 pe 12 pe 12 21 22 22 22 22 0KKKKKp12 e p 12 e() sp 1 rp 1 cp 1 R
154
Overall mass matrix described in Eq. (2.17) is given as
MMM 123 M (B2) where
(()MM11 M 12 0 0 se11 s L se 21 22 11 12 MMMMse1122 se se se 0 21 22 0MMM0se22 se () s R 11 000MMre1 () r L 21 000Mre1 000 0 M 1 000 0 000 0 000 0 000 0 000 0 000 0
00 00 00 00 00 00 12 M000re1 22 11 12 MMre12 re M re 2 0 0 21 22 MMMre22 re () r R 0 0 M 11 12 2 00MMMce11 () c L ce 21 22 11 00MMMce112 ce ce 00 0M21 ce2 00 00 00 00 00 00 155
000 0 000 0 000 0 000 0 000 0 000 0 000 0 M 3 M00012 ce2 22 MMce2 () c R 0 0 0 11 12 0MMpe11 () p 1 L M pe 11 0 21 22 11 12 0MMMMpe11 pe 11 pe 12 pe 12 00MMM21 22 () pe12 pe 12 p 1 R
Overall gyroscopic matrix as described in Eq. (3.10) is given as
GGG 123 G (B2) where
GG11 () G 12 0 0 se11 s L se 21 22 11 12 GGGGse1122 se se se 0 21 22 0GGG0se22 se () s R 11 000GGre1 () r L 21 000Gre1 000 0 G 1 000 0 000 0 000 0 000 0 000 0 000 0
156
0000 0000 0000 12 G000re1 22 11 12 GGre12 re G re 2 00 21 22 GGG00re22 re () r R G2 0000 0000 0000 0000 0000 0000
00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 G3 00 0 0 00 0 0 0G11 () G G 12 0 pe11 p 1 L pe 11 21 22 11 12 0Gpe11 G pe 11 G pe 12 G pe 12 21 22 00 Gpe12 GG pe 12 () p 1 R
In equations (B1), (B2) and (B3), sub-matrices for the first planet (i 1) are shown while they can be expanded to the sub-matrices for planets i (iN2, ) using the same structure provided in these equations.
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APPENDIX C
Elements of the Forcing Vector
The forcing vector in Eq. (2.26) was given as
4N F=()tt Fk () (C1) k 1
It is formed by 4N components Fk ()t , each defined by a particular gear mesh based on its phasing relationship. The first four components that are associated with the meshes of the first planet (i 1) are defined as follows:
()L 0 fsp 0 0 f ()R 0 sp 0 0 0 0 0 0 F ()tket ()L () 0 F ()tket ()R () 0 1 sp sp1 , 2 sp sp1 , 0 0 0 0 ()L 0 fsp1 0 0 ()R 0 fsp1
158
0 0 0 0 0 0 ()L 0 frp 0 0 ()R 0 frp F ()tket ()L () 0 F ()tket ()R () 0 3 rp rp1 , 4 rp rp1 . 0 0 0 0 ()L 0 frp1 0 0 ()R 0 frp1
Here, sub-vectors f are 6x1 vectors defined in Sections 2.2.1 and 2.2.2 and 0 is a zero vector of dimension 6. The remaining 4(N 1) terms corresponding to planets 2 to N can also be written in the same fashion.
159