Hua Xia

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UNIVERSITY OF CINCINNATI

Date:

I, ,

hereby submit this original work as part of the requirements for the degree of:

in

It is entitled:

Student Signature:

This work and its defense approved by:

Committee Chair:

11/12/2010 1,173

2-Nov-2010

Xia Hua

Master of Science

Mechanical Engineering

Hypoid and Spiral Bevel Gear Dynamics with Emphasis on

Gear-Shaft-Bearing Structural Analysis

Teik Lim, PhDTeik Lim PhD

Xia Hua

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Hypoid and Spiral Bevel Gear Dynamics with Emphasis on

Gear-Shaft-Bearing Structural Analysis

A thesis submitted to the

Division of Research and Advanced Studies

of the University of Cincinnati

in partial fulfillment of the requirements

for the degree of MASTER OF SCIENCE

in the Program of Mechanical Engineering

of the College of Engineering and Applied Science

November 2010

by

Xia Hua

B.S. Zhejiang University of Technology, Zhejiang, P.R. China, 2007

Academic Committee Chair: Dr. Teik C. Lim

Members: Dr. Ronald Huston

Dr. David Thompson

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ABSTRACT

Hypoid and spiral bevel gears, used in the rear axles of cars, trucks and off-

highway equipment, are subjected to harmful dynamic response which can be

substantially affected by the structural characteristics of the shafts and bearings. This

thesis research, with a focus on gear-shaft-bearing structural analysis, is aimed to develop

effective mathematical models and advanced analytical approaches to achieve more

accurate prediction of gear dynamic response as well as to investigate the underlying

physics affecting dynamic response generation and transmissibility. Two key parts in my

thesis are discussed below.

Firstly, existing lumped parameter dynamic model has been shown to be an

effective tool for dynamic analysis of spiral bevel geared rotor system. This model is

appropriate for fast computation and convenient analysis, but due to the limited degrees

of freedom used, it may not fully take into consideration the shaft-bearing structural

dynamic characteristics. Thus, a dynamic finite element model is proposed to fully

account for the shaft-bearing dynamic characteristics. In addition, the existing equivalent

lumped parameter synthesis approach used in the lumped parameter model, which is key

to representing the shaft-bearing structural dynamic characteristics, has not been

completely validated yet. The proposed finite element model is used to guide the

validation and improvement of the current lumped parameter synthesis method using

effective mass and inertia formulations, especially for modal response that is coupled to

the pinion or gear bending response.

Secondly, a new shaft-bearing model has been proposed for the effective

supporting stiffness calculation applied in the lumped parameter dynamic analysis of the

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CONTENTS

Chapter 1. Introduction ....................................................................................................... 1 1.1 Literature Review...................................................................................................... 2 1.2 Motivation, Objectives and Thesis Organization...................................................... 4

Chapter 2. Finite Element and Enhanced Lumped Parameter Dynamic Modeling of Spiral Bevel Geared Rotor System ..................................................................................... 6

2.1 Introduction ............................................................................................................... 6 2.2 Proposed Dynamic Finite Element Model ................................................................ 7 2.3 Proposed New Lumped Parameter Synthesis Method for Existing LumpedParameter Dynamic Model and Its Difference from the Old Lumped ParameterSynthesis Approach ...................................................................................................... 13

2.3.1 Spiral Bevel Gear 14-DOF Lumped Parameter Dynamic Model .................... 13 2.3.2 Proposed New Lumped Parameter Synthesis Method in Spiral Bevel Gear ... 16 14 DOF Lumped Parameter Model........................................................................... 16 2.3.3 Difference Between Old Lumped Parameter Synthesis Approach and Proposed

New Lumped Parameter Synthesis Approach .......................................................... 28 2.4 Comparison Results and Discussions ..................................................................... 28 2.5 Conclusion .............................................................................................................. 35

Chapter 3. Effect of Shaft-bearing Configurations on Spiral Bevel Gear Mesh andDynamics .......................................................................................................................... 36

3.1 Introduction ............................................................................................................. 36 3.2 Mathematical Model ............................................................................................... 37

3.2.1 Mesh Model ..................................................................................................... 37 3.2.2 Spiral Bevel Gear 14-DOF Lumped Parameter Dynamic Model .................... 38 3.2.3 Finite Element Modeling of 3-bearing Straddle Mounted Pinion Configurationfor the Effective Lumped Stiffness Calculation........................................................ 41

3.2.4 Axial Translational Stiffness Model Refinement ............................................ 44 3.3 Comparison of 3-bearing Straddle Mounted Pinion and 2-bearing OverhungMounted Pinion on Gear Mesh and Dynamics ............................................................. 46

3.3.1 Analysis on Equivalent Shaft- bearing Stiffness Models and Pinions Lum pedShaft-bearing Stiffness Matrices of Two Pinion Configurations .............................. 49 3.3.2 Comparison on Gear Dynamics ....................................................................... 52 3.3.3 Effect of 2-bearing and 3-bearing Configurations on Mesh Model ................. 57

3.4 Conclusions ............................................................................................................ 70 Chapter 4. Conclusions ..................................................................................................... 72 BIBLIOGRAPHY ............................................................................................................. 74

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LIST OF FIGURES

Figure2.1 Dynamic finite element model of spiral bevel geared rotor system ................... 8 Figure2.2 Spiral bevel gear pair dynamic model ............................................................. 11 Figure2.3 Spiral bevel gear 14 DOF lumped parameter dynamic model ....................... 16

Figure2.4 Static finite element modeling of 3-bearing straddle mounted pinionconfiguration ..................................................................................................................... 19 Figure2.5 Axial translational stiffness model ................................................................... 20 Figure2.6 A design of beam with lumped mass ................................................................ 21 Figure2.7 Beam with lumped mass model of pinion with integrated shaft ....................... 23 Figure2.8 Dynamic mesh forces ....................................................................................... 30 Figure2.9 Dynamic mesh forces ....................................................................................... 31 Figure2.10 Dynamic mesh forces ..................................................................................... 32 Figure2.11 Dynamic mesh forces ..................................................................................... 34 Figure2.12 Dynamic mesh forces ..................................................................................... 34 Figure2.13 Dynamic mesh forces ..................................................................................... 35

Figure3.1 Tooth load distribution generated from quasi-static three-dimensional finiteelement tooth contact analysis program ........................................................................... 38 Figure3.2 Spiral bevel gear 14 DOF lumped parameter dynamic model ........................ 41 Figure3.3 Static finite element modeling of 3-bearing straddle mounted pinionconfiguration ..................................................................................................................... 44 Figure3.4 Axial translational stiffness model ................................................................... 45 Figure3.5 3-bearing straddle mounted pinion (upper) and 2-bearing overhung mounted

pinion (lower) .................................................................................................................... 49 Figure3.6 Finite element model of 3-bearing mounted pinion (left) and finite element model of 2-bearing mounted pinion (right) ...................................................................... 52 Figure3.7 Comparison of 2-bearing and 3-bearing configurations on dynamic mesh

force .................................................................................................................................. 53

Figure3.8 Comparison of 2-bearing and 3-bearing configurations on modal strain energydistribution ........................................................................................................................ 54 Figure3.9 Comparison of 2-bearing and 3-bearing configurations on dynamic bearingload ................................................................................................................................... 56 Figure3.10 Comparison of 2-bearing and 3-bearing configurations on pinion response 57 Figure3.11 Effect of pilot bearing position on mesh point for 3-bearing case ................. 58 Figure3.12 Effect of tapered roller bearing position on mesh point for 3-bearing case .. 59 Figure3.13 Effect of tapered roller bearing position on mesh point for 2-bearing case .. 60 Figure3.14 Comparison of 2-bearing and 3-bearing configurations on mesh point ....... 61 Figure3.15 Effect of pilot bearing position on line-of-action vector for 3-bearing case . 62

Figure3.16 Effect of tapered roller bearing position on line-of-action vector for 3-bearing case ...................................................................................................................... 63 Figure3.17 Effect of tapered roller bearing position on line-of-action vector for 2-bearing case ...................................................................................................................... 64 Figure3.18 Comparison of 2-bearing and 3-bearing configuration on line-of-actionvector ................................................................................................................................. 65 Figure3.19 Effect of pilot bearing position on mesh stiffness for 3-bearing case ............ 66

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Figure3.20 Effect of tapered roller bearing position on mesh stiffness for 3-bearing case........................................................................................................................................... 67 Figure3.21 Effect of tapered roller bearing position on mesh stiffness for 2-bearing case........................................................................................................................................... 67 Figure3.22 Comparison of 2-bearing and 3-bearing configuration on mesh stiffness .... 68

Figure3.23 Comparison on dynamic mesh force without considering the difference of mesh stiffness .................................................................................................................... 69 Figure3.24 Comparison on dynamic mesh force considering the difference of meshstiffness .............................................................................................................................. 69

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Chapter 1. Introduction

Hypoid and spiral bevel gears are widely used as final set of reduction gear pairs

in the rear axles of trucks, cars and off- highway equipment to transmit engines power to

the drive wheels in non-parallel directions. The hypoid and spiral bevel gear dynamics

becomes more and more significant for the concern of noise and durability, because

under dynamic condition the mesh force acting on gear teeth are amplified which

potentially reduces the fatigue life of gears and the large dynamic force can be

transmitted to housing which causes structure-born gear whine. Accordingly, it is needed

to perform in-depth investigation on hypoid and spiral bevel geared system dynamic

response and resonance characteristics to form a deeper understanding in the physics

controlling dynamic force generation and transmissibility to achieve superior design for

quiet and durable driveline. Though it is the fact that much is known about dynamic

characteristics in parallel axis gear system, research on the dynamics of nonparallel axis

geared systems such as hypoid and spiral bevel gears is not mature.

Most previous analytical work mainly focuses on gear mesh modeling and its

application to analyze gear pair dynamics, nonlinear time-varying gear pair dynamic

analysis considering gear backlash, time-varying mesh characteristics, mesh stiffness

asymmetry effects and friction, coupled multi-body gear pair dynamic and vibration

analysis and so on. Very little amount of attention is given to the gear-shaft-bearing

structure of the geared rotor system. The goal of this thesis is to gain a better

understanding on the effect of gear-shaft-bearing structural design on hypoid and spiral

bevel gear system dynamics and to establish new computational models more accurately

accounting for gear-shaft-bearing dynamic characteristics.

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1.1 Literature Review

Dynamics of parallel axis geared rotor system has been studied extensively[1-14].

Papers by Ozguven and Houser[1] and Blankenship and Singh[2] provide a

comprehensive review of mathematical models used to investigate dynamics of parallel

axis geared rotor system. Among these studies, a special attention has been paid on gear-

shaft-bearing structure rather than the dynamics of the gear itself. In 1975, Mitchell and

Mellen[3] indicate the torsional-lateral coupling in a geared rotor system by conducting

experiment study. In 1981, Hagiwara, Ida[4] analytically and experimentally studied the

vibration of geared shafts due to run-out unbalanced and run-out errors and it is observed

that both torsional and lateral modes could be excited by gear errors and unbalanced

forces. In 1984, Neriya, Bhat and Sankar[5] studied the effect of coupled torsional-

flexural vibration of a geared shaft system on dynamic tooth load by using lumped

parameter dynamic model in which equivalent lumped springs were used to represent the

flexibility of shaft-bearing structure. In 1985, Neriya, Bhat and Sankar[6] used finite

element method to model the geared rotor system and introduced the coupling between

torsion and flexure at the gear pair location. In 1991, Lim and Singh[7] developed linear

time-invariant, discrete dynamic models of a generic geared rotor system based on their

newly proposed bearing matrix formulation by using lumped parameter and dynamic

finite element techniques to predict the vibration transmissibility through bearing and

mounts, casing vibration motion, and dynamic response of the internal rotating system. In2004, Kubur and Kahraman[8] proposed a dynamic model of a multi-shaft helical gear

reduction unit formed by N flexible shafts by finite elements. This model has an accurate

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representation of shafts and bearings as well as gears, which is used to study the influence

of some key gear-shaft-bearing structure parameters.

Though large numbers of research has been done on parallel axis gear dynamics,

the research on dynamics of right-angle geared rotor system such as bevel and hypoid

gear is still scanty. In recent years, a group led by Lim[15-19] began to develop the

dynamic model of right-angle hypoid and spiral bevel geared rotor system and analyze

the dynamic characteristics of hypoid and spiral bevel geared rotor system. In one of the

study, Cheng and Lim[15] developed the single-point gear mesh-coupling model based

on both unloaded and loaded exact gear tooth contact analysis. This mesh model is thenapplied to develop multiple degrees-of-freedom, lumped parameter model of the hypoid

and spiral bevel geared rotor system for linear time-invariant and nonlinear time-varying

analysis. In 2002, Wang, H. and Lim[16] developed a multi-point gear mesh-coupling

model based on Cheng single point gear mesh -coupling model and applied it to dynamic

analysis of hypoid and spiral bevel geared rotor system. In the same year, Jiang and

Lim[17] formulated a low degrees of freedom torsional dynamic model to analyze the

nonlinear phenomenon through both analytical and numerical solutions. In 2007, based

on the low degrees of freedom torsional dynamic model, Wang, J. and Lim[18] extended

Jiangs work and further investigated the influence of time -varying mesh parameters and

various nonlinearities on gear dynamics. In 2010, through developing various more

accurate high degrees of freedom lumped parameter dynamic models, Tao and Lim[19]

examines torque load effect on gear mesh and nonlinear time-varying dynamic responses,

coupled multi-body dynamics and vibration, influence of the typical rotor dynamic factor

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on hypoid gear vibration, effect of manufacturing error or assembly error on gear

dynamics and the interaction between internal and external excitations.

1.2 Motivation, Objectives and Thesis Organization

From above literature review, it could be observed that most of the research on

hypoid and bevel geared rotor system dynamics is concerned with gear mesh dynamics so

the flexibility of gear-shaft-bearing structure is simply represented by equivalent

supporting springs or even ignored in much study only focusing on the effect of gear

mesh characteristics. Very little attention has been paid to the detailed modeling and

analysis of gear-shaft-bearing structure for the concern of dynamics of the whole geared

rotor system. Therefore, in this thesis, an attention will be given to the gear-shaft-bearing

structural analysis to achieve more accurate prediction of gear dynamic response and to

investigate the effect of shaft and bearing design on gear dynamics.

Chapter 1 presents the general introduction, literature review, motivation and

objective for this thesis research. It discusses current progress in gear dynamics research

and the limitations of the research on hypoid and spiral bevel gear dynamics. The

discussion further illustrates the objectives of this thesis, which is to perform study on

dynamics of hypoid and spiral bevel geared rotor system with emphasis on the gear-shaft-

bearing structural modeling and analysis.

Chapter 2 proposes a finite element dynamic model of hypoid and spiral bevelgeared rotor system to fully account for dynamic characteristics of gear-shaft-bearing

structure. In addition, the proposed finite element dynamic model is used to guide the

improvement of the existing lumped parameter dynamic model using effective mass and

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inertia formulations, especially for modal response that are coupled to the pinion or gear

bending.

Chapter 3 proposes a new shaft-bearing model for the effective supporting

stiffness calculation for the lumped parameter dynamic analysis of the hypoid and spiral

bevel geared rotor system with 3-bearing straddle-mounted pinion configuration. In

addition, two typical gear-shaft-bearing configurations used in automotive application are

compared for their different contribution to the hypoid and spiral bevel gear mesh and

dynamics. Parametric study is also performed to analyze the effect of gear-shaft-bearing

configuration on gear mesh and dynamics.Chapter 4 gives a summary of the significant achievement of this thesis research

and the recommendations for future work.

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Chapter 2. Finite Element and Enhanced Lumped Parameter Dynamic

Modeling of Spiral Bevel Geared Rotor System

2.1 Introduction

Along with the operating speed of geared rotor system growing higher, the

dynamics of geared system becomes more and more significant for the concern of noise

and durability, because under dynamic condition the mesh force acting on gear teeth are

amplified which potentially reduces the fatigue life of gears and the large dynamic force

can be transmitted to housing which causes structure-born gear whine.

Dynamics of gear systems have been studied extensively [1-14]. Though it is the

fact that much is known about dynamic characteristics in parallel axis gear system,

research on the dynamics of nonparallel axis geared systems such as hypoid and spiral

bevel gears is not mature. In recent years, a group led by Lim [15-19] began to develop

the dynamic model of spiral bevel geared rotor system and analyze the dynamic

characteristics of spiral bevel geared rotor system. In one of the study, Cheng and Lim

[15] developed the single-point gear mesh-coupling model based on the exact spiral bevel

gear geometry. This mesh model is then applied to develop multiple degrees-of-freedom,

lumped parameter model of the spiral bevel geared rotor system. Later, based on this

model, Tao and Lim [19] investigated the influence of various gear system parameters on

dynamic characteristics of the spiral bevel geared rotor system. However, due to limiteddegrees of freedom, the lumped parameter model may not fully take into account the

shaft-bearing dynamic characteristics and also the lumped parameter synthesis method

used in this model is not mature.

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In this paper, two modeling methods of spiral bevel geared rotor dynamic system,

i.e. the finite element dynamic modeling and the enhanced equivalent lumped parameter

synthesis, are introduced and compared. This first objective of this paper is to develop a

dynamic finite element model which could better take into account and describe the

shaft-bearing dynamic characteristics than the multiple degrees-of-freedom, lumped

parameter dynamic model [15]. The second objective is to develop a more accurate

lumped point parameter synthesis method fully considering the shaft-bearing structural

characteristics in existing lumped parameter model [15] and compare with the proposed

dynamic finite element model.

2.2 Proposed Dynamic Finite Element Model

As shown in Figure2.1, the mass/inertia of the pinion head and ring gear is

separately lumped at one node and the two nodes have mesh coupling between them.

The mass/inertia of the differential is lumped at one node. The pinion shaft and gear shaft

are modeled with beam elements, for which consistent mass matrix is used. The bearings

are modeled as stiffness matrices according to a bearing stiffness formulation[21,22]. The

engine and load are separately represented by one node. All nodes of the system

respectively have 6 DOFs except for the two nodes representing the engine and load

which only have torsional DOFs. The system totally has 17 nodes and accordingly

922*115*6 DOFs.

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Figure2.1 Dynamic finite element model of spiral bevel geared rotor system

The stiffness and mass matrices of each beam element are determined and

assembled to form stiffness ][ spK and mass ][ sp M matrices of pinion shaft and stiffness

][ sgK and mass ][ sg M matrices of gear shaft. Overall shaft stiffness and mass matrices

of the system are then assembled as ]][][[][ sgsps K K DiagK and

].][][[][ sgsps M M Diag M

The engine and load are separately connected to one node at pinion shaft and one

node at gear shaft with torsional springs. The stiffness matrices of the torsional spring

elements used to connect the engine and pinion shaft and to connect the load and gear

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shaft could be written in terms of individual torsional spring stiffness as ][ tspK and ][ tsgK ,

both of which are 7 by 7. The overall stiffness matrices of torsional spring elements of

the whole system could be written as ]].[][[][ tsgtspts K K DiagK The overall mass

matrices of engine and load of the whole system could be written in terms of torsional

moment of inertia of engine and load L E I I , as ].[][ , L E L E I I Diag M

In industry, pinion shaft is usually supported by 2 or 3 bearings and gear shaft is

usually supported by 2 bearings. Suppose that the system has a total of n bearings, the

overall bearing stiffness matrix of the whole system could be written by assembling the

individual bearing element stiffness matrices )1]([ ntoiK bi as

].][][][][[][ 321 bnbbbb K K K K K

The gear stiffness coupling matrix which represents the mesh coupling between

the two nodes representing pinion head and ring gear could be derived from the free

vibration equations of motion of spiral bevel gear pair. The dynamic model of the spiral

bevel gear pair is shown in Figure2.2. The pinion and gear, which are both built as rigid

body, are connected by linear gear mesh spring and damper. Using a quasi-static three-

dimensional finite element tooth contact analysis program[23,24] and concept of contact

cells[15], the averaged mesh point, averaged line-of-action, averaged mesh stiffness and

loaded transmission error are obtained to represent the mesh spring connecting point,

mesh spring direction , mesh spring stiffness and transmission error excitation between

pinion and gear. Pinion and gear are both allowed to move in 6 directions so the gear pair

dynamic system totally has 12 degrees of freedom. The generalized coordinates of pinion

and gear are separately expressed as

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T gzgygxggg pz py px p p p pg z y x z y xq },,,,,,,,,,,{}{ . The undamped free vibration

equations of motion for this gear pair dynamic system could be expressed as:

0

0

00

0

0

0

0

0

0

0

0

gmgxmgmgymgzgz

gmgzmgmgxmgygy

gmgymgmgzmgxgx

gzmgg

gymgg

gxmgg

pm pxm pm pym pz pz

pm pzm pm pxm py py

pm pym pm pzm px px

pzm p p

pym p p

pxm p p

y pnk x pnk I

x pnk z pnk I

z pnk y pnk I pnk zm

pnk ym

pnk xm

y pnk x pnk I

x pnk z pnk I

z pnk y pnk I

pnk zm

pnk ym

pnk xm

(1)

where, ),,( lzlylx nnn is the line-of-action vector, ),,( lmlmlm z y x ),( q pl is the

mesh point vector. q pl , refers to pinion and gear local coordinate systems

respectively. mk is mesh stiffness. p is relative displacement between pinion and gear

along line-of-action and is expressed as:

px pm pz py pm pz pz pm py px pm py

py pm px pz pm px pz p py p px pgxgmgzgygmgz

gzgmgygxgmgygygmgxgzgmgxgzggyggxg

n yn xn xn z

n zn yn zn yn xn yn x

n xn zn zn yn zn yn x p

(2)

Combining equations (1-2), a clearer equation of motion could be obtained as:

0}]{[}]{[ pg pg pg pg qk qm (3)

here,

],,,,,,,,,,,[][ gzgygxggg pz py px p p p pg I I I mmm I I I mmmdiagm (4)

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}{}{}{}{

}{}{}{}{

][

gT

gm pT

gm

gT

pm pT

pm

pg

hhk hhk

hhk hhk

k (5)

Here, }{ ph and }{ gh are the coordinate transformation vectors between the spiral

bevel gear line-of-action direction and generalized coordinate directions for pinion and

gear separately. They are expressed as:

),(},,,,,{}{ q plnnnh lzlylxlzlylxl , (6)

),(},,-{},,{ q pln yn xn xn zn zn y lxllyllzllxllyllzllzlylx . (7)

Figure2.2 Spiral bevel gear pair dynamic model

The gear mesh stiffness matrix ][ pgk and the mass matrix ][ pgm of the gear pair

can be obtained from Equations (3-7). The overall gear mesh stiffness and mass matrices

of the whole system could be obtained as ]][[][ pg pg k DiagK and

]][[][ pg pg m Diag M .

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The mass and stiffness matrices of the whole dynamic finite element system are

derived as ][][][][ , L E s pg M M M M , ].[][][][][ tsbs pg K K K K K

The system proportional damping is assumed in this model as

][])[][]([][ pgmtsbss K K K K C (8)

where, s is the system damping ratio, m is the mesh damping ratio.

The excitation of the whole system could be written as

)()(]}{}{[)}({ t e jck hht F mmT

g p

(9)

The equation of motion of the whole spiral bevel geared rotor system could be

expressed as

)}({)}(]{[)}(]{[)}(]{[ t F t X K t X C t X M . (10)

The direct method is applied here to calculate the steady state forced response as

)}({)]([)}({ 1 t F H t X . (11)

The dynamic response of pinion head and ring gear could be derived from )(t X as

}.{}, g p X X The dynamic transmission error is expressed as

}.}{{}}{{ gg p pd X h X h (12)

The dynamic mesh force in line-of-action direction is expressed as

)()( 00 d md mm ck F .(13)

where, mk is mesh stiffness; mmm k c is mesh damping; 0 is loaded

transmission error.

The given spiral bevel geared rotor system in Figure2.1 is an example used to

explain proposed dynamic finite element modeling theory. The same theory could be

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applied to spiral bevel geared rotor system with other kinds of pinion or gear

configurations.

2.3 Proposed New Lumped Parameter Synthesis Method for Existing Lumped

Parameter Dynamic Model and Its Difference from the Old Lumped Parameter

Synthesis Approach

2.3.1 Spiral Bevel Gear 14-DOF Lumped Parameter Dynamic Model

The spiral bevel gear 14-DOF lumped parameter dynamic model[15] used in

this study comprises of a spiral bevel gear pair, an engine element and a load element

as shown in Figure2.3. Engine and load respectively have 1 DOF which is torsional

coordinate. Pinion and gear are both modeled as rigid body which separately have 6

DOFs. Torsional springs are used to connect pinion and engine as well as to attach

gear and load. Pinion and gear have mesh coupling. mk is the averaged mesh stiffness

and TE is the static transmission error. Since pinion and gear are built as rigid body,

their mass and inertia are lumped at each lumped point. Lumped shaft-bearing springs

are connected to each lumped point of pinion and gear to support pinion and gear.

The equation of motion could be expressed as:

)}({}]{[}]{[}]{[ t F qK qC q M

(14)

The generalized coordinates are expressed as:

T L

T g

T p E qqq },}{,}{,{}{

(15)

T lzlylxllll z y xq },,,,,{}{ . (l = p, g) . (16)

The lumped mass matrix is described as:

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],,,,,,,

,,,,,,[][

Lgzgygxgzgygx

pz py px pz py px E

I I I I M M M

I I I M M M I diag M

(17)

]][][[]][[]][[][ tsgtsp pgll K K DiagK DiagK DiagK

(18)

Here, ][ llK is the lumped shaft-bearing stiffness matrix of pinion and gear.

][ pgK is the gear mesh coupling stiffness matrix. ][ tspK is the coupling stiffness

matrix of the torsional spring used to connect pinion and engine. ][ tsgK is the

coupling stiffness matrix of the torsional spring used to connect gear and load.

The damping [C] is assumed to be system proportional, which is expressed as:

]][[]])[][[]][[(][ pgmtsgtsplls K DiagK K DiagK DiagC (19)

where s is system damping ratio and m is mesh damping ratio.

The force vector )}({ t F at the right side of Equation (14) is,

)()(]}{},{[)}({ t e jck hht F mmT

g p

(20)

Here,

}{ ph

and }{ gh are the coordinate transformation vectors between the

spiral bevel gear line-of-action direction and generalized coordinate directions for

pinion and gear separately. They are expressed as,

},,,,,{}{ lzlylxlzlylxl nnnh , (21)

},,-{},,{ lxllyllzllxllyllzllzlylx n yn xn xn zn zn y . (22)

Here {nlx , n ly , n lz } is the line-of-action vector; {xl , y l , z l } is the mesh point

vector; l = p, g refers to pinion and gear local coordinate systems seperately.

The dynamic transmission error d is solved in frequency domain and

expressed as,

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}{}{ gg p pd qhqh

. (23)

The dynamic mesh force along line-of-action direction is expressed as:

)()(00

d md mmck F

.(24)

Here, mk is mesh stiffness; mmm k c is mesh damping; 0 is loaded

transmission error.

The deficiency of this model lies in that it is a lack of a fully validated method

to synthesize the lumped point parameters, i.e. the lumped shaft-bearing stiffness

matrix ][ llK , lumped mass/inertia of pinion pz py px pz py px I I I M M M ,,,,, and lumped

mass/inertia of gear gzgygxgzgygx I I I M M M ,,,,, , which is key to representing shaft-

bearing structural dynamic characteristics. It may cause inaccurate dynamic response

prediction if the lumped point parameters are not well determined.

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Figure2.3 Spiral bevel gear 14 DOF lumped parameter dynamic model

2.3.2 Proposed New Lumped Parameter Synthesis Method in Spiral Bevel Gear

14 DOF Lumped Parameter Model

The basic idea of proposed lumped parameter synthesis method is to

approximate the continuous parameter models of pinion and gear to lumped

parameter models while having the same 1st

order pinion and gear bending modes.

2.3.2.1 Equivalent Lumped Shaft-bearing Stiffness Calculation

Static finite element model of 3-bearing straddle mounted pinion

configuration is shown in Figure2.4. The reason to do this static finite element

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modeling is to calculate the pinions equivalent shaft -bearing stiffness relative to

the lumped point. The pinion with integrated shaft is modeled with several

uniform cross-section beam elements. Bearing is modeled as bearing stiffness

matrix calculated following a bearing stiffness formula[21,22].

Add a unit load at lumped point and then the equation for this static finite

element model could be expressed as:

}{}{}{ S RP (25)

Here ,{P} represents the external load exerted at all the nodes; {R}

represents the reaction load at all the nodes; [ S] is the assembled stiffness matrix;

{ } represents the displacements of all the nodes.

A more detailed equation could be drawn from (25) as:

S

F

SSSF

FSFF

S

F

S

F

SS

SS

R

R

P

P(26)

Here, P F means the external load exerted at the nodes at the part of pinion

with integrated shaft. P S means the external load at the nodes at the bearing outer

races. RF represents the reaction load at the nodes at the part of pinion with

integrated shaft. RS represents the reaction load at the nodes at the bearing outer

races. F represents the displacement of the nodes at the part of pinion with

integrated shaft. S represents the displacement of the nodes at the bearing outer

races.

Since the reaction load is only exerted at the nodes at the bearing outer

races and the nodes at the bearing outer races are fixed, RF and S in equation (26)

could be set to be zeros,

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0

0 F

SSSF

FSFF

SS

F

SS

SS RP

P

.(27)

Thus, (28) could be drawn from (27) as:

F FF F PS 1 . (28)

The lumped point displacement { l1 } could be got from {F }. The

relationship among the unit external load at the lumped point { lP1 }, the

displacement of the lumped point{ l1 } and the equivalent shaft-bearing stiffness

relative to the lumped point ][ llK could be expressed as:

}{}{ 11 llll K P . (29)

Following above procedure, by adding a unit load in other five directions

separately to the lumped point, the lumped point displacements corresponding to

each unit load could be calculated and obtained, which are written as

)6,5,4,3,2(}{ ili . The unit load at the lumped point in each of other 5 directions

could be written as )6,5,4,3,2(iP li . Similarly, the following formulation

could be obtained as:

)6,5,4,3,2(}{}{ iK P lillli (30)

Combining (29) and (30),

]][[][ 654321654321llllllllllllll K PPPPPP (31)

So, the equivalent shaft-bearing stiffness relative to the lumped point

][ llK could be calculated as:

1654321654321 ]][[][

llllllllllllll PPPPPPK (32)

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Figure2.4 Static finite element modeling of 3-bearing straddle mounted pinionconfiguration

However, the equivalent shaft-bearing stiffness calculated from static

finite element model may not accurately describe the equivalent axial translational

stiffness. So the axial translational stiffness model of 3-bearing straddle mounted

pinion configuration shown in Figure2.5 is developed in order to refine the axial

translational stiffness described by equivalent shaft-bearing stiffness ][ llK

calculated from static finite element model. In Figure2.5, K b1 and K b2 are axial

translational stiffness of bearing1 and bearing2. K s1 is shaft axial stiffness from

load point to center of bearing1. K s2 is shaft axial stiffness from center of bearing1

to center of bearing2. K c is additional cascade stiffness with bearing2 to represent

the shaft-bolt-york between the center of bearing2 and inner race of bearing2. K hb

is housing bolt stiffness.

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Figure2.5 Axial translational stiffness model

The axial translation stiffness of ][ llK calculated from static FE model

does not take K c and K hb into account. The refinement should be made according

to Figure2.5 in the following way. Before doing finite element calculation, the

cascade stiffness K s3 should be added into the axial translation stiffness of

bearing2 K b2. After doing static finite element modeling, the temporary equivalent

shaft-bearing stiffness is obtained. Then the temporary equivalent shaft-bearing

stiffness should add K hb into its axial translation stiffness to get the eventual

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equivalent lumped shaft-bearing stiffness of the 3-bearing straddle mounted

pinion.

The equivalent lumped shaft-bearing stiffness of other pinion and gear

configurations could be calculated in the similar way[19].

2.3.2.2 Effective Lumped Mass and Inertia Calculation

The first step is to generate the first bending mode shape functions of

pinion with integrated shaft and gear with integrated shaft. The Initial Parameter

Method[20] used in this paper to calculate first bending mode shape function is

described using the coordinate system I defined below as Figure 2.6. This methodhas been proved to be valid for dynamical calculation for beam with arbitrary

peculiarities and different boundary conditions.

Figure2.6 A design of beam with lumped mass

In Figure 2.6., the dotted line at y=0 which is the left end represents an

arbitrary type of support. Transverse displacement 0 z , angle of rotation 0 ,

bending moment 0 M and shear force 0Q at y=0 are called initial parameters.

State parameters transverse displacement z(y), angle of rotation )( y , bending

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moment )( y M , shear force )( yQ at any position y may be presented in the

following forms (Bezukhov et al, 1969; Babakov, 1965; Ivovich, 1981)[20].

)]([)]([)]([11

)()()()()(

22

2

302000

iiiiiiii y yk U J y yk V z M k y yk V R

k EI k

EI k

kyV Q

EI k

kyU M

k

kyT kyS z y z

(33)

)]([)]([)]([11

)()()()()(

22

20000

iiiiiiii y yk T J y yk U z M k y yk U R

k kEI

EI k

kyU Q

kEI kyT

M kySk kyV z y

(34)

)]([)]([)]([1

)()()()()(

22

0002

0

iiiiiiii y yk S J y yk T z M k y yk U R

k

k kyT

QkyS M EIk kyV EIk kyU z y M

(35)

)]([)]([)]([)()()()()(

22

002

03

0

iiiiiiii y yk V J k y yk S z M y yk S R

kySQk kyV M EIk kyU EIk kyT z yQ

(36)

Where M i = lumped masses (note: M 0 = bending moment at x=0)

Ji = moment of inertia of a lumped mass

R i =concentrated force (active or reactive)

yi = distance between origin and point of application R i or M i

zi, i = vertical displacement and slope at point where lumped

mass M i is located

S(y), T(y), U(y), V(y) = Krylov-Duncan functions

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)sin(sinh2

1)(

)cos(cosh2

1)(

)sin(sinh2

1)(

)cos(cosh2

1)(

kykykyV

kykykyU

kykykyT

kykykyS

k= 4 2 EI m

, m is line density of the uniform beam, is radian

natural frequen cy, E is Youngs Modulus, I is rotary inertia of the cross -sectional

area.

This theory could generally be applied to the pinion and gear of spiral

bevel geared rotor system. Here, take an overhung mounted and simply supported

pinion for example as Figure 2.7. The pinion is modeled as a uniform beam with a

lumped mass at the lumped point a .

Figure2.7 Beam with lumped mass model of pinion with integrated shaft

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Accordingly, transverse displacement z(y), angle of rotation )( y ,

bending moment )( y M , shear force )( yQ at any position y of the pinion model

shown in Figure2.7 could be expressed by using the Initial Parameter Method[20]

as:

)]([)()]([)(1

)]([1

)]([11

)()()()()(

22

2

212

302000

a yk U a J a yk V a Mzk EI k

c yk V Rk

b yk V Rk EI k

EI k kyV

Q EI k kyU

M k kyT

kyS z y z

(37)

)]([)()]([)(1

)]([1

)]([11

)()()()()(

22

21

20000

a yk T a J a yk U a Mzk kEI

c yk U Rk

b yk U Rk kEI

EI k kyU Q

kEI kyT M kySk kyV z y

(38)

)]([)(

)]([)()]([1

)]([1

)()()()()(

2

2

21

0002

0

a yk Sa J

a yk T a Mzk c yk U Rk b yk U Rk

k kyT

QkyS M EIk kyV EIk kyU z y M

(39)

)]([)(

)]([)()]([)]([

)()()()()(

2

221

002

03

0

a yk V akJ

a yk Sa Mzc yk S Rb yk S R

kySQk kyV M EIk kyU EIk kyT z yQ

(40)

The boundary condition could be described as:

0)(;0)(;0)(;0)(;0;0 00 d Qd M c zb zQ M .

Substitute the boundary condition into (37-40) and get the following

equation.

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0)]([)()]([)(1

)()()(

22

2

00

abk U a J abk V a z M

k EI k

k kbT

kbS zb z

(41)

0)]([)()]([)()]([1

1)()()(

22

1

200

ack U a J ack V a z M

k bck V R

k

EI k k kcT

kcS zc z

(42)

0)]([)()]([)()]([1

)]([1

)()()(

22

2

102

0

ad k Sa J ad k T a z M k

cd k T Rk

bd k T Rk

kd EIkV kd U EIk zd M

(43)

0)]([)()]([)()]([

)]([)()()(22

2

12

03

0

ad k V akJ ad k Sa z M cd k S R

bd k S Rkd U EIk kd T EIk zd Q

(44)

Displacement and angle of rotation at y=a are expressed as:

k kaT

kaS za z)(

)()( 00 (45)

)()()(00

kaSk kaV za (46)

Therefore, the homogeneous system of equations is obtained. If and only

if the following determinant, which represents the frequency domain, is zero, the

system has a non-trivial solution.

0][ 4321T r r r r (47)

where,

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T

kaSabk JU EI k

kaT abk MV EI k k

kbT

kaV abk JU kEI

kaSabk MV EI k

kbS

r

00

)()]([)()]([)(

)()]([)()]([)(

2

2

4

2

2

3

2

1

(48)

T

EI k

bck V

kaSack JU EI k

kaT ack MV EI k k

kcT

kaV ack JU kEI

kaSack MV EI k

kcS

r

0

)]([

)()]([)()]([)(

)()]([)()]([)(

3

2

2

4

2

2

3

2

2

(49)

T

cd k T k

bd k T k

kaSad k JSkaT ad k T k M

kd EIkV

kaV ad k JSk

kaSad k MT kd U EIk

r

)]([1

)]([1

)()]([)()]([)(

)()]([)()]([

)(

2

2

2

22

2

3

(50)

T

cd k S

bd k S

kaSad k kJV k

kaT ad k MSkd U EIk

kaV ad k JV k kaSad k MSkd T EIk

r

)]([

)]([

)()]([)()]([

)(

)()]([)()]([)(

22

2

2223

4

(51)

Multiple solutions of k which are expressed as k 1, k 2, k 3, k 4, k 5 could be

solved from above equation. k 1, the smallest value of k, is for the first bending

mode. Substitute the value of k 1 to the equation. After cleaning, then 0 z , 1 R , 2 R

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could be expressed in terms of 0 . Substitute the relationship ),( 00 f z

)(),( 0201 f R f R to (37,38) to solve mode shape function z(y), )( y .

Then according to balance of kinetic energy at the first bending mode, the

first equation could be expressed as:

)(5.0)(5.0

)(5.0)(5.0)()(5.0

22

222

0

a I a z M

a J a z M dy y z ym

effectiveeffective

d

(52)

Where, effective M and effective I are pinions effective mass and effective

moment of inertia that need to be solved.

As for the model in Figure2.7, the lumped stiffness relative to Point a and

the first bending natural frequency could be obtained as 22][ aK and 1 . As the

continuous parameter model in Figure2.7 and its equivalent 2DOF lumped

parameter model should have the same first bending nature frequency 1 . The

second equation could be expressed according to 1 as:

00

0][

21

effective

effectivea I

M K (53)

According to equation (52) and (53), the effective mass effective M and

effective moment of inertia effective I could be obtained.

Then, in equation (17), the lumped mass and inertia of pinion could be

express as:

effective pz px M M M , effective pz px I I I (54)

total py M M , torsion py J I (55)

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Where, total M is the total mass of pinion. torsion J is the torsional moment of

inertia of pinion. Note, x is in horizontal direction, y is in axial direction, z is in

vertical direction.

torsiontotal J M , are directly used for py py I M , since pinion does not have

torsional and axial translational deformation when the geared rotor system is

excited at relatively low frequency.

The lumped mass and inertia of pinion or gear with other kinds of

configurations could also be calculated by following the procedure above, which

is not explained in detail here.

2.3.3 Difference Between Old Lumped Parameter Synthesis Approach and Proposed

New Lumped Parameter Synthesis Approach

The new lumped parameter synthesis approach and the old lumped parameter

synthesis approach have the same process of equivalent lumped shaft-bearing

stiffness calculation. While, the old lumped synthesis approach simply treats the total

mass/inertia of pinion or gear as lumped mass/inertia, and by contrast, the new

lumped synthesis approach calculates and uses the effective mass/inertia of pinion or

gear as the lumped mass/inertia.

2.4 Comparison Results and Discussions

First of all, by using exactly the same spiral bevel geared rotor system, the

proposed finite element dynamic model and the old lumped parameter dynamic model are

compared on dynamic mesh force. Three different cases are taken for example here. In

Case 1, pinion and gear are both overhung mounted and simply supported, which

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corresponds to Figure2.8. In Case 2, pinion and gear are both overhung mounted and

flexibly supported, which corresponds to Figure2.9. In Case 3, pinion is straddle mounted

and flexibly supported while gear is overhung mounted and flexibly supported, which

corresponds to Figure2.10.

From the comparison results, it could be easily observed that dynamic mesh

forces of two models are different at some modes. In Figure2.8, dynamic responses

cannot match at Mode a and Mode b, and by observing the mode shapes of Mode a and

Mode b of old lumped parameter model, Mode a and Mode b are both coupled to

component 5 and 7, which are pinion bending components. In Figure2.9, Mode a, Mode band Mode c of old lumped parameter model fail to match finite element dynamic model.

The mode shapes of the three modes show that they are all coupled to pinion bending,

which are represented by component 5 and 7. In Figure2.10, Mode a of old lumped

parameter model matches very well with finite element model while Mode b and Mode c

of old lumped parameter model show certain discrepancy with finite element model. It

could be observed from the mode shapes that Mode a is not coupled to pinion bending

represented by component 5&7 or to gear bending represented by component 11&13,

Mode b is coupled to large pinion bending and Mode c is coupled to large gear bending.

Three cases show the same phenomenon that dynamic responses of finite element

dynamic model and old lumped parameter dynamic model may not match well at the

modes that are coupled to pinion bending or gear bending.

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0 500 1000 1500 2000 2500 3000 3500 400010

-1

10

0

101

102

103

10 4

105

Frequency(Hz)

M a g n

i t u

d e

( N )

Figure2.8 Dynamic mesh forces , dynamic finite element model

, old equivalent lumped parameter model

a b

a. b.

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0 500 1000 1500 2000 2500 3000 3500 400010 -1

100

101

102

103

104

105

Frequency(Hz)

M a g n

i t u

d e

( N )


,old equivalent lumped parameter model

Figure2.11, Figure2.12 and Figure2.13 show the comparison of finite element

model and new lumped parameter model on dynamic mesh force separately for Case 1,

a b c

a. b. c.

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Case 2 and Case 3. All of the three cases show that two models have reasonably close

dynamic responses. Especially at low frequency, two models almost show perfect match.

In the old lumped parameter model, the lumped parameter synthesis method

simply treats total mass/inertia as lumped mass/inertia which leads to inaccurate

representation of shaft-bearing dynamic characteristics and leads to inaccurate modal

responses that are coupled to pinion or gear bending. In the new lumped parameter

model, by using the effective mass/inertia instead of total mass/inertia, the shaft-bearing

dynamic characteristics is more accurately considered and the modal responses that are

coupled to pinion or gear first bending show better match with finite element dynamicmodel.

However, at higher frequency range, finite element dynamic model and new

lumped parameter dynamic model still show certain minor discrepancies which may be

caused by the following reasons.

(a). The process to calculate effective lumped shaft-bearing stiffness and effective

mass/inertia may not be perfect, in which minor computational errors may exist.

(b). Since new lumped parameter synthesis approach is developed based on the

first bending mode of pinion and gear, the new lumped parameter model cannot

accurately predict modes that are coupled to more complicated pinion or gear bending at

relatively high frequency range.

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0 500 1000 1500 2000 2500 3000 3500 400010-1

100

101

102

103

104

105

Frequency(Hz)

M a g n

i t u

d e

( N )


, equivalent lumped parameter model

2.5 Conclusion

A finite element dynamic model of spiral bevel geared rotor system is proposed in

this study, which could better account for shaft-bearing dynamic characteristics than

existing lumped parameter model. The finite element dynamic model is also used to

provide guide and reference for the enhancement of equivalent lumped parameter

synthesis theory to be used in existing lumped parameter model. Dynamic responses of

two models have been compared and show good consistency at relatively low frequency.

Both models could be used not only to predict the dynamic response of the spiral bevel

geared rotor system, but also to help engineers figure out the best designs from the

viewpoint of vibration and noise.

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Chapter 3. Effect of Shaft-bearing Configurations on Spiral Bevel Gear

Mesh and Dynamics

3.1 Introduction

Dynamics of gear systems have been studied extensively [1-19]. It is known that

spiral bevel gear dynamics may not be accurately predicted by ignoring the flexible

components such as shafts and bearings. In industry, different kinds of shaft-bearing

configurations of rear axles exist. For example, pinion could be overhung mounted with 2

bearings which is typically used in light or medium duty rear axle, while pinion could

also be straddle mounted with 3 bearings which is typically used in the heavy duty rear

axle. The effect of shaft-bearing configurations on spiral bevel gear mesh and dynamics

therefore needs attention. In this study, a new shaft-bearing model has been proposed for

the effective supporting stiffness calculation for the lumped parameter dynamic analysis

of the spiral bevel geared rotor system with 3-bearing straddle-mounted pinion

configuration. Also, the 3-bearing straddle mounted pinion configuration and the 2-

bearing overhung mounted pinion configuration are compared on dynamic

characteristics, i.e. natural frequency, dynamic mesh force and dynamic bearing force,

and on mesh model parameters, i.e., mesh point, line-of-action vector, mesh stiffness,

using 14-DOF lumped parameter dynamic model and quasi-static three-dimensional

finite element tooth contact analysis program. Moreover, parametric study of bearing

position and bearing type is performed to analyze the effect of shaft-bearing

configuration on spiral bevel gear mesh and dynamics.

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3.2 Mathematical Model

3.2.1 Mesh Model

Mesh model is the basis of the spiral bevel gear dynamic model. The key step

to develop the spiral bevel gear dynamic system is to effectively model the gear pair

meshing relationship. In this paper, a theory[15] of synthesizing the lumped mesh

model based on the tooth load distributions generated from quasi-static three-

dimensional finite element tooth contact analysis program[23,24] is applied to

calculate the mesh point, line-of-action vector, mesh stiffness and static transmission

error.

The contact zone shown in Figure3.1 is divided into N grids. For each grid i, r i

(rix, r iy, r iz) is the position vector; n i (n ix, n iy, n iz) is the normal vector; f i is the load.

Static mesh force could be computed as:

222

111

,,, z y xtotali N

iiz zi

N

iiy yi

N

iix x F F F F f nF f nF f nF . (1)

The line-of-action vector could be calculated as:

total z ztotal y ytotal x x F F nF F nF F n / , / , / . (2)

The mesh position could be calculated as:

x z y y x z N

ii

i

N

iiy

F xF M zF yF M x f

f r y / )(, / )(,

1

1 (3)

where, iyixixiy N

ii zixizizix

N

ii y r nr n f M r nr n f M

11

, .

The mesh stiffness could be expressed as:

0 / Ltotalm eF k (4)

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where, e L is loaded translation transmission error and 0 is unloaded translation

transmission error.

Figure3.1 Tooth load distribution generated from quasi-static three-dimensional finiteelement tooth contact analysis program

3.2.2 Spiral Bevel Gear 14-DOF Lumped Parameter Dynamic Model

The spiral bevel gear 14-DOF lumped parameter dynamic model[15] used in

this study comprises of a spiral bevel gear pair, an engine element and a load element

as shown in Figure3.2 Engine and load respectively have 1 DOF which is torsional

coordinate. Pinion and gear are both modeled as rigid body which separately have 6

DOFs. Torsional springs are used to connect pinion and engine as well as to attach

gear and load. Pinion and gear have mesh coupling. K m is the mesh stiffness and TE isthe static transmission error, which are actually time-varying. Since pinion and gear

are built as rigid body, their mass and inertia are lumped at each lumped point.

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Lumped shaft-bearing springs are connected to each lumped point of pinion and gear

to support pinion and gear. The equation of motion could be expressed as:

}{}]{[}]{[}]{[ F qK qC q M . (5)

The generalized coordinates are expressed as:

T L

T g

T p E qqq },,,{}{ , (6)

T lzlylxllll z y xq },,,,,{}{ . (l = p, g) . (7)

The mass matrix and stiffness matrix are described as:

],,,,,,,,,,,,,[][ Lgzgygxggg pz py px p p p E

I I I I M M M I I I M M M I diag M ,

]][][[]][[][ tsgtspll K K DiagK DiagK

(8)

][ tspK is the coupling stiffness matrix of the torsional spring used to connect

pinion and engine. ][ tsgK is the coupling stiffness matrix of the torsional spring used

to connect gear and load. ][ llK is the lumped shaft-bearing stiffness matrix of pinion

and gear calculated through shaft-bearing stiffness models which would be described

in detail later. The damping [C] is assumed to be component proportional.

The force vector {F} at the right side of Equation (5) is,

T Lmgm p E T F hF hT F ],,,[}{ . (9)

Here, T E and T L are torques exerted on the engine and load. F m is the dynamic

mesh force in line-of-action direction. h pF m and hgF m are equivalent mesh forces and

moments exerted on the pinion and the gear in generalized coordinate directions, and

h p and hg are the coordinate transformation vectors between the spiral bevel gear line-

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of-action direction and generalized coordinate directions for pinion and gear

separately. They are expressed as,

},,,,,{lzlylxlzlylxl

nnnh

, (10)

},,-{},,{ lxllyllzllxllyllzllzlylx n yn xn xn zn zn y

. (11)

Here {nlx , n ly , n lz } is the line-of-action vector; {xl , y l , z l } is the mesh point

vector; l = p, g refers to pinion and gear local coordinate systems seperately.

If the model is nonlinear time-varying, the dynamic transmission error d is

solved by numerical integration in time domain and expressed as,

T gz pygygxgggg

T pz px p p p pd R z y xh z y xh }, / ,,,,{},0,,,,{

.

(12)

Here, R is the gear ratio.

If the model is reduced to linear time-invariant, the dynamic transmission

error d is solved in frequency domain and expressed as,

}{}{ gg p pd qhqh

. (13)

If the model is nonlinear time-varying, the dynamic mesh force F m can be

expressed as:

cd d mcd m

cd c

cd d mcd m

m

bif cbk

bbif

bif cbk

F

000

0

000

)()(

0

)()(

. (14)

If the model is reduced to be linear time-invariant, the dynamic mesh force is

expressed as:

)()( 00 d md mm ck F . (15)

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Here, K m is mesh stiffness; C m is mesh damping; 0 is unloaded transmission

error; bc represents gear backlash.

Figure3.2 Spiral bevel gear 14 DOF lumped parameter dynamic model

3.2.3 Finite Element Modeling of 3-bearing Straddle Mounted Pinion Configuration

for the Effective Lumped Stiffness Calculation

As shown in Figure3.3, static finite element model of 3-bearing straddle

mounted pinion configuration is developed based on static finite element model of 2-

bearing overhung mounted pinion configuration[19] to calcul ate the pinions

equivalent shaft-bearing stiffness relative to the lumped point. The pinion with

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integrated shaft is modeled with several uniform cross-section beam elements.

Bearing is modeled as stiffness matrix calculated according to the bearing stiffness

formulation[21,22]. The model totally consists of 9 nodes, 5 uniform cross-section

beam elements, and 3 bearing elements.

Add a unit load at the lumped point in one direction and then the equation for

this static finite element model could be expressed as:

}{}{}{ S RP (16)

Here ,{P} represents the external load exerted at all the nodes; {R} represents

the reaction load at all the nodes; [ S] is the assembled stiffness matrix; { }

represents the displacements of all the nodes.

A more detailed equation could be drawn from (16) as:

S

F

SSSF

FSFF

S

F

S

F

SS

SS

R

R

P

P(17)

Here, P F means the external load exerted at the nodes at the part of pinion with

integrated shaft. P S means the external load at the nodes at the bearing outer races. RF

represents the reaction load at the nodes at the part of pinion with integrated shaft. RS

represents the reaction load at the nodes at the bearing outer races. F represents the

displacement of the nodes at the part of pinion with integrated shaft. S represents

the displacement of the nodes at the bearing outer races.

Since the reaction load is only exerted at the nodes at the bearing outer races

and nodes at the bearing outer races are rigidly fixed, RF and S in equation (17)

could be set to be zeros,

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0

0 F

SSSF

FSFF

SS

F

SS

SS

RP

P(18)

Thus, (19) could be drawn from (18) as:

F FF F PS 1 (19)

The lumped point displacement { l1 } could be got from {F }. The

relationship among the unit external load at the lumped point { lP1 }, the displacement

of the lumped point{ l1 } and the equivalent shaft-bearing stiffness relative to the

lumped point ][ llK could be expressed as:

}{}{ 11 llll K P . (20)

Following above procedure, by adding a unit load in other five directions

separately to the lumped point, the lumped point displacements corresponding to each

unit load could be calculated and obtained, which is written as )6,5,4,3,2(}{ ili .

The unit load at the lumped point in each of other 5 directions could be written as

)6,5,4,3,2(iP li . Similarly, the following formulation could be obtained as:

)6,5,4,3,2(}{}{ iK P lillli (21)

Combining (20) and (21),

]][[][ 654321654321llllllllllllll K PPPPPP (22)

So, the equivalent shaft-bearing stiffness relative to the lumped point ][ llK

could be calculated as:

1654321654321 ]][[][llllllllllllll PPPPPPK . (23)

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Figure3.3 Static finite element modeling of 3-bearing straddle mounted pinionconfiguration

3.2.4 Axial Translational Stiffness Model Refinement

The equivalent shaft-bearing stiffness calculated from static finite element

model ][ llK may not accurately describe the equivalent axial translational stiffness.

So the axial translational stiffness model of 3-bearing straddle mounted pinion

configuration shown in Figure3.4 is developed based on the axial translational

stiffness model of 2-bearing overhung mounted pinion configuration[19] in order to

correct the axial translational stiffness described by equivalent shaft-bearing stiffness

calculated from static finite element model ][ llK . In Figure3.4, K b1 and K b2 are axial

translational stiffness of bearing1 and bearing2. K s1 is shaft axial stiffness from load

point to center of bearing1. K s2 is shaft axial stiffness from center of bearing1 to

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center of bearing2. K c is additional cascade stiffness with bearing2 to represent the

shaft-bolt-york between the center of bearing2 and inner race of bearing2. K hb is

housing bolt stiffness.

Figure3.4 Axial translational stiffness model

The axial translation stiffness of ][ llK calculated from FE model does not

include K s3 and K h. The refinement should be made according to Figure3.4 in the

following way. Before doing finite element calculation, the cascade stiffness K c

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supporting gear. A bearing stiffness formulation[21,22] is applied here to calculate

stiffness of these bearings. As for pinion, D refers to Bearing#1 to pinion back side

distance. L refers to Bearing#1 to Bearing#2 distance. S refers to Bearing#0 to pinion

back side distance and this is only applicable to 3-bearing mounted pinion. As for gear, D

refers to Bearing#3 to ring gear back side distance. L refers to Bearing#3 to Bearing#4

distance.

Table 1. System Parameters

Gear ParametersPinion Gear

Number of teeth 14 45Offset (m) 0 0Pitch angle (rad) 0.391 1.282Pitch radius (m) 0.067 0.215Spiral angle (rad) 0.478 0.478Face width (m) 0.063 0.063Type Left Hand Right HandLoaded side Concave Convex

Shaft Parameters3-brg Pinion Shaft 2-brg Pinion Shaft Gear Shaft

Outer diameter(m) 0.09 0.09 0.12Inner diameter(m) 0 0 0D(m) 0.028 0.028 0.026L(m) 0.115 0.15 0.055S(m) 0.1Backcone thickness(m) 0.01 0.01 0.048Youngs modulus 2.07e11 2.07e11 2.07e11 Poissons ratio 0.3 0.3 0.3

Bearing Parameters

Bearing#0 Bearing#1 Bearing#2Kxx (N/m) 8.599e9 8.823e9 8.599e9Kxy (N/m) 0 1.095e2 -1.671e2Kxz (N/m) 4.236 1.277e1 4.236Kxx (N/rad) 4.101e -1 2.457e-1 4.101e-1Kxz (N/rad) -1.521e8 1.452e8 -1.521e8Kyy (N/m) 0 8.887e8 1.721e9Kyz (N/m) 0 -2.138e1 1.73e1

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Figure3.5 3-bearing straddle mounted pinion (upper) and 2-bearing overhung mounted pinion (lower)

3.3.1 Analysis on Equivalent Shaft-bearing Stiffness Models and P inions Lumped

Shaft-bearing Stiffness Matrices of Two Pinion Configurations

Static finite element models of 2-bearing mounted pinion and 3-bearing

mounted pinion are shown in Figure3.6. The pinion with integrated shaft is modeled

with several uniform cross-section beam elements and the bearings are modeled with

linear springs. The finite element model of 3-bearing mounted pinion consists of 9

nodes, 5 uniform cross-section beam elements and 3 linear spring elements. The finite

element model of 2-bearing mounted pinion consists of 7 nodes, 4 uniform cross-

section beam elements and 2 linear spring elements. The axial translation stiffness

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models for two kinds of configurations are identical as Figure3.4, since the pilot

bearing of 3-bearing configuration cannot stand the axial load.

It could be predicted that the lumped shaft-bearing stiffness of two

configurations will be different. As for the 2-bearing mounted pinion, the equivalent

lumped shaft-bearing stiffness could be derived as:

078432038100637509802081404

038100078432081404003163750

6375008140409644889154689

9802003189150901318788

0814046375046898788096448

][

E ...-.- E .-

. E . E ..-.

.- E . E ..-.

.-.-.- E ..

E .-... E .

k k k k k

k k k k k

k k k k k

k k k k k

k k k k k

K

ll z z

ll x z

ll zz

ll zy

ll zx

ll z x

ll x x

ll xz

ll xy

ll xx

ll z z

ll x z

ll zz

ll zy

ll zx

ll z y

ll x y

ll yz

ll yy

ll yx

ll z x

ll x x

ll xz

ll xy

ll xx

ll

.

As for the 3-bearing mounted pinion, the equivalent lumped shaft-bearing

stiffness could be derived as:

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07140.502771052040441207294.7

02771.007140.507294.78690.05204.0

5204007294.710585.153.1493.12

44128690.0531409015190.84

07294.75204093.1290.8410585.1

][

E ..-.- E -

E E -

.- E E -

.--.- E .

E -. E

k k k k k

k k k k k

k k k k k

k k k k k

k k k k k

K

ll z z

ll x z

ll zz

ll zy

ll zx

ll z x

ll x x

ll xz

ll xy

ll xx

ll z zll x zll zzll zyll zx

ll z y

ll x y

ll yz

ll yy

ll yx

ll z x

ll x x

ll xz

ll xy

ll xx

ll

Certain stiffness elements change significantly from 2-bearing to 3-bearing

configuration. They arell

xxk ,ll

zzk ,ll

x xk ,ll

z zk ,ll

x zk andll

z xk .ll

xxk ,ll

zzk are horizontal

and vertical translational stiffness, which becomes larger. ll x xk ,ll

z zk are both bending

stiffness, which also becomes larger. ll z xk ,ll

x zk are both representing the coupling

between translation and bending, which becomes smaller. The significant change of

these stiffness elements may lead to the change of modal frequency and dynamic

response.

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Figure3.6 Finite element model of 3-bearing mounted pinion (left) and finite element model of 2-bearing mounted pinion (right)

3.3.2 Comparison on Gear Dynamics

Here, two configurations are compared from the viewpoint of the dynamics of

spiral bevel geared rotor system, by using the 14-dof lumped parameter dynamic

model and it is assumed that the dynamic system parameter affected by

2-bearing and 3- bearing configurations only lies in pinions lumped shaft -bearing

stiffness.

As shown in Figure3.7, the dynamic mesh forces of 2-bearing case and 3-

bearing case show obvious difference, including certain main peak. For example, by

changing 2-bearing configuration to 3-bearing configuration, the peak at about 800

Hz is shifted to the left and the peak amplitude is increased. Thus, it could be

concluded that effect of 2-bearing configuration and 3-bearing configuration on

dynamic mesh force could be significant.

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500 1000 1500 2000 250010

2

103

104

freqenceny(Hz)

D y n a m

i c M e s

h F o r c e

( N )

Figure3.7 Comparison of 2-bearing and 3-bearing configurations on dynamic mesh force

, 2-bearing case; ,3-bearing case

Figure3.8 shows the comparison on system modes. As for 2-bearing

configuration case, at low frequency range, there are many pinion bending modes

which have large pinion bending strain energy, such as Mode 6, Mode 7, Mode 8 and

Mode 9. While at high frequency range, there are few pinion bending modes. By

contrast, as for 3-bearing configuration case, at low frequency range, there are few

pinion bending modes, while at high frequency range, there exist pinion bending

modes which are dominated by pinion bending strain energy. Thus, the effect of 2-

bearing and 3-bearing configurations on dynamic system modes is significant and

pinion bending modes are at lower frequency for 2-bearing configuration case. This

phenomenon may be caused by the increased lumped shaft-bearing bending stiffness

from 2-bearing configuration to 3-bearing configuration.

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Figure3.8 Comparison of 2-bearing and 3-bearing configurations on modal strain energydistribution

(a) Description of x-axis in (b) and (c); (b) 2-bearing configuration case; (c) 3-bearingconfiguration case

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 1(0.0 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 2(16.3 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 3(47.3 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 4(421.2 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 5(430.3 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 6(673.8 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 7(793.9 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 8(1252.9 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 9(1739.2 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 10(1754.8 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 11(1920.2 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 12(1975.2 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 13(4062.8 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 14(4139.6 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 1(0.0 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 2(16.3 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 3(47.3 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 4(420.4 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 5(430.3 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 6(583.8 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 7(620.2 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 8(678.5 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 9(830.2 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 10(1273.4 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 11(1920.2 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 12(1970.7 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 13(3145.5 Hz)

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Mode 14(3246.8 Hz)

(a)

(b)

(c)

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0 1000 2000 30000

100

200

300

400

freqenceny(Hz)

M a g n

i t u

d e ( N m

)

0 1000 2000 3000

0

10

20

30

40

50

60

freqenceny(Hz)

M a g n

i t u

d e ( N m

)

Figure3.9 Comparison of 2-bearing and 3-bearing configurations on dynamic bearing

load (a) Bearing1:Dynamic bearing moment around z-axis; (b) Bearing2: Dynamic bearing

moment around x-axis; , 2-bearing case; ,3-bearing case

As shown in Figure3.10, generally, pinion response in 2-bearing configuration

case is larger than pinion response in 3-bearing configuration case, especially in low

frequency range. And the change of pinion response matches the change of system

modes. This is because, by changing 2-bearing configuration to 3-bearing

configuration, horizontal, vertical translation and bending stiffness increase

significantly and pinions translational and bending motion is large ly restrained.

Therefore, pinions vibration could be significantly reduced through changing 2 -

bearing configuration pinion to 3-bearing configuration pinion.

(a) (b)

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0 1000 2000 30000

2

4

6x 10

-6

freqenceny(Hz)

M a g n i t u

d e

( m )

0 1000 2000 3000

0

2

4

6x 10

-6

freqenceny(Hz)

M a g n

i t u d e ( m

)

0 1000 2000 30000

0.5

1

x 10-4

freqenceny(Hz)

M a g n

i t u

d e ( r a

d )

0 1000 2000 3000

0

0.5

1

x 10-4

freqenceny(Hz)

M a g n

i t u

d e ( r a

d )

Figure3.10 Comparison of 2-bearing and 3-bearing configurations on pinion response(a) Horizontal x-axis translation displacement; (b) Vertical z-axis translation

displacement; (c)Rotation displacement around x-axis; (d) Rotation displacement

around z-axis , 2-bearing case; ,3-bearing case

3.3.3 Effect of 2-bearing and 3-bearing Configurations on Mesh Model

The difference of shaft-bearing compliance between two kinds of

configurations may lead to the difference of mesh model. By using Quasi-static

Three-dimensional Finite Element Tooth Contact Analysis Program, the effect of

shaft-bearing configurations on mesh model, i.e. mesh point, line-of-action, mesh

stiffness, is studied here.

From Figure3.11, it could be concluded that for 3-bearing configuration case,

the effect of pilot bearing position on mesh point is very slight.

(a) (b)

(c) (d)

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0 10 20 30-0.064

-0.062

-0.06

-0.058

-0.056

-0.054

-0.052

RollAngle in One Mesh Cycle(Degree)

M e s h

P o

i n t X C o o r d

i n a

t e ( m )

0 10 20 300.174

0.176

0.178

0.18

0.182

0.184

0.186


M e s h

P o

i n t Y C o o r d

i n a

t e ( m )

0 10 20 30-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0


M e s

h P o

i n t Z C o o r d i n a

t e ( m )

Figure3.11 Effect of pilot bearing position on mesh point for 3-bearing case , S =0.002 inch; , S =1.18 inch. , S =2.36 inch.

, S =3.36 inch; , S =4.36 inch; , S =5.36 inch;

From Figure3.12 and Figure3.13, it could be concluded that both for 3-bearing

configuration case and 2-bearing configuration case, the effect of tapered roller

bearing position on mesh point is very slight.

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0 10 20 30-0.064

-0.062

-0.06

-0.058

-0.056

-0.054

-0.052


M e s

h P o

i n t X C

o o r d

i n a

t e ( m )

0 10 20 300.174

0.176

0.178

0.18

0.182

0.184

0.186


M e s

h P o

i n t Y C o o r d

i n a t e ( m

)

0 10 20 30-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0


M e s

h P o

i n t Z C o o r

d i n a

t e ( m )

Figure3.12 Effect of tapered roller bearing position on mesh point for 3-bearing case

, L =5.965 inch; , L =4.445 inch. , L =2.925 inch.

0 10 20 30-0.064

-0.062

-0.06

-0.058

-0.056

-0.054

-0.052

RollAngle in One Mesh Cycle(Degree) M

e s h P o

i n t X C o o r d

i n a

t e ( m )

0 10 20 300.174

0.176

0.178

0.18

0.182

0.184

0.186


M e s

h P o

i n t Y C o o r d

i n a t e

( m )

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0 10 20 30-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0


M e s

h P o

i n t Z

C o o r d

i n a t e

( m )

Figure3.13 Effect of tapered roller bearing position on mesh point for 2-bearing case

, L =5.965 inch; , L =4.445 inch. , L =2.925 inch.

Comparison of 2-bearing and 3-bearing configurations on mesh point in

Figure3.14 shows that there exists certain small influence of 2-bearing and 3-bearing

configurations on mesh point. In other words, by changing 2-bearing configuration to

3-bearing configuration, the mesh point is influenced a little and obviously this

influence comes from adding a pilot bearing to the rear end of pinion rather than

changing the distance between the tapered roller bearings.

0 10 20 30-0.064

-0.062

-0.06

-0.058

-0.056

-0.054

-0.052


M e s

h P o

i n t X C o o r d

i n a

t e ( m )

0 10 20 300.174

0.176

0.178

0.18

0.182

0.184

0.186


M e s

h P o

i n t Y C o o r d

i n a

t e ( m )

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0 10 20 30-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0


M e s h

P o

i n t Z C o o r d

i n a

t e ( m )

Figure3.14 Comparison of 2-bearing and 3-bearing configurations on mesh point , 2-bearing configuration case; ,3-bearing configuration case.

Figure3.15 shows that, as for 3-bearing configuration case, there is little effect

of pilot bearing position on line-of-action vector.

0 10 20 300.2

0.22

0.24

0.26

0.28

RollAngle in One Mesh Cycle(Degree) L i n e - o

f - a c t

i o n

X D i r e c

t i o n a

l C o s

i n e

0 10 20 300.5

0.52

0.54

0.56

0.58


f - a c

t i o n

Y d i r e c

t i o n a

l c o s

i n e

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0 10 20 300.76

0.78

0.8

0.82

0.84


f - a c

t i o n

Z d i r e c t

i o n a

l c o s i n e

Figure3.15 Effect of pilot bearing position on line-of-action vector for 3-bearing case , S =0.002 inch; , S =1.18 inch., S =2.36 inch.

, S =3.36 inch; , S =4.36 inch; , S =5.36 inch;

It could be concluded from Figure3.16 and Figure3.17 that, both for 3-bearing

configuration case and 2-bearing configuration case, the effect of tapered roller

bearing position on line-of-action vector is very tiny.

0 10 20 300.2

0.22

0.24

0.26

0.28

RollAngle in One Mesh Cycle(Degree ) L i n e - o

f - a c

t i o n

X D i r e c

t i o n a l

C o s

i n e

0 10 20 30

0.5

0.52

0.54

0.56

0.58


f - a c t

i o n

Y D i r e c

t i o n

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