NASA

Contractor Report 4334

AVSCOM

Technical Report 90-C-019

Vibration Transmission Through

Rolling Element Bearings

in Geared Rotor Systems

Rajendra Singh and Teik Chin Lim

Ohio State University

Columbus, Ohio

Prepared for

Propulsion Directorate

USAARTA-AVSCOM and

NASA Lewis Research Center

under Grant NAG3-773

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Division

1991)

ACKNOWLEDGEMENTS

We are very grateful for the financial support provided throughout this Ph.D.

research by the NASA Lewis Research Center, Department of Mechanical Engineering,

and Graduate School of the Ohio State University. Appreciation is also extended to J.S.

Lin, D.R. Houser, J.J. Zakrajsek and F.B. Oswald for providing some of the

experimental results used in Chapters III-V.

iii

PRECEDING PAGE BLANK NOT FILMED
PA_ _ _/ INTENT!ONAt:LYBI_A/_
TABLE OF CONTENTS

ACKNOWLEDGEMENTS ................................................................. i ii

LIST OF SYMBOLS ....................................................................... i ×

CHAPTER

I INTRODUCTION .................................................................... 1

1.1 Introduction ...................................................................... 1

1.2 Literature Review ............................................................... 6

1.3 Scope and Objectives ........................................................... 7

II BEARING STIFFNESS FORMULATION ....................................... 12

2.1 Introduction ...................................................................... 12

2.2 Literature Review ................................................................ 14

2.3 Assumptions and Objectives ................................................... 16

2.4 Bearing Load-Displacement'Relations ........................................ 18

2.5 Development of Bearing Stiffness Matrix [K]b m ............................ 24

2.6 Numerical Estimation of [K]bm ................................................ 3 l

2.7 Validation of Proposed Model ................................................. 33

2.8 Parametric Studies ............................................................... 36

2.9 Conclusions ....................................................................... 46

FR2CED;_;G 7A _-_IE_k;_i"X NO-_ f._LMED
III BEARING SYSTEM STUDIES .................................................. 51

3.1 Introduction ...................................................................... 51

3.2 Literature Review ................................................................ 53

3.3 Assumptions and Objectives ................................................... 54

3.4 System Goveming Equations .................................................. 56

3.4.1 Method A: Lumped Parameter Model ................................ 56

3.4.2 Method B: Dynamic Finite Element Formulation ................... 57

3.4.3 Other Methods ......................................................... 58

3.5 Bearing System Stability ....................................................... 60

3.6 System Reponse ................................................................. 63

3.7 Example Case I: Rigid Shaft andPlate System .............................. 71

3.7.1 Vibration Models ...................................................... 71

3.7.2 Bearing Transmissibility ............................................. 75

3.8 Example Case II: Rigid Shaft and Plate Supported on Flexible Mounts ... 80

3.8.1 Vibration Models ....................................................... 80

3.8.2 Frequency Reponse .................................................... 84

3.9 Example Case m: Experimental Study ....................................... 88

3.9.1 Physical Setup .......................................................... 88

3.9.2 Bearing Analysis ....................................................... 88

3.9.3 System Study ........................................................... 91

3.10 Concluding Remarks .......................................................... 96

IV GEARED ROTOR SYSTEM STUDIES ........................................... 97

4.1 Introduction ...................................................................... 97

4.2 Literature Review ............................................................... 101

4.2.1 Casing andMount Dynamics ......................................... 101

4.2.2 Gear Dynamics ......................................................... 102

4.2.3 Rotor Dyamics ......................................................... 102

4.3 Assumptions and Objectives ................................................... 103

4.4 Theory ............................................................................ 105

vi
V

4.4.1

4.4.1.1

4.4.1.2

4.4.1.3

4.4.1.4

4.4.1.5

4.4.1.6

Method A: Lumped Parameter Model ............................... 105

Equations of Motion ........................................ 105

System Kinetic Energy ..................................... 107

Shaft Stiffness Matrix ...................................... 107

Gear Mesh Stiffness Matrix ............................... 110

Flexible Mount Stiffness Matrix .......................... 111

Bearing Stiffness Matrix ................................... 112

4.4.2 Method B: Dynamic Finite Element Formulation ................... 113

4.4.3 Other Methods .......................................................... 113

4.5 Example Case I: Single-stage Rotor System with Rigid Casing

and Flexible Mounts ............................................................ 114

4.5.1 Vibration Models ....................................................... 114

4.5.2 Eigensolution ........................................................... 118

4.5.3 Transmissibility ........................................................ 126

4.6 Example Case II: Geared Rotor System with Rigid Casing

and Flexible Mounts ............................................................ 131

4.6.1 Bearing Analysis ....................................................... 131

4.6.2 Vibration Models ....................................................... 133

4.6.3 Eigensolution ........................................................... 136

4.6.4 Transmissibility Spectra ............................................... 140

4.7 Example Case III: Geared Rotor System with Rigidly

Mounted Flexible Casing ....................................................... 144

4.7.1 Physical Setup .......................................................... 144

4.7.2 Vibration Models ....................................................... 146

4.7.3 Casing Response ....................................................... 147

4.8 Concluding Remarks ........................................................... 153

STATISTICAL ENERGY ANALYSIS

5.1

5.2

............................................ 155

Introduction ...................................................................... 155

Modal Analysis of Gear Casing and Mounts ................................ 156

5.2.1 Finite Element Model .................................................. 156

5.2.2 Experiments and Modal Validation ................................... 158

vii
5.2.3 ParametricStudies .....................................................

5.3 Justificationfor UsingSEAApproach .......................................

5.3.1 ModalDensities ........................................................

5.3.2 LiteratureReview ......................................................

5.4 ExampleCaseI: CouplingLossFactor of Plate-Cantilevered

Beam System ....................................................................

5.5 Example Case II: Coupling Loss Factor of Circular

Shaft-Bearing-Plate System ...................................................

5.6 Example Case Ill: A Circular Shaft-Bearing-Plate System .................

5.6.1 Theory ...................................................................

5.6.2 Validation and Parametric Studies ....................................

5.7 Example Case IV: A Geared Rotor System ..................................

5.7.1

5.7.2

5.7.3

5.7.4

Assumptions ............................................................

Coupling Loss Factor ..................................................

Vibroacoustic Response ...............................................

Experimental Validation and Parametric Studies ....................

160

165

165

166

167

169

175

175

182

188

188

190

192

194

5.8 Concluding Remarks ............................................................ 200

VI CONCLUSIONS AND RECOMMENDATIONS ............................... 201

6.1 Summary ......................................................................... 201

6.2 Future Research .................................................................. 205

LIST OF REFERENCES ................................................................ 206

viii
LIST OF SYMBOLS

Ao

A(¢o)

<A2>

As

ao,ai

[c]

[C]b

C

DOF

dG

dp

dbi

unloaded distance between the inner and outer raceway groove curvature

centers

loaded distance between the inner and outer raceway groove curvature centers

of j-th ball

accelerance transfer function

mean square acceleration spectra

casing plate surface area

shaft cross sectional area

locations of the outer and inner raceway groove curvature centers

system damping matrix

bearing damping matrix

wave speed

degree of freedom

gear diameter

pinion diameter

inner raceway diameter of bearing

ix
dbm

dbo

EI

EC

EU

Es ,Ec

e(t)

P W

Fjba(t)

Fjbm

Fjsa(t)

Fjsap

Fjsm

{f(t)}a

{f)ap

{f}bm

bearing pitch diameter

outer raceway diameter of bearing

shaft diameter

flexural rigidity of the shaft

Rayleigh dissipation function

kinetic energy

potential energy

total vibratory energy level in the internal (s) subsystem and external (c)

subsystem (Chapter V)

static transmission error

generalized force in Lagrangian formulation

alternating bearing force in the j = x, y, z or r direction

mean bearing force in the j = x, y or z direction

applied altemating force on the shaft, j = x, y, z or r

complex Fourier coefficient of Fjsa(t), p = 1, 2, 3, ....

applied mean force on the shaft, j = x, y or z

generalized alternating applied load vector

complex Fourier coefficient of {f(t)} a, p = 1, 2, 3, ....

mean beating load vector
lif0}ca

{f(t)ls

{f(t) }sa

If}sm

G

H k

h

Ijc

IzL

IzM

Ijp

Ijc

tjs

{I}

Kn

[K]

[K]v

[K]bm

altemating casing load vector

total shaft load vector

alternating shaft load vector

mean shaft load vector

bearing outer ring geometrical center

nonlinear functions, k = 1,2,3,...,V, given by equations (2.12a,b) and (2.13)

geometrical thickness

mass moment of inertia of the gear about j = x, y, z axis

mass moment of inertia of the load about z axis

mass moment of inertia of the motor about z axis

mass moment of inertia of the pinion about j = x, y, z axis

mass moment of inertia of the casing about j = x, y, z axis

mass moment of inertia of the shaft about j = x, y, z axis

Lumped inertia row vector

rolling element load-deflection stiffness constant

system stiffness matrix

mount stiffness matrix

proposed bearing stiffness matrix of dimension 6

×i
[K]bms

[K]h

[Kh

[K]:

kTM

kTL

kbiw

^

k biw

ks,kc

kh

kvj

LA

LV

Le

Mjba

Mjbm

a matrix of dimension 5 as a subset of [K]bm with last row and column

excluded

gear mesh coupling stiffness matrix

shaft stiffness matrix

shaft segment stiffness matrix

effective torsional stiffness of the driving shaft coupling

effective torsional stiffness of the driven shaft coupling

bearing stiffness coefficient, i, w = x, y, z, 0x, 0y, 0 z

estimated bearing stiffness coefficient, i, w = x, y, z, 0x, 0y, 0 z

shaft (s) or plate (c) wavenumber (Chapter V)

gear mesh stiffness

mount stiffness coefficient, j = x, y, z, 0 x, 0y, Oz

frequency bandwidth-averaged mean square acceleration in dB re <A2>ref= 1.0g 2

spatially and frequency bandwidth averaged mean square mobility level (dB re

<V2>ref = 1.0 m2/N2s2)

shaft segment length

shaft length

alternating bearing moment about j = x or y direction

mean bearing moment about j = x or y direction

xii
Mjsa

[M]s

[_c

mR

n_

m-s

msej

Iml

Ng

n

nc

n s

p(t)

%

Q

qw

{q(t)}a

{q}ap

ahemating shaft bending moment about j = x or y direction

system mass matrix

shaft mass matrix

casing mass matrix

rotor mass

casing mass

shaft mass

shaft lumped mass

lumped mass row vector

number of gear teeth

rolling element load-deflection exponent

casing plate modal density (see Chapter V)

number of shaft segments (Chapter IV) or shaft modal density (Chapter V)

normalized dynamic transmission error

resultant normal load on the j-th rolling element

acoustic source directivity (Chapter V)

generalized displacement vector in Lagrangian formulation

generalized alternating displacement vector

complex Fourier coefficient of {q(t)} a, p = 1, 2, 3, ...
{q}b

Iq}ba

{q}bm

{q(t) }ca

Iq(t) Jsa

{q}sm

R(to)

Re

--4

Rj

rL

q

S

Sc

lT]b

a'm(O

Tjsm

t

Ujca(t)

total bearing displacement vector

altemating bearing displacement vector

mean beating displacement vector

alternating casing displacement vector

alternating shaft displacement vector

mean shaft displacement vector

force or moment transmissibility transfer function or room constant (see

Chapter V)

real part of a complex number

j-th bearing coordinate vector, j = 1, ns+l, ns+2, 2ns+2

bearing radial clearance

pitch radius (roller) or radii of inner raceway groove curvature centers (ball)

room surface area (Chapter V)

total surface area of the casing plate in example case III (Chapter HI)

bearing field transfer matrix

applied altemating torque on the shaft, j = x, y or z

applied mean torque on the shaft, j = x, y or z

time

altemating casing translational displacement in the j = x,y or z direction

xiv
Ujsa(t)

Ujsm

{u(t)}ca

{ u(t) }sa

v(o )

<V2>

X

Z

z¢

z

z o

_o

_Xj

O_

[_ja

[_jm

AFrbm

altemating shaft translational displacement in the j = x,y or z direction

mean shaft translational displacement in the j = x, y or z direction

casing alternating displacement vector

shaft alternating displacement vector

mobility transfer function

spatially and frequency bandwidth averaged mean square mobility

shaft (s) or plate(c) alternating velocity in the j=x,y,z direction

solution to the problem H k = 0, k = 1, 2, 3, ..., V

total number of rolling element

driving point shaft impedance (Chapter V)

driving point plate impedance (Chapter V)

number of loaded rolling element

characteristic hnpedance of the surrounding medium (Chapter V)

unloaded bearing contact angle

loaded j-th rolling element contact angle

average absorption coefficient (Chapter V)

alternating bearing angular displacement about the j = x or y direction

mean bearing rotational displacement about the j = x or y direction

incremental mean radial bearing force

×V
A rm

Ao_

_SBj

6X

_Sjm

_j

aja

(5)_j

(a)zj

*j

{_}j

rls,qc

qsc

K

{_t}p

Hs

incremental mean radial bearing displacement

frequency bandwidth

resultant elastic deformation of the j-th ball element

resultant elastic deformation of the j-th roller element

error vector

mean bearing translational displacement in the j = x, y or z direction

resultant elastic deformation of the j-th rolling element

altemating bearing translational displacement in the j = x, y or z direction

effective j-th rolling element displacement in the radial direction

effective j-th rolling element displacement in the axial direction

stability functions given by equation (3.6) and Table 3.1

mode shape vector corresponding to the j-th natural frequency

dissipation loss factor for the intemal (s) or extemal (c) subsystems

coupling loss factor

radius of gyration

complex Fourier coefficient of the principal coordinate vector, p = 1, 2, 3, ...

extemal power input to the intemal (s) subsystem

net power transfer from the internal (s) subsystem to external (c) subsystems

material density

xvi
Ojca(t)

Ojsa(t)

Ojsm

{O(t)}Ra

{0(t)}ca

{0(t)}sa

(5

(5c

fiz z

fizzM

fizzL

O)

(o h

tOjd

(.D O

03p

vj

altemating casing angular displacement about the j = x, y or z coordinate

alternating shaft angular displacement about the j = x, y or z coordinate

mean shaft angular displacement about the i-axis; j = x, y, z

rotor alternating angular displacement vector

casing altemating angular displacement vector

shaft alternating angular displacement vector

Rayleigh damping matrix proportionality constant

casing plate radiation efficiency (Chapter V)

mean rotational speed of the shaft

mean rotational speed of the driving shaft

mean rotational speed of the driven shaft

bandwidth center frequency (Chapter V)

gear mesh frequency

undamped natural frequency, j = 1, 2, 3, ...

damped natural frequency, j = 1, 2, 3, ...

fundamental frequency

excitation frequency, p = 1, 2,3, ...

bearing load angle

angular distance ofj-th rolling element from the x-axis

xvii
;j

[IT

II

()

()

( )*

(^)

modal damping ratio

transpose of a matrix or vector

magnitude or determinant

first time derivative

second time derivative

complex conjugate

estimation based on simple models

xvlil
CHAPTER I

INTRODUCTION

1.1 INTRODUCTION

Noise and vibration generated by the rotating mechanical equipment including

geared drives have always been a problem in the implementation of new technology in

automobiles, rotorcrafts and industrial machines [1-6]. Recently, the need for reliable

vibration/noise prediction methods have been found to be crucial as faster and lighter

machines are being designed [6-8]. In most of these rotating systems, structure-borne

noise paths through bearings, which support the rotating shafts on flexible or rigid

casings, are dominant [5,6,9]. Hence, in order to obtain reliable mathematical

prediction of the overall dynamic system, a complete understanding of the vibration

transmission mechanism through bearings, and the role of bearings as a dynamic

coupler between the shaft and casing, is critical.

Current beating models, based on ideal boundary condition or purely translational

stiffness element description, cannot explain how the vibratory motion may be

transmitted from the rotating shaft to the flexible casing and other connecting structures

in rotating mechanical equipment [10-15]. These simple models are only adequate for

the free and forced vibration analyses of the rotor dynamic system enclosed in a rigid

casing. For example, a vibrational model of a rotating system based upon the existing

bearing models can only predict purely in-plane type motion on the flexible casing plate,
givenonly thebendingmotionontheshaft. However,experimentalresultshaveshown

that thecasingplatemotion is primarily flexural or out-of-planetype [9,16,17]. This

paradoxis essentiallydueto anincompleteunderstandingof the bearingasvibratory

motiontransmitterinrotatingmechanicalequipment.

The main focus of this researchis to clarify this issuequantitatively and

qualitativelybydevelopinganewmathematicalmodelfor theprecisionroiling element

bearings, and extend the proposed bearing formulation to examine vibration

transmissibility in rotating mechanicalequipmentthroughseveralexamplecasesof

bearingsystemsandgeared drives. The superiority of the proposed model compared to

simple models is also demonstrated in these example cases. A typical shaft-bearingcasing-mount system is shown in Figure 1.1. The rigid or flexible shaft may be

subjected to forces and/or torques and supported by a bearing on a flexibly or rigidly

mounted casing. Here, the vibration transmission is from the shaft to the casing and

mount through the bearing system. Figure 1.2 shows a typical rolling element bearing

subjected to forces and moments due to the rolling element deformation. The bearing is

free to rotate about the axis perpendicular to the bearing plane, and hence does not

transmit any dynamic moment about this axis. However, dynamic moments about the

other two orthogonal axis exist which have not been considered in simple bearing

models. Finally, a generic geared rotor system consists of a motor, spur gear pair,

flexible shafts, load, flexible couplings, rolling element bearings, flexible casing and

mounts as shown in Figure 1.3a is also considered. The system is excited by the gear

kinematic transmission error at gear mesh frequency. A single-stage rotor system with

rotating mass unbalance excitation as shown in Figure 1.3b is treated as a special case of

Figure 1.3a. Further description of each system will be presented in later chapters.
3

Force I Rigid or

Flexible

Shaft

I

i

Rolling

Element

Bearing

Rigid or

Flexible

Casing

 leMo t IILLILLLLILLLILklllli

Rigid or " Z

.V//////////////,4 -_/////////,/i.

Figure 1.1 Schematic of a typical shaft-bearing-plate-mount system
I

Radial Force

_ Moment

nt

(a)

Moment

, .-----I_---

Axial Force

(b) (e)

(d) (e)

Figure 1.2 Schematic of typical rolling element bearings. (a) Bearing forces and

moments. (b) Deep groove ball bearing. (c) Angular contact ball bearing.

(d) Taper roller bearing. (e) Cylindrical miler beating.
Driven Shaft [_

Flexible Coupling

Rigid or Flexible Casing

--" Pinion

-- Gear

Rolling

Element

"-- Bearing

222

Ff_

Driving Shaft

N

(a)

_ Rolling

Element

Bearing

Casing d Flexible I [ "°'°rI_oxiUl_S_ _iup_ni_o_

,..//././//////_

____ Flexxble Motmts _

Co)

Figure 1.3 Generic rotating mechanical systems. (a) A geared rotor system. (b) A

single-stage rotor system
6

radial

include

resulted

1.2 LITERATURE REVIEW

Simple bearing models either assume ideal boundary conditions for the shaft or

time-invariant translational springs in the axial and radial directions [10-15]. The ideal

boundary conditions for the shaft are typically simply-supported for short beatings,

clamped for long bearings or free (for torsional motion only) [10-12]. Formulas for the

stiffness coefficients derived from the force-displacement relation commonly used by

bearing designers [18,19] are given by Harris [18], Gargiulo [14] and White [20]. In

1982 Rajab [21,22] realized the limitations of the simple models and philosophically

proposed two additional stiffness coefficients which couple the radial and rotational

bearing degrees of freedom, given radial and moment about the axis transverse to the

line of action. In 1988, Young [23] extended Rajab's [21] analyses to

mean axial force while retaining other features of Rajab's model. This

in bearing stiffness matrix of dimension 3.

Simple beating models are widely used in vibration models of rotor dynamic

systems which typically exclude casing and mount dynamics, to calculate critical speeds,

dynamic stability, and responses due to shaft excitations such as mass unbalance and

gear transmission error [10-15]. In most cases, the vibration transmission through

bearings is not the primary issue, and thus the bearing models tend to be

simplified. None of the current models studied [20-22,24] can fully explain

vibration transmission through bearings in systems similar to Figure 1.1.
Publicationsonthemodalanalysesof geareddrivesandsingleor multiple stage

rotor systems indicate that very little has been reported directly on the force

transmissibility through bearings,and the dynamic effects of bearing, casing and

mountson theinternalrotatingsystem[10-13,15,25-29].A comprehensivereviewof

theavailablevibrationstudiesof casingandmountshasbeengivenby Lim andSinghin

1989[6]. In thesestudies,thedynamicinteractionbetweenthecasing-mountsystem

andtheinternalrotatingsystemisoftennotincorporated,andinfew instanceswhenthis

interactionis modeled,onlypurelyradialand/oraxialforceonthebearingsareincluded.

Such models still do not explain how the vibration is transmitted from the

shaftto thecasing. A morecomprehensivereviewof therelevantstudieswill begiven

in eachchapter.

1.3 SCOPE AND OBJECTIVES

A newmathematicalmodelfor theprecisionrolling elementbearingin Figure 1.2

is developed and incorporated in linear time-invariant discrete and broad band vibration

models of Figures 1.1 and 1.3. This study proposes a comprehensive bearing stiffness

matrix of dimension 6 which explains the vibratory motion transmission through the

bearings and allows for the study of overall geared rotor system dynamics. The lumped

parameter and dynamic finite element techniques are used to develop the discrete

vibration models while statistical energy analysis method is used for the broad band

vibration models. Experimental validation is also included in each chapter, where the

driving point and cross point accelerance and mobility levels predicted by theory are

compared to experiments.
The specific objectives of this research are grouped as follows: bearing

stiffnessformulation,bearingsystemstudies,gearedrotor systemstudies,andstatistical

energyanalysis.Eachchapteris selfsufficientsinceit is writtenin ajournalpaperstyle.

Accordingly,adetailedproblemstatementis alsoincludedin eachchapter.

a. Beating stiffness formulation: Since simple bearing models used in rotor dynamic

analyses are inadequate in explaining the role of bearing as a vibratory motion

transmitter, this study resolves this issue by proposing and developing a new rolling

element bearing stiffness matrix which is suitable for the analysis of the vibration

transmission through either ball or roller bearing. A numerical scheme is also developed

to compute the stiffness coefficients and indicate the existence of solutions to the

nonlinear algebraic bearing equations describing the bearing load-displacement

relationships. The proposed bearing model is partially verified by comparing with

published analytical and experimental results. In addition, the character of the bearing

stiffness matrix and its sensitivity to various bearing parameters will be discussed.

(Chapter II)

b. Bearing system studi_: The specific objectives of this chapter are to incorporate the

proposed bearing matrix developed in Chapter II in linear discrete vibration models of

the bearing system as shown in Figure 1.1 using lumped parameter and dynamic finite

element methods to compute the eigensolution and forced harmonic response, and to

evaluate the dynamic stability. The vibration transmission through beating is also

predicted for several example cases considered previously [14,20,24] and an

experimental setup [17]. The advantages of the proposed formulation compared to the
9

simple models is demonstrated by comparing their predicted transfer functions. The

theory is also validated by comparing analytical predictions with experimental data on a

shaft-bearing-plate system. (Chapter III)

c. Geared rotor system studies: Current geared rotor system vibration models exclude

the effects of casing and mounts, and do not address the overall system behavior. The

objectives of this chapter are to incorporate the proposed bearing matrix in the discrete

vibration model of the generic geared rotor system of Figure 1.3 and conduct overall

system studies by calculating eigensolutions and forced harmonic responses with

emphasis on the prediction of vibration transmission through rolling element bearings.

The effects of casing and mount dynamics on the internal rotating system is also

evaluated. Example cases which include a single-stage rotor system with flexible shafts

supported by two identical rolling element bearings on rigid casing and flexible

mounts,and a spur gear pair with motor and load inertias attached to two flexible shafts

supported by four rolling element beatings on rigid or flexible casing and compliant or

massive mounts will be studied analytically and/or experimentally. Also, the advantages

of the proposed formulation as compared to simple models of geared drives will be

demonstrated. (Chapter IV)

d. Statistical energy analysis: At very high frequencies, the narrow band approach using

the lumped parameter or dynamic finite element model may not be adequate due to the

high structural modal density. To overcome this problem, statistical energy analysis

method is used to predict the vibratory energy transmission in and noise radiation from a

geared rotor system as illustrated in Figure 1.4a. The proposed bearing matrix is again
lO

motor

Structure borne energy

transmission

.... I_ Noise radiation

Energy transmission

through bearings

Energy transmission

through mounts

(a)

Input Power

Shaft-beating system

(bending modes) Energy Dissipation

_ Energy Transfer

Plate-mount system V

(flexural modes) Energy Dissipation

(b)

Figure 1.4 Vibration energy transmission in a geared rotor system. (a) Structure-borne

noise paths. (b) Statistical energy analysis model.
11

incorporatedin the vibratory energymodel. In this method,only the mean-square

spatiallyaveragedresponseoveronethird octavefrequencybandwidthsarepredicted

which involves solutionto a setof algebraicequationsobtainedthroughthevibratory

energy balanceof eachsubsystemshown in Figure 1.4b. Severalexamplecases

includingaplate-cantileveredrectangularbeam,circular shaft-bearing-plate system and a

geared rotor system are chosen to demonstrate the salient features of this technique.

(Chapter V)
CHAPTER 2

BEARING STIFFNESS FORMULATION

2.1 INTRODUCTION

Current rotor dynamic models describe precision rolling dement beatings either as

ideal boundary conditions for the shafts [10-12], or as purely translational stiffness

elements [ 13-15]. Such simple bearing models may be adequate for the free and forced

vibration analyses of the rotor dynamic system enclosed in a rigid casing. But these

mathematical models cannot explain how the vibratory motion may be transmitted from

the rotating shaft to the flexible or rigid casing and other connecting structures. For

example, a vibration model of a system similar to Figure 2.1, based upon the existing

bearing models, can only predict purely in-plane type motion on the flexible casing plate

given only the bending motion on the shaft. However, experimental results have shown

that the casing plate motion is primarily flexural or out-of-plane type [9,16,17]. This

paradox is essentially due to an incomplete understanding of the bearing as vibratory

motion transmitter in rotating mechanical equipment including geared drives where

structure-borne noise paths through bearings are often dominant.

This chapter clarifies this issue qualitatively and quantitatively by developing a

new mathematical model for precision rolling element bearings. A schematic of a

generic system with a flexible shaft rotating at speed f_z and subjected to mean load

vector {f}sm={Fwsm,Twsm}, w = x, y, z, flexible casing and mount is shown in Figure

12
13

X

U

Y

U xsm

__ 0 xsm

Rolling

Element

Bearing

z zsm

Flexible

i Casing

Figure 2.1 Schematic representation of the vibration transmission problem. Here the

flexible shaft is subjected to mean forces Fws m and torques Tws m where w

= x, y or z, is the direction and subscript m and s implies mean and shaft

respectively. Also, 0 is the angular displacement and u is the translational

displacement.
14

2.1; the shaft is supported on one of the following bearings: deep groove ball bearing,

angular contact ball bearing, thrust ball bearing, straight roller bearing or taper roller

bearing. A new bearing stiffness matrix [K]b m will be proposed which is expected to

demonstrate a coupling between the shaft bending motion and the flexural motion of the

casing plate. It will be shown that the translational bearing stiffness coefficients

currently used in rotor dynamic models are a small subset of the proposed [K]b m.

Several example cases are employed to validate our theory. Our bearing model can be

easily incorporated in analytical or numerical models typically used for the dynamic

analyses - this will be the basis of Chapters III and IV of this report.

2.2 LITERATURE REVIEW

The ideal boundary conditions for the shaft have typically been assumed to be

simply-supported for short beatings, clamped for long beatings or free (in the torsional

mode only) [10-12]. In other cases, researchers describe the bearing as time-invafiant

translational springs with stiffness coefficients kbr r and/or kbz z in the radial and axial

directions, respectively [13-15,20]. Formulas for such nonlinear stiffness coefficients

are given by Harris [18] and Gargiulo [14]; these are derived from the radial or axial

mean force-displacement equation commonly used by the precision rolling element

bearing designers [18,19]. Their derivations neglect the effects of radial clearance and

mean bearing force vector {f}bm on the load distribution and hence are applicable only

for constant load angle VI of 180 degrees. White refined these formulations by using a

finite difference approximation for the computation of stiffness coefficients for radial

ball and roller beatings, and by including the effects of radial clearance and force on the
15

load angle'_t I [20]. Even with these refinements the mathematical model is still

incapable of predicting the total vibration transmission across bearings.

In 1982 Rajab [21], realized the limitations of the current simple theory and

philosophically proposed two additional stiffness terms kbr 0 and kb00 which couple the

relative radial and rotational bearing displacements between the inner and outer rings,

given the mean radial load and moment about the axis transverse to the radial line of

action. In 1988 Young [23] extended Rajab's [21] analyses to include the mean

axial force Fzb m, and then used a discrete summation over all of the loaded

rolling elements to obtain bearing forces and moment instead of the integral form

while still retaining other features of Rajab's model. This resulted in a 3x3

bearing stiffness matrix. Some of the salient features of Rajab's [21] and

Young's [23] models are summarized in Reference [22].

Experimental determination of the bearing stiffness coefficients has been strictly

limited to the translational coefficients kbr r and kbz z. A method for the measurement of

in situ bearing stiffness under oscillating loading conditions has been given by Walford

and Stone [30]. Recently, Kraus et al. [24] designed an in situ measurement test stand

to determine the translational bearing stiffness from measured vibration spectra, in

conjunction with the single degree of freedom system theory. They determined the

effect of preload, bearing release and rotational speed fl z on kbr r and kbz z. Their results

show that kbr r and kbz z are essentially linear and the effect of O z is negligible when a

high preload is applied on the beating.
16

2.3 ASSUMPTIONS AND OBJECTIVES

Due to the following key differences, a separate formulation of [K]bm for both ball

and roller type rolling element bearings is required: (i) ball bearings have elliptical

contacts and roller types have rectangular contacts between the inner race, rolling

elements and outer race when loaded, and (ii) the loaded contact angles o_j of the ball

types may change but _j in the roller type remains relatively constant [31]. Each bearing

is characterized by its kinematic and design parameters such as unloaded contact angle

o_o, radial clearance rL, effective stiffness coefficient K n for inner ring-single rolling

element-outer ring contacts, angular misalignment, preloads, radius of inner raceway

groove curvature center for ball type and bearing pitch radius for roller type [18,19,31].

It is expected that [K]bm is given in terms of these parameters.

The mean bearing displacements {q}bm as shown in Figure 2.2 are given by the

relative rigid body motions between the inner and outer rings. The total bearing

displacement vector is given as {q]b={q}bm+{q(t)}ba where {q(t)}ba is the fluctuation

about the mean point [q}bm during the steady state rotation. Accordingly one must

consider time varying beating stiffness coefficients. However in our analysis, such time

varying bearing stiffness coefficients are neglected by assuming very small vibratory

motions i.e. {q}ba _ {q}bm, and high bearing preloads. Consequently, only the mean

beating loads and displacements are included in the derivation of [K]bm. The basic

load-deflection relation for each elastic rolling element is defined by the Hertzian contact

stress theory [18,19,32], and the load experienced by each rolling element is described

by its relative location in the bearing raceway. Further it is assumed that the angular

position of each rolling element relative to one another is always maintained due to the

rigid cages and pin retainers. Secondary effects such as centrifugal forces and
17

Y

_ym

ym

F

bym

bym

F_._ M bym

dbo dbm dbi :

F xm (_ xm

M bxm _ xm

Outer Ring of raceway diamater dbo

Ring of raceway diameter d bi

• Rolling Element of diameter D

F

bzm

h,. _.- Z v w v

8

zm

Figure 2.2 Rolling element bearing kinematics and coordinate system. Here the

following nomenclature is used: dbo is the outer raceway diameter, dbm is

the bearing pitch diameter, dbi is the inner raceway diameter, _/is the

angular position of roiling element, _Swm is the mean translational

displacement, _-m is the mean angular displacement, Fwb m is the mean v

bearing force, and Mpb m is the mean bearing moment where w = x, y, z,

and p=x, y, are the dir-ections.
18

gyroscopic moments on the bearing are ignored as these effects are evident only at

extremely high rotational speeds. Tribological issues [32,33] are beyond the scope of

this study and hence our analysis assumes bearings to be unlubricated.

The specific objectives of this chapter are to: (i) propose and develop a new rolling

element bearing stiffness matrix [K]b m which is suitable for the analysis of the vibration

transmission through either ball or roller beating, (ii) develop a numerical scheme to

compute [K]b m and discuss the existence of solutions to the nonlinear algebraic beating

equations describing load-displacement relationships, (iii) verify our proposed model by

comparing its predictions with published analytical and experimental results [14,20,24]

for the translational stiffness coefficients kbxx, kbyy and kbz z, (iv) relate [K]b m to

various kinematic and design parameters, and perform parametric studies to investigate

the effect of unloaded contact angle ¢to and preloads, and (v) characterize the nature of

[K]bm and recommend its usage. Finally it should be noted that dimensionless

parameters will not be used here as the metric units are invariably employed to specify

bearings [32].

2.4 BEARING LOAD-DISPLACEMENT RELATIONS

In this section, the relationships between the beating forces {Fxbm, Fybm , Fzb m }

and moments {Mxbm, Mybm } transmitted through the rolling element beating, and the

bearing displacements {q}bm as given in Figure 2.2 will be derived for both ball and

roller beatings. The mean applied loads {f} sm at the shaft as given in Figure 2.1 and

bearing preloads generate the mean beating displacements {q}bm and loads {f}bm.

These displacements {q}bm are used to derive the resultant elastic deformation _5(Vj) of
19

thej-th rolling elementlocated at angle_/j from the x-axis. From the ball bearing

kinematicsshownin Figure 2.3,8B(_j) is

=_A(vj)-A o , 5Bj>0

5B(_t j) }o , 5Bj<0

A(_j) = _/(5" )2zj+ (5")_j

(2.1a)

(2.1b)

(8*) zj = A o sin 0% + (5) zj ; (5*) rj = A o cos ot o + (5) rj (2. lc)

where A o and A are the unloaded and loaded relative distances between the inner a i and

outer ao raceway groove curvature centers. Similarly for the roller bearing kinematics

shown in Figure 2.4 for otj=0_ o, 8R(_j) is

(08 . .sin a. >0

)rjC°saj+(f)zj J , 8Rj (2.2)

5 R(lltj) = 5R j

, <0

Note that in equations (2.1) and (2.2) 5Bj <_0 or 5Rj < 0 implies that the j-th rolling

element is stress free. In both equations (2.1) and (2.2), the effective j-th rolling

element displacements in the axial (5)z j and radial (5)rj directions are given in Figure 2.5

in terms of the bearing displacements {q}bm-

(5)zj=Szm +rj{_xmSin(vj)-_ymCOS(_j)} (2.3a)

(5)r j = 5xm cos_j + 8y m sin _j- r L (2.3b)
20

OuterRing(o)

j-th Ball

InnerRing(i)

BearingCenterline

I //

L/

i+0_ o

i i/,

r

Figure 2.3 Elastic deformation of rolling element for non-constant contact angle _j

given by the change in the distance between the inner a i and outer ao

raceway groove radius curvature centers due to the mean bearing loads or

displacements.
21

( )zJ

OuterRing(o)_ /_ Rj

j-th Roller _ _,//_

g.Z////l l

Inner Ring (i) _ _.x._[

r °°ll

II II

Figure 2.4 Elastic deformation of rolling element for constant contact angle (zj = czo

given by the change in the relative position of the inner and outer raceways

due to the mean bearing loads or displacements.
22

r.

xm j

ym

G

Z

ym rj

_x

Figure 2.5 Decomposition of the effective radial (_)rj and axial (_)zJ deformations of

the j-th rolling element in terms of the mean bearing displacements {q}bm.

Here G is the beating outer ring geometrical center.
23

where rj is the radial distance of the inner raceway groove curvature center for the ball

type or is the pitch bearing radius for roller type. Equations (2.1)-(2.3) in conjunction

with the Hertzian contact stress principle [ 18,19,32] stated as follows yield the loaddeflection relationships for a single rolling element.

_in. (2.4)

Qj=Kn j

where Qj is the resultant normal load on the rolling element, and K n is the effective

stiffness constant for the inner race-rolling element-outer race contacts and it is a

function of the bearing geometry and material properties [18,19,31]. Note that the

exponent n is equal to 3/2 for ball type with elliptical contacts and 10/9 for roller type

with rectangular contacts. Previously, we have mentioned that the loaded contact angle

c_j for the roller beating remains unchanged from the unloaded position o_o, but on the

other hand _j may alter in the ball bearing case. The sign convention is such that o_j is

positive when measured from the bearing x-y plane towards the axial z-axis as shown in

Figures 2.3 and 2.4, and negative otherwise. For the ball bearing of Figure 2.3, the

loaded contact angle o_j is

A o sin 0% + (_5) zj

tall (o_j) = A o cos 0% + (8) rj

(2.5)

where (5)zj and (5)rj are given by equations (2.3a) and (2.3b). It is appropriate here to

note that Rajab [21,22] and Young [22,23] in their derivation of the bearing stiffness

model used an expression similar to equation (2.2) but with _xm=_ym=Szm=rL--0 in
24

Rajab's analysis and fixm=_ym=rL=0 in Young's analysis for both ball and roller

bearings. Since always otj is given by equation (2.5) irrespective of the formulation and

since equation (2.2) is valid only if otj=o%, their ball and roller bearings analyses are

in error. Expressions similar to equation (2.1)-(2.5) with minor differences

have also been used by Eschmann et al. [31], Jones [34] and Davis [35], but their

intentions were to calculate static bearing forces rather than to derive the bearing stiffness

models for vibration transmission analysis.

2.5 DEVELOPMENT OF BEARING STIFFNESS MATRIX [K]b m

Our proposed bearing stiffness matrix [K]b m is a global representation of the

bearing kinematic and elastic characteristics as it combines the effects of z number of

loaded rolling element stiffnesses in parallel given by Sj > 0. First, we need to relate the

resultant bearing mean load vector {f}bm to the bearing displacement vector {q}bm" This

can be achieved through vectorial sums Qj (_wm,[_pm; w = x, y, z and p = x, y) in

equation (2.4) for all of the loaded rolling elements which lead to the following bearing

moments {Mwb m } and forces {Fwb m } as follows

M

xbm

M

ybm

M Am

z [ sin _l/j ]

=y_rjQjsin aj/ -c°svjt

J [ o ]

Fybm = Qj

F zbm J

cos O_j cos Xl/j

cos_j sin _j

sin t_.

J

(2.6a)

(2.6b)
25

Replacing Qj ando_jin equation(2.6) in termsof [Swm,_pm} yields the following

explicit relationshipsbetween{f}bmand{q}bmfor ball bearings

n

M um ao}

Mybm = Kn 2 2

Mzb m J _J[AosinO_o+(8)zj] +[A o coSOto+(8)rj]

sin _tj ]

r.j {A o sin 0% + (8) zj } 1 - cos gtj

t 0

(2.7a)

n

Fxbm { .J[Ao sin ot o + (8) zj]2 + [Ao cos O_o + (8) rj] -Ao}

FY bm = Kn _ 2

Fzbm j _¢/[A o sin o_o + (8) zj]2 + [A o cos ot o + (8) rj]

[A o cos Oto+ (8)rj] cos _/j /

[A o cos Oto+(8)rj] sin _ttj

[A o sin ot o + (8) zj] J

(2.7b)

and similarly for roller bearings

M xbm }

M ybm

L Mzbm

[FxumI

Fybm

F zbm

Z

=K n sino_ o_rj

J

Z

= Kn E {(8)rj

J

{ (8) rj cos a o + (;5) zj

cos o_o + (8) zj sin 0%} n {

sin lit j ]

sin°t°}n J-c°s_I/Jlo

c°s_°c°sgtj ]

cos o_o sin _j

sin ot o J

(2.8a)

(2.8b)
26

where (8)r j and (_5)zj are functions of {Swm,_pm} as defined by equation (2.3).

Approximate integral forms of equations (2.7a,b) and (2.8a, b) are often used instead of

the summation forms to eliminate explicit dependence on _j, especially in the case of

only one or two degrees of freedom bearings [18,32]. For instance Rajab [21,22] chose

the integral form representation but made a mathematical error in constructing the

integrand.

Now we define a symmetric bearing stiffness matrix [K]b m of dimension 6 from

equations (2.7a,b) and (2.8a,b) and by assuming that {q}ba <<{q}bm

[K]

bm

_Fwb m OFwb m

28.1m _im

_M w bm 3M wbm

_im _[_im

{q}bm

; w, i = x, y, z (2.9)

Here each stiffness coefficient must be evaluated at the mean point {q}bm- Explicit

expressions for the ball bearing stiffness are as follows; note that [K]b m is symmetric

i.e. kbiw=kbwi .

n Aj (_5" 2 }

- 2_1/j "_j__-Xo + j-(5*)rj kbx× = Kn _ (Aj Ao )n cos )rj A 2 2

j A3.

J

(2.10a)
27

kbxy = K n Z

J

I1

(Aj- Ao) sin nAj (5") 2 + A2 _ (5,);rj t_tjcos_tj -Aj-A o ' J

X3

J

(2.10b)

(Aj- Ao)n (5")ri nA. 1t

, J

) zj cos - A (5 _/j Aj o

kbxz=Kn Z A 3

j J

(2.10c)

z rj (A j - A o)n (5*) rj (5*) zjSin

kbx0 =KnE" A 3

j J

nAj

rj (A j- Ao )n (5") rj I nAj

kbx0y= KnE A 3

j J

(2.10e)

A 2 * 2

kbyy Kn_ (Aj-A°)nsin2_j_ Aj-Ao + j-(_ ) = A?

j J

(2._o0

nAj 1} z (Aj-Ao)n (5*)rj(5*)zjSin _j -Aj-A o

kbyz = Kn Z A3.

j J

(2.10g)

k

z

byOx = Kn E

J

nA. }

n 5" 2 J 1

r.(A-A o) ( )rj(5*)zj sin _Jj A-A o

j J J

A?

J

(2.10h)
28

---_nY--, A3

kby0y j J

nA. }

J .

r.j(Aj - Ao )n(_*) rj (_*) zj sin Vjcos Wj 1 Aj - A o

(2.10i)

{ nAj(_*)_j+ A2_(_,)_j} z %-A°_n -x]:_o J

kbzz = K n '_

j A 3

J

(2.10j)

kbz0x = Kn X

J

rj (Aj - Ao )n sin Vj t Aj(_5* 2 A 2. zj}

n )zj + -(8")

A.-A o J

l

A3

J

(2.10k)

z rj (Aj - Ao )n cos _j

kbz 0 = Kn X

;}

Aj- A o J

A3

Y j

(2.10/)

= Kn 2,1

b0x0 x

J

r 2

n Aj (5") zj r 2J(A j- Ao )nsin2Wj Aj-A o

A3

J

(2.10m)

k

b0x0y

z r2(Aj- A°)nsin vj v ( _zj--i_:_:5, ° A

J

A 3

J

(2.10n)
29

r2. (A.-A

z j J

kb0 Kn A3.

J j

kbi0z =kb0i0z =0

; i=x,y,z

(2.10o)

(2.10p)

where (5)zj, (8)rj and Aj are defined by equation (2.1).

stiffness coefficients kbi w = kbwi are given explicitly as

And the roller bearing

Z

_in- 1 2

k bxx = nK n cos 2 0to Z Rj COS lit j

J

Z

_ n Kn cos2aoZ 5 n- 1

kbxy 2 Rj sin 211/j

J

Z

n K sin 2or oZ _Sn-1 k bxz - 2 n Rj cos _ttj

J

Z

n [in- 1

kbx0x- _ K nsin 20_ oZ rj Rj

J

sin 2_j

Z

k bx0 - nK2 n sin 2o%_rj_5 n-1 Rj

Y j

COS

Z

kbyy = nKn c°s2 0to_ 5n-lRj sin 2 _j

J

Z

n K n sin 2c_ o _ 8n - 1

kbyz- 2 Rj sin _j

J

(2.11 a)

(2.1 lb)

(2.11c)

(2.1 ld)

(2.11 e)

(2.11f)

(2.11g)
30

n

kby0x = _K n

z

sin 2a o _ rj _n-Rj

J

1

sin 2

_gj

k __ __

by 0

Y

Z

4 Knsin 2ao_ rj6 n-Rj

J

1

sin 2Vj

z

5n- 1

k bzz = n Kn sin 2 c_o _ Rj

J

z

kbz0x=nK n sin20_oE rj_in- 1

J Rj sin _tj

kbz0y=-nK nsin20_ oE rj_n-1

J Rj cos _lfj

z

o 8n- 1

kb0x0x =nKnsin20_ Erj 2 Rj sin2wj

J

z

kb0 0y = _ n2 K n sin 2 ao_ r j2 _iRj n - 1 sin

x J 2_rj

z

n-|

kb0y0y -- nK n sin 2aoE rj 2 8Rj cos21gj

J

(2.1 l h)

(2.1 li)

(2.1 l j)

(2.1 lk)

(2.11/)

(2.11 m)

(2.1 ln)

(2.11o)

kbiOz = k b 0.0 = 0 • i=x,y,z (2.11p)

1 Z

where 5r_j is defined in equation (2.2). It should be noted that all stiffness terms

associated with the torsional degree of freedom 13zm are zero due to the fact that an ideal

bearing allows free rotation about the z-direction. Also, the translational stiffness

coefficients kbii, i=x,y,z for _ym=Szm=_xm=_ym=0, 8xm=_zm=_xm=_ym= 0 or

_xm=Sym=_xm=_ym--0 are equivalent to the bearing stiffness coefficients commonly
31

used by investigators [14,15,20]. The nature of these and other features of [K]bm will

be discussed later in Section 2.9.

2.6 NUMERICAL ESTIMATION OF [K]bm

The coefficients kbi w can be computed by one of the following two methods: I.

directly compute kbi w given mean beating displacement vector {q}bm employing

equations (2.10a-p) and (2.11a-p), or II. numerically solve the nonlinear algebraic

equations described by equations (2.7a,b) and (2.8a,b) to obtain {q}bm from {f}bm,

and then evaluate kbi w per method I. Note that {f}bm may be functions of the mean

shaft loads, bearing preloads, and shaft and casing compliances depending on the

configuration and flexibility of the rotating mechanical system. If the bearing system is

statically determinate, then {f}bm may be computed explicitly in terms of {f}sm and

preloads using the force and moment equilibrium equations. Conversely for an

indeterminate system, appropriate field equations for the shaft and casing plate are

needed in addition to the equilibrium equations to obtain {f}bm which must also include

shaft and casing compliances. Calculations of {f}bm and {q}bm in this case are

simultaneous, which may be extensive especially when the system is very flexible, and

may even require discretization using finite element or lumped mass technique.

However, in many real machines the in-plane stiffness of the casing plate which

supports most of the mean bearing load is much higher than the bending stiffness of the

shaft. Hence the casing in-plane stiffness term may be neglected without contributing

any large error to {f}bm [18,31]. And only the Euler's beam equation for a statically

indeterminate shaft is used along with the nonlinear bearing load-displacement equations

(2.7a,b) and (2.8a,b).
32

Method I is computationally direct and needs no discussion. But method II deals

with as many as 10 N nonlinear algebraic equations for N bearings if the casing

flexibility is neglected. One must choose an appropriate numerical method as the

nonlinear algebraic equations must be solved iteratively [36,37]. In addition, the

available numerical methods need a prior knowledge of the approximate location of the

solution vector being sought and hence one must be careful in interpreting the numerical

results. In this study, we adopted the Newton-Raphson method for its good

convergence characteristic [36,37]. To implement this method, equation (2.6) for each

bearing is rearranged as

{t{ } {sinai}f°t H1 Mxbm - _rj Qj sino_j =

H 2 = Mybm j -c°slltj 0

i i: Fromojc°s°jsin :

H 5 Fzb m J sin otj

(2.12a)

(2.12b)

where H1,H 2 ..... H 5 are functions defined for computational reasons. For an

indeterminate system, there are additional functions H6,H 7 ..... H V from the field

equations. Using Taylor's series, any function H k in equations (2.12a,b) can be

expanded about the solution vector X = {q }bm for a statically determinate system and X

T T IT = [ {q} bin' {f} bm for a statically indeterminate system as follows by neglecting

second and higher order terms.
33

v _H k

H k(X+SX) _- H k(X)+ Z_ _Xj

j J

; k=1,2,3 .... V (2.13)

The solution for the incremental vector 5X can be obtained by setting Hk(X+SX) = 0

per equations (2.12) and (2.13) which yields a set of linear algebraic equations. This

vector 5X is added to the previously computed vector X given by Hk(X) = 0 for the

next iteration until the convergence criterion, say that fX is within a specified tolerance,

is satisfied. Our proposed numerical scheme can be summarized as follows: (i) guess

bearing displacement vector {q} bm and/or load vector {f}bm, (ii) compute fX and check

against a specified tolerance, (iii) add fix to the previous solution vector X and repeat

steps (i) and (ii) until the convergence criterion is satisfied. We have found that a few

initial guess trials are required in most cases to obtain reasonable results.

2.7 VALIDATION OF PROPOSED MODEL

In order to validate our theory we compare the translational stiffness coefficients of

the proposed bearing matrix [K]b m with published analytical and experimental results

[ 14,20,24]. First we apply our theory to predict the nonlinear axial kbz z = kbzz(Szrn)

and radial kbr r = kbrr(frm) stiffnesses as shown in Figure 2.6. Our predictions are

found to be within 2% of Gargiulo's [14] formulas which are commonly used for both

ball and roller bearings.

For the second example case, we consider the ball bearings used by Kraus et

a1.[24] for an in-situ determination of the beating stiffness. Using their bearing design

parameters, we compute radial stiffness coefficient kbr r as a function of the axial preload

Fzb m. Excellent comparison between theory and experiment is seen in Figure 2.7.
34

E

° .,-.I

"-d

oF..1

.<

13

Proposed Theory

Gargiulo

L i

)00 0.025 0.050

Axial Deflection (mm)

¢a

i

o_

0.6

Proposed Theory

o Gargiulo

L.I I'1

0.4 _R u u

oiler

I

0.2

%00 .... oo_5.... 0.050

Radial Deflection (mm)

Figure 2.6 Comparison between the proposed theory and Gargiulo's formulas [14] for

axial kbz z and radial kbr r stiffness coefficients of ball and roller bearings.
35

120

s0

ID

-- 40

°w,,_

/

/

Proposed Theory

O Experiment (0 rpm)

lx Experiment (1000 rpm)

I |

0 20 40 60

Axial Preload (kg)

Figure 2.7 Comparison between the proposed theory and the experimental results of

Kraus et a1.[24] for kbr r as a function of the mean axial preload.
36

Finally, wecompareour resultsfor thenonlinearradial stiffnesskbrr with those

reportedearlierby White [20] for bothball androllerbeatings.We notediscrepancies

in Figure2.8betweenourtheoryandWhite'sresults.In orderto explainthesewenow

A

define k brr using the finite difference approximation which was also used by White:

A

k brr -- AFbrm / ASrm -_ Fbrm / (Srm-rL). Now a good match is evident in Figure 2.8

A

between our k brr values and the data given by White. However, the correct formulation

is obviously given by our proposed theory which is based on the analytical partial

derivatives kbr r = bFbr m / _Snn as the displacement _Srmmay be large.

2.8 PARAMETRIC STUDIES

The proposed matrix [K]b m includes a coupling between the casing flexural

motion and shaft bending motion which is reflected by some of the dominant offdiagonal, kbx0y, kby0x , kbz0x and kbz0y, and rotational diagonal, kb0x0 x and kb0y0y ,

stiffness coefficients; these are labeled as 'coupling coefficients' for discussion

purposes. Such stiffness coefficients are investigated further by varying preloading

conditions and unloaded contact angle ty_o for both ball (set A) and roller (set B) beatings

whose design data are listed in Table 2.1.

The coupling coefficients given a constant mean radial displacement _rm (radial

preload), as shown in Figures 2.9 and 2.10 for both ball and roller bearings

respectively, are found to increase as t_o increases and reach a maximum when t_o is

near 90 °. On the other hand, the radial translational stiffness coefficients in the x and y

directions are found to decrease as _ increases. These observations imply that for deep

groove ball type or straight roller type bearing (s o = 0 °) the radial stiffness coefficients

kbr r are dominant, but for angular contact ball type or taper roller type bearing (ct o > 0 °)
37

r,o

IO

300

200

100

0

Proposed Theory

.......... Estimated

[] White

.43.... f _ Cl _-- IO----C1-

i

0.000 0.025

Radial Deflection (mm)

0.050

r._

r_

"d

4o0

(a)

200

d

/

Proposed Theory

.......... Estimated

[] White

0

0.000 0.025 0.050

Radial Deflection (mm)

(b)

Figure 2.8

A

Comparison between the proposed theory kbr r, estimated k brr, and White's

analytical results [20]. a. Ball bearing, b. Roller beating.
38

Table 2.1 Design parameters for typical ball and roller bearings used for parametric

studies

Parameters Set A (ball type) Set B (roller type)

Load-deflection exponent n

Load-deflection constant K n (N/m n)

Number of rolling element Z

Radial clearance rL (mm)

Pitch radius tt(mm)

Ao (ram) t

3/2 10/9

8.5 E9 3.0 E8

12 14

0.00005 0.00175

19.65 21.25

0.05

t Unloaded distance between inner and outer raceway groove curvature centers (see

Figure 2.3)

tt Equivalent to rj for roller bearings and rj-Ao/2 for ball bearings given in equation

(2.3)
39

20O

-g

0

0

• - . • .

45 90

4O

= 20

r_

m

i • i -

00 45 90

t_

i

0 45 90

Unloaded Contact Angle (deg)

Figure 2.9 Dominant stiffness coefficients of ball bearing set A for 0 ° < txo < 90 ° and

given a constant mean radial bearing displacement 3xm = 0.025 mm.
40

¢._

_D

0.25

_ [ __-" ..... "".-..,,.,k bxz I

0.00

0 45 90

_-. 0.2

0.1

0.0

0 45 90

Z

o

4

0

0 45 90

Unloaded Contact Angle (deg)

Figure 2.10 Dominant stiffness coefficients of roller bearing set B for 0 ° < O_o _<90 °

and given a constant mean radial beating displacement 5xm -- 0.025 mm.
41

the coupling terms are more significant. Note that in Figure 2.10, all the stiffness

coefficients are zero at oto = 90 ° for the roller type. This is due to the fact that in the

thrust roller bearing, radial flanges are included to resist the roller motion in this

direction which is not modeled here, and hence these stiffness coefficients must vanish.

In addition, thrust roller beatings are designed to carry axial loads [18,31]. On the other

hand, ball bearings have finite stiffness coefficients at o_o = 90 ° due to the curvature of

the raceway which provide some resistance to the radial preloads. In general, the trends

in both ball and roller bearing stiffness properties are similar when each is subjected to

mean radial displacement or preload.

In the case when the bearings are subjected to mean axial displacement (axial

preload), as shown in Figure 2.11 for the ball type and Figure 2.12 for the roller type,

the number of nonzero stiffness coefficients are less than those seen for the radial

preload only. Again, it is observed that both ball and roller bearings display similar

trends. Over mid to high o_o values, the coupling coefficients are found to be

significant. The translational stiffness coefficients are relatively constant except for the

axial stiffness which increases as oto increases. This is expected due to the inclination of

the rolling element line of contact from the x-y plane which increases elastic support in

the z-direction. At o_o = 0 °, all the stiffness coefficients for roller bearings are zero as

there is no constraint in the axial direction. In real beatings such a constraint is provided

by the axial flanges [18,31], however this bearing is not designed to carry any axial

preload.

Results for the misalignment in ball and roller bearings simulated by specifying a

mean bearing angular displacement 13ym are shown in Figures 2.13 and 2.14

respectively. The dominant stiffness coefficients are the same as those seen for the
42

0

0 45 90

,-- 0.2

i 0.1

0.00

• • | • • • • °

45 90

v

t_

._,-i

e0

0 45 90

Unloaded Contact Angle (deg)

Figure 2.11 Dominant stiffness coefficients of ball bearing set A for 0 ° < a o _<90 ° and

given a constant mean axial beating displacement _Szm= 0.025 mm.
43

O0 90

"_ 0.4

kbOyOy ; kbOxO x

o

0.2

r_

0.0 0 45 90

Z

o

8

0

0

i i i i • I i

45 90

Unloaded Contact Angle (deg)

Figure 2.12 Dominant stiffness coefficients of roller bearing set B for 0 ° < txo < 90 °

and given a constant mean axial bearing displacement _zm = 0.025 mm.
44

E

Z

o,)

i1)

° ,...i

.,,...a

o.o

e-

_o

Z

Z_

t/l

e_o

e-,

Z

Z_

r._

_O

ee_

cD

1.8

0.9

k bzz

0.0

0 90

0.5

0.4

0.3

k bxx

kbyy

............... . . . . . ._kbxz

45

k

bOyOy

kbo xO,,

.2 , i , • , |

0 45 90

0

0

......... kbxOy ; kbyO,x ,

45 90

Unloaded Contact Angle (deg)

Figure 2.13 Dominant stiffness coefficients of ball bearing set A for 0° < oto < 90 ° and

given a constant misalignment _ym = 0.015 rad.
45

_' 0.3

Z

¢,¢)

o

m

45

90

05

0.I

o.oo

45

90

%

45

90

Unloaded Contact Angle (deg)

Figure 2.14 Dominant stiffness coefficients of roller bearing set B for 0 o < % < 90 °

and given a constant misalignment j3ym = 0.015 rad.
46

radial preload case. For ball bearings, most of the stiffness coefficients remain constant

for 0 ° < ct o < 90 °. On the other hand, the stiffness coefficients for roller bearing have

trends similar to those found for the radial preload cases.

From the detailed parametric studies, it is concluded that the nature of [K]bm is

dictated by the bearing type, 0to and preloads. Also, the coupling coefficients are not

negligible in most cases as assumed previously by many investigators.

2.9 CONCLUSIONS

Results of Section 2.8, which show similar trends for some of the cases, imply

that there may be a systematic approach to characterize the proposed beating stiffness

matrix [K]b m. From the kinematic and geometrical considerations, it is always possible

to impose any bearing displacement vector {q}bm which denotes relative rigid body

motions between the inner and outer rings as long as the roiling element is still within

the elastic deformation regime. On the other hand, an arbitrary application of {f} bm may

not produce a singular displacement response from the bearing due to its kinematic and

geometrical constraints. Hence, we compute [K]b m and {f}bm by systematically

varying {q}bm. The results of all possible forms of [K]bm are listed in Table 2.2 and

2.3 for ball and roller bearings respectively. Also included here are the current beating

models which are based on the translational spring descriptions; these models do not

show any coupling. Note that the exact values of the stiffness coefficients are not given

as these depend on specific parameters; therefore only the dominant kbi j terms are listed

for all possible bearing load configurations along with the corresponding {q}bm and oto.

Also, note that not all combinations of the bearing loads are possible which complicates

bearing stiffness calculations further, especially for the numerical method II. Tables 2.2
47

Table 2.2 Comparison between the proposed and current ball bearing stiffness

coefficients.(p = x, y; i = x, y but i ¢ p)

Mean Mean bearing displacement

bearing_-t

loads Oto-=0° 0°<O_o<90 ° O_o--90 °

Dominant stiffness coefficients

A

current t proposed ttt

Fpm 8pro - - kpp

Fzm 8zm _zm - kpp,kzz

Fzm - _ 8zm kzz

Mpm [3pm - _ _

Fzm,Mpm - _ 8zm,13pm kzz

Fxm,Fym 8xm,Sy m - - kpp

Fpm,Mpm _pm,_pm - - kpp

Mxm,Mym [ipm,_pm -

Fpm,Fzm, 8pm,Szm, 8pm,Szm, _Spm,Szm, kpp,kzz

Mim _im _im _im

Fzm,Mxm, 5zm,[3xm, 8zm,[_xm, _zm,_xm, kpp,kzz

Mym _ym 13ym _ym

{f] m combinations of {q }m kpp,kzz

kxx,kyy,kzz,k0x0x,k0y0y,kz0i

kxx,kyy,kzz,k0x0x,k0y0y,

kx0y,ky0x

kxx,kyy,kzz,k0x0x,k0y0y

kxx,kyy,kzz,k0x0x,k0y0y,kiz

kxx,kyy,kzz,k0x0x,k0y0y,kz0p

kxx,kyy,kzz,k0x0x,k0y0y,

kxy,k0x0y,kz0x,kz0y

kxx,kyy,kzz,k0x0x,k0y0y,

kx0x,ky0y,kpz,kz0i

kxx,kyy,kzz,k0x0x,k0y0y,

kxy,kxz,kyz,k0x0y

kxx,kyy,kzz,k0x0x,k0y0y,

kx0y,ky0x,kpz,kz0i

kxx,kyy,kzz,k0x0x,k0y0y,

kxy,k0x0y,kz0x,kz0y

all non-zero except 0 z terms

l- Ideal boundary condition models used to describe the bearing are not tabulated.

tt Here the subscript b which implies bearing has been omitted for brevity.

ttt All terms associated with 0 z are zero because of the free rotation about the z axis.
48

Table2.3 Comparisonbetweentheproposedandcurrentroller bearingstiffness

coefficients.(p= x, y; i = x, y but i _:p)

Mean Meanbearingdisplacement Dominantstiffnesscoefficients

bearing_t , , , , ,

loads _o---0 ° 0°<0_o<90 ° 0Co=90o current t proposedttt

Fpm _pm - - kpp kbp p

Fzm - _zm - kI_zz kxx,kyy,kzz,kOxOx,kOyOy,

kxOy,kyOx

Fzm - - _zm k_ kzz,k0x0x,k0y0y

Fzm,Mpm - - 8zm,_pm kzz kzz,k0x0x,k0y0y,kz0 p

Fxm,Fym 8xm,Sy m - - kpp kxx,kyy,kxy

Fpm,Fzm - _pm,_zm - kpp,kzz kxx,kyy,kzz,k0x0x,k0y0y,

Mim _im kx0y,ky0x,kpz,kz0i

Fzm,Mxm , - _ 5zm,_xm , kzz kzz,k0x0x,k0y0y,k0x0y,

Mym _ym kz0x,kz0y

{f} m combinations of {q}m kpp,kzz all non-zero except 0 z terms

t Ideal boundary condition models used to describe the bearing are not tabulated.

tt Here the subscript b which implies bearing has been omitted for brevity.

ttt All terms associated with 0 z are zero because of the free rotation about the z axis.
49

and 2.3 should provide some insight to the solution of the nonlinear algebraic bearing

load-deflection equations which requires a prior knowledge of the type of solution

being sought as outlined earlier. In most practical problems, mean bearing loads are

typically known. This knowledge can be combined with Table 2.2 or 2.3 to formulate

the nonlinear load-deflection equations in the simplest form by deleting all of the zero

displacement terms.

Tables 2.2 and 2.3 show that the coupling coefficients kbx0y, kby0x, kbz0x, kbz0y,

kb0x0 x and kb0y0y are found to be dominant in most of the ball bearing cases, and only

in some of the roller bearing cases. This is essentially due to the curvature of the

raceway in ball bearing which invariably causes the rolling element to orient itself such

that 0 ° < o_j < 90 ° which generates ball loads in the z direction as well. However, in the

roller beating case where 0tj = O_o, the same phenomenon does not occur when 0co = 0 °

or 90 °, and the coupling coefficients are seen only when a.o _ 0 ° or 90 °. In fact for the

0 ° and 90 ° unloaded contact angle cases, the stiffness coefficients associated with x and

y directions and those associated with the z, 0 x, and 0y directions do not exist

simultaneously; the former is dominant when o_o = 0 and the latter prevails when 0_o =

90 °. Another ease of interest here is the ease when bearing loads are complex as given

by the last row in Tables 2.2 and 2.3 where all of the bearing stiffness coefficients

unrelated to the rotational degree of freedom 0 z exist. Solution to these cases may

require a large number of iterations.

In summary, we have developed a comprehensive bearing stiffness matrix from

the basic principles which includes all possible rigid body degrees of freedom of a

bearing system. This matrix has been validated partially using several analytical and

experimental examples. Further validation of [K]bm is not possible as coupling
50

coefficients are never measured [24,30]. Nonetheless, our theory is general in nature

and is applicable to even those configurations which may be different from the generic

case shown in Figure 2.1. Further research is required to incorporate tribological issues

[32,33] in this formulation. However the proposed stiffness matrix in its present form,

unlike the current models, is clearly capable of explaining the nature of vibration

transmission through bearings - this is the subject of Chapters III and IV of this

report, which will also include further comparisons between theory and experiment.
CHAPTER III

BEARING SYSTEM STUDIES

3.1 INTRODUCTION

Current bearing models [10-15] can not explain how the vibratory motion may be

transmitted from the rotating shaft to the casing and other connecting structures in

rotating mechanical equipment. For instance, experimental results [9,16,17] have

shown that casing plate motion for a system similar to Figure 3.1 is primarily flexural or

out-of-plane type given only the bending motion on the shaft. Using existing vibration

models, only in-plane type motions on the casing plate are obtained. Such limitations

associated with current bearing models have been discussed thoroughly in Chapter II of

this report. Also in Chapter II, a new mathematical model for the precision rolling

element bearings has been developed in order to clarify this issue qualitatively and

quantitatively.

This study extends the proposed bearing formulation and demonstrates its

superiority over the existing models in vibration transmission analyses. A schematic of

a generic system with a flexible shaft rotating at constant speed _ z, flexible casing and

mount is shown in Figure 3.1. The shaft is supported by a rolling element bearing

which is modeled by a stiffness matrix [K]b m of dimension 6 as proposed in Chapter II.

The excitations at the rotating shaft are given in terms of an alternating load vector

{f(t)}sa = {Fjsa(t),Tjsa(t) }T = {f(t)}s- {f}sm; j=x,y,z, where Fjsa(t) and Tjsa(t) are the

51
52

0

Uxsa

Fxsa

ysa

x

_0xsa

Flexible Rolling

Element Txsa _ Bearing

F_a_Uz_ _ _z

_0 I _]-_z=t}zsm zsa zsa

Y Flexible

Casing

llllllllllllllllillll IIIIIIIID

IIIFlexibleMountIIIII

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIS

////////////////,4- _5..///////////.

Figure 3.1 Schematic representation of the vibration transmission problem. Here the

flexible shaft is subjected to alternating forces Fisa(t ) and torques Tisa(t )

where j = x, y or z, is the direction and subscrii_t a implies alterna_ting.

Also, 0 is the angular displacement and u is the translational displacement.
53

alternating force and torque respectively, {f(t)} s is the total load vector of dimension 6,

{f}sm represents the mean load vector, and superscript T implies the transpose. In the

vibration analysis, {f}sm and bearing preloads are not included as they do not appear in

the governing equations of the linear vibration model but are used for computing [K]bm.

The effect of bearing coupling coefficients, which are off-diagonal and rotational

diagonal terms of [K]bm as described in Chapter II, on the eigensolution, forced

vibration, and vibration transmission through bearings is evaluated. Our theory will be

illustrated and validated through 3 physical system example cases; experimental

verification is also included.

3.2 LITERATURE REVIEW

The existing bearing models which assume either ideal boundary conditions [10-

12] for the shaft or translational stiffness elements [13-15] have already been discussed

in Chapter II. Various formulas for estimating translational stiffness coefficients

commonly used by researchers have been compared with our proposed [K]bm

formulation. These simple bearing models are widely used in vibration models of the

rotor dynamic systems, which typically exclude casing and mount dynamics, to calculate

critical speeds, responses due to shaft excitations such as mass unbalance and gear

transmission error, and dynamic stability [10-15]. In most of these cases, the vibration

transmission through bearings is never or not the primary issue, and thus the

beating models tend to be simplified. None of the current models studied [20-22,24]

can fully explain vibration transmission through bearings in systems similar to

Figure 3.1. In 1979 White [20] evaluated the roiling element bearing vibration transfer
54

characteristics using a two degrees of freedom (DOF) vibration model of the system

shown in Figure 3.1. His formulation is based on only the radial bearing stiffness

coefficient kbr r . He concluded that an increase in preload increases kbr r and system

natural frequencies. He also found that the effect of bearing nonlinearity is negligible at

higher preloads. In 1987 Kraus et al. [24] proposed a single degree of freedom model

for a similar physical system (with a very compliant mount) to estimate kbr r from

measured vibration transmission spectra. In both of these studies, the coupling

coefficients of [K]bm are not included.

In 1982 Rajab [21] philosophically proposed a bearing stiffness matrix which

consists of kbr r, kbr 0 and kb00 coefficients. Some of the key features of his model are

also summarized in Reference [22]. This model is in fact a subset of our [K]bm as

shown in Chapter II of this report. He incorporated his bearing model in a

system study using a commercial structural synthesis program [38]. However,

based on our study we have inferred that he incorrectly synthesized the system

model given the plate experimental modal data, shaft finite element model and

analytical bearing model. Moreover, an error was found when he converted kbr0

and kb00 coefficients to "effective stiffness coefficients" which he claimed to

couple the shaft bending motion to the plate out-of-plane motion. Also, this

method excludes the bearing rotational degree of freedom, which from our study

was found to be important.

3.3 ASSUMPTIONS AND OBJECTIVES

Linear discrete vibration models of the generic system shown in Figure 3.1 are

used to incorporate [K]bm and to characterize the vibration transmission through rolling
55

elementbearings.The stiffnesscoefficientsof [K]bm are evaluated using the analytical

expressions presented in Chapter II of this report. Effect of the gyroscopic

moment on the shaft dynamics is not included. Since the bearing system is statically

indeterminate, the direct stiffness formulation technique is used to obtain the system

governing equations as opposed to the flexibility formulation. The governing equations

for the system vibration model can be given in the matrix form as

[M] {_(t) } a + [C] {gl(t) } a + [K] {q(t) } a = {f(t) } a (3.1)

where [M], [C] and [K] are the system mass, damping and stiffness matrices

respectively, and {q(t)} a and {f(t)} a are defined as the generalized alternating

displacement and applied load vectors respectively. Due to the linearity of the vibrating

system, mean shaft loads {f}bm and preloads do not directly affect the dynamic

response of the rotating system and hence are excluded from equation (3.1). However,

{f}bm and bearing preloads are assumed to be constant to ensure a tirne-invariant [K]bm

matrix which depends only on these mean loads or on the mean deflection operating

points. Accordingly, only the alternating shaft loads {f(t)}sa in Figure 3.1 which

represent typical machine excitation due to the kinematic errors, mass unbalances and

torque fluctuations are included in the forced vibration problem. The energy dissipation

associated with the rolling element bearings is assumed to be an energy equivalent

viscous damping matrix [C] b = g [K]bm where _ is the Rayleigh damping matrix

proportionality constant. Dynamic instabilities due to the oil whirl phenomenon and

asymmetry of rotating elements [11,12] are clearly beyond the scope of this study and

hence are not considered here.
56

Thespecificobjectivesof thischapterareto: (i) incorporatetheproposedbeating

matrix [K]bm,developedin ChapterII of this report,in the linear discretevibration

modelof therotatingmechanicalequipmentasdescribedby equation(3.1)usingboth

the lumpedparameteranddynamicfinite elementmethods,(ii) evaluatethedynamic

stability of theproposedbearingsystemmodelusingtheLiapunov'ssecondmethod,

(iii) calculateeigensolutionand forced harmonic responses,and predict vibration

transmissionthroughrolling elementbeatingsfor threeexamplecases,(iv) demonstrate

the advantagesof our formulationover theexisting modelsby Krauset al. [24] and

White[20], and(v) validatetheproposedtheoryby comparinganalyticalpredictionwith

experimentaldataonananalogoussystem.

3.4 SYSTEM GOVERNING EQUATIONS

3.4.1 Method A: Lumped Parameter Model

The proposed bearing matrix [K]b m can be easily implemented in equation (3.1).

Note that, the coupling coefficients of [K]bm provide the capability to predict casing

rigid body angular 0jca(t), j=x,y,z, and translational Ujca(t ) motions given only the

unidirectional transverse shaft forces. Hence we can couple the shaft motions to the

motions of a casing of a system similar to Figure 3.1 but with rigid shaft and rigid

casing using a lumped parameter model. The bearing preloads can now be included in

the mean shaft load vector {f} bm by a direct vector addition as the rigid shaft can be

assumed to be a single lumped mass for this purpose. An alternating displacement

T T

vector {q(t)}a = {{q(t) } sa, {q(t)} ca is defined where {q(t)}sa = {Ujsa(t),0jsa(t)} T

iT,

= }T and {q(t)}ca {Ujca(t),0jca(t) , j=x,y,z, are the shaft and casing alternating
57

displacementvectorsrespectively.Thegoverningequationsof motionfor thisgeneric

vibrationmodelwith DOF= 12aregivenby equation(3.1)with

[ ] [ ]

[0] [K ] bm bm

[ M] s ; [K] = _ [K ] bm [ K ] bm + [K] (3.2a,b) [M] = [0] [ M] c v

= _ {f(t)} sa} (3.2c,d) [C] = cr [El ; {f(t)}a [ {0}

[K]

bm

kbxx kbxy

k

byy

symmetric

k bxz

k byz

kbzz

k bx0 x k bx0 y 0

kby0x kby0y 0

kbz0x kbz0y 0

kb0_0 x kb0x0y 0

k i:_ yOy 0

0

(3.2e)

where the stiffness matrices [K]bm and [K] v pertain to the bearing and mount

respectively, and the matrices [M] s and [M] c are diagonal shaft and casing mass matrices

respectively; each matrix of dimension 6. Specific examples of this method along with

the eigensolution and forced response studies will be presented in Sections 3.7 and 3.8.

3.4.2 Method B: Dynamic Finite Element Formulation

Consider the dynamic finite element method of incorporating [K]bm in equation

(3.1) especially when shaft and casing plate are elastically deformable over the

frequency range of interest. This method is different from the lumped parameter

formulation of Section 3.4.1 which assumes non-compliant shaft and casing. For
58

example, if the flexible casing plate is considered to be very large compared to the

bearing dimensions, then the beating nodal point on the shaft can be coupled to only one

bearing nodal point on the plate as shown in Figure 3.2a. Accordingly, the present form

of [K]bm is implemented in the finite element model as a generalized stiffness matrix like

the lumped parameter model. On the other hand, when the flexible casing plate

dimensions are f'mite and of the order of bearing dimensions, then several bearing nodal

points are considered as shown in Figure 3.2b. The discretization philosophy here

assumes that a relative displacement vector, given by the difference between the

averaged displacement vector of bearing nodal points on the plate and the displacement

vector of a bearing nodal point on the shaft, is equivalent to the actual rigid body bearing

motion. Accordingly, we divide the bearing stiffness coefficients equally among all the

generalized stiffness elements connecting the bearing nodal points on the plate to a single

bearing nodal point on the shaft. In the limit, where all the bearing nodal points on the

plate are collapsed to a single nodal point, [K]bm is recovered as in the first method.

Our finite element formulation uses conventional structural elements typically available

in commercial software programs [39] - this will be illustrated in Section 3.9. Other

features of this method are similar to those discussed earlier in Section 3.4.1.

3.4.3 Other Methods

Alternate methods of incorporating [K]b m in equation (3.1) such as finite

difference which is similar to method B, flexibility, component mode synthesis and

transfer matrix formulations are also possible. In the flexibility formulation, the bearing

flexibility matrix can be obtained by inverting a subset of the bearing stiffness matrix

[K]bm s which excludes zeroes corresponding to 0 z angular direction from 1K]bm. In the
59

q}

Point

J

Flexible shaft

(a) Single Beating Nodal Point on the Plate

Fleble?'/////J

(b) Multiple Bearing Nodal Points on the Plate

Figure 3.2 Discretization method for implementation of [K]bm in finite element model

of a system similar to Figure 3.1. Here {q(t)}sa and {q(t)}ca are the

alternating shaft and plate displacement vectors at bearing location

respectively. Subscript a implies alternating component.
6O

transfer matrix method,the field matrix [T]b for a bearingcanbe easilyrelatedto

[K]bms.

{ q(t)} sa_ _{q(t)} ca

{f(t)}sa j =[ T] b[{f(t)}ca [I] [ K]- 1 ]

= bms (3.3a,b)

; [r] b [01 [I1

where {{q(t)}T,tf(t)}L} sa T and {{q(t)}T,tf(t)}T} ca T are now the state vectors

at bearing locations on the shaft and casing plate respectively. Equation (3.3) can now

be integrated with transfer matrices of the shaft and plate which are well documented in

References [40,41]. Direct application of these alternate methods [40-43] are beyond

the scope of this paper and are left for further research.

3.5 BEARING SYSTEM STABILITY

The stability of the proposed linear, non-gyroscopic model of a bearing system

similar to Figure 3.1, which is governed by equation (3.1) with {f(t)} a = {0}, can be

determined using several techniques such as Liapunov's stability method, RouthHurwitz criteria, or from the direct evaluation of system eigenvalues. Here the

Liapunov's second method is used for its simplicity when applied to such a vibration

model [44,45]. If the system matrices [M], [C] and [K] of equation (3.1) are always

symmetric and positive definite, then the system is asymptotically stable per Liapunov.

The first condition is directly satisfied since [M], [C] and [K] are symmetric. Further,

since [M] is diagonal and consists of only positive entries, it is clearly positive definite.

For [K], the positive definite test can be performed by evaluating its principal minor
61

determinants which is demonstrated here for the generic lumped parameter model.

Consider the decomposition of [K] given by equation (3.2) into a product of 3 matrices

-[I] i 0 l[ K'bm [I] [0] [K]v ]t ii [0] [I] I

where the square submawix [0] of the appropriate dimension consists of only zero

entries, [I] is an identity matrix of the same dimension, and [K]b m and [K] v have been

defined in Section 3.4.1. The determinant of [K] is the product of the determinants of

the three matrices on the right hand side of equation (3.4)

= ItKavI[tKlbm[ (3.5)

If [ K]wj I > 0, w,j = 1,2 ..... P and P = 1,2 .... ,12, [K] is positive definite. Other

principal minor determinant with P<12 can be obtained by excluding the stiffness

coefficients which are not entries in the principal submatrix of equation (3.5). Since

I [ K v ] I = kvx'kvy'kvz'kv0x'kv0y" kv0z > 0, it implies that equation (3.5) is positive

only if [ K ] is positive. We may recall that [K]bm has zero entries in the last row bm

and in the last column corresponding to torsional 0 z angular direction which forces the

bearing system to be serrJdefinite. Now define a new matrix [K]bms of dimension 5 as

a subset of [K]bm with these zeroes excluded. If [K]bm s has positive principal minor

determinants, then this system is dynamically stable because it will consist of stable

oscillations superimposed on the mean shaft rotational motion Oz _: f2z(t). Further, it

follows that [C]bm s which is proportional to [K]bm s is also positive definite if [K]bm s is
62

positivedef'mite.Theresultingequationobtainedfromtheexpansionof thedeterminant

of [K]bmsin termsof its entrieskbwj, w,j=1,2,3,4,5,is givenasfollows in termsof the

stability functionsOj, j = 1,2,3,4.

[ K]bms [

2 04(55,34,45,35) {kb24O4(11,23,12,13)+kbl4O4(22,13,12,23)}+

2 04(44,35,45,34) {kb25O4(11,23,12,13)+kbl5O4(22,13,12,23)}+

2 04(24,35,25,34) {kbl4_4(25,13,23,35)+kbl5O4(12,34,24,13)} -

kbl5 kb22kb34_3(14,35,15,34)+ kb24kbl5 kbl4 03(23,35,33,25)+

2 kbl4 04(25,45,24,55) _4(13,23,12,33)+ { kbl4 kb25}2 kb33+

• 1(1,2)_2(45,34,35,44,55)+ _1(1,3) _2(45,24,25,44,55)+

kb15_1(2,3) O3(14,45,15,44) + kb11{ 04(24,35,25,34) }2 +

{kbl4} 2 { kb2303(23,55,35,25)-kb22 _1(3,5) } +

_1(4,5) { kbll O1(2,3)+ O2(23,13,12,33,22)} +

2 kbl5_4(24,45,25,44)_4(23,13,12,33)+

2 kb15O4(14,25,15,24)• 4(23,34,24,33) (3.6a)

01 (WI,WII ) = kbwlWi kbwilwii _ { kbwlWI I } 2 (3.6b)

02(wI,wII,WlII,WIV,WV) = 2kbwl kbwli kbwiu-kbwlv { kbw m } 2-kbw V { kbwll (3.6C)

03 (wI,WlI,WIII,WIV) = 2kbwl kbwii - kbwm kbwiv (3.6d)

04 (wI,wII,WIII,wIV) = kbwI kbwu - kbwlli kbwiv (3.6e)

where wj, j = I,II,III,IV,V, are the dummy variables and each wj may represent either a

single number or a set of two numbers in equation (3.6a). The principal minor

determinants of [K]bm s can also be derived from equation (3.6) by excluding the
63

appropriatestiffness coefficients which are not entries of the particular principal

submatrix. Hence,the beatingsystemis stableif eachprincipal minor determinants

derivedfrom equation(3.6) is positive. The stability of the proposedbearingmodel

given in Chapter II of this report can now be verified using these conditions.

Inequalitiesassociatedwith the stability criteria for thesemodelsaresummarizedin

Tables3.1and3.2for ballandroller bearingsrespectively.Theseinequalitiesarisedue

to thefact that weareyet to imposeany restrictionson thesestiffnesscoefficients. In

Chapter II of this report, these stiffness coefficients are given as functions of

bearingkinematicanddesignparameters,andhenceany coefficientcannot assumean

arbitraryvalue asit is relateduniquelyto othercoefficientsthroughtheseparameters.

Extensivenumerical studiesperformedover a wide rangeof theseparametershave

indicatedthatthebearingmodelsproposedin ChapterII areindeedstableprovidedthe

preloadsare sufficiently large to avoid the clearancenon-linearity. Figures 3.3-3.5

illustrateexamplesof thesebearingsystemstabilitystudiesfor precisionroiling element

bearingswhosedesigndataaregivenin Table3.3. In all of thesefigures,thestability

functions_j givenin Tables3.1and3.2arefoundto lie within thestableregion.

3.6 SYSTEM RESPONSE

Theeigensolutionof thelinear,non-gyroscopicundampedsystem,formulatedby

setting{f(t)} a= {0] and[C] = [0] in equation(3.1)givenby methodA or B, yieldsreal

valued natural frequenciestoj, j=1,2,3 ..... and the modal matrix [U] = [{_}1, {_}2,

.... {_}j .... ] for the stable system. Since the system is proportionally damped, the

modal damping ratio is _j = _ _2/2 and the damped natural frequency is j--
64

Table 3.1 Bearing system stability criteria for the proposed ball bearing model

(j=xorl, yor2; p=xorl, yor2 but p#j)

Proposed beating model t

see Table 2.2

Stability criteria tt

kxx,kyy,kzz,k0x0x,k0y0y,kz0p

kxx,kyy,kzz,k0x0x,k0y0y,kx0y,ky0x

kxx,kyy,kzz,k0x0x,k0y0y

kxx,kyy,kzz,k0x0x,k0y0y,kpz

kxx,kyy,kzz,k0x0x,k0y0y,kxy,

k0x0y,kz0x,kz0y

kxx,kyy,kzz,k0x0x,k0y0y,kx0x ,

ky0y,kjz,kz0p

kxx,kyy,kzz,k0x0x,k0y0y,kxy,

kxz,kyz,k0x0y

kxx,kyy,kzz,k0x0x,k0y0y,kx0y,

ky0x,kjz,kz0p

O1(3,p+3) > 0

O1(1,5) > 0 ; O1(2,4) > 0

always stable

Of(p,3) > 0

O1(1,2) > 0 ; O1(3,4 ) > 0 ;

k3301(4,5)+O2(4 5,3 4,3 5,4 4,5 5) > 0

kl lk2201(3,4)+kj32kppk44+k142k22k33 > 0;

O1(j,3 ) > 0 ;

{k3301(1,4)O1(2,5)O5( p p+3,j+3 j+3,p p,4 4,5 5)}-

{kj3 k 3 p+3} 2- {05( j j+3,p p,1 1,2 2,j+3 j+3)} > 0

O1(4,5) > 0 ; O1(1,2 ) > 0 ;

kll O1(1,2)+O2(2 3,1 2,1 3,2 2,3 3) > 0

Ol(p,j+3) > 0 ; O1(j,3 ) > 0 ;

kjjOl(3,p+3)+O2(3 p+3,j 3,j p+3,3 3,p+3 p+3) > 0

all non-zero except 0 z terms [K] bms[ > 0

t

tt

Here the subscript b which implies beating has been omitted for brevity.

Stability functions • 1 and • 2 are defined by equations (3.6b) and (3.6c) and • 5 is

given by • 5 ( wi,wli,Wlil,Wlv,w V ) = {kbw I} 2 kbwi I _ kbwrri kbwi v kbwv
65

Table 3.2 Bearing system stability criteria for the proposed roller bearing model

(j=xor 1, yor2; p=xorl, yor2 but p_j)

Proposed bearing model 1

see Table 2.3

Stability criteria it

kxx,kyy,kzz,k0x0x,k0y0y,kx0y,ky0x

kzz,k0x0x,k0y0y

kzz,k0x0x,k0y0y,kz0j

kxx,kyy,kxy

kxx,kyy,kzz,k0x0x,k0y0y,kx0y,

kyOx,kjz.kzOp

kzz,k0x0x,k0y0y,k0x0y,kz0x,kz0y

all non-zero except 0 z terms

always stable

O1(1,5) > 0 ; O1(2,4) > 0

always stable

O](3,j+3) > 0

O1(1,2) > 0

O](p,.j+3) >0 ; O](j,3) >0 ;

kjjOl(3,p+3)+O2(3 p+3,j 3,j p+3,3 3,p+3 p+3) > 0

O1(3,4) > 0 ;

k3301(4,5)+O2(4 5,3 4,3 5,4 4,5 5) > 0

I [K] bms[ > 0

t Here the subscript b which implies bearing has been omitted for brevity.

tt Functions O 1 and O 2 are defined by equations (3.6b) and (3.6c)
66

z

0

o

k_

,l,,,a

o _,,,4

°,,,_

,.Q

4OO

20O

1

STABLE

S 1(roller)

UNSTABLE MAR GINAL

Sl(ball)

!

0.025 0.050

Radial Deflection (mm)

(a)

Z

O

o

o_..I

2000

100(3

0

-IOOO 1

u.O00

l S3(ball)J

STABLE

S3(roller)

UNSTABLE

MARGINAL

!

0.025

Radial Deflection (mm)

0.050

(b)

Figure 3.3 Plot of stability criteria functions for ball and roller bearings subjected to

mean bearing radial deflection _Sxm. (a) S1 = (I)1(2,4) and $2 = _1(1,3).

(b) $3 = kbl 1 (/)1(3,5) + (I)2(35,13,15,33,44).
67

¢'t

Z

O

"___,

°_..i

5o0 1'

STABLE

250

0

UNSTABLE

I

-250 _ '

t).000

S 1, $4 (roller) ....

sl, s4 Coal0

MAR GINAL

|

0.025 0.050

Axial Deflection (mm)

Figure 3.4 Plot of stability criteria functions for ball and roller bearings subjected to

mean bearing axial deflection 8zm. Here, S1 = _1(2,4) and $4 = _1(1,5).
68

Z

0

o

.,,._

1000

STABLE S2(ball)__/S2(r°ller)

/ / Sl(roller)

sl(t_) L/

UNSTABLE MAR GINA

-1000

u.u000 0.0025 0.0050

Angular Deflection (rad)

(a)

Z

O

.t....a

°....q

,.Q

2O00

1000

l _S3CoaU)

STABLE

S3(roller)

UNSTABLE _ MARGINAL

,

-1000 '

u.u000 0.0025 0.0050

Angular Deflection (rad)

(b)

Figure 3.5 Plot of stability criteria functions for ball and roller beatings subjected to

mean angular misalignment _ym. (a) S1 = O1(2,4 ) and $2 = O1(1,3 ). (b)

$3 = kbl 1 O1(3,5)+O2(35,13,15,33,44)
69

Table3.3 Designparametersfortypicalballandrollerbearingsusedfor systemstudies

Parameters Ball type Rollertype

Load-deflectionexponentn

Load-deflectionconstantKn(N/mn)

Numberof rolling elementZ

RadialclearancerL (mm)

Pitchdiameter(mm)

Ao(ram)t

Unloadedcontactangle¢zo

3/2 10/9

1.0E9 1.0E8

12 14

0.00005 0.00175

40.05 38.00

0.05

40° 15°

t Unloadeddistancebetweeninnerandouterracewaygroovecurvaturecenters
70

given by _jd = _j -- _j . Free vibration response due to the initial conditions is

not considered as only the steady-state particular solution corresponding to sinusoidal or

periodic load vector {f(t)} a is of primary interest. Define excitation by the Fourier series

expansion as {f(t)}a = _ {f}ap ei°_pt where COp = P0_o, coo is the fundamental

P

frequency, and {f}ap is the complex Fourier coefficient load vector. The steady-state

particular solution {q(t)}a is given by the normal mode expansion technique [40,42,43]

as

{q(t) } a = [U] I_p {_t} p

i_pt) {_}T{ f}ap

e ;_JP= 0 2-j 0 2+ i2_j0_jC0p

; p=l,2... (3.7a,b)

An alternate approach would be to assume the harmonic solution for the alternating

displacement as {q(t)} a = Z {q}ap ei°pt- Substituting this and {f(t)} a definition into

p

equation (3.1), we get

Adj [- o_2[M] + _L[K] 1 " _k=l + io¢.Op (3.8a,b)

{q}ap I [_ C0p

where the operator Adj refers to the adjoint of the dynamic stiffness matrix. Features of

this method are summarized in References [40,42,43]. Since the vibration transmission

across the beating is the primary issue, we now define sinusoidal load transmissibility

R(pc0) terms between two arbitrary locations I and II as
71

afwh,fjii (_p) = [fwla(C0P)[ [fjHa(°_p)[

; w,j-- 1,2 ..... 6 (3.9)

where fwIa and fjlIa are components of the dynamic load vectors at two arbitrary

locations I and II respectively. The accelerance A(O_p) and mobility V(0_p) transfer

functions with motion at location I due to an alternating force or torque fjlIa applied at

location II on the shaft are

[ Clwia (03p) [ (3.10a)

AqwI.'f_,(_P)= [fjlIa(C°p)[

1 A(_p) ; w,j= 1,2 ..... 6 (3.10b)

V(COp)- irOp

where /i wI a is a component of the acceleration vector at location I.

response functions can also be defined in a similar manner [40,42,43].

Other frequency

3.7 EXAMPLE CASE I: RIGID SHAFT AND PLATE SYSTEM

3.7.1 Vibration Models

Consider the mechanical system shown in Figure 3.6a which is assumed to be

freely suspended or softly mounted such that [K] v = [0]. A ball bearings (see Table

3.3) with constant axial preload is supporting a short rigid shaft subjected to a mean

torque Tzs m _=Tzsm(t) and a sinusoidal radial force Frsa(t) = Frsal e ic°°t applied very

close to the bearing. A lumped parameter model with DOF = 12 is proposed in Figure

3.6b. Conversely, the same system has also been analyzed by Kraus et al. [24] using a

simple vibration model with DOF = 1 as shown in Figure 3.6c with only kbr r
72

(a)

Fxsa(t)

Y

X

Z

Rigid Casing

Ball Bearing

Rigid Shaft

Free boundary condition

(b)

Rigid Shaft {

Ball Bearing {

Rigid Casing {

{f(t)}sa_ _{fft)}sm

I

DOF = 6

DOF = 6

(c)

Rigid Shaft {

Ball Bearing {

Rigid Casing {

S fxsa (t)

[ ms ] --- ¢.Uxsa(t)

[ mc 1----_ u xca(t)

Figure 3.6 Example case I: freely suspended rigid shaft, ball bearing and rigid plate

system subjected to alternating radial force Frsa(t ) applied at the shaft. (a)

Physical system. (b) Proposed multi-degree of freedom vibration model

with DOF = 12. (c) Simple model by Kraus et al. [24] with DOF = 1.
73

coefficient. Thebearingstiffnessmatrixfor anaxiallypreloadedball bearingin Figure

3.6b hasnon-negligible stiffnesscoefficients kbxx, kbyy,kbzz, kb0x0x, kb0y0y,kbx0y

andkby0xwhich arefunctionsof themeanaxial preloadasgivenin ChapterII of this

report. The system matricesof equation (3.2) can be modified for this case by

suppressingother bearing stiffnesscoefficients and [K]v. It can beeasilyobserved

from equations(3.1)and(3.2)that5 setsof uncoupleddifferentialequationsexist. The

simplestthreesetsarehomogeneousand pertainto therigid body torsionalmotions

0za(t)of theshaftandcasing,andaxialvibrationUza(t)of theshaft-casingsystemwhich

areof no interesthere.Theremainingtwo setsarealmostidenticalandassociatedwith

either{Uxa(t),0ya(t)}Tor {Uya(t),0xa(t)}Tdegreesof freedomfor rigid shaftandcasing.

If thecoordinatesystemis chosensuchthatFrsa(t)line of actioncoincideswith thexaxis, then the steady-statesolution to the set of differential equationsin terms of

{Uya(t),0xa(t)}T is trivial. Hence,the problemreducesto a semi-definitevibration

systemwith DOF = 4. Accordingly,rewrite [M]s, [M]c and[K]bmin equation(3.2)in

termsof thedisplacementvector{q(t)}a = {Uxsa(t),0ysa(t),Uxca(t),0yca(t) }Tas

Eros01Lmc 0] [M] = ; [M] = ; [K]

s 0 I s c 0 I c

bm

kbxx kbx0y

k k

bX0y b0y0y

(3.11)

First two eigenvalues corresponding to the rigid body motions in x and 0y directions are

zero. The dimension of equation (3.11) is further reduced to DOF = 2 by defining

relative motions 8xa(t) = Uxsa(t) - Uxca(t) and 13ya(t) =0ysa(t) - 0yca(t) which turn out to

be the bearing rigid body motions.
74

f'mo° t't (3.12a)

m c Ic

Ym = "m s + m c , 7I is + ic [C] = o[K] bm (3.12b)

It may be noted that purely translational 8xa(t) model by Kraus et al. [24] shown in

Figure 3.6c constitutes a subset of equation (3.12) with kbx0y = 0. Eigensolution of

equation (3.12) with [C]=[0] yields the following natural frequencies 01j and modes

(*}j

J BI_B 2 mskb0y 0 011,2 = 2Tram S3Clis ; Bl=Tm +_/'ilskbx x (3.13a,b) Y

B2 =_( 7ram skb0 y 0y _ ,YiI skbxx) 2 +4Ym m s_/ii sk bX0y2 (3.13c)

{qi}T = 1 { 1 , B 3 } (3.14a)

1,2 JYmm s + 7ii sB2

_/m m skb0y0y - y ilskbx x + B2

= (3.14b)

B 3 2Yi k bx0

Y

On the other hand, the eigensolution of the single degree of freedom system is given by

_ =_/kbxx27m m--s and {$}_={ are_m1 ms } where subscript 2 and superscript ^

chosen to indicate that this solution essentially estimates ol2 and {_}2 in equations
75

A A

(3.13) and (3.14). Since _2 does not include kbx0y and kb0y0y, co 2 < ¢.02 as evident

from Table 3.4 for 3 different axial preloads. This natural mode is dominated by fixa(t)

as indicated by equation (3.14). And, the flu'st mode {9}1 which is predominantly

13ya(t) is also affected by the axial preload.

Table 3.4 Bearing stiffness coefficients and undamped natural frequencies (Hz) of

example case I t

Proposed model Simple

Axial Bearing stiffness coefficients (DOF = 2) model

preload , , , (DOF=I)

A

Fzbm(N ) kbx x (N/m) kbx0y (N) kb0y0y (Nm) co1 co2 co 2

115 1.84 E7 -3.05 E6 1.36 E4 156 372 341

190 2.13 E7 -3.12 E6 1.70 E5 191 395 367

285 2.43 E7 -3.09 E6 2.02 E5 221 416 392

t Other system parameters are: ms=10.0kg, Is=0.025kgm 2, Ym=0.4,'Ys=0.3,a=lE-6s.

3.7.2 Bearing Transmissibility

The forced harmonic response of equation (3.12) can be obtained using the

dynamic stiffness approach given by equation (3.8)
76

(kbOyOy_- 032_I Is) Ym Fxsal eica °t

_Jxa(t) = (kb0y0y X 2)( _,- 032ymm s) (k

2 (3.15a)

- 03oTI Is kbxx - bx 0y _')

i¢0ot

bx0 k Ym Fxsale

Y (3.15b)

_ya(t)= (kb0y0y 2 s)( bxx 032ym m sl-( X0y )

-k

It can be seen from equation (3.15) that Fxsa(t) not only excites 8xa(t) but _ya(t) as well,

which is not predicted by Kraus et al. [24]. The steady-state solution for this

simple model is given by the following; compare it with equation (3.15a).

it0ot

Ym Fxsal e

_xa(t) = (kbxx__ 032oYmms)

(3.16)

Both models are used to determine the load transmissibility magnitude terms R(03o)

which are computed using {Sxa(t),13ya(t)}T, [K] and [C]. Dynamic bearing force Fxba(t )

and moment Myba(t) magnitudes excited by the shaft force Fxsa(t ) are given by force

transmissibility RFxba,Fxsa(03o) and moment transmissibility RMyba,Fxsa(03o)

respectively.

RF F (too) =

xba _ x$a

ym ._/( 1 +02032) 2

2 2

B4+(l+ _ 030) B

(3.17a)

_/ (1+o 2032)2 0 B4+(l+o 2032) (B 5 + B6) +B 7
77

RMyba, Fxsa (('00) =

YmYi IsO_ 2 J(1 + (_2_2o) k2bx0y

(3.17b)

(l+cj2m2)2B4 +(1+_ mo)(B5 +B 6)+B 7

B4 = (kb0y0ykbxx- k2bx0y] 2

(3.17c)

]35 = C°o"[IIs 2 k bxx ( kbxx _,li s _o2 + 2k2xoy - 2k bxx kbo y Oy) (3.17d)

B6=°3oq(mmskb0y0y 2 kb0 Ymm s°_o2 + 2k X0y - 2kbxxkb0 y 0y y0y

B

= 2e°4ym m sYI Is _ 2kbxxkb0y0y-°)20('/1Iskbxx + _'mm skb0y0y) } +

4

_oYm m sYi is { Ymm sYi isO_4_ 2kbx 0 y(1-0"2°)2) } (3.17f)

Only the force transmissibility, as given below, is predicted by the simple model [24];

compare it with equation (3.17a).

^ kbxx'Ym _/(1 + _20)2o)

RF F (too) = t-

,b,' "'" 4(kbx x -- 0_2ym m s)2 + ( kbxxO_ )

(3.18)

Figure 3.7 compares equations (3.17) and (3.18). Our model predicts higher

RFxba,Fxsa(_O) and 0)2 than the simple model due to the additional constraints imposed

by [K]bm. Also, it is clear that the simple model can not predict dynamic moment

transfer through the bearing. The bearing transmissibility functions R(e%) predicted by

our model for 3 different axial preloads are shown in Figure 3.8. Note that the resonant
78

10'

o_..q

o,..q

..0

10 _

10'

10 °

10"

10 -2

10

0

R 1(proposed)

Rl(Kraus)

R2(proposed)

t

25O 500

Excitation Frequency (Hz)

Figure 3.7 Bearing transmissibility spectra Rfwba,Fxsa(O)o) for example case I. Here,

RI: force transmissibility with fwba = Fxba and R2: moment transmissibility

with fwba = Myba, as predicted by our model with DOF = 2 and the simple

model by Kraus et al. [24] with DOF = 1.
79

o,-.I

°,.,_

o ,...-i

[..,

O

kr.,

102

10'

10 °

10"

10.2

i

10"

0

,,,,A

|

| / '_

i(

II

II

If

II

!

V Preload = 115N

Preload = 190N

Preload = 285N

, , . I , , I

250 500

Excitation Frequency (Hz)

(a)

10'

v

10 o

10"

|0"7

0

Figure 3.8

Preload = 115N

Preload = 190N

Preload = 285N

,A ,_, A

l\ / \t \ ,,,'/' '\ ',,

/ 7-./2__ -.\\

/// .....

f _ IF[

250 500

Excitation Frequency (Hz)

(b)

Effect of mean axial preload on the bearing transmissibility spectra

Rfwba,Fxsa((.Oo) defined by equation (3.17) for example case I. (a) Force

transmissibility with fwba = Fxba. (b) Moment transmissibility with fwba =

Myba.
80

amplitudes vary depending on the amount of axial preload and the resonant frequencies

increase with increasing preloads as expected.

3.8 EXAMPLE CASE II: RIGID SHAFT AND PLATE SUPPORTED ON

FLEXIBLE MOUNTS

3.8.1 Vibration Models

The physical system of example case I is modified to include flexible mounts, [K]v

[0], and mean radial shaft force Frs m _ Frsm(t ) and Frsa(t) = Frsalek°o t as shown in

Figure 3.9a. The ball bearing (see Table 3.3) is also preloaded in the axial direction.

This is modeled using lumped parameter theory with finite mount stiffness ceefficients

kvj, j = x, y, z, 0 x, 0y or 0 z as illustrated in Figure 3.9b. Since the rigid shaft

assumption still holds, Frs m = Frb m is applied directly on the bearing in a manner similar

to the axial preload. The bearing matrix [K]b m in Figure 3.9b has non-negligible

stiffness coefficients kbx x, kbyy, kbzz, kb0x0x, kb0y0y , kbx0y , kby0x kbx z and kbz0x

which are functions of the mean bearing load vector {f}bm = {f}sm or the mean bearing

displacement vector {q}bm = {q}sm - {q}cm as given by Chapter II of this report.

Conversely, White [20] has investigated this problem using a simple model with DOF =

2 as shown in Figure 3.9c. It may be noted that his bearing system model did not

include the effect of axial preload.

Like example case I, the governing equations (3.1) and (3.2) can be modified and

reduced to 4 uncoupled sets of differential equations. The first two sets associated with

0ysa(t) and 0yca(t ) are homogeneous. The third set is similar to equation (3.11) but with

x and y subscripts interchanged, and two mount stiffness coefficients kvy and kv0 x

included. However, it is still homogeneous and therefore has only trivial steady-state
81

(a)

Fxs a (t) Rigid Casing

Fxsm Ball Bearing

Z

Rigid Shaft

(b)

(c)

Figure 3.9

Rigid Shaft {

Ball Bearing {

Rigid Casing {

Flexible Mount {

{f(t)}sa¢ t{f(t)]sm

! [M]s t-''_ '{q(t)lsa DOF=6

[K]bm_ _ [C]b

[ [M]¢ t---l{q(t)]ca DOF=6

[K]v _ _[_ [C]v

fsxa (t)

RigidShaft { [ ms [..._.Uxsa(t )

Ball Bearing { kbrr_ +Cbrr

Rigid Casing { [ me [--- ¢ u xca(t)

FlexibleMount{ kw_ + c w

Example case II: rigid shaft, ball beating and rigid plate system supported

by flexible mounts and subjected to alternating radial force Fxs_(t ) applied at

the shaft. (a) Physical system. (b) Proposed multi-degree of freedom

vibration model with DOF = 12. (c) White's vibration model [20] with

DOF = 2.
82

solution. The final set corresponds to {q}a(t) = {Uxsa(t), Uzsa(t), 0ysa(t), Uxca(t), Uzca(t),

0yca(t)} T with [M] s, [M] c, [K]bm and [K] v of equation (3.2) reduced to

[K]

bm

S 0 0

[M] = m s 0

S !

L 0 o i s

k bxx k bxz k bx0 y

k bxz k bzz k bz0 y

kbx0y kbz0y kb0y0y

; [K]

v

mc 0 0

[M]c= 0 mcO

0 0 I c

kv X

0

0

0 0

kvz 0

0 kv0y

(3.19a,b)

(3.19c,d)

The vibration model by White [20] may be formulated by retaining only two equations

corresponding to Uxsa(t) and Uxca(t) and excluding all bearing and mount stiffness

coefficients except for kbx x and kvx.

Analytical eigensolution of the undamped system is not possible since it requires

solving for the zeroes of a 6-th order polynomial in co2. Therefore, this problem is

solved numerically using an eigenvalue routine [42]. Using the same system parameters

as in example case I with 3 different Fxs m, natural frequencies and modes are found as

given in Tables 3.5b and 3.5c for both our and White's models. Corresponding bearing

mean loads {Fxbm,Fzbm,Mybm} T and relevant bearing stiffness coefficients computed

using the method derived in Chapter II are listed in Table 3.5a. Tables 3.5b and 3.5c

indicate that only the In:st and fifth modes of our model are predicted by White's model;

here superscript ^ is again used to denote estimation based on the simple model. The

first natural frequency 0) 1 predictions by both models are very similar. But 0) 5

prediction, whose mode is similar to the second mode of example case 1, indicates a few

discrepancies. White's model also underestimates this natural frequency due to the
83

Table 3.5 Results of example case IIt

(a) Computed bearing mean loads and stiffness coefficients

Shaft

mean load

Fxsm

(N)

Bearing loads tt Bearing stiffness coefficients

A A

Fzbm Mybm kbxx kbxz kbx0y kbzz k(b_? (N) (Nm) (N/m) (N/m) (N) (N/m)

kb0y0y

(Nm)

44 198 0.62 2.26E7 5.84E7 -2.90E5 8.50E7 -3.10E3

94 216 1.16 2.64E7 1.14E7 -2.33E5 8.55E7 -1.36E4

122 229 1.38 2.93E7 1.40E7 -1.98E5 8.60E7 -2.19E4

1.71 E4

1.72E4

1.74E4

(b) Undamped natural frequencies (Hz)

Mean Proposed model (DOF=6)

shaft load

Fxb m (N) 011 012 013 °14 015' 016

Simple model (DOF=2)

A ^

011 015

44 93 100 111 287 350 607

94 96 100 115 300 351 614

122 97 100 117 308 364 620

97 321

97 345

98 362

(b) Modes of vibration ttt

Proposed model (DOF=6)

{d_}l {(D}2 {_}3 '[_b}4 {_b}5 {_}6

Simple model (DOF=2)

A A

0.217 0.031 -0.030 -0.130 0.180 0.044

-0.035 0.203 0.003 0.022 -0.036 0.236

1.648 0.084 5.966 -0.354 -1.245 -0.068

0.171 0.084 -0.071 0.107 -0.139 -0.031

-0.027 0.194 0.008 -0.018 0.028 -0.165

0.167 0.009 0.917 4.648 3.295 0.073

0.218 0.229

0.187 -0.178

1' Other system parameters are: ms=10.0 kg, Is=0.025 kgm 2, mc=15.0 kg,

Ic=0.03 kgm 2, ts=lE-6 s, kvx=lE7 N/m, kvz=lE7 N/m, kx0y=l E5 Nm.

_t Fxb m =Fxsm

t_ These are for mean shaft load Fxs m = Fxb m = 94 N.
84

incompletebearingstiffnessmodelemployed.Hereagainour predicted modes include

8xa, 8za and 13ya displacements of shaft and casing components which are not

considered by White's model.

3.8.2 Frequency Response

The forced harmonic response solution is also determined numerically using the

dynamic stiffness approach outlined in Section 3.6. Driving point (with j = s) and cross

point (with j = c) accelerance spectra AqwjaFxsa(C0o) are given in Figure 3.10 with qwja =

Uxj a, Uzj a or 0yja for both models. Here, we observe that White's [20] model

overestimates the magnitudes of the accelerance and can not predict, unlike our model,

axial Uza(t ) and 0ya(t ) angular motions on the shaft and casing. Figure 3.11 shows the

bearing transmissibility spectra RfwbvFxsa(C0o) for fwba = Fxba, Fzxa or Myba which

indicate that transmissibilities corresponding to Fzx a and Myba are not predicted using

White's model. Such loads also serve as mechanisms for vibration transm]::_on

through the bearing to the casing, in addition to Fxb a. The mount transmissibility

spectra Rfwva,Fxsa(C0o) as shown in Figure 3.11 indicate that Fzv a and Myva are also

transmitted to the mounts in addition to Fxv a due to casing motions in x, z and 0y

directions. The effect of mean radial bearing force Fxb m on the load transmissibilities

R(O_o) through the bearing is shown in Figure 3.12 for 3 different mean loads. We

again observe that the resonant amplitudes and frequencies are mean load dependent

through [K]b m. Similar trends show that an increase in mean load raises the resonant

frequencies although the effects are not as pronounced as those found in example case I.
85

e_0

°,..*

10'

10'

10'

100

10-_

10_

:

0

!

White

J

(a) Shaft

I

500 1000

Excitation Frequency (Hz)

10'

102

10'

V

_.)

'_ 10'

e-, 10"

102

l0 -_

0

I

Ii - Proposed

(b) easing

I

500 1000

Excitation Frequency (Hz)

e_0

o

.,..1

e-3

10_

10'

100

10"

10-_.]_

II

'°"t/

10_,±

0

(c)

I Casing Shaft

i

500 1000

Excitation Frequency (Hz)

10'

10'

l05

o

t_

10'

10-'

10.5

10 0 500 1000

Excitation Frequency (Hz)

Figure 3.10 Accelerance spectra Aqwja,Fxsa(O)o) for example case II as predicted by our

formulation and White's model [20]. (a) Driving point accelerance with

qwja = Uxsa" (b) Cross point accelerance with qwja = Uxca" (c) Accelerance

with qwja = Uzsa for shaft and qwja = Uzca for casing. (d) Accelerance with

qwja = 0ysa for shaft and qwja = 0yca for casing.
86

.4

"5

°_

e-.

L_

©

k_

10'

10 _

10'

!

10 o J

10"

10 _

10 "_

0

Proposed

Whim

(a) Bearing

• i ....

500 1000

Excitation Frequency (Hz)

o

10:

10'

10°

10 "_

10-:

10"

10-_

0

Proposed

•_ White

(b) Mount

.... i ....

500 1000

Excitation Frequency (Hz)

_._ 1°' I

._..,

,o,I

"_ ,°'[/

T

10"-- "

10 _

(c)

10'

"_ 10"

_ 10._

_ 10" t

,o-,-V ---7-. .

0 500 1000 0

Excitation Frequency (Hz)

!

k Beating

ii I

500 1000

Excitation Frequency (Hz)

Figure 3.11 Bearing Rfwba,Fxs_(COo) and mount Rfwva,Fxsa(¢0o) transmissibility spectra

for example case II as predicted by our formulation and White's model

[20]. (a) Beating force transmissibility with fwba = Fxba. (b) Mount force

transmissibility with fwva = Fxva- (c) Force transmissibility with fwba =

Fzb a for beating and fwva = Fzva for mount. (d) Moment transmissibility

with fwba = Myba for beating and fwva = Myva for mount.
87

10'

._ 10'

10 °

o 10"

]0 -_

10 .3

0

(a) Mean Load = 44N

I Mean Load = 94N

_,__ - Mean Load = 122N

500 000

,.Q

. ...,d

E

e_

b.

o

0

102

(b)

lO'

IO°F

10":

,o,/

b'

10 _ rO

_m

'1

10 _

0

_fi Mean I.x)ad = 44N

Mean Load = 94N

Mean Load = 122N

!

5OO 1000

,F.,I

co

E

O

Figure 3.12

10'

10 °

10"

10 "*

10"

10"

]0 -_

0

(c)

t

r /#

i at

: /#

F/

Mean Load = 44N

.

"

500 1000

Excitation Frequency (Hz)

Effect of the mean radial bearing force Fxb m on the bearing transmissibility

spectra Rfwba,Fxsa(_O) for example case II. (a) Force transmissibility with

fwba = Fxba" (b) Force transmissibility with fwba--Fzba. (c) Moment

transmissibility with fwba = Myba.
88

3.9 EXAMPLE CASE III: EXPERIMENTAL STUDY

3.9.1 Physical setup

The final example case examines the experimental setup of Lin [17] as shown in

Figure 3.13a. This system is similar to Figure 3.1 and consists of a 159mm long x

25ram diameter non-rotating shaft supported by two rolling element bearings of 25mm

bore x 51mm outer diameter. One is supported on a rectangular plate of approximate

dimensions 762mm x 457mm x 9ram and the second is rigidly connected to the base.

The plate is also bolted to a massive base structure. Excitation force Fy s which consists

of a mean Fysm = 445N via a preloaded spring and an alternating Fysa(t ) component

applied transversely at the free end of the shaft using a vibrating shaker. Driving and

cross point accelerance spectra are measured at the shaft and on the plate respectively.

Further details of this experiment are summarized in Reference [ 17].

3.9.2 Bearing Analysis

Initially, only the static analysis is performed to obtain [K]bm for this experimental

system using the method proposed in Chapter II of this report. The static analysis

neglects plate flexibility; this assumption is valid since the bearing mean loads are

sufficiently low and do not deflect the plate. The shaft-beating system is statically

indeterminate as shown in Figure 3.13b. The mean force Fysm on the shaft produces

mean bearing load vector {f}bm = {0,Fybm,Fzbm,Mxbm,0} T which depends on the

mean bearing displacement vector {q}bm = {0,Sym,Szm,[_xm,0} T" The proposed

bearing matrix [K]b m includes stiffness coefficients kbxx, kbyy, kbz z, kb0x0 x, kb0y0y,

kbx0y, kby0x, kby z and kbz0x which are direct functions of {q}bm- In contrast, the
89

j FT Analyzer r----fSignal _Conditioners ] - 7

t Acce lerom

Bearing •

_ Plate _

fi

/i

//

//

l/

/J

Force Transducer

Shaker I I _2;alt orl

IPower

IAmplifier |

Shaft Support

(a)

y

Fysm

I..-I 1 :ll: 12

Mxblmj_ _ _ Mxb2m

(b) Fzblm _q_ ;- Flexible Shaft

Bearings t. Fzb2m

Fyblm Fyb2m

Applied Force

and Preload F

I

I

z

Bearing Model

@

--- Bearing Stiffness Element

/k Simply-Supported

A Clamped

_Measured Response

(c)

l igure 3.13 Example case III. (a) Schematic of the experimental setup consisting of an

overhung shaft, 2 ball bearings and a rectangular plate [17]. (b) Static

model of shaft and bearings used for computing [K]b m. (c) Finite element

model of the experimental system shown in (a). The generalized stiffness

matrix elements for the bearing model are shown by dashed lines,

connecting 1 node on the shaft to 4 nodes on the plate.
90

co_wentional bearing models include only kbxx, kbyy and kbz z coefficients. From Figure

3. ! 3b, the force and moment equilibrium equations for this system are

Fybl m + Fyb2 m - Fysm = 0

Myblm + Myb2m - Fyb2m/1 + Fysm(/1 +/2) = 0

Fzblm + Fzb2m = 0

(3.20a)

(3.20b)

(3.20c)

Since the shaft-bearing system is statically indeterminate, bending theory for the shaft

and rigid body motion constraint in the z direction are used to estimate stiffness

coefficients

E1 _y2m - EI _ylm + Fyblm(/13/6) + Myblm(/12/2) = 0

El _x2m - El _xlm + Fyblm(ll2/2) + Myblm/1 = 0

_Zl m - _z2 m = 0

(3.21a)

(3.21b)

(3.21c)

Additionally, 6 nonlinear algebraic equations defined by the mean bearing loaddisplacement relations as given in Chapter II are required. These nonlinear algebraic

equations are solved using Newton-Raphson method [36,37]. Since mean loads on

each bearing are sufficiently large, the bearing stiffness coefficients for both bearings are

ahnost identical, as listed in Table 3.6 along with other system parameters.
91

"l_able 3.6 Design and estimated parameters for two identical rolling element bearings

used in example case HI

Load-deflection exponent n=3/2

Load-deflection constant Kn=6.92E9 N/m n

Number of rolling element Z= 10

Radial clearance rL=5.0E-5 mm

Pitch diameter=-38.1 mm

Unloaded contact angle C_o=0°

Aot--0.05 mm

11=41 mm

/2=84 mm

kbx x =1.44E8 N/m

kbyy =3.69E8 N/m

kbyz=2.04E8 N/m

kbzz=1.72E8 N/m

kbx0y=-2.56E5 N

kby0x=3.52E5 N

kbz0x=4.02E5 N

kb0x0x--4.19E4 Nm

kb0y0y=l.02E4 Nm

t Unloaded distance between inner and outer raceway groove curvature centers

3.9.3 System Study

We incorporate the proposed rolling element bearing stiffness matrix [K]b m in a

fi_ite element model which includes shaft and plate dynamics, using the formulation

given in Section 3.4.2. The finite element model shown in Figure 3.13c is implemented

with a commercial software [39]. The shaft component is modeled using 2 noded

Timoshenko beam elements with axial degrees of freedom in addition to the bending

motion. The plate model is constructed using 4 noded quadrilateral plate elements with

shear deformation and rotary inertia effects. Each node has 3 translational and 3

rotational degrees of freedom. Four generalized stiffness matrices corresponding to the

first bearing, each matrix being equivalent to 1/4 [K]bm, are used to couple the single

shaft node to 4 plate nodes. The second bearing connects one end of the shaft to a

grounded node. The boundary conditions for the plate along the perimeter are chosen to
92

be a combination of ideal clamps, Uxa(t) = Uya(t) = Uza(t) = 0xa(t) = 0ya(t) = 0za(t) = 0,

and simple-supports Uza(t) = 0 as shown in Figure 3.13c in order to represent the

physical model as much as possible. Here, the energy dissipation is assumed to be

given by the modal damping ratio _ = 0.03. A sinusoidal force Fysa(t) = FysalelC°o t is

applied at one end of the shaft to simulate the experiment.

Over the frequency range of 400Hz to 2000Hz, Figure 3.14 compares the driving

point accelerance spectra Auysa,Fysa(t.Oo) = ti ysa/Fysa. The simple theory shown here

represents the conventional way of modeling bearings while the other features are

exactly the same as in proposed model. Our predictions match measured spectra very

well. Conversely, the simple model predicts slightly higher accelerance amplitude and

lower resonant frequency in the vicinity of 800Hz due to the incomplete bearing model

used. Cross point accelerance spectra Auzca,Fysa(COo) = fi zca/Fysa are shown in Figures

3.15a and 3.15b where t] zca is measured for 2 different locations on the plate as shown

in Figure 3.13c, and excitation Fysa(t) is once again applied transversely at the shaft.

Here, each predicted accelerance spectrum has been averaged over 4 points in the

immediate vicinity of the measured location. Reasonable comparisons between the

proposed model and experiment are seen. Here, the discrepancies are primarily due to

physical setup complexities and the limitations associated with the finite element model

in describing some of these. In Figure 3.15, the simple model is not included because it

predicts exactly zero out-of-plane or flexural motion of the plate. Next, the cross point

mobility level L V is defined by averaging mean square mobility spatially over the entire

plate and over a frequency bandwidth Ac0. This level is directly related to the structureborne noise or vibratory energy transmitted through the bearing.
93

e.a3

o

10° Proposed Model /

/

/ Simple Model

Experiment [Lin]

Frequency (kHz)

Figure 3.14 Driving point accelerance spectra A., p (030) yielded by the proposed

uysa,- ysa

model, simple model and experiment by Lin [ 17] for example case HI.
94

10'

,1.)

°4

¢--

_h

10"

10 1

l0 -2

10 "_

0.4

?l I A A .,

,'t I

,4 Proposed Model

Experiment lLin]

i i , ,

1.2

Frequency (kHz)

2.0

(a)

10'

10°

10 _

10 a

10-_

0.4

I

I

I

t

I I Proposed Model

Experiment [Lin]

! | ,

1.2 2.0

Frequency (kHz)

(b)

Figure 3.15 Cross point accelerance spectra Auzca,Fysa(t.0o) yielded by the proposed

model and experiment by Lin [17] for example case III. In this case, the

simple model predicts zero response /izc a at the plate. (a) Point C1 (see

Figure 3.13c). (b) Point C2 (see Figure 3.13c).
95

L v = 10 log 10{ < V2zca { 1 ZZRe r: lt zca __ca >}=101°gl0 2ScA°_A°_Sc \ xsa xsa

dB re <V2>ref = 1.0m2]N2s2 (3.22)

where ( )* implies the complex conjugate, Sc is the plate surface area and Re{ } implies

the real part of the complex number. Table 3.7 compares L v predictions by proposed

and simple models with experimental data. It can be seen from this table that the

proposed model predicts the experiment quite well and the simple model fails to predict

any plate vibration. We can therefore conclude that our model is indeed valid for

vibration transmission analyses.

Table 3.7 Predicted and measured cross point mobility level L V as def'med by equation

(3.22)

1/3 Octave band

center frequency (Hz)

Experiment (dB) Proposed Simple

[Lin] model (dB) model (dB)

400 - 102 - 105 -oo

500 -92 -96 -_

630 -95 -94 .oo

800 -88 -97 .oo

1000 -87 -95 -oo

1250 -97 - 108 -oo

1600 -108 -115 -oo

2000 - 106 - 107 .oo
96

3.10 CONCLUDING REMARKS

A new mathematical model for precision rolling element bearing has been

developed and incorporated in linear system dynamic models using lumped parameter

and finite element modeling techniques for the vibration transmission studies of a

generic single shaft-bearing-plate system. Stability studies indicate that the bearing

system is dynamically stable for most of the practical designs. Through 3 example cases

including one experimental study, we have shown that our proposed vibration model is

clearly superior to the models currently available in the literature. The current models

tend to underestimate the resonant frequencies and force/moment transmissibilities, and

overestimate the accelerance amplitudes as compared to our models. The proposed

model also predicts how the vibratory bending motion on the shaft is transmitted to the

casing, illustrated through coupling coefficients of the proposed beating stiffness matrix

[K]bm. Finally, the forced response trends indicated that increase in the mean bearing

loads increases system resonant frequencies. We are extending this model to predict

vibration transmission in rotating equipment with multiple shafts, bearings and gears.

Other applications are evident as our theory is general in nature. However, it is

restricted to linear systems. Bearing non-linearities are being examined in a parallel

study [46].
CHAPTER IV

GEARED ROTOR SYSTEM STUDIES

4.1 INTRODUCTION

The focus of this chapter is on the dynamic analysis of geared rotor system shown

in Figure 4.1 which includes a spur gear pair, shafts, rolling element bearings, motor,

load, casing and flexible or rigid mounts. For this purpose, discrete vibration models

are developed and used to predict vibration transmission through bearings. Also, the

effects of bearing, casing and mount dynamics on the internal rotating system dynamic

characteristics are investigated. Traditional analysis approaches [ 15,25-29] in the gear

dynamics area concentrate on the internal rotating system and exclude dynamic effects of

casing and flexible mounts. Moreover, simple bearing models are typically used which

assume either ideal boundary conditions on the shaft or translational spring elements.

Some of the limitations associated with current bearing models have been discussed

thoroughly in Chapters II and III of this report. For instance, simple bearing

formulations can not explain how the vibratory motion may be transmitted from the

rotating shafts to casing and support structures, and moreover can not predict the effects

of bearing, casing and mount dynamics on the internal rotating system adequately.

Chapter II of this report also presents a new mathematical model for the precision

rolling element bearings to clarify this issue qualitatively and quantitatively, and Chapter

Ill utilizes the proposed bearing stiffness matrix [K]bm formulation to analyze the

97
98

x

z

Y

Driven Shaft

f2

zM

Motor [

Flexible

Coupling

Rigid or Flexible Casing

-- Pinion

Gear

Rolling

Element

"-- Bearing

Driving Shaft

Figure 4.1 Schematic of a geared rotor system with flexible shafts and rigid spur gear

and pinion, and supported by 4 rolling element bearings on a flexibly

mounted casing. The shaft at the motor end is driven at mean speed f_ zM

which in turns drives the load at mean speed _ zL" This physical system is

symmetric about a plane intersecting both the driving and driven shafts.

The y axis is assumed parallel to the gear mesh line of action in the pressure

angle direction at mesh point. This schematic is used for example case II

and III.
99

vibration transmission problem in a genetic shaft-bearing-casing-mount system. Our

bearing formulation has already been validated through several example cases.

This study extends the proposed beating formulation of Chapters II and III to

examine vibration transmissibility in rotating mechanical equipment through two generic

systems which are a spur gear pair drive and a single-stage rotor system with rolling

element bearings as shown in Figures 4.1 and 4.2 respectively. It should be noted that

Figure 4.2 can be treated as a special case of Figure 4.1 - this will be illustrated later.

The physical systems are assumed to be driven at mean speed g) z = f2zM in Figure

4.1 and f2 z in Figure 4.2; subscript M refers to the motor end. Each shaft is supported

by two identical rolling element bearings which are modeled as stiffness matrices [K]b m

of dimension 6 as proposed in Chapter II of this report. These generic systems may

be excited by motor and/or load torque fluctuations, rotating mass unbalances or gear

kinematic transmission error e(t) [29,47]; here e(t) is defined as the deviation of the

relative gear-pinion angular position from its relative ideal location. Typical excitation at

the shaft is defined by an alternating load vector {f(t)}sa = {Fjsa(t),Tjsa(t)}T = {f(t)} s _

{f}sm; j=x,y,z, where Fjsa(t) and Tjsa(t ) are the alternating force and torque respectively,

{f(t) }s is the total load vector of dimension 6, {f} sm represents the mean load vector,

and superscript T implies the transpose. In the case of the geared drive, {f(t)}sa consists

of e(t) at the mesh point. Additionally, in the case of the single-stage rotor system,

transverse forces Fxsa(t) and Fysa(t) due to mass unbalances are of interest. Note that in

the dynamic analysis {f}sm and bearing preloads are not included as they do not appear

in the governing equations of the linear vibration model but are used for computing

[K]bm. Other effects such as bearing coupling coefficients, motor and load inertia, and
I00

X

Rolling

Element

Bearing

Casing I

I IFlexible

Coupling

Rotor S_ _i

Flexible Load

Flexible Mounts Yf//JJ'/J'//L4"

l-igure 4.2 Schematic of the single stage rotor system with flexible shaft rotating at

mean speed f2 z and supported by 2 identical rolling element bearings on a

flexibly mounted rigid casing. This physical system is symmetric about a

plane intersecting the shaft axial axis and parallel to the x-z plane. This

schematic is used for example case I.
101

casing and mount dynamics are considered as the emphasis is on the overall system

behavior.

4.2 LITERATURE REVIEW

The available literature on bearing models commonly used for internal rotor and

gear dynamic analyses has already been discussed in Chapters II and III of this

report. Although there are numerous publications [10-13,15,25-29] on the modal

analyses of geared drives and single or multiple-stage rotor systems, very little has been

reported directly on the force transmissibility through bearings, and the dynamic effects

of beating, casing and mounts on the internal rotating system.

4.2.1 Casing and Mount Dynamics

A comprehensive review of the available vibration studies of casing and mounts,

without the internal rotating system, has been given by Lim and Singh in 1987 [6].

Other related publication not included in Reference [6] are by Gaul and Mahrenholtz [48]

in 1984, and Smith [49] in 1988 who have developed lumped parameter models of

flexible machine foundations, excluding the internal rotating system dynamics. Their

studies report rigid body translational and rotational vibration responses of the casing

when excited by forces and/or moments. However, these studies are not specifically

directed towards geared rotor systems, and dynamic interactions between internal

rotating system and casing-mount system are not incorporated. In addition, Lim et al.

[50] in 1989 have also performed a dynamic finite element analysis of a helicopter

gearbox excluding the internal rotating system. Results again show that casing rigid
102

body vibration modes are dominant over the lower frequencies for a flexibly mounted

c_ising, but numerous casing plate elastic modes are observed at higher frequencies.

4.2.2 Gear Dynamics

Current gear dynamic models include only the internal rotating system and simple

bearing models, and typically exclude casing and mount flexibilities [15,25-29]. In few

instances casing and mounts have also been included, but restricted to cases where only

purely radial and/or axial force on the bearing, and unidirectional vertical/horizontal rigid

body motion on the casing are modeled, as summarized in Reference [6]. Such models

still do not explain how the vibration is transmitted from the gear mesh to

the casing and into the machine foundation as witnessed in previous experiments

[9,16,17].

4.2.3 Rotor Dynamics

Similar to the geared rotor models, most of the existing rotor dynamic models

concentrate on the internal rotating system and address issues related to stability, critical

speeds and rotating mass unbalance response [10-13]. But a few investigators have

included support flexibility models [51-53]. For instance Lurid and Wang [51] in 1986

proposed an impedance matching approach to reduce the large degrees of freedom

(DOF) required in such models. Using one example case, they reported that support

foundation has little influence on the internal rotor resonances, but then cautioned that

this might not be true for other systems. Vance et al. [52] in 1987 incorporated

measured support foundation parameters in a transfer matrix model of the rotor dynamic

system, and concluded that the omission of support flexibility may miss some of the
103

rotor critical speeds of interest. Earles et al. [53] in 1988 reported, based on a finite

element model of the support foundation, that the rotor critical speeds shift by including

support flexibility. In all of the above mentioned studies, the overall dynamic behavior

of the rotor system is only investigated partially. Moreover, only a subset of journal

bearing [K]bm coefficients is included which can not predict completely the vibration

transmission through beatings.

4.3 ASSUMPTIONS AND OBJECTIVES

Linear discrete vibration models of the geared and single-stage rotor systems

shown in Figures 4.1 and 4.2 are used to incorporate [K]bm similar to Chapter III. The

stiffness coefficients of [K]bm are evaluated using the analytical expressions presented

in Chapter II. Each rotating shaft is modeled as an Euler beam in the lumped parameter

model and as a Tirnoshenko beam in the dynamic finite element model but the effect of

the gyroscopic moment on the shaft dynamics is not included. The casing is assumed to

be rigid in the lumped parameter model and flexible in the dynamic f'mite element model.

The rigid casing assumption is valid for many practical designs as it may be massive and

the rotating speeds may be sufficiently low to avoid significant elastic deformation. The

governing equations for both discrete vibration models can be given in the general

matrix form as

[M] {_(t) } a + [C] {el(t) ] a + [K] {q(t) } a = {f(t) } a (4.1)

where [M], [C] and [K] are the system mass, damping and stiffness matrices

respectively, and {q(t)} a and {f(t)} a are defined as the generalized alternating
104

displacement and applied load vectors respectively. Like Chapter III of this report,

{f} bm and bearing preloads are also assumed constant to ensure a time-invariant [K]b m

matrix. Energy equivalent viscous damping matrix [C]b = cr [K]bm is assumed for the

energy dissipation mechanism in the bearing where c is the Rayleigh damping

proportionality constant. Other features of the proposed theory given in Chapter III of

this report are retained.

The specific objectives of this study are to: (i) incorporate the proposed bearing

matrix [K]b m, developed in Chapter II, in discrete vibration models of generic geared

system of Figure 4.1 as described by equation (4.1) using lumped parameter and/or

dynamic finite element method, (ii) conduct overall system studies by calculating

eigensolutions and forced harmonic responses, (iii) evaluate the effects of casing and

mount dynamics on the internal rotating system, and predict vibration transmission

through bearings, (iv) examine the following 3 example cases: I. single-stage rotor

system with flexible shaft supported by two identical rolling element bearings on rigid

casing and flexible mounts as shown in Figure 4.2, II. spur gear pair system with motor

and load inertias attached to two flexible shafts supported by four rolling element

bearings on flexibly and rigidly mounted rigid casing as shown in Figure 4.1, and III.

case H with flexible casing and rigid mounts, and (v) demonstrate the advantages of our

formulation over the existing vibration models.
105

4.4 THEORY

4.4.1 Method A: Lumped Parameter Model

4.4.1.1 Equations of Mgfion

Both physical systems shown in Figures 4.1 and 4.2 are discretized using lumped

parameter technique to yield equation (4.1) through the Lagrange's equations of motion

[45].

d| 3ET( 3Eu]_ _ET 3Eu _EC-F ; w= 1,2,3 .... (4.2)

dt _.3_lw 3_lw ) 3qw + 3qw + 3_lw w

where E T and E U is the kinetic and potential energies respectively,

1

EC=-_-{_I}T[c]{_I} is the Rayleigh's dissipation function and Fw is the

generalized force. The total system potential energy E U and kinetic energy E T are

obtained by adding the energy of each system component which are derived in

subsequent sections. Equation (4.2) when applied to Figure 4.1 will lead to equation

(4.1). It is then rearranged in terms of the following {q} to obtain the partitioned

system matrices

T

{q(t)} a = { {u(t)} T {u(t)} T {0(t) }TRa {O (t) } Tca {0(t) } T} sa (4.3a)

[M] =DIAG{ {m] T {m}Tc

[K]=

[K]ll [K]12 [K]13

[K] 21 [K]22 [K]23

[K] 31 [K] 32 [K] 33

{I} TR {I} T {I }Ts } (4.3b)

[C] = o [K] (4.3c)
106

where {0(t)}Ra (subscript R indicates rotor) consists of the alternating angular

displacement of all rotors including gears, motor and load; {m} T and {I} T are the

lumped mass and inertia row vectors respectively; and the operator DIAG { } transform

the row vector into a diagonal matrix with components of the vector corresponding to

the diagonal elements. The stiffness sub-matrices [Kiwi = [K]j T, w,j = 1, 2, 3,

consist of the appropriate terms corresponding to the partitioned {q(t)}a" For example,

T }T [K]I1 couples { {u(t) }T {u(t) } ca degrees of freedom to itself, and [K]12 provides

sa

T }T T {0(t)}T }Tdegrees of a coupling between { {u(t) }T {u(t) } ca and { {0 (t) } Ra ca

freedom.

Now assume that the rotary inertia of each shaft lumped mass is negligible. The

system matrices given in equation (4.3) are therefore simplified by eliminating {0(t)}sa

degrees of freedom from {q(t)} a in equation (4.3a). The resulting mass and stiffness

matrices are

[M]=DIAG{ {m}Ts {m}T {I}R { (4.4a)

[K] = I

-1

[K] 11 - [K] 13[KI 33 [KI 31

-1

[K] 21 -[K] 23tKlr _ 33 [K] 31

[K]12 - [K] 13[K]313 [K] 32

[K] 22 - [K] 23[K] ;13 [K] 32

(4.4b)

This analytical lumped parameter formulation will be used for example cases I and II. It

may be noted that if the effect of the rotary inertias is required, equation (4.3) can be

used instead.
107

4.4.1.2 System Kinetic Energy

A flexible shaft of length L s with rigid rotor of mass m R is subdivided into n s

number of segments of equal length Le = Ls]n s with lumped masses at both ends of each

segment as shown in Figures 4.3a and 4.3b for a single-spur gear mesh geared drive

and single-stage rotor system respectively. Each lumped mass has 3 translational and 3

rotational degrees of freedom as illustrated in Figure 4.3c. The total system kinetic

energy E T is given by 2 { el} T[ M] { _1}. The mass matrix [M] is diagonal and consists

of the lumped masses and inertias.

4.4.1.3 Shaft Stiffness Matrix

Using the direct stiffness approach, the stiffness matrix [K]

e

of dimension 12

s

e

corresponding to the alternating displacement vector {q(t) } sa = {Uxj(t)' Uxj+l(t)' Uyj(t),

Uyj+l(t), Uzj(t), Uzj+l(t ), 0yj(t), 0yj+l(t), 0xj(t), 0xj+l(t), 0zj(t), 0zj+l(t)}T, of a generic

shaft segment is given by

[K0u ]e

e

[Ku0l s

e

[K00] s

(4.5)

and the non-zero elements of [K uu ]e, [ K e eT e u0]s=[K0u]s and [K00] s of dimension

6 are given by the following equations (4.6), (4.7) and (4.8) respectively.

kll = k22 = k33 = k44 = -k12 = -k21 = -k34 = -k43 = 12 EI/L 3e

k55 = k66 = -k56 = -k65 = AE/L e (4.6)
108

Bearing

1

Driving Shaft

Pinion

Bearing

ns+l

Driven Shaft

• • •

n s +2 2n s+2

Bearing m Bearing

Gear

(a)

Rotor

Bearing m Bearing

1 ns+l

• • • ql

Flexible Shaft

(b)

Uxj U xj+l

_ z UzJ Uzj+l

y Uyj 0z j Uyj+ 1 0zj+ 1

(c)

Figure 4.3 Lumped parameter model of the flexible shafts with rigid rotors. Each shaft

is supported by 2 identical bearings on both ends. (a) Internal rotating

system of a typical geared rotor system of Figure 4.1. (b) Internal rotating

system of a typical single-stage rotor system of Figure 4.2. (c)

Discrefizafion of a shaft segment and the degrees of freedom associated with

each lumped mass.
109

k l 1 = -k22 = k33 = -k44 = k12 = -k21 = k34 = -k43 = 6, EI/L 2 (4.7)

kl 1 = k22 = k33 = k44 = 4 EI/CLe ;

k12 = k21 = k34 = k43 = 2 EI/Le ;

k55 = k66 = -k56 = -k65 = GJ/L e

(4.8)

where E is the modulus of elasticity, G is the shear modulus of elasticity, I is the

moment of inertia, and J is the polar moment of inertia of the shaft. The lumped

stiffness matrix [K] s corresponding to {q(t)}sa = {Uxj(t), Uyj(t), Uzj(t), 0yj(t), 0xj(t),

e

0zj(t)} T, j = 1,2,3 .... for the shaft is constructed by the superposition of all [K] s

matrices and merging terms associated with each degree of freedom.

[Kuu]s [Kue]s] [K]s= [K0u]s [K00] s

(4.9)

T

The non-zero elements of [Kuu] s, [Ku0] s = [K 0u] s and [K00] s of dimension 3(ns+l)

are similarly given by the following equations (4.10), (4.11) and (4.12) respectively.

kll = k22 = k3ns+l,3ns+ 1 = k3ns+2,3ns+ 2 = 12 EI/L3e ;

k33 = k3ns+l,3ns+ 1 = AE/L e

kwi+3,w I = kwi,wl+ 3 =-12 EI/L3e ; w I = 1,2,4,5,7,8 ..... 3ns-1

kwn,w n = 24 EI/L 3e ; wII = 4,5,7,8,10,11 ..... 3ns-1

kwlii+3,Wli I = kwm,wm+ 3 = -AE/L e ; wii I = 3,6,9 ..... 3n s

kwiv,wi v = 2 AE/L e ; Wlv = 6,8,12 ..... 3n s

(4.10a)

(4.10b)

(4.10c)

(4.10d)

(4.10e)
11o

kl 1 = k22 = -k3ns+l,3ns+l = -k3ns+2,3ns+2 = 6 EI/L 2

-kwi+3,w I = kwi,wi+ 3 = 6 EI/L2e ; w I = 1,2,4,5,7,8 ..... 3ns-1

(4.11a)

(4.1 lb)

kll = k22 = k3ns+l,3ns+ 1 = k3ns+2,3ns+ 2 = 4 EI/L e ;

k33 = k3ns+l,3ns+ 1 = GJ/L e

kwx+3,w I = kwbwi+3 = 2 EI/L e ; w I = 1,2,4,5,7,8 ..... 3ns-1

kwli,WH = 8 EI/L e ; Wli = 4,5,7,8,10,11 ..... 3ns-1

kwm+3,w m = kwm,wtu+3 = --GJ/L e ; wii I = 3,6,9 ..... 3n s

kwiv,wi v =2GJ/L e ; wlV = 6,8,12 ..... 3n s

(4.12a)

(4.12b)

(4.12c)

(4.12d)

(4.12e)

Note that here [K] s is only for a shaft, of dimension 6(ns+l ). The same formulation

should be applicable to a geared system with multiple shafts where each shaft stiffness

matrix is still given by [K] s. Accordingly, the shaft potential energy Eus is given as

1 T (4.13)

EUs= 2" {q} sa [K] s{q }sa

Formulations for the motor and load rotary inertias, and flexible coupling torsional

stiffness will be given in the later sections on example cases.

4.4.1.4 Gear Mesh Stiffness Matrix

For the generic geared rotor system with a spur gear pair, the driving and driven

shafts are coupled via a linear, time-invariant spur gear mesh stiffness k h _ kh(t ). Now

we define the gear mesh coupling stiffness matrix [K]h as follows where coupled
111

torsional and translationalmotions of the gearand pinion aredefined by {UyGa(t),

UyPa(t),0zGa(t),0zpa(t)}T

[K] h

k -k h

-k h

khd G

h

k

h

khd G

2 2

khd P khd P

khd G khd P

2 2

khd G khd P

2 2

2

khd G khdpd G

4 4

khdpd G khd2p

2 2 4 4

(4.14)

where dG and dp are the gear and pinion diameters respectively; subscript G and P refer

to gear and pinion respectively. Here, the y axis is taken to be parallel to the gear mesh

force which is along the pressure angle direction at the mesh point. The potential energy

EUh in this case is

EUh = 2{UyGa' 1 UyPa' 0 0zpa}[K] zGa' h U a] u yPa

0zGa

0zpa J

(4.15)

4.4.1.5 Flexible Mount Stiffness Matrix

The flexible mounts are represented by a diagonal stiffness matrix [K]v

corresponding to {q(t)}ca which consist of effective stiffness coefficients kvw, w = x, y,

z, 0 x, 0y, Oz. This modeling procedure assumes that no coupling exists between the
112

casing rigid body degrees of freedom due to the flexibility of the mounts. Accordingly,

one has

Euv = _-{q} [Klv {q}ea (4.16)

4.4.1.6 Bearing Stiffness Matrix

The proposed bearing stiffness matrix [K]b m corresponding to the bearing degrees

of freedom {Swa(t),13wa(t)} T, w = x, y, z, has been formulated in Chapter II. The

corresponding potential energy Ebb is

1 _waj

waj' Dwaj} [K] bm[ Dwa j

(4.17)

Here, {Swaj(t),13waj(t)} T may be expressed in terms of (q(t)]sa j and {q(t)}ca through a

coordinate transformation for the j-th bearing located at R j -- {xj, yj, zj} from the

casing center of mass. For the geared rotor system, four bearings are denoted by j = 1,

ns+l, ns+2 and 2ns+2 as shown in Figure 4.3a, while for the single-stage rotor system

two bearings are denoted by j = 1 and ns+l as shown in Figure 4.3b. Assuming the

small angle approximation, {Swaj(t),13waj(t)}T for the j-th bearing is

-1 0 0 0 0 0 1 0 0 0

0 -1 0 0 0 0 0 1 0 -z.

J

0 0 -1 0 0 0 0 0 1 Yi

0 0 0 -1 0 0 0 0 0 1

0 0 0 0 -1 0 0 0 0 0

0 0 0 0 0 -1 0 0 0 0

zj -yj

0 X.

J

--X. 0

J

0 0

1 0

0 1

(4.18)
113

4.4.2 Method B: Dynamic Finite Element Formulation

The dynamic finite element technique of incorporating [K]bm in equation (4.1) is

employed especially when the casing plate is elastically deformable over the frequency

range of interest, as discussed previously in Chapter III. In this method, [K]bm is

implemented in the dynamic finite element model as a generalized stiffness matrix. This

stiffness element provides the only coupling between the flexible shaft and casing at the

appropriate nodal points on both components corresponding to the bearing locations.

Additionally, the gear mesh stiffness k h in Figure 4.1 behaves like a linear translational

spring. Since the coupling between torsional and transverse shaft vibrations is due to kh

oJdy, the corresponding portion of the stiffness matrix is also given by equation (4.14).

Our finite element formulation uses conventional structural elements typically available

in commercial software programs [39] for the shaft, casing plate and mounts - this will

be illustrated for example cases I, II and III. Other features of this method are similar to

those discussed earlier in Section 4.4.1.

4.4.3 Other Methods

Alternate techniques of formulating the geared rotor system dynamic problems

such as finite difference (which is similar to method B), flexibility, component mode

synthesis and transfer matrix formulations are also possible. It may be noted that

Berman [54] has philosophically proposed application of the component mode synthesis

to geared problems. Other researchers [40,41] have utilized the transfer matrix method

in the rotor dynamic problems. However, such models need to be modified or extended

in order to solve the overall dynamic problems discussed in this research. Direct
114

application of these alternate methods [40-43,54] are beyond the scope of this

report and are left for further research.

4.5 EXAMPLE CASE I: SINGLE-STAGE ROTOR SYSTEM WITH

RIGID CASING AND FLEXIBLE MOUNTS

4.5.1 Vibration Models

Consider the single-stage rotor system shown in Figure 4.2 where two axially

prcloaded identical deep groove ball bearings support a flexible rotating shaft; see Table

4.1 for bearing parameters. The shaft is assumed to be subjected to a mean torque Tzs m

Tzsm(t) and harmonic excitation forces {Fxsa(t) = Fxsal(D.z) e i'ozt, Fysa(t) = Fysal(f2 z)

ei(f_zt-n/'2) }T due to the rotating rotor mass unbalance. The rotor is fixed to the center of

the shaft which coincides with the rigid casing centroid. The shaft is assumed to be

transversely decoupled from the motor and load due to the flexible torsional couplings.

A lumped parameter model with ns+2 lumped masses is developed according to

the theory given in Section 4.4.1. The bearing stiffness matrix for an axially preloaded

ball bearing has significant stiffness coefficients kbx x, kbyy, kbz z, kb0x0 x, kb0y0y , kbx0y

and kby0x which depend on the mean axial preload. Since there is no external Fza(t )

force on the system and no coupling between the torsional and transverse motions of the

shaft, the dynamics associated with {Uzsaj(t),Uzca(t),0zRa(t)} T degrees of freedom (here

subscript j is a dummy index to identify the shaft lumped masses) are decoupled from

the others and have trivial steady-state particular solution. The system matrices of

equation (4) can now be simplified by suppressing such degrees of freedom. Also, it

can be easily observed from equations (1) and (4) that 2 sets of uncoupled differential

equations exist. One set has {q(t)}a = {Uxsaj(t),Uxca(t),0yRa(t),0yca(t)} T which are

excited by Fxsa(t) and the other set consists of {q(t)} a = {Uysaj(t), Uyca(t), 0xRa(t),
115

Table 4.1 Design parameters for deep groove ball bearings used for example cases t

Load-deflection exponent n

Load-deflection constant K n (N/m n)

Number of rolling element Z

Radial clearance rL (mm)

Pitch diameter (mm)

A o (mm) tt

Unloaded contact angle %

3/2

7.5 E9

12

0.005 - 0.02

38.5

0.0625

23 ° - 47 °

These bearings are currently being used in a NASA gear test facility [55]. Also see

example cases II and III.

tt Unloaded distance between inner and outer raceway groove curvature centers.
116

0xca(t)} T with Fysa(t). The steady-state solution to these two sets will always result in

the synchronous whirling motion of the internal rotating shaft. However, since the two

sets of equations are similar and independent, only one set is considered in the following

analysis. Hence, the problem reduces to a 2 dimensional vibration system with DOF =

ns+4 which consists of a flexible shaft with rigid rotor vibrating transversely and

coupled to a rigid casing and a flexible mounts through two bearings. Accordingly, [M]

in equation (4.4) is rewritten in terms of the alternating displacement vector {q(t)}a =

{u xsaj(t),Uxca(t),0yRa(t),0yca(t) }T as

[M] =DIAG {msc j ,m c, lyR, lyc} (4.19)

where m and Iy are the lumped mass and inertia about the y axis respectively, and the

subscript e refers to the shaft segment. To define [K] in equation (4.4), the nonzero

elements of [KIll, [K]12, [K]22, [K]23, [K]I3 and [K]33 are given by equations (zL2O),

(4.21), (4.22), (4.23), (4.24) and (4.25)respectively.

kll = kns+l,ns+ 1 = 12 EI//L 3e + kbxx ; kns+2,ns+2 = kvx + 2 kbx x

kns+2,1 = kns+2,ns+ 1 = kl,ns+2 = kns+l,ns+ 2 = -kbx x

kwl+l,w I = kwi,wi+ 1 =-12 EI/L3e ; w I = 1,2,3 ..... n s

kwll,Wi I = 24 EI/L3e ; Wli = 2,3 ..... n s

(4.20a)

(4.20b)

(4.20c)

kns/2,1 = -kn ,s/2+2,1 = 6 EI/L 2e

k12 = -z 1 kbx x - kbx0y

; kns+2,2 = (z 1 + Zns+l ) kbx x + 2 kbx0y

; kns+l,2 = -Zns+l kbxx- kbx0y (4.21)
117

kll = 8 EI/Le ; k22 = kv0y+2 kb0y0y+ (z2 + z 2 s+l)kbxx + (Zl+Zns+l) kbx0y (4.22)

kl,ns/2 = kl,ns/2+l = 2 EI/L e ;

k21 = -z 1 kbx0y -kb0y0y ; k2,ns = -Zns+l kbx0y -kb0y0y (4.23)

kll = -kns+l,n s = 6 EI/L 2 + kbx'Oy ; kns+2,1 = kns+2,n s = -kbxOy ;

kns/2+l,ns]2+ 1 = -kns/2+l,ns] 2 = 6 EI/L 2e (4.24a)

k12 = -kns+l,ns_ 1 = -kns/2,ns/2_ 1 = kns/2+2,ns/-2+ 2 = 6 EI/L2e ; ns>2 (4.24b)

kwm-l,Wli I =-kwm_l,wiir_2 =-kwRl+ns/2,wur_2+ns/2 = kwRl+ns/2,WlXl+ns/2 = 6 EI/L2e ;

ns>-6, wIII= 3,4 ..... ns/2 (4.24c)

kl I = kns,n s = 4 EI/L e + kb0y0y

kwlv,wi v = 8 EI/L e ; ns>2, wlV = 2,3 ..... ns-1

kwv,wv_ 1 = 2 EI/L e ; kwv_l,wv = 2 EI/L e ;

ns>2, w v = 2,3 ..... n s (w v _ ns/2+l )

(4.25a)

(4.25b)

(4.25c)

T

Due to the symmetric nature of [K], [K] wj = [K]jw" The generalized load vector

{f(t)} a = {0 ..... Fxsa(t), 0 .... } consists of only the x component of the rotor mass

unbalance force. The other set of linear governing equation may be obtained by

interchanging subscripts x and y in the above formulation. Our proposed vibration

model can also be readily reduced to the conventional rotor dynamic models, excluding

the gyroscopic moment, internal structural damping, dissimilar bearings and other

secondary effects, by retaining only kbx x, kbyy and kbz z in our bearing formulation.
118

4.5.2 Eigensolution

The natural frequencies 00j and modes _j of Figure 4.2 with system parameters

given in Table 4.2 are obtained numerically. Initially, we check for 00j convergence rate

as shown in Figure 4.4. through 004 and 005 by varying the number of shaft segments

ns; it may be noted that 001,002 and 003 behave like 004, and o36 trend is very similar to

005. Beyond n s = 4, there are no noticeable changes in the natural frequencies. These

predictions are now compared with two simple lumped parameter vibration models and a

dynamic finite element model (FEM) in Table 4.3. One of the simple models excludes

the effect of casing and flexible mounts, and both employ a conventional bearing model

with only kbx x incorporated as described earlier in Section 4.5.1. The dynamic finite

element formulation models the flexible casing constructed with four noded quadrilateral

plate elements - a description of the methodology will be presented later in example case

llI. Our proposed theory and FEM are in excellent agreement with each other as the

deviations are within +4% for first 6 natural frequencies. The corresponding mode

shapes _j are shown in Figure 4.5. Note that for each mode in the x-z plane given here,

there is a complementary mode in the y-z plane which will have the same natural

frequency if kvy=kvx, kv0x=kv0y and Ixc=Iy c. Using the exactly same parameters,

results obtained from two simple models are also compared in Table 4.3. The simple

model with casing and mount dynamics included predicts lower natural frequencies as

compared to our prediction except for t_ 5; here the symbol ^ implies estimation using a

sinlple formulation. This model gives a rough approximation for most of the modes, as

evident from Table 4.3. Further, the simple model without casing and mounts can not

predict all the modes below 2 kHz. In addition, t_ 3 shifts while other modes remain
119

Table 4.2 System parameters of example case I: single-stage rotor system shown in

Figure 4.2

Rotor mass m R (kg) and inertia IyR (kgm 2)

Shaft mass rn s (kg) and length L s (m)

Casing mass mc (kg) and inertia Iyc (kgm 2)

Shaft flexural rigidity EI (Nm 2)

Bearing axial preload Fzb m (N)

kbxx, kbx0y, kb0y0y "_

kvx, kv0y

5.0, 1.24E-2

1.03, 4.2E- 1

148, 4.27

1.626E3

238

1.37E8,-1.96E6, 4.27E4

4.0E8, 1.8E7

t Computed kbw j as proposed in Chapter II. Only relevant bearing coefficients are

listed.

Table 4.3 Comparison of natural frequencies in Hz for example case I

Simple Models

Proposed

lVlode Theory FEM A 1% without casing with casing A2%

1 97.2 97.2 0.0 69.5 69.4 28.6

2 262 262 0.0 not predicted 262 0.0

3 309 309 0.0 297 294 4.9

4 332 334 0.6 not predicted 329 1.5

5 1405 1365 -2.9 1442 1443 -5.7

6 1755 1690 -3.8 1452 1453 14.0

At% = 100 x (FEM -Proposed Theory) /FEM

A2% = 100 x (FEM - Simple Model)/FEM
120

b_

5

_9

Z

3

2

0

0

I i I I | I I I I

10 20

Number of Shaft Segments, n s

Figure 4.4 Natural frequencies 014 and 015versus the number of shaft segments, ns for

example case I.
121

_)1

I-I

E--" _ j

_2

_3 _4

_5 x ¢>6

hL Z v

Figure 4.5 First 6 mode shapes of example case I: single-stage rotor system including

rigid casing and flexible mounts.
122

nearly the same. These observations are consistent with the findings reported earlier

[51-531.

We examine the effect of casing mass mc (and inertia Iyc *_ mc) while retaining

other parameters of Table 4.2. The fundamental frequency 031 (f'trst transverse mode of

the shaft-rotor system) and 033 (second transverse shaft-rotor mode) are quite insensitive

to rn c. On the other hand, 035 and 036 are proportional to (rnc) -1/2 for a light-weight

casing given by mc/(ms+mR) < 1, but are invariant for rnc/(ms+mR)> 1 as shown in

Figure 4.6a. The converse is seen for 032 and 034 which are nearly constant for

mc/(ms+mR) < 1, but are proportional to (rnc) -1/2 for mc/(ms+mR) > 1. These results

imply that the single degree of freedom approximation can be applied to 035 and 036 in

the region where rnc/(ms+mR)< 1, and similarly to or2 and 034 in the other region.

2

Next, we vary mount stiffness coefficients kvx (and kvey = L s kvx) to simulate

the effect of mount flexibility on the system. Here o.B and 036 are found to be unaffected

by kvx due to the fact that the corresponding modes are predominantly shaft bending

motion type as shown in Figure 4.5. Figure 4.6b indicates that 031 and 033 are

proportional to (kvx) 1/2 for soft mounts (kvx/kbx x << 1) but are insensitive to kvx for

stiff mounts (kvx/kbx x >> 1). Conversely, 032 and 034 are proportional to (kvx) 1/2 for

stiff mounts but remain nearly invariant for soft mounts. Similarly, each mode in its

appropriate region can be assumed to behave like a single degree of freedom system. In

Figure 4.6b, unlike Figure 4.6a where only one transition point at mc=ms+m R exists, 2

transition points are found. These are located below and above kvx/kbx x = 1, and

increase with increasing shaft and bearing stiffnesses.

Now, we examine the range of bearing preload Fzb m from 100 to 5000N on 03j. It

is seen that 03j increases with increasing Fzb m due to an increase in the magnitudes of the
123

v

Z

lO'

10°

10 -t

10 2

10-'

Massive

(0 6

........ _5

(1) 4

°°,, . ..., .,

10' 10' 10 _

Normalized Casing Mass, m c/(m s +m R)

(a)

10'

10'

tq

%

2;

10'

10 '

10'

10 °

10'

Soft Mount

0) 4 ._._

Stiff

I _" Mount

10"' 10 "_ 10' 10"

Normalized Mount Stiffness, k vx/k bxx

...) ........

10' 10'

(b)

Figure 4.6 Effect of casing mass mc normalized with respect to shaft and rotor masses,

and mount stiffness kvx normalized with respect to kbx x on system natural

frequencies c0j for example case I.
124

bearing stiffness coefficients kbw j as shown in Table 4.4 for the two extreme Fzb m

values; this observation is similar to those found earlier in the bearing system studies of

Chapter III. Note that 0) 2 is the most insensitive to Fzb m, and 0)3 and 0)4 show a

moderate variation. But 0)1 increases by about 45% due to larger bearing motions as

compared to 0)2, 0)3 and 0)4 modes. The remaining two modes of interest are also

affected significantly as large motions across the bearings are again found in these

modes. Unlike the trends associated with variations of m c and kvx, here each 0)j

approaches an upper bound solution, as shown in Figure 4.7 for 0)1, if Fzb m is

sufficiently high to stabilize magnitudes of kbw j terms.

Table 4.4 Effect of bearing preload Fzb m on 0)j (Hz) for example case I

Preload Fzb m (N) 0)1 0)2 0)3 0)4 0)5 0)6

100 89.4 262 305 331 1366 1706

5000 130 263 321 346 1855 1968
125

m

0.14

0.12

0.10

I

0.08 ' '

0 2500

Beating Preload (N)

5000

Figure 4.7 Effect of bearing preload Fbz m on co1 for example case I.
126

4.5.3 Transmissibility

The forced harmonic response due to the rotating rotor mass unbalance force

Fxsa(t) = Fxsal(f2z) e if_zt is obtained using the dynamic stiffness technique given earlier

in Chapter III. Also, recall from Chapter III the definition of sinusoidal load

transmissibility functions R(_ z) between two arbitrary locations I and II with load

vectors fwla(t)and fjlia(t)as R(f2z)= [fwia(f_z)l/I fjna(f_z)[ where in this case

fjlla(t) = Fxsa(t). The bearing force RFxba,Fxsa(_'2z) and moment RMyba,Fxsa(_'2z)

transmissibility spectra are compared in Figure 4.8 for our formulation and simple

model, given the same system of Table 4.2, both models differ only in [K]bm

formulation. Note that the results presented are for only one bearing due to symmetry of

the problem. We observe that the simple model predicts lower RFxba,Fxsa(f2z) than

proposed model except in the vicinity of co 1. On the other hand, RMyba,Fxsa(g2z) is not

predicted at all by the simple model. This component of the bearing transmissibility

spectra is primarily due to coefficients kbx0y and kb0y0y which are obviously not

included in the simple model. Our predictions also show that higher modes over 250-

350 Hz contribute significantly to the bearing transmissibility which are not seen in the

force transmissibility spectra yielded by the simple model.

Now we compare the mount load transmissibility in Figure 4.9. We observe that

the trends of mount force transmissibility RFxva,Fxsa(f2z) spectra predicted by both

models are similar, although the simple model predicts slightly lower amplitudes above

co1. The simple model again can not predict the moment transmission RMyva,Fxsa(_-2z)

across the mounts. Unlike the bearing transmissibility spectra, here the effects of _3

and _4 on RMyva,Fxsa(_2z) are more significant than ¢1 and t_2. This is mainly due to

the large angular motions on the casing for t_3 and ¢4.
127

E

0

100

10

1

.1

.01

0

Proposed Theory

", .......... Simple Model

ii

i

i !

i I

I'

i !

t 1

/ ;

41

, i

250 500

Frequency (Hz)

(a)

10 °

r_

_D

fi

o

10 "l

10 .2

10"

|0"

0

, J ,

250

Frequency (Hz)

500

(b)

Figure 4.8 Bearing transmissibility for example case I. (a) Force transmissibility

RFxba,Fxsa(_Z). (b) Moment transmissibility RMyba,Fxsa(_Z) which is not

predicted by simple model.
128

10 _

._ 10'

10°

o 10"

102

0

1ii Proposed Theory

_ .......... Simple Model

;: J,_

i

250 500

Frequency (Hz)

(a)

10'

.g.._ 10 o

o ,,,,,q

r_

_ 10"

b"

10 _

0

Proposed Theory

10 .3

0 250 500

Frequency (Hz)

(b)

Figure 4.9 Mount transmissibility for example case I. (a) Force transmissibility

RFx,,a,Fxsa(f2z). (b) Moment transmissibility RMyva,Fxsa(_'_z).
129

10 _

" 10'

"_ 10 °

10"

I02

10 -3

0

1.0E6 N/m

4.0E8 N/m

1.0E9 N/m

250

Frequency (Hz)

5OO

(a)

102

10'

10 °

"_ 10"

r_

10 -_

O 10-'

10"

10-'

0

f \.l.0E9 N/m

4.0E,8 N/m

i

250

Frequency (Hz)

500

(b)

Figure 4.10 Effect of mount stiffness kvx on the force transmissibilities for example

case I. (a) Bearing. (b) Mount.
130

102

r_

etO

10'

10"

10"

10"

10 .3

0

1.48E1 kg

1.48E2 kg

1.48E3 kg

250 500

Frequency (Hz)

(a)

102

t_

.,-.4

E

b-,

o

10'

10°

10 -I

10' I

10-' "

0

/", rl 1.48E1 kg

] ' 1.48E2 kg

_ .......... 1.48E3 kg

• , I , i ,

250 500

Frequency (Hz)

(b)

Figure 4.1 1 Effect of casing mass m c on the force transmissibilities for example case I.

(a) Beating. (b) Mount.
131

Next, we investigate the effects of rnc and kvx on the bearing and mount

transmissibilities.Figure4.10ashowsthatessentiallythesamebearingtransmissibilities

areobtainedfor threekvxvalues. In thecaseof themounttransmissibilityasshownin

Figure4.10b,theamplitudesin generaldecreasewith lowerkvxexceptnearresonances.

The casingmassmc alsodoesnot affect thebearingtransmissibility significantly as

shownin Figure4.1la. Themounttransmissibilityamplitudesincreasewith lowermc

asshownin Figure 4.11bexceptnearresonancesagain. It may benotedthat similar

trendsareobservedin thebearingandmountmomenttransmissibilityspectra.

4.6 EXAMPLE CASE II: GEARED ROTOR SYSTEM WITH RIGID

CASING AND FLEXIBLE MOUNTS

4.6.1 Bearing Analysis

Now consider the geared rotor system in Figure 4.1 with flexibly mounted rigid

casing whose bearing and other system parameters are given in Tables 4.1 and 4.5

respectively. Four ball bearings which support the two shafts are subjected to identical

mean axial displacement Szm. The spur gear pair drive is driven by a mean torque Tzs m

Tzsm(t) which also generates mean radial bearing force Fybm j and moment Mxbmj.

The stiffness matrix [K]bm for each bearing under these loads has significant

coefficients kbxx, kbyy, kbz z, kb0x0 x, kb0y0y, kbx0y, kby0x, kby z and kbz0x. A set of

governing nonlinear algebraic equations, consisting of 3 bearing load-displacement

relations for each bearing as given in Chapter II and from the shaft bending theory is

given by

Mxbmwi =TzsmLs/(8 dG)-2EI(2_xmwi+[_xmwli+3(Symwi-15ymwli)/Ls)/Ls ;

Mxbmwli=-Mxbmwi ; for Wll=ns+l if wi=l, and Wli=2ns+2 if wi=ns+2. (4.26)
132

Table 4.5 System parameters of example case II: geared rotor system shown in Figure

4.1 t

Gear and pinion masses mp = m G (kg)

Gear and pinion rotary inertias IyG=Iyp,lzG=Izp (kgm 2)

Number of gear/pinion teeth

Gear/Pinion Pressure Angle (degrees)

Shaft mass m s (kg) and length Ls (m)

Casing mass m c (kg) rotary inertias Ixc,Izc (kgm 2)

Shaft flexural rigidity EI (Nm 2)

Effective Torsional Stiffness kTM,kTL(Nm/rad )

Motor and Load inertia IzM, IzL

Gear mesh stiffness k h (Nhn)

Mean axial bearing displacement 6zm (N)

Mean input torque Tzb m (Nm)

kbyy ' kby0x ' kb0x0x tt

kvy, kv0 x, kv0 z

0.5

1.5E-4, 3.0E-4

28

20 °

2.8, 2.54E-1

77, 1.5, 1.9

1.25E4

6.05E3, 2.45E,4

1.00E-l, 3.35

1.0E8

6.00E-4

72

9.7E8, 6.0E5, 1.0E6

9.0E8, 1.4E7, 2.4E7

t The intemal rotating spur gear pair is currently being used in a NASA gear test

facility [55].

tt Only relevant bearing stiffness coefficients are tabulated.
133

Due to the physical symmetry, we assume 5ymw I = 5ymw n and _xmw I

simplify the nonlinear algebraic equation set for each bearing to

= -[3xmwi I which

Mxb m = Tzs m L s / (8 dG) - 2 E113xm / L s ; Fybm = Tzsm/d G (4.27)

where Mxb m and Fbyrn are expressed in terms of a constant _izm, and variables 8y m and

13xm. Solution to these 2 equations may then be used to compute kbij(6zm,Sym,13xm)

directly.

4.6.2 Vibration Models

A lumped parameter model of Figure 4.1 is developed in accordance with Section

4.4.1. Each shaft is divided into n s number of segments as shown in Figure 4.3a. The

system matrices of equation (4.4) are simplified by neglecting the longitudinal shaft

motion and casing rigid body degrees of freedom Uzca(t), Uxca(t) and 0yca(t). Both

driving and driven shafts are assumed to be coupled to the motor and load respectively

through flexible torsional couplings. Therefore, only the motor IzM and load IzL rotary

inertias are considered, and the exterior portion of the shafts beyond the flexible

couplings are modeled as purely torsional stiffness elements which are then combined

with the flexible coupling stiffnesses. The system is excited by the static transmission

error e(t) at the mesh point which generates gear and pinion force Fh(t) = k h e(t), parallel

to the line of action, and torque ThG(t) = 1/2dGkhe(t) or Thp(t) = 1/2dpkhe(t), about the

axial z-axis. The mass matrix [M] in equation (4.4a) in terms of {q(t)} = {Uysaj(t),

Uyca(t), 0xGa(t), 0zGa(t ), 0xPa(t), 0zpa(t), 0zMa(t), 0zLa(t), 0xca(t), 0zca(t)} T is
134

[M] = DIAG {msej,mxc,IxG,IzG,Ixp,Izp,IzM,IzL,Ixc,Izc } (4.28)

The matrices [K]ll, [K]12, [K]13, [K]22, [K]23 and [K]33 define the symmetric [K] of

dimension 2ns+ 11 in equation (4.4b). Nonzero elements of [K] 11 of dimension 2ns+3

are

kns+2,ns+ 2 = k2ns+2,2ns+ 2 = kll = kns+l,ns+ 1 = 12 EI/L3e + kbyy

kwl+l,w I = kwi,wl+ 1 = -12 EI/L3e ;

kwi+ns+2,wl+ns+ 1 = kwl+ns+l,Wl+ns+ 2 =-12 EI/L3e ; w I = 1,2,3,...,2n s

kwli,Wi I = 24 EI/L 3e ; kwli+ns+l,wu+ns+l = 24 EI/L 3e ;

wi! = 2,3 ..... ns/2,ns/2+2 ..... n s

kns/2+l,ns/2+ 1 = 24 EI/L 3e + k h ; k3ns/2+2,3ns/2+2 = 24 EI/L 3e+ k h

kns/2+l,3ns/2+2 = k3ns/2+2,ns/2+l = -k h ; k2ns+3,2ns+3 = 4kbyy + kvy

kwll/,2ns+3 = k2ns+3,WlI I = -kbyy ; Wli I = 1, ns+l, ns+2, 2ns+2

(4.29a)

(4.29b)

(4.29c)

(4.29d)

(4.29e)

The nonzero elements of [K] 12 of 2ns+3 rows and 8 columns are

k3ns/2-1,1 = -k3ns/2+3,1 = kns/2,3 = -kns/2+2,3 = 6 EI/L 2 ; k2ns+3, 7 = 4kby0x ;

-kns/2+l,2 = k3ns/2+2,2 = kh/(2dG) ; kns/2÷l,4 = -k3ns/2+2,4 = kh/(2d P) (4.30a)

kwiH,7=-kby0x+zwmkbyy ; kwili,8=-Xwnikbyy ; WliI=l,ns+l,ns+2,2ns+2 (4.30b)

The nonzero elements of [K]13 of 2ns+3 rows and 2n s columns are
135

kll = kns+2,ns+1= 6 EI/L2e+ kby0x

2

k3as/2+2,3ns12+ 1= -k3ns/2+ 1,3ns/2 = kns/2+ 1,ns/2+ 1 = -kns/2+ 1,ns/2 = 6 EI/L e

k2ns+3,1 = k2ns+3,ns = k2ns+3,ns+l = k2ns+3,2n s = -kby0x

k12 = -kns+l,ns-1 = kns/2,ns/2-1 = kns/2+2,ns/2+ 2 = kns+2,ns+ 2 = 6 EI/L2e ;

-k2ns+2,2ns-1 = -k3ns/2+l,3ns/2_l = k3ns/2+3,3ns/2+ 2 = 6 EI/L2e ; ns>2 (4.31b)

-kwlv,WlV-- 1=kwlv,WlV+ 1=kwlv+ns/2+ 1,wlvns/2+ 1=-kwlv+ns+ l,Wlv-- l+ns =6 El/L 2e ;

k wlv+ns + 1,w/v+ 1+ns=-kwiv+ns/2+ l,wiv+ns/2-1 =kwlv+3ns/2+2,WlV+3ns/2+ 1=6 EI/L 2e ;

-kwlv+3ns/2+2,WlV+3ns/2_ 1 = 6 EI/L2e ; ns>6 ; wlV = 2,3 ..... ns/2-1 (4.31c)

; kns+l,ns = k2ns+2,2n s = ---6 EI/L2e + kby0x ;

.

(4.31a)

where nonzero elements given by equations (4.31b) and (4.31c) exist only if ns>2 and

ns>_6 respectively. The nonzero elements of symmetric [K]22 of dimension 8 are

kll=k33=8EI/Le ; k22=kTM+khd2/4 ; k25=-kTM ;

k44=kTL+k hd2P/4 ; k24=k42=-khdGdp/4 ; k55=kT M ;

k77 = 4kb0x0 x + X (zj) 2 kbyy + kv0 x ; k88 = kv0 z + E (xj) 2 kbyy

k46 = -kTL

k66 = kvL

(4.32)

where kTL and kTM are the effective torsional stiffnesses due to shaft and torsional

coupling respectively at load and motor ends, and the summation Z is over the 4 ball

bearings denoted by j = 1, ns+l, ns+2 , 2ns+2. The nonzero elements of [K]23 of 8

rows and 2n s columns are

kl,ns/2 = kl,ns/2+l = k3,3ns./2 = k3,3ns/2+ 1 = 2 EI/L e

k7,wv = + Zwii I kby0x - kb0x0 x ; k8,w v = -Xwn I kby0x

(4.33a)
136

wiii = 1 (w v = 1),ns+ 1(wv=ns),ns+2(w v =ns+ 1),2ns+2(wv=2ns) (4.33b)

Finally, the nonzero elements of symmetric [K]3 3 of dimension 2n s are

kll = kns,n s = kns+l,ns+ 1 = k2ns,2n s = 4 EI/L e + kb0y0y

kwvI,wv I = kns+wvi,ns+wv I = 8 EI/L e ; ns>2, wvI = 2,3 ..... n s

kwvii,wvli_ 1 = kwvlr__l,wvii = kns+wvu,ns+wvli__ 1 = 2 EI/L e ;

kns+wvli__l,ns+wvi I = 2 EI/L e ; ns>2, wvi I = 2,3,...,n s (Wvi I _ ns/2+l )

(4.34a)

(4.34b)

(4.34c)

As discussed earlier, our proposed vibration model again differs from the conventional

spur gear pair dynamic models essentially due to [K]bm formulation. The simple model

with casing and mount dynamics can be obtained from our proposed model by retaining

only kbxx, kbyy and kbz z in [K]b m.

4.6.3 Eigensolution

The natural frequencies 03j and modes _j of Figure 4.1 with system parameters

given in Tables 4.1 and 4.5 are computed using the proposed theory, a dynamic finite

element model, and the simple theory with and without casing and mounts. The FEM

model includes casing flexibility - this will be described later in example case II. Our

proposed theory differs from FEM by less than +4% for the first six 03j and is within

+10% for 037 and 038 as shown in Table 4.6a. The corresponding natural modes are

described in Table 4.6b. Simple models in general predict lower natural frequencies and

deviate substantially except for 031 and 603. The simple model without any casing
137

Table 4.6 Eigensolution of example case II

a. Comparison of natural frequencies 0_j in Hz

Proposed

Mode (_j Theory FEM Al %

Simple Models

A

without casing with casing A2%

1 0 0 0.0

2 31.0 31.0 0.0

3 480 480 0.0

4 526 525 -0.2

5 563 564 0.2

6 790 760 -3.9

7 1147 1093 -4.9

8 1208 1100 -9.8

0 0 0.0

29.8 29.1 6.1

not predicted 480 0.0

not predicted 510 2.9

not predicted 560 0.7

666 650 14.5

837 809 26.0

694 681 38.1

A a% = 100 x (FEM - Proposed Theory) / FEM

A2% = 100 x (FEM - Simple Model) / FEM

b. Summary of mode shapes

Mode _j Description

1

2

3

4

5

6

7

8

Motor-gear-pinion-load system rigid body torsional motion

Motor-gear-pinion-load system torsional motion

Casing rigid body rotational motion 0xc a

Casing rigid body translational motion Uyca

Casing rigid body torsional motion 0zc a

First shaft coupled transverse-torsional motion

Second shaft coupled transverse-torsional motion

Shaft transverse motion
138

dynamics as outlined here is in excellent agreement with the results reported by

Kahraman et al. [15]. This provides further validation for our theory.

The effect of casing mass m c (and inertias Ixc, Izc o, mc) is shown in Figure 4.12a

for 033, 0) 5 and 0)7- Here 032 and 036 are unaffected by rn c. On the other hand, 033 is

always proportional to (mc) "1/2 except when mc/(ms+mG+mp) << 1. This

proportionality feature, similar to the single degree of freedom theory, is also seen for

607 and 608 for a lightweight casing, and 604 and 035 for a massive casing. It may be

noted that the trends of 0)8 and 034 are similar to or-I and 035 respectively although they

are not included in Figure 4.12a. Unlike example case I where the transition point is at

mc/(ms+mG+mp) = 1, two transition points are observed here due to the effects of the

gear, pinion and casing rotary inertias.

The effect of mount stiffness kvy (and kv0 x, kv0 z _ kvy ) on 0)j is given in Figure

4.12b. We again find that 602 and 606 are not affected by a variation in kvy. Also, two

transition points are found which separate a region of constant 0)j from the region where

60j ,_ (kvy) 1/2 for some of the natural frequencies. Here 038 is similar to 037 which is

proportional to (kvy) 1/2 for stiff mounts but is insensitive to kvy for softer mounts. The

converse is seen for or2 with a lower transition point. On the other hand, 035 *" (kvy) 1/2

in the region between the two transition points, and is constant elsewhere. Finally, 0) 3

is seen to be always increasing with kvy.

The efi'¢ct of mean bearing displacement 15zm is summarized in Table 4.7 for two

extreme _Szm values. Only the natural frequencies associated with the shaft transverse

and/or torsional motion are sensitive to _Szm or preloads Fzbm(Szm ). The rate of

increase for 0)j is high for low 8zm, but the rate decreases with a higher _Szmas observed

earlier in example case I due to the stabilization of kbij.
139

10'

10'

g

m

_ 10"

N

Z

10-2 ........ • ........ * ........ |

10" 10' 101 10 2 10'

Normalized Casing Mass

(a)

o

o

$

Z

10'

10°

10-t

10"2 ........ • ........ , ........ • ....... ° .......

10" 10.2 10" 10° 10' 102

Normalized Mount Stiffness

(b)

Figure 4.12 Effect of casing mass rnc (normalized with respect to shaft and gear

masses) and mount stiffness kvx (normalized with respect to kbxx) on the

system natural frequencies cojfor example case II.
140

Table 4.7 Example case II: effect of mean bearing displacement 8zm on c0j (Hz)

5zb m (mm) tO1 032 ¢-03 ¢-04 ¢-05 036 ¢-07 ¢-038

0.06 0 29.9 480 520 562 764 852 852

1.00 0 31.0 480 526 563 791 1180 1215

4.6.4 Transmissibility Spectra

The response due to the static transmission error excitation e(t) at the mesh point is

computed using the same dynamic stiffness technique used for example case I. Here,

only the fundamental harmonic of e(t) at mesh frequency 03h is included. All

uansmissibility functions are normalized with respect to the magnitude of the gear mesh

force Fh(t ) = k h e(t). Figure 4.13a and 4.13b compare the bearing force RFyba,Fh(03h)

and moment RMxba,Fh(C.0h) transmissibility spectra as predicted by our formulation and

simple model. It may be noted that the transmissibility spectra for all four bearings are

similar. Although simple model compares reasonably well with our proposed theory for

RFyba,Fh(_h), it is not capable of predicting RMxba,Fh(_h) , as also seen previously. In

the case of the mount transmissibility shown in Figure 4.13c, only the net moment

Mzva(t) is transmitted. The resultant vertical force Fyva(t ) and moment Mxva(t ) are

negligible due to the force and moment cancellations at the mount feet. The Fourier

spectrum of the normalized dynamic transmission error p(t) = [yp(t)-

YG(t)+(dp0p(t)/2)-(dGOG(t)/2)]/e(t) is shown in Figure 4.13d. Only two modes, t)2

which is predominantly torsional vibration of the shafts and <_6 which is a coupled
141

10'

o_'_ o

10

,.Q°_._

10.2

o 10'

o

10-'

0

Proposed Theory

Simple Model

500 1000

Frequency (Hz)

(a)

"_" 10"

°,,-i

10 .2

"_ lO'

10_

E

o

10"

0

Proposed Theory

500 1000

Frequency (Hz)

(b)

,.-, 10'

E

:_ 10 °

o_..4

o,--i

az 10'

"_ 10'

[._ 10 -_

= 10"

o

10'

o

Proposed Theory

Simple Model

I i •

500 1000

Frequency (Hz)

(c)

1.4

1.2

1.0

Proposed Theory

Simple Model

.8 " J "

0 500 10013

Frequency (Hz)

(d)

Figure 4.13 Comparison of transmissibility and normalized dynamic transmission error

spectra for example case II. (a) Bearing force transmissibility

RFyba,Fh((.0h). (b) Bearing moment transmissibility RMxba,Fh(O) h) which

is not predicted by simple model. (c) Mount transmissibility

RMzba,Fh(O)h). (d) Normalized dynamic transmission error p(c0h).
142

10

0 500

10" -f

10-'

lOOO

6E-2 mm (c)

6E-1 mm

I

4

°_,,I

10 "_

10"

= 10"

O

l0 ._1 !

0 500

Frequency (Hz)

1000

Figure 4.14 Mount transmissibility studies for example case II. Effect of (a) casing

mass mo (b) mount stiffness kvx and (c) mean bearing displacement 5zm

on RMzba,Fh(O)h).
143

10'

._ 10"

,. 10 .2

10-'

0

10 _

0

(a)

L

"_// 'L,'

6E-2 6E-1 mm \_/

i

500 1000

,_10'

[ (b)

 l°il

lO [I

_ 10"

10 _,

0

6E-2 mm

6E-1 mm

!

500 1000

1.2

1.0

i

08 i

0

6E-2 mm

6E-1 mm

f

(c)

500 t CX)O

Figure 4.15 Effect of mean bearing displacement _izm on (a) bearing force

transmissibility RFyba,Fh(03h), (b) bearing moment transmissibility

RMxba,Fh(f.0h), and (c) normalized dynamic transmission error p(c0h) for

example case II.
144

transverse-torsional mode, affect p(c0h) significantly. The sixth mode produces a higher

p(03h) amplitude at o h than _2- Additionally, the simple theory underestimates p(c0h) as

compared to the proposed model.

The effects of rn o kvx and 8zm on the transmissibility spectra and P(Ola) are given

in Figures 4.14 and 4.15. Mount transmissibility RMzva,Fh(C0h) is lower for a heavier

casing, more compliant mount and higher 8zm as shown in Figure 4.14. In addition,

the effects of m c and kvx on RMzva,Fh(C0h) are more prominent than those of _Szm. The

bearing transmissibility spectra and p(tOh) are affected only by 8zm as shown in Figure

4.15, and are virtually insensitive to rn c and kvx for the parameters given in Tables 4.1

and 4.5. Vibratory force and moment transmission through the bearing are slightly

lower for a larger 8zrn due to the additional constraint provided by higher kbi j values on

the rotating shafts. Conversely, normalized dynamic transmission error p(coh) increases

if a larger bearing preload is specified.

4.7 EXAMPLE CASE III: GEARED ROTOR SYSTEM WITH RIGIDLY

MOUNTED FLEXIBLE CASING

4.7.1 Physical Setup

The final example case examines the NASA gear test facility as shown in Figure

4.16 [55]. The system parameters are equivalent to those in Tables 4.1 and 4.5 except

for the flexible steel casing of approximate dimensions 0.33 x 0.28 x 0.25 m and plate

thickness of 0.006m. Rigid mount feet attach the four corners of the bottom casing plate

to a massive foundation. High precision gear and pinion are used which are identical

with 0.006m facewidth, 0.089m diameter and 1:1 ratio. Four axially preloaded high

precision deep groove ball bearings are being used to support 0.03m diameter shafts of

length 0.254m on the flexible casing. The input and output shafts are only coupled
145

TORSIONAL TEST -- BEARINGS

GEARBOX

SPEED

INCREASER

/

(a)

(b)

Figure 4.16 Example case Ill: (a) Schematic of the NASA gear test facility. (b) NASA

spur gear pair system supported on rigidly mounted flexible casing [55].
146

torsionally to the rest of the gear test facility through flexible couplings, and the geared

system is driven by a 149kW DC motor. Vibrational level at various positions on the

casing plate are measured using PCB 303A and Endevco 2271 accelerometers over the

operational speed range of 2250-5750 rpm which corresponds to the gear mesh

frequency coh range of 1050-2683 Hz.

4.7.2 Vibration Models

A dynamic f'mite element model of Figure 4.16b is developed using a commercial

software [39] as outlined in Section 4.4.1. The magnitudes of kbi j, which are included

in the FEM model through a 6 dimensional generalized stiffness matrix element, are

computed using the two nonlinear algebraic equations (4.27) which neglect the static

elastic deformation of the casing plate. The shafts axe modeled using 2 noded

Timonshenko beam elements with axial degrees of freedom capability. Four noded

quadrilateral plate elements with shear deformation and rotary inertia effects are used to

construct the flexible casing. The shafts and torsional couplings exterior of the test

gearbox are modeled as torsional stiffness elements only, and the gear, pinion, motor

and load are incorporated in the FEM model using generalized mass and inertia

elements. Gear mesh coupling between the gear and pinion is described by a generic

stiffness matrix of dimension 6, similar to [K]bm. Free rotational 0za(t ) boundary

condition is specified at the motor and load inertias, while ideally clamped Uwa=0wa=0,

w=x,y,z, boundary points at the comer of the bottom plate are assumed to simulate the

rigid mount feet.

Natural frequencies and modes predicted by a similar FEM model have already

been verified by comparison with experimental modal analysis in an earlier publication
147

by us [50]. Hence, eigensolutions will not be discussed here. But, it may be noted that

the conventional geared rotor dynamic models with simple bearing models and without

casing dynamics are not valid for high excitation frequencies (beyond the first casing

plate elastic mode). For the forced response study, only the fundamental harmonic e%

of the static transmission error excitation e(t) = 3.5 sin(0_t) gin computed from the gear

tooth profile and geometry [46]. About 110 dynamic degrees of freedom are selected to

minimize computational effort while still maintaining sufficient accuracy. Modal

damping ratio _ = 0.05 is assumed over the frequency range of interest.

4.7.3 Casing Response

Several locations on 3 different casing plate surfaces as shown in Figure 4.17 have

been chosen for the experimental validation of our theory. Figure 4.18 compares

predicted and measured mean square acceleration spectra at coh over 0 kHz to 4 kHz

even though measurements have been conducted only between 1 kHz to 3 kHz. In

general, good agreement is found between experiment and theory. Some discrepancy is

observed above 2.4 kHz which is due to the limitation of the FEM model in the high

modal density regime. It may be noted that the simple theory which utilizes only

conventional bearing models cannot predict flexural vibration of the casing plate as

shown in Figure 4.18. A broad band vibratory energy comparison is achieved by

averaging the mean square value of the acceleration over one-third octave bands, L A in

dB per unit frequency bandwidth Ao3, which is defined as follows

L

A'dB=101ogl0[(2_-_)ZRe{Act) <(AA*)A2>ref}]

; <A2>ref = 1.0 g2 (4.35)
148

Y

Z

Bearing Plate 18(c): ii x

Driving Shaft

/ p>,.----....._.,,<,,_d

Driven Shaft Mount Feet

Figure 4.17 Measurement locations on the casing plate for example case 11I.
149

v

o

°..._

0

<

ea_

v

0

0

0

<

O

<

Figure 4.18

10'

10 _

10'

10 °

10'

10 -_

10"

10 _

0

10 _

A Experiment

/ , -- Proposcd"llaeory

20OO 400O

I0'

10°

10'

10 -_

10"

10"

0

10'

- A t, /k Experiment

_Proposed Theory

i

2000 4000

10' (c)

10" _ A Experiment

_Pmposed Theory

10 ....

0 2000 4000

Frequency (Hz)

Comparison of casing flexural vibrations for example case Ill with 8zm =

0.08 mm. (a) Side plate supporting the bearings and facing the

dynamometer (location 18a). (b) Top plate cover (location 18b). (c) Side

plate with no bearings (location 18c).
150

where A* implies the complex conjugate of the acceleration. Our predictions are within

+5.0 dB of the measured values as shown in Table 4.8.

The sensitivity of results to [K]bm is evaluated next. Recall that [K]b m is

computed based on the knowledge of the mean axial displacements _zm applied to the

beatings. This parameter has been assumed to be a constant in our analysis although its

exact value for the NASA experimental setup is not known. Using 2 reasonable values

of [izm, we obtain a range of acceleration spectra in Figure 4.19. Almost all the

experimental data are now within this range. This explains experimental scatter

observed in Figure 4.18 as experimental _Szrn may vary slightly from one steady-state

speed to another during testing.

In example case II, we found that only a few coefficients in [K]bm are necessary

depending on the excitation. Accordingly, we had used only kbyy, kby0x and kb0x0 x.

But in the present FEM model of case III, we must include the entire symmetric [K]b m

matrix of dimension 6 with no simplification at all. Now if we omit the off-diagonal

term kbx0y which is not directly excited by e(t) in the y-direction of Figure 4.1 while still

retaining other kbi j coefficients, no significant changes in our predictions are seen.

Similarly, if the off-diagonal terms kbzy and kbz0x which are related to the shaft axial

degrees of freedom are also neglected, predicted acceleration spectra vary slightly.

tIowever, if kby0x and kb0x0 x which constraint the bending motion of the shaft excited

by e(t) are assigned zero values per simple theory, zero out-of-plane acceleration levels

are observed. This confirms that the vibration transfer through the bearings, from the

shaft bending motion to the casing flexural motion, is highly dependent on these two

terms.
151

Table 4.8 Comparison of L A (dB) for example case III.

Center

Frequency (Hz)

18(a)

Location in Figure 4.17

A

18(b) 18(c)

1250

1600

2O00

250O

Theory

Experiment

Theory

Experiment

Theory

Experiment

Theory

Experiment

5.9

4.0
152

10'

10 _

10'

10°

10"

-_ ]0"

O

< 10"

10 _

10"

0

10 _

10'

_o 10 °

cO 10'

.,=.,

0 10 .2

o

o

¢_}

< 10-'

10 _

lO -j

o

lO'

10'

10'

¢: 10 °

O

10"

o 10 -_

¢J

¢J

< 10"

10 _

10 .5

0

Af V _ _ _.i f_'_ _._ \

_ /.,a-, j, a a #wff \v/'-

_/--//-'\\/ a Experiment

// 2E-2 mm

/ 2E-I mm

I

/

/ J

2000 4000

/--/

//_z A Experiment

/

/t_ / • 2E-2 mm

/// 2E-1 mm

f/ . ,

2OOO 4OOO

/ 2E-1 mm /

/

/ . ° , i , ,

2OOO 40OO

Frequency (Hz)

Figure 4.19 Sensitivity of predicted casing acceleration to preload _zm for example case

III. (a) Side plate supporting the bearings and facing the dynamometer

(location 18a). (b) Top plate cover (location 18b). (c) Side plate with no

bearings (location 18c).
153

4.8 CONCLUDING REMARKS

A genericgearedrotorsystemmodelhasbeendevelopedusinglumpedparameter

anddynamic finite element techniques which incorporates a new mathematical model for

precision rolling element bearings proposed earlier in Chapter II. This system oriented

model includes internal rotating system, rolling element bearings, flexible torsional

couplings, motor and load, flexible or rigid casing, and compliant or massive mounts.

The discrete shaft model excludes the effect of gyroscopic moment, but includes

torsional, flexural and longitudinal motions. In example cases I and II, only flexural

and torsional motions are predicted. The effects of mount stiffness, casing mass and

bearing preload on the overall dynamic behavior have been investigated through 3

example cases of single-stage rotor and geared rotor systems excited by rotating mass

unbalance at shaft frequency and kinematic transmission error excitation at mesh

frequency respectively. The results indicate that our proposed model is more accurate

than conventional models given in the literature. For instance, we are able to

predict bearing and mount moment transmissibilities and improve casing flexural

vibration prediction significantly using our theory; measurements made on case III

validate our formulations partially. Conversely, the conventional models are unable to

account for any bearing moment transmissibility, and consequently predict zero flexural

response on the casting.

Natural frequencies of the example case III geared rotor system increase with

higher mount stiffness and bearing preload, and lower casing mass. Transition mass

and stiffness points which separate regimes of a natural frequency diagram are

predicted. Through extensive parametric studies, we find that bearing transmissibilities

may be reduced by using a higher bearing preload. However, this may result in higher
154

dynamic transmission error for the spur gear pair, mostly due to the coupled flexuraltorsional motion. Mount transmissibility is affected by changes in casing mass, mount

stiffness and mean bearing axial displacement, but bearing transmissibility is most

sensitive to the mean bearing axial displacement. Similar conclusions can be drawn

regarding the other two example cases.

Our theory, though restricted to the linear and time-invariant dynamic system, is

comprehensive. It can be used for analysis as well as design studies of other rotating

mechanical systems with multiple shafts and gear pairs or multi-staged rotors.
CHAPTER V

STATISTICAL ENERGY ANALYSIS

5.1 INTRODUCTION

It has been shown in Chapters II-IV, using classical lumped parameter and dynamic

finite element techniques, that the proposed bearing model is more accurate than

existing models for predicting vibration transmission through bearings in a geared rotor

system. Although the proposed model has been shown to be reliable up to a moderately

high frequency, it is conceivable that this model is inadequate at very

high frequencies where the modal density is high. Classical vibration models do not

predict modes accurately in this frequency regime, and even if it is possible to do so by

employing closely spaced nodal points, such models require a significantly large

computational effort. Moreover, the vast amount of predicted response spectra at many

spatial points would be difficult to interpret. Accordingly, asymptotic or statistical

methods must be adopted; typical techniques include the statistical energy analysis [56-

60], asymptotic modal analysis [61-63] and asymptotic analysis using infinite system

impedances [64].

This study concentrates on the development of a broad band vibratory energy

transfer model for a geared rotor system with the proposed bearing model using the

statistical energy analysis (SEA) method. This method has been applied successfully to

a wide variety of structural dynamic and acoustic systems with large number of modes

155
156

[56-60,65-69]. However several unresolved research issues still exist [57,61,62,70,

71]. The specific objectives of this study are to: (i) conduct modal analysis of a geared

rotor system with flexible casing and mounts, (ii) investigate the feasibility of applying

SEA to this problem, (iii) analyze the following 4 example cases using SEA: I. a platecantilevered rectangular beam, II. case I with circular shaft-bearing system replacing the

cantilevered beam, III. a circular shaft-bearing-plate-mount system, and IV. a simple

geared rotor system, and (iv) perform parametric studies to examine the characteristics

of vibratory energy transfer through bearings, and mean square vibroacoustic response

of the casing. The first and second example cases are revised and extended versions of

a study performed by Lyon and Eichler [56,58]. The later two example cases are taken

from Chapters HI and IV where these systems have been studied at low frequencies.

Experimental validations are also included.

5.2 MODAL ANALYSIS OF GEAR CASING AND MOUNTS

Analytical and experimental modal analyses of a real gear casing and mounts have

been conducted to investigate the feasibility of using SEA. Natural frequencies c0j and

modes qbjare calculated using a commercial finite element method (FEM) program [39],

and predictions for a rigidly mounted, stiffened gearbox are verified by the experimental

modal analysis (EMA).

5.2.1 Finite Elemen! Model

The rectangular gearbox as shown in Figure 5.1a is approximately 0.254m x

0.279m x 0.330m (10.0in x ll.0in x 13.0in), and all of its plates are 0.006ha (1/4 in)

thick made of 1020 steel except the regions near the bearings which are 0.025m (1.0in)
157

OUNTS

(a)

_ Flexible III \

mount _ _ 0.0064m thick

aluminium fuselage

sheet with dimensions

0.7620m x 0.6640m

supported in between

gearbox and mounts

(b)

Figure 5.1 Gearbox with rigid and flexible mounts.
158

thick. There are four circular holes for the bearings, two at each side plate supporting

the shafts. Figure 5.1b illustrates the 0.254m (10.0in) tall flexible mount frame which

is constructed from eight 0.006m (114 in) thick, 1020 steel angle beams with three

different lengths. Fuselage sheet, as shown in Figure 5. lb, is attached horizontally to

the flexible mount structure. The casing is supported at each comer of the base plate for

all mounting conditions, and the mounts are attached to a rigid foundation.

Two FEM models of the rectangular gearbox without its spur gears set, shafts and

bearings are developed for the rigidly and flexibly mounted casings. These FEM

models consist of four-noded quadrilateral plate elements with bending and membrane

capabilities for the housing and attached fuselage, and two-noded shaft element with

shear deformation and rotary inertia capabilities for the flexible mount skeleton and

housing plate stiffeners. The boundary conditions are: (i) zero displacements and

rotations at each comer of the base plate for the rigid mount, and (ii) similar conditions

at each foot of the flexible mount. The interfaces between adjacent housing plates are

assumed to be continuous. About 100 dynamic degrees of freedom are specified to

reduce computational effort while still maintaining sufficient accuracy. Natural

frequencies are computed up to at least 4 kHz to cover the gear mesh frequency regime.

5.2.2 Experiments and Model Validation

Modal experiments have been performed on a NASA high precision gearbox with

the spur gear set, shafts, and ball bearings installed. An approximate configuration of

the NASA gearbox is shown in Figure 5.2. The nominal dimensions of the gear

housing have been given in the previous section. The variable center distance gear-shaft

pair is supported by four ball bearings. Four side plates and a base plate are welded
159

WVVvVV 

Speed Increaser Gearbox .__ Dynamometer

PZ>Z,':///

 cr°ph°n 8 m

SECTION

A-A 11'

0. 7

,0.006 m thick

1020 steel plate

J_

I

V

A

Stiffener

SECTION B-B

0.279 rn

0.006 m

I

I

I

!

........ I ..............

II I II

II I II

II I I I

II I I I

I! I I I

II I II

I I

L --_" B

r, 0.330 m

c--

0.140 m

|- - _

.J

J"i

A

!

Figure 5.2 NASA gear test facility [55].
160

together while the top plate is bolted to the side plates. The housing plates are stiffened

internally, and the gear housing system is mounted rigidly to a massive foundation.

Dynamic transfer functions are obtained only on the exterior of the gear casing structure

using the GENRAD 2515 system [72]. For these experiments, 154 measurement points

have been selected in the direction transverse to the plane of the casing plates with the

reference point being approximately near the center of the top plate to avoid nodal points

of interest. Natural frequencies and modes are then estimated using the MODAL PLUS

program [73]. Here, the exponential method has been used to extract modal parameters

and generate analytical functions for the transfer functions, while the circle fit method

has been used to construct the modal vectors.

Figure 5.3 compares predicted and measured t.oj, and FEM is found to be in good

agreement with EMA. For each mode _j, two simplified illustrations are shown in

Figure 5.3: (i) mode shape of the three visible plates ,and (ii) mode shape of the three

nonvisible plates in an approximate isometric view. The higher modes, not shown here,

are also given by similar combinations of plate flexural motions. Comparison between

theory and experiment for the higher modes is made on the basis of number of modes

within each one-third octave bands over 500-4000 Hz in Figure 5.4a, because of the

high number of participating modes observed. The results again indicate that FEM is in

good agreement with EMA.

5.2.3 Parametric Studies

The rigidly mounted gear housing is observed to possess only the elastic modes of

the casing plates. On the other hand, FEM model of the flexibly mounted gear casing

indicates that the first six modes are translational and rotational rigid body modes of the
161

031 = 501 Hz o 2 = 598 Hz

(497 Hz) {584 Hz)

033 = 627 Hz 034 = 752 Hz

(639 Hz) (781 Hz)

035 = 889 Hz

(877 Hz)

Figure 5.3 Mode shapes of the rigidly mounted, stiffened NASA gearbox as predicted

by FEM (EMA results are given in parenthesis). Bold sign implies larger

amplitude.
162

0

0

m

12

I0

8

6

4

2

0

0.50

measured

predicted

(EMA) - - - -

(FEM) ,"_,',"

S J

0.63 0.80 1.00 1.25

One-Third Octave Band

1.60 2.00 2.50 3.15 4.00

Center Frequency (kHz)

(a)

25

20

¢J

0

0

L_

IO

z

flexible /_k

...... rigid / _A

J _

i _'J

5

L'S-' .....

.063 . I . 160 .25 .4 .63 1.0 1.6 2.5 4.0

One-Third Octave Band Center Frequency (kHz]

(b)

Figure 5.4 Number of modes within each one-third octave bands for the NASA

gearbox. (a) Predicted and measured number of modes for rigidly mounted

casing. (b) Number of modes for rigid and flexible mounting conditions.
163

the casing as shown in Figure 5.5; for example, 031 = 54 Hz corresponds to the casing

vibrating in the Y-direction as shown. In addition, 03j are considerably lower, by

approximately one order of magnitude as compared to those of the rigidly mounted gear

casing. These rigid body vibration modes result from the complex elastic deformations

of the flexible mount skeleton and fuselage sheet. The casing plate natural frequencies

are also lowered, especially the first few, when the box is mounted flexibly. Figure

5.4b compares the number of modes of the flexibly mounted gear casing to that of the

rigidly mounted one. High modal counts above 1 kHz are seen.

The introduction of gear casing plate stiffeners as shown in Figure 5.2 does not

change the nature of the mode predictions: Also, the natural frequencies for this case

only vary slighfly; lower modes are affected more by the stiffeners than the higher ones.

Note that the numbers of modes in each 1/3 octave band over 400-4000 Hz range remain

nearly the same as evident from Table 5.1.

Table 5.1 Number of modes for the stiffened and unstiffened gearboxes mounted

rigidly.

1/3 Octave Band Center

Frequency(kHz) 0.4 0.5 0.63 0.8 1.0 1.25 1.6 2.0 2.5 3.15 4.0

Stiffened 0 1 2 3 1 5 7 6 10 1 1 1 1

Unstiffened 1 3 2 1 2 6 8 7 1 1 10 1 1
164

Z X

0

Z

0) 3 = 72 Hz 0,)4= 93 Hz

0x 0y

Figure 5.5 Rigid body modes of the flexibly mounted gearbox as predicted by FEM.
165

5.3 JUSTIFICATION FOR USING SEA

5.3.1 Modal Densities

Analytical and experimental modal analyses of the NASA gear casing clearly show

that modal density tends to be high at higher frequencies. Although the analysis is

restricted to a gear casing system, it is reasonable to assume that similar results are valid

for other rotating mechanical system housings. Accordingly, SEA can be used

justifiably.

Next, consider the modal densities of a shaft or beam ns and rectangular plate nc,

given for bending motion with simply-supported boundary conditions [56]

j 2

Psnhs A c 3Pc(1-I.t 2)

ns=Ls 4EI s03 2 ; nc= _- (5.1)

where p is the material density, A c is the plate surface area, L s is the shaft length, E is

the modulus of elasticity, I s is the area moment of inertia of the shaft, 03 is the

bandwidth center frequency, h is the plate or shaft thickness, _ is the Poisson's ratio,

and the subscripts s and c denote shaft and plate respectively. For typical numerical

values given later in Section 5.4 we find that: (i) nc = 0.227, and (ii) n s = 0.012 at 100

Hz, n s = 0.004 at 1000 Hz and n s = 0.002 at 4000 Hz. Note that nc >> n s as expected.

Although the shaft has a fairly low number of modes in the frequency range of interest,

SEA is still valid since the plate modal density n c is very high.
166

5.3.2 Literature Review

A majority of publications are on the application of the SEA procedure to

dynamic systems with high n such as structural-acoustical interactions in a fuselage

[56,65,74], sound transmission through panels [60], and vibratory energy transmission

in mechanical equipment [56]. Of interest here are analytical or experimental estimation

of SEA parameters for simple structural systems described by a flat plate, a cylinder

and/or a shaft [56,58,59,66]. In these studies, structural connections are often assumed

rigid such as in the ideally welded case.

Lyon and Eichler [56,58] in 1964 and Lyon and Scharton [59] in 1965 developed

analytical expressions for the coupling loss factor _ in several connected structures,

such as a plate bonded to a cantilevered beam [56,58]. Here r1 was derived assuming a

semi-infinite beam attached to an infinite plate and by further assuming that only a

dynamic moment coupling at the joint can describe the motion/force transmission

phenomenon. This problem is re-examined in this study and is then extended to a

circular shaft-bearing-plate-mount system. In addition, Lyon and Eichler [56,58] also

developed SEA models of two structures inter-connected through a single (scalar)

stiffness element. Typical examples include two longitudinal rods connected via a linear

stiffness element, two discrete masses coupled by a linear spring, and a plate attached to

a single degree of freedom resonator [56]. A similar analysis on the longitudinal

vibration of linearly coupled rods was performed by Keane and Price [69] in 1987.

Loss factors of typical line or point connected structures such as a plate welded to a

cylinder, cross beams and two perpendicular plates bolted or welded together, have been

calculated assuming ideal rigid joints [56,58,59,75,76]. But a compliant bearing system

problem is yet to be analyzed.
167

5.4 EXAMPLE CASE !: COUPLING LOSS FACTOR OF PLATECANTILEVERED BEAM SYSTEM

First, we attempt to rework the plate-cantilevered beam problem of Lyon and

Eichler [56,58] as shown in Figure 5.6. Only flexural motions of the plate and beam are

considered in this case. Accordingly, Lyon and Eichler [56,58] developed an

expression for the coupling loss factor qsc which describes the vibratory energy transfer

between the beam(s) and the plate(c) due to a moment coupling at the joint

tom s + Zs Re (5.2)

where 1_='_Is/Asr- is the radius of gyration, c=_/E/p s is the wave speed, m is the mass

and Re( ) implies the real part of a complex variable. The driving point moment

impedances for the plate Z c and beam Z s are [58,77]

16p c hc_¢2c c 2 PsAs KsCs 2 2 k s

Zc(to) = 4i ; Z s(to) = to (I-i) (5.3)

to (1- _ ln(kchs) )

where k is the wavenumber and h is the plate or beam thickness. Here, note that Z s of

equation (5.3) differs from Lyon and Eichler's expression [56,58] in the sign of the

imaginary part which is probably a typographical error. They assumed that IZcl>>lZsl,

_:s=_:c, Cs=Cc and ps=Pc, and simplified equation (5.2) to yield a frequency invariant

expression for qsc as

rls c = Ws/(4Ls) (5.4)
168

y

Response

Ls

_.0_ _v_g___c

(a) Finite System

Semi-inf'mite Shaft _K.. Shaft

"_Response

-,,Rigid Connector

N_. ¢ Dr!.ving

"_ +Plate _F P°mt _

(b) Infinite Systems

Figure 5.6 Schematic of example case I: a plate-cantilevered beam. (a) Experimental

finite System. (b) Theoretical infinite system.
169

where W s and L s are the beam width and length respectively as shown in Figure 5.6.

However, our calculations indicate that IZcl>lZsl but not IZcl>>lZsl for the parameters

used by Lyon and Eichler [58], as shown in Figure 5.7a. Using equation (5.2), lqsc is

rccomputed and compared with the approximate model given by equation (5.3) in Figure

5.7b. Experimental results given by Lyon and Eichler are also plotted in Figure 5.7b. It

can be seen that our calculation is better than earlier prediction. At low frequency, a

discrepancy is observed between theory and experiment which is primarily due to the

low modal density in this regime. The presence of a low natural frequency may be due

to the compliant epoxy bond between the beam and plate. However, above the

threshold frequency where many modes participate, shown as a vertical line in Figure

5.7b, the slope of the least square straight line fit on the experimental data is nearly the

same as the predicted rlsc.

5.5 EXAMPLE CASE II: COUPLING LOSS FACTOR OF CIRCULAR

SHAFT-BEARING-PLATE SYSTEM

Next, we modify Figure 5.6 by inserting a ball beating between the circular shaft

(which replaces the beam in Figure 5.6) and the rectangular plate. Again, a semi-infinite

shaft and an infinite plate are assumed. For SEA, we reduce the system to a plate

subsystem and a shaft-bearing subsystem. The coupling loss factor rlsc is still given by

equation (5.2), but Z s must be modified to account for the compliant bearing.

Consider a shaft with boundary conditions shown in Figure 5.8. The bearing end

is subjected to zero transverse velocity Vysa(t,z=0-)=0 and a sinusoidally varying

moment Mxsa(t,z=0-)=Mxsa ek°t. Here, z=0- refers to the junction point between beating

and plate, and z=0 + is the junction point between the bearing and shaft. In the

frequency domain, using the definition of bearing force vector
170

0.6

10

Proposed

i i | i i i ! ,

100

Frequency (Hz)

(a)

1000

10-1

O

¢,9

r_

O

et0

.9

O

r,.)

10 -2

Low modal

density

I

I

A i

I

I

A

A Experiment

Least Square Curve Fit

of Experimental Data

Proposed

Approximate Model

A

10 -_ ' ' .......

10 2 10

1/3 Octave Band Center Frequency (Hz)

(b)

Figure 5.7 Comparison between Lyon and Eichler's approximation [56,58] and our

proposed formulation for exanaple case I. (a) Comparison of IZc[/iZc+Zsl.

(b) Comparison of predicted qsc with experimental results given by Lyon

and Eichler [56,58].
171

Mxsa(O.), z=O-) = Mxs a

v ysa(r.o, z= 0-) = 0

"2

Bearing

Semi-infinite Shaft

Z

z=0

Figure 5.8 Boundary conditions for example case II: semi-infinite shaft-bearing

system.
172

{f(c0)}b=[K]bm{V(Ol)}b/i_ in terms of the velocity vector {v(o_)} b and the proposed

bearing stiffness matrix [K]bm, Vysa(O.l,z=0-)=0 is shown to be equivalent to

Fysa(00,0+)=(kby0x/kb0x0x)Mxsa+Vysa(O3,0+) { (kby0x/kb0x0x)2kb0x0x-kbyy }/(ioi).

The governing equations for Mxsa(e0,z) and Fysa(O_,z ) of the shaft in terms of Vysa(O,z)

are

d 2v El d3vysa EI s ysa s

Mxsa = - i0_ dz 2 ; FY sa = "io_ dz 3 (5.5)

In general, the bounded solution for Vysa(CO,z) is

Vysa(O_,z) = { b 1 e -jksz + b2 e -Iqz } (5.6)

Using equations (5.5) and (5.6), we enforce the boundary conditions for the shaft at

z=0 + to obtain the following closed form solution for the coefficient vector

{b}={bl,b2} T.

2

c0k by0 x co k by0 x

i03 k kbyy

ie0Mxs a k b0,_0 x s EIk 2s kb0x0x

- 2 + M •

bl EIk s 3 by0x 2i k 2 xsa '

El ks(1 + i)- 2ik +

byy k b0 x0 x

olk by0 x CO k by0 x

icok _-

kb0x0x s E_ kbyy kb0x0x

b2 = 2ik 2 Mxsa (5.7)

by0 x

EIk 3(1 + i) -2ik

s

+

byy k

b0x0x
173

Hence, the bearing-shaft impedance Z s is then given by

Mxsa(¢0, O) kb0x0x

Z s(C0) = = (5.8)

dv dv

ysa (60, 0-) ysa (60 0 +) dz k by 0xV ysa(¢°" 0+) + k b0 x0x dz '

i03+ M

xsa

Although Mxs a appears in equation (5.8), Z s is independent of the magnitude of Mxs a

dv ysa

since Vysa(00,0 +) and _(c0, 0 +) are linearly proportional to Mxs a in equation (5.5).

Using equations (5.2), (5.7) and (5.8), rlsc is computed for several bearings

whose stiffness coefficients are tabulated in Table 5.2. Figure 5.9 compares these

where the material and geometrical properties of the shaft and plate are equivalent to

those used by Lyon and Eichler [56,58] in example case I. For a very soft bearing, Z s

is dominated by the bearing parameters, and hence IZcl>>lZs I is valid. This implies that

lqsc reduces to the frequency invariant expression given by equation (5.4) as shown by

set A in Figure 5.9. On the other hand, rlsc for a rigid beating (say set B or C) is

typically smaller than alsc for a soft bearing given by A. The extreme values of rlsc, as

kbi j or bearing preload becomes very large, depend on the relative magnitudes of the

bearing stiffness coefficients as shown in Figure 5.9 for sets D and E. Such deviation

for a stiff bearing from set B is primarily due to the presence of off-diagonal stiffness

coefficient such as kby0x. However it is found that rls c for most stiff bearings will

approach set E in Figure 5.9 which is obviously identical to the prediction for the

example case I using equations (5.2) and (5.3).
174

10 .2

O

0

011

O

r,.)

.......... ...........................................

A

"''°'_ ...... B

............. C

....... D

E

10 -3 , , , , ....

10 2 10 3

1/3 Octave Band Center Frequency (Hz)

Figure 5.9 Coupling loss factor _lsc used in example case 11 for various bearing

stiffness coefficients listed in Table 5.2.
175

Table 5.2 Typical bearing stiffness coefficients of Figure 5.9 for example case II.

Set kbyy (N/m) kbyOx (N) kbOxOx (Nm)

A Ve_ CompfiantBearings(kbu < 1E3)

B 1E5 5EA 2E4

C 5E5 8E4 5E4

D 1E8 1E7 IE6

E 1E8 3E7 1E7

5.6 EXAMPLE CASE III: A CIRCULAR SHAFT-BEARING-PLATE

SYSTEM

5.6.1 Theory

Consider a circular shaft-bearing-plate-mount system similar to example case II but

with a shaft of finite length as shown in Figure 5.10a. Note that this system has been

analyzed earlier using the deterministic vibration modeling technique for low frequency

response• Recall the unconstrained end of the non-rotating circular shaft is subjected to

a harmonically varying force Fys(t) = Fysa etc°t + Fysm, where Fysm is the mean force•

From Chapter III, we know that the longitudinal and torsional motions of the shaft, and

the in-plane vibration of the plate can be neglected. Hence, the shaft bending vibration

and plate flexural motion are of interest here. Also, the previous deterministic vibration

models indicate that the coupling between these two motions is mainly due to the

dynamic moment at the bearing provided the longitudinal shaft vibration is not excited.

Accordingly, two subsystems which can be easily identified using SEA are the

transverse modes of the shaft-bearing system and the flexural modes of the plate-mount
176

Flexible Shaft (s) __pplied Force

Mounts

(a)

Input Power

Shaft-bearing system

(bending modes)

E s (co), n s (co)

I-Isc

Energy Dissipation

Energy Transfer

Tic

Energy Dissipation

Plate-mount system

(flexural modes)

Ec (co), n c (co)

Figure 5.10

(b)

Example case III: (a) schematic of the circular shaft-bearing-plate-mount

system and (b) A SEA model of the shaft-bearing-plate-mount system.
177

system as shown in Figure 5.10b. We now proceed to derive rlsc governing the

vibratory energy flow between these two subsystems using the driving point junction

impedance method suggested by Lyon [56].

Since the rectangular plate dimension is much larger than the bearing dimension,

the plate is again assumed infinite. Consider the vibrational power flow Flsc from the

shaft-bearing subsystem (subscript s) to the plate (subscript c), due to the uniform

external Gaussian random force over a frequency bandwidth Am with center frequency

(.,0,

Es (co) E c(ta) )

I-Isc(Co) =cOrlsc(CO) ns(CO) ns(CO) nc(CO)

(5.9)

where nj and Ej (j = s, c) refer to subsystem modal density and total vibratory energy

respectively. Since the plate is assumed to be reasonably well damped and geometrically

large, equation (5.9) is approximated assuming nc>>n s or Ec/ne<<Es/n s to yield

rI_c(CO)

rl sc(C0) = cO Es(C0) (5.10)

For the shaft, E s = ms<V2> where m s is the shaft mass and <V2> is mean square shaft

transverse velocity averaged over Ac0 and shaft length L s. Using the expression for Hsc

given by Lyon and Eichler [58] as discussed in the previous section, rlsc in terms of Z s

and Z c is identical to equation (5.2) which is rewritten here for the circular shaft case as

rlsc(C0)- 4EI coL s Re ( 31Zc+ZZcs (5.11)
178

For an infinite steel plate of thickness h o the point moment impedance Z c is given by

equation (5.3); it is rewritten as

D

4 Eh 3c {1 4i (2-9-0-kcds) } 1

Zc(m)=3co_l-_2 ) ----_-In ; kcds<<l (5.12)

The shaft-bearing impedance Z s in equation (5.11) is derived next by solving the

boundary-value problem for the system shown in Figure 5.1 la. The free end of the

shaft has a vanishing dynamic shear force Fysa(0a,z=Ls)=0 and bending moment

Mxsa(O,z=Ls)=0. The bearing end is similar to Figure 5.8. Following the same

argument used previously for example case II will lead to the two boundary conditions

described by Mxsa(O,z=0+)=Mxsa and Fxsa(tO,z=0 +) = (kbx0y/kb0y0y)Mxsa +

Vysa(C0,0 +) { (kbx0y/kb0y0y)2kb0y0y-kbxx }/(it0). Governing equations are still given

by equation (5.5), but the solution for Vysa(C0,z ) is assumed to be of the following form

[77]

Vysa(t,z ) = { b 1 e -jksz + b2 e jksz + b3 e -ksz + b4 e ksz } (5.13)

Using equations (5.5) and (5.13), the boundary conditions yield the following algebraic

problem

[B]{b} = Mxs a { kby0x/kb0x0x, i_/(EIk_), 0, 0} T (5.14)
179

M xsa(O.), z=0-) = Mxs a

Vysa(O_,z= 0-) =0

Bearing

M xsa(O, z=L s) =0

F ysa(O), z=L s ) = 0

Flexible Shaft

z=0

v Z

(a)

V ysa (CO,z=-Ls) = 0

dvysa (o.),z_-_-L s) =0

dz

7J

,_ Flexible Shaft

Bearing

M xsa(O), z =0 ) =0

Fysa(O), z = 0) = Fxs a

-""----.-I_ Z

z=-Ls z=O

(b)

Figure 5.11 Boundary conditions for example case III: finite shaft-bearing system.

(a) Moment applied at the bearing end. (b) Force applied at the free end.
180

The nonzero elements of coefficient matrix [B] of dimension 4 are

lkby0x ik_

• Y0x 1 3 ik + .... , B = Elk 3 ik + ;

BII=-_ Elks- byy kb0x0 x 12 s- byy kb0x0 x

i 3 by0x kby0x

B 13 = _ Elk s - k byy + ; B 14 Elk 3

kb0,0x = s-kbyy + kb0x0x---- ;

B21=B22=-B23=-B24=l ; B31=-iB41=ie -jksLs ; B32=iB42=-ie jksLs

B33=-B43=-e -ksLs ; B34=B44=e ksLs (5.15)

Both [B] and {b} = {b 1, b2, b 3, b4} T call be easily obtained numerically. The bearingshaft impedance Z s is still given by equation (5.8).

The same procedure may be applied to obtain the driving point force impedance for

a harmonically varying transverse force Fysa(t,z--0). Note that the origin is redefined at

the forcing point as shown in Figure 5.11 b for convenience. The boundary conditions

dv ysa (co, - L s ) =0.

are Fysa(C0,z=0)=Fysa, Mxsa(C0,z=0)=0, Vysa(0_,z=-Ls)=0, and dz

These conditions at z---L s can be rewritten for z=-Ls + like the previous case to obtain

dv

Mxsa(°_'z=-Ls)=-(kb0x0x ysa (co, + dz - Ls)+kby0xVysa(f°'-Ls+))/(i_)

dv ysa + + .

Fysa(O),z=-Ls)=-(kby0x T(_,- L s ) + kbyyVysa(0_,-Ls))/(1c0 )

(5.16)

These prescribed boundary conditions again yield a set of algebraic problem similar to

Equation (5.14). The nonzero elements of the coefficient matrix [B] of dinaension 4 are
I81

Bll =-B2! = B12 = B22=-B13 = iB23 =-B14 =-iB24 =-1.0 ;

B

B 31 = ( - jkskb0x0x + kby0x +EIk 2 e ik'L' s)

B 32 = (jkskb0x0, + kby0x + EIk 2 iksL s)e- ';

33 (-kskb0x0x+kby0x - EIk2 kaL* = s)e ;

B 34 = (ksk b0x0 x + k by0 x -EIk 2 -ksLs s )e ;

B41 = (ik skby0x - k - iEIk 3 )eik,L, . byy s ,

B42=(_ikskby0x_kbyy+iEI k3s) e-ik_Ls

B43 = (kskby0x- kbyy +EIk 3s) ek'L* ;

B44 = (- kskby0x - kbyy_Eik 3S) e -k,L s (5.17)

Tile right hand side vector {b} of the algebraic problem is {0,FysaO/(EIk3s),0,0} T.

Force impedance at the driving point is then given by Zs(O,z=0)=Fysa/Vysa(C0,0).

Accordingly, the input power is I-Is = (1/2) F 2 Re{(1/Zs)* } where Re{ } is the real

ysa

part of the complex variable and ( )* implies the complex conjugation.

We can now compute the vibratory energy transfer Ilsc through the bearing and

steady-state subsystem energy levels E s and E c by applying the energy balances to both

subsystems shown in Figure 5.10b; here tics = rlsc ns/nc.

rls +71 sc(O)

- lq sc (o) 11c + rl cs(o) c(0 )J

(5.18)
182

Es(°)) = °3(fisT1 c + _cr}sc+rlsrl cs) '

Hsrlsc

Ec(m) = c°(qsTI c + Bcrlsc + Bsq cs) (5.19)

Since E = m <V2>, the following velocity levels may be obtained at any center

frequency _ from either equation (5.18) or (5.19)

2 ( )

<Vc> m s lq sc(m)

2 m +r I (m)

<V s > c rl c cs

(5.20)

5.6.2 Validation and Parametric Studies

In order to validate our SEA formulation, we compare the mean square mobility

level of the plate with experimental data reported earlier in Chapter III. Note that

although all nonzero bearing stiffness coefficients kbi j are computed and given in Table

3.6, only kbyy=3.69E8 , kby0x=3.52E5 and kb0x0x=4.19E4 are used as they appear to

be the most significant ones according to the proposed theory. Using equation (5.19),

<V_> is computed and compared with experimental results in Figure 5.12. Theoretical

predictions for three values of dissipation loss factor rls=rlc_'-rls(0) ) are given since the

choice of structural damping is critical to the SEA analysis. It can be seen from Figure

5.13 that the experimental data are approximately bounded by qs=0.0003 and rls=0.03.

Here "qs is also assumed to be frequency invariant which may not be valid in the

experiment. Accordingly, comparison between theory and experflnent is deemed io be

excellent.

Further comparison between theory and experiment can be made for the case of a

semi-infinite shaft considered in the example case II. Using equations (5.2), (5.7) and
183

_D

;>

cD

°_..¢

O

-50

- 1O0

-150

IO0

o Lin's Experiment

Theory

rl s = 0.0003

O O

r1 s = 0.003

1"1s = 0.03

| | i i i i i | I i , • , I • •

1000 10000

1/3 Octave Band Center Frequency (Hz)

Figure 5.12 Comparison between theory and experflnent [17] for exanlple case Ill with

"3

a finite shaft. Here plate mobility level in dB is 101oglo<Vc> re

<V2>= 1.0(m/s) 2 for Fysa = 1.ON.
184

-so _- o Lifts Experiment

_ Theory

_-' = 0.003

J -_ool n _- 0.03

-15 O0

1/3 Octave Band Center Frequency (Hz)

Figure 5.13

Comparison between theorY and experhnent, shaft. [17] for example case tII.

14ere theorY considers a semi-infinite
185

(5.8) for "qsc, the mobility levels are computed and are found to be given by straight

lines as shown in Figure 5.13. These lines represent the asymptotic behavior of the

system when the shaft is very long i.e. L s _ o.. Also, note that Figure 5.13 is

consistent with the trends reported by Lyon and Eichler [58] for the plate-cantilevered

beam problem. Again, most experimental data are bounded within the range given by

rls = 0.0003 to 0.03.

Consider the finite shaft length Ls=l.32 m of high modal density n s. In Figure

5.14 rlsc is compared for a finite shaft and semi-infinite shaft. It can be seen that the

result for the semi-infinite shaft follows the average values of the finite shaft.

Next, the effect of bearing preload or stiffness coefficients on the mean-square

velocity response of the plate-mount system is evaluated. Equation (5.20) is used to

predict the curves shown in Figure 5.15 while keeping other system parameters the

same. The bearing coefficients used in this analysis are from Tables 3.3 and 3.4. Three

bearing preloads used here are l15N, 190N and 285N which are referred to as low,

medium and high preloads. We observe minor changes with preload except in the

vicinity of 4 kHz.
186

10"1

10-2

10"3

10 -4

•_ 10 -5

o

106 Finite Shaft

Semi-Infinite Shaft

10 -7 , , , i I ,

0 5000

Frequency (Hz)

| I

10000

Figure 5.14 Predicted coupling loss factors rlsc for a semi-infinite and a finite shaft in

example case III.
187

10 1

10 0

10-1

._IA_ 10 -2

> >

vlv

10-3

10-4

- Low Preload

Medium Preload

High Preload

i • ,w | a | | | | * * * •

10-5

100 1000 10000

1/3 Octave Band Center Frequency (Hz)

Figure 5.15 Effect of bearing preload on the normalized casing plate response (example

case III).
188

5.7 EXAMPLE CASE IV: A GEARED ROTOR SYSTEM

5.7.1 Assumptions

As the final case, we investigate a generic geared rotor system with 4 bearings as

shown in Figure 5.16a. This system is not only a natural extension to example III, it

can also be treated as an approximate model of the NASA gear test facility in Figure 5.2;

for gearbox details, refer to Chapter IV. The intent of the SEA analysis is to predict

asymptotic casing plate response and radiated sound pressure from the gearbox, on an

order of magnitude basis. Accordingly, the following assumptions are made:

1. Consider the case of 4 identical rolling dement bearings and 2 identical shafts

carrying gears.

2. The vibratory source associated with the static transmission error excitation is at the

gear mesh. The net exchange of energy through the gear mesh coupling is

neglected and the vibratory energy is assumed to flow outward from the gear mesh

source to the casing through shafts and bearings. The source is assumed to be far

away from the bearings.

3. Only shaft bending vibration is coupled to the casing plate flexural motion.

Torsional modes are not included since they are relatively low in number compared

to the shaft bending modes. In steady-state, both shafts have equal amount of

vibratory energy (E s) associated with their bending motions which are uncoupled

from the external motor and load.

4. The portion of torsional vibratory energy which is transmitted to the load is

incorporated in the dissipation loss factor (qs) of the shaft-bearing subsystem.

Here rls is assumed to be given by qs=yqc_:qs(00) where tic,tic(tO ) is the loss
189

ate

Flexible

Shaft

Gear

X

. / Response

apur ./

Pinion

Shaft

Rigid Mounts

(a)

M xsa(¢.o, z=0-) = Mxs a

V ysa(¢.0, z=0-) = 0

Bearing

Flexible Shaft

dz

V,

[_ Bearing

Z

z=0 z=L s

(b)

Figure 5.16 Example case IV: (a) a generic geared rotor system with casing and mounts

and (b) boundary conditions for a finite shaft-bearing system.
190

.

°

7.

factor of the casing; ), is expected to be higher than one due to additional losses at

the bearings and load.

Bearing dimensions are assumed to be much smaller than the wavelength on the

casing plate. Also, bearings are not expected to be exchanging vibratory energy.

Modal densities of shaft n s and casing plate n c are given by equation (5.1).

A diffuse vibration field is assumed for the extemal casing-mount system. The

energy dissipation mechanism for this system includes acoustic energy radiated

from the plate in addition to the structural damping.

Application of the SEA principle will result in 2 subsystems, one intemal

consisting of 2 shafts and 4 bearings and one external consisting of 2 casing plates and

mounts similar to Figure 5.10b. The internal subsystem in this case can store the total

vibratory energy of 2 shafts. Vibratory energy transfer as viewed through Figure 5.10b

represents the algebraic sum of the energy transfers through 4 bearings.

5.7.2 Coupling Loss Factor

The coupling loss factor rls c between the intemal (shafts-bearings) and external

(casing) systems is derived, based on the formulation given by equations (5.11) and

(5.12). This implies that vibratory energy transfer is associated with only the dynamic

moment at the bearing, and the casing plate is infinite with respect to the bearing

dimensions. The driving point hnpedance of the shaft Z s is derived next given the

boundary conditions shown in Figure 5.16b. At the left bearing end (z=0-),

Vysa(t,o,z=0-)=0, and Mxsa(tO,z=0-)=Mxs a which are equivalent to Mxsa(t,o,z=0+)=Mxsa,
191

andFysa(co,0+)=(kby0x/kb0x0x)Mxsa*Vysa(co,0+) { (kby0Jkb0x0x)2kb0x0x-kbyy }/(ico).

L + with zero translational and angular velocities is described The other bearing end at z= s

at z=L- as

S

Mxsa(co,z=Ls)=_(kb0x0x dv ysa _(co, L s ) +kby0xVysa(co,Ls))/(ico)

dv ysa (co, L s ) + kbyyVysa(co,Ls))/(icO) Fysa(co'z=Ls)=-(kby0x (5.21)

Using equations (5.5) and (5.13), the above boundary conditions are evaluated to

formulate an algebraic problem of the type given by equation (5.14). The nonzero

elements of the coefficient matrix [B] are

/ i

lkby0x 1 / 3 . by0x 1 3

BII = "_ lk ; B12='_ Elks-ikbyy+ k-_0x; _ ' s-ikbyy + kb0x0 x

IE k2 / k2

i 3 by0x _ i/ _,3 . + by0x ) .

B13 _ lk kbyy+ , 15 ='-_--t:tKs--K _" ,

s-- kb0x0x) 14 _ byy kb0x0x J

B21=B22=-B23=-B24=1 ;

B31=(-jkskb0x 0x+kby0x+Elk2 s)e-ik'L'

B32 = (jkskb0x0x +kby0x +Elk2s) eik'L_ ;

+k -Elk )e _ ksL s ;

B33 = (- kskb0x0 x by0 x

B 34 = (kskbox0x +kby0x - Elk2s )ek_Ls ,"

B41=(ikskby0 -kbyy-iElk3)e-ik_Ls .

x S
192

- k + iEIk 3s) eiksL_ ;

B42 =(- ik skby0x byy

B43=(kskby0x-kbyy +EIk3s) e-ksLs ;

B44 (- kskby 0 kbyy - Elk 3 ksLs = x- s ) e (5.22)

The right hand side vector {b} of equation (5.14) is Mysa { kby0x/kb0x0x,

ito/(EIk_),0,0 } T, and the shaft impedance Z s is evaluated numerically using equation

(5.8).

5.7.3 Vibroacoustic Response

Consider the total energy dissipation by the casing as the sum of the energy terms

associated with structural damping dissipation and energy escape via sound radiation.

Hence the total dissipation loss factor tic T for the casing of area A c and mass mc is

z o Ac ¢_c(0_)

r I cT(tO) = 1] c + tO m c (5.23)

where zo is the characteristic impedance of the surrounding medium, and ¢_c is the

radiation efficiency of the casing. In this analysis, two radiation efficiency models are

used: (i) _cl = 1.0 for an ideal radiator as for many gearboxes the measured radiation

efficiency has been found to be close to unity [78], and (ii) t7c2 for a shnply-supported

rectangular plate [79]. Several investigators have used _c2 successfully ha SEA

applications [60,66,671].
193

Since this SEA model is very similar to the one developed for example case Ill,

solutions given by equations (5.19) and (5.20) are still valid here provided the existence

of multiple paths, as opposed to only one path in example case III, is recognized. The

mean-square velocity levels of the shafts-bearings system <Vs2> and casing-mount

system <Vc2> are

2 ns (rlcT +4tics)

< V s > (co) =

m s co(rl srl cT + 4rl cTrlsc

2 4Hs risc

< V c > (co) =

m c co( _1srl cT + 4rl cT a] sc

+4rls rl cs)

+4rlsn cs)

(5.24)

Power injected into the system is developed by examining the internal static transmission

error excitation e(t)=eh(co)e k°t. For the spur gear pair, the gear mesh frequency

c0(rad/sec)=Ng_zMn/30 where Ng is the number of gear teeth and f_zM(rpm) is the

driven shaft speed. By definition, the input power by the gear mesh elastic force

Fh(t)=khe(t)=eh(co)e k°t is given by I-ls(co)=l/2 (kheh) 2 Re { (lfZs(co))* } . At a very high

frequency, it is reasonable to assume that the boundaries of the shafts do not affect the

gear mesh source regime. Hence the infinite shaft assumption should hold.

Accordingly, Z s can be obtained analytically using the driving point shaft impedance

given in Reference [77]; however, the gear mass mg must be included.

Z s(co) = 2PsAsC s (1 + i) + icom g (5.25)

Sound power W radiated from the casing is computed using the following equation

where A c is the total casing surface area
194

W(o) = zoAc< V2c(O)> Oc(O) (5.26)

Assumingasourcedirectivity Q(c0)associatedwith gearedsystemmountingcondition,

thesoundpressurelevelLp(dBre20_Pa)in thefarfield at distancer from thecasingis

. Q(c0) 4

L r,(O) = L wCC0)+ 10 log 10t + R--7 ) (5.27)

where R(_)=_S/(1-_) is the room constant, _(_) is the average absorption

coefficient, S is the room surface area and L w (dB re 1.0pW) is the sound power level.

5.7.4 Experimental Validation and Parametric Studies

The SEA formulation for this example case is verified by comparing results with

vibroacoustic responses measured on the NASA gearbox [55]. A detail description of

the experimental setup and its system parameters have been given in Chapter IV.

Additionally unweighted Lp at r = 0.38 m, directly above the surface of the top cover

casing plate as shown in Figure 5.2, has been measured with a B&K type 2230 sound

level meter. Predicted and measured Lp are compared in Figure 5.17a for y=10, Oc=Oc2

and bearing preload 5zm=0.04mm. The second term in equation (5.27) is dominated by

Q/(47tr 2) since 4/R << Q/(4rtr 2) in this case due to _ = 1.0 and room surface area S

being very large. Predictions are found to be within +10dB of the measured values for

typical structural dissipation loss factor 0.004_<rlc<0.04. Figure 5.17b shows the

comparison of spatially averaged mean-square casing acceleration level (0_2<V_>)

between theory and experiment. Experimental curve represents the averaged value of

the measurements made at 3 casing plate locations (top plate, side plate with
195

_D

>

.._

0

r,O

120

8O

40

10 2

A

vlc= 0.004 _^/_

NASA ,'""-"--'// \

Experiment ,," j 1

• | i i I • • • | •

10 3

113 Octave Band Center Frequency (Hz)

(a)

= 0.04

10 4

0

° _,,,q

,<

Figure 5.17

10

.1

.01

10 2

q c = 0.004

NASA Experiment _l _!

I ¥

/

tic = 0.04

i I i • . • . . I i

10 3

| I • • • • •

10 4

1/3 Octave Band Center Freqeuncy (Hz)

(b)

Comparison between theory and NASA experiment [55] for example case

IV (7=10, Q=2, cc=oc2, 8zm=0.04mm). (a) Sound pressure level Lp.

(b) Spatially averaged casing acceleration.
196

bearings and side plate without bearings). Again, predicted acceleration spectra, which

are similar to that found for the sound level, are found to be in reasonable agreement

with the measured spectrum given the appropriate values of tic and y. In general, we

observe that the response level decreases with increasing tic. Also, comparisons

suggest that tic in this system may be frequency dependent.

Next we vary Y but keep tic=0.02 constant in Figure 5.18. Comparison between

theory and measurements also indicates that T=10 is the best fit for the experimental data

especially at the higher frequencies. Here, radiation efficiency t_cl and bearing axial

preload 8zm=0.04mm have been specified.

Now we investigate the effects of casing plate radiation efficiency c_c and bearing

preload or mean axial displacement _zm on Lp. Figure 5.19 compares Ocl and CYc2.

Based on the comparison with experiment it seems that the radiation efficiency of the

NASA gearbox is better modeled with Ccl although the measured Lp curve is mostly

between (Ycl and (Yc2 curves. Differences between _cl and (Yc2 are significant dt lower

frequencies, but the variation never exceeds 10dB over the entire frequency range of

interest. It may be noted that since the acoustic energy radiated W(o_) is significantly

smaller than the energy dissipated by the system, virtually no change is found in the

predicted casing acceleration spectra by varying t_c. The effect of bearing preload on Lp

is shown in Figures 5.20a and 5.20b for Crc2 and _cl respectively. Figure 5.20

indicates that Lp is lower when the bearing preload is increased. A similar effect is seen

for the casing acceleration level which is consistent with the deterministic model

prediction of Chapter IV.
197

>

r._

"0

o

oO

120

8O

7= 10

7= 100 /

/

NASA Experiment /

4010 2 ' , , ,... i'03 ....... 10 4

1/3 Octave Band Center Frequency (Hz)

Figure 5.18 Effect of T-=ris/qc on predicted Lp for example case IV (tic=0.02, Q=2,

_c=(Ycl, 8zm=0.04mm).
198

_D>

m.

o

60

120

80

40

10 2

NASA Experiment

°c1 

| | • • ' • | , | i | | | i | i •

10 3 10 4

1/3 Octave Band Center Frequency (Hz)

Figure 5.19 Sound pressure level prediction with two a c formulations for example

case IV: acl = 1.0 (ideal radiator) and (rc2 for simply-supported plate

(y=10, Q=2, _1c=0.02, gzm=0.04mm).
199

tD

-3

t-q

o

120

_2aTl

80

40

10 2

=0.01 mm

NASA

Experiment

_2aTl

J

J

=0.1 mm

| | | | | | i I | i | i • • • , o

10 3 10 4

1/3 Octave Band Center Frequency (Hz)

(a) Oc=_c2

ID

.3

o

120

_ZITI

80

40

10 2

=0.01 mm

NASA

Experiment

_zin

S

S

S

=0.1 mm

p | • • • | i • | • • • | J | I •

10 3 10 4

1/3 Octave Band Center Frequency (Hz)

(b) (_c=_c 1

Figure 5.20 Effect of bearing axial preload 8zm on Lp for example case IV (tic=0.02,

Q:2, 7=10).
200

5.8 CONCLUDING REMARKS

The vibration transmission through bearing has been analyzed using the SEA

technique. A new procedure has been developed to compute the coupling loss factor

which relies on the solution of the boundary value problem at the bearing-casing

interface. This scheme incorporates the bearing stiffness matrix developed earlier as a

part of the deterministics vibration models in Chapters H-IV. Experimental validations

verify the proposed theory even though a very simple model for the geared rotor system

has been presented. A more detailed SEA model including energy sharing between

subsystems is required to analyze this system. This is left for future research.
CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

6.1 SUMMARY

A new precision rolling element beating model is proposed for the analyses of the

vibration transmission through bearings and overall system dynamics in rotating

mechanical equipment including geared rotor systems. Current beating models, based

on either ideal boundary conditions or purely translational stiffness element description,

can not explain how the vibratory motion may be transmitted from the rotating shaft to

the casing and other connecting structures in rotating mechanical equipment. For

example, a vibration model of a rotating system based upon the existing beating models

can only predict purely in-plane type motion on the flexible casing plate given only the

bending motion on the shaft. However, experimental results have shown that the casing

plate motion is primarily flexural or out-of-plane type. Chapter II clarifies this issue

qualitatively and quantitatively by developing a new mathematical model for the

precision rolling element beatings from basic principles. A comprehensive beating

stiffness matrix [K]bm of dimension 6 is proposed which clearly demonstrates a

coupling between the shaft bending motion and the flexural motion on the casing plate.

A numerical scheme which involves a solution to nonlinear algebraic equations is

proposed for the estimation of the stiffness coefficients given the mean bearing load

vector. And, a second method which requires the direct evaluation of these stiffness

201
202

coefficients given the mean bearing displacement vector is also discussed. Some of the

translational stiffness coefficients of the proposed beating matrix have been verified

using available analytical and experimental data. Further validation of [K]bm is not

possible as coupling coefficients are never measured. Also, parametric studies on the

effect of unloaded contact angle, preload, or bearing type are included. These results

lead to a complete characterization of the bearing stiffness matrix.

Chapter Ill extends the proposed bearing stiffness formulation and demonstrates

its superiority over existing models in vibration transmission analyses for a generic

single shaft-bearing-plate-mount system. The beating stiffness matrix [K]b m is

incorporated in discrete system models using lumped parameter and finite element

modeling techniques. Shaft, plate and mount flexibilities are also included in such

models. The stability issue associated with the proposed bearing model is addressed

analytically using Liapunov's stability method and the system is found to be dynamically

stable provided the preloads are sufficiently high. Eigensolution and forced harmonic

response to the following rolling element bearing system example cases are obtained

using our formulation and results are compared with the predictions yielded by the

current vibration models: I. rigid shaft and plate system freely suspended, II. rigid shaft

and plate supported on flexible mounts, and III. an experimental setup consisting of a

flexible shaft, two ball bearings, a rectangular plate and the supporting structure.

Analytical results indicate that our proposed model is indeed capable of predicting plate

rigid body angular motion or plate flexural motion as excited by shaft motion. Such

predictions are not observed in simple vibration models. Also, lower degrees of

freedom models, developed by several previous investigators tend to underestimate the

resonant frequencies and force or moment transmissibilities as compared with our multi-
203

degreeof freedom models. Comparisonsbetweenour model and the available

experimentshavebeenfoundto bereasonablygood.

The overall dynamicsof a gearedrotor systemwhich includesa spurgearpair,

shafts,rolling elementbearings,prime moverand load (attachedto the gearedrotor

systemthroughflexible torsionalcouplings),rigid or flexible casing,andcompliantor

massivemounts is studiedin ChapterIV. Linear time-invariant, discretedynamic

models of a generic gearedrotor system with proportional viscous damping are

developedusinglumpedparameteranddynamicfiniteelementechniqueswhicharethen

used to predict the vibration transmissibility through bearingsand mounts,casing

vibrationmotion, anddynamicresponseof theinternalrotating system.Theproposed

bearingformulationis alsoincluded.EachrotatingshaftismodeledasanEulerbeamin

thelumpedparametermodelandasaTimoshenkobeamin thedynamicfinite element

model,but thegyroscopicmomentis notincluded.Eigensolutionandforcedharmonic

responsestudies due to rotating massunbalanceor kinematic transmissionerror

excitation for the following examplecasesareobtainedusing our formulation and

comparedwith simplemodelscurrentlyavailablein theliteratureand/orexperiment:I. a

single-stagerotor systemwith flexibly mountedrigid casingconsistingof two bearings

asaspecialcaseof thegearedrotor system,II. a spurgearpairdrive supportedby four

bearingsinstalled in a flexibly mountedrigid casing,andIII. an experimentalsetup

consistingof high precisiongearandpinion,andfour identicalroiling elementbearings

containedin a flexible casingmountedrigidly on a massivefoundation. In example

casesII andIII of ChapterIV, thegearmeshstiffnessis assumedto belinearandtimeinvariant. Analytical predictionsshowthatour theoryis indeedcapableof predicting

bearingandmountmomenttransmissibilitiesin additionto theforcetransmissibilities.
204

Also, flexural vibrationsof thecasingplatearepredictedwell asour theoryis in good

agreementwith measurementsmadeoncaseIII of ChapterIV; suchpredictionsarenot

seenin simplemodels.

In ChaptersII-IV, thedeterministicvibrationmodelsof gearedrotor systemwith

proposedbearingstiffnessmatrix incorporatedhavebeenshownto bereliable up to

moderatelyhigh frequencies.But thesemodelsareinadequateatvery high frequencies

when the modal density is high. Classicalvibration modelsusually do not predict

modesaccuratelyin this frequencyregime,require large computationaleffort and

producespectraatmanyspatialpointswhicharedifficult to interpret.Accordingly,we

haveusedthe statisticalenergyanalysis(SEA) methodto predict the mean-square

vibratory responseof internalandexternalsubsystemsin a gearedrotor system.The

feasibility of applying SEA to this systemis alsoinvestigatedby performing modal

analysisof a gearbox. Four examplecasesare analyzedusing SEA: I. a platecantileveredrectangularbeam,II. caseI with circularshaft-bearingsystemreplacingthe

cantileveredrectangularbeam,Ill. acircularshaft-bearing-plate-mount system,andIV a

simple gearedrotor system. In the first two examplecases,we have revised and

extendedLyon andEichler'splate-cantileveredrectangularbeamproblem[56,58] to

improve the coupling loss factor prediction and to formulate the vibratory energy

transferproblemthroughrolling elementbearings.Thethird andfinal examplecases

computethe systemresponsespectraandcomparethemwith measurements.Good

agreementis found betweentheoryand experimentprovided proper values of the

dissipationloss factorsand bearingpreloadsareused. The NASA gearboxradiation

efficiencyisfoundto benearlyunityat higherfrequencies.
6.2 FUTURE RESEARCH

Severalareasof potential research problems based on the present study of the

vibration transmission through bearings, are identified as follows:

o Analyze vibration transmission through hydrodynamic bearings, using the

proposed work on rolling element bearings. Develop an experimental methodology

to estimate the bearing stiffness matrix and transfer properties. Also, develop

bearing diagnostic techniques using vibration transmission theory proposed in this

dissertation.

° Extend the proposed overall geared rotor system vibration model with spur gears to

helical, bevel and worm gear drives. Develop vibration models using the proposed

bearing model to analyze multiple transmission paths. Improve gear diagnostic

procedures using analytical bearing transfer properties.

, Refine the current statistical energy analysis formulation of a generic geared rotor

system to investigate the vibratory energy transfer among smaller subsystems and

to include the effects of rotating shaft torsional modes. Generate analytical and

experimental schemes to predict coupling loss factors for typical shaft-bearing-plate

system. Finally, establish gearbox design methodology for reduced airborne and

structure-borne noise.

205
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E-5716
Report Documentation Page

1. Report No. NASA CR-4334 2. Government Accession No. 3. Recipienfs Catalog No.

AVSCOM TR-90-C-019

5 Report Date

November 1990

4. Title and Subtitle

Vibration Transmission Through Rolling F:len]ent Bearings in Geared

Rotor Systenls

7 Author(s)

Rajendra Singh and Teik Chin Lira

9. Performing Organization Name and Address

Ohio State University

Department of Mechanical Engineering

Columbus, Ohio 43210

12. Sponsoring Agency Name and Address

Propulsion Directorate

U.S. Army Aviation Systems Command

Cleveland, Ohio 44135-3191

and

NASA Lewis Research Center

Cleveland, Ohio 44135-3191

6. Performing Organization Code

8. Performing Organization Report No

None (E-5716)

10

tl.

13

Work Unit No.

505-63-51

Contract or Grant No.

NAG3-773

Type of Report and Period Covered

Contractor Report

Final

14. Sponsoring Agency Code

15 Supplementary Notes

Project Manager, James J. Zakraisek, Propulsion Systems Division, NASA Lewis Research Center. This report is

based on a portion of the Ph.D. thesis of Teik Chin Lira.

16. Abstract

A new mathematical model is proposed to examine the _,ibratic, n transmission through rolling clement bearings m geared rotor

systems. Current bearing models, based on either ideal boundary conditions fl_r the shaft or purely translational stiffness element

description, cannot explain how the vibrator), motnm may be transmitted from the rotating shaft to the casing. For example, a

vibration mtKtel based upon the simple bearing formulations can only predict purely in-plane type motion on the flexible casing plate

given only bending motion on the shaft. However, experimental results have shown that the casing plate motion is primarily flexural.

Tiffs study el:miles this issue qualitalively and quantitatively by developing a comprehensive bearing stiffness matrix of dimension 6

to model the precision rolling clement bearings using basic principles. The proposed bearing stiffness matrix is partially verified

using available analytical and experimental data. and is completely characterized. This study extends the proposed bearing

tbrmulation to analyze the overall geared rotor system dynamics including casing and mounts. The bearing stiffness matrix is

included in discrete system models using lumped parameter and/or dynamic finite element techniques. Eigensolution and lorced

harmonic response due to rotating mass unbalance or kinematic transmission error excitation for the following examples are

computed: 1. single-stage rotor system _ith flexible shaft supported by two bearings on rigid casing and flexible mounts, It. spur

gear pair system with motor and load inertias attached to two flexible shafts supported by four bearings on flexibly mounted rigid

casing, and Ill. case II with flexible casing and rigid mounts. In several of these examples, analytical predictions compare well with

measured data, validating the proposed lormulation. Numerical predictions show that the proposed theory is capable of predicting

bearing moment transmissibilit) in addition to force transmissibility. A statistical energy analysis model combined with the bearing

stiffness matrix is developed to predict the high frequency asymptotic dynamic behavior. The mean-square spatially averaged

',ibroacoustic responses are calculated fi_r several example cases. A physical gearbox is also analyzed to demonstrate Ihe salient

features of the proposed technique. Good agreement is tbund between theory and experiment.

7 Key Words (Suggested by Author(s))

Vibration transmission

Bearings

Geared systems

18. Distribution Statement

Unclassified- Unlimited

Subject Category 37

19. Security Classif (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of pages

232

22. Price*

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