Analysis of high-speed rotating systems using Timoshenko beam theory in conjunction with the transfer matrix method

Rochester Institute of Technology

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5-1-1989

Analysis of high-speed rotating systems using

Timoshenko beam theory in conjunction with the

transfer matrix method

Beth Andrews O'Leary

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Recommended Citation

ANALYSIS OF HIGH-SPEED ROTATING SYSTEMS USING

TIMOSHENKO BEAM THEORY IN

CONJUNCTION WITH

THE TRANSFER MATRIX METHOD

Beth Andrews

O'Leary

A Thesis Submitted in Partial Fulfillment oftheRequirements fortheDegreeof

MasterofScience

in

Mechanical

Engineering

Approved by: Professor .

s>.crn^i

J.S. Torok (Thesis

Advisor)

C

Doctor

G.H.Garzon

Professor

H.A. Ghoneim

Professor

Professor

R.B. Hetnarski

B.V. Karlekar (Department

Chairman)

DepartmentofMechanical

Engineering

Collegeof

Engineering

ANALYSIS

OF HIGH-SPEED ROTATING

SYSTEMS

USING

TIMOSHENKO BEAM THEORY IN

CONJUNCTION WITH

THE TRANSFER MATRIX

METHOD

I,

BethAndrews

O'Leary,

hereby

grantpermissionto theWallace Memorial

Library

ofthe Rochester Institute of

Technology

to reproduce _my thesis in whole or in part.

Any

reproductionwillnotbe forcommercialuseor profit.

ROCHESTER INSTITUTE OF TECHNOLOGY

-This volume is the _property of the Institute, but the

literary

rights of the author must be respected. Passages must notvbe copied or _closely paraphrased without the previous

written consent of the author. If the reader obtains _any assistance from this volume he must give proper credit in his

own work.

This thesis has been used

by

the

following

persons, whose

signatures attest their acceptance of the above restrictions.

ACKNOWLEDGEMENTS

Numerous people have supported and encouraged me

during

the research and

developmentofthis thesis. Inparticular,Iwouldliketo thank

My

advisor, Dr. Joseph S.

Torok,

_{for challenging} me to excel beyond my expectations. His insightand guidancehelpedtoproduceasuccessfulthesis.

My

first mentor in Rotor

Dynamics,

Dr. Guillermo

Garzon,

for our _many

conversations andhiscontinual encouragement.

My

committee_members, Dr.

Garzon,

Dr. GhoneimandDr. Hetnarski fortheir

detailedreview of_mythesisandfortheirconstructive comments and_challenging

questions.

My

parents,

Thorp

and

Dorothy

Andrews,

fortheir_{faith in my determination}and

theircontinual support.

My

husband,

Kevin

O'Leary,

for

drawing

_many ofthe figuresinthe thesisand

ABSTRACT

Higher_operatingspeedsandincreased sensitivityofmodern electro-mechanical systems

require

improved

methodsforthecomputation of critical speeds and systemresponseof flexible rotating shafts.

Many

high-speedsystems _generally containdisks with masses

approaching the mass of the shaft. These observations emphasize the importance of

including

theeffects of_rotatoryinertiaandsheardeformationoftheshaftintheanalysis. Traditional theory, which models a massless _shaft, would be inaccurate for these

systems.

An analysis of flexible rotor systems has been performed _using the Transfer Matrix

Method. Although the method is well

known,

the present _study utilizes Timoshenko

Beam

Theory

in the construction offieldmatrices, whichrelate statevectors at adjacent

nodes ofthe system. This approach takes into considerationthe effects oftransverse

shear and _rotatory inertia. Also included in the model are gyroscopic effects of the

spinningdisks. Theseeffects are _generallyneglectedinclassicalrotordynamictheory.

A general model wasdeveloped fortheanalysis oftypicalconfigurations in which the

shaft is simply _supported, and can _carry an _arbitrary number of disks. Numerical

simulations were performedforcomparision with classical results. Thesecase studies

show agreement with what is to be expected

by introducing

the greater

flexibility

of

TABLE

OF CONTENTS

List of

Figures

vi

List of

Symbols

ix

1

INTRODUCTION

1

2 BACKGROUND 3

3 THEORY AND PROGRAM DEVELOPMENT 6

3.1 Transfer Matrix Method 6

3.1.1 Natural Frequencies 20

3.1.2 Mode Shapes 22

3.1.3 Forced Response 27

3.1.4 Description ofGeneral PointandField Matrices 31

3.2 Shaft Motion 34

3.2.1 Formulation of Field Matrices

Using

35 Bernoulli-Euler Beam

Theory

3.2.1.1 Field Matrix

[FXJ

35

3.2.1.2 Field Matrix

[FyJ

39

3.2.2 FormulationofField Matrices

Using

Timoshenko 44

Beam

Theory

3.2.2.1 Field Matrix

[Fx]

44

3.2.2.2 Field Matrix

[Fy]

61

3.3 Disk Motion 71

3.3.1 Moment EquilibriumEquations

Relating

to 72 DevelopmentofGyroscopic Coupleand

Rotatory

Inertia

3.3.1.1 Transition Matrix 75

3.3.2 Mass Unbalance 82

3.3.3 Formulation of Point Matrices 85

4 RESULTS 97

4.1 Figures 101

5

CONCLUSIONS

108

6

RECOMMENDATIONS

110

References Ill

List

of

Figures

3.1.1 _{Typical System Configuration} 1

3.1.2 Simple

Spring-Mass

System UtilizedtoIllustratethe 6 TransferMatrixMethod

3.1.3 Free

Body

DiagramofSpring-MassSystem 7

3.1.4

Relationship

Between StationsWith RespecttoMasses 12

3.1.5 DescriptionofFields With RespecttoMasses 13

3.1.6

Relationship

BetweenFieldsWith RespecttoMasses 13 Fora

Simply

Supported ShaftWithoutan

Overhang

3.1.7 IdentificationofStationsForaMulti-Disked

Simply

15 Supported ShaftWithOverhangs

3.1.8

Multi-Disked,

Simply

SupportedShaft 22

3.1.9 First

Bending

Mode For Figure 3.1.8 22

3.1.10 Second

Bending

Mode For Figure 3.1.8 23

3.1.11 Third

Bending

Mode For Figure3. 1.8 23

3.2.1 Typical System Configuration With Coordinate Axes 35

3.2.2 Free

Body

DiagramofShaft Element in X-Z Plane 36

Using

Bemoulli-EulerBeam

Theory

3.2.3 Free

Body

DiagramofShaftElement in Y-Z Plane 39

Using

Bemoulli-EulerBeam

Theory

3.2.4 Free

Body

DiagramofShaftElement in X-Z Plane 45

Using

Timoshenko Beam

Theory

3.2.5 Shaft Element in X-Z Plane Subjectedto

Bending

Moment 46

3.2.6 ShaftElement in X-Z Plane Subjectedto

Bending

₄₇

MomentandShear Deformation

3.2.7 Free

Body

DiagramofDifferential ElementofShaft ₄₈ intheX-Z Plane

3.2.8 Free

Body

DiagramofShaft Element in Y-ZPlane 62

3.2.9 Shaft ElementinY-Z Plane Subjectedto

Bending

Moment 63

3.2.10 Shaft Elementin Y-ZPlaneSubjectedto

Bending

64 MomentandShearDeformation

3.2.1 1 Free

Body

DiagramofDifferential ElementofShaft in Y-Z Plane 65

3.3.1 Localand_{Global Coordinate Systems For}

Whirling

Disk 73

3.3.2

Relationship

BetweenGlobalandLocal Coordinate 76

SystemsintheX-ZPlane

3.3.3

Relationship

Between Globaland LocalCoordinate 77 SystemsintheY-Z Plane

3.3.4

Rotatory

Inertias andGyroscopic Couples

Acting

onDisk 81

3.3.5

Relationship

Between Centerof

Gravity

ofaDisk 82 andCenterofRotation

3.3.6 Forces

Acting

onDisk DuetoMassUnbalance 84

3.3.7 Free

Body

DiagramofForces

Acting

onDisk in theX-Z Plane 85

3.3.8 Free

Body

DiagramofForces

Acting

onDiskin theY-Z Plane 86

3.3.9 Free

Body

DiagramofMoments

Acting

on Disk in X-Z Plane 87

3.3.10 Free

Body

DiagramofMoments

Acting

onDiskin Y-Z Plane 88

3.3.11 Gravitational Force

Acting

onAngled System 93

3.3.12 Mass MomentsofInertia ForaDisk Without Mass Unbalance 94

3.3.13 Location ofMass Unbalanceon aDisk 95

4.1.1

Simply

Supported Shaft With Three Nested Disks 97

4.1.2

Simply

Supported Shaft With

Overhanging

Mass 98

4.1.3 Case

Study Utilizing

Bernoulli-EulerBeam

Theory

andthe 101 Transfer Matrix Method Without GyroscopicCouple

4.1.4 Case

Study

Utilizing

Timoshenko Beam

Theory

andthe 102 Transfer MatrixMethod Without GyroscopicCouple

4.1.5 Natural Frequencies for TimoshenkoandBernoulli-EulerBeam 103

Theories WithandWithout GyroscopicCouple

4.1.6 Non-SynchronousMotion Foran

Overhanging

Disk 104

Utilizing

TimoshenkoBeam

Theory

4.1.7 Case

Study Utilizing

Bernoulli-EulerandTimoshenko 105 BeamTheoriestoDetermine Forced Response Due to

MassUnbalance Without Gyroscopic Couple

4.1.8 ForcedResponseof

Overhanging

DiskDueto 106

MassUnbalanceWithGyroscopic Couple

Utilizing

Timoshenko Beam

Theory

4.1.9 EffectofGravitational Force on

Overhanging

Disk 107 Vs. OrientationofShaft

A.l SystemConfiguration 113

A.2 Flowchart

Deriving

Global Transfer Matrix 114

A. 3 Flowchart

Deriving

Natural Frequencies From 116 Global Transfer Matrix

A. 4 Flowchart

Deriving

Mode Shapes 117

List

of

Symbols

A cross-sectional area oftheshaft

[A]

overalltransfermatrix_{for left overhanging}section of shaft

[A']

[A]

matrix modifiedtoinclude

boundary

conditions atstationa

[A(j]

displacementtermsfrom

[A]

matrix

[ASJ

slopetermsfrom

[A]

matrix

a station_containingleft

bearing

[B]

overalltransfermatrixformiddle sectionofshaft nestedbetween the

bearings

[B']

[B]

matrix modifiedtoinclude

[A']

matrix

[B"]

[B']

matrix modifiedtoincludethe

boundary

conditionsat stationb b station_containingright

bearing

(3j

anglebetweencenterof_gravityandYaxis at stationi

[C]

overalltransfermatrixforright_overhangingsectionfo shaft

D shaftdiameter

{

d0

}

displacementvectorat station0

E elastic modulus ofshaft

q distance fromcenterof rotation tocenterof_gravityofdiskat stationi

Fg

gravitationalforce actingondisk

[F]j

fieldmatrixforshaftelement

Lj

[FXJ

fieldmatrixfor X-Zplane

[Fy]

fieldmatrixfor Y-Zplane

G shear modulus ofshaft

g gravitationalconstant

H angular momentumofdisk

I area moment ofinertiaofshaft

Isx>

Isv

mass momentsofinertiaof shaft abouttheXandY_{axes,respectively}

It

1,

It

2 transversemass moments ofinertiaofthediskaboutitscenter of mass

i stationlabel

ks

shaftform factor

Lj

lengthof shaft element

\\

,

^2, ^-3,

A.4

rootstocharacteristic equation

M couples_actingondisk

M(j

mass ofdiskwithout massimbalance

Mj

unbalanced mass at stationi

Mx

moment_actingaboutXaxis

My

moment_actingaboutYaxis

mj totalmassofdiskat stationi

including

massduetoimbalance

mj+i totalmass ofdiskatstationi+1

including

massduetoimbalance

m-_i total mass ofdiskatstationi-1

including

massduetoimbalance

ms mass of shaft element

n laststation_{in rotating}system

Q. _rotatingspeedofdisk

(0 whirl

frequency;

natural

frequency

ofthesytem

coxvz

angular_velocityvectorofdiskin XYZcoordinate system

fxvz)' absolute_velocityvectorofdiskin

(XYZ)'

coordinate system

Pa

reactionforceat

bearing

'a'

Pjj

reactionforceat

bearing

'b'

[P]

j point matrixformass_mj

Qx>

Qv

parameter substitutionsutilizedin Timoshenko

Theory

for XandY

directions

R parameter substitution utilizedinTimoshenkoBeam

Theory

r radius ofdisk

{

S

}

L_j state vectorto theleftofstationi

{

S

}

^i

state vectorto theright of stationi

{

S

}

L-.i state vectorto theleftofstation i-1

{

S

}

Rj_

_i state vectorto theright of station i-1

{

S'

}

state vector

{

S

}

modifiedtoincludereactionforceat

bearing

'a'

{

S"

}

state vector

{

S'

}

modifiedtoinclude thereactionforceatstationb

{

Sx

}

statevector

describing

X direction

{

S

y

}

state vector

discribing

Y direction

t timeatwhichforcedresponsewillbe determined

[Tx], [Ty]

transitionmatrices_relating

(XYZ)'

coordinate systemtoXYZ

coordinate system

x anglebetweentheshaft andhorizontal surface

6X

slopein X-Zplane

9y

slopein Y-Zplane

9X

angular_velocityofdiskin Y direction

6V

angular_velocityofdisk in X direction

0 angular acceleration ofdisk in Y direction

9 angular accelerationofdiskin X direction

Uy

massunbalancein Y direction

Ux

massunbalancein X direction

[U]

globaltransfermatrix

Vx

force in X direction

Vy

force in Y direction

w displacementutilizedtoexplaintheTransferMatrix Method

x displacementinX direction

Chapter

1

INTRODUCTION

Amodel_{is developedfor}arigid

bearing,

flexibleshaft, nonsynchronous,rotatingsystem

defined in both transversedirections. Thegeneral model can be described as a_simply supported shaft with_overhangs,on which _anymultiplenumberofdiskscan beattached.

The analysis allowsforthevariation ofthe shaft_{diameter along}thelength ofthe shaft.

Nonsynchronousmotion, whichisgeneric, takesplacewhenthewhirl _velocityand spin

velocity are not equal. Synchronousmotion can be derived from informationobtained

fornonsynchronousmotion.

/////> /<w/7

I

Figure 3.1.1: TypicalS_ystemConfiguration

The modelis analyzedforthe naturalfrequencies alongwith their_{corresponding}mode shapes and/or the forced responses _using the Transfer Matrix Method. This method consists of the calculation of a series of relationships between the field and point matrices. The field matrix describes the motion of the shaft and the point matrix

describes themotionofthediskormass.

Boundary

conditions_relatingtodiscontinuities

in the shaft (ie.

bearings,

the ends ofthe _shaft) are taken into account in the general procedure.

The derivationoffield matrices is outlined, forcomparison, using twodifferent beam

which_{is incorporated}inthe_{traditional analysis,} relatesthe stiffnesses betweenvarious

sections of the shaft, but considers the shaft to be massless. The Timoshenko Beam

Theory

includesthecentrifugal

force,

_{rotatory inertia}and sheardeformation ofthe shaft,

along with the stiffnesses ofthe shaft. The

flexibility

inherentto this

theory

tends to lowerthe natural

frequencies,

since itreduces the overal stiffness ofthe shaft. Such a

modificationbecomes_veryimportant_{for high-speed rotating}shafts,inwhichthemassof

theshaftapproachesthemass ofthedisk.

Thepoint matrix canbeassembled

by

_calculatingthecentrifugalforceofthe_{disk along}

with the moment and gyroscopic couple _actingon the disk. Thegyroscopic couple is determined fromthemoment equations andis afunctionoftheradialmass moment of

inertiaofthe

disk,

thewhirl_velocityandthespin velocity. Thegyroscopic couple raises

thenaturalfrequenciesofthesystemdueto_{its tendency}tostraightenthe shaft.

Analyzing

a_rotatingsystemfornatural

frequencies,

modeshapes andforcedresponsesis

essential in

determining

a proper design configuration for the system and in

troubleshooting

existingsystems. Thenatural

frequency

ofthesystemmust notbe in the proximityof

forcing

frequenciesthatdrivethevarious components ofthesysteminorder

to avoid resonant behavior. A forced response curve indicates the

frequency

range

Chapter

2

BACKGROUND

Theearliest reference tovibration of_rotating shafts was

by

Rankine

[1],

whoin 1869

definedthecritical speeds or naturalfrequenciesof such a system. His model consisted

of a uniform shaftthatdisplaced fromstatic equilibrium and was considered stable _only

uptoitsfirstcritical_speed, an undetermined_stabilityat critical speed and unstable above

critical speed. In 1894 Rayleigh

[2]

developed an approximate _energy method to

determinethefundamental

frequency

duetolateralvibrationof a_non-rotating shaft. This

method,whichisthebasisoftheFinite Element

Method,

minimizesthe total _energy of

thesystem

by

_{first assuming}a single shapefunctionandthen_obtainingthe fundamental

frequency. Thesuccess ofthemethoddependedupon_choosinga propershapefunction

thatcorrespondedwiththemodeshape and matchedits

boundary

conditions.

Timoshenko

[3]

appliedRayleigh's Methodto_rotatingshafts andinvestigatedtheeffect

of sheardeformationon the natural frequenciesofa shaft. Jeffcott

[4]

first described

whirl with his rotating system of a singleunblancedmass situated on a massless shaft

betweentworigid

bearings,

nowknown as aJeffcottrotor. Heconsideredthe shaftto

displace ina plane andtoprecess at an angular_velocityequalto therotationalspeedofthe

shaft. He concluded that the whirl amplitudeincreased while _approaching the critical

speed and decreased beyondthe critical _{speed; thereby, claiming}that the system was

stableabovethecritical speed.

Prohl

[5]

first proposed the use of the Transfer Matrix Method for lateral dynamic

analysis. He dividedthe rotorinto discrete masses and thereby, considered the shaft

massless. Gyroscopicmoments were alsoincluded in theanalysis. Computationswere

quite tedious without computers.

Therefore,

the models had to be kept as basic as

possible. Pestel and Leckie

[6]

described the formulation ofthe field matrix in the

transversedirectionusing Timoshenko Beam Theory. In theTransfer Matrix

Method,

thefieldmatrixdescribesthemotionoftheshaft.

simply supported, overhanging shaft with gyroscopic moments. The

flexibility

coefficientswere

determined

by

defining

_{thedisplacementorrotationat one station} using

Bernoulli-Euler Beam

Theory

duetoa unitforceor moment_acting onthe system at an

adjacent station. This method can be used for systems with multiple

disks,

but it

becomes cumbersome for more than three disks. Examples are charted

describing

forward synchronous motion. Eshleman and Eubanks

[8]

studied the effectofaxial

torque on the critical speeds of a simple system _using partial differential equations.

Included inthe_studyweretheeffects oftransverseshear,rotatory inertiaandgyroscopic

couple. Themathematical model waskeptsimplifiedinordertoanalyzetheeffects ofthe

various parameters.

Using

Bernoulli-EulerBeam

Theory,

whichrefers toa massless shaft, Ruhl

[9]

studied

the _stability of_rotating shaftsdue toa mass unbalance _using the Transfer Matrix and

Finite Element Methods. Ruhlwasthefirstto_studytheuseofthefiniteelement method

for _{modeling rotating} systems.

Bearing

effects were included in the _model, but

gyroscopic couple_{rotatory inertia}and sheardeformation were notincluded. Theeffect

ofresidual shaftbowonthe unbalanced responseofaJeffcottrotor was analyzed _using

differentialequations

by

Nicholas,

Gunterand_{Allaire [10]. Residual bow may be due}to

various_effects, such as a gravitationalforce.

Damping

forceswereincluded inthestudy.

The studywas conductedtodeterminepossibleimprovementsto the

balancing

technique.

Nelson

[11]

wasthefirstto_study theuseofTimoshenko Beam

Theory

todeterminethe

shape functionof a_rotatingshaft,which wasthenutilizedin theFinite Element Method

todeterminethenaturalfrequencies ofthesystem. Previousanalyses had includedthe

study oftheeffects of_{rotatory inertia}and gyroscopic couple _using finiteelements, but

hadnotincludedsheardeformation. Hisresults were comparedtoclassicalTimoshenko

Beam

Theory

analyses for non-rotating and _rotating shafts. Rao

[12]

published an

analysis ofcritical

bending

speeds and forced responses of a _simply supported shaft

usingtheTransfer Matrix Method. He assumedthe shafttobe massless and without a

gyroscopic couple.

Benson

[13]

modeled a clampedoverhung diskwith

"active"

and "passive" gyroscopic

couples. Active gyroscopic couples are

forcing

functions due to disk skew. Passive

of angular momentum ofthedisk. Rieger

[14]

describedthenon-synchronous motionof

a_{clamped,overhungdisk using}theMethodofInfluenceCoefficientsandBernoulli-Euler

BeamTheory. Rao

[12]

analyzeda simplified model ofthenon-synchronous motion of a

clamped, overhung disk usingBemoulli-Euler Beam

Theory

in conjunction with the

Transfer MatrixMethod. Bothauthors plotted thenaturalfrequencies as afunctionof

whirl

frequency

_{parameter versus rotational speed parameter.}

Thepresent analysis utilizes the typical _setupof a_simply supported shaftwithmultiple

overhanging disks as well asdisks nested betweenthe bearings. The analysis, which

appliesthegyroscopic coupletoeachdiskand assumes non-synchronousmotion,utilizes

thepower oftheTransfer Matrix Methodandthemuchimprovedcomputational speed

and_abilityofthecomputer.

Finally,

the analysisincludesthe mass,rotatory inertiaand

shear deformation ofthe _shaft, which is of practical importance forsystems driven at

very high speeds. These high speed systemshave disksand shaftsofcomparable_mass,

Chapter 3

THEORY

AND PROGRAM

DEVELOPMENT

3.1

Transfer Matrix Method

TheTransfer MatrixMethod isadiscretizationprinciplethatcanbeusedtodeterminethe

natural

frequencies,

mode shapes and forced responses ofa _vibrating system. The

method consists of

defining

the

boundary

conditions at one end and _appending to it

information pertaining to the system ateachdefined increment along the length ofthe

shaft,untiltheopposite endisreached. Thesystem

information,

referredtoas thestate,

isthe

displacement,

slope, moment and shearforceat each

boundary,

shaftsection and

disk. This information istransferredfromone sectionto thenext_adjoiningsection until

anoveralltransfermatrixhas been formulated.

The Transfer Matrix Method can be applied to _{any linear} system. The method is

demonstrated usinga simple spring-mass systemwith a

forcing

function. Thissystemis

presented in figure 3.1.2. Vierck

[17]

presents an analysis of a spring-mass system

withouta

forcing

function.

I

i>

R

v

V,

R

R

l-H^-

? _{F cos}

l

t

TTT)

^/sss^ssJ>jZi/s^J

DZH

0

y

Stations arelocated at changesin equilibrium. The terms 'R' and

L'

refer to the right

andleftof eachstation. Three _stations,whoselocationsare indicated

by

thenumbers

0,

1 and

2,

arerepresentedinfigure 3.1.2. Astatevectorisa columnvector_containing,in

this case,the

displacement

andforceontheright and leftof each station. Statevectors are relatedtoone anotherthroughpoint andfieldmatrices,whichdescribethemotion of themasses and _springs,respectively. Afree

body

diagramofthesystemis developed in figure 3.1.3to

facilitate formulation

ofthepoint andfieldmatrices.

,R

^->VW^-

v>W/u

F cos

2

1

%

X,

%

X,

Figure3.1.3: Free

Body

DiagramofSpring-MassSystem

Theforces anddisplacements actingoneach component are summarized from thefree

body

diagramandexpressedinmatrixform inthe

following

steps.

1)

Sprine 1:

xV

x0

+

tf

=

Fc

(3-1.1)

(3.1.2)

1

0

1

r _{I R} X

IpJ

(3.1.3)

whichisoftheform

{S}\

= [F]*

{S}?

R

(3.1.4)

2)

Mass 1:

xl

-xl

-R

F^

= _m-x

lM

(3.1.5)

(3.1.6)

whereinthecase of sinusoidal motion of_m\

Xj =

Ai

sin cot

"2 2

x- =

-A! co sin cot = -co _xx

(3.1.7)

(3.1.8)

Substituting

x^ intoequation3.1.6obtains

F*

=

-mjco^j +

F^

(3.1.9)

Equation3.1.5 and3.1.9canbe re-_{written as}

1

0

-tr^co 1

f),

(3.1.10)

whichisoftheform

3)

Spring

2:

*5t

xi

+

F?

Fo

=

F

R

(3.1.12)

(3.1.13)

Equations3.1.12and3.1.13 canbere-written as

* X

F L

2

"i

k2

1 .

x

F R

(3.1.14)

whichisoftheform

{S)L2

=

[F]2{S}Ri

(3.1.15)

4)

Mass 2:

x!

b2

x^

F^

=

m2x2

(3.1.16)

(3.1.17)

whereinthecase ofharmonicmotionof_m2

x2 =

A2

sin cot

2 2

x2

=

-A2 co sin cot = -co _x2

(3.1.18)

(3.1.19)

Substituting

X2intoequation3.1.17 obtains

F?

=

Equation3.1.16and3.1.20canbere-_writtenas

1

0

-m2co

1

Fl

(3.1.21)

whichisoftheform

{S}2

=

[P]2

(S>2

(3.1.22)

Thestatevectorsare relatedthroughthepoint andfieldmatricesinthe

following

manner:

{S}?

{S}?

=

[FMSh

[PMS}^

[F]2{S}!

[P]2(S}2

[P]i[F]i{S}0

[F]2[P]i[F]i{S}0

[P]2[F]2[P]i[F]i{S}0

(3.1.23)

(3.1.24)

(3.1.25)

(3.1.26)

Equation3.1.26can bewrittenas

X R 1

"l

k2

1 i -L

ki

.F, 2 -m2co

1

1 . -mjco 1 1

(3.1.27)

Aglobaltransfermatrixisformulateduponmultiplicationofthepointandfieldmatrices.

R

X

< . =

F

₂

m-co

1-i

(m2

+ m1)o)

01201-0)

k2ki

k-

k-k2

k2

mi mi mi

+ - ₊

Ui

k2

k- j

b2+

1

The

boundary

condition of_xR0=_{0issubstitutedinto state}

vector

{

S

}R0

toobtain

x R 2 m2miC0

k2ki

_1_

ki

4 ( mjco _i

kik2

k2

m2

m2

vt\\

\ *!

co2+ 1

1 /

(3.1.29)

Forthe

homogeneous

solution,

F^

equalszero,whilefortheparticularsolution,

FR2

is

equalto theappliedforceFcosQt. Thenatural

frequency,

_co,ofthesystemis determined

fromthehomogeneous solutionoftheequation

0

=

m2m_{i co}

k2ki

m V M m2 mi k coz + 1 l /

(3.1.30)

Forthe nontrivial_solution,

F0

cannot equal _{zero;therefore,} the termsinthebracketmust

bezero. Thenatural

frequency,

_co,can nowbe solved_usingthequadraticformula.

To determine the forced response, the

forcing

function,

for instance

Fcos2t,

is

substitutedinequation3.1.29for

FR2-F cos Qt =

m2m1co

k2ki

^m2

m2 m-^

+ +

k2

k a* coz + 1 l /

F0

(3.1.31)

The

force,

F0,

can be calculated from equation 3.1.31 utilizing the natural

frequency

determined fromequation3.1.30. Nowthat

F0

is

known,

theforcedresponses_xR2and

In general,fora_{rotating shaft, thesysteminformation is writtenin matrixformdefined}

as field matrices and pointmatrices. A station is located at _anychangein equilibrium such as a _mass,

bearing,

coupling

or

boundary

(see figure 3.1.4). A state vector is a

column vector

containing

displacement,

slope, moment and shear force for station i.

Adjoining

state vectors arerelatedtoone anotherthrougheitherfieldmatrices orpoint matrices. The field matrix [F]- _for the shaft element

Li

describes the equilibrium

conditions for

L{

or, in thetraditional theory, the stiffness matrix forthe shaft section.

The point matrix[P]- describesthe equilibrium conditionsfor mass i_or, in _general,the

mass matrixforthedisk.

In_general,

SRi

i-1

m(i-l)

t

Li

SLj

SR

mi

t

m(i+l)

Figure 3. 1.4:

Relationship

Between Stations With RespecttoMasses

whereLrefersto theleftof a station andRrefersto therightofa station.

The relationship betweenthestatevectors

{S}^

and [S)i_iRisgiven

by

{S}iL= [F]i{S}i_iR

(3.1.32)

The_{relationship between}thestate vectors {S}-Land {S)iRisgiven

by

{S}iR=[P]i{S}iL

(3.1.33)

A)

If the length of the shaft

begins

with a mass such as in the case of an

overhanging

shaft

mass 1 mass

2

mass

3

k-

field

1

JC

field

2 H*

field

3

Figure 3.1.5: DescriptionofFields With RespecttoMasses

let

{

S

}

_jL=

{

S

}

₀=

starting

boundary

conditions

(3.1.34)

{S)lR =

[P]i{S}0

(3.L35)

{S}2L =

[F]i

{S}iR =

[F]i [P]i

{S}0

(3.1.36)

{S}2R = [P]2{S)2L=

[P]2[F]l[P]l

{S}0

(3.1.37)

{S}3L = [F]2{S}2R=

[F]2[P]2[F]i[P]i

[S}0

(3.1.38)

This transferof state information is continueduntil a

bearing

is encountered. Thestate

vectorismodifiedatthe

bearing

_accordingto therequired

boundary

conditionsof zero

displacement,

continuous slope and moment and a change in shear force due to the

reactionforceofthebearing.

B)

Iftheshaftdoesnotcontaina mass at

{

S

}0

such asisthecase of a_simply

supportedshaftwithoutan_overhangasin figure

3.1.6,

then

mass 1 mass

2

field

1

[

field 2

+

field 3

Figure 3.1.6:

Relationship

BetweenFields WithRespecttoMasses

{S)iL=

[F]i{S}0

{S)iR= [P]i{S}iL=

[P]1[F]1{S}0

(S}2L=

[F]2

{S)iR=

[F]2 [P]i [F]! {S}0

{S)2R=

[P]2

{S}2L=

[P]2

[F]2

[P]l

[F]i

{S}0

(3.1.39)

(3.1.40)

(3.1.41)

(3.1.42)

Thistransferof stateinformationiscontinueduntila

bearing

isencountered, atwhichthe state vectorismodified_accordingto the

boundary

conditions specified atthebearing.

Specifically,

the

boundary

statevector {S

}0

canbe definedas

{S}0

=

w0

w0

e0

. .

e0

0

V0.

.

o

.

for an _overhanging shaft

(3.1.43)

(S0}

=

w0

0

e0

M0

. <

e0

>

0

.V0.

V0.

for a _simply supported shaft

(3.1.44)

{S0>

=

W0

0

e0

M0

=

0

.V0. .V0.

for

a clamped end

(3.1.45)

The clampedend condition would existin a system that had two _{bearings very} close

together, thereby,causingtheslope oftheshaft atthatpointto_{be essentially}zero.

Allmatrices are shown as4x4and column vectors as4x 1 forease of explanation. The

analysis was _actuallycarried out forthe general case of a 17 x 17 matrix and a 17 x 1

vectortobe

discussed

in section3.1.4.

Further discussionon the specifics oftheTransfer Matrix Methodpertains toa _simply supported shaft with _overhanging

disks,

the same system that serves as the general

model.

Simply

Supported

Shaft With Overhangs:

1

2

a a+1

b

b+l n

7\

7\

Figure3.1.7: IdentificationofStationsForaMulti-Disked

Simply

Supported Shaft With Overhangs

In general,it follows fromtheabove that

{S}aL=

[A][S}0

(3.1.46)

{S}bL= [B]{S}aR

(3.1.47)

{S}n

=[C]{S}bR

(3.1.48)

where

[A],

[B]

and

[C]

are theoverall transfer matrices forthe

left,

middle and right

sections,respectively. Forexample,

[A]

=

[F]2[P]2[F]l[P]l

(3.1.49)

Thestate vector andthematrix

[A]

are reformulated atthefirst bearing. Thestatevector

{

S

}0

isredefinedtobeafunctionofdisplacement_only,through the

following

steps:

{s

=

[A]

{S}0

=

0

6a

Ma

? =

-.Va.

An A12

A13

A14

A21 A22 A23 A24

A31 A32 A33 A34

A41 A42 A43

A44j

w.

e.

0

0

J

(3.1.50)

The firstrow of equation 3.1.50 shows that thedisplacement vanishes at the

bearing,

thereby

resultingintherelation

Anwo

+

A129o

=0

(3.1.51)

whichcanberewritten as

e0

=

fAnl

K^XIJ

wr

(3.1.52)

Thiscanbesubstitutedinto

{

S

}

₀togive

(S}0

=

w0

<

11

A12

0

0

(3.1.53)

{si

=

0

fAn

^22

l32

>42

VM2/

(W0}

=

0

A'2i

A' 31

A' 41

(w0>

(3.1.54)

Now {S}aR={S}aL₊

Pa

(3.1.55)

where

Pa

is the reaction force at the bearing. The slope and moment are continuous acrossthe

bearing

and arethereforenot modified.

{

S}aR_{can bewritten} as

{S}*

=

0

0 0

w0

A'21

0 0 0

0

A'31

0 0 0

< >

0

A'41

0 0

1

^a.

=

[A'] {S*}

(3.1.56)

Thisequationissubstitutedintotheequationfor

{

S

}

5L

togive

{S}bL =

[B]

[S}aR =

[B] [A]

{S'}

=

[B']

{S'}

(3.1.57)

Asimilarprocedureiscarriedoutforthesecond

bearing

as wasdoneatthefirstbearing.

The statevector

{S'}

andthematrix

[B']

willbemodified atthesecondbearing.

(S)b

=

f \

0

eb

Mb

. =

[vbj

B'n B'l2 B13 B14

W

B21 B22 B23

B24

0

B31 B32

B33

B

₃₄

0

B'41 B?42 B>43

B44_

P,

(3.1.58)

indicating

thatthedisplacementatthesecond

bearing

iszero. Fromequation3. 1.58, it

followsthat

B'nwo

+

B'14Pa

= 0

(3.1.59)

which canberewritten as

Pa

= "

a ₁₁

VB'l4y

Wn

(3.1.60)

This canbesubstitutedinto

{S'}

as

IS'}

=

wc 1

0

0

B'n

B'

14J

(3.1.61)

which willthenproduce amodified

[B']

matrixcalled [B"].

{S)t

=

0

\D _14/

{w0)

=

0

B21

B31

B41j

Now {S}bR={S}bL₊

Pb

(3.1.63)

where

Pb

isthe reaction force at the bearing. The slope and moment are continuous

acrossthe

bearing

and_{therefore, are not modified.}

{

S)bR

canbewritten as

(S)b

=

0

0 0

Wo

ln

0 0 0

0

$3!

0

0 0

"

0

J41

0

0

1

Pb.

[B"]

{S"}

(3.1.64)

Thisequation canbe substitutedintoequation3.1.48 for

{S}n

{S]n

=

[C]

{S}bR =

[C] [B"]

{S")

=

[U]

{S"}

(3.1.65)

The _ntb disk is atthe end ofan _{overhanging shaft,} whose

boundary

conditions reflect

zero momentand shearforceand_continuityof slope anddisplacement.

Thus,

{S}n

= Wr

en

o

o

u

u12

u13

u14

U2i u22

u23

u24

u31 u32 u33

u34

u41

u42 u43

u44_

wc

0

0

Pk

(3.1.66)

The

[U]

matrixisaglobal matrixthatdescribesthemotion oftheentire_rotatingsystemin

the transversedirections. Thematrix was developed

by

_{relating adjoining} state vectors

and appropriate

boundary

conditions through point matrices and field matrices in a

systematic approachthatbeginsatstation 1 and ends at station n. The onlyunknownin

the

[U]

matrix is_CO, the whirl frequency. Whirlfrequencies that maintain _{equality in}

3.1.1

Natural

Frequencies

In a typical _environment,

disturbances

aretransferred to the _rotating system from its

surroundings. These_vibrations,for_example,can resultfrom an_operating

frequency

of

an_adjoiningcomponentorfrom floorvibrations. When a

forcing

frequency,

caused

by

the

disturbance,

equals the natural

frequency,

resonance will occur. An undamped, resonant mode willdisplacewithan_{arbitrarily large}amplitude. It is

important,

therefore,

to define the natural _{frequencies of a} system to determine if resonance, and the

subsequently largeamplitudes,willbeavoided. Thisanalysis ismost_usefullyconducted

during

thedesignof the_rotating system.

Theoverall transfermatrix

[U]

isutilizedto findthe naturalfrequencies ofthe system.

Since the moment and shear force are equal to zero in the state vector,

{S}n,

the

following

equations canbewritten:

woU3i+PbU34=0

(3.1.67)

w0

U41

+

Pb U44

=0

(3.1.68)

Thisiswritteninmatrixformas:

U31

u34

u41

u44_

w.

(3.1.69)

Forthesystemtobe physicallymeaningful,_w0and

P\j

cannotbothequal zero. Thusfor

a nonzero solutionofequation

3.1.69,

thedeterminantofthecoefficientmatrixmustbe

zero:

det

U31 u34

u41

u44_

=

0

(3.1.70)

Fora2x2matrix,thisrequires

U3iU44-U34U4i=0

Equation 3.1.71 is referred to as the

frequency

equation. _{The only} unknown in this

equation is CO,the whirl_{velocity (this is} shown in later derivations).

Every

solution of

the

frequency

equation willbea naturalfrequency.

Dueto the_complexityofthe

frequency

equation

3.1.71,

thenaturalfrequenciescannotbe

solvedfor

directly,

but must be determined_numerically through an iterative process.

Thisprocess entails_makinganinitialguessforco and then_{checking if}thesolutionofthe

frequency

equationiszero within sometolerance. If it isnot_zero, anotherguessismade

andthesolutionisagain checkedtobezero. Ifthesolutionto the

frequency

equation had

been _approaching zero,butthen changed_{directions resulting in}

increasing

_magnitudes,

then the solution to the

frequency

equation is within this range of guesses. Linear

interpolation isutilized withinthisrangetodeterminethefinalsolution. Anewguessfor

the next natural

frequency

isbegun at a small _{increment away from}this region. The

process continues as

before,

beginning

withincrementalstepsinthe newguesstowarda

new minimum. This scenario continuesuntil theentire

frequency

range ofinterest has

3.1.2

Mode

Shapes

The mode shapedescribesthedisplacementconfiguration ofthesystem. The shapecan

be utilizedto determineiftheperformance ofthe system will be acceptable. The first

bending

motioninforwardwhirl produces themost severedisplacementonthesystem.

This _severitydecreasesas thenumber ofbends increase inthe shape. Forthe

following

multi-disked, simply supported, symmetric systemthe

bending

mode shapesare shown

below:

I

~?F

1

I

1^

I

Figure 3.1.8:

Multi-Disked,

Simply

Supported Shaft

Figure 3.1.10: Second

Bending

Mode For Figure3.1.8

Figure 3.1.11: Third

Bending

Mode For Figure3.1.8

The Transfer Matrix Methodwillbeutilizedtodeterminethemode shapes associated with

the natural frequencies. Each mode shape represents a collection of relative

displacement,

w0, at station zero can _{arbitrarily be} set to unity. The only other

unknownsinthestate vectors atthe boundariesare

Pb,

Pa

and

0o

which canbesolved in

terms of w0. At this_point, it is assumed that a natural

frequency

has already been

determined forthe _{only unknown, co, in}the transfermatrices. Thisvalue ofcowill be

utilizedinequations

3.1.66,

3.1.60 and3.1.52tofind

Pb,

Pa

and

9,

respectively,and in

thepoint andfieldmatricestofindthedisplacementsateachstation. Itshouldbepointed outthatthedisplacementsatthebearingsare_alreadyspecifiedtobezero.

Sincetheshearforceiszero at {S

}n,

thelastrow of equation 3.1.66canbe writtenas

woU4i+PbU44=₀

(3.1.72)

or

Pk

= -

'LV

vU44

J

w

(3.1.73)

Also,

equations 3.1.52and3.1.60canbe written as

e0

=

-t A "\

All

wr

V^12/

(3.1.74)

Pa

= "

'bV

VB'l47

wn

(3.1.75)

Pk

= -

'LV

vU44y

(3.1.76)

e0

=

-All

12/

(3.1.77)

Pa

= "

VB'l4V

(3.1.78)

Themode shapes are determined

by

first_substitutingthevaluesfor

Pa,

Pb, 0o,

and_w0

intothestate vectors and_substitutingcointothefieldmatrices and point matrices andthen

solving forthedisplacementattheleft (for_consistency)ofeachstation.

Forexample,

{S}2L =

[F]i[P]i

{S}0

(3.1.79)

Denote

[F]i[P]i

as

[FP]i

sothat

w 9

M

V

=

[FPli

w0

e0

o

o

(3.1.80)

1 _{1 L}

W

9

M

V

=

[FPL

f \

1

An

A12

0

.

0

.

> wr

(3.1.81)

fromwhich

w2

FPn

-

'AiT

vAi2J

FP₁₂ Wr

(3.1.82)

This procedureis thenrepeatedforeach station. The displacementvalues can then be

3.1.3

Forced

Response

Theactualdisplacementduetoapplied

forcing

onthesystemcanbe determined usingthe

Transfer Matrix Method. This displacement isreferred toas the forcedresponse. The

investigatedmodel hasbeen developedtodeterminetheforcedresponseduetoa mass

unbalance and a gravitational force. Asimilaranalysis canbe carriedout for any force

application sincetheoverallTransfer Matrixwasdevelopedas ageneral 17 x 17 matrix.

The specific values ofthe load would remain to be defined in the 17th column ofthe

respectivefieldor point matrices.

Asystemforce isincorporated inthe transfermatrixasanextracolumn appendedto the

point_matrix, ifit is aforce on the

disk,

orthe field matrix,if it is aforce on the shaft.

Unity

isappendedto the statevectortocompensateforthisextracolumn. Forexample,

thepointmatrixanditsassociated state vectorsfora4x4matrixis

{S)iR= [P]i{S}iL

(3.1.83)

-w R

9

< M =

V

.

1

. i

Pll Pl2

Pl3

Pl4

0

P21 P22 P23

P24

0

P31 P32

P33

P34

0

P41

P42 P43

?44

Uco2

0

0

0

0

1

w

9

M

V

1

(3.1.84)

mwhichUco2refersto thecentrifugalforce duetoa massunbalance.

Equation 3. 1.84isoftheformofastandardlinear

system

[A]{x)=[B]

(3.1.85)

where

[A]

containsthemass andstiffnessmatrices,

{x}

isthestatevectorand

[B]

isthe

forcing

function.

Solving

for

{x}

defines the particular solution for this equation.

Theoveralltransfermatrix can nowbewritten as

{S}n

=

wE

en

o

o

1

u u12

un

u14 u15

U21 u22 u23 u24 u25

U31 u32 u33

u34 u35

u41 u42 u43 u44 u45

0

0

0

0

1

W0

0

0

Pb

1

(3.1.86)

The

displacement,

w0, andthe

bearing

reaction

force,

P^

can be deduced

directly

from

3.1.86. Since themoment and shearforceare zero at station_n, the

following

equations

canbewritten:

U31wo

+

U34Pb

+

U35

= 0

(3.1.87)

Equation3.1.87canberewrittenas

Pk

=

'-U35-U3iw0^

u

34 J

(3.1.88)

and

U41

_w0+

U44

Pb

+

U45

=0

(3.1.89)

whichcanberewrittenas

(-UuPk-IL,^

wc

=

44 rb~

u45

u

₄₁

(3.1.90)

w0

=

-U44

1

vU41y

u35

u31

77- ~

77-(w0)

u34

u34

'U45^

vU4iy

(3.1.91)

fUu-UU,^

(VAA

w =

vU4iy vU34y vU41y

1

+ 44

fu

\U4iy A

fTT

u31

vU34y

(3.1.92)

Equation3.1.92canthenbesubstitutedback intoequation3.1.88 tofind Pb.

Now

0o

and

Pa

can bedeterminedfromequations 3.1.74and3.1.75.

9r

fAnl

vAi2y

w

(3.1.93)

Pa

= "

a ₁₁

VB'l4/

wr

(3.1.94)

Theforcedresponses are nowdetermined inthe samemannerasthemode shapes. The

whirl

frequency,

_co,is substitutedintoeachfieldand pointmatrix. Thedisplacementor

forcedresponse atthe _{left (for consistency)}of each station canbe determinedfrom the

relationshipsthatcomprisetheTransferMatrix Method. Forexample, thedisplacement

3.1.4

General

Description

of

Point

and

Field Matrices

A_rotating system canbe

described

ingeneralterms

by

a 17x 17matrixformulatedfrom

point matrices andfieldmatrices. The state vector canbeexpandedtoinclude motionin

theXandY

directions,

respectively. These directionsare orthogonal toone another and

also to the shaft. (See figure

3.2.1)

The motion in the X and Y directions can most

generally be described

by

terms_{containingsin(cot)} and_cos(cot)factors. Thestatevectors

forthegeneral matrix aredefinedas

{S,.c}

=

Xc

'x,c

M_y.c

-V_x,c

and

{Sx,s}

=

xc

'x,s

M

-V

y.s

x,sj

(3.1.95)

<Sy,c>

-Yr

'y,c

M_x,c

y.c

and

*Sy,s>

9_y.s

M

V

x,s

y.sJ

(3.1.96)

where 'c' and 's'

representcosine and sine. Thenegative signs associatedwith

Vx

andY

are conventionthat yields positive coefficients in the field and point matrices and are

carried throughout the entire analysis. The moment

My

is in the state _vector,

{Sx},

because themomentis_actingonthesystemintheX-Zplane,but its directionis_alongthe

Yaxis. Thissymbolconvention also appliesto themoment

Mx

being

inthe statevector

{Sv}- Thesestatevectors, along witha_unityrowtocompenstaefor

forcing

functions,

{S}

=

Xc

9

_x,c

M

y.c

-V

x,c

Xs

9_x,s

M

y.s

-V

x,s

-Y,

(3.1.97)

9_y.c

M_x,c

T

y.c

"Ys

9,

'y.s

M

x,s

V_y.s

1

Theoveralltransfermatrixforthegeneral solutionis derived ina similarmanner asthe4

x 4 matrix. The general solution requires _transferring all information in the form of

matricesinsteadof equations. Forexample,equation 3.1.51 inthe4x4matrixiswritten

as

Anw0

+

A1290

= 0

(3.1.98)

Inthegeneralsolution, thissame_{step in}theTransfer Matrix Methodiswritten as

[Ad]{do}

+

[As]{So}

=

{0}

(3.1.99)

where

{do}

isthe4x 1 displacementvectorat station0and

{S0}

isthe

Inexpanded

form,

equation3.1.99

becomes:

Ai,i

A15

A19

A1>13

'Xc

Ai,2

A1>6

A1>10

A1>14

6x,c

A5,i

A5>5

A5>9

A5ii3

A5,2

A56

A5)10

A5)U

6x,s

Ag.i

Ac,

5

A99

A913

-Yc

A9,2

A96

A910

A9U

6y.c

^13,1

Ai3)5 A]3>9

A13i3_

."YSj ₀ .Ai3,2

A13>6 A1310

A1314_

,6y,s.

=

{0}

(3.1.100)

Further _complexity arises in the solution ofthe equations. For _{example, to} solve for

{

S

}

₀inequation3.1.99,theinversesofthematrices mustbe found.

{S0}=-[As]-l[Ad]{d0}

(3.1.101)

This complexity iscarried through themodel until theoverall transfermatrix is found.

Toobtainthenaturalfrequencies ofthegeneral _system,avaluemust befoundthat will

ensure that thedeterminantof an 8 x8 subset oftheoveralltransfermatrix

[U]

is zero.

The8x8 matrixisobtained

by

_requiringthatfourmoment equationsandfourshearforce

3.2

Shaft Motion

Themotion oftheshaftis

determined

by

thenaturalfrequenciesandtheforcedresponses

arededucedfromthefieldmatrix as part oftheTransferMatrix Method. Themechanics

thatgovernthemotion canbe described

by

eithertheBernoulli-EulerBeam

Theory

orthe

TimoshenkoBeamTheory. TheBemoulli-Euler Beam

Theory

is thetraditional_theory usedinrotordynamic analysisduetoitssimplicity. Itneglectsthemass oftheshaft and only relates shaft sections to one another through stiffness values. The Timoshenko Beam

Theory

includes,

_alongwiththeshaft_stiffnesses, shear

deformation,

_{rotary inertia}

and centrifugalforce. Theseadditional featuresincreasethe_complexity oftheproblem

3.2.1

Formulation

of

Field

Matrices

Using

Bernoulli-Euler

Beam

Theory

In this _section, the general 17 x 17 field matrix (see equation

3.1.97)

will be defined

usingthe

Bernoulli-Euler

Beam Theory. Themechanicswillbe described intermsofthe

state vectorfortheX

direction,

{

Sx

}

_, andthestatevectorfortheY

direction,

{

Sy

}

. The coordinate system will be defined as

having

its origin at the farthest left point in the

system with theZaxis

being

positiveto the_{right along}thelengthoftheshaft,theXaxis

being

positiveinthe upward vertical direction andtheY axis

being

positive out ofthe

page. Infigure

3.2.1,

the coordinateaxisisshown on atypical system.

H

/

'

H-.

-y^

/\

Figure 3.2. 1: Typical System Configuration WithCoordinate Axes

3.2.1.1 Field Matrix

[Fx]

The relationship between displacement x, the slope

9X,

themoment

My

and the shear

*>

z

Figure 3.2.2: Free

Body

DiagramofShaft Element in X-Z Plane

Using

Bemoulli-Euler Beam

Theory

The sign convention for moment is defined as positive and for shear is defined as negativeforthisshaft element. Thiscorrespondsto traditionalmatrix notationforease of

solution. This isvalidifa negative

Vx

shearforceis maintainedthroughout the entire

transfermatrix. Theparameters on thisfree

body

diagramthathavenotbeendefinedin

thissection are

i=station atlocationi

i-1 =_stationlocated beforestationi

R=_{theright sectionadjoining}a station

L=_theleftsection_adjoininga station

Lj

=lengthof shaftlocatedbetweenstationsi-1 andi

The

following

general relations

describe

therelativedisplacementandrelativeslope of a beamelement x = 9 = ML2

2EI

+ VL3

3EI

ML + VL2 EI

2EI

(3.2.1)

(3.2.2)

These equations arefromBemoulli-Euler Beam

Theory

for acantilever beam. The E representstheelastic modulus ofthe shaftmaterial andtheIrepresentstheareamoment ofinertiaofthe shaft. Fora round shaft (whichisthe_onlycross-section consideredin

this analysis), I equals n

D^/64,

in which D is the shaft diameter.

Using

these beam

theory

equations, plus _{compensating for} a change in slope at x^.j, the

following

equationsfor displacementand slopecanbe formulated.

X; =

X?-l +

6*

-_1

Li

+

M^j

'l?^

2EI

+

V

X.l \J

( L3

"

3EI-

(3.2.3)

9V

'X,li

ex,i-l

+ M

l

r

Li.

y,i

I

Eli

+ V

f L2

X,l

2EI:_\J

(3.2.4)

The shaftdiametercan_{vary along}its

length;

thereby,thearea moment ofinertiamustbe

defined for each shaft section 'i'. In a_rotating system, achange in the shaft diameter

may be required to accommodate such components as _motors, bearings and collars.

Tapered beams are analyzed

by

discretizing

the shaft into numerous sections. Each section would contain a _subsequently smaller

diameter, thereby

_{approximating} the continuityofthe taperedbeam.

The equations for shearand moment are written based on the equilibriumconditions

M^i

=

M^i_i

-V^i_i ^

(3.2.5)

yL._v vR- _{i or} -VL- = ₁

x,i v x,i-l U1 Y x,i y x,i-l

(3.2.6)

The moment and shear equations are substituted into the displacement and slope

equations toobtain x-L

and

9X

-L

as afunctionoftheright sectionofstation

'i-1'

only.

Theseequationsbecome

xi

=

4-i

₊

0x,i-i

Li+

(MRi_!

-v;ti_! L^

^L2

^

v2EIiy + V R x,i-l

f

j2

\

X; =

4-1

₊

li-i

L;

+ MR i_i v2EIiy - V R x,i-l v6EIiy

UElJ

(3.2.7)

(3.2.8)

and

ei,i

=

eVi

+

(My.i-i

VR,i_i

L;)

r t . \

vEIi, + VR 1 T v x,i-l v2EIiy

(3.2.9)

eiti

=

eRi_i

+

MRi_i

vEIiy

VR-

-v _x,i-l

( L2

^

2EIJ

(3.2.10)

Inmatrix

form,

theseequationsbecome

X

L

6x

? =

My

[-VxJ

i

1

Li

0

1

<

L2

^

(

L3

^

v2EIiy

I

Eli

J

1

v6EIiy

^

,2 >

0

0

0

0

0

Equation 3.2.1 1 canberepresentedintheform:

{SxhL =

[FJi

{SX}RM

(3.2.12)

3.2.1.2 _{Field Matrix [Fy]:}

The relationships between

displacement

y, the slope

9y,

themoment

Mx

and the shear

force

Vy

_actingon a shaft element are embodiedinafree

body

diagram depicted in figure

3.2.3.

Y

4

V

R

y.i-i

i-1

i,^

<s>

+

t

Y

i

X

R

-?

Z

l-l 'i-1

Li-Figure 3.2.3: Free

Body

DiagramofShaft Element in Y-Z Plane

Using

Bemoulli-EulerBeam

Theory

Thesignconventionformoment and shearisdefinedas positiveforthis shaftelement.

theX-Zdiagram. Notethatapositiveretation of the shaft produces adisplacement inthe

negative direction. In the X-Z plane, a positive rotation produces a positive

displacement.

From

Bemoulli-Euler

Beam

Theory,

the general equations canbewritten as

y =

-6V

=

ML/

2EI

ML

EI

VLJ

3EI

VL2 2EI

(3.2.13)

(3.2.14)

Using

thesebeam

theory

_equations, inconjunctionwiththeequilibrium equationsderived

fromthe free

body

diagram,

the

following

equationsfor

displacement,

_slope, moment

and shearforcecanbewritten. Thesehavetheform

y\ =

y*-i

-<i-i

^

-Miti

(

t2

^

2EI V^M/ + VL y.i

(

l}

^

ey,i

~

6y,i-l

+

Mx,i

vEIiy

-

VL-2EIi,

& =

MRi_i

+

VRi_i Lj

VL = VR ^ vy,i vy,i-l

(3.2.15)

(3.2.16)

(3.2.17)

(3.2.18)

The shear and moment equations can be substituted into the displacement and slope

equations,whichwillresult inthe displacementand slopeintheleft sectionofstationi

being

afunctionoftherightsection ofi-1 only.

0y,i

9y.i

~

ey,i-i

+

(MRi_i

+

VRi_i

Lj)

fLi

vEIi,

- V

y.i

(

L2

^

v2EIiy

e?fi-i

+

MRi_i

vEIiy

+ VR- 1

^ v

y,i-i

(

L2

^

2EIi

(3.2.19)

Inmatrixformthese_{equations,along}withthemomentandshearequations, become: f *\

Y

L

9y

=

Mx

IVy.

i

1

-Li

-=

0

1

0

0

0

0

' L2

1

.2ElJ

(

L ^

Eli

1

0

v^M/ ( L3

1

,6ElJ

2ElJ

Li

1

M,

R

(3.2.21)

i-1

The displacement equation is multiplied

by

a negative one to produce positive

coefficientsforease of solution. This isvalidifa negative_{y displacement is}maintained

throughout theentiretransfermatrix. Thematrix equation3.2.21 becomes:

-Y

Mv

(

t2

A

(

T3

^

1

Lj

0

1

0

0

0

0

L

2EIiy

Eli,

1

0

6EI

Li

2

^

2EIi

Li

1 -Y

9,

M,

yJ i-1

(3.2.22)

Equation 3.2.22can bewrittenintheform

{Sy}-L =

[F]y>i

{Sy}Ri_i

(3.2.23)

The

displacement,

slope,momentand shearcan beexpandedand writteninthe general

x=

Xccos cot +

xs

sincot

(3.2.24)

ex

=

6xc

cosm +

9xs

sinm

(3.2.25)

My

=

Myccoscot

₊

Myssincot

(3.2.26)

vx

=

Vxc

cos cot +

Vxs

sincot

(3.2.27)

and

y=

yccoscot +

ys sincot

(3.2.28)

9y

=

9yc

coscot +

9ys

sin cot

(3.2.29)

Mx

=

Mxc

coscot +

Mxs

sincot ₍₃₂

30)

Vy

=

Vyc

cos cot +

Vys

sincot

(3.2.31)

in which

'c'

refersto _cos(cot) and 's'

refersto sin(cot).

Naturalfrequenciesofsimplified_models,thatcontain equivalent stiffnessesintheX and

Y

directions,

canbe determined from 4x4 field and point matrices formulated in one

direction only. The simplified model assumes that theslope anddisplacementofeach

disk and shaft element are equivalentin both theX and Y directions.

Therefore,

one

directioncanbeeliminatedfromtheanalysis.

The generalmodel presentedinthis

investigation,

whose state vectoris represented

by

equation

3.1.99,

wouldberequiredtodeterminethenaturalfrequenciesof a systemwith

anisotropic bearings and todetermine theresponse dueto a

forcing

function.

Bearing

forces canbe modeledinthis analysis

by

_modifyingthe fieldmatrices _surrounding the

bearing

station to accommodateforthe stiffness of the bearing.

Utilizing

the general

modelfor analyzinga systemdoesnotimposerestrictionsthatwouldlimitthescope of

theproblem.

The

[Fx]

and

[Fy]

fieldmatrices willbeformulated intoan overall 17x17generalfield

^X.C

R

^x,s

Sy.c

=

Sy,s

.

1

. i-1

[Fx]

0

0

0

0

0

[Fx]

0

0

0

0

0

[FyJ

0

0

0

0

0

[Fy]

0

0

0

0

0

1

^x,c

Sx,s

sy.c

^y.s

[

1

(3.2.32)

3.2.2

Formulation

of

Field Matrices

Using

Timoshenko Beam

Theory

A general 17 x _{17 field} matrix will be defined from the mechanics associated with

Timoshenko

Beam Theory. This

theory

includes,

inaddition to thestiffnessofthe shaft,

the effects of shear

deformation,

rotary inertiaand centrifugal force. The coordinate

system willbethe same as thatdefinedfortheBemoulli-Euler

Theory

shownin figure

3.2.1. Theorigin willbe located atthefurthestleftpointinthesystemwith theZaxisin

the direction of the _shaft, the X axis oriented _vertically and the Y axis oriented

horizontally. The field matrix will be derived in terms ofthe state vector for the X

direction, {Sx},

andthestate vectorfortheY

direction,

{Sy

}.

3.2.2.1 Field Matrix

[Fx]

The displacementx,theslope

9X,

themoment

My

andtheshearforce

Vx

are related as

depicted in the free

body

diagram in figure 3.2.4. Also shown in the diagram are the

centrifugal

force,

_msco^xdz, andthe_rotary

inertia,

Isyco29x

dz. Thesign conventionin

the free

body

diagramdefines the moment as positive andthe shear as negative. This

convention is consistent with the conventions associated with the Bemoulli-Euler

Y

Figure3.2.4: Free

Body

DiagramofShaft Element in X-Z Plane

Using

Timoshenko Beam

Theory

Fora circular shaft_section,

Is

yiscalculatedas

IS)y

= (l/12)mS)i(3a2+

Li2)

(3.2.33)

inwhich

'a'

is theradiusoftheshaft[18].

To define and understandthe equation for shear

deformation,

the

following

free

body

+-z

Figure 3.2.5: ShaftElement in X-Z Plane Subjectedto

Bending

Moment

Figure 3.2.5 shows a shaft element subjectedtopure

bending

only, dueto the moment

My

= EI (d9x/dz). This is_the fundamentalrelation in Bernoulli-Euler Beam Theory.

Line a', that passes through the end of the bent shaft, is perpendicular to the

cross-sectional face oftheend oftheshaft. Line

b',

which indicates theposition of an

unbent_shaft,isparallelto theZaxis.

A negative shearforce on the shaft elementproduces apositive displacement at z-L,

which, alongwith thedisplacement due to a

bending

moment, produces anet positive

<>

Figure3.2.6: Shaft Element in X-Z Plane SubjectedtoBendingMomentandShear

Deformation

Theorientationofthecenterlineoftheshaft, along linea",changeswithout _anyrotation

occurring at x. Line b'remains parallelto theZ axis.

9X

is theslope duetoa

bending

moment , VX/GA is the slope due to shear force and dx/dz is the total slope of the

centerline oftheshaft.

Theparameters in figure 3.2.6aredefinedas

G=_shearmodulus

GAS

=_{shearstiffness}

As

=

A/Kg

A= cross-sectionalarea oftheshaft

iq=form factorthatdependsontheshape ofthecross-_section

The

relationship

including

_shear

deformation

can nowbewritten as

dx

Vx

dz

GAC

(3.2.34)

Rearranging,

theshear_{force is deducedas}

VY

=

GAC

dz

(3.2.35)

The generalfree

body

diagramforthefieldmatrixin theX-