Current Flexible Rotor-Bearing System Balancing Techniques Using Computer Simulation

Rochester Institute of Technology

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Theses

Thesis/Dissertation Collections

1975

Current Flexible Rotor-Bearing System Balancing

Techniques Using Computer Simulation

John Kendig

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CURRENT

FLEXIBLE ROTOR-BEARING

SYSTEM

BALANCING

TECHNIQUES

USING

COMPUTER SIMULATION

by

CURRENT FLEXIBLE ROTOR-BEARING SYSTEM

BALANCING

TECHNIQUES USING

COMPUTER

SIMULATION

by

John

R.

Kendig

A

Thesis Submitted

in

Partial Fulfillment

of

the

Requirements for

the

Degree

of

MASTER

OF SCIENCE

in

Mechanical

Engineering

Approved by:

Dr.

Dr.

Dr. _.

Dr.

Dr.

(ThesisAdvisor)

(External Reviewer)

(Assistant Provost RIT)

(DepartmentHead)

(Dean CollegeofEngineering)

DEPARTMENT

OF MECHANICAL

ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER,

NEW

YORK

FOREWORD

When the thesis topic was formulated in 1972, it was felt that a

comprehen-sive study of current high-speed flexible rotor balancing techniques should

be made. In conjunction with Dr. N. F. Rieger, Gleason Professor at Rochester

Institute of Technology, a three point study was decided upon for the thesis,

the three points being:

1. Study the modal balancing method of Bishop, et al,

2. Write a computer program to apply the modal technique,

3. Make a computer simulated comparison of the modal and influence

coefficient technique.

In the attached thesis dissertation, the three points of the original plan are

covered. The points have been studied in the following manner:

1. Study of the modal techniques of Bishop, Federn, Kellenberger, and

Moore and Dodd;

2. Modal balancing programs written including BAL, MBAL, and Modal

I and II;

3. Comparison of three rotor systems-an undamped steam turbine,

a dal'T!ped steam turbine, and a gas turbine

.

The following dissertation is submitted in partial fulfillment of the require

-ments for the Master of Science degree in Mechanical Engineering from

Rochester Institute of Technology.

Respectfully submitted

John

TABLE

OF

CONTENTS

ABSTRACT 7

LISTOF ILLUSTRATIONS 9

LIST OF TABLES 11

NOMENCLATURE 12

INTRODUCTION: OBJECTIVES 15

SURVEY OFTHE LITERATURE 17

Rigid

Body Balancing

23

Flexible Rotor

Balancing

29

Modal

Balancing

30

Influence Coefficient Approaches 43

Other Flexible Methods 46

MethodComparisons 47

EQUATIONSOF MOTION 49

DevelopmentofEquations 49

Modal

Balancing (Bishop

andGladwell) 52

MethodofMinimum Planes 60

Exact Point-Speed Influence Coefficient

Method(LundandRieger) 63

Generalized Equations Solutions 69

ANALYTICAL BALANCING SIMULATION EXPERIMENTS 72

Modal Methodof

Bishop

andGladwell 73

Simultaneous Modal MethodofKellenberger 80

InfluenceCoefficient Method (LundandRieger) 80

Balancing

Programs 82

Major Programs 82

Auxiliary

Programs 84

Examples 84

Flexible Rotor in Flexible - Undamped Bearings

-Planar Unbalance 85

Flexible Rotor in Flexible- Undamped Bearings

TABLE OF

CONTENTS

(Cont.)

Flexible Rotor in Flexible

-Damped Bearings

-PlanarUnbalance 99

Flexible Rotor in Flexible - Damped Bearings

-Spatial Unbalance 105

Rigid- _{Flexible Rotor in Flexible}- _{Damped Bearings} 116

RESULTS.... , 131

CONCLUSIONS 133

RECOMMENDATIONS 135

ANNOTATED BIBLIOGRAPHY 136

APPENDIX A: Goodman's Influence Coefficient Method 147

APPENDIX B: Modal

Balancing

Program"BAL"

152

APPENDIX C: Modal

Balancing

Program "MBALI"

1 68

APPENDIX D: Modal

"Averaging

Program"MODALI"

177

APPENDIX E: Modal"Averaging"

Program "MODALII"

186

APPENDIX F: Data ManipulationProgram"MANI"

196

APPENDIX G: Computer Graphics Program"SHAPE"

200

APPENDIXH: Computer GraphicsProgram"SHAPEII"

204

APPENDIX I: Computer Graphics Program"AMP-SPD"

ABSTRACT

As

the

cost of

machinery has

risen and

the

need

for

dependability,

safety

and

increased

per

formance have in

a similar manner

increased,

the

needs of

industry

for

viable

flexible

rotor

balanc

ing

techniques

has

no

less increased. Various flexible

rotor methods

have been

advocated

for

su

percritical shafting,

but few

studies or comparisons

have

appeared

in

the

open

literature.

Among

the

procedures

for

balancing

large

and/or

high

speed

(supercritical)

rotors are the "N"

modal

method of

Bishop

and

Gladwell,

the

"N

+

B"

modal of

Fedem,

the

"N" and "

+

B"

simultaneous

modal method of

Kellenberger,

and

the

influence

coefficient method of

Lund

and

Rieger.

Each

of

the

aforementioned

balancing

techniques

is

examined andexplained

in

detail. The

first

known

modal

balancing

programs are

listed

and

described.

Using

these programs as a

basis,

the

influence

coefficient method of

Lund

and

Rieger is

compared to the modal methods of

Bishop

and

Gladwell, Fedem,

and

Kellenberger. The

companies are madewiththe aid ofa

Prohl

based

unbal

ance response computer program.

The

rotor systems used

for

the comparison are

flexible

shafts,

some mounted

in damped

bearings,

and some mounted

in

undamped

bearings.

One

sample sys

tem exhibits rigid

body

behavior

in

addition

to

flexible

behavior.

These

examples

form

the

basis

ofthe

first known

direct

computer

based

comparison

between

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Figure 24 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29 Figure 30 Figure 31 Figure 32 Figure 33 Figure 34 Figure 35 Figure 36 Figure 37 Figure 38 Figure 39 Figure 40 Figure 41 Figure 42 Figure 43 Figure 44 Figure 45 Figure 46 Figure 47 Figure 48 Figure 49 Figure 50 Figure 51 Figure 52 Figure 53 Figure 54

LISTOFILLUSTRATIONS

Jeffcottrotor system withdeflectioncaused

by

unbalance. Cross-sectionofJeffcottrotor.

Phaseangle_{in vicinity}of a critical speed.

Rotor behavior below,at.andabovea critical speed. Systemresponsein_vicinityofacriticalspeed.

Relative PositionsofCenter-of-Gravity,equilibriumcenter,and

bearing

centerasafunctionof speed. Bromberg's displacementconstruction.

Vector diagramofdisplacement

balancing

method. Rathbone's

balancing

procedure.

Thearle'smethod oftwo-planebalancing. Rotoraxisdeflectionshapes.

Failureof2-plane balancetobalance flexiblevibrations.

General surveyofthefirst four CHARACTERISTIC SHAPESofa turbogeneratorshaftinsoft-isotropicbear ings.

IllustrationoftheNmodalmethod. Argand diagramof rotor vibration.

Argand diagramwithnon-negligiblevibrationinother modes.

Typicalvector'sforvibration atfirstchosen speed(near but belowthefirstcritical speed).

Vectorrepresentation ofconditionsatthesecond chosen speed(nearrunningspeedandbelowthesecond

criticalspeed).

MooreandDodd'smodalAVERAGINGprocedure. Influencecoefficient principles.

Signconventionfortheequation of motion. Criticalspeed effects.

Deflectionoftherotoraxisinthe vicinityof a critical speed.

Modificationofthemass axis

by

addition of adiscrete

balancing

mass at z=_z Applicationofthemodal

balancing

method of_BishopandGladwell.

Samplerotordeflections.

SamplerotorAVERAGINGprocedure. Modal AVERAGINGprocedure. Modal BALANCINGprocedures. Steamturbine.

Steamturbinerotor model.

Planarunbalance weightdistribution. Originalunbalance response. Originalrotordeflectionat2,300rpm. Originalrotordeflectionat5,500rpm. Originalrotordeflectionat10,000rpm. Nmodalmethod,firstmode removed. Nmodal method,second mode removed. Nmodal method, thirdmode removed. Nmodalmethod,finalbalance.

Simultaneous Nmodalbalance.

Influencecoefficientbalance,3-speed.

Spatialunbalancedistribution. Originalunbalanceresponse.

Originalrotorresponseat2,300rpm. Originalrotorresponse at5,500rpm, Originalrotor response at10,000rpm.

SimultaneousNmodalbalance.

Simultaneous Nmodal witha newN balance distributionapplied. Simultaneous Nmodal witha newN 2 balance distributionapplied.

Simultaneous Nmodal with2-planelow-speedbalanceappliedafter,with modaltrimming. SimultaneousNmodal with2-plane balanceat10,000rpmafter,with modaltrimming. Influencecoefficient3-speedbalanceusing5-planes (1-9-14-17-25).

Originalunbalance response. Originalrotordeflectionat2,300rpm. Originalrotordeflectionat2,700rpm. Originalrotordeflectionat5.500rpm. Originalrotordeflectionat8,000rpm. Originalrotordeflectionat10,000rpm. Originalrotordeflectionat1 1,500rpm.

Argand diagram: original response of

bearing

atStation 3. Arganddiagram: original response of

bearing

atStation 23. Nmodalbalance.

Simultaneous Nmodalbalance.

Nmodalbalanceat undamped critical speeds. Originalunbalance response.

Originalrotordeflectionat2,300rpm. Originalrotordeflectionat2,700rpm. Originalrotordeflectionat9,000rpm. Originalrotordeflectionat12,000rpm.

Nmodalbalance:firstmode removed at2,700rpmusing Station14

Rotordeflectionat2,700rpm afterfirstmode removal.

Rotordeflectionat2,700rpmafter second moderemovalat12.000rpmusing Stations1and25 Rotor deflectionat9,000rpm after second mode removal at1 2,000fpm_usingStations1and25.

Rotor deflectionat1 2,000rpmafter second mode removal at1 2,000rpm_usingStations1and25

Modal AVERAGING:4-plane(1-7-16-23)at2,700rpm.

Modal AVERAGING:5-plane(1-7-10-16-23)at2,700rpm.

Modal AVERAGING:4-plane (1-7-16-23)at9,000rpm.

Modal AVERAGING:5-plane(1-7-10-16-23)at9,000rpm.

Modal AVERAGING:4-plane(1-7-16-23)at1 1,500rpm. ModalAVERAGING:5-plane(1-7-10-16-23)at1 1.500rpm. Influencecoefficientbalance: firstmode at2300rpm_usingStation 14.

Influencecoefficientbalance: 2-plane(7-16)at2,700rpm. Influencecoefficientbalance: 2-plane(1-23)at9,000rpm.

Influencecoefficientbalance: 3-plane(8-14-21)at2,300rpmand9,000rpm. Influencecoefficientbalance: 3-plane(8-16-21)at2,300rpm and9,000rpm. Influencecoefficientbalance: 4-plane(1-7-1623)at2,700rpm and9.000rpm. Gasturbinerotor model.

Firstsystem critical at7,051.87rpm(no damping). Secondsystem critical at9,592.82rpm(no damping). Thirdsystem critical at24,946.4rpm(nodamping). Originalrotordeflectionat7051.87rpm(with damping). Originalrotordeflectionat9,592.8rpm(with damping). Originalrotordeflectionat24,946.6rpm(with damping). Originalrotor response.

Original dampedrotordeflectionat7,200rpm. Original dampedrotordeflectionat10,400rpm. Original dampedrotordeflectionat25,450rpm.

Rigid-body

2-plane balanceusingStations2-₁6at500rpm. Rotor deflectionat7,200rpm.

Rotor deflectionat10,400rpm. Rotor deflectionat25,450rpm.

ModalAVERAGINGappliedto_{rigid-body balanced}rotorat25,450rpm_{using 5-planes (1}-2-13-16-21 ).

Rigid-body

balanceplus3-plane (1-2-16)influence balanceat1 0,400rpmand25,450rpm.

Rigid-body

balanceplus3-plane(1-2-21)influence balanceat10,400rpm and25,450rpm

Rigid-body

balanceplus4-plane(1-2-_{16-2 1}

)

influence balanceat1 0,400rpm and25,450rpm.

Rigid-body

balanceplus5-plane(1-2-13-16-2 1)influenceat7,200rpm,1 0,400rpm and25,450rpm.

Rigid-body

balancep'lus5-plane

(1-2-13-16-21)

influenceat10,400rpm,25,450rpmand66,000rpm.

Rigid-body

balanceplus7-plane (1-2-8-1 1-13-1621) influenceat 10,400rpm, 25,450rpm, 50.000rpm. and66,000rpm.

R9ure 111

Rigid-body

balanceplus 7-plane

(1-2-8-11-13-1621)

influence at7,200rpm, 10,400rpm, 25,450 rpm. and66,000rpm.

Figure 112 Influencecoefficient _{balance using 7-planes}

(1-2-8-11-13-16-21)

at 10,400 rpm, 25,450 rpm, 50.000 and66,000rpm_{(no rigid-body balance).}

Figure 113 Influencecoefficientbalanceusing7-planes(1-2-8-_{1 1}

LIST OFTABLES

Table I Comparisonofthenumber of required planes.

Table 2 Computationoftheoveralleffectoftheweights addedmostly forthesecond mode.

Table 3 Computationofthe overalleffect oftheweights added_{mostly for}the thirdmode.

Table 4 Gasturbinerotor modeldimensional data.

NOMENCLATURE

change of masseccentricityas aresultof addition ofdiscreteweightadditions

A cross-sectionalarea

A.B.C, rotor vibrationcoordinates,i.e.OA.OB.etc A'.B'.C.

A1.A2.B1...

b viscous

damping

coefficient of elementi

b.b'.b"

viscous

damping

coefficient

b criticalviscous

damping

coefficient

B.B'.B viscous

damping

coefficient

c

| hysteretic

damping

and stiffnesscoefficient of elementi

cc'.c"

linearstiffness coefficient

C ij influencecoefficient of response of planeitoeffect at plane

j

C masscenter of shaft slice

C_r modalconstant(s)formode r

D.D'.D"

hysteretic

damping

and/or stiffnesscoefficient

e.e'

eccentricityofmasscenterfromgeometric center

e complex statementofmasseccentricity

ei eccentgeometricricityof mass centerfromcenteratplaneiincomplex notation

er complexeccentricityexpressibleinmodalform forthe rthmode

E geometriccenter of shaft section

El flexuralrigidity

FK

,F

bearing

forces inx andydirections,respectively

g forceofgravityconstant

I . moment ofinertiaabouttheiaxis

K_j stiffness of

bearing

i

m mass of shaft section

m.r. weightinthe ithplanewithamagnitude of m and at aradialposition of rfromthegeometric center

M mass center of shaftsection,i.e.thecenter ofgravity

M

bending

moment

N generalizedcoordinatesfortherth mode

0 geometriccenter-lineformed

by

the

bearing

centers

q; unbalance

forcing

functionofelementi

S shearforce

,

tj.Tj

trialweight atplanei

Uj.U'j unbalance_actingatplanei

Ur characteristic mode ofvibrationformoder

V

;j vibrationat planeiunder conditionor speed

j

V complex motionof shaft slice

V r complexrepresentationofshaft motion expressibleinmodaltermsfortherthmode

w speed ofshaft rotation,radians per second

w_; ith system natural resonant

frequency

W distributedmassof shaft

x Xcoordinateofgeometric center of shaftsection

x'.y'

XandYcoordinatesofdeflectionof

bearing

support

x",y"

XandYcoordinatesofdeflectionof rotorshaft

Xr modal representation ofdeflectionof rotorin XZplanefortherth mode

y Ycoordinate ofgeometric center ofshaftsection

z*

normalizingconstant

Z

b Z dimension frompositionb.about which momentsaretakenfor

bearing

i

Z axisabout which rotor rotationandmotion occurs

Zj.Zj

axial position of planei

a _tj influencecoefficient relation reactionatitoadisturbanceat

j

/3 phaseangleof masscentertoelasticcenter

y phaseangleofreaction

Y weight perunitvolume

0 (z) characteristic modalfunctionforthes principalmode,

being

afunctionof z 5

8 s(z|

)

deflectionand angularpositionof planei intheprincipal mode x,i.e.a complex statement ofthelocationof thecenterofgravityof the ith section inthe sth mode

2

r modalfunctionfor rth mode

INTRODUCTION: OBJECTIVES

Theyear 1882markedthe yearthat Gustav de Laval built his first impulse turbine, for use ina cream separator. From 1884to 1889 de Laval producedturbineswith capacities _{ranging from 1}

hp

at 100.000rpmto 100 kilowattsat6,000

rpm.The importanceofthese dates is notthat

they

mark anintroductionof an impulseturbinewith commercial appli cations,butrather,thesignificance residesinthefactthat these,as well asthefirst deLavalturbineof1882were pieces ofrotatingapparatusdesignedandbuilttobeoperated at rotational velocitiesinexcess ofthefirst

bending

critical speed ofthe system;

they

were supercritical systems.

So

long

as

turbomachinery

or otherformsof_{rotatingmachinery}remained belowseventy-five percent ofthefirstcritical

speeditwas possibletoachieve_{satisfactory levels}of residual unbalance

by

_{using the}classicaltwo-planelow-speed bal ance which compensatesforthe _rigid-bodyeffects of unbalance vibration.

However,

by

operatinga rotatingsystem to

within seventy-five percent ofits first flexible critical speed orbeyond, the residual mass eccentricities

(unbalance)

no longerexhibit a constant_relationshipwiththerotorresponse,butratherbecomesvariable,dependenton speed,because

ofthe elasticbehaviorofthe system.

Withtheinception of elasticbehavioroftherotorsystem andtheattendantlossofaconstant unbalance- _{rotorresponse}

relationthe rigid-bodybalance is incapableof_{correcting for}unbalance overthe entire speed range,because itsbasisis founded onthe residual unbalance

bearing

a constant_{relationship to} the rotor response. Hencewith thefailureofthe

two-plane

balancing

schemeto_successfullycope withthesituation offlexibleshafting,it becomes necessaryto

develop

new

balancing

approaches which compensate fortheflexible characterdemonstrated nearthe system flexural natural frequencies (system flexiblecritical speeds).

Inthe

intervening

years since 1919,whenJeffcottintroduced thefirstaccurate analyticalformulation ofthe unbalance

response of a simpleflexible rotor, numerous methodsfor

balancing

flexible rotors have beenproposed and/orutilized.

Someofthese methods have met with success and others with obscurity.Two methods, inparticular, haveappeared, both

incidentally

intheearlysixties,whichhave beenmorewidelyadvocated and studiedthananyothers.Thetwometh

ods are the modal method of

Bishop

and Gladwell and the influence coefficienttechnique.

Exceptforthe limited comparisonbetween these methods made

by

theArmour Research Foundation in 1962

(39),

no

comparisonof_{any form} seemstoexistbetweenthe two methodsin theopenliterature. In lightoftoday'srequirements

for highspeed _rotatingequipment,it is imperativefrom economic,safetyanddefenseviewpointsthatthe_rotatingappa

ratus produced be freeof unwanted unbalance vibration and effects. Itis. therefore, thepurpose ofthisdissertationto studyin-depththosemethods of

balancing

flexiblehighspeed_machinerywhich are mostwidelyusedin

industry

anddis tributed in theopen literature.

In linewiththisin-depthstudy, acomparisonofthemethodsismadeusinganalyticalcomputersimulationsbaseduponthepro

ceduresdemonstratedinreference(91).Discussionsofthecomputer programs produced

during

thisresearch, strengthsand weak nessesofthe method, and an analysis oftheresults are presentedinthisreportaswell as thevarious procedures utilized in its

production.

Thisfirst known comprehensive study and comparison ofthe modal versus influencetechniques is presented infulfill

The resulting equation.

mx +

bx

+ ex =

mew cos wt,

yieldsthesolution

mew

-bt

x =

Ae

2m

sin(q

-J

T~~2

2~

t + oi

)

+

V(c-

mw

)

+

b

w

2

JT

2

2

' cos (wt-

B)

_,

where

tan

3

=

bw

;

q

=

y4mc-t

2m

c-mw

andA anda are _abritraryconstants.

Similarly

in the _{y direction}the motion isdescribed

by

-bt

1

2m

1

y

=

A

e sim

(qt

+ 0(

)

+ mew sin

(wt

-3

)

Tl

2

2

2

2

(c-mw

)

+b w

The firsttermsofthesolutionsdenotethetransientshaftbehaviorwhichdiesoutintime

leaving

thesteady-state motion

with a vibratory amplitude of

2

mew

-l/

Fl

2

2 I

y

(c-mw

)

+

b

w

which is caused

by

the

forcing

action ofthe masseccentricity,

Thepathofthemass centerinthesteady-stateconditionisthatof a circular orbit which reachesitsmaximum amplitude

when

2c

w =

4mc

-

2b

Thephase angle ofthe displacementofthe masscenter relativeto theelastic centerisgiven

by 0.

which canbeseento

be 0when wiszero,or_verysmall.When mw2= c,/3= _{7i72, orthedisplacementnowlagsthemass center}

by

90.This is

thesituation atthecritical speed ofthesystem.Abovethisvalueofw,ficontinuestoincrease invalue untilit becomestc at_{very high}valuesof w.Thischange ofphaseoccurs at_everycritical speed of asystem,goingthrough theentire phase change in a _very small range ofspeeds as can be seenfrom Figure 3.

180

150*

120

90"

% 60

y

(

1

1 a

'

I

- _b/bc

|

* _0.025

1

30*

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1,8 2.0

SPEED

*i/r

Figure 3 Phaeaanglainvicinityof a critical apaad.

The change in the response oftherotor system with speed,i.e. the phase-speed relationship, is illustrated in Figure

4.

which demonstratesthe eccentricity- _{displacementchanges with speeds.}

(a)

w < _{w_; $} * 0

r

(b)

w

-wr; 6 * ir/2

In theabsence of

damping,

which provides a convenient means of

demonstrating

themass-displacement-speed relation

ship,andintheneighborhood ofthecritical speed,w =Jc/m ,theequation of motionleads toa solutionwhichincreases

continuallywith speed and, therefore, approachesinfinity.

The inclusionof

damping

preventstheamplitude_{from approaching}

infinity

andkeeps itwithinfinitebounds. Jeffcpttob

served that the

damping

isnot zero: hence, it is possibletooperate the system beyond thecritical speed.

Jeffcott's formulationwhen rewrittenintermsof vibrationlevelsofforcesshowstherotorincreases inamplitudeandforcesasthe critical speedisapproached,butis low below 90percent,or above110percentofthecritical.

Figure 8 Syatamraaponaa_{In vicinity}ofa criticalapaad.

Experimentalverification ofJeffcott'sanalysis was made

by

Muster

(78)

andthe results areshown in Figure 6.

Extension oftheJeffcottanalysistoinclude flexible anddampedbearingswas made

by

Smith

(103)

who modifiedthe

equations of motion in fixedaxesto be

mx - Fx

-mw

(g

cos wt -

h

sin wt) =

₀

mx

-Fy

-mw

(g

sin wt +

h

cos wt) =

₀

wheregandhare out-of-balancedisplacementswhich are constantin_rotatingcoordinates,and Fxand

Fy

arethebear

ing

forces inthe xand _y directions,respectively.The forces Fxand

Fy

include bothelastic and

damping

_{effects,applied} through the bearings,and_anyother external

disturbing

forcesapplied otherthanthrough the

bearings,

forexamplethe

viscousforceson the rotor caused

by

_rotating in a viscous environment.

Forthecase ofthe Jeffcott rotor, the system, ortotal deflection

(x,

y)isthe sum oftwo

independent

_{parts,one of}

which

(x\

y')isthedeflectionofthe

bearing

support andtheother

(x",

y")

being

the

deflection

ofthe rotorshaft With thisconventionthe

bearing

characteristicsare

Fx

= -b'x' -c'x

*

p

T

X

+

a

A

+

X

EQUILIBRIUM CENTER

+

X

a o

. Strabostopic Motion Picture

*

/

Relative positions of center of _gravity

(?),

equilibrium center

(a)

and

bearing

center (0).

NOTE:

Center-of-grav1ty

and equilibrium center fixed in the disk; the

bearing

center is fixed in space.

ECCENTRICITY I

"critical

SPEED OF ROTATION

where c'

and b'

aretheelasticand

damping

_{coefficients, respectively}ofthe bearings.

The

shaft can

be

characterizedas

Fx

= -b" (iH

+ wy") -c"x"

Fy

-p (y" _{-wx") -c"y"}

where b"

and c"

describethe shaft

damping

and elastic coefficients.

Describing

thebearings and shaft as above leadstothe equations of motion ofthe form

x +

(j

+n)

x _{+ wny +} jjx =

₀

y +

(j

+n)

y

- _wnx

+ py =

₀

with:

j

= _m

(

c +c"

)

b"

=

stationary

damping

coefficient_{representing the} effect of

damping

thebearings:

=

rotating

damping

coefficient_representingthe n

-m

(

C' +c"

)

effectof

damping

intheshaft;

y

= m

(

c'

+c")

=overall stiffnessof_mounting

In the absence ofinternal

(rotating)

damping

theequations are uncoupled,

x +

jx

+ ux =

₀

y

+

jy

+ ^iy =

₀

Thecriticalspeed ofthesystemisgiven

by

w=TfiTwhichisthesame asfortheundamped system.Inclusionof

damping

(stationary-bearing)

gives riseto large unbalance inducedvibrations at thecritical speed butare finite in value.

Smith investigated various combinations of symmetrical shafts, asymmetrical shafts, symmetrical bearings and asym

metrical bearings. Further expanding Jeffcott's work,Smith also investigated multimass systems as well as rigid rotor systems.

When referringtotheassumptionof viscousdamping, Smithwrote, "Theassumptionastoviscouslaw ofdamping,.is

Rigid

Body lancing

Eventhough Jeffcottcontributed theanalytical developmentofthe response of aflexiblerotor,lowspeed or rigid

body

balancing

continuedto preoccupy writers inthe

balancing

field,

and many practicaltwo-planetechniqueswere formu latedanddisseminated. Priorto 1

929,

when_sufficientlyaccuratephase measurement equipmentbecameavailable

(90),

itwas_necessaryto_relyupon methods_{based solely}upondisplacementmeasurements.

Bromberg (23)

elaborated on one

such techniquewhich is illustrated in Figure 7.

Figura 7 Brombarg'a diaplacomantconstruction.

Inthis

figure,

0 isthecenter ofthevector plane _representing a measuringplane oftherotor, VectorOPrepresentsthe

unbalance force: vectors

OA, OB,

and OC represent theresponse amplitudes

(forces)

ofthebearings to a trial weight placed _{successively in}threeangular positionsinthe

balancing

plane.Vectors

OA',

OB'andOCaretheresultant vectors obtained

by

_addingOP to

OA,

OB and

OC,

.respectively.

Triangle ABC isequal and parallelto triangle

ABC; translating

A'B'C ontoABC ispossible

by

moving inthedirection

anddistanceas given

by

AA,

B'B, orCC.

Similarly,

0canbetranslatedto0'

by

movementinthedirectionanddistance vectorsA'A, B'B orCC. Thesubsequentvector.

00'

isthe necessary balanceweight and location

being

equal to -OP.

While in use, thevectorsOP,

OA',

OB'

and

OC

are not

known,

themagnitudes,butnot angular orientations of

OA,

OB andOCare known. Also it isknown that

OA',

OB'

andOC are some multiples of

OA,

OB and OC such that

and

OA'/OB'/OC =

m/n/p

O'A/0'B/O'C =

m/n/p

Thedirection and length of

00'

can bedetermined if it is realizedthat 0'liesat the intersection of:

a. The locus of points whosedistancesfrom A and B have the ratio m/n.

b. The locus of pointswhosedistances from B and C havethe ratio n/p.

c. The locus of points whose distancesfrom Cand A

have

the ratio p/m.

Graphically

it ispossibletosolvethisproblemand

Bromberg (23)

elaboratesonthemethod and gives some

A _shortcoming ofthis technique is that it must be performedtwice in ordertoobtain thecorrect balance

weightand location. Thisisbecauseeach graphical application yieldstwopossible solutionsand, therefore, in ordertodeterminewhich solutioniscorrect, itmustbeperformed a secondtimewithdifferentdata.The

solution which repeatsisthecorrect solution.The minimum number of_measuringrunsisthenfour,usgally

one original response andthreetrial weight runs.

Anexample ofthe straightdisplacementmethod is illustrated in Figure 8,inwhich thevectorsOA, OB and

OC representthe

bearing

amplitudes _{corresponding to} the original unbalance condition and trial weights placedsuccessively in each oftwo angularlocations, thesecond

being diametrically

opposed tothefirst.

Figura 8 Vactor diagramofdlaplacamant

balancing

method.

VectorOB canbeconsideredtobe_thesum oftheoriginal unbalance vector,

OA,

andthechangeintheun

balance level, AB;similarly, vector OC isthesum of vectorsOAandAC. If thetrialweight wasthesamefor both trial weight runs,then thechangeinthe amount of unbalance willbeidenticalwiththe magnitudes of vectorsABandAC

being

equal.ThismeansthatOA isthemedian oftriangleOBC. Althoughtheangular re

lationsare unknown,it ispossibletoconstructtriangleOBC.

Using

Euclidiangeometry,OAcanbe doubledto

produce

OD,

which canbeconsideredthediagonal of parallelogram OBDC.

Constructing

theparallelogram yieldstriangle OBC.

Balancing

isthen accomplished

by increasing

themagnitude ofthecorrection weightinthe ratio ofOA/AB = _{OA/AC,and theangle of}correction

being

angleOAB,counterclockwisefromthe locationofthefirsttrial weight location.

As in the case of Bromberg, this method can lead to the alternate solution deduced from parallelogram

OB'DCas given in Figure 8. It istherefore necessary torepeattheoperationand produce anadditionaltwo solutions. The common solution yieldedfrom the twooperationsisthe correct balancesolution.

Thismethodof

balancing

isappliedtoeachofthe two

balancing

planesinturn,afterwhichtheprocessisre peated inan iterative fashion until a satisfactory unbalance vibration level is obtained.

Theaccuracyofthis

balancing

approachto two-planerigid

body balancing

and itsacceptance_{is amply}illus trated

by

its inclusion inthe

balancing

literatureand its

being

advocatedasasimple and_veryeffective bal ancingtechniqueaslate,and asrecentlyas

1957,

when

Hill,

BarkerandMurtland

(7)

published arevised ver sion ofthe amplitude

balancing

method.

Theadvent ofaccurate phase measuringequipment _{ultimately led}

RathboneJ90)

tointroduce thefirstpub

re-sponseto theaddition ofaknowntrialweight at a knownangularlocation. Because boththe magnitude and

angularposition of each ofthesevectorsis

known,

it ispossibleto

immediately

determinevector

AB,

repre senting the change insystem responseto the trial weight.

CORRECTION 4

Figure 9 Rathbone'a

balancing

procedure.

As inthecase ofBromberg'sconstruction, themagnitudeofthefinalcorrectionmassis found fromthe_mag

nitude ofOA/AB. withtheangular position

being

thevalue of angle

OAB,

rotatedcounterclockwisefromthetrial

position. Additionofthecorrection mass allowstheother end oftherotorsystemto_{be similarly}corrected in likemanner.

Ifafter

balancing

each end ofthe rotorinturn it doesnot produce_satisfactorybalance

levels,

theprocedure canbe

iteratively

applieduntil_satisfactorybalancelevelsare achieved. Rathbonealsodemonstratedthe ap plicability ofthe method toasymmetric bearings.

A_shortcomingof_{consideringonly}one ofthe two

balancing

planesat atimearisesfromthelackoftotalin dependenceofthevibration inone plane as relatedtoanother. Inotherwords,ineachandeverysystemthe addition of a weightinone plane willhavean effect of some proportion on_everyother plane ofthesystem.In practice this means thata balance weight added in onebalance plane toannul thevibration ofthe corre sponding

bearing

will produce a changeinthevibration oftheotherbearing. Ifthisattendant alteration ofthe

vibration levelswassuchthatattenuation ofthevibrationat one

bearing

forcedamplification ofthevibration attheotherbearing,then the techniqueofconsideringonlyone plane at atimecanleadtodifficult

balancing

situations.

Thearle

(106)

recognizedtheshortcomingsofconsideringone planeat atimeinthe

balancing

procedure.To correct forthiscondition heauthorizeda

balancing

procedure whichsimultaneously balanceda rotorintwo

Themethodintroduced

by

Thearle is depicted in Figure10,inwhich vectors

OA

andOBrepresentthemagni'

tudeandangular placement oftheoriginal

bearing

vibrations,toone scale, as well astheoriginal effective unbalance atbearings A and

B.

respectively, toanother scale.

Figure 10 Thearle'*method oftwo-plane balancing.

Ifa trialweight ofknown magnitude and angular positionisplacedinthe

balancing

planecorrespondingto

bearing

A.theresultant

bearing

responsecanbeplotted as vectors

OA,

, at

bearing

A,

and

OB,

, at

bearing

B. The effectoftheadditional weight isequivalenttovector

AA,

and

BB,

, at bearings Aand B,respectively.

Similarly,

ifatrialweightisaddedinthe

balancing

planeof

bearing

B,theprocedure canberepeatedtopro ducevectors OA2,

OB2,

AA2 and BB2, which represent the_{corresponding} effectsat

bearing

B.

It ispossibleto annul thevibrations at bearingsAand B _{simultaneously if}values of andfican befound

suchthat

c< AA +

BAA

=

-OA

1

2

oi BB + 3BB = _-OB

1

2

Because there aretwo equations and two unknowns a solution ispossible, which is,

by

vector algebra

OA

(BB

)

-AA

(OB)

<* =

2

2

AA

(

BB

)

-

AA

(BB

)

2

1

12

AA

(OB)

-0A(BB

)

3

=

1

1

AA

(BB

)

-

AA

(BB

)

2

1

12

Itcanbe deducedthatthechangein

bearing

vibrationalso reflects anequivalentchangeintheresidual mass

unbalance;the trial weightswill, therefore,berelatedto theoriginal unbalance

(UA,

UB

)

inproportionto the

-U oi =

A

~t~

A

3

=

B

B

representtherelationsbetweentheoriginal mass unbalancesandthetrialweights(TA,

TB).

Solutionofthese vector relationsyields the proper magnitudes and positions ofjhe balance weights in terms ofthetrial

masses.

Thecalculationsinvolved in Thearle'sprocedure are easiertocomprehendif

they

are written intermsofin fluencecoefficientsalongthelines derived

by

Lord Rayleigh in hisTHEORY OF SOUND

(1894)

, inwhichthede flectionor reaction at one point of a systemis defined intermsof animpressed disturbanceat another point.

Rewriting

equations

(a)

interms of influencecoefficients results in

- =

oru,

+

cxou

V._ al

1

a2

2

AO

_ = o< h ₊ oi

u

VB0

h11

h22

whereV

QA BQ isthe originalvibration ofthebearings Aand B, respectively.U1 and

U2

aretheeffective

unbalances,in

balancing

planes 1 and

2,

respectively; theinfluencecoefficients, a

ci , representingtheeffect of an impressed force in plane

i,

as measured at

bearing

c. Thevalues ofthevariousterms aredetermined fromtrial runs; thefirst run yieldstheoriginal vibrationvectors

Vao

andUbo ; thesecondandthirdrunsare

trialweight runs with atrial weightfirst inonebalanceplaneandthen theother.Thetrialweight runsyield changesinthe measured

bearing

vibrationswhich canbeusedto determinetheinfluencecoefficients.Ifvec

torV_A1 indicatesthevibration at

bearing

Awhenatrialweight represented

by

vectorT, isappliedinbalance plane 1, thenthe corresponding influence coefficient, aa1 canbe calculated

by

V

- _V

OC =

Al

AO

Al

T

1

In likemanner all oftheinfluencecoefficientscanbedeterminedwhich allowsforthesolutionoftheequiva lent unbalances, Ui and U2. The balance weights arethen simply-U

, and-U2

So

long

as rotorvelocities remain well belowthefirstflexural critical speed ofthe rotor-bearing system,or

unbal-ance, experiencetwofundamentalcharacteristicmodesof vibration which aretermedthefirstandsecond ri

gid

body

critical speeds.Figure 11 diagramsthenature ofthesevibrations.The firstrigid modehastheentire

axis oftherotortranslatedawayfrom theaxisofrotation(centerlinesofthebearings). Thesecondvibrational formisoneinwhichtheaxis oftherotorisinclined,which makestherotorappearto"tumble"inspace trac

ing

twocones in space, one at eitherend ofthe rotor.

~T

FIRST RIGID MODE

T

Figure 11 Rotoraxiadeflectionahapea.

Thefirstrigidmode, ortranslatorymode,isexcited

by

anyformof mass unbalance_{existing in}therotor,trans

lating

inthedirectionofthegreatestunbalanceforce. Becauseofthe_sensitivityofthismodetoforceunbal ance, it isalsoreferredtoasa staticunbalancemode,which can

be

corrected

by

_{considering onlythe}sum

mation offorces _acting on the rotor.

Smith

(103)

investigatedrigid

body

motionand ascertained that thefirstrigid mode will occur at an angular

velocity of

w

K

B

where

KB

isthestiffnessofthebearingsand Ixisthemoment ofinertia aboutthex axis.Thisexpression is validonly fora symmetricalshaftin symmetrical bearings. Smithalso investigated other shaft and

bearing

Smith alsodetermined that the second rigid

body

vibrational mode occurs at

w =

where the terms are definedas previously.

Also,

it should be noted that the secondvibrational criticalfre quency depends upon the moment ofinertia about thez axis, which isthe axis of rotation ofthe system.

Thissecond rigidmode,ortheconicalmodeisstimulated

by

theactions of unbalanced momentsintherotor system.Correction forthisvibration must,therefore,takeintoaccountthemomentsinthesystem and isre ferredto as moment ordynamic balancing.

Inthecase ofthe

translatory

modeitispossibletoannulthesystemvibration

by

_placinga single weightany where_alongtheaxial lengthoftherotor and at an angular position andofsufficient magnitudeto_satisfythe requirement thatatthe bearings

^Forces

=

₀

Similarly

forthe conical modetwo correction masses can be utilizedto guarantee

yjMoments

=

₀

aboutthe

bearings, thereby

_yielding a_satisfactorybalance. As isobvious,itispossibletocombinethese two operations and with the use oftwo

balancing

planes guarantee low speed balance

by

considering

y,Forces

=

₀

^Moments

=

₀

atthebearings. It isexactlytheseconditions whichthelowspeed,two-plane,rigid

body

balancing

methods consider.

Flexible Rotor

Balancing

Rotorsystems which possess rotors which behaveas rigidbodies

by

definitionsexperience no,ornegligible, distortionoftherotor shaft._{The corresponding}unbalancevibration, even attherigid

body

critical speedsisa resultof a constant unbalance-motion relationship. Inother words,therotor will always vibrate suchthat the "high"

sideofthe rotatingshaft correspondstotheplaneof unbalance oftherotor; attachment of

balancing

weights

diametrically

oppositeto the "high" side will reduce thevibration ofthesystem.

Referring

back to Jeffcott'smodel, the maximum unbalance amplitude corresponds tothe point of effective unbalance.

(See

Figure

4.)

Referenceto Figures 3 and4showsthatthisconstantcoincidenceoftheunbalancedisplacement is lost in the vicinityoftheflexuralcritical speed. Notonlyistherea phasedifference betweentheunbalance andthe displacement but theshaft distorts, reaching its maximum amplitude at the critical speed. As the system

speed approachesanyoftheflexuralcriticalspeeds,thesystem'samplitude-phase cycle

is

repeated.Typical deflection shapesforthe firstthree modes of vibration are sketched in Figure 1 1. Becauseofthe distortionofthe rotor with speed,it becomes necessary forthebalance weightstovary with speed so asto

Illustrated in Figure 1 2aretherotordeflectionshapes atthefirstthreeflexi6lerotor criticalspeeds.Theserepresenttheaffects of

an unbalance, _{U. acting}attherotormidspan.Inordertobalancethefirstmode_{it becomes necessary}toplaceweights,each ofa

magnitude ofU/2,ineach ofthe two

balancing

planes.Theresultistheeliminationofthevibrationinthefirstmodeandthealtera

tionofthe modeshapesas shown inFigure 1 2.

?ORIGINAL VIBRATION

VIBRATION AFTER BALANCING FIRST MODE

Figure 12 Failureof2-plane balancetobalance flexiblevlbrationa.

Ifthe unbalance U isa assumedtoact atthemidplane oftherotor,whichifuniform and symmetrical,then theoriginal

unbalance willbeat a modein thesecondmode and willnotexcitethatmode

(1,

3, 5, 7..). Theaddition oftwobalance

planesateither endoftherotorinwhichbalance weights are addedtocorrectthefirstmodewill affecttheunbalance

distributionandit becomesapparent,referringtheFigure 12,that theaddition ofthebalanceweightswill now excitethe

second mode of vibration.

Itispossibletocorrectthisvibrationifbalanceweights

W3

and

W4

are added asindicated in Figure 12andproportioned

according to therelativedistortionoftheirplanes ofattachment._{Although the}second mode vibrationhasnowbeencor

rected,thecorrection weightsforthesecond mode will reintroduce an unbalanced conditioninthefirstmode as well as

all higher modes.

Recorrection ofthefirst mode will likewise introduceanew unbalance conditioninthesecond mode, and soonandso

on. This is_especially true if not_onlyflexible modes mustbe considered butalsothe rigid

body

modes.

Smith

(103)

and Rathbone

(90)

both _early recognized the

inadequacy

of two-plane

balancing

for flexible systems.

Althoughtwo-planecorrectioniseffective ata given

balancing

speed,it issometimesimpossible foratwo-planebalance

tocorrect arotor systemfor itsentirespeedrange,especiallyifthesystemexperiencesrigid

body

_{effects which mustalso}

beconsidered. Ifthesystem considered isan unsymmetricalrotorand/orunsymmetrical

bearings,

then asSmith

(103)

statesit is not possible to balancethe rotor unlessthe unbalance masses areentirelywithin the correction planes.

Modal

Balancing

Asa rotor system increasesinangularvelocityitwill executecertain naturalformsof vibrationwhichwill achievelarge

amplitudes ofdeflectionandforceatthesystem crtiticalspeed,bothrigid and_{flexural. Typical deflection}shapesof a uni

instrumentation

it is possibleto determine a system crtitical speed and a rough approximation of

its

deflection shape. Armedwith such

knowledge,

it ispossibletoformulateand_applya rotor

balancing

techniguewhich accountsfortheflex ible as well asthe rigid behavioroftherotor system.

Such a

balancing

system was proposed

by

Federnandhas beenreferencedinmuchofthe

literature,

forexample(31,39,

78). An intuitive stepwise approach,

Federn's

method is based upon thetwobasic equationsof mechanicstoremovethe rigid

body

effects, i.e.

^Forces

=

₀

^Moments

=

₀

Subsequent inclusionofthesefactors intheflexiblerotor

balancing

processallowsfor

balancing

theflexiblerotor system without

upsetting itsrigid

body

balance.

Figure 1 3referstothis techniqueanditsapplication.Asa rigid

body,

therotor canbe balanced intwoplanestoannulthe effects of thetranslatory and conical modesas shown previously.Iftheplanes 1 and5are usedforthisstep,therelative positions ofthecorrection weights will be shown in Figure 13

(a)

and(b),

Balancing

thefirst flexuralmode canbeaccomplished

by

theuse of a single weight at mid-span,plusthe

inclusion

oftwo endweightsso proportioned as nottoupsetthelowerspeed corrections.

Similarly

thesecondmode canbecorrected

by

theuse oftwoweightsas illustrated in Figure 13

(d)

withthe twoend masses proportioned sothatnolowerspeed ef fectsare introduced. Theprocess is_similarlyrepeatedfor all other modes ofimportancealways_using (N +

2)

balance weights forthe Nth modeto be corrected,so that unbalance in modes N -1,N-?, ... 1 will not be introduced.

Knowledgeofthedeflectionshapeallows each ofthe necessaryweight

distributions

tobecalculated.Theresultantdis tributionsare oftheproperrelative magnitudes and angular orientationsfor balancing: however,thefinalmagnitudes and positions must bedetermined

by

using the calculated distributions as sets oftrial weightsand then

deducing

thefinal weights_accordingto thechangeinvibrationlevels intherotorsystem,such asexemplified

by

themethod ofFigure9.

Removaloftheeffects of each mode in turn resultsin thesuccessful reductionof rotor vibrationthroughout the speed range ofthe system as

long

as all modes of major consequences are considered.

Bishop

[\0)3_{extendedJeffcott'sanalysistocoverthevibrations}

ofuniform symmetricalcontinuousshafts underthein fluenceofdistributed internaland external

damping

and_springforcescarriedinsymmetricalidealbearings.

Recognizing

thattheresponseof a rotor systemtocentrifugal unbalanceforcesoccurs intermsofthenatural_vibratorymodes ofthe system, theequations ofmotion were solved intermsof itsprincipal natural modes of vibration.Thistechniqueof "modal analysis"

wasthenformulated

by

Bishop

andGladwell

(13)

totakeintoconsiderationtheeffects oftheaddition of smalldiscrete masses. Inthismannerthemodification ofthemodalseries, i.e. theunbalancedflexuralvibrations of ashaft,couldbeex pressedintermsofdiscretemass,or

balancing

units.Withtheestablishment ofthese relationsitwas possiblefor

Bishop

and Gladwell to examinethe rigid

body balancing

procedure and to formulate and present a stepwise

balancing

pro cedurebased upon the modal analysis ofthesystem interms ofitsorthogonal modes of vibration and response.

Briefly,theconcept of modal

balancing

which

Bishop

andGladwell developed isbasedupontheideathatanyvibration of_anyrotor at_anyspeed canbeexpressedas acombination orsumoftheeffectsofeach principalmodeof vibrationatthatspeed.Also,ata critical speedtheprincipal mode ofvibration _{corresponding}tothatspeed willbe sufficientlyamplified soastobethedoninant modeofvibration, with all other mode effects

being

negligible

by

comparison.Therefore, it ispossiblewith ajudiciousselection of

balancing

planestoremoveeach principalmodeinturnthroughout thespeedrange ofthesystem,andinso

doing

successfully bal

ancetherotor.Thus, becauseofthe principaloforthogonality,removal ofoneor more principal modes willnot affectanyother mode,and removal of all modes withinthespeedrange will removetheprimemodes of vibration,

leaving

onlysmallresidualeffects fromthoseprincipal modes notremovedwhich lieoutsidetherealm of operation.

TURBOGENERATOR

I

-O

DEFLECTION CAUSED BY STATIC UNBALANCE (AT SLOW ROTATING SPEEDS).

(A)

L._.i

LL.

ANGULAR DEVIATION CAUSED BY DYNAMIC UNBALANCE MOMENT (AT SLOW ROTATING SPEEDS).

(B)

V-SHAPED MODE: BENDING DUE TO A SYMMETRICAL

INTERNAL MOMENT AT HIGH ROTATING SPEEDS (NEAR THE SECOND FLEXURAL CRITICAL SPEED).

(C)

CO

S-SHAPED MODE: BENDING DUE TO AN ANTISYMMETRICAL

INTERNAL MOMENT AT HIGH ROTATING SPEEDS (NEAR

THE SECOND FLEXURAL CRITICAL SPEED).

(D)

If it isassumedthat thevibrationalbehaviorof a rotor system undertheeffects of centrifugal unbalanceforces

is

expres

sible_solelyintermsofitsnatual flexuralprincipalmodesthenit ispossibletowritethe condition of

balancing

toeliminate the"S" principal mode,denoted

by

6S (z)

asa solution to a series of simultaneous equations which relatethedisplacement

of

balancing

planesz

f andtheassociatedbalance weights, m_, r_, ,to theeffective unbalance distribution

Us *,

whichis

expressiblein terms identicalto thatofthe natural modetowhichitpertains and stimulates. Underthese definitionsthe

balancing

procedure is mathematically expressible as

0

(z

)

0

(z

)

11

12

0

(z

)

2

1

OOOOOOOOOQi

V VZ / OOOOOOOOO

s

1

0

(z

)

1

s

0

(z

)

s s

m r

1 1

m r

2*2

m r

s s

0

-U*

s

withthe condition that

0

(z

)

₀

(z

)

.oooo

0

(z

)

11

12

1

s

0

(z

)

s

1

0

(z

)

s s

Ascanbeseenfromtheseequations,it_{is necessary}touseN balanceweighttoremovethe Nthnatural mode of vibra

tion. Forexampleif inFigure 1

4,

the planardeflection shapesillustratedarethenormal modes of a rotor andit is desired

toremovethe thirdmodeofvibration, thenit isnecessary tousethreebalanceplanes. Ifthesmallletters inthediagram

are giventobethedeflection inthe correspondingbalanceplane, such as "b"

being

thedeflectionof planez2in thefirst

mode (Si

(x)),

thenthe

balancing

condition becomes

m r

1

1

m r

2 2

m r

3 3

with

a

d

g

b

e

h

c

f

i

Unlesstheexact nature ofthenatural modes,Qi (z),are knownas well astheexactexpression oftheunbalance,thepre cedingequations cannotbesolved.

However,

a relativebalanceweightdistributioncanbe

found

by

setting

U3*

equalto

anyconvenient numericalvalue,such as 1, 100,etc.Theresultantsolutions of m,^,_{m2 r2}and

m3r3

arethentherelative magnitudes ofthe balance weights whichcanbeutilized astrial weightsforthesystem.

Noting

thechangeinvibration levels atthebearingswhentheseweightsare added allowsthecorrectmagnitudes and angularlocationsofthebalance

weightstobedetermined

by

anyconvenientmeans,such as Figure 9. Insertionofthethencalculatedweights removes the effects ofthethird mode without_affectingthefirst or second modes, which are assumedtohave been previously balanced.

-TV

Figure 14 Illustrationof_{the N}modalmethod.

TherequirementofNweights or planestoremovetheNthnatural modehasresultedinthisprocedure

being

termed the "N modal

approach."

Toexperimentallyvalidatethe modal approach prescribed

by Bishop

and

Gladwell. Parkinson,

_Jacksonand

Bishop

(Ijj,

]6)

performeda series of

laboratory

experiements onaseries ofsmall,

long,

slender,

flexible

shafts ofuniformand non

uniform circularcross-section,supportedin ball bearingswhichapproximatecloselytoideal

bearings

of_{the pinned,}fora singlebearing,or clamped,foradouble

bearing,

end condition.

By

so_selectingtheshafts and

bearings

the characteristic functions0 j

(z),

ofthe principal modes of vibration could becalculated and comparedto theexperimentally derived functionobtained

by

electromagneticallyexciting the

Comparisonofthe

theoretically

calculated andthenon-rotatingexperimental characteristic

functions

by

the investigators ledtotheconclusionthat themode shapesderived

by

either methodwere sosimilarthatdiscrepanciescouldbe ignored, and that the rotors exhibited _nearly ideal modeshapes. Based upon thecloseness of_{the experiementally determined} mode shapesto theirorthogonal calculated counterparts, itwasfurtherconcluded that thecharacteristic

(orthogonal)

functions fora real rotor system couldbe determined

by

excitingthesystem at itsresonantfrequency;theresultantde flection shape wasthen thedesiredcharacteristicdeflection function, Si (z).

Therefore,

theactualbehavioroftherotor system couldbe used inthe

balancing

procedure rattierthan the calculatedfunctions.

Arelated

finding

wasthatwith thesystemstested,the

damping

was of negligible proportions anddidnot altertheshaft vibrationsfrom those ofthe

theoretically

undamped system.

Determination ofthe angular location of the plane of unbalance was _{experimentally} achieved

by

_scribingthe shaftat speeds which were equalincrementsabove andbelowtheresonant

frequency;

theplane of unbalance wasthenmidway between the two marks, andthe radial plane ofbalance weight attachment was 180 degrees opposite.

*

Determinationoftheactualbalance weights wasa trialand error procedure basedupon a heuristictechniqueuntil a minimum vibration levelwas achieved. Formodes otherthanthefirst,theproportions ofthetrailweights were obtained from the simultaneous relations given previously.

Acknowledgmentofthefactthatreal rotorsmaynot possess modes of vibration neartheirresonantfrequencies inwhich distortion inextraneousmodes

is

negligibleled

Bishop

andParkinson

(1_8)

toinvestigatemethods of modeseparation, or isolation. Theculmination oftheirwork wastheadaptation ofthe polar plottechniquetographrotor amplitude-speed-phase relations. Figures 1 5and 16 illustrate thistechnigue.Thecomplex vibration, orthree-dimensional spatialvibration of a rotor systemmaybeconsideredtobeoftheforma=_{X+ iY,which correspondto theprincipal planes ofsymmetry} ofthe rotor system.

Plotting

a, the rotoramplitude and phase angle at a _measuring plane, on anArgand diagram for knownspeeds and speedincrements,w,it ispossible toproducea polar plot oftherotormotion,asin Figure 15. From the resultinglocusof pointsit ispossibletoconstructacirculararc which willjointhepointssoplotted Theresultingarc willassume a circular

form,

as in Figure 1

5,

ifthe resonantfrequenciesofthesystem are well separated.

IMAGINARY

0 REAL

argef

\

PLANE OF BALANCE WEIGHT ATTACHMENT. /or/1

"

IMAGINARY

'

0

\

\

REAL

fS#arg

A

Figure 16 Argand diagramwith non-negligible vibrationinothermodes.

Investigation oftheplot produced

by

such a method will yield _{information regarding}characteristicsoftherotor system

which are applicableinthe modal

balancing

procedure. Forexample,if thespeedincrementsare equalthroughthecriti

cal speed range, then thepoint on the plot_{corresponding}tothegreatest changein phaseangle,with referenceto the

changeinspeed, correspondsto theresonant natural

frequency

Ofthesystem.InFigure15,thecritical speed would coin cide, orbevery nearlyequalto the pointdenoted as

wt

on the diagram.

Referenceto thespeed-phase anglediagramofFigure 3revealsthat theplaneof unbalanceleadstheplane ofmaximum

displacement

by

90 degrees;the

balancing

planehencewill

lag

thedisplacement

by

90degrees.Therefore, ifadiameter

to thecircleoftheplotisconstructed,thenthe

balancing

plane will

lag

thedisplacement

by

90 degrees

(see

Figure 15).

Ifa suitabletrial

balancing

weightdistribution,such as wouldbeobtainedfromthecharacteristicdeflectionshape atthe

critical speed,isattachedtotherotorintheindicated

balancing

plane, then

by

observingthechangeinthesize oftheplot

ofthe Argand diagram, it is possibletogenerate a set of

balancing

weightsto annulthesystem vibrations.

Theresonantfrequenciesofthesystemmaynot._{however, be widely}separated and/ortheresponse oftherotorsystem

tomodesofvibrationotherthantheprincipalmode of vibration under consideration_maynotbeof negligible proportions.

Aspointed out

by

theauthors

(18)

thissituationdoesnot produce anyinsurmountablecomplications; in

fact,

the only

change willbethat thecircular plot

joining

the locusof pointsintheArgand diagramwillbeoffsetfromtheorigin ofthe

diagram,asinFigure 16. In Figure 1

6,

thecriticalspeed andthe

balancing

operation,previouslydetailed,isunaltered ex

ceptfortheshiftawayfromtheorigin.Thisshift, exemplified

by

00' ofthediagram, isa measure oftheeffects of other vibrational modes on the response ofthe system inthevicinity of a criticalfrequency.

Verification ofthe material presented on modeisolationwasaccomplished

by

means ofexperimentsusingsome ofthe

laboratory

shafts of

(1J5, 16)

and an actual 6500

-hp

motor shaft.

Extension oftheArgand responsediagram to industrial rotors, and oftheapplication ofthemodal

balancing

method of

Bishop

andGladwellto thesame rotors, was accomplished

by Lindley

and

Bishop

_{(17). TheeffectivenessoftheArgand}

diagramto thedeterminationof critical speeds aspresentedin

(18),

as well asthesimilitude oftheexperimentalplotsto

the predicted plotsof

(18)

was illustrated from thetests doneon a 200 megawattturbo alternator.

Theapplicationofthemodaltechnique tobalance thesame200megawatt generatorrotor_{is described in detail. The de}

flection forms, 91

(x),

fortheactual

balancing

process, since some

knowledge

ofthem

is

necessary forthe procedure,

"It maybe argued thatthese deflection formsare known, and inthe absolute sensethis istrue. Butwithlargerotorsdesigncomputationsofthecritical speeds are certainto be madeand.withthe moderncomputing machinesavailable,littleextra effortisrequiredtoobtain additionalinformationaboutthemodalshapes.Theseare calculated ones and are notnecessarily the true ones; butforthefirstand second critical speedstheestimated modal shapes shouldbesuffi cientlyaccurateforthe purpose required...

"

4

Furtheremphasisisplaced uponthissubjectlater inthesamework when theauthors state"thaassertionthat (it ises sential to know the modal shapes, and

they

must be determined experimentally) seems unjustified, since calculated modal shapes have hitherto proved_satisfactory

enough." 5

An empiricalmethod of

determining

theactual rotordeflectionshapes

is, however,

mentioned that

being

themethod of

(H>)

inwhich a massis"traversed"

along the axis oftherotor;the resultingchangeinthe

bearing

response,ifplotted, will yield a very accurate approximation ofthe actualdeflection shape.

A

difficulty

encountered with application ofthe modal methodtolargeindustrialrotorsarisesfromthefactthatuniikea

laboratory

model rotor, such as

(15, 16)

, large industrial rotors

do

notnecessarilyexhibit

truly

in-phase

bearing

reac tionsattheodd criticals

(1,3,

5, ...),nor will

they

exhibit

truly

out-of-phase

bearing

vibrationattheeven criticals

(2,

4, 6,

...).The lackofthisideal

bearing

behaviormeansthatat a given criticalspeed,iftheattendantcharacteristicmodeisre moved, therewill still be measurable

bearing

vibration. As a result,in the

balancing

procedureit is necessary toreduce the

bearing

vibrationstoan _acceptably small amount.

Often inthebalanceweight calculationsdifferent sets and positions ofthecorrection masses will beobtainedforeach

bearing

considered.This in practice necessitates a compromisebetweenthedifferent balanceweightdistributions.One such means of compromiseis

by

faking

theaverages ofindicatedcorrections.Theconceptisillustratedin Figures17and

1

8,

which represent the methodforthe odd and even modes, respectively.

-r- J-A

OA

-vector giving amplitude and direction of vibration at _bearing A.

OB - _vector

giving amplitude and direction of vibration at bearing B.

Dotted Lines - _vectors

showing effect of added trial mass.

Mass required to balance out OA - _{7 1/2 x} trial mass and rotated through angle

Q^

.

Mass required to balance out OB - 9 _x trial mass and rotated through angle

By

b. OA. - _vector

giving amplitude and direction at bearing A.

OB. - _{vector giving amplitude and} direction _at bearing B.

Mass added - ₈

xtrial.mass rotated through angle e~ +

Q,

I

17 Typicalvector'sforvibrationatflratchosen

OA - _vector

giving amplitude and direction at bearing A.

OB

-vector giving amplitude and direction at bearingB.

Dotted Lines

-vectors showing effect of added trial couple. Couple required to balance out the OUT-OF-PHASE vector, AB - _{3-1 x trial}

couple rotated through angle e. OA,,

OB,

- _vectors

showing the IN-PHASE vibration to be balanced out _by themasses: M. near mid-span on one side.

M_, M- _{diameterically} opposite to M,, but M is near one end of the rotor, and M, is near the other end. Position and magnitude of M-, M_, and_M, choosen

so that they have little influence on the first and second modal balance.

Vectorrepresentetionofconditionsatthasecond choaen apeed(nearrunningspeedandbelow thesecondcritical apeed).

Final

balancing

oftherotor_alongmodallines isaccomplished

by

adistributionofbalanceweightsaccordingto thenext highestmode of vibration outside ofthesystem overspeedrange.Determinationofthis

balancing

massdistribution along standard modal lines, i.e.so as nottoupsetthelowerbalancedmodes, allowstherotorfo beabalancedatoperational

speed and further reducethe system vibrations overthe level previouslyattained.

Parkinson (81)examinedthebehaviorofa symmetrical shaftinasymmetricbearingsinmodalterms.Heestablishedthat theshaft executes characteristicmodesof vibrationineach principal planeof thebearing.Thesemodesareidenticaltothemodes character

izing

thesame shaftsupported insymmetrical bearings,whose stiffnesses equalthestiffnesses oftheasymmetricbearinginthe correspondingprincipal plane.The_resultingvibrationinone modeina given principal plane wasthereby foundnottoaffectany other modeinthesameplaneorinthe otherprincipal planebecauseoftheconditionoforthogonality.Parkinsonwasabletoextend

themodal

balancing

approachtocoverasymmetric

bearing

effects.Thealterationof theproceduretocorrectforasymmetricsup port conditions_{involves removing}eachcharacteristic modeinturnaddingone additionalbalanceplaneforeach successive mode encountered. Inother words ifthecritical speed oftherotorsystem are suchthat

w,

< w

,

*< /2<w2'<Wj

...

wheretheasteriskimpliesthe 7Zplane andtheotherspeedsdenotetheXZplane, thenitispossibletoremovetheeffectsof wi withonemass, wi* withtwo,W2 withthree,andsoon.Theconditionsfor_determiningthebalancingmasses arethesame simul

taneousrelationsand non-zerodeterminantconditions which mustbe fulfilled fora symmetrical shaftin symmetricalbearings.

Themodal analysis of rotor systems which are comprised of uniform asymmetric shaftssupported insymmetricalbear ingswasalso investigated

by

Parkinson

(82),

withthe

feasibility

of applicationofthemodal

balancing

techniqueasthe

objective. So

long

asthe shaft issupported in ideal bearings,the shaftwill execute characteristic modes ofdeflection which will bethesame inboth principal planes oftheshaft,thoughthecritical speedsfortherth mode.9_{r (z)}. will be different ineachoftheprincipal planes.Thecritical speeds ofthesystem willoccurinpairs,namely wf<w*

, fortherth

mode where_wr isthecritical speed_{corresponding to the}plane of principal stiffness withthelowestvalue offlexural ri

gidity, and w*

istheremaining plane of principal stiffness.

Balancing

such a system

by

use ofthemodal approachispossiblethrough threerotorspeed runs; one willbetheoriginal unbalance responseofthe rotor; theothertworuns are with trial

balancing

distributions in each oftwodifferentradial

planes.

By

means of some simple vector algebra and construction of some graphicaloperations, it ispossibletodeter

mine the

balancing

massesnecessary toattenuate therotor vibrations.

Further investigationto establishthe limitations, ifany,of modal

balancing

wasconducted

by

Parkinson

(84),

whoin vestigatedtheeffectsofmassiveflexiblebearings. Priorto thisinvestigationthemountings were assumedtobemassless

supports _{exhibiting only}stiffness characteristics, i.e. massless springs. So

long

asthe

"spring

bearing"approximation wasused thecondition oforthogonality couldbe.derivedand applied, namely

0 (z) 0

(z)dz

.

₀

, r

j

r s

Introductionofthe massive supports, however, leads theequations of motiontoproduce an altered_{orthogonality}rela

tion. If

ty

(z)

isthedeflection shape ofthe rth mode ofthe rotor

bearing

system, r

.1 n

ty

(z)ip

(z;dz

+X)

CMipriPsi

=

₀

, r

J

s r

i

=1

o

results,whereCisanarbitraryconstant, Mi isthemass ofthesupportat z=

z\ and

Pj

_(r) and

Pj

(s) arethedeflections ofthe

bearing

support as z =

Zj

inthe rth and sth modes, respectively._{The meaning}ofthisequationisthatnolongerare

themodal methodisthelinearindependenceofthedeflectionshapes

i^

(z),

ty2

(z)

which canbe deduced fromexper

imental observation ofthe shaft as itvibrates at speeds in the_vicinityof _{its corresponding}critical speed.

Normally

after_applyingthe modaltechnique toallcharacteristicfunctionswithinthespeedrange,thevibration ofthero torwillbeat a_sufficientlylow leveltobeacceptable.

Sometimes, however,

ifthenexthighercritical speed lies justout side oftherotor speedrange,oriftherearetwomodes situated closetogether,the resultingbalancedrotordoesnotex

hibita_satisfactorystate ofbalance. MooreandDodd

(34)

published a practical modalbasedtechniquetoalleviate sucha

condition.Thetechniqueis basedupon a graphical solutionto the

balancing

problem. Theconstruction so usedisshown

inFigure 1 9. Thisparticulartechniqueisbaseduponthe realizationthatat a given speedforwhich

balancing

is desired.

suchas at an _operatingspeed,the residual vibration ofthe rotor willbe intheextraneous modes ofthe system not al

readyremoved. Thesemodes willbe botheven and odd modes.

Thence,

iftrialweights are addedfortheremovalofthe nexthighereven mode andtheresultsrecorded,as well astheresults_{from attaching}trialweightsforan odd mode, it is

possibleto annul the

bearing

vibrations

by

_usinga certain combination oftheseweights and positions. This particular combination is the solution derived from the construction of Figure 19.

The balanceachieved

by

this techniqueof modal

balancing

isaccurate_{only for}a single speed and will not_normallyannul

bearing

vibrations over a wide speed range; it is meant_only as afinal balance technique tobeusedincases of rotors

which are not_{satisfactorily balanced}

by

normal modal methods.

Industrialapplication ofthe "N"

modal method_alongthe lines formulated

by Bishop

andGladwellwas_successfullyand

"intuitively"

demonstrated

by

Grobel

(46)

who stated that the modal method was

being

_successfully employed at

General Electric

Corporation, Schenectady,

New York,as_earlyas 1950on large turbine-generatorrotors of _{up to} 100

tonsweight and between

bearing

spans of80 feet.Thoughpriortotheformualizedpresentation ofthemodal

balancing

procedure

by Bishop

and Gladwell

(13),

it was nonethelessthesame basic modal procedure.

Prause, MeachemandVoorhees

(75)

demonstratedthesuccessful_{applicability}ofthe

Bishop

and Gladwell "N"

method to the

balancing,

and the proposed production balancing, of a 28-foot helicopter power-transmission shaft _operating beyond its fifth critical speed.

Mooreand Dodd

(35)

notedthatinlarge industrialrotorsthereare casesinwhich a rotor will experience non-negligible

rigid

body

vibrations, inwhich caseforsuccessful balancetobeachieved a lowspeed, rigid

body

balancemust befirst

performed.

Kellenberger

(54),

recognizing the possible non-negligible rigid

body

effects on a rotating system modified the modal

analysis of

Bishop

andGladwelltoincorporatea rigid

body

two-planebalance. Theconcept ofKellenberger'sformulation

isthe modal approach ofFedern, butwith an analytical modal analysisstyle _{development along}the linesof

Bishop

and

Gladwell.

Theanalytical basis ofKellenberger's development isthemodalformulationof

Bishop

and Bladwellwiththeaddition

of

Z

Forces

=

₀

E Moments