Machine tool spindle design

Rochester Institute of Technology

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Theses

Thesis/Dissertation Collections

2000

Machine tool spindle design

Jamie Hoyt

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MAC~ETOOL

SPINDLE

DESIGN

•

Dy

Jamie

A.

Hoyt

A Thesis Submitted in

Partial Fuifiiiment of the

Requirements for

i:he

MASTER OF SCiENCE

iN

MECHANICAL ENGiNEERING

Approved

by:

Dr.

Wayne W. Walter

Department ofMechanical Engineering

Dr.

Hany A. Ghoneim

Department ofMechanicai Engineering

Dr.

Kevin

B.

Kochersberger

Department of Mechanical Engineering

Dr.

Satish Kandiikar

Department ofMechanical Engineering

Thesis Advisor)

DEPARTMENT OF MECHANICAL ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

Permission granted

Tide of Thesis: Machine Tool Spindle Design

i,

iamie

Hoyt, hereby grant permission to the Waiiace Library of the Rochester

Institute of Technoiogy to reproduce my thesis in whole or in part. Any reproduction will

not be for commercial use or profit.

Date:

~

/17

/oc

Abstract

Inmodern machinetoolapplicationstheperformance of amachinetoolisjudged byits abilitytoproduce work-pieces_accuratelyandefficiently. Thestiffness ofthe

machinetool spindlehasa profound impactontheoverall machineperformance. The work presentedhereprovides atoolformachinetoolspindle designersto_develop spindlesthatare_sufficiently stiffto meettheirneeds. Theanalysis presentedhereis divided intothreemain sections.

Thefirstportionisa static analysis. The static analysis calculatesthelateral deflection ofthe _{spindle-bearing} system. AMatlabprogramwasdevelopedthataiiows theuserto enterthespindle parametersintoabatch fileandobtaintheplotsofthe

deformedshapeofthespindle.

Thenext portionisadynamicanalysisofthe spindle. Thisportionincludes both themodesof vibrationandtheforcedresponse. Themodalanalysistreats thespindle as a

continuousEuier-Bernouiii beam. Anumerical methodfor_handlingthestepsintheshaft and applied_boundaryconditionswasdevelopedthatcouldbeextendedto_manyother applications inrotordynamics. AMatlabprogram wasdeveloped forthedynamic

analysis. Thisprogram provides adesignerwithplotsofthemode shapes andforced responsegiventhespindledesignparameters.

The final sectionisanoptimizationofthespindle design. Givenconstraintson thelocationand stiffness ofthesupport_bearings, aMatlab programwillreturn valuesfor

theseparameters_{resulting in}thespindle configurationthatpresentstheminimum

Table ofContents

Page

I

TableofContents 2

ListofTables 3

ListofFigures 4

Nomenclature 6

1.0 Introduction 10

2.0 StaticAnalysis 15

2.1 DeflectionofElasticShaft 17

2.2 DeflectionofBearings 26

2.3 MatlabSolution 27

3.0 Dynamic Analysis 34

3.1 ModalAnalysis 35

3.2 Matlab SolutionforMode Shapes 61

3.3 Forced Response 66

3.4 MatlabSolution for ForcedResponse 70

4.0 OptimizationAnalysis "4

4. i Optimization Model 74

4.2 TheConstrained Steepest Descent Algorithm 81

4.3 The Matlab Solution forOptimization 83

5.0 Conclusions 87

References 90

AppendixA : Batch_{Fiie Template} 91

AppendixB:Matiab Programs for StaticAnalysis 93

AppendixC: Matiab Programsfor Dynamic Analysis 107

List ofTables

Page

3.1 Comparison ofResonantFrequencies (FEAvs. _Analytical) 66

4.i TableofDesignVariables 76

4.2 OptimumValues for Static Analysis 84

ListofFigures

Page

2.i Static SpindleModel 16

2.2a Elastic DeflectionofSpindle Shaft 18

2.2b DeflectionofSpindleBearings 18

2.3 ModelofUniform Beam 19

2.4 TransformationofBeamSegment 22

2.5 Shearand_BendingMoment DiagramforaBeamSegment 23

2.6 Batch_File,Static Analysis 28

2.7 MaltaRepresentationofSpindle_Geometry 29

2.8a DeflectionContributionofElasticShaft 30

2.8b DeflectionContribution ofSupportBearings 30

2.9 Total DeflectionofSpindle 31

2.10 ComparisonofTotalSpindleDeflection (FEAvs. _Malta) 33

3.1 DynamicSpindleModel 36

3.2 LateralVibrationofanEuler-Bernoulli Beam 37

3.3 Sample Euler-BernoulliBeam 40

3.4 ModeShapes for SampleBeam 43

3.5 Stepped Euler-Bernoulli Beam 45

3.6 Sample Stepped Euier-BernouiiiBeam 49

3.7a _BoundaryConditions for Section i 52

3.7c _BoundaryConditions forSection3 53

3.7d _BoundaryConditions forSection 4 53

3.8 Free_BodyDiagram, Joint 1 55

3.9 Free_{Body Diagram,}Joint2 57

3.iO FreeBodyDiagram, Joint3 59

3.11 Batch_File,Modal Analysis 62

3.12 Mode 1 Comparison 63

3.13 Mode 2 Comparison 64

3.14 Mode 3 Comparison 65

3.15 BatchFile,Forced Response 7 i

3.16 ComparisonofForced_Response,FEAvs. Analytical 73

4.1 OptimizationModelofSpindle 75

Nomenclature

Symbol

A Crosssectionalarea ofbeam_[in2]

A Gradientvectorof_inequalityconstraint[dependent on_constraint]

ai Locationofdrive_pulley[in]

a? Locationofrear_bearing[in]

a3 Locationoffront_{bearing [in]}

&4 Locationofgaugeline_[in]

bk Locationofkthjoint inspindleshaft_[in]

c Gradientvector_[in/in]

d Vectorofdesignchanges_[unitless]

gi(x) Ithinequalityconstraint[dependenton_contraint]

D Distancebetweensupportbearings_[in]

E Young'smodulus_[psi]

f Quadraticsubproblem_[unitless]

FQ Cuttingforce_[Ibf]

Fcub Unbalanceforcedueto_cuttingtool_[Ibf]

Fd Driveforce_[Ibf]

Fde Equivalentdriveforce_[ibf]

Fdub Unbalanceforce duetounbalance ofdrive_puiiey[ibf]

f(x) Costfunction_[in]

Kf Lateralstiffness offront_{bearing [Ibf/in]}

Kfmax Maximumlateral stiffness offront_{bearing[ibf/in]}

Kg Torsional stiffness offront_{bearing[in-ibfj}

K; Generalizedstiffness_[ibf/in]

Kr Lateral stiffnessof rear_{bearing[ibf/in]}

Kfmax Maximum lateralstiffness ofrear_{bearing[ibf/in]}

M(x) Bendingmomentin spindle shaft_[in-ibfj

MappHed_Externally applied moments_[in-ibfj

Mb Reactionmoment atfront_{bearing[in-lbfj}

Mbe Equivalentreactionmoment atfront_bearing[Ibf]

M; Generalizedmass_[lbf-s2/in]

Mk Moment inducedatkth joint inspindle shaft_[in-lbfj

Mt Momentat guageline dueto_cutting forceandtoollength _[in-ibfj

OH _Cantilever, distancebetween front_bearingand gaugeline_[in]

q,; Generalized coordinate_[in]

Qi Generalizedforce_[Ibf]

R _Penaltyparameter_[unitless]

Rf Reactionforceatfront_bearing[ibf]

Rfe Equivalentreactionforceatfront_{bearing [Ibf]}

Rr Reactionforceatrear_{bearing [ibf]}

Rre Equivalentreactionforce atrear_{bearing [ibf]}

T _{Kinetic energy[in-ibfj}

tj Stepsize[dependentondesign_variable]

[Ty] Transformationmatrixbetweenith andjth beam segments _[unitless]

ti Lengthof_cuttingtool_[in]

u Strain energy_[in-lbfj U _{Potential energy[in-ibfj}

Uj Lagrangemultiplier_[unitless]

V Maximum constraintviolation[dependenton_constraint]

V(x) Shear force inspindle shaft_[ibf]

Vk Shearforce inducedatkthjoint inspindle shaft _[ibf]

x Axial position_alongshaft_[in]

x Vectorofdesignvariables

x0 Fractionofmoment exerted_byfront_{bearing[unitless]} yb Elasticdeflectionofspindle shaft_[in]

y* Elastic deflectionofspindle shaft[in._]

6f Deflectionatfront_bearing[in]

5q; Virtualdisplacement_[in]

6r Deflectionatrear_bearing[in]

8wj Virtualwork_[in-lbfj

Ei Convergencecriteria_[unitless]

82 Maximumallowable constraintviolation _[unitless]

<p _{Descent function}

[in]

p Mass_{density[Ibf-sTin4]}

1,0 Introduction:

Great demandsare placedonthecapabilitiesoftoday'smodernmachinetools to

producepartsthat are_{dimensionaiiy}correctwith_{increasingaccuracy} andthroughput.

Someofthe machinetoolcomponentsthatimpactthe_accuracyandthroughputofthe

machine arethedrivesystems,way systems,controlandfeedbacksystems,andfinally

themachinetoolspindle. Themachinetool spindleistheelementofthemachinethat

either supportsthework-pieceorthe_cuttingtool. Inadditionto_beingasupport

structure,the spindle alsorotatesathighrates of speedto providerelativemotion

betweenthework-piece andthe_cuttingtool. Thereforethe spindlehas adirect impacton

boththe throughput(materialremoval_rate), andthe_accuracyofthefinishedpart.

AccordingtoLewinschai(1985),themostcommon requirements ofa machine

toolspindle are:

High running accuracy

Highspeed_capability

Greatstiffness

Lowand even_runningtemperature

Minimumneedofmaintenance

Often inmachinetool spindlestheseparameters will conflict witheach other. Inorderto

achieveahigherspeed_capabilitythedesignermusttradeoff spindle stiffnessfor speedor

visa-_{versa. Thespindle designer}

must_carefullyweightherequirements oftheuserto

determinethebestpossiblebalanceoftheseparameters.

Thegoal ofthisresearchisto provideatoolfora spindledesignertoaid inthe

evaluation ofthespindle stiffness. High_runningaccuracy, high_operatingspeed capability,lowand even_{runningtemperature,} and minimum need of maintenance are

typicallyfunctionsofthebearing's_{geometry, manufacturing, lubrication,} and methodof

mounting. Ifthe spindledesignerisableto_quantifythestiffness requirementsforthe bearinghecanthenwork withthe_bearingmanufacturertoselecttheproperbearings for

theapplication.

Al-Shareefet ai. ₍₁₉₉₀₎developeda quasi-static methodof_analyzingmachine

toolspindles. Theiranalysistakes theamplitude ofthedynamic forcesand appliesthem

toa static modelofthe_{spindie-bearing} system. Forthe static analysisthedeflection

contributionofthespindle shaft andthe deflectioncontribution ofthe spindlesupport bearingsare superimposedtoobtainthe totaldeflectionofthe system.

Thestatic analysis ofthespindle shaft assumesthata steppedflexibleshaftis

pinnedinthelocationofthesupportbearings. Theanalysis ofthisflexibleshaft consists

ofatransformationfromastepped shafttoauniformshaft. Thistransformationyielded additionalshear and_bendingmoments ateach ofthejoints intheshaft. _{The resulting} uniformshaftwasanalyzed_using classicalmechanics.

The deflectioncontributionofthespindlesupportbearingsassumes arigidshaft

supported_bylinearsprings. Thereactionforcesyieldedthedeflection at each ofthe

springs. _Essentially,thedeflectioncontribution ofthebearingsis a straight line fit betweenthe_resultingdeformedpositions ofthe springs.

Inadditionto thestaticanalysis an optimization ofthedeflectionatthe end ofthe

spindie was presented. Theoptimizationanalysis consisted_primarilyof_varyingthe

spindle designparameters and _lookingattheeffect onthe_resultingdeflectionatthe

spindie gaugeline. Plotswerepresented_iiiustratingtheeffectofthevariation ofthese

parameters. The_followingconclusionsweredrawn fromtheseplots.

Inthedesignof a spindiethereexistan optimumratioofthe_{bearingspacing}

to the_overhangofthespindie. Astheflexurai stiffnessincreasesandtheratio

offronttorear_bearingstiffnessdecreasestheoptimum_{bearing-overhang}

decreases.

A dimenskmless flexurai stiffness_(Kf(OH)3/EI)of greaterthan5resultsin

minimum deflectionatthe_cutting tool. Thedeflectionattheend, or gauge

line, ofthespindieis _verysensitivetotheflexuraistiffnessformagnitudes

lessthan5.

Havingmorethan3 stepsinthe shaftis desirable for_obtaining minimal

deflectionvalues.

The_{magnitude, position,} anddirectionofthe_drivingforce_greatlyeffectsthe

deflection atthegaugeiine. Foreachscenariothereexists anoptimum

locationofthedrivepulley.

In Lewinchai₍₁₉₈₃₎ a similar_studyonthevariationof spindiedesignparameters

waspresented. Plotswere generatedthatillustratedtheeffect ofthe_bearing

spacing-overhangratio onthespindie stiffnessforsupportbearingsof_varyingstiffness. From

theseplotsit couldbeconcludedthat for_verystiff supportbearingstheoptimum_spacing

betweenthebearingsbecomesshorter. Itcouid alsobeconcludedthatifthespindiehas a

longoverhangthestiffnessofthebearingshasalesserimpactonthestiffness ofthe

spindle.

Otherworkintheoptimumdesignofmachinetoolspindleswasalso done in

Montusiewicz et ai.(1997). Inthisworka modelofa machinetooi spindie supported_by

hydrostatic bearingswas presented. Tne_studyconsisted of_applyingafour-stage

muiticriterionoptimization _strategyto a staticmodei ofa spindie. Theobjective ofthe

analysiswastoreducetheradialandaxial deflectionofa_spindie,thetotalmassofthe

spindie,thetotaipowerlossofthe_bearings,and_finaiiythesizeofthebearings. The

analysisdividesthespindie systeminto foursubsystems. Eachofthesesystems are

optimized_locally,and_finaiiyintegratedtoprovide agiobal optimization. Theoutcome

ofthisanalysiswas a computeraidedoptimumdesignpackage. Thispackage aiiows

spindiedesignersto_{interactiveiy}designanoptimum _{spindie, inputting}requireddesign

variablesthroughout theoptimization process.

Aqualitative dynamicanalysisofa machinetooispindie waspresented in

Al-Shareefet ai.(i99I). _{Traditionally}inthedynamicanalysisofmachinetooispindiesthe

firstmodeisthought toberesponsibleforpoor_cuttingquality. Thepurpose ofthiswork

wasto assessthisassumption. Therewas concernthat thiswouldnotbethecase since

therangeof_operatingfrequenciesforagiven spindleoftenexcitethe highermodes. The

first fourmodesforanexamplespindieweresolvedfor analyticallyand comparedto

experimentairesults. Themodal analysis presented ignores_dampingandrotational

betweenthenon-rotationai naturalfrequenciesandtherotational criticai speeds. _By

lookingattheindividual modeshapes_theyfoundthat thefirstmodecontributedthemost

to thedeflectionatthe tooi towork-pieceinterface. Aiiothermodesinthe _operating

frequencyrangeexhibited nodalcharacteristicsatthisinterface. Sincetheexcitation

forcewouldbe exertedhere_theyconcludedthat thefirstmode wouldindeed bemost

accountableforpoor_cutting quaiity. However_they also notedthatatthehighermodes

therewassignificantdeflectionatthe locationofthesupportbearings. Thiscouidresult

inthedegradationofthese_bearingandan eventualloss of spindle stiffness.

Someotherworks, _pertainingmore_generallyto thefieldof rotor_dynamics, were

also researched. Twooftheseworksdeal_primarilywiththeextension ofthe conventional

transformationmatrix_(CTM)technique. Intheworkdone_byCurtietai. ₍₁₉₉₃₎an

expressionforan8x8 dynamicstiffness matrix ofa_{rotating Timoshenko}beam is

derivedand reiatedto theconventional4x4dynamic stiffnessmatrix. Thisprovidesfor

theinclusionofanisotropic supports.

Inworkdone_{byMurphy (1993)} apolynomialtransfermatrix wasdeveiopedto

replacetheconventionaltransfermatrixformodal andforcedresponse analyses. The

advantageofthepolynomialtransfermatrixis anincrease incomputational speedof3.5

to 100timesovertheconventionaltransfer matrix. Exampleproblems wereanalyzed

usingboththeCTMandPTMmethods asweiias afiniteelementanalysis. Theresults

forailthreecases wereidenticalandthespeedofthePTMmethodwas_considerably

faster.

2.0 Static Anaiysis:

Thestatic anaiysiscalculatesthelateral deflectionofthespindie. Figure 2.1 illustratesthemodel under scrutiny. The _followingassumptionswere_necessaryto

performtheanaiysis:

1. The spindie shaftis assumedtobe anEuier-Bernouiii Beam.

2. Thespindie issubjectedto a_{cutting force,}adrive_force, andthereactionforcesatthe bearings. The drive forcemustbeappliedbehindtherearbearing.

3. Thetorsionaland axialdeflectionsofthespindie shaft are neglected.

4. Thecenteriineofthespindie shaftis_exactly iniinewiththecenteriineofthe_bearing bores. There isno contributionto thelateraldeflection dueto _{manufacturing}

misalignment.

5. Thespindie_housingandthe_cuttingtooiarebothassumedtohaveaninfinite

stiffness.

6. Itisassumedthat thespindie issupported_byonlytwobearings. Thisiscommonfor

mostmachinetoolspindles. _{Manufacturability}precludestheuse of morethantwo

bearings inmostspindies.

7. Thecontributionoftransversesheardeformationto theoveralllateral deflection is assumedtobenegligible. Itwas observedina_studyconducted_byAl-Shareefand Brandon,that thecontributionof sheardeformationis dependentontheratiobetween thelengthofthespindie andthe spindienose overhang. Thesheardeflection for

shortspindieswithsmall overhangscontributes moreto the overall

Figure2. i SpindieModel

deflectionthanlonger,more sienderspindies. _{A variety}of spindies were analyzedinthis

studyand amaximum contributionof12percent wasfound(Ai-Shareefet_{ai., 1990)}

Superpositionwas employedtocalculatetheiaterai deflectionofthe spindie. The

elasticdeformationofthespindie _{shaft, ys} andthedeflectionofthespindiebearings, yb

were superimposedtocalculatethe overalldeflectionofthespindie (seefigs. 2.2aand

2.2b). Equation 2.1 givestheoveralldeflectionofthespindie.

v. _{=ys+yb (2.1)}

2.1 Deformation ofElasticShaft:

Fortheelastic contributionofthe spindie shaftAi-ShareefandBrandonproposea

methodto transform thestepped spindie shafttoauniformshaft(Al-Shareefet. _ai, 1990).

Thisapproach willbeemployedinthisanalysis. Whentheshaftistransformedthereisa

moment,M*and shear_force,\\inducedateach_{step in}theshaft(fig. 2.3). Inaddition

theappliedforcesand reactions mustbetransformedintoequivalentforcesappliedto

beamsegmentswithlarger_bendingmoments ofinertia. Theseequivalentforcesare

noted_usingthesubcript"e"(i.e._Fj -> F<je).

The deflectionoftheuniformbeamcanbe_easilyanalyzed_usingconventional

beam_theoryand _{singularity functions. The singularity}functionswill berepresented_by

expressions in<>. Ifthevalueoftheexpression withinthesebracketsis lessthanzero

thefunctionbecomeszero(i.e. <2-4>2

=_{0). If}

thevalue oftheexpressionisgreaterthan

zero, thefunction simplybecomestheexpression withinthebrackets(i.e. <4-2>" = <4-2f).

Theshear_{force, V(x)}oftheuniformbeamcan befoundtobe:

a1 _H|

Figure 2.2a Elastic DeflectionofSpindieShaft

'////

Figure2.2b DeflectionofSpindie Bearings

b4 m b5

1 X] Oi XI OJ \1 O* XI to XI DO XI TOW. XI tJIl-1 X]

11 1 1~1

1 1

VI_{i_} V2i _V3, V4i V5i _{V6, Vn-2,} Vi

^M2^M3^M4^M^Mr^

311 X

I

"J

XSxVx

Rre

a2 y

X>xV\.

Rfe

a3

Mbe

1

a4+ti

Figure2.3 Model ofUniformBeam

Vix)=

F(x-u1)D+Fc(x-a,-tl)D

-Rfe(x-a>r+iv*(*-b*r

Themoment ofthe_{beam, M(x)}becomes:

M(x)=]V(x)dx+MappI,eJ

M(x)=

Fde{x-a1)1

+_Fc{x-a4

-ttf -R^x-aJ

+

Vt{x-bkf

(2.2)

*=i

Theslope ofthe_beam,8(x)becomes:

i

0(x)= --JM(x)dx

(2.3)

W

(2.5)

0(x)= El

^{x-aiY+^(x-at-tiy-^(x-a^-^(x-a,y

+M{x-<*2) +1i

Integratingtheslope ofthebeamyieldstheelasticdeflection, ys(x):

y,(*)=

El.

(2.6)

{t-f(x-a$

o '6 o

-^(x-atf+t^ix-bj+t^x-btf ' (2.7)

o _i=i 6 _i=1 2

Mu , ,2

+

-f-(x-a3) +qlx +q2

Theintegration_constants,qi &q2 canbe found_byapplyingthe_{followingboundary}

conditions:

ys(x=

a2)=_v

Solvingfortheintegrationconstants yields:

o2 =

~%a2-a;f -

^-(a2

6 _*=i o _k=l 2

(2-3)

1 <fe

6 fc3 ~{2 -i>3)-TL{3

t=i 6

+

f^L-b.)2-(a,-b.)2)

(2.9)

I S 2

Thederivationforthemoments and shearforcesinduced, andtheequivalent

applied forceswhenthestepped shaft istransformedintoauniformshaftwillnowbe

presented. The derivationbegins_{by looking}attheinternalshearand_bendingmoments

foran_arbitrarysegmentinthe steppedbeam_(Fig2.4). Anillustrationofthe shearand

bendingmomentdiagramsis also offered(Fig. 2.5).

Fromtheshear and_bendingmomentdiagrams itwasfoundthat:

V{x)=

Vl=Vr (2.10)

and

M(x)=M,-V,x

FromCastighano's Second Theorem:

dU

m7=y

(2.11)

(2.12)

and

*>;r t

ou

dM

6 _(2.13)

Thestrain_energy,U forone-dimensional_{bendingisknowntobe:}

'7777777777777777777777777777777'/. NJ/

Figure2.4Transformationof aBeamSegment

V

/lx

J

Figure2.5 Shearand_{BendingMoment Diagrams foraBeamSegment}

r M 2

H^n* _<214>

o

Itshouldbenotedthat thisexpressionforthe strain_{energy does}notinclude any

contributiondueto transverse shear_{deformation. Substituting}equation(2.₁₄₎into

equations(2.₁₂₎and_(2.13),witheqns. (2.₁₀₎and(2.1₁₎fortheoriginalbeam segment

(priorto thetransformationto theuniform _shaft), yieldsthe_{following y}and6:

*

EI\ 2 3

J

i f _yp }

e=\Mrl--^-\ (2.16)

El\

}

^

Similarly, whentheanalysis isrepeatedforthesegment afteritstransformationthe

deflectionand _slope,y*

and8* arefoundtobe:

v*=-\MZ-J.2Li (2.17)

' EI*

{

j

0*

= -\\MJ^-\ (2.18)

EI

[

J

Thedifferences in_yand8 mustbecompensatedforwiththeinducedshearforceand

bending moment.

[EI EI

}\

)

A^=^J

l

Umi-LtL-).

f2.20)

[/ /

J[

J

^

Therefore:

i

\}Mf

Vf\_\

VJT]

[EI

ET)\

J

[ET}{

j

[EI EI

)[

J

}[

J

Multiplyingeqn. _{(2.22) by}-1/2and_adding eqn. _(2.21)yields:

(2.22)

V =/

1

1_

/ r (2.23)

Substitutingeqn. _(2.23)into eqn. _(2.21)yields:

MM=r 1 J.

i ?wr (2.24)

Thisanaiysis canberepeatedtofind aninduced shear and_bendingmomentat each_{step in}theshaft. Theinduced forceand moment nowbecomeappliedforcestoa

beamsegmentwith amomentof inertia ofI Thisanalysiscanbeextendedtoshowthat aii appliedforcesmustbe scaled_byafactorofIn/I. WhereInisthemomentofinertiaof theuniformbeam(largestmomentofinertia inthestepped _shaft),andIisthe momentof

inertiaofthe segmentthat theforce is appiiedto. Thereforetheinduced forcesand

momentsbecome:

vk=i

- fd{x -a,

>*

- fc{x -a4

-ttf (2.27)

Mr _{=RXbk-a2Xbk-a2y+Rf(bk-a3)(bk-a3y}

-fc{K-<**-ti)(bk-a4-tl)-mb(bk-ay

The cutting, driving, and reactionforcesfromthestepped spindieshaft must also

bescaledtoprovide equivalent appliedforcesontheuniformshaft. The_scalingofthese

forcesyieids:

E*=i~Fd (2.29)

Ifi

K=-rK (2₃₀₎

*Rr

*Rf

Mbe=-^Mb (2.32)

2.2 Deflection ofBearings:

The deflectioncontribution ofthespindlebearingswascalculated_byassuming

thatthe spindieisa rigid shaft supported_bytwoflexiblebearings(Figure2.2b). The

cuttingforceFc,andthe_drivingforce_Fdwereusedtosoive forthereactionsatthe

bearing. Thetworeaction_forceswereusedtocalculate6rand_5f, thedeflectionsat

the twobearings. Thedeflectioncontributionofthebearings isa straightlinethrough

8rand8f.

(Sr-Sflx-a2)+Sr(a3-a2)

-t \ k^u)

{<*3-<*z)

Where

S_f

8 -

fcfo

(234)

(a3

fAas -a2)+mb-fc(p4 +tl-a3)-(fc+_{fd\a3_a2)}

(a3-a2)Kf

mb=fMA+d-<**K (236)

2.3Matlab Solution:

Aprogram wasdeveloped_{using Matlab}toautomatethestatic anaiysisofthe

spindie shaft. Theusermust_simplyenterthe_{geometry,loads,} and support parameters intoa spreadsheet called aiSbatchfile"

A copyofthebatch fiietemplateispresented in Appendix_{A. The Matlab programming}code usedtoautomatethestatic anaiysiscanbe found inAppendix B.

Anexampie oftheanaiysisfora simpie spindieispresentedhere. Figure2.6 illustratesthebatchfiieforthe static anaiysis. Uponthe completionofthebatch fiiethe

program wiiireadthefiieand report a geometricrepresentationofthe spindle. Theplot illustratesthe_geometryoftheshaft aswellasthelocationsofthe_{bearings,cutting force,}

anddriveforce(see_{figure2.7). This feedback}allowstheuserto_easilycheckfor mistakesinthebatchfiie. Withailtheinformationcorrecttheprogramcalculates and reportsplots ofthedeflectioncontribution oftheelasticshaft(figure_2.8a)andthe

deflectioncontribution ofthebearings (figure 2.8b). _Finaiiytheprogramreportsa piot of

thetotaideformationofthespindle(see figure2.9).

Batch File:

Geometry:

NumberofSectjons(#):

Section Length OuterDiameter InnerDiameter Area MomentofInertia (#) fin-) (ml (inl fin"21 fin "41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3 2.25 2 0.83449 0.47265783

3 2.375 2.125 0.88357 0.560861726 3 2.5 2.25 0.93266 0.659419991

3 2.625 2.375 0.98175 076890787 3 2.75 2.5 103084 0.889900605 3 2.875 2625 1.07992 1022973438

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Bearings:

LatStiffnessofRearBearingQbAri): 100000 LatStiffnessofFront:_BearingQb/n): 500000

Fractionof mom. onFront Bearing; 0.1 LocationofRear_{Bearing (in.)} 75 LocationofFront_Bearing(in.) 13.5

Pulley:

Locationof_Pulley(in.)

Static Belt Tension (lb):

~x-nK$M-v<>:..-'.;;:

.

Tool:

'i^ml^i^M^^^M: "zM^ffi&ai&m,

StaticCuttingForce fib): _j 300

W^^^^^^t^^^- I 330

t^^^-f^U^i^f'^^-.-^j'

-','''.- *&>*-?

^33:33-^mmSi^timW^i *#>^flB*5SS4S3i

LengthofTool(in) 1 2

Speed:

Material Prooerties:

Modulusof_Elasticity(psi):

Figure2.6 Batch FileforMatlab Solution

inputdimensions

7

-6

-

-5-4

-3

-1 1 1 1 T 1 1 1 1

-OD

ID

? D Pulley

-e- -v- Rear

Bearing

+ -*- Front

Bearing

~

Tool

-r *

1

...

,_,

1 1 1 1 1 1 1

8 10

x,(in)

12 14 16 18

Figure2.7MatlabRepresentationof_Geometry

x 10 Deflection Contribution ofElastic Shaft

Figure 2.8a Deflection ContributionofElastic Shaft

x 10 Oeflection Contribution ofBearings

-i 1

Figure 2.8b DeflectionContributionofSupportBearings

x 10 Combined Spindle Deflection

0

Figure2.9 Total DeflectionofSpindle

Inordertoconfirmtheresultsoffered_bythe_program,afiniteelementanalysis of

thesample spindlewas performed_usingAnsys. Thespindlewas modeled_using

one-dimensional _linearlyelasticbeamelements. Thebearingswere modeled_{using linear}

springelements. _{The cutting}forcewastransformedinto aforcemomentcouple and

applied attheendofthespindle shaft inorderto accountforthetoollength. Figure2. 1 0

comparesthedeflectionsoftheshaft_{using both}methods. Itisclearfromtheplotthat

thereisanexcellentcorrelationbetweenthe twoanalyses.

1.00E-02

8.00E-03

=- O.OOE+00₊

>-StaticDeflection Comparison

*

? FEA

Matlab

X(in)

Figure 2. 10 ComparisonofTotal SpindleDeflection(FEAvs.Matlab)

5.0 DynamicAnalysis:

Thedynamicanaiysisforthespindie shaft consists oftwoportions. Thefirstpart

oftheanalysisisthemodaianalysis. The beamistreatedas a continuous systemforthis

portion oftheanaiysis. The second partoftheanaiysis solvesforthedeflectionofthe

spindie_bymeansof modal superposition. The_following assumptions were madein

ordertoperformtheanaiysis:

1. The spindie shaftisassumedtobeanEuler-Bernoulli Beam.

2. Thespindie issubjectedto a_cuttingforce(FcSin(rj>ct)), adrive force_{(FdSin(Qdt)),}

unbalanceforces_{(FcubSin(cot)}& (FdubSin(ot)), andthereactionforcesatthebearings.

Thedriveforcemustbeappliedbehindtherearbearing. _{The cutting}forceanddrive

forceareassumedtobe harmonic.

3. The massesofthe_puiieyand_cuttingtooiare assumedtobeconcentrated. Themass

ofthe_puiieyisassumedtobeconcentrated atthecenteriine ofthepuiiey. Themass

ofthetooiisassumedtobeconcentrated attheend ofthespindie shaft. Thispointis

oftenreferredtoasthegaugeiine.

4. Thereis no unbalance excitationintroduced_bythe spindle shaft.

5. Therotational affects ofthespindie shaft are neglected.

6. Thetorsionaiand axiaideflectionsofthespindle shaft are neglected.

7. Thecenteriine ofthespindle shaftis _exactlyiniinewiththecenteriine ofthe_bearing

bores. Thereisno contributionto the lateral deflection dueto_{manufacturing}

misalignment.

8. The spindie_housingandthe_cuttingtooiarebothassumedtohaveaninfinite

stiffness.

9. Itisassumedthat thespindieis supported_byonlytwobearings. This iscommonfor

most machinetooispindles. _{Manufacturabiiitytypically}precludestheuseofmore

thantwobearings inmost spindies.

10. Thecontribution oftransversesheardeformationto theoveralllateral deflection is

assumedtobe negligible.

1 i. _Dampingisneglectedinthedynamic anaiysis.

Themodeiscrutinizedinthedynamicanaiysisis_very simiiarto themodei usedinthe

static analysis. Onemajordifference istheuse ofatorsional _springtorepresentthe

torsional stiffnessofthefront supportbearings. Inadditionthemasses ofthe_puiieyand

cuttingtooiareincluded. Seefigure3.1 forthedynamic modei under scrutiny. 3.1 Modal Anaiysis:

Thefoundation forthemodal anaiysisisthe derivationofthewave equationfor theiateraivibrationof acontinuousEuler-Bernoulli beam. Figure 3.2 representsthefree bodydiagramof an_differentialelement ofanE-Bbeam. _ApplyingNewton'ssecond iaw

tothebeamelementitcanbeshownthat:

~^_dx =-pA~^_at (31)

and

V^

OX

bn

bk

b2

b1

3

FdSin(wdt)

Md

"X

Kr

FdubSin<wt)

/?77777

mrrw fmrm

a3

a4

^ Mt

Tool i_~.

FcSin(wct)

FcubSin(wt)

TL

Figure3.1 Dynamic Spindie Modei

y

">

Figure3.2 Differential ElementofanE-B Beam

Itcan alsobeshownfrom StrengthsofMaterialsthat:

Substitutingeq. 3.3into3.2yieids:

M=_EI^4

(33)

ox-Finaliy, substitutingeq. 3.4into 3.1 and_rearrangingyields:

V=_{EI^4 (3.4)}

d"v hJd"v

-+^r^r=0 0.5)

dr pA dx*

The_followingharmonicsolutiontoeq. 3.5was assumed:

y(x,t)=y(x)smo)t _(3.6)

Substitutingtheassumed solution(eq. _3.6),intothedifferential equation _(3.5)yieidsthe

followingforth-order differentialequation:

dx4 P*y

=0 (3.7)

where:

P=B*EL _(3.8)

p

EI

Itcanbeshownthat thegeneral solutionto the_precedingforth-order differential equation

is:

y(x)=A_coshfix_+Bsinhfix₊C_cosfix+Dsinfix (3.9)

Equation 3.9representsthewaveequationforan_{E-B beam. The}modeshapesfora

beamcanbefound_{bysubstituting}values for_pthatcorrespondto theresonant

frequencies. Theconstants_A,B,C, andDcanbe solvedfor_byapplyingthe_boundary

conditionforthebeam.

Asystematic method_involving numerical methods wasdevelopedto soiveforthe

resonantfrequenciesandtheir _{corresponding}mode shapes. Thismethodisnot exclusive

to the spindie probiem athand. Itcanbeextendedto thelateralvibrationof_many

Euier-Bernouliiproblems. Listed belowarethe stepsto thismethod:

1. Establishtheboundaryconditionsforthesystem.

2. Collectthesystem ofequationsintomatrixform.

3. UsingGaussianElimination numerically reducethematrix.

4. _UsingtheBisectionMethodora comparable root_findingmethod soiveforthe

resonant_frequency, p.

5. BacksubstitutetofindtheconstantsA,B,C andDforthebeam segment.

Figure 3.3 representsa simplebeamusedtoillustratethisapproach. Tne beamunder

scrutiny here isauniformE-Bbeam fixed atbothends. Thefirst_stepistofindthe

boundaryconditions. Sincethebeam is_fixed-fixed,thedisplacementandrotationatx=

0,iarebothequaltozero. Expressed mathematicaiiy:

K0)=0 _(3.10)

7(0)=0 _(3.11)

y(i)=v _^3.iz)

7(0-0 _(3.13)

x=u

SubstitutingEq. 3.9 intoEqs. 3.10-3.13yieids:

V(U)=_{A + C} =₀

y [V)=B_+jj=_v

y(l)- _a

cosn(p/)+ d_smn(/)+c_cos(/?/)+ u_sm(fit)=_v

V-iV

(j.io;

y(/)=A

sinh(/>/)+B_cosh(/?/)

-C_sin(fii)+D_cos(/w)=₀ (3 ₁₇₎

Thenext _stepistocollectthissystem offourequationsinto matrixform. ThisyieidsEq.

3.18:

10 10

0 10 1

cosh(/3?) sinh(yS0 cos(/#) sin(/?/)

sinh(>9/) cosh(y9/) -sin(yS7) cos(/37)

\A 0

B

< ?=

0

>

0

> 0

(3.18)

Step (3)reducesthematrixineqn. _{3. 18 using}Guass-Jordanelimination. Thereduced

systemis illustrated ineqn. 3.19:

0

1

sin(/ff)-sinh(/ff)

cosOGf/)-cosh03T) 0 0 0 cos0G0cosh(y3/)-l

1 0 1

0 1 0

0 0 1

rA"

V

B 0

. =< o}

D

K. J L0J

(3.19)

Thereduced system can_beusedtosoivefortheresonantfrequencies, pj

(cos(fil)cosh(fil)

IfDwasequaito_zero,then_AB,and Cwouldaisoequal zero. Thiswould notbea

meaningfulresult. Thereforeitcanbeconcludedthat:

This iswheretheroot_finding method suggested_{in step(4)}comesintopiace. The

roots ofeq. 3.21 ieadto theresonantfrequenciesofthesystem. _Solvingfortheroots

yieids:

fiti,fi2l,fi3l=_{4.7,7.8.1 1.0}

After solvingfortherootsthe_{final step is}tobacksubstitutetoobtainthe

constantsA3,C andD. Beginthe substitution_byassumingthatD=i. Working

backward fromDitcanbeshownthat the_remainingconstants are:

sinh(ffi)-sin(/7/)

cos(p/)-cosnij&/)

B=_-l

.

_ sin(/ff)-sinh(/?/) cos(/)

-cosh(/?/)

Substitutingtheseconstantsintoeq. 3.9yieidsthemode shapeforthesamplebeam. The

equationforthemode shapesbecomes:

sin(/y)-sinh(/?y/)

cos(B

7l)

-coshifi,1)

J }

j=U,.. _(3.22)

cosifijl)

-cosh(yff;7)

Figure 3.4illustratesthefirstthreemodeshapesforthesamplebeam.

Themethoddescribedherecanbeappliedtofindthemode shapesforail uniform

E-Bbeam problems. However ifthebeam isstepped, asisthecasewiththespindie

shaft, thereneedstobea setof_boundaryconditionsforeachbeam segment. Thisleads

First Three modes for Sample Seam

-0.5

V

/

\

/

\

w>

mode 1

- - -

-mode 2

mode 3

-1.5

x _{(norm alized)}

Figure3.4Mode Shapes for SampleBeam

toa_veryiargesystem of equations. Asidefromtheproblem of_havinga_verylarge

system,thenumberof stepswouidchange fordifferent spindies. Thiswouidmake

automation_verydifficult. Atransformationmatrix wasdevelopedtohandlethestepsin

the shaft. Thetransformationmatrix relatestheconstantson one sideofa_steptothe

constants ontheotherside ofthe step. Thismakesthenumber of equations inthesystem

independentofthenumberof stepsintheshaft.

The deveiopmentofthistransformationmatrix_{beginsby looking}atan_arbitrary

stepinanEuier-Bernouiiibeam (see figure3.5). Inorderfor_continuitytoexistthe

deflection, slope, moment,and shearforceatthejointmustbethe samefor both beam

segments.

yi\i)=

y2Kl) v>-<;

JV(/)=

Jy(/) (3-24)

{EIl^Ml={Ell^Ml (3.25)

ax" ax"

mm={EI),qM _0.26)

ax

' ax

Substitutingeq. 3.8yieids:

ax cosn(pj/)

-+-yjj smh(yDj/)+_C,cos(y&,/)+_{uK sm(p,/)}=

A2cosh(/y)+_B2sinh(/y)+_C2cos(J32l)+_{D2 sin(fi2l)}

fit(AsinK^,/)+_5,cosh(/9/)

-C,sin(#/)+_Dxcosifij))= fi2

(A2

-C2sm(fi2l)+_D2cos(B2l))

fi^A,

-Dxsin(#0)=

fi2{A2

44

(3.27)

(3.28)

3

/

/

\

\

\

El El

Figure3.5 _StepinE-BBeam

/V(4 sinh(#/)+_{fi,cosh{#/)+C, sin(#/)}

->,cos(#/))=

/?23(^2smh(&/)+2cosh(p2l)+_C2sin(fi2l)

~D2cos0?2/))

Thesystem offourequations and eight unknowns canbecoiiectedintomatrix form (3,30)

0091(0,;) smh(/y) cos(/y) sm(AO -cosh(/M) -sdnfa(p,i)

smh(#/) codtfyW) -saAfi,}) ooe<J,l) -smh(J?,/) -cosb^,/)

-cos(p,/)

,2

-oos(JJ)

(&\0i (Ethfii (EJ\J: (EI\P?

srrin^i) coshil,!) snu^i) -cosf,^,ij --

-siring) -

-r-ooA^fi^j -

-r-smtn^j 3o^l^)

(7).tf. _{(EI\fii {EI\B{ (EI).^}

'a 0

A 0

Q 0

A

.=.

0

0

5, 0

c2 n iA, 0

(331)

UsingGauss-_{Jordanelimination foiiowed}

bybacksubstitutiona_relationshipcanbe found between_Ai-Di andA2-D2. Two ratios, Ri and_n,weredefinedto_simplifythe

reiationsmp.

^ = (^h

(0,

(3.32)

1=^ _(3.33)

A, = tl/?1

+1i(cosh0g1/)cos(^2/)-/?1

sinh(#/)siiih(2/)K

\rR2

+\]

+-* _{'{cosK^Osin^/?,/)-^,sinh(#/)cosh(y92/)}82}

_

K

^{coshOgjOcosOg,/)

R}

-Cli?1

~1i{eo8hOg1Osin(^2/)-^l

2

E1^1

J{sinh()g/)sirt(/y)

-i?,cosh(BJ)cos(B2l)p2

+**-*

'{,

+

2

C, = tli?1

'fa

-cos(y9I/)cosh(y92/)}^2

+

M_ziJ^?]

+

^

{cos(#/)cos<2/)

+-l^^^s(^1/)sin(^/)-i?Jcos{^1/)sin<y92/)}D2

(3.34)

(3.35)

(3.36)

Vr2 -ii

- i-J J

{#,

L 2 1 (337)

~^Rl2+%lco^OshX^/)

-sin</y)sin</y))C2

+*li?1

'{ft,

Thecoefficientsfromeqns. 3.34-3.37canbecollected intoatransformationmatrix_[Tj,

suchthat:

(3.38)

Theuseofthe transformationmatrix canbe iiiustrated_byexpandingthe sample

beamproblemtoinciudestepsinthebeam (see figure 3.6). _Applyingthe_boundary

conditions wouid resultinthe_following system of equations:

'

Ax

'K

< =

{Tj

IA,

10 10 0 0 0 0

0 10 10 0 0 0

0 0 0 0 _{eosh0ff3/3) sinh(^3/3) cos(B3l3) sm(fi3l3)}

0 0 0 0 _{sinh(&/3) cosh(/?3/3) -sin(j03/3) cos(jff3/3)}

'K

}A]

\B>]

c3

A;

Iftransformationmatriceswere not_used,the_{only way}tosolvethe systemofequations

wouidbetorelate_A>-Di to_{A3-D3by including}the_continuityequations. Thiswouid

increasethe size ofthe systemto 12equations and 12unknowns. Itwouid also makethe

Figure3.6SampieSteppedEuier-BernouiliBeam

size ortne systemaepenaent ontnenunroer or steps intnesnan. 1nis in-turn wouia

make automation moredifficult. Ifthe transformationmatrices wereusedthe system of

equationswouidbereducedto4equations and4 unknowns,regardiessofthenumber of

stepsinthe shaft. Thefirsttwoequationsinthesystembecome:

10 10

0 10 1

K\T]

B3

c3

>=_{< (3.40)}

The lasttwo equationswiiibethe same as representedineqn. 3.39. Oncethe system of

equationsis deveiopedsteps3-5ofthepre-described method canbeusedtosolveforthe

resonantfrequenciesandtheir_{corresponding}modeshapes.

The_five-stepprocess andtransformationmatrixcannowbecombinedand

appiiedtofindthefrequenciesand modes shapes ofthespindledepicted inFigure 3.1.

Inordertoencompassaii ofthe_externally applied_boundaryconditionsthebeam mustbe

divided into foursections. Figures3.7a-3.7ddepictthefoursubdivisions. Thefirst

sectionis betweentherearfreeend andthedrivepuiiey. Thesecond section is between

the_puiieyandtherear supportbearing. Thethirdsectionisbetweentherearandfront

supportbearings. The forthandfinalsectionis betweenthefront support_bearingandthe

cuttingtooi. Therewiii be fourconstantsforeachofthefoursectionsforatotalof

sixteenconstants.

Beginningwiththefreeend of section _one,theshearforceand_bendingmoment

at x=₀

arebothequaltozero.

Expressedmathematicaiiy:

V,(0)=EI^-=0 _(3.41)

ax

and

M,(0)=EI^-=₀

(3.42)

Substitutingeqn. 3.9intoequations3.10and 3. 1 1 and_settingx equaltozero yieids:

5,-D, =0 _(3.43)

and

?~ s\

Atthejunction betweensections 1 and2there arefour_boundaryconditions. The

firstthreeconditions involvethe_deflection, slope and_bendingmomentatthejoint

betweensections i and2. Sincethereare no_externallyapplied_moments, andthe

structureiscontinuous,the_{deflection, slope,}and_bendingmoment atthejointmustbe

equai for bothsections. Therefore:

Mai)=yiUh) (345)

JV(,)=

JV(a.) (346>

Eiyx \a1)=

niy2 \a,) (j.<w;

Substitutingequation3.9intoequations3.45-3.47yieids:

A, _cosh(/5b,)+_{Bi sinh(Sal)}+_C5cos(pat)+_{Dt sin(fial)}

-A2coshOfibj)

-B2sinh(fiax)

-C2cos(fiax)

-D2sin(fiax)=0

Atsinh(/x7j)

-+-B. _cosh(y(5bi)

-C,sin(/3ar.₎+_{D, cosOSa,)}

-,42sinh(Bax)-B2 coshfjSa,)+_{C2sin(fiax)-D2cos(J3ax)}=₀

(3.48)

(3.49)

VI

4

3

V2

Md /

/

Figure 3.7a_Boundary Conditionsfor Section 1

V2(a1) ms*\<

Md

a1

///////

a^

V3(a2)

V3(a3)

Kr

/////////

a3

\1'

V4(a3)

X Kf.Ktt

Figure 3.7c_BoundaryConditions forSection 3

V4(a3)

Kf.Ktt

/////////

Mt

a3

Figure 3.7d_BoundaryConditionsforSection 4

Axcosh(/xjt)+_{.81sinh(/x*5)-C5 cosOSa.)-D, sin^)}

-A2coshfjfifatj)

-B2sinh(y8afj)+_C2cosO^)+_D2sm(Bax)=0

The forth_boundaryconditionatthisjointisaffected_bythemassofthepuiiey. The mass

ofthe_puiieyintroducesan external shearforce. Figure3.8 iiiustratesthefree_body

diagramatthejoint. Theshearforce introduced_bythemassisequaltotheD'Aiembert

forceassociatedwiththe_puiieymass.

Therefore:

Vm =

mpy=

-mdm2y2(ax) (3.51)

Forequilibrium atthejoint:

rry \ -w t / \ rr /** J*>\

y^ai)-y2(ql)=

vm (i.oz)

Ely^(ax)-Ely2^(ax)=

-mda}2y2(ax) (3.53)

Substitutingequation3.9intoequation3.53:

Aisinh(/w,)+Bxcosh(pax)+_C5sinf/Sz,)

-D,cosGSa,)

-4[sinK^I)-^^cosh(^0]-57[cosh(/fa1)-^^sinh(/b1)] (3.54) p EI

-fi'EI

^^cos(fiax

-^

fi3EI ' 2 x fi3EI

+C2[sin(^I)--f

cos^^+Atcos^)--!_sm(p\)]=0

Thefirstthree_boundaryconditionsforthejointbetweenthesecondandthird

sectionarethesame asthe_boundaryconditionsbetweenthefirstandsecondjoint.

Therefore:

A2cosh(

fi(a2

-a,))+_B2swh(fi(a2

-ax))+C2cos(fi(a2

-a,))

+_D2sin(fi(a2 -ax))

-A3cosh(fi{a2

-ax))

-B3 sinh(fi(a2

-a,)) (3.55)

-C3

-D3sin(fi(a2

-ax))=0

V1(a1)

V2(a1)

/

/

Vm%

Figure3.8Free_BodyDiagramofJoint 1

a2 %\wi\p\a2-ai))+n2cemj>[a2 ~ax))-K.2$m(p(a2-a,jj

+_D2cos(fi(a2

-ax))-A3 sinh(fi\a2 -ax))

-B3 cosh(/5(a2

-ax)) (3.56)

+

C3

-D3 cos(B(a2

-a.₎₎=0

A2

-a,))+_{52 sinh(p(a2} -a,))

-C2cos(p(a2 -a,))

-D2sin(/?(a2 -a,))-/43cosh(/?(a2-a,))-53sinh(p(a2 -ax)) (3.57)

+

C3

-a,))=₀

Fortheforth_boundaryconditionatthisjointthe shearforceintroduced_bytherear

support_bearingmustbeaccounted. Figure 3.9iiiustratesthefree_bodydiagramatthe

joint. The shearforceintroduced_bythe_bearingisproportionaitothe shaft's

displacementatthejoint.

rr

r^-s \ ?<* ^C\

vkr=Kry3ia2) (xrt)

Forequilibriumatthejoint:

rr / \ rr s \ rr s*y J?C\\

EIy2!,,(a2)-EIy3"!(a2)=_KTy2(a2) (3.60)

Substitutingequation3.9intoequation3.60:

A2sinh(fi(a2 -ax))+B2_cosh(fi\a2_-ax))+_C2sin(p(a2

-ax))

-D2cos(B(a2-ax))-A3[swh(fi(a2 -ax)) + -^-cosh(fi(a2 -a,))]

fi'EI

-53[cosh(/?(a2 -ax)) + -Smh(B(a2 -ax))]+C3[sm(B(a2 -ax))

K* _U

fi'EI

+-^-cos(>9(a2-aI))]+D3[cos(B(a2 -ax))^-sin(fi(a2 -ax))]=0

fi EI _fi EI

Thefirsttwo_boundaryconditionsforthejointbetweenthe third andfourth

sectionare comprisedofthe_continuityconditions(y3=y4andy3'=y4?).

\ /*i /o\

v^a2)

V3(a2).

Vkr si/

rigure3.9Free_BodyDiagramofJoint2

Therefore.

A2cosh(/?(a2

-a,))+_B2sinh(p(a2

-ax))+_C2cos(/?(a2 -a,))

+_{D2 sin(p(a2 -ax))-A3cosh(/?(a2}

-aj)-53 sinh(/>(a2 -a,)) (3.62)

-C3

-ax))-D3 sin(fi(a2 -ax))=Q

A2 sinh(/>(a2

-a,))+B2cosh(fi(a2 -a,))

-C2sin(^(a2 -a,))

+D2cos(B(a2 -ax))-A3 sinh(/?(a2 -ax))-B3cosh(/J(a2 -a,)) (3.63) +

C3

-ax))

-D3cos(/?(a2

-a5))=0

Thethirdandforth_boundaryconditions areinfluenced _bythe_bendingmoment and

shearforceassociated withthe torsional andiateraistiffnessofthe frontsupportbearing.

Figure 3.10illustratesthefree_bodydiagramatjoint3.

Forequilibrium atthejoint:

rr/ \ rr / \ rr f+ s~

*~\

^v3 ^3 )

-r,iy4 ia3)=_a,

v4 ia3 ; vj _od;

and

i /

"* \* / ^ /I jCiCi

^3l3;-Aj4(a3;=iWtr/ lJOOJ

EIy%"{a3)-EIy4'(a3)=Kfy4(<*3) (3 67)

Substitutingeq. 3.9into eqs. 3.65and 3.67yieids.

A3sinh(/?(a3-a2))+B3cosh(pi>3 -a2)) +C3_sin(^(a3 -a2))

-D3cos(B(a3-a2))-A4[smh(B(a3-a2))+^-cosh(>9(a3 -a2))]

ATy

54[cosh(^(a3-a2))+

^-sinh(y9(a3

-a2))]+_{C4[sin(y?(a3-a2))}

+

^-cos(^(3

fi LI '

fi EI

V3(a3) M3(a3)

M4(a3)

Mkft

V4(a3)

\y Vkf

Figure3. 10Free_BodyDiagramofJoint3

ana

A3cosh(p(a3 -a2))+B3sinh(fi(a3

-a2))

-C3cos(/?(a3 -a2))

-A_sin(/?(fl3 -a,))-4[cosh(/?(a3-a2))+^-sinh<Aa3 -a,))]

-54[anh(#a, -fll))+-^7Cosh(>9(fl7 -a^M+CJcos^^, -a,))

^'^

+_-^sinO?(a, -a,))]+Z)4[sra(jS(a,_{-a7))--^-cos(/?(a,-a,))]}=0

/?'/ " "

B"EI

Thefinaltwo_boundary conditions arerelatedto the_cuttingendofthespindie.

Thefirstoftheseconditions relatestothe_bendingmoment. Sincethe_rotaryinertiaof

the_cuttingtooiisneglectedthemoment attheendofthespindieis equaltozero.

Therefore:

Ml(0)=EI^-^-=_{0 (3.70)}

dx"

A4cosh(p(a4

-a3))+_B4sinh(p(a4 -a3))

-C4cos(fi(a4

-a3))-D4sin(/?(<z4

-a3))=0

The last_boundaryconditioninvolvestheshearforceattheend oftheshaft. The shear

forceisequaito theD'Aiembert forceassociatedwiththemass ofthetooi.

Therfore:

V(a4)=

mty(a4)=

-mp

2y(a4

Substitutingequation3.9:

A4[smh(P(a4

-a3))+

^-cosh(y?(a4

-a3))]+_Bs[cosh(8(a4

-a,)) fi El

2 2

+

^-sinh(/?(a4

C4[sin(y?(a4

%-cas(P(a4

fi EI fi'EI

m,(o" .

i r-*

fi'EI

-D4[cos(fi(a4-a3))^wi(P(at-a3))]=0

Inordertosoivetheset of simuitaneous_equations,theset ofsixteenequations and

sixteen unknowns were collectedinto a matrix.

3.2 Matiab SoiutionforModeShapes:

Aprogram wasdeveloped_usingMatlabto automatethemodal anaiysis ofthe

spindie shaft. Dataiscoiiected and entered intoaspreadsheet. Thissheet acts asthe

batch fiie forthemodaianalysis. Muchlikethe static_anaiysis,theusermust enterthe

geometry,mass_information, and support parametersintothebatchfiie. _{A copy}ofthe

batch fiietemplateispresentedin_{Appendix A The Matlab programming}codeusedto

automatethemodai anaiysiscanbefound inAppendix C.

Anexample oftheanaiysisfora simpie spindie ispresentedhere. Figure 311

iiiustratesthebatch fiie forthemodai anaiysis. The "grayedour"

information doesnot

pertainto themodal analysis. Uponthecompletionofthebatch filetheprogram will

readthefiie and report ageometricrepresentationoftheinformation. Withaiithe

informationcorrecttheprogram calculates andreportstheresonantfrequencies forthe

sample spindie. Thesample spindiewasalso modeled_{using Ansys.} Acomparison

betweentheFEAand analytical resultsforthefirstthreemodesispresentedin Figures

3. 12through3. 14.

Batch File:

Geometry:

NumberofSections(#): _ZJ

Section Length Outer Diameter Inner Diameter Area MomentofInertia

(#) On) On) On) 0n~2) 0n~4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3 2.25 2 083449 0.47265783

3 2.375 2.125 088357 0.560861726

3 2.5 2.25 0.93266 0.659419991

3 2625 2.375 098175 0.76890787

3 2.75 2 5 1 03084 0.889900605

3 2.875 2.625 1 07992 1.022973438

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Bearings:

Lat. StiffnessofRearBearing(lb/In): Lat StiffnessofFrontBearing(lb/in): Tor^tiffnessofFrontBearingOn-_lb)

LocationofRear_{Bearing (in.)} LocationofFront_{Bearing (in.)}

Pulley:

Locationof_{Pulley (in.)}

Massof Pulley(Ib-^gVinfr Tool: 100000 500000 7.5 13.5 4.5 J m i33.33 emmmmsb 0025879917 ^ttR MaRaseSc^:uJi^:."ftwe*>fe.); :

LengthofTool(in) | 2I

Speed:

Material Properties:

Modulusof_Elasticity(psi):

Density(lb^n~3):

30000000 0.289

Figure3.11 BatchFileforSample Spindle

-0.2

--0.4

Mode 1 Comparison

(FEA vs. Closed _Form)

X (in)

Figure 3.12 Mode 1 Comparison

0.9

0.8

-0.7

0.6

>- _0.5

-0.4

Mode 2 Comparison

(FEA vs. Closed _Form)

0.3

0.2

CF Solution

FEA Solution

0.1

10

X _(in)

15 20

Figure 3.13 Mode 2 Comparison

Mode 3 Comparison

(FEA vs.Closed _Form)

0.8

-0.6

-0.4

-0.2

--0.2

--0.4

2<?

-0.6

CF Solution

FEA Solution

-0.8

X _(in)

Figure3.14Mode3 Comparison

Theresonantfrequenciesforthefirstthreemodes are comparedintable3.1. Itis

clearfromthe table that the twomethodscorrelate_{veryclosely for}thetwomethods.

Table 3.1 ComparisonofResonant Frequencies (FEAvs. _Analytical)

Mode FEA Analytical

j

1 19.26 19.17 0.4672897

2 61.28 61.72 0.7180157

3 95.57 100.33 4.9806425

3.3 Forced Response:

Theforcedresponseofthespindleiscalculated_using a numericmodal

summationprocedure. Thedevelopmentoftheforcedresponsebeginswiththeequation

of motionfora_beam,Dahlehet. aL (1989).

[/v"(x,of

Thenormalmodesforthe_{beam,<|>i(x),} must_satisfythe_followingequation:

(EI6)"-a>Mx)t'i=0 _(3.74)

Inadditiontoeqn. _3.74, sincethenormalmodes areorthogonal_theymust also _satisfythe

followingequation:

i&rfjdx- 0 for *

j 0

Thesolutionto theforcedresponsecanberepresentedinterms_<j>,(x) as:

y(x,t)=

2>,(x)<7,(0

(3.75)

(3.76)

Where_q,(t)isthegeneralized coordinate. Thegeneralized coordinatecanberealized

usingtheLagrange Equation. Lookingfirstatthe_{kinetic energy}yields:

T=

\]y\x,t)m{x)dx

2 0

Substitutingeqn. 3.76for_y(x,t)yields:

T=

^HHaiaj>JMj*"(x)dx 2', ;

T=

\ZMa

(377>

Wherethegeneralized_mass,Mjisdefinedas:

Ml=^2(x)m(x)dx _(3.78)

o

Thepotential _energy,Ucan_{be defined}as:

U=

^EIy"2(x,t)dx

2

0

2 .

Wherethegeneralized _stiffness,Kjequals 1 _'

J 0

^4w

K,=\EI[<p"(x)]2ax _(3.80)

0

Ifeqn. 3.74issubstitutedintoeqn. 3.79itcanbeshownthat:

tf=

Ageneralized_force,Qicanbe definedby lookingattheworkdone_bya virtual

displacement,6qi.

(3.81)

<*, =

J/(*,o2>^*

rearranging:

<^,=2>,a (3.82)

where:

ft=J/(x,fM(*>*

FromtheLagrange Equation:

fdT\

dt

8T dU

(3.83)

(3.84)

Substitutingforthekinetic_energy, potential_energy,andthegeneralizedforceyieldsthe

followingdifferentialequation.

<ii+a>i2<li=T

j/(*,0*,(x>fc

j<f,2(x)m(x)dx

(3.85)

where:

. (3.86)

Themodel ofthespindleassumes foursimpleharmonicloads. Theharmonic

loadsincludethedrive_force,cuttingforce,unbalanceofthe_pulleyand unbalance ofthe

cuttingtool(see fig. 3.1). Allfouroftheforces are assumedtobeinphasewith each

other and oftheform:

f(x,t)=

F(x)sm(cot) (3.87)

Eachoftheforcesare appliedtoa single point. _Assumingtheforce isapplied at x=

x,,it

canbe_{describedusing}thedeltadiracfunctionas:

f(x,t)=_{Fsin(a>t)S(x-x0) (3.88)}

Bydefinitionthedeltadirac function isequaltozeroforall x not equaltoXo. Furtherit

canbeshownthat:

]F(x)S(x-xo)dx

Substitutingthis_{relationship into}eqn. 3.85yields:

4t^=4WW

(3.90)

j<f>2(x)m(x)dx

Assumingthe_followingsolutionto eqn. 3.90:

qi(t)=

q,sin(0Jt) (3.91)

yields:

ft =

M"f

\co2

-G)2)\<t>2mdx

Thedenominatorofeqn. 3.92mustbe broken down forthefoursections ofthespindle

andeach ofthesegments_(steps)inthe shaft_described inthemodal analysis.

a, =

tl^K

(3.93)

W

J

(Ak

sin(#x))2

dx

n=\ k=\ o

Forthisanalysis_onlythesummation ofthefirstfourmodeswere utilized. After

thefirst fourmodesthedifference betweentheresonantfrequenciesand thedrive

frequenciesbecome largeand q,approacheszero. Thereforethe_steadystate response

becomes:

Y=

Mx +_<t>2<l2 +_&<73 +_Ma (3-94)

The_deflections, Ywerecalculatedforeach ofthefourexcitationforcesand superposed

toyieldthetotalforcedresponse:

y,=y*+Ym+Y*+YA* (3-95>

3.4 Matlab Solution forForced Response:

Aprogramwasdeveloped_{using Matlab}toautomatethecalculationoftheforced

responseforthespindle shaft. Themagnitude andfrequencyoftheexcitationforces is

enteredintoabatchfile. In additionto theloadinformationtheprogramreadsthe first

fourmodescalculatedinthemodal analysisprogram. The Matlab programmingcode

usedto automatetheforcedresponsecan befoundin Appendix C.

Anexampleoftheanalysisforasimple spindleispresentedhere. Figure 3.15

illustratesthebatchfileusedforthisexampleproblem. The"grayed

out"

information

doesnot pertainto thisanalysis. It shouldbenotedthattheprogramwillnotfunction

Batch File:

Geometry:

NumberofSectjons(#): ₃

Section Length

Of) (in)

Outer Diameter

On)

Inner Diameter Area

fin) (in~2)

MomentofInertia fin-4) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3 a25 2 0.834486 047265783

3 2.375 2.125 0.883573 0.560861726 3 2.5 Z25 093266 0.659419991

3 2.625 2.375 0.981748 0.76890787

3 2.75 2.5 1030835 0.889900605 3 2.875 2 625 1079922 1.022973438

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Bearings:

Lat StiffnessofRear_Bearing(ItMn): Lat StiffnessofFront_Bearing(lb/In.): Tor StiffnessofFrontBearing(in-_lb):

10000C

Locationof_{RearBeanng(in)}

LocationofFront_{Bearing (in.)}

Pullev:

Locationof_{Pulley (in.)}

MassofPulley(\b-s~2Ari

Harmonic Drive Force (lb): DriveFrequency(Hz):

PulleyUnbalance (rb-s~2):

Tool:

MassofTool Oj>^n):

Harmonic_CuttingForce (lb):

CuttingFrequency(Hz): Tool Unbalance (lb-s"2): LengthofTool(in) Speed:

Spindle Shaft Speed (Hz):

Material Properties:

Modulusof_Elasticity(psi):

Density(lb/fn'3): 30C 133.33 0.004559453 16.666665671 30000000 0.289

Figure 3.15 BatchFilefor Sample Spindle Forced Response

properlyifthemodalanalysisfromsection3.2isnot completedfirst. Thesamplespindle

wasalsomodeled_{using Ansys.} AcomparisonbetweentheFEAand analytical resultsfor

theforcedresponseispresentedin Figure3.16. ThecomparisonbetweentheFEAand

analytical responsesshows aclose correlationbetweenthetwomethods. Thereisa6.5%

difference inthedeflectionatthe toolforthe twomethods.

0.01

Dynamic Response Comparison

0.008

->-0.006 +

0.004

4-0.002 +

-0.002

FEA Response

-Analytical Response

-0.004

-0.006

-0.008

-0.01

X (in.)

Figure 3.16 ComparisonofForced Response FEAvs. Analytical

4.0 Optimization Anaiysis:

Theoptimizationanaiysis consistsof_minimizingthedeflectionofthespindie

shaft atthegaugeline (see figure4. 1). Thisanaiysisbuildsuponthestatic asweiiasthe

dynamicanaiysis. Optimal parametersare offeredfor bothcases. The_foiiowing

assumptions_appiyto theoptimizationanaiysis:

i. The designvariablesforthisanaiysis arethe iaterai stiffness andtheposition ofthe

twobearings. Allotherparametersare assumedtobeconstant.

2. Eachdesign iteration isapproximated_usingtheTayiorseriesexpansion. This

approximationisrequiredtodefineaquadratic_programming subprobiem.

3. Theoptimization point_mayor_maynotbetheglobal minimum. Howeverthevaiues

assurealocalminimum.

4.i Optimization Modei:

Thedevelopmentoftheoptimizationproblem restsin_minimizingacost_function,

f(x),where xisthedesignvariablevector. Fortheoptimizationofthemachinetooi

spindiethecost_function,f is definedasthedeflectionatthespindle's gaugeline.

/(i)=

^(a4) (4i>

Givenvaiuesforthedesign_parameters,avaluefor_yt(a4) canbeobtained_numerically

usingtheMatlab routinesdevelopedinChapters 2& 3. The design variabies,x arelisted

intable4. 1. Theremainder ofthespindie designparametersareassumedtobe fixed.

Thisisa_fairlyaccurateassessment sinceforan_existingspindiedesigntheother

parameterswouid_{significantly influence}the_supporting components (i.e. gearboxand

spindiehousing).

Tabie4.1 TableofDesignVariables

Design Variable Vector Parameter

xfl) a(2), postion of rear_org

x(2) a(3),position offront_brg x(3) 1Kf, lateral stiffness offrontbra.

x(4) jKr, lateralstiffness of rear_brg

Generaiconstrainedoptimumdesign definesthe_{followingequality}and _inequality

constraints respectively:

g,.(x)<u _v^.j;

Forthisoptimizationproblemthereexists no_equality constraints. Thefollowing

equationsdefinethe_inequality constraints.

xx>ax+D (4.4)

x2<a4-GH (4.5)

x3<KfimK V)

*4 <Krnax (4.7)

Tosummarizethese_constraints,thefirstconstraint(eqn._4.4)stipulatesthat thelocation

oftherear_bearingmustbe beyondthelocationofthe_puiieybya_distance,D. Thisis

requiredtoensurethat the_pulleyis"outboard"ofthesupportbearingsandthereis

sufficient _spacingto accommodatethewidth ofthe_puiieyandthewidth ofthebearing.

Thesecondconstraint(eqn. _4.5)requiresthatthere existasufficient _overhangto

accommodatefeaturesinthespindie shafttoaccept and supportthe tooi. Tnethirdand

forthconstraints (eqns. _4.6-