machine systems torsional vibration of ronald l. eshleman.pdf

8/11/2019 Machine Systems Torsional Vibration of Ronald L. Eshleman.pdf

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TORSIONAL VIBRATION OF MACHINE SYTEMS

by

onald L. Eshleman, Ph.D, P.E.

a s

a s s

Ronld L. Eshlmn s drctor o thVbrton Insttut d vbrton

con

sultnt to ndstry Hs rs xprtsnclud the dsgn nd devlopmnt omchnery nd ult dgnoss nd corrcton He hs publshd xprmntlnd nlytcl work n th lds torsonl vbrtons, rotor dynmcs shocksolton, nd vhcl dynmcs Dr shlm s th techncl dtor th hk Vbr Dge mothly jor

nl contnng bstrcts of publshed tcls d rports, r

v cls nd ns. H s ctv n th Acoustcl Socty Amrc nd the Amrcn Scty o Mchncl Engnrsnd serves s chrmn Amrcn Ntonl Stndr Insttute Commte S nd s vc chrmn InterntonlStndrds Orgnzton Commtte TC 108

ABSTRACT

This papr is cncd with trsinal vibrtin in thdsin,

dvlpmnt and ult diansisand

orrctinf

mchin systms Th physica bhvir f torsinl vibrtinsin shatin systms and surcs f trsinal xcitatin r r-viwd Th cncpts of trsinal ntural uns, criialspds rsnan, and rspns rcs trsss and tins

ar explaind Th llwing mthds r dling and al-culating natural uncis nd rspnss ar dscribd:lumpd and ntinuus parmtr mdl th Hlzrmthd th mdal rspns mtd and drct intratin fuatins f mtin T slutin f a transint trsinal vibr-tin prblm f a sycrnus mtr driv systm is xplaindTh masurmnt f trsinal vibrtin i th th ld andt labratry is dscrbd mnitring and prblm crrctnar mphszd Systms sujctd t trsna vbrtins arusd t lustrat masuring tchnius

INTRODUCTION

All rtating machinry undrgs s r f trsialvibratin In sm css th virtin cann b dtctd with-

ut spcial masuring upmnt vn whn t lvl scls t dstructv amplitud Mny spcs f trsina vibr-tin ar anlgus t sht vibrtin atr t twisting stissand plar mmnt f inrtia v n substutd r th shatbnding stiss and diamtral mmnt f inrtia rspc-tivly Trsinal vibrtin can b snsd y s lvl andvrtin prcptbl t tch whn smtig mdats in-tractn f th vbrtn wth th grund r gr th rcupling jaws unad ar sts tht ar usd t ar spds fpwr transmissin sysms allw th ntrctn f trsnalvbratn with th grund slidr crank mhansms n ngsd cmprssrs prvid th path t th grund i rcprctin macs If gar bxs andr rciprctn macs r

part fa

drv train, thrr, xcss nis and vibrtion nindicat trubl

Sht vibrtin critri ar usualy uantid in trms ofmtins i, displacmnt vlity r acclratin Motinsar rary a con with trsinal vibrtin hwvr bcausthy d nt act syst nctin Rthr it is th strsss thtdtrmin th structural intgrity and lif f cmponnts Tr-sinal vibrtry mtins unlk sht whirin alwys prdustrss rvrsls thrb using tigu Compnnt tlrtlss rvrsd strss than stady strss In additin strs ncntrtin ctrs assciatd with mchind mbrs rducth tivnss f ladbarin atrial ur iustrat

th twistin f a sh and th trsinal md ha of an -tric mtrprsr ytm uln n th r trinallws >r mislignn in ssmbly Nt th md shp

it shws tht th stiss f th uplng is much lss than thtf h thr sh sctins

Althugh th trsinl vibrtin shwn in Fiur 1 cursin th rm f a singl harmni cmpnent, this not usuyth cas with rl systm.Trsional vibrtion usully taks thrm f cmplx vibrtin sinl hvin any drnt -quncycmpnnts. Shck abrpt startupsand unlad-ing f g

ar teth au transint torsinal vbrtions in sm

systms, and startup f synchrnu ltri mtr systmsn aus trsinal rsnan. Rndm torsinal vibrtin frtating mhins is rlativl mmn du to ar in-curais

and ball barin dc. Slxitdvibrtin tht

s, h aditin f nrgy t sytm by its mtin unti irchsa

limit cyl r dstruts is nt cmmn in pwrtransmision and genera

ionsystms; eltrialmechanical in-tractin can brin n this prblm. Shar ilurs in shascausd by trsinal vibrtin usuy ur nar a strss cn-centrtion point such as a keyway or the pint at which theshtdiamtrchangs brupty. Trsinal iu is manstby ar war, gar tth tigu ky iu, and shrinkfslippag Nis tht ccurs whn gar tth unlad indicats asvr trsinal vrtin prblm

Allwal lvls f trsinal vibratin hav nt t bnsabshd r gnra machin systms bcaus s many vr-abls ar nvvd Allwabl trsina vibrtin lvls dpndn th amunt f stad tru bing carrid by th pwrtransmissin cmpnnt Trsinal mtins such as sh d-

ctins pr s ar nt rstritd by nctin strsss mustthus b calculatd and cmpt ilur anlysis cnductdTlrab mtins n mst systms ar dpndnt upn thstiss and rlatv twist acrss th cmpnnt: n dgr ftrsinal vibratin amplitud is usualy unsatisfactry but typ-cally an rdr f magnitud ss s satsfactry Trsinal vibr-n vs usually cat b cntrlld ad rducd y d-crasg th surc xctatn bcas t s typcally ducd byt mac funct xcptins icud gar tth inac-curacs and ngin baancng gas ad inrtia rcs Damp-g dvics can b usd t dssipat rgy; hwvr tungby stss and nrtia rarrgmt th ssm sms tb th bs mthd fr rducg trsa vibrtin


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PROCEEDINGS OF THE SIXTH TURBOMACHINERY SYMPOSIUM

MODEL

MODE SHAPE

!1 PUMP ORCOMPRESSOR

i

gure 1. Torsional Vibration an Electric Motor -Compressor Ut

This papr is an rviw f trsinal viratin tchnlgyand includs nginring guidlns r applying th tchnlgy t sling prblms Th physical natu f trsinl vibrtin i dscribd as wl as mthds r calculatin and masurnt tchnius r masurmnt and anlysis ultdiagnsis and crrctin, and imprtant dsign cnsidrtinsPractical cass inlving lctric mtrs ngins and cmrssrs r rviwd

PHYSCL SPECTS

Physiclly trsinal virtin is simir t latral vibrtin r sh twist has bn substitutd r shat bnding.Trsinl xcitatins and strss and motin mchaniss including critical spds simpl and cmplx harmnic otinstransint and slfxcitd mtins ar dscrid in this sctin

Torsoal Ectatos

Tabl shws som surcs of torsinal virtion xcittion It should b notd tht most of ths sourcs ar rlatdto th work don y th machin and thus cannot rvdor rducd Crtain construction and installation soucs ar rnout coupling unbalanc coupin mislignnt ndgar toth mchining rrrs can b rducd an or il

lr passing xcitations can charctrid y th numbr of

BLE SOURCES OF TORSOAL VBTO EXTAO

Surc

Mchncl

Gar Runu

Gar Tot Mahining Tlrancs

Cuplng Unbalac

!Hs Jon

Cuplng Mslgnmn

Sysm Funcn

Syncrnus Mr Saup

urg

cny f Impllr Scrlls

Blad Passing

Mr r Gap ccnrcy

ngn Frng Ra

PupsCprssrs and Dsrs

Mr r Turbn Drvn Syss

ngn Gard Sysms S Cuplngs

ngn Gard Syss S uplng

Amplitud% Avrag

Toru

040

1040310

510

1530

50 rgrar

runcy

X 3 X RPMN f Gar Tth X RPM

RPM X, 4 X 6 X RPM

N f driv lmntsx

RPM

2 X slip ny

Dpnds n Surg Ra

N Blads X RPM

Dpnds n ngnDsgn and OprangCndns

N ansX

RPMN ans PM

N f Pls r Blads RPM

Dpnds n ngnDsgn and OprangCndns can bn and n X PM

Dpnds n ngnDsgn and OpangCndns


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TORSIONAL VIBRATION OF MAHINE SYSTE.S 5

bads vans n th wh th quny f xitatinuas th numbr f bads tims shat sd Th amitudf uating tru is tn givn in trms f rntag favrag trqu gnratd in th systms In r oxass ngins r xaml th muti uny ompo-nnts dnd n ngin dsign and r outut Th rla-tinshi bn gas harmnis and inrtia torqu haronishas a baring n th v f torsina xitation Prssur un-

baan xitatin is a ntin f int rssur sur ndxtnt f asing distortin. T harmnis of torsina xita-tin m sug a sok nditin ar dndnt highrtn nrma prssur r a short ti and rat f rssurinras and dras.

Trsnl Rspns

Th omonnts f a tyial rtating mahin ar on-nctd ithr dirty by coupings and shas r indirty bygars, bts andor othr sdhanging dvis.

Whn sub-jd t xtatio sours ths assiv poar omnt ofinrtia and xib ompnnts absorb nd dissiat nrgyand rdu a uniqu att of torsiona mti ald r-sos Rsons is ratd to th dsign of th mahin i it is a idiation f th dions and strss to

hih thsys is subjtd Hroni nlysi n b usd to radown rsos ito ru f idivid si urvs ah sinurv has th oing rrtis tud has ith r-' to th rsons urv d quy Ths r iallow th urvs to b rrd to

h orii rsons uvPhas angs n ot obtain i hroni nlysis cssry to rostru th rpons urv

As with othr {s f vibrtio torioa vibration inahi ous in hartrsti tts ald od iur. Th od sa is dndn on torsiol stiss l-in d) o ot of inrtia bis nd dpibinsrd of th vrious oonnts orisi th sys-m Th vibrtion quny at hih th od sh ois ad th ntural quny h ntal uny oi-ds ith an xtation quny a st of rsonn xistsAt rsonan th syst di alon ontros th otionsnd strsss in th systm

It hs b notd tht oonnts of rottin hinsaus xitatios Tal l t qunis tht r ultis ofsha sd it is r this raso that virtion ods xitd by quis othr th th uny of th shaspd h a shatindd xittion orrsponds to th -quny of any mod of viration a sial ondition of ro-nn ald ritial sd xists Fiur 2 istrts thritial sd of a mahin Ddin o th xitationsours Tal , ritial sds n ou t any shaftsds, artiulary in riratin mahins Th omi-tion of multil xitations nd ultil natural qunisthr unis r shon in iur 2 rodu so ayrtial sds that it is on diult to find a s ortinasd ran. Not hovr that not a ritia sds arritia" baus of inhrnt systm daming and th v ofxitation. Although xrin with si ahins nabsth nginr to idnti iortant ritia sds it is bst tohk th rsns strsss to dtrmin whthr or not amahin must b rdsignd r saf ratin. Nt as thatth natual fruncis shon r th systm in igur 2 do nthang with main sd This is unik atra vibratin.

Th torsina vibratin rsns f machin systms isusualy ompiatd by a arg numbr f critia sds. n-ik atra mtins hwvr instabiitis and sfxitdvibratins ar rar with trsina vibratin. In sm cass t

u:a

aQ>

CRCALSPED

3rdNATURAFREQUENCY

d ATURAFREQUEY

- _s

AURAL

REUENY

SHAF OAIG SPEED (P)

Fgur Tsl Crtl Sps Sn Systs

ouing of lrhia onts suh s indutiomto driv systs an u xitd virtions

Tosional ysts ar susti\ to ilus as a rsult oftrait xittio A rid tartin quy o rquirdto rvt ovrhtin tri otor indis indus lrstss. In addition statu ith synronous sd mtordrivs an

disstrous if tsio rsoa is xitd Thsonan is th rsult of thi of h ynhrnous otorxition nd th syst torsional naturl qy Th syn-hrou otor hs lr ulati torq duin strtu.Th quny of thi ulsation is qul t ti th sli -quny d it varis o ti th in quny at tartuto zro t synhronous sd Any torsional riti sds

ithi this quy ran ay xid duin th sartuyl vn thouh th xitation is trnsint th usatintoqu y lw strong nouh to aus sha ilus r sv-rl strtu yls In xtry ritil situations iu anur durin sing startu. Sohr [l] has analyzd suh ailur and doumntd th orrtiv ation takn A transintrsos analysis ondutd prio to quimnt instaationaos dtrmination of rquird syst tiu if. Dughty[2] has ubshd thniqu to dtrmin th sd at whiha systm ust b rtd throuh ritial sd to avoidunatab strsss

CLCLTIONS

Cauations bsd on mathmatia mods whthr m-

px or simp an b f vau in dsign, dvmnt andut diagnsis f mains has bn shwn tat vibratinis dndnt n th mass asti and damping prrtisof th mahin Sourcs f vibratin cal a fring functin,inud rircating mass and bad assing. T mass andastic prprtis dtrmin th natua fruncis f tsystm and thus th critica spds. Daming in th rm fat dissipatin cntrs t vibratin vs. Damping isusuay mst ctiv at rsnan th v at whic afring funtin uas a natura frunc. Tab 2 andiur 3 shw sm f t nmncatur usd in tsinavibratin acuatins. is assumd t b much argr than that is is statinary and vibrat.


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PROCEEDINGS OF THE SIXTH TURBOMCINERY SYMPOSIU1

Jo

Fiure 3. Modelin Torsionl Virtin. Multimss mdelb mll moel

MODELING

The st iortnt artf solvin vibrtinble ithe selectin f theticl

odel; the roess is le"oelin Tw bsi errrs

re

onl ssoitwi h

the lutin f virtion pble nueril error n elin errr The bet theticl slutin nnot veroe elin errr It is iortnt to btin the best el vilble; if n prxite solutin is desire the el berewrk so tht soluion is esier rxite eths inite whe her nswers re hih r lw The ethds ue to

olve rle dictte the thetil l tht must euse

theticl slutin invlves lultin the response fthe sste to iven exittin

xcitatin -

Se phsil nsiertins re irtnt in einSstem ss ing n rin nction re bsolute vlue in cultins f vibrtion r nturl equencies nyquntit is nelete terere it ets the results Stiess is reltive untity tht is the stiesses f jent eberscn e cpre t eterine iple el If one tinessis n rer f gnitue lrer thn the cent ne it n einre Fr instne etile el f rcpressrsyste Fiue l is swn in Figue 3 If the stiess f ecuplin is n rer f nitue lwer thn te djentsht te siplie tw inerti el f Figure 3 n eused Fruls nd uielines fr delin trsinlvibrtins cn e fun in Nestries [3]

Methods Soluton

Te e used slve ny virin prble dependsn e type f prle time lltd n e ury re

uire The accuracy f he calculad respnse and f henaurl freuenc epen n the me f e pysical prble an e nuercal echniue use n e analycal del

Nuericl clcultins cn be e wit slide le eskclcultr esk cuter r hihsee iitl cputer Teethd use is electe n the bsis

prble plexit

cur reuireecnics n

tie

The t f cputer time ust e weiged inst tecot f n f curse use f cine des n cpleteleliinte he use f n sie prring nd t input/utut ust be hnle

Fr sile singleegreefee sstes Fiue 3bthe ruls iven in Tle n be used t clulte thetrinl nturl euence n respse Fr re cplte sstems Fiure 3 the Hlzer et [4] is cnluse t clulte the sste nturl euenies

Te Hlzer e h utilzes trnsfr tricies prptethe ects f ssu nd knwn bunr conditins crss

n N sttin sste mel In he Hlzer eth the inepenen vrile is equeny whic the syste eshpe is clulte the e she stise te unrycnitin the ssume euenc i nturl euenc Ineect the etrintin f nturl euenies cn be trilnderrr ress This trilnerrr rcess cn e reuedt n iterton ress b ssuing ze true t the left enf tpil trsinl del; t the rih end f the del thetrue is ltte resiul nctin Fiure 4 Becuse nlthe reltive vlues f ngulr displeent re iprtnt in thede she he islceent t e le en is ssue t bene The nturl euencie re tine b prptin theiniil vlue rss the trnsr triies the initil shaf sttin t the nl F) sht sttion s shwn seticlin equtin 1

()

Prctil sstes hve n sttion in the e; it is therere nt rtil to ultipl the trnsr tricies Instethe iitl r esk uter is pred to trnsfer theeects f the initil nditins r n ssue equencrss the el he e shpe of the syste is eterinedwhen the equency used prpagae he en coniinsarss the el is he nural equeny

The ritil sees f trsionl syste re tine th ntul equenies n the escrition f te torinlrcin henen In ur strkecyle intel coustin

enine the enine rnksht kes tw revlutins er ccleThe engine cliner ressue Pt is peridi in tw enginerelutins; e inertil ets re peridic in ne enine revlutin Te theticl reltinship [] f the peridic sand neria rue is shwn n equatin ; e fundaenalper crrespnds w revlutins f he rank

T(wt}=P(t)rsinwt [1+ y coswt]

+ m(

[

r . . 3r .2 rw sm wt-sm 2wt-sm3wt] 2


5/10


6/10

18 PROCEEDINGS OF HE SIXH URBOMCHINERY SY!vtPOSIU\1

w

:.wc T

I

F Sctc Dtt Ntul FJcs Rsdl TrJ-FJy Dia

rtor sped, and system od shs.The

modldampig

valus sed in this aalysis are a ntion f the mod shapesf t systm 1odal analysis is shwn scmatialy in Fiure

. The mdes re oupled nd the vibration cotrib-

ued m moe is alulatd as if i wer e a singl-

dg

re e o

f r

d om

syste. At


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TORSIONAL VIBTION OF MCHINE SYSTEMS 19

Figure 6 Torsional Vibration Test Facility (Courtesy ovjoy, Ic

tined rm strin gages. The rt of nulr motion is measured and interted by the unit shown in Figure 7 Th nulr displeent n be measued t the end f a sh withnders nd t i ermedia e points with a gear-magne ic pickup arrangemen shown in Figur 6 n ithr side of theupling. In t h ndl point of torsionl ibrtion deis ten at the oupin; ouplin hubs herer re exeenlctions r msurin torsionl ibrtion. Ntul equenyand mode shape lcultins hould

be made eore

snsors aremountd t oid plaing the pikup t node. Trsionl ibrtin n e mesued t many intermdite points in a systeso lon s snsor n e pled on gear r some otherprtrudin emen . On enginer wrapped haksa bldund sh ollr

The brod b a n d trsionl ibrtin signal must b enlyzd r individul hrmni mponnts. Waverm

analysis fr torsional vibratin problms is more dicult thanr latral vibration problems. In fct, a waeanalyzer is neces-sary r diagnosti work on tsionalibration problems.Fig-ue Sa showsa torsional response signture f the test cility(Figur 6). The engine was runing at 2,000 rpm under

minimum load. The vibration responses to the ne-half rdrand sondrder engine exitatins are evident at fequenciesof 16 Hz and 666 Hz respectively. The 33 H rst ordersignal is dueto mass unbalance in the system. The analog signalin Figure S is an instantaneos trqu measurement. Theomplex signature rects thedifult waverm analysis. Thissignal an b proessed in a trsional vibratin analyzer unitshown (see Figue7) to tain its harmnic ontent.

DESIGN CONSIDERATINS

The llowin riteri re usully onsidered in th desinof poer trnsmission systems: ntion structurl intritysty li ost nd mintinbilty. Torsionl ibrtion is aftor in strutual integrity (peak-lad capability), l ftiguedue to trsional stress reversals), a d cost. Complex ibratindampers or absorbers can be ostly. Des ig criteria fr tor-sionl vibrtion re untied in terms of stresses: both thestrsses ttributbl to norml peklod mhine unction nd

he strsses risin om torsionl ibrtion must be onsid-erd.

rocedur

s

dsinprodure

>r torsionl ibrtio nlysis n bebased on measuemn s or mthmtis (se Fiure 9. Thesys m mponents i., mo r (drive elment) nd om-prssr pump, o vhi (driven elment) r seletd nth bsis of nti. The seletion of intrmedite mponntssuh s upns blts and gear boxes is bsd n nrml ldpity (depending on the typ f euipmnt srie tris inluded).

If he sys em is to be analyzd thmtily a mdl isprprd (see Fiur 3. Th inrtis nd stisss f l

Figure 7. Torsional Viration Facility Control an d Instrumentation Panel ourtesy of Lovoy Inc.).


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TORSINAL IBRTION OF MACHINE SYSTES 21

ompoet li. If eessary, components are modied to

lowerstresses. Suh modicatio anbe simple orcomplcated

dependig onthe desig constraints. Itis not always possi-

ble to add inertia or hage the stiess of a ompoent to

orret aprblem; i suh ases eitheradamper which dissi-

pates eergy asheat or a absorer hich manages orrear-

rages theappliation ofenergy, must be addedto the sst

These predures ar usually costly but oe necessary

Cas Hisry a Mr-Drivn Cmprssr

Aniductiomotor drive reiprocatinompressor sys-

temwa to bedesigned to operateat three speeds: 1,170 rpm

1, 470 rpm, ad 1 70 rpm Sigicant exitation ocured in

the reciproati ompressor (Table 3) at 2 4, ad 8 times

runispeed. Tble4 show sdtr the modein Fiue 3.

The stiesso f the oriinl oulig isvery much lower thn

other stiessvlue ithe system. Itis r this reso tht

th mode l be siplified s shon in Fiure 3b the rst

turl equey n be lculted usin equtio (6 The

inertia valuesr the model ar the sum of alinertias oneah

side of the exible elemet, the uplig The smpld

model has the llowig haratests:

]

301 lb s mor 000 lin/d oupln)

] 09 ln s (ompressor

fn 396 Hz

6

Th

ntulequenyof 39.6 Hz i oseto the 39z -

itationned om he seod orer virionexction t

1,70 rpm(le3). Therelve torion vn mued

cro hi coupling opion of th ni t , 170rp

w2.26 deree. Muliplctionof he orionl viriony

heouln te gveavlue Jr viroryorue of250

lnTheek lod orue reuireet out 4000 ln(Jrthunt and the coupln lodrtn 7, 000 !in; oth

he nturl equency clulton nd th reone meure-

met showed tht the coupln would

stir oupli (k = 349 370 linrd ith slihtlylrer rtin (8400 !bin s used ote tht the simplenturl equny rmul is still lid beuse the ouplinstiess is n order of mnitude less thn ny other stess i

TA BL E 3 MOTOR-COMPRESSOR FORCINGFREQUENCIES

rSpeed Frueny

rpm) rder Hz

1 70 8 233

4 17

2 58

1470 8 196

4 98

2 49

1170 8 156

4 78

2 3

the system This oupli hd hlf ierti lue of 47 lbise The lulted turl euey s 91 H lose tothe 98 H exittio otined om the uth order t 1470rpm Mesuremets reeled 15 deree reltie torsiolibrtion ross the oupli hih oresponds to ibrtorytorue of 914 lbin The pe torue s thus redued to 4914!bi hih proed to be stistory for mhie opertio Itis oten diult to selet oupli hin both the properlod rti d stiess tht permits opertio t umber of

speeds or r re of speeds

FAULT DIAGNOSIS AND CORRECTION

The solutio of torsiol ibrtio problem is sometimesobious more ofte it is ot This setion desribes methodsused to dinose d orret torsiol ibrtio problems

Diagnsis

The ult dinosis tehniues used to study torsiolibrtio problems re ot unlie those employed i lterlibrtion Some uderstndin of the phenomen ssoitedith torsionl ibrtion is essetil model suh thtshon in Fiue 3 oen proides isiht ito problemKolede of exittio equeies nturl frequeies nd

mode shpes n be pplied to the dinosis nd re espellyhelpl i lotin sensrs hen mesurements re eessryto obtin dt In mny ses torsionl ibrtio must

be

mesured t poit remote om here the fiure urrd It istherere usel to o the mod shpe so tht the test luesn be used to lyz the ilure ode of the uit

Ccion

ost torsionl ibrtios rse buse the system hs lre number of ritil speeds Problems re orreted byhin the ritil speed of he system so s to lter itstosiol respose Critil speeds re hed preduelled tui by lterin stiess or inerti I t is osiolyeessry to dd dpin d bsrptio deies to the systeto redue torsionl stresses mpin deies lleite stressby their motio d re thus more etie whe used isystems tht operte er or t ritil speeds Dmpi deies should e used spily hoeer beuse they reostly Corretio by reduin exittion is prbbly the bestorreion method i eerl but is the lest etie r torsiol ibrtio pblems The reson is tht most torsionlibrtion oriintes in mhine ution Torsiol ibrtioorretion therefore ioles modifition of the system rtherth redution of exittio

Cas Hisory f an Engin Crankshaf Failur

A smll soln eie ws diretly oetd to rep-rti ompressor Whener wo reiprtn hnesre oneted morethnusul ttentio should be pid to torsl brtos ths se rdom rsh lures ereour i the eie ritil speed lysis f the systembee t ws ssembled hd idited tht the eie s notu e ritil speed o respose lysis hd beeduted t tht time hoeer so it ws the st step i thedss the brti poblem mdl lysis f thesystem ws duted usi model of the eieompressosystem [] lysis f the stresses i the shft reeledtht ml stess leels (iure 0 uld y s muh s5% depdi o the etti f th le betwee theee d the mpress shfts dm ul ett ued beuse ti oll hd bee used t et the uits ths ws the soure of the dm rshtflues The pblem s sld by pl the shfts du


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22 PROCEEDINGS OF THE SIXTH TURBOMACHINERY SYMPOSIUM

BLE 4. MOOCOMPESSO MODEL

Cpn CmpnN. Dsrpn

0 Mr Araur1 CuplgHalf2 CuplgHalf3 Cmprsr Cyldr4 Cmprssr Cyndr Cmprsr Cylndr6 Cmprssr Cyldr

2

i19

Ua1

16_

machine systems torsional vibration of ronald l. eshleman.pdf

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