Steam Turbines - Cambpell

GER-800

PROTECTION OF STEAMFROM AXIAL

TURBINE DISKVIBRATION

By

Wilfred Campbell

TANGENTIAL VIBRATIONOF STEAM TURBINE BUCKETS

By

Wilfred Campbelland W. C. Heckman

Reprintedfrom

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

PAPER NO. 19~2OAND 19~5

GENERAL ELECTRIC

WHEELS

CONTENTSPageNo.

PART I Purposeandscopeof the investigation 7

PART II Expositionof thenatureandtheoryof vibration in turbine wheels. . . 17

PART III — Method of designand testing for the protection of turbine bucketwheelsfrom axial vibration 43

DIscussIoN---~Protectionof steamturbine disk wheels 71

TANGENTIAL VIBRATION OF STEAM TURBINE BUCKETS 77

DISCUSSION— Tangentialvibration of steamturbine buckets 85

WILFRED CAMPBELL

WILFRED CAMPBELL

Wilfred Campbellwasborn in Manchester,England, in 1884 andservedan apprenticeshipwiththe Lancashireand Yorkshire Railwayfor sevenyears. During the latter part of his apprenticeshiphewasselectedas oneof two to attend the ManchesterMunicipal School of Technologywhere hereceivedseveralprizesof distinction for his ability in design. Later he was employedas a draftsmanby the NortheasternRailway Company.

In 1907 he cameto America,and for a year was assistantto the mastermechanicat the ArnoldPrint Works, North Adams, Mass. In 1908 he was employed by the General Electric Companyas adraftsmanin the Direct-currentDepartment. During the warhe wasassociatedwith Mr. C. E. Eveleth,now managerof the SchenectadyWorks, and Dr. Irving Langmuir of the ResearchLaboratory, inconnectionwith the design of submarinedetectorsby the GeneralElectric Companyfor the UnitedStatesGovernment.

Since 1919 Mr. Campbell has beenan engineerin the Turbine EngineeringDepartmentof theGeneral Electric Companywhere he has had chargeof the investigationof turbine vibrationsof allsorts. In recognitionof his outstandingcontributions to turbine design,he recentlyreceivedanawardfrom the CharlesA. Coffin Foundation.

The resultsof this exceptionallyvaluable researchwerepresentedby Mr. Campbellat the springmeeting of the American Society of Mechanical Engineers,May a6 to 29, 1924. The paper wasreprinted in full in the GENERAL ELECTRIC REVIEW, June,July, and August, 1924.

Mr. Campbell’s untimely death occurred at Schenectadyon July 7th, after anattackof acuteappendicitis. The loss to the turbine art will be appreciatedby all who are familiar with his work.The breadthof his scientific knowledge is well illustrated by this paper, althoughit was only one ofmany interests. His ready ingenuity was supported by a keen intuition which was invaluable toan engineer of his type. His genial good nature and unfailing optimism endearedhim to all hisassociates. His integrityand fearlessnesswon the confidenceof turbine operatorsall over the land toanunusualdegree. His intenseinterest in all modern inventions gavehim confidencethat no problemwasbeyondsolution.

FrancisC. Pratt, vice-presidentof the General Electric Company, in charge of engineering,incommentingupon Mr. Campbell’sdeath,made the following statement:

“Mr. Campbell’s suddendeath was a greatshock to his many friends and associatesin thisCompany. At the time of his death he was the engineer in chargeof investigationsof turbine wheelvibrations. Whenhe undertookthis importantwork in 1919, it wasthought by someengineersthatthe limits in the size of individual turbine units had been reached. Mr. Campbellhada raregeniusfor analyzingandsolving abstruseproblems of a mechanical nature and did not sharethese adverseopinions regarding the limitations in the size of steam turbines. He discovered the cause of thetroublewith largeturbine wheelsandbuckets, devised remarkablyingeniousmethodsof detectingandmeasuringthe phenomenaconnectedwith them and provided means for overcomingthem. He hasmadea fine and lastingcontributionto theindustryandhis work will be long rememberedby engineerswho designandoperatesteamturbines.”

The Protectionof SteamTurbine Disk WheelsFrom Axial Vibration*

By WILFRED CAMPBELL

TURBINE ENGINEERING DEPARTMENT, GENERAL ELECTRIC COMPANY

Preparedwith the co-operationof A. L. Kimball, Jr., ResearchLaboratory, and E. L. Robinson,Turbine EngieeeringDepartment; both of General Electric Company

PART I

Oneof themost important featuresin thedesignand manufactureof a steamturbine is the elimination ofthe possibility of vibration occurring at the various natural frequenciesof its disk wheelsand buckets. Toaccomplishthis purposethe GeneralElectric Companyhasfor severalyearsmadean extensiveinvestigationofvariousforms of vibration and waveswhich mayexist in steamturbine disk wheels. Oneof theresults wasthedeterminationof the lawsof dangerouscritical speedsthat must heguardedagainst,togetherwith other minorresonantconditions that it is advisableto avoid. Testing machineswith completeauxiliary apparatusweredevelopedfor theverification of predictedfrequenciesandcritical speeds,and a programfor theroutine testingof turbine wheelswas instituted. This resultedin thedeterminationand adoption of theprocedure,necessaryin all cases,for the definite protection of steamturbine bucket wheels from axial vibration, as justified byseveralyears of successfulmanufacture—AUTHOR.

PURPOSE AND SCOPE OF THE INVESTI-GATION

The purposeof this article is to presentthemain features of the work done by theGeneralElectric Companywhich led to thesolution of the problem of vibration ofturbine disk wheels, andto describethe waywheels are designedand tested in order toinsurefreedomfrom vibration.

The investigationwas undertakenin orderto account for wheel failuresof a peculiaranderratic nature which could not he explainedon the basis of high stress alone. Thenumberof failures was small consideringthetotal number of wheels in operation. Thesefailtires were not confined to any single typeof machine,hut they did showa preferenceingeneralfor thin wheelsof largediameter.

That this difficulty has actually heenovercome with no major alteration in theturbine is emphatically brought out byresults obtained by the General ElectricCompanyin the past three yearsfrom its useof disk wheelsproperlydesignedandtested.

Up to the end of 1923, this company hadmanufacturedand installed over 9000 steamturbines aggregating in total generatingcapacity more than 15,000,000 kw. Thisinvestigation is chiefly concernedwith largesize machines, that is, of over 5000 kw.capacity. Before the year 1919, the GeneralElectric Company had manufactured andinstalled in operating plants a total of 227

*This article was presentedas a paper at the Spring meet-ing of she American 5ocicty of Mechanical Engineers, May26-29, 1924, at Cleveland, Ohio.

turbinesof ratings exceeding5000 kw. eachThe total generatingcapacity representedbythese machines was over 2,404,000 kw.Since the year 1919, when this particularinvestigation was started, there have been206 more turbines installed exceeding 5000kw. each, increasing the total of generatingcapacityof thelargersizeturbines to 5,864,500kw. at the end of 1923.

TABLE IFOR TURBINES OF OVER 5000 KW. IN-

STALLED BEFORE MARCH 1, 1924Numberof wheels installed 4399Numberof wheels tested (standing) 3596Number of wheels rotated in wheel testing

machine 320Numberof testsin wheel testing machine 405Numberof wheels testedin customer’splants

(standing) 1683Number of machines investigated in custo-

mer’s plants 291Number of machines tested under load in

customer’splants 24Number of wheels replaced to avoid possible

trouble 497Numberof wheelstunedfor vibration 212

Table I shows the magnitude of theinvestigation, giving a few figures as to thenumber of machines and the number ofturbine wheelsinvestigated. The capacityofthe testing machinesnow in operationis suffi-cient to provide for the testing of 600 wheelsannually, under all conditions of speed.

That the methods developedare effectiveis attestedby the successfulelimination of allserious wheel and bucket troubles due toaxial vibration from the operation ofrecentlybuilt turbines.

7

PROTECTION OF STEAM TURBINE DISK WHEELS PART I

TRAVELING WAVE IN A THIN DISK MODEL OF A TURBINE WHEELHigh speed motion picture of a 4-node wave traveling backward one-half revolution in the wheel which is clamped

stationary. Alternate exposures have been omitted in this reproduction

8

PART I PROTECTION OF STEAM TURBINE DISK WHEELS

Fig. 1. Broken Turbine Bucket Wheel 9th Stageof l5000-kw.,1800-r.p.m., 9 stageTurbine. This was caused by axial

vibration of a type which has been eliminated by theinvestigation describedin later parts of this article

Fig. 2. Detail of the Fracture in Fig. 1, Showing CharacteristicFatigue Failure

Fig. 3. Fatigue Crack which Started at a Hole Fig. 4. Small Turbine Wheel Broken by Accidental Over speeding,2nd Stage of 500-kw., 3600 r.p.m., 3-stageTurbine

9

PROTECTION OF STEAM TURBINE DISK WHEELS PART I

HISTORICAL OUTLINE

Beforediscussingin detail thenatureof thevibrations to which disk wheelsaresubject,abrief narrative of the work of investigationwill be presented. This seemsnecessaryinorder to give a proper perspective of theproblemasa whole.

Growth of CapacityThe design of turbines with one row of

bucketsperwheel took placelong prior to theentry of America into the war. Thesedesigns used higher linear bucket velocitiesandwere producedin unprecedentedquanti-ties during the war period. The rapid

7

increases of turbine capacity which tookplace at the sametime were in alarge degreeaccomplishedby the use of larger diametersintroducedto give greaterbucketspeeds,andby using longer buckets. The mechanicalpossibilities were pushedto the limit. Whilesomeimprovementsin steamconditionscameat the sametime, theimportant thing to noteis that the real period of increasedcapacitydue to improved thermal processesoccurredlater and is still going on whereas,almost atone leap, the early designswere pushedto thelimit from the point of view of structuralstrength.

Severalprinciples of design pointed in thedirection of light wheels. The maximumwheel stressis at the bore and this could bereduced by using lighter and thinner diskshaving less centrifugal bursting tendency.In fact, thesewere cut down in thicimessasmuch ascould be without creating a newmaximum in the web due to the pull of the

Fig. 5. Fatigue Failure which Did Not PassThrough Holes;11th Stageof 30,000-kw., 1500-r.p.m., 12 stage Turbine

Fig 7. Turbine Wheel Failure as a Result of FatigueBending; 3rd Stage of 6000-h.p. Turbine

Fig. 6. Detail of Fig. 5, Shoa~ngCharacteristicFatigue Failu,e

10

PROTECTION OF STEAM TURBINE DISK WHEELS

buckets. And not only did the desire forconservative stresses point toward lightwheels, but also the desire for a stiff rotor.Heavy wheels are accompanied by lowercritical shaft speeds. The unquestionedadvantageof a stiff shaft, when possible,alsodictated light wheels. These various influ-enceswereperfectlynatural at the time. Thesubjectof vibration hadnot beenbrought toprominence. It washazyand uncertainandno difficulties had been experiencedon thisaccount. The plain path of reasonseemedtobealongthelines indicated.

Types of FailureIn order to visualize the sort of difficulty

which led to the presentinvestigation, it willhewell to examineanumberof failures.

Figs. 1 and2 showa breakwhich originatedin a small tappedhole anti passedthrough alargesteam balancehole. This was a vibra-tion fatigue failure. An examination of thefractured surface shows the characteristiccentral line and progressivecurves. Fig. 3shows anothercrack discoveredbefore com-plete rupture in the samewheel.

Fig. 4 shows a wheel which completelyburst. Note that the line of fracture haspassedthrough every one of the holesin theweb of the wheel. This wasdue to accidentaloverspeed

Figs. 5 and6 showa typical fatiguefracturewhichdid not originateat ahole.

Fig. 7 shows a break in which cracksoriginatedat more than onehole. Figs. 8 and9 show details of this fractureat eachof thetwo holes. Although Fig. 7 bearsa resem-blancein its completenessof failure to Fig. 4,the type of fracture is entirely different andcharacteristicof afatiguefailure.

In other casesthe only damagewas loss ofbuckets. Fig. 10 showsa wheel from which anumberof bucketshavebeenbroken. In thiscasethe breaks were due to axial vibration.

In Fig. 11 various failures occurring in thebucket dovetail al-c shown. In eachcasethemarking indicates vibration in an axialdirection.

Fig. 12 shows another class of turbinetrouble in which a diaphragmhasbeenscoredin two diametrically opposite spots by arubhing wheel. Fig. 13 is a section of thesame diaphragm at the point of greatestrubbing together with a profile of the wheelshowing the shapetakenby all of the buckets.

Theseexampleswill selve to illustrate thevarious types of failure. Most of them wereplainly due to vibration resulting in fatigue.A few were clearly the result of accidentaloverspeedsuchas Fig. 4. In certain casesthefractures avoided holes, but in general there

PART I

Fig. 8. Detail of Fig. 7, Showing How the Crack Started at One Hole

Fig. 9. Detail of Fig. 7, Showing How the Crack Started at Another Hole

11


was a distinct affinity for holes owing un-doubtedly to the higher localized stressesaround them.

Hole StressesThe first remedialmeasureusedwasto give

immediateattention to the localized stressesin the neighborhoodsof the steam balanceholesand other discontinuities in the webs ofthe wheels.

It is not the objectof this article to discusshole stresses,but it is necessaryto note thatthey becomeserious only in connectionwithvibration, by constituting a place for afatigue crack to start. If a wheel is properlyprotected from vibration, there will be norepeated stresses at any point Ratherelaborate experimentshave shown that thereinforcementof holes will serve to increasethe resistanceof a wheel to vibration in caseit should be necessaryto design with theexpectation of fatigue stresses. But thegreater safety lies in guarding against thestresses themselves by the precautionarymeasuresdevelopedfor usein manufacture.

Bucket LacingThe early experimentson the vibration of

bucketsindicatedthat alacingwire parallelingthe shroud band and connecting differentgroups of buckets would remove varioussecondary vibrations. Although the funda-mental period was not much influenced thisexpedient was actually used in a number ofturbines. The lacing wire, however, failedto removethe causeof the troubleandbecameitself anotherhazard. During these experi-mentsa taut wire attachedto the endof eachbucket raised the frequency thus illustratingthe similar effectdueto centrifugal force.

The Idea of Wave MotionAbout the same time, certain types of

vibration of standing wheels were investi-

PART I

Fig. 10. Rootsof Buckets Broken by FatigueBending Due to Vibration; 12th Stageof

10,000-kw., 1500-r.p.m., 12-stageTurbine

7th stagebucketbroken at neckof dovetail12,530-ksv.. 1500-r.p.m.,8-stage

6th stagebucket broken at neck of dovetail12,500-kw., 1800-r.p.m.,9-stage

9th stagebucket broken at neck of dovetail 18th stagebuckets brokenat root of blade and at neck7500-kw.. 1800-r.p.m..S-stage of dovetail. 20,000-kw., 1800-r.p.m.. 23-stage

Fig. 11. Details of Bucket Failure Due to Axial Vibration

12

PART I PROTECTION OF STEAM TURBINE DISK WHEELS

gatedby meansof sand pictures. The usualform was a series of vibrating segmentssymmetrically arranged about the circum-ference and extending into the web butseparatedby radial lines of quiet called nodalradii or nodes. These will he discussedindetail later.

Windage TheoryIn connection with the study of strains

about holesin wheel webs, a seriesof India

nlbber wheels as shown in Fig. 14 had beenmade andphotographedby meansof instan-taneouselectric sparkswhile rotating at highspeed. In order to produce representativestressesthroughout the rubber disks, metalweights were attached about the circumfer-ence, shapedto simulate the loading due toturbine bucketblades.

In running thesewheels it was discoveredthat above certain definite speeds their

circumferences developed a form of wavemotion as shown in Fig. 15. This wasexaminedby meansof an intermittent sparkeither synchronizedwith thespeedof rotationor adjustedto occurat slightly greateror lessfrequencies. The shapesof the waves andtheir rates of progresswere thus examinedvisually, and it was found that thesewavesprogressedaround the wheel in the directionof rotation, but at a less speedthan thespeedof the wheel, that is, relative to the wheel

itself the wave was travelling backward,seeminglydriven by thewindageencounteredlike thefluttering of aflag.

This gave rise to the so-called windagetheory that the waves were developedanddriven backward in the wheel by the atmos-pherein which it wasrevolving. Investigationof this theoryled to the rotationof paperdisksin a vacuum. The wave motions, clearlyobservablein an atmospheredisappearedas

Fig. 12. Turbine Diaphragm which has Suffered Rubbing in Two DiametricallyOpposite Regions; 17th Stage of 30,000-kw., 1800-r.p.m., 17-stageTurbine

Fig. 13. Drawing Showing Depth ofRubbing at Worst Parts of Fig. 12.All ofthe buckets on the wheelwererubbed alike to the shape shown

13


the atmosphere became rarefied. rfheseexperiments were immediately extended toexceedinglythin steeldisks which were foundto behave in a similar manner. Since thewaves could not exceed the speed of theactuating wind without encounteringresist-ance,they could only be supportedin wheels

whose natural frequencies corresponded towaves travelling at less speedthan the speedof rotation. It was therefore made a condi-tion of design of turbine wheels that thenatural wavespeedshould exceedthe speedofrotation whereverpossible. This resultedin ageneralthickeningof all wheelsbeingdesignedfor turbines constructedat that time to givethem agreaterrigidity to withstand the sup-posedlydetrimentaleffect of the wind action.

Windage Theory AbandonedOscillograph coils were placed in two large

turbines and during operation completesurveys of these machines showed thatseveral stagesdeveloped wave phenomenaof the same general characteristics. Tur-bine wheels were thus shown capable of

supporting travelling waves. In theseturbines the waves appeared only whenthe turbine carried more than a certaindefinite load and died out when the loadwas removed. However, waves were foundto be travelling in the wheel in a direc-tion opposite to the rotation of thewheel and at a higher speed, whichcould not be explained by the windagetheory.

PART I

Fig. 14. India-rubber Wheels Used to Examine the Strains d.oout Holes in the Web

14

PROTECTION OF STEAM TURBINE DISK WHEELSPART I

Wave PhenomenaRecords Made from a Thin SteelDisk

The first oscillograms taken in which a1-evolving coil was used were made with athin sheet metal disk, Fig. 16 is typical ofthe type of oscillograph record obtained.The upper curveA is takenby acoil station-ary in spaceandoppositethe rim of the disk;it shows that there is a wave motion in thedisk. The smoother portion of the curvecorrespondsto the part where the wheel diskis most remote from the coil, and the moredisturbed portion indicates that the wheelrim is in close proximity. These moredisturbed portions, which may be calledbeats, occur at a much lower rate than thespeedof rotation so that the correspondingwave crest producing them must haveprogressedaroundin thedisk itself.

CurveB is takenby a coil madeto revolvewith the wheel but mounted on a separatearm. When this coil passednearthe support-ing pillow blocka voltagewasinducedwhichmakes the long narrow lines in the curve,there being oneof thesefor eachrevolution.The V-shapedpoints in this recordshowthata transverse motion is taking place in thewheel disk, resulting in changeof clearancebetweenthe disk and the revolving coil. Itcan he shown that this is not in any sensesynchronous with revolutions but simplyrecords a transversevibration of the wheelitself.

Curve C is taken from the 40-cycle alter-nating-current line and is here used as atiming wave. Thesmall interruptionsin thiswave are made by a contact on the shaftcarrying the model steel disk and they corre-spond therefore to revolutions of the disk.

The recordshown in Fig. 17 wasmade by athin steel disk revolving slowly. This diskwas vibrated by means of an alternating-current magnet carried on a rotating armat the samespeed. The upper record A wasobtained by a fixed coil while the lowerrecord B is a 40-cycle timing wave. Theirregularities in the record obtained by the

Fig. 15. Photograph by an Instantaneous Spark of anIndia-rubber Wheel Exhibiting Wave Phenomena

at a Speedof 650 r.p.m.

Fig. 16. One of the First Oscillograms Recording DiskVibrations. Trace A was recorded by a stationary coil.Trace B by a coil revolving with the disk, and Trace C is a40-cycle timing wave. Model disk wheelofthin sheetmetal

Fig. 17. Record of 6-Node Vibration in a Thin Steel DiskExcited by an Alternating-current Magnet Revolvingwith the Wheel. Trace A was recorded by a stationarycoil. Trace B is a

4Ocycletiming wave

15


TABLE II

EARLY SUMMARY OF WHEEL AND BUCKET TROUBLES

PART I

The methodsby which such troubles have been eliminatedare describedin Parts II and III of this article

stationary coil are causedby the passageofthe rotating exciting magnetof which, it willbe noted, the electrical frequencyis one-halfthe mechanical frequency. This record,briefly, shows wheel vibration of the 6-nodetype including both forward and backwardwaveswhichwill beexplainedin the nextpartof thisarticle. This recordis reproducedin thefifth edition of Stodola’s “Steam Turbines.”

Recognition of Wheel Critical Speeds

SubsequentlyTable II wascompiled whichdefinitely establishedthe importance of thewave stationary in space, th~j~~~avepro~~jn backward in tj~j~eL~L.tj-~e~twhi~eirotatesforwar~

~alledawheelc”~ Inmaking astatementof theimportanceof thisphenomenon,it should not be assumedthatit is the only type of wheel vibration of aseriousnature. This type of vibration causesby far the largestnumberof failures, in fact,so large a fraction that breaks causedbyothertypesof vibration may fairly be treatedas exceptional.

At this time the importance of obtainingtest data on full-sized wheels under actualoperating conditions was first fully appre-ciated. This resulted in the design andconstruction of the first wheel-testingmachine.

Rating

35000—1500--2015000—1800—915000—1800—95000—1800—73000—Variable—4

30000—1500—12

20000—1800—12

30000—1800—1730000—1800—1745000—1200—217500—1500—85000—3600—5

30000—1800—1730000—1800—1715000—1800—2310000—1800—935000—1500—2230000—1500—20

Stage

199933

11

10

13172122

1112218

1714

Date TroubleOccurred

191819181919192019211921

119171919

1192019181918191819191919192019201920192119211921

Trouble

WheelWheelWheelWheelWheelWheel

Buckets

BucketsBucketsBucketsBucketsBucketsBucketsBucketsDovetailBucketsBucketsBuckets

Nodes

444664

8

84464864868

Backward OperatingSpeedof Wave Speed

R,P.S. R.P.S.

25.6 2527.9 3029.5 3028.5 304S.1 4825.2 25

30.2 30

30.1 3032.6 3019.8 2025.4 2557.4 6029.6 3028.8 3029.5 3030.6 3024.2 2525.9 25

16

PART II

EXPOSITION OF THE NATURE AND THEORY OF VIBRATION

IN TURBINE WHEELS

To illustrate standingvibrations in turbinedisk wheels, the following method wasused.A turbine wheelwasmountedin a horizontalposition on a stub shaft. An electromagnetwasclampedwith its polesclose to the edgeof the wheel. Onpassingan alternating cur-rent through the coils of this magneta seriesof pulls was exertedon the wheel tending todeflect it in adirection transverseto theplaneof the disk. The frequency of these pulls istwice the frequencyof the alternating currentused becauseevery complete electric cyclecorresponds to two current pulsations inthe magnet, and an electromagnetexerts apull when current flows in either directionthrough the coil. The alternating-currentgenerator was driven by a variable-speeddirect-current motor by meansof which thefrequencyof themagnetpull could be variedover a wide range.

Sand PicturesSandwasscatteredover the wheel surface

andthe frequencyof the magneticpulls wasvaried until a particular frequency wasreached at which the wheel responded.Fig. 18 showsa casewherethewheel vibratedin four segments. In this vibration eachsegmentspringsup anddown, scatteringthesandover to the quiet or nodal zoneswherethere is no up-and-down motion. If, how-ever, the frequency of the deflecting pullsof the magnet is altered evena very smallamount, the vibration immediately dies out,althoughthe magnitudeof the impulsesof themagnetremainsthe sameasbefore. On rais-

ingthe frequencyof themagneticpullsanotherpoint is found at which the wheel responds.It vibrates in segments,as before, but withSix nodalradii or nodesequallyspacedaroundthe wheel circumferenceinstead of four.

Figs. 19 and20 illustrate a 6-nodevibrationand showthat its location is not necessarilydependenton the position of a seriesof sym-metrical discontinuitiessuchas the steambal-ance holes. Not only does the disk wheelrespondwhen the pull frequencycorrespondsto four or to six radial nodes, but it mayrespondreadily to frequenciescorrespondingto 8, 10, 12, or evena largernumberof nodes,the number of nodes always being evenbecausefor everysegmentwhich springsup-ward during a vibration, the segmentnext toit on the other side of a nodal line mustSpring downward. These photographsillus-trate the caseof a small wheel with shortbuckets.

Figs. 21, 22, 23 and24 showcasesof 4, 6, 8and 10 nodalvibrations for the caseof a diskwheelcarryinglongbuckets,thetotal diameterof wheel andbucketsbeingover 8 ft. Thiswheel wasphotographedwith a layerof paperon the bucketsto hold thesand. In the casesof four and six nodes it is seen that theregionsof amplitude largeenoughto movethesanddo not extendso deeplyinto the wheel asin the wheel with short buckets, while inthe casesof eight andtennodesthesandfigureis confinedto the bucketzoneentirely.

Thefollowing generalobservationsmay bemade on this type of vibration in which seg-mentsaroundthe edgeof thewheel springup

17


and down, being separatedfrom eachotherby radialnodal lines:

(1) Every disk wheel respondsreadily tovibrations of four, six, eight, etc.radial nodes,eachtype of vibrationhaving its own characteristic fre-quency.

Fig. 20. 6-Node Sand Picture Showing Independence ofPattern from Hole Location. Compare with Fig. 19

(2) The higher the number of nodes thehigherthe frequencyof the vibrationand the less easily is the vibrationexcited.

PART II

(3) The higher the number of nodes themore difficult it is to force the sandfigurestowardsthecenterof the disk.

(4) Both the disk wheel and the bucketsvibratetogetheras a continuousdiskand mustbe treatedas aunit in thistype of vibration.

Vibrations may also take place with twonodes, as will subsequently be discussed.This type exerts a couple on the shafttransverseto its length, while the typesdescribedare balancedin their reaction onthe shaft.

Fig. 18. 4-Node Sand Picture Made by Vibration of a Wheel Fig. 19. 6-Node Sand Picture Made by Vibration of a Wheelwith Short Buckets with Short Buckets

Fig. 21. 4-Node Sand Picture Made by Vibration of a TurbineWbeeLwith Long Buckets Covered with Paper. The

Active Region Extends into the Wheel

18

PART II PROTECTION OF STEAM TURBINE DISK WHEELS

Many other types of vibration exist, in-cluding concentricring nodes and combina-tions of ring and ~ A hybridform resulting from a combinationof six- andtwelve-noderadial types is shown in Fig. 25.These types of vibration are not readilyexcitedanddo not enterinto this discussion,

Fig. 24. 10-Node Sand Picture Made by Vibration of a TurbineWheel with Long Buckets Covered with Paper. The

active region is confined to the buckets

becausethey havenot beenfound to be thecauseof serioustrouble.

Effect of Centrifugal Force on Vibration Frequency

After the naturalvibration frequenciesof aturbine disk wheel when not rotating aredeterminedas described, a question whicharisesis the effect upon thesevibration fre-quenciesof the rotation of the wheelat high

speed. The frequency of a given type ofvibration is determined by two factors,(a) the stiffness and (b) the massof the vi-brating body. Thestiffer the bodythe fasterit vibrates, and the more massiveit is theslowerwill it vibrate. Now centrifugal forcehas no effecton the massof the wheel,but it

Fig. 25. ComplexSand Picture with 12 Nodesat the Edge and6 NodesNear the Center Made by Vibration of a Thin

Steel Plate. This is a rare type of motion

has a powerful stiffening effect. This forceacting radially outwardaround the edge ofthe wheel stiffens it and raisesits vibrationfre~uency. This may be compared to theraising of the vibration frequencyof a kettledrum by drawing the membrane outwardaround the edgesby the tightening screws.Thereforeit may be inferred that centrifugalforce raisesthe naturalvibration frequenciesof a turbine diskwheel.

Fig. 22. 6-Node Sand Picture Made by Vibration of a Turbine Fig. 23. 8-Node Sand Picture Made by Vibration of a TurbineWheel with Long Buckets Covered with Paper Wheel with Long Buckets Covered with Paper

19

PROTECTION OF STEAM TURBINE DISK WHEELS PART II

It is well known that a particle of massmwith an elasticsupport, of suchstiffnessthata force R5 is required to produceunit deflec-tion, will have a natural frequencyof vibra-tion, f.~,expressedby

is = ~ I ~y

2ir ~ mlithe sameparticle is supportedin another

manner with an elastic factor R~,its newfrequencywill be - -

I ‘Rfc=

l~7

’Z~~fl’4

l

Fig, 26. 4-Node Standing Vibration. The diagram representsthe developed edgeof the wheel during three

successivephases

Now when both stiffnessesact at oncethefrequencywill be

1 /R~+R~fr~ m

Suppose R5 to represent the stiffnessfurnishedby elasticsupportsandR~thestiff-nesscontributed by centrifugal effects. As-suming the latter proportional to the squareof the speed,N5, in revolutions per second,this proportionality may be expressedby theuse of an arbitrary coefficient B defined bythe relation

R, = B (4~.2mAT5

2)

Making useof this relation andeliminatingR5by theuseof equation(1) the frequencyofthe particle, fr, due to the combinedeffectsofstiffnessand rotation mayhe written

f = ~J2 + BNS2 (5)

This formula, herederivedfor the caseof aparticle, hasbeenjustified many hundredsoftimes for usewith a completeturbine bucketwheel by actual measurementas describedinlater sectionsof this article. Stodola’ arrivedat thesameconclusionon theoreticalgrounds.

The speed coefficient, B, varies with thedesignof the wheel andthe type of vibration.If thevibrating sectorsextenda considerabledistanceinto the wheel so that the deflectioncurveextends well toward the wheel center,B hasa lower value than when most of thebending of the wheel is near its edge, as inthe case of a larger number of nodes. Thevalue of the speed coefficient is generallyfrom 2 to 3, and a coefficient as small asunity is rare.

‘SchweitzerischeBauzeilang, May, 1914.

Traveling WavesThus far disk-wheelvibrations with radial

nodes and the effect of centrifugal force onthesevibrations havebeendiscussedin somedetail. The type of vibration which hasbeenfound to beresponsiblefor seriouswheel fail-

(l) ures will now be taken up. This type ofvibration results when, instead of thewheelsvibrating in segmentswith stationary radialnodes, a wave train travels around thewheelcircumference.

(2) Before consideringtraveling waves, a dia-grammatic representationof radial nodalvibrations of a turbinewheelwill bepresented.Fig. 26 representsdiagrammatically the edgeof a turbine diskwheel, andshowsthe curvesassumedby it when the wheel is vibratingwith four nodes. The drawingshowsthe edgeof the wheel developedas thoughall pointsalong the entire circumferencecould be seenat once. Evidently thetwo endsof eachcurvecorrespondto thesamepoint on thewheelandare, therefore,numberedidentically.

(3) CurvesI, II, and III showthreesuccessivestagesone-quarterof a completeperiodapart.The point on the wheel edge marked P ischosenhalf-way between nodal points andvibrates through the maximum amplitude.The points 1, 2, 3, etc.,remainstationaryasthey lie in the quiet nodal radii betweenthevibrating segments. First the wheel edgeisbentasshownby the full curveI; one-quarterof a period later the edge becomesstraight

(4) asshownin curve II, but theportionsbetweennodal points have a rapid motion which car-ries them over to the maximum deflectioninthe opposite direction in curve III, one-quarterof a vibration period later, or one-halfa period from the initial position. At the

1~’1

Fig. 27. 6~NodeStanding Vibration. The diagram representsthe developed edgeof the wheel during three

successivephases

end of a full period, the shapeevidently isagain the same as it was at the start asindicated by the full line curve I. Fig. 27shows the same sequencefor a 6-nodevibra-tion.

Fig. 28 showsthe developededgeof a wheelin a similar manner and representsthe caseof 4-nodetraveling wavesinsteadof standingvibrations. The difference betweenthe caseof traveling waves and standing vibrationsis seento be that thenodalpoints 1, 2, 3, etc.,

20


move along the edge of the wheel instead ofremaining at fixed points.

Curve If shows the wheel shapeafter thenodalpoints 1, 2, etc.,havemovedone-quarterof a wavelength to the right, and curve IIIshows the shape after another one-quarterwavelength motion where the nodes havemoved to the right one-half a wavelengthin all. At this instant the shapeof the wheelis the same as for the easeof the standingvibrations previously considered. The differ-ence lies in the motion only. In the case ofthe standing vibration the nodesare station-ary. In the caseof the traveling waves thenodesaremoving to the right. Fig. 29 showsthe samesequencefor a six-nodevibration.

Comparisons Between Standing Vibrations andTraveling Waves

The following comparisonsmay be madebetweenthe standingvibrations andthecorre-spondingtravelingwavesfor agivendiskwheel:

(a) In each case there must be an evennumber of nodes, that is, for everyupward portion of the deflectioncurve thereis acorrespondingdown-ward portion becauseof the conti-nuity of the circumference.

(b) In standing vibrations the nodes arestationary in the wheel; in travelingwaves they movearound it. In thefirst case we have true nodes in thesense that they representparts ofthe wheel which are always quiet sothey may be observed by the eye.In the secondcasewe havetravelingnodes; every part of the wheel edgevibrates and no quiet zonescan be

I It ~ —~ —

Fig. 28. 4-Node Traveling Wave. The diagram representsthe developed edgeof the wheel during three

successivephases

seen. A rapidly moving travelingwavecan be seenby the eyeonly bymeansof instantaneousillumination.

(c) The frequency of vibra~~f~qy.~y~l~deofa ~ndTsk~heel~oJ~stai~I~vibrationor for traveling~is the same. This important ~5oint~ll~j~s~tly be explained. Aknowledge of it is requisite to thedetermination of the velocity of a

traveling wave from the standingvibration frequency. For instance,turning to Figs. 26 and 28, it is seenthat if the vibration frequenciesofeachpoint on the rim are the sameineach case, the traveling wave mustmove to the right one whole wave-length while the standing vibrationgoes through one comulete cycle.~ travelj~veerseconde uals the ~iZiT7l~r o corn letefit rations o the corres ondin stand-

wave er secondmult , e

~toacon~te wave.

Fig. 29. 6.Node Traveling Wave. The diagram representsthe developed edgeof the wheel during three

successivephases

(d) For the standing vibration the ampli-tude of the particles varies alongtheedge of the disk from zero at thenodal points to the maximumvibra-tion amplitude at points half-waybetweenthe nodes. For the travel-ing waves, all particles around theedge of the wheel vibrate in turnthrough the sameamplitude.

(e) For astandingvibration all of thepar-ticles along the edge of the wheelvibrate in the sametime phase,thatis, all particles vibrate together sothat eachcomes to restat the sameinstant and each has its maximumvelocity of motionat thesameinstantduring the vibratory motion. Fortraveling waves, however, the par-ticles along the wheel edge do notvibratein time phasebut vibrateoneafter another in turn, successivelycoming to rest and successivelyacquiring their maximum velocityof motion during vibration. Sincethey all vibrate one after anotherthrough the sameamplitude a waveshaperesults of constantamplitudetraveling aroundthe wheel edge.

To sum up the last two paragraphs: for astanding vibration, the particles along theedgeof a wheel all vibrate in the sametimephase,but their amplitudesvary successivelybetween nodes; for a traveling wave allparticlesvibrate throughthe sameamplitude,but their time phasesvary successivelyalongthe wheel edge.

21

PART II

Relations BetweenStanding Vibrations and Travel-ing Waves

A well-establishedprinciple of wavemotion

wave trains travel~g~,jnopposite ctwa~e7!27l~7~T~that[Tl~e ~ standi~,~fration. A

~ princi~T~ob-served when two stones are thrown on thesurfaceof apond giving riseto two outspread-ing wave trains. Midway betweenthe twostonesthetwo identicalwavetrainsapproach-ing from oppositedirections are superposedupon eachother. Thereresults in this regiona series of standing vibrations of the surfaceof the pond with stationary nodal pointsbetween them. The particles of watervibrate up anddownwith the samefrequencyfor the standingvibrationsas for eachof thewave trains of which these vibrations arecomposed.

This illustration taken from mechanicshasother parallels. The sound vibrations in anorganpipewith fixed nodalpointsaresimilarlyexplained by a combination of oppositelymoving sound waves. In long-distanceelec-tric transmission lines standing vibrationswith fixed nodesbetweenthem may also beproducedby the combination of two oppo-sitely movingwave trains.

A considerationof Fig. 30 will be useful asan illustration. A, B, C, andI) showsucces-sive stagesof a standing vibration for eachquarterof its period, resultingfrom thesuper-position of two identical wave trains movingin oppositedirections. The crossesrepresentthe wave progressingtoward the right, whilethe circles show a leftward-moving wave ofequal amplitude. When the stage E isreached,the cycle is completedand the de-flection curve is the sameasat thefirst stageA. In the stageA thetwo oppositelymovingwaves are exactly superposed upon eachother so that they add.

When eachtraveling wavehasmovedone-quarter of a wavelengthas shown in B theycancel, so thereis zero up or down displace-ment at all points, resulting in the straightline. After the secondquarter of a wave-length of motion both wavesareagainsuper-posedso that the displacementsadd, but thedisplacementsare all oppositeto thoseshownin A. In stageB the deflectionscancelagainand in stage E after each wave train hasmoved a completewavelengththe deflectionsadd again, giving the original deflectioncurve.

The importantpoint to be understood isthat the natural frequencyof vibration of theparticlesis the samefor a standingvibrationas for a travelingwave. It has alreadybeenexplained that the frequency of a particledependsonly on its massanda stiffnessfactor’representedby the restoringforce perunit ofdisplacementwhich, for isochronousvibra-tions,isthesamefor eachunit of massthrough-out the entire structure. The principles ofelasticity show that these proportional re-storing forces, acting upon the various par-ticles of unit mass,dependon the shapeof

ç.. ~ -.-. ,.,-*, .----~

..,. 7~’S~. ~~

/‘

~ ::.::1...

.:~ ...

C ...~ ~‘

~__~ ~ ~

~D

,... ~. ~

E ~ ~

Fig. 30. Composition of Two Equal Waves Traveling inOpposite Directions to Form a Standing Vibration

deformation only. Since the shape is thesamefor eithertype of motion, it is seenthatthe vibration period of each particle is thesamein eithercase.

This explanation showshow the speedof awave train in a turbinewheel with a particu-lar number of nodesmay be calculated fromthe standingvibration frequencyof the wheelwhenvibratingwith thesamenumberof nodes.

Wave Speeds

~fhe~ntmjh~W~etrain of a particular number of wavelepgt.~~ at


22


one~particuiar chq.w.dcrjstjc sleed oytly,,. Justasa4-nodestandingvibration respondsat onefrequency so also the 4-nodewave trains ofwhich the standing vibration is composedmust travel aroundthe wheelat oneparticularspeed. So also for wavetrains of 6, 8, or 10nodes, etc. Each has one particular speedwith which it musttravel in the disk wheel.

Detection of the Presenceof Traveling Waves in aRevolving Disk

Fig. 31 representsthe developededge of aturbine wheel carrying a 4-nodewave train,which moves to the right with a certainparticular velocity. C5 is a small magneticcoil fixed to the stationary diaphragm soit can register the to-and-fro motions of thewheel in an oscillograph by meansof the in-ductive effect of the wheel as it approaches

~CR

Fig. 31. Developed Edge of a Wheel Carrying a4-Node Wave

(C~is a stationary magnetic exploring coil and C, a similarcoil revolving at the samespeedasthe wheel.)

and recedes from the magnetic coil duringvibration. A coil C,. is attachedto an armwhich is carried around with the revolvingdisk wheel, so it also can register the vibra-tion frequency of the revolving wheel in anoscillograph.

When the wheel is stationary, both coilsregisterthe samefrequency. Assumethat awave travels around the wheel 25 times asecond. The frequency registered by eachcoil would be 2X25=50 cycles per second,becausefor the caseof four nodesshown thewheel carriestwo completewaves.

Assumenow that the wheel is revolving at10 r.p.s. in the direction in which the wavemoves. Sincethewave alwayshasa definitespeed in the wheel, the coil carried aroundwith thewheelshould registeralmost the samefrequencyas before. The frequencywould beexactly the same were it not for centrifugalforce, the effect of which we havealready dis-cussed. Thewheelis stiffenedby it so that thevibration frequenciesareraised andthe wavespeedsareincreased. The effect would not bevery great at 10 r.p.s. Assume,for example,that the wave train travels around the wheel

26 times a secondinstead of 25 times due tothis cause. The revolving coil will then regis-ter 2X26=52 cycles per second when thewheel is revolving 10 r.p.s.

On the other hand, the stationarycoil nowregistersa higher frequencythan it registeredwhen the wheel was stationary. The wavetrain on the wheel is carried forward by thewheel motion at a speedof 10 r.p.s. besidesits natural speed in the wheel of 26 r.p.s.,so the wavepassesthe fixed coil at a speedof10+26=36 r.p.s., and the frequency regis-tered by this coil is 2 X36 = 72, because,thewheel carries a train of two waves in thecase assumed. Therefore,the forward-trav-eling wave registers a frequency of 52cycles per secondon the moving coil and 72cycles per second on the stationary coil,whereaswhen the wheel was stationary bothcoils registered50 cycles persecond.

Now considera casewherethe4-nodewavetrain is moving backward in the wheel whilethe wheel is revolving at the same speedof10 r.p.s. The effect of the centrifugal forceis the sameasbefore so the wavemust travelin the wheelwith the samespeedof 26 r.p.s.asbefore, but in the oppositedirection. Thefrequency recordad by the revolving coil is2 X26 = 52 cyclesper second, the sameas forthe forward-traveling wave, since this coilrecordsthesamefrequencyfor the samewavespeedwhetherthe wavemovespastit forwardor backward, The effect upon the frequencyrecordedby the stationary coil, however, isdifferent. Since this is a backward-travelingwave, the forward motion of the wheel of 10r.p.s. allowsthe waveto travel backwardpastthefixed coil with a speedof only 26—10= 16r.p.s. In other words, the forward motion ofthe wheel subtracts from the backwardmotion of the waveas measuredby the fixedcoil. The frequency registeredby the fixedcoil is, therefore,2>< 16 = 32 cyclespersecondfor the 4-nodebackwardwave train.

Again, take the case of this disk wheelcarrying.bothwavetrains simultaneouslyandalso revolving at 10 r.p.s. For the forwardwavetrain it will berecalledthat therevolvingcoil registers 52 cycles per secondand thefixed coil registers 72 cycles per second;while for the backward moving wave trainthe revolving coil again registers 52 cyclesper second and the fixed coil registers 32cyclespersecond. The revolving coil registersonly onefrequencyof 52 for eitherwavetrainseparately or for the combination, hut thefixed coil registers72 for the forward-travelingwaveand32 for the backward-travelingwave,

23


and both 32 and 72 simultaneously for thetwo waves superposed,that is, for the vibra-tion in the wheel.

To sum up these last two paragraphs,itmay he said that when a disk wheel carryinga standing vibration is revolved so that theradial nodal lines arecarried aroundwith thewheel, the frequency recorded by a coilcarriedaroundwith thewheel slowly risesdueto centrifug~dforce as the speedof the wheelincreases. The frequencyrecordedby a coilfixed on the diaphragm,so that it does notrevolve with the wheel, registers two fre-quencies, the higher frequency due to theforward-movingcomponent wave train andthe lower frequencyof the backward-movingcomponentwave train. Thesetwo frequen-cies diverge moreandmoreas thewheelspeedis increased.

Frequency-Speed Diagram

Thesefactsareshowngraphically in Fig. 32which gives a diagrammatic representationthat hasbeenfound to be very useful.

The vertical scalerepresentsthe frequencyregistered in an oscillographby the magneticcoils, and the horizontal scale the rotationalspeedof the disk. Themiddle curvegivesthevariation of frequencywith speedas recordedby the revolving coil. The upperand lowercurvesshowthe two frequenciesregisteredbythe fixed coil, the uppergiving the frequencydue to the forward componentwave and thelower the frequency due to the backwardcomponent wave train of the 4-node vibra-tion. When the wheel is at rest the figureshowsthat both coils register50. As thespeedof the wheel is raisedto 10 r.p.s. therevolving-coil freqtlency rises to 52 and the two fre-quencies recorded by the fixed coil diverge,the upper rising to 72 and the lower fallingto 32.

The gradual rise of frequency of the wheelas its speedis increasedis expressedby equa-tion (5) previously derived.

Jr = \/f52 + BN

52

Theuppercurveshowshow the frequencyof aforward-movingwave train, asmeasuredat afixed point, rises relatively to the frequencydetectedby the revolving coil, becausethiswave train is carried forward by the wheel,and is thus passing the fixed coil at a higherspeedthanit would were the wheel not rotat-ing. This rise in frequency is measuredbythenumberof wavelengthspersecondthat thewave train is carried forward by the wheelrotationwhich equalsthe product of the num-

herof waveson thewheeirim, ).~n,bythenum-berof revolutionsnor secondof the wheel, N

5.

This product )/2

nN. is the frequencyin excessof that of the wheel as measuredby the re-volving coil, that is, in excessof f~. If H isthe higher frequencyrecordedby the station-ary coil and representedby the upper curve,then

H fr+ ~2

nNs (6)

In the sameway the lower curveshows howthefrequencyof thebackward-travelingwaveas measured at a fixed point is decreasedbecausein this casethe wavemotion is oppo-site in direction to the motion of the wheel.

Thus if M is the lower frequencyrecordedbythe stationary coil and representedby thelower curve,

M = Jr — (7)

If the backward-movingcomponentwavetrain is absent, only the upper frequencyisregistered by the fixed coil correspondingto a forward-moving traveling wave. If theforward-moving component wave train isabsent,only the lower frequency is recordeddue to the backward-traveling wave. Thefrequency recorded by the revolving coil,however, is always the samewhether one orboth of the componentwave trains exist andin whateverrelative amplitudesthey exist.

It is therefore evident that by the use oftwo exuloring coils asdescribed,onerevolvingwith the wheel andthe other being fixed inspace, the presenceof a forward- or a back-ward-traveling wave train or both can bedetected.

10 20Spead Rev. per 5ec.

Fig, 32. FrequencySpeed Diagram for 4 Nodes

24


The first observationof traveling waves ina turbine wheel was made by meansof fixedoscillograph coils installed in the dianhragmsof an operating turbine in 1919. Early in1920 during the investigation of anotherturbine, several stageswere equipped withtwo such coils 30 deg. apart. One of thesestages yielded the records reproduced inFig. 33.

Observation of Traveling Waves by Means of TwoFixed Coils 30 Deg. Apart

The upper curveof Fig. 33 is producedbythe 25-cycle alternating-current generatorbeing driven by this turbine. Since the gen-eratorhastwo poles, it revolvesoncefor every

cycle, and this alternating-currentfrequencycurve marks off the generatorrevolutions onthe film. The two lower oscillograph curvesare the recordsof the two fixed coils, 30 deg.apart. Time is measuredto the right. Theupper of thesetwo curves is the recordof thefirst of the two fixed coils. Thelower curve isthe record of the second coil, set in thediaphragm 30 deg. beyond the former so agiven point on thedisk wheel reachesthis coilsomewhatlater than the first one,

The recordsof thesetwo coils showa closecorrespondence,both having ahigh-frequencyoscillationwhich goesthroughalow-frequencypulsation in amplitude giving the effect ofbeats. Furthermore, the upper curve liesbehind the lower one by about one-quarterof the distance between the low-frequencyamplitudepulsationsor beats. Sincethe timerecorded by the aniplitutIe pulsations isfour times asgreatas that by which the upper

recordlies behind thelower, dueto the 30 deg.betweenthe coils, the most likely explanationis that there are high spots on the wheel,4X30 deg., or 120 deg., apart, which causethis amplitude pulsation. This can he macicclear from a considerationof Fig. 34.

The wheel revolves in a counter-clockwisedirection as shown by the arrow. The twocoils marked1 and2 are30 cing. apart,andarefixed in space. The diagrammatic oscillo-graph record shows the heatsrecordedby thetwo coils, No. 1 lagging behind No. 2 by one-quarterof a period. Since it takes four timesas much time for successivehigh spots toreach a given coil as for a given high spot topass from one coil to the other, these high

spots must be 4X30 deg., or 120 deg., aparton the revolving disk. The inferenceis thatthesehigh spotsarewavecrestscorrespondingto a train of threewaves, 120 deg.apart,andthat the waves causethe disk wheel to ap-proachand recedefrom the recording coilsperiodically. When the wheel is close to acoil, the oscillograph respondsstrongly, andwhen it recedestheoscillograph respondslessstrongly.

If there is such a wave train the questionnow is to determine its d5rection and speed.The high spotsevidently move in aclockwisedirection opposite to the wheel rotation,becausethey reach coil No, 2 beforethey reachcoil No. 1, asshownin Fig. 33 wheretherecordof coil No. 2 showsa time lead over that ofcoil No. 1. This meansthat the wavewhichcausestheselow periodpulsations is travelingin the wheel against the direction of wheelrotation, and that this wave train travels

PART II

F

C

Fig. 33. Oscillograph Records Made by 17th Stage of 20,000kw,, 1500rpm, 23’StageTurbine

(The upper curve is a 25-cycle timing wave, The other two curves were made hy stationary coils 30 deg. apart.This is a case of 6-nodeforward- and backward-traveling waves.)

25


backwardevenfaster than the forward rota-tional speedof the diskwheel. Therefore,tofind the backwardspeedof this wavetrain inthe wheelit is necessaryto addthespeedwithwhich it travels backwardpast the fixed coilsto the forward rotational speedof the wheel.From Fig. 33 it is seenthat 63/b high spotspassa given coil for nine revolutions of the wheel,orsincethewheel revolves25 timespersecond,6 1/2X25/9=18 1/18 high spots per second.Sincetherearethreewavelengthson the wheelcorrespondingto the threehigh spots120 deg.apart, 18 1/18 high spots per secondcorre-sponds to a wave speed past the coils of18 1/18÷3 = 6 1/54 r.p.s. Thewheelrevolvesforward 25 r.p.s. so the wave speed in thewheel=25+6 1/54=31 1/54 r.p.s.

The conclusion is thus reached that thiswheel carriesa train of threewaves 120 deg.apart (correspondingto 6 nodes)andthat thiswave train moves backward in the wheel311/54 r.p.s. which is 6 1/54 r.p.s. fasterthan the wheel revolvesforward, so the wavepassesthe fixed coils 6 1/54 r.p.s. in a back-ward direction.

Thus far there has been no attempt toexplain the causeof the superposedhigherfrequencywhich showsstrongly in the recordof Fig. 33. This higher frequencyis found tohave60 1/2 periodswhile the wheel revolves9 times, or a frequency of 60 1/2X25/9=168 1/18 cyclespersecondas measuredon thefixed coils, since the disk wheel revolves 25times a second. This frequency is due to aforward-moving wave train of exactly thesame type as the backward-moving wavetrain, that is, a train of three waves 120 deg.apart,or a 6-nodewave train. If sucha wavetrain registers a frequencyof 168 1/18 cyclesper secondon the fixed coils, its speedpastthese coils must be 168 1/18÷3=56 1/54r.p.s., becausethere are three wavelengthson the wheel rim. Since this wave train isassumedto move forward and the wheel isalso moving forward, its speedin the wheelmustbe less than that registeredby the fixedcoil by an amount equal to the wheel speed,that is, 25 r.p.s., becausethe wave train iscarriedforward by the wheel rotation. Thisgives a wave speedin the wheel of 311/54r.p.s. But this is exactly the characteristicspeedof a train of threewavesas it checkswith the speed of the backward-travelingwave train of this type already found. Itwill berecalled(fromthefirst paragraphunderthe heading“Wave Speeds”) that a giventype of wave has a definite speedin a diskwheel which revolves at a given speed,and

that this wavespeedis the samewhether thewave train travels forward or backward inthe wheel. This coincidence is thereforestriking evidence of the truth of the state-ment that the high frequencyregistered wascausedby a forward-moving wave train ofsix nodes, that is, of the sametype as thebackward-movingwavetrain.

There is further evidence that the wavetrain recordingthe higherfrequencyis movingforward and the wave crests are 120 deg.apart as in the caseof the backwardwaves.

COIl

Record of No.1

Record of No.2-—-~fi~k~ 4Fig. 34. Diagram Showing 6-Node Forward and Backward

WavesSuch as are Recorded in Fig, 33

From a close examinationof the film recordof Fig. 33 it will be seen that the higherfrequencyof coil No. 1 leadsthatof the lowerrecordby afraction of aperiod. (Thepolarityof the two coils happensto be oppositein thisrecord.) This meansthat the disturbanceproducing this harmonic moves forwardbecauseit reachescoil No. 1 beforeit reachescoil No. 2. Furthermorethis leadis asbeforeabout one-quarter of a complete period.This again correspondsto waves which are4X30 deg., or 120 deg.,apart.

As to the relative amplitudesof thesetwowave trains, a casual inspection of the filmshown in Fig. 33 might lead one to believethat the amplitude of the forward wavetrainproducingthe higher frequencywas as greatas that of the backwardwave train. It isnecessaryto keepin mind, however,that in anoscillographrecord the amplitude is depend-

26


ent on the induced voltage which in turndepends on both the amplitude and thefrequencyof the vibration so that higher fre-quencieshaveamplitudesrecordedwhich aremagnified in proportion to the increaseoffrequency. For instance, since the higherfrequency is about nine times as great asthe lower in Fig. 33, the higher frequencywould be expected to be amplified aboutnine times as much as it should be com-pared with the lower frequency recorded.In all probability the amplitude of the

backward-traveling wave is greater thanthat of the forward wave. There are otherreasonsfor believing this, to be consideredlater.

The film record in Fig. 33 which has justbeenanalyzedis the onewhich is reproducedon page916 of the fifth edition of Stodola’sbook on SteamandGasTurbines.

Fig. 35 shows the frequencyspeeddiagramfor this wheel for six nodes. A and B corre-spond to the high and low frequenciesrecordedby the film shownin Fig. 33, that is,168 1/18and 18 1/18 cyclespersecondor tothe wave speeds56 1/54 and 6 1/54 r.p.s.,becausein the case of six nodesthe wheelcarries three complete waves. C equalsthe value of ,fr, the frequency of the wheelitself rotating at the normal runningspeed of 25 r.p.s. This may be calcu-

latedby formulas derived from equations(6)and (7).

~=H— ~/~nN5f,.=M+Y

2nN

5Thus f,.=1681/18—75=931/18

f,.= 181/18+75=931/18

(8)

(9)

Supposethe 6-nodewave train should stillpersistwith the speedof thedisk wheelwhose6-node characteristicsare shown in Fig. 35reducedfrom 25 r.p.s. to 15 r.p.s. Then thefixed coil would recordthe frequenciesA’ andB’ insteadof A andB. If therewerearevolv-ing coil, it would record the frequencyC’instead of C. If the wheel were brought torest with the wave still persisting,both coilswould record the same frequency or thestandingfrequencyE for six nodes. It cantherefore be seen that if the standingfre-quencyB is measuredfor a given numberofnodessucha diagramcanbe constructedto afair approximation,becausean approximatevalueof the speedcoefficientcanbe assumed.For dependableresults, however, rotationaltests are necessarywith the wheel-testingmachinewhich will be describedin Part IIIof this series. The frequency-speeddiagrammayhaveon it curvesfor all of theusualtypesof radial nodal vibrations as, for instance,4, 6, 8, and 10 nodes.

Fig. 33, the record just discussed,is therecordof the 17thstageof a23-stage,20,000-kw, turbine. This recordwastakenwhile themachinewas carrying 13,000-kw. load. Fig.36 showsa recordof the samewheel,butwithonly 5000-kw. load. No vibration phe-nomenadevelopedat this load. The jaggedand irregular record repeatsexactlyfor eachrevolution. It may be regardedas the wheelautograph, and due to slight irregularitiesin the rim oppositewhich the coils areplaced.Theseare magnifiedbecauseof the high speed.Fig. 37 shows where the load has beenraisedto 9000kw. Not until the loadreaches11,000kw. asshownin Fig. 38 do thevibrationphenomenadistinctly develop. Fig. 39 showsthe record at a 20,000-kw. load or full load.The wave phenomenawhen once developedappear to remain about the same up to20,000-kw. load.

Observation of Traveling Waves by Means of TwoFixed Coils 90 Deg. Apart

Fig. 40 gives the frequency-speeddiagramfor the 17th stageof a 23-stage,1800-r.p.m.,15,000-kw. turbine, where wave motion in awheel was detected by means of two fixed

305pe~d Rev, per So~c,

Fig. 35. FrequencySpeed Diagram for 6 Nodes,Points A, B and C were determined from the

record in Fig. 33

27


coils on the diaphragm. In this casethe coilswere 90 deg.apart. Fig. 41 shows the recordfrom which the diagram was made. Thiscasediffers from theonepreviouslydescribedin that the higher frequencyrecordeddue tothe forward-moving wave develops onlyafter the load is removed. The removal ofload is shown by the upper curve whichregisters the electrical frequency of thegeneratorbecominga straight line, remember-ing that time is measuredto the right. Thehigherfrequencydevelopsin aboutonesecond

Fig. 40. Frequency-SpeedDiagram for 4 Nodes.17th Stage of 15,000-kw., l800-r.p.m.,

23’StageTurbine

(Points A and B were determined from the recordin Fig. 41.)

of time, correspondingto 60 alternating-current cyclesof the uppercurve. As this isa 4-pole machine two alternating-currentcycles correspondto one revolution, and 30r.p.s. is the running speedof themachine.

This is known to bea caseof four nodesorof two waves180 deg. apart,becausefrom therecord it appearsthat it takeshalf as long fora high spot to go from onecoil to anotherasfor two successivehigh spotsto passonecoil.The high spotsare twice as far apart as thecoils, that is, 2X90 deg. = 180 deg.

Anothergood checkis obtainedby the useof equations(6) and(7) derivedin connectionwith the frequency-speeddiagram.

H =f,.+3/2nN.

Mfr3~flNs

H-MH — M = nN5

, or n= N

Subtracting,

N5

25

From the frequency-speeddiagramshownin Fig. 35 correspondingto the film in Fig. 33previously discussed, H = 168 1/18 cyclesper see,, M = 18 1/18 cycles per see,, andN

5=25 r.p.s.

Thus from equation (10)

H—M 1681/18—181/18

n=—-——=——--— —=6nodesFor the caseshown in Figs. 40 and 41 an

exact analysisis difficult becausewhere thehigher frequencycomesout clearly thema-chine has doubtless increased slightly inspeed, due to the sudden dropping of the6500-kw. load, There can be no doubt,however,that the following interpretationisvery close to the truth.

H = 130~/~cyclespersec.,M = 103z~cyclespersec.,N5 =30 r.p.s.

H—M130)4—103/2

4 nodeN

5— 30 — S

“Feathering” Action of Buckets a Possible Cause of

Traveling Waves

Thus far nothinghas beensaid about thecause of vibration in the two cases justdescribed. Further study brought out thefact that wavesof this sort rarely occur inturbine disk wheels and the phenomenonisconfined to unusually thin types of wheelsin which waves are easily built up. Afterthese films were analyzed, a satisfactoryexplanation of the causewas sought. Re-ferring to Fig. 42 in which the circumferenceof the wheeldisk is formedinto a waveshape,the relative angular twisting of the twobucketsA andB will be somewhatasshown.It will be seen that the axial componentfrom the energyleft in the steamat thepointof leaving the bucketsas shown at C and Dwill be greaterat C than at B, both reactionsbeing towardthe left in the diagram,

Assuming the wheel to be stationary inspace and the wave form to move upwardin the wheel to a new position asindicated by the dotted line, it will be seenthat the bucketsA and B aremoved to thepositionsshowndotted, This meansthat theforceC, which is largerthanB, is operatingonthe bucketA in thedirectionthatA ismovingdue to the wave transition. At the samein-stant the force D is opposing the motionof bucket B moving toward the rightdue to the wave transition, However,since the forces acting in the direction of

(10) bucketmotion, as at C, aregreaterthan theforces opposingthe bucket motion, as at D,

PART II

U,(U

“U

,~2

Co a so LI ZO 2

Spe.eUl P~v.ptrSec.

29


Fig. 41. Oscinograph Records Made by 17th Stage of 15,000 kw., 1800-rpm., 23-Stage Turbine

(The upper curve is the 60-cycle line current. The other two curves were made by stationary coils 90 deg. apart.4-node forward- and 1 ackward-traveliug waves are indicated.)

30

Steam Turbines - Cambpell

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