vibrationsof a spinning disk

8/12/2019 Vibrationsof a Spinning Disk

1/9

he Vibrations o n Xpinni7zg Disk

By H. LAMB,.R.S., and R. V SOUTHJVELL.*

(Received April 6, 1921.)

1 This pap er tre ats of th e transv erse vib ratio ns of a circu lar disk ofuniform thickness rotating about its axis with constant angular velocity.The problem has a practical bearing, as throwing light on the occasioi~alfailu re of tu rb ine disks. I an impulse turbine blades are fitted to the rimof a th in d isk, and steam is ad m itt rd to them from nozzles which are usuallyarranged symnietrically, bu t not always continuously, round the periphery.The failures appear to be sometinies due to the Wades conling in contact withthe adjacent parts. This points to transverse vibrations, and there can belit tle dou bt th at th e phenomenon is one of resonance between the periodicforces exerted by th e stea m jets an d th e p eriods of free vibratio n of the disk.?

Fo r a thorough investig ation it would be necessary to asc ertain, toareasonable deg ree of app roxim ation, th e f ree periods of a sym inetrical diskof an y given profile. This niay form th e theme of a subsequ ent inve stiga-tion by one of th e autho rs. I n the meantime the stltdy of th e simplercase now discussed illu stra tes th e p hysical aspects of th e problem, and hassuggested an important simplification in method.

2. The special difficulty of th e problem arises from the fact t h at centrifuga lforce and flexural rigidity are both operative, and that the normal modesof vibratio n dep end on the ir relative importance. The v ibration s ofa non-rotating disk were fully discussecl by Kirchhoff in two well-known papers.:I n the other extreme, where the disk is so thin and the rotation so rapid th atth e flexural forces are negligible, the problem ad m its of a very simp lesolution, as follows

W hen an elast ic disk rotates in its own plane, in relative equilibrium, theprincipal tensions are along and perpendicular to the radius vector. If n be

M y attention was .drawn to this quest ion by Prof. Gerald Stoney, who showed mesome pret ty experiments on a rotat ing india-rubber membrane. This led to variouscalculations. I found subsequently that Mr. Southwell had become interested in thesame problem, and a conlparison showed that the methods and tlie results were closely

parallel. This pap er is accordingly a joint production, bu t it should be sta ted th at theimpor tant remark tha t the formulap2=pI3 p ,2of 54 gives not merely an approximationbu t a lower liriit to the frequency is due to Mr. Southwell.-H. L.

t A paper by K. Baum ann, read before the Ins titu tio n of Electrical Engineers onMarch 15 1921,gives examples of failures which ha ve been trace d to thi s source.

U eb er die Schw ingun gen einer elastischen Scheibe, Crelle's Jour na l,' vol.40(1850),an d Pog g. Ann.,' vol. 81 (1850).


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7 3he Vibrations of a Spinning Disk.

the radius, p the density, and w the angu lar velocity of rota tion , th eir valuesa t a distance r from the centre are*

I = A a 2 - y2 pw2, Q = (Ana-B.1.2) paa, 1 )respectively, where

= + 3+o) , B = + 1+3u), ( 2 )

denoting Poissoii's ratio. These values arc obviously unaffected bya smalldisplacement (7c) iiormal to the plane of the disk.

Th e equ ation of transverse rnotion of an el em en t of area whose polarco-ordinates are T, 8 ) is, unde r th e condition stated,

i~~~ al apl.sec. = ' ~ ~ s ea T + 7?;QW se,,

Assl~liiing hat w varies as cosse cospl t , where s is integral, we find

This is readily integrated by a series. Assuriiing

we have

or, if we write

Ck+a

k - ~ ~ ) k + / ~ t + 2 )C,, ( / c + 2 - s ) ( k + 2 + s ) (9)The solution which is finite for 7 = 0 begins with Hence; insertingthe factors temporarily oniitted,

( s - n ( s+ ? L +2 ) 7.

The series in { is hypergeometric. I n Gauss's notation,w = c P 11 cos eospit, 11

a

where = h s-nt), @ = ( ~ + m , + 8 ) , y = s+1 . ( l a >

* Love, Elasticity, 1 2 (b) .


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7 Prof H. Lamb and Mr. V. S o ~ ~ t h w e l l .

The expansion is obviously convergentfor 1 < a hut siricer = a P, (13)

i t becomes, by a known theorern, logaritll~riically nfinile a t th e edge1 = aunless it terminates.* He nce we niusb have

where is a positive integer . Thus

A rbi tra ry con stants may of course be added to8 and tTh e mode of vibration th us found is characterisecl by L nodal circles and

s eq uid ista nt nodal cliameters. Th e corresponding frequen cy is deter-mined by

= ( S + ~ ~ L ) ( S + ? ~ L + ~ ) I - ~ ~ I ~ , (16)o 2

where A a nd B have the values (2). Th at this ~ .a tio houlcl be indep er~d eiltof t he d iam eter ancl thickiless of (,he disk might, have been anticipated hornthe theory of dimensions.

Th e modes for which s = 0 involve displacem ent of t he ce ntreof the disk,?and are thctefore uot relevant to the case of a disk moui~ tedon a shaft.Th e lat ter rem ark applies also to th e mocle whereo = 1,?2= 0 ; the disk the11rem ains plane i n form, h ut is slightly untru e to the axis. The rilost inte r-esting modes, an d the orips l~ lo steasily realised exp erim er~ta lly, re those inwhich there are no 11odal circles. Putting: z = 0, we have

with

W h e n s = 2, we have two nodal diaineters a t right angles, and

For a n india-rubber mem brane we may pu t a = practically, whence

p w = 1.5.

Cf. Rayleigh, 'Theory of Sound, 336.

t t may be noticed that if we put v/a=sin + the formula 15) may in this case bewritten

U C P I 1 COS ) COS pit

where P, is the zonal harmonic of order n.


4/9

The ibr tiolzs o a pinning Disk.

W rit ing (16) in the formpI2 = Xw2,

the values of X for various m odes ar e given in th e following table, on th ehypothesis a = 0.3, which corresponds fairly to th e case of a steel disk. Th emode for which n = 0 s = 2, is th e rnost imp ortant from th e practical pointof view, since i t is th e gra ves t ty pe of free v ibration if we exclude (for th ereason already given) the case of n = 0, s = 1.

It is to be observed, tliroughout this paper, th at th e vibrations consideredar e vibrntiorls re lativ e to th e undistul.bed disk- in par ticula r, th e nodal linesin the modes investigated are in reality lines of rest in the disl i , and areaccordingly carried round by the lotation. The results may, however, be

state d in a different form. By superpo sition of two norm al modes of th esame type, with a suitabl e difference of phase, we can con struc t wave-formswhich travel round the disk w ithtli angular velocity + p l / s . The precedingresults would, iu fact, have been unaltered i f we had assumed that zo variesas cos pit T s 8 ins tea d of co spl t coss8. The corresponcling angular wave-velocities i n space are w + P ~ / s . Thus, in the case Y = 0, s = 1 w h e ~ el = wby (16) the wave-velocities are 2 0 and 0 In the la t ter a l ternat ive , themembrane merely rotates in a fixed plaue sliglitly inclined to that which

we have talten a s th e plane of reference.3 The preceding may be regarded as results which are approximated to asth e speed of rotation is increased. I n th e other extreme, where th e influenceof the rotation is negligible, ancl th e flexural rigidity predom inant, we m ayhav e recourse to th e investigatio n of Kirchho ff. The results a re of th e form

where E is Young s modulus, h is th e half-thickness, arid isa numericalcoefficient depen ding on th e partic ular mode of vibration and (to someex ten t) on th e value of o The resu lts are actually presented by Xirchhoffsom ewh at differently. H e gives table s of frequencies of various modesrelative to th at of th e gra vest mode of all asa sta nd ar d, first on Poisson shypothesis of a = 16 and secondly for c = . The gravest mode in question


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7 6 Prof. H. Lamb and Mr. R. V. Southwell,

has two nodal diameters and no nodal circle. The frequency for this isgiven by Kirchhoff in semi-vibrations per second, viz.,

and

for a and espectively.

An independent calculation for the case of a 0.3 gives the followingva lue s of th e coefficient p in (21). The nu mb er of nodal diam eters isdenoted , as before, by s, an d th a t of n oda l circlesIjy IL.

4. When the centrifugal tensions and the flexural stiffness are botheffective, the d ificultie s of a n exa ct solution are m uch increased. Th eequation of motion is

i z l u ~ U Q a'l/. EILp - = - - p,- --- v 4 w ; t . a . I-,,,an d the boun dary conditions are as given in liayleigh's Th eor y of Sound,

215. If we assume that 71 . varies as cos se co sp t , as before, th e equationca n be integ rate d b y a series, bu t tile relation lnetween successive coefficients

consists of t h ~ erms, and th e deduction of general results would not appea rto he easy.

Ap prox ima tions which ar e ad eq ua te fro111 th e p ractica l po int of view can,however, be obtained by Rayleigh's nlethod of an assumed type,? at allevents, for th e ruore i m po rta nt mocles. Th e p rocedure is equivalent to this,that we calculate the potential and kinetic energies 11 th e basis of a nassumed for111of 111 and determine the frequency so that the mean energiesof th e two lrirlds sha ll be equal. Since the frequen cy th us deterlrlined is

statioriary for slight variations of the type, a good approxim ation canbefound in th is way if th e typ e be suitably chosen. I t is known, moreover,that, in th e case of th e grav est modeof all, the frequency obtained will be an

The auth ors are indebted to Miss B S. Gough for assistallce in tlie ra the r laboriouscalculations.

t 'Theory of Sound,' 5S88,89.


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277he Vibrations o f a pinning Disk

upper l imit to the true value. I n the present problem, th is sta tementobviously applies also to the gravest mode having an y assigned num bers ofnodal diameters.

The po tenti al energy of th e centrifugal forces is

where P and Q have the values (I), and the integration extends over the areaof t he disk.

The pot en tial energy of flexure is, in C artes ian co-ordinates,

To transform to polar co-ordinates we have

1 a 2 ~ aw\ aZw1 ;l 1 l owq2 -'ufXr yy ---- -- - + - - (28)kaFae ~2 a s a+ s a? .The kinetic energy is

T dr. 29)h 27If, the n, we assume

211 4 T, 0) cospt,the method referred to leads to a result which may be written

where T 4 )differs from T in that awlat is replaced by 4.I n estimating in this way the frequeucy of the grav est mode for a given

value of s it is sometimes convenient to adopt for C a form containing avariable parameter, and to adjust this afterw ards so as to make t he fractionin (31) a minimum. This reduces the upp er limit referred to. This plan isadopted, and illustrated by numerical examples, in 5 below. The work issomewhat intricate, but the formula (31) suggests a simpler process, which isprobably adequate for practical purposes, having regard to the fact th at th eelastic constants E and c or a given disk are not likely to be know11 to ahig h degree of accuracy .

I t is evident th at if precisely th e righ t form be given toC the formula(31)

will be exact. Now, on account of th e stat ion ary pro per ty referred to, th etrue value of 4 would give an approximate value forpl viz.,

* Theory of Sound, 214.


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278 Prof. H. Lamb and Mr. V. Southwell.

I n the same way th e tru e value of would give an approximate value forp~ ,viz.,

Moreover, by Rayleigh's theorem, these estimates will in the case of thegrav est mode hav ing an assigned num ber of nodal diameters be too high.The second member of the equation 31), when ha s th e correct value, istherefore greater than p12 pa2 . Henc e in the case supposed the formula

where pl a n d p ~ ,re th e values of p on th e tw o ex trem e suppositions of infinitethinness, and infinitely slow rotation, respectively, gives a n approximationwhich is of th e nat u re of a loz er inzit to the tru e value.

Th e prop erty here inferred nlay be generalised as follows. If th e restoringforces which coiltrol th e vibration s of a n elastic sys tem canbe separated intotwo or more groups wh ich affect the potential energy inde pen den tly, arid ifth e gravest frequency of vibration be found on the assuniption t ha t eachgrou p a cts sep arately (th e inert ia being ~ulc han qed ), he n the sun1 of tliesquares of th e frequencies th us found is less tha n tlie square of th e grea testfrequency which can occur when al l the groups a ct s imultaneou sly;i.e. i t i sless th an th e square of the gravest frequency natural to the system.

A s a numerical example, suppose that

in C.G.S. units . I th e gravest mode with two nodal diameters we have, onthe hypothes is th a t 4

from (19) and (22). The formula 34) gives

as a lower lii-nit to th e frequency.5. W e proceed to illustrate th e applicatioil of Rayleigh's m ethod to th e

problem.

See footnote on p. 272.


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7 9he Vibrations o f a Spinning Disk

where is a constant to be afterwards chosen so as to mak e the inferredfrequency a minimum. W e find

s S ) A-s2Bs + l s + 2

+P2 (i 44s2+4s+4)4-s2(b+2)B)] cos2pt, 3 7s + 2 ) . s+S)

v, nEk3 [ 8 ~ + 1 ) , 8 ~ + 2 1 - ~ )S ~ S - 1 ) + 2 @ s s P - 1 )3 1- u2 a2

+ s 1 ) s 2 - 2) cos2pt, 38)

Equating the mean value of V1 V2 to that of T, or expressing that thetotal energy is constant, we have

where L, M, N, I m 72 have th e values given helow. W e have to choose soas to niake this fract ion a m inimum .

The station ary values of p 2 ar e givenby the quadrat ic

In - n ~ ~ ) p ~ - l NnL- 2 mM) p2 +LN- M 2 0 , 4 1 )and the corresporitling values of /3 by eit he r of th e equations

W e write for shortness

so that 5 l ike w is th e reciproc al of a time. The coefficients in 4 0 ) havethen the values

We have specially in view th e caseof s 2. Adop ting the value for CTwe have


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28 he Vibrations o f a pinning Disk

Hence

A s a partial verification of the method we note th at when 0 thequad ratic 41 ) reduces to

I 2 2

1 5 11- 1 9 7 1 2 f + 2 1 2 9 9 2c c 0, 51)th e lower roo t of which is

C10.8955.

This makes

This is higher tha n the correct value 22), as was inevitable, but the erroris less tha n per cent.

To illustra te th e case where th e effects of centrifu gal force and flexuralrigidity are comparable, take th e num erical data given a t the end of 4 .These make

w 9.86960 x lo4, c 1.97847 x l o 4 , 53)

The qua dratic 41) becomes

The lower root isp2 4.50890 x lo5 ,

whence, for conlparison with 35 ),

This is probably very close to the true value. I n any case the upper andlower limits which we have ob tained differ olily by ab ou t per cen t.Forpractical purposes the approxim ation 3 5 ) would probably be sufficient in allcases, for reasons given above. It is, of course, obtained w ith mu ch lesslabour than 57).

The value of l corresponding to 56 ) is -0.169.

vibrationsof a spinning disk

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