TUOINICIL .iPORT67-74 OSD

OXON MLmiRANE FIREQUIiNCIES

FOR SPHERICAL SihLL VIBRATIONS

ByEdward W. Ross, Jr.

A!!

.. . ., . 5!

__- May 1967

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Offic~e of the

SScientific oirecthe

BestAvailable

Copy

Di•tribution of this

~unztis unlim~ited AD_______

i II

ThQCANICAL REPORT67-74-MV '

II

by

Edward W. Ross, Jr.

" I• ~May 1967

U.S. ARMY NATI(X LAWAMORIE SNatick, Massachusetts 01760

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i

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• ! • -. _ •-• ... .... ... .... . .. ..... .. .......

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1. Introduction 1

2. Solutions of the *1embrane System .

3. rj-Ahavior of Membrane Solutions as L-- 0

4. Natural Frequencies for a Free-Edged Hemisphere 7

5. Discussion Io

6. References

I.

a '~1

II

I:n thiis paper the rambrane solutions for non-sTr•etric vibration.i ,ieai dome are studied. 7ne solutions are written in a con-

* vznien: £orm, and it is shoun how these reduce to the finiliar nem-•Sbrane 4id i-n.xtensional solutions in zhe static limit. This informza-t 1io.1 Ce-bules one tv perceive several difficultiAes in finding inexten-sio:nal frequencies by numerical means and to sioggest ways around these4il-ficultics. Also membrane frequencies arc found for circurxferenti alwave •.uýbers up through eight by direct calcUlation and compared withSCV~r.:t dizferent approximations. One type of approximation, due toJefir.rys _nd Jeffreys, gives quite good results. It is seen thatea..... • :ork agrees well with these results except for a few frequen-Ces tich ir'e not. found at all in the present calculations.

I''

i -I

'4 •. . .

4 ' iI IiI iii iii I*I i I I i i i ii i

• . ..;'.-n. i.c i-halv i%.r 4k) tc-n1:; .%; p-1 '•!n c, i : •r i •:t "

of- n '' s;-eis. In paper we 5-,7A. I ocus Oa.

t .;,- on ii ric vtbrazioa c.f thlin spherical. 's. ;

Z:Z-$.sia- thin-shell "1t.,ory t=e zover:.inn system of equatiofls Is o0'

order, and solutions in the form oi Associated Legendre Functic_...

.vw oeen zound by R.ainins and Iildinson1 and Prasad2 . Four of these

Zre of bending type and can often be approximated with 1t'ae ail

.. ~~ 1..- ,yaltotic methods used by the writer 3 ". Although we shall '-4 c

sor•e re•.aris about these solutions, our main concern here is with the

r~n~nl; •our solutions, which are of membiane type.

We b.iAll first wr_ te down these fowr solutions in various forms that

are uaseful arid shall describe some aproximations that are applicable in

several limiting cases. Then in Section 3 we shall present fornulas

shod-ins how these dynamic solutions reduce to the familiar stiztic men-

bra-ne arid inextenrsional solutions as the dimensionless frequer.cy,p,-,

tends to zero. It is helpful to have this information, in particular,

to 1noiv utat linear conbinations of the Lcgendre Function solutions

approica the inextensional solutions asSŽ•O , before one tries to

deduce inextensional frequencies and =odes from the Legendre Function

solutions. This new information should help in evading the difficultytL-.t hia.; heen experienced (see It•'z) in makirn: these co puta ions.

zit 4 - t'i~tijss r-,eujt.s aoutt ri.eubranc natural frequencie.s for hc.ai.:- :-

Pheres xi :h free-edges. These results are compared " Ui each other and

also with the earlier ca!culations of NJahdi and Xzlnins7 . The results

are discussed in Section 5.

i

Sa-'.o' -willb deiýr&:ted1S3Ldk

"-;n -vrcýS of .Usociated Legendre Functions by Na-hd-z and iairnns"Ai S........ ~c •1,:!.iy ~if edvrslo~ns ci2 '.lelr sQ!utions, w-hic~h are ccon-

* ::n.' •,. 5&Aciyin.; •htv 2i:•ii as O..i

. .4... ,,..4 "

V. =-M 411 csc -. , V csc0,3 12 2

',Y A dZ-~ 1 0

Here i is the cir•.mferential wave number (a non-negative integer for -7:

shells of revolution) and

S"--

k n (3/2)2 +.• ( 1: + "3( 3/ 2 +2 I + W,

ihere W is Poisson's Ratio. The functions Z and are re±ae to - ,

Associated Legendre Functions by e.g.

Zn (cos ) (2 Cm aPn-n+1) . Ccos )

Ss-z P i'1I+-a~na ti-'n- =+I; ( 4; .- CSoNs /

wh er (...) irotes the hypergeometric function. We shall also use

• i.- rcpt•.2ntation

2

- -~ ~ ~!

4 2 : r: 4..- -; ; -----,•.ii m i i i i • i lll II I I IIIi i i li , ,ll

cr :0. ..*.! a;:-, O - by a i-'-ta I ~ :~opcr*& of

A~ pair oif Zsynptotic for;.u1Ia --or 2--cosjS) whe n ;-rer.

.4- ...-. ntzcirua o

(s)~~ x~~~n 4\1 Sire adJcrys.F.ite rv t

.Izcrci fr3ua~Lt X

(n --.

=c co csz

24 Co

I!

: ••,; •.:t i; he: io~a.-~r c .. c (n•O the torsic :,2a e ::.brn

. . , are :;.01 nceszarily accurate approxia,2ons to

01.:; c _iCOre te e...bra.. e Plus 'ending) sys *-.a

Si. = ., ect oE *,he nc:rbrane approxiraation uras also s.-o m to

t.za~ oi f zh*.v occurrenc- of a transizion poin: (of infinite

ordirŽ) ai tit asy .otziC approximations to -he bending solut'ons when

The sin~vuarity in (2) I.&Cen--- is evidence that this diffi- ,

,,::1 r.-,sts in ihe non-symmetric case. 11ovaiver, the analogous

Cua: ion for k, (3), contains no such singularity, and we conclude

Mri.ý rorsio=nal rembrane solutions S' and Sm3", are free of this de-

f zc t at I;

mSeco,-n the asymptotic formulas (7) and (8) for Zn are valid .iten

n and m are both large and are of course equally applicable to the

nc cionZ-. These asymptotic representations suffer from a transi-;

tion point at "t, where

sin = =/n (9) . •.

It is worth duelling on the differences between this transition -

puint ond •he one described above, ihere-.-. =I.

The moa. obvious difference between the two kinds of transition

-2oints is this. The transition point for•--=l shows itself as a

sin.ularizy in the relation (2) between n and•R and does not involve

w, h.tereas the pre ;ent transition point, is determined by the relation

(9) between m and n and does not show itself as a singularity of any

4

.•_ m ml mm~ mmml mm Ilmr mmmml iN~e m m rm| m m m m mlmmm ml m m ml mm m mmlm m mm i I

V~ 5:~ I.I tat Zile. ~ ....

. ..v .. ,.-I Liut t.r.I gonilctc bye.'tc.= does not, nd (ii)

•• . " - " ] , : .. 2i'u-•ar1'' t :A[ ~o ints ,-ivvn by (9). W• c•-nc. ude t]at thc,.

: i.. .. :X.'..Jn tequations lor2.=1 im.IIes that .- me •o • -.

'... ouions are then poor approximations to complete solutior•s.

. .:senc- of such a singularity at 4=X means that the present transi-

tion 'xoint does no: significantly affect the accuracy of tae -.:brzne

zi s approximations to co=plete solutions but merely a .s

:'e aiccuracy of the asymp-totic for=ulas, (7) and (8), as appruxitzions

.to the ... brane solutions. Alternatively, we may say that, when•=i,

benrin; effects intrude into the me brane equations, or that there is

•c-: xci:-.;e of bending and stretching encrgies, but bending does not

a:' L ;Irat happens Mic • shortcouings of the approximations

(7) •rd (8) near O can be overcorme entirely within the scope of the

:ez' rznc theory. w;ithout considering bending effects.

. v'or of .lembrane Soluziov-s azý 2-N0.

We consider first the passage to the static lirit,<O- In this

case n->i and k-> 1I and

sin V 8 1.2+ =--; -+1- 1-Coso]

'2 / (1 ÷ m (r ÷ cos) tanl}

Q =(-in") d Q10 ( x)/dx , x coS

x x in (I + x)/(-x) -1

0 vL givS the solutins N.= (jz=,2,3,4) to the statical membrane

.... ' ith the follohrinj; dilpl~acements:

S

v -

o- sin. ; -I(

"4 th4 +a

Aoj; + .I.

U 1 Y1CS n- a

2 2

1.V Sint =~4 ~i

U3 -V3 r

(1 + coi~u$ +[,~o~o/1cs)

Cot

+ +U 3

-P - GC' snZal-

a - t ints.ii zo fo ~or ra / 2 -nd -,hc, s o iu i 0

viblraiio-.41 s.Iulo S." art: foun by co~mozrin.- be-'-.3

... 'ZVI: Sze' 4~ 1.~ Tic results are

~4 ý3- 4

z-nd :

, s'; s I

i-=(- -z )C

-77 (m+1

A-j

4.:~ ~~~zu~cesfor a 'Frac-Llze- ~evisphere

7n:..;, seczion we shzul ;se t~he £o~ sof Secti.on to obtain

k3 t: he natuc ral irequecies 2'or a i~ pwia

7

-rz

Zl kCos)

~-h rcccen~tio %'5) is uscd, Lie frccucncy cor4ition car, be-

bv an Kan'.

~~ ~ ~ýCsulzi s vý:ry ca2-.il dcri-1ed fro= 1) f

~. ~ c:. I!. ~~r .criu xt~cions i I ) are ox.

-~ ai; (S) .:duczz an nroxim~ately toI+

xi -e- l c--nnaz hý satlsfid for ral, non-zaro val eso7-

~c;~~~e .:_ r ql:ncicez (i.e., no natuiral

of tipe) can bz! fou~nd rzena. Since thIis haS

r Zvci -ey cr . -:.s~s Of t'n 7z.aDrane Solutions, uit =,ý

ts:"z It tdezcx n,:z precludca Zt` occurzeace Of lbending andin

:.23rz. :t~n -~,nar -)z: s 'z riA-r out the transitional Z::

n'-I.

- ~ z to .X. is obtained i:f weCOiŽ

n , s: i ~ -t :c r e.;-ion 0f1or Z, (w .-

* .2,7 :. .;tc.'nr~ or by usi~ntr. ta

d. -(n-znI)cSCI• P -(n+l)cos,% Pn

it, h..:Ltlr with the asymptotic relation for Z * The frequency equation 'n

assu:aes different forms in three regions, as follows:

-12 x -nM -M -()14

) k) mn k! 2

M JnXI Ik(s(CI-ah- 2ta4Tr(k-m+1) T

NC X~n )C -1Cm ~ n n Nk~~K)0CsK

Natural f•- .encies nay be estimated easily In the various regions •° !;

w ith the uid of ths formla. The reslts of these comuttions

are compared with thei calculationas bae on the accurate frequency !b:

equation (15) in Tabie 1.I

-12

A further simplitfication Is possible when •)2or ;.•

(k+;, >(A1"÷2 >>= ý4"7

"-in which case (19) becomes : ;

ta-ha) kt+lr) takn-a+)al) 0 01n-.-:

whence !,

k + ss2j m (20)l

eqain (1 I') i L Ta -(21)

£~2t

A ute smlfctini osil hn h)ao

2

:;:c J and L are sufficiently large positive integers, . hen .WTh' n

-•e h from (2) and (3)

Cor.Oinjn- these with (20) and (21) we find two families of natural

fr,,:uc~ic•. j'iven byi

$tz2•J. ÷ -+ ) I 2 f 1A (22)

.= (-2L +,m L2(l ÷,)3-.12 (23)

T.:e first of these families is associated wi'h stretching (torsionless)

n-cbrane =odes, the second with torsional modes. The frequency for each-.

branch depends linearly on m with different slopes for the two families.

T.hese predictions are: shown in Table I and Figure ±.

5. Discussion

We shall cozment first on the behavior of the solutions asQ..?. 0 "

and then on the natural frequency calculations.

The formulas (10)-(13) show clearly that in the limit as ?.-.o

the dynamic membrane solutions in the form of Legendre Functions are .

equivalent to the familiar static membrane and inextensional solutions.

'We &ind that, as , two linear combinations of the dynamic iolutionas

tend toward the static, inextensional solutions and two other combina-

t:-ons tend to the static zembrane solutions. These linear combinations

are given in (10)-(13).

We shall now point out several of the pitfalls that one may en-

c.u.te- in tryiiV to calculate the inextensional frequencies for a

fr.•.-cX:.ed dcize from the gnenral solution. The gener.al solution with

"i L ......c.,;c e r -actions in the fo,,

= C P1 (coS'6) + Ck P., OCCOS

+ C. p (cosu; + Cb, b (cosd)2

,..:ied expre 4io Zor the other variables. H!ere n d kand

'.•-: --cts • he Le.endre Rnutions conaecte~d with the =emb-nh e

s3:-zions, and b. and b 2 are the degrees connected with the berdin ,,t

soluz:io.s. 7hese solutions were given explicitly by Kalnins and '

vow the inextensional frequencies are very low, i.e., 4.

=0 ( h2 / )'4. 2

w h is the thickness and R the radius of the shell. We have pre- "

viously poinzed ou;5 that, when,.is this s=a/l, the bending solutions

.. -e nearly statical and are both of ef.c-cffect type. Also, we see

fýroz (2 and (3) that

J~ 1 . -

It is • anco Venient property of the Associated Legendre Function

that, e.g.. P. m(cos:S) vanishes identically when n is an Integer and

"3, n. Since iuextensional solutions occur only when m!e2, we see that 4"

;P. •(cosi) and P. 2(cosii) will be very small for the inextensional modes. i

This =y rake it hard to evaluate Cn and Ck numerically. This difficullty -,

c-an ba e'vaded by uring the solutions Zn, Z m1 which do not become s=l"

when and k are nzar unity, i.e., we may write the general solution as

,a ZC 2 (cos4) + CI zk ?i(cosS)

+ C b1, (cosI) + C P N(cos6) (2•)bbb2 b2

4.-/

--------------------------- 4 -

h,.'cver, we are not out of the ..oads yet, for it is only a ve.ry

linear combination of Z and zm that leads to an inexten-fl k ;

sionai solution. If w;. rewirite (25) as

-W, Z I+ Zk + -tk)(L - l)

4 Cb Pb ( cosf) + Cb Pb (Cos$;)-2 2

wt" :-,-t fr,; (12) that tihe firt:t becomes... b4COm. in-xt"-ni.ona. l :1!;

" "*--•,.• bl| [�." : .,', i.I ¢|[:.*|l~l,, lurioll.! • lUtiol. In urdul r tu

obtain a 4ede which is predominantly inextensional we must haveC,

Cn a •e. 1 'k%

It is plain that, if the constants are evaluated numerically with

insufficient accuracy, so thatCn IandC ' are not quite equal, we may

b, accidentally introducing a little bit of the membrane solution.

Even a little trace of the membrane solution is enough to destroy the

iný!xtensional property of the solution and prevent one from obtaining

inextensional frequmncies. It is possible that this difficulty can

be overcome by careful calculation. If not, it may be necessary to

set C *-C. "throughout the calculation, temporarily discarding onen F

inmbrane boundary condition. After an inextensional frequency has

been found, the discarded boundary condition can be used to calculate

the difference C'-C', which Is initially taken as exactly zero.

The results for membrane natural frequencies of a free-edge hemi-

sphere are given Table I and Pi,:,ure i. The gross features are these:

2 2(i) Mien m , there are no membrane frequencies.

-, 2S•WhenQ> n , the simple asympstotic estimates (22) and (23)

are fairly reliable. These predict two families of frequencies whi-ch

/*1

-- 1 Vý, t.eslope-s of thie lvg fam.ilics, being diif ersn

., i..ti,.s arc less accurz:e near points u.bere two Iaýzquency

lines cross Jhan els.,.1here, and the errors in the asyz-pt-ti¢ c

¢e~i-.ts change sign at such points.

(iii If Si and n2Mire of comparable size, the picture is more

co.-plicated. When Z.and 'mare both large, the most important analyti- <

cal features are the n-audk- transition lines, i.e., the locus of.

1..•. 1 i" A. ll:. 3- ;1 Jul h to rt-:.p.*- I i v. . 'Slat.:.,- . rt" :I.,,are in

Hi -ure i. elow tile .k-line the approximate frequency condition,

ki7), involtzs no oscillatýry functions, and tihre is merely a single

frýcquency for each ; Between the nl-and k-transition lines the Ere-

qucnlcy cordition,(1S), contains an oscillatory function of k, indicatinr •

that there is one family of frequencies, namely those of torsional

type. Abovc the n-line crA families of fr,_-quencies are found, cor- ...

rc_,ponding to the two oscillatory functions that occur in the fre-

quency condition (19).

IntcrestinZ information is revealed by comparisons among the

various approximations given in Table I. First, we see that the approx-

irate frequency equations, (17)-(19), giive generally good agreement

with the more accurate frequency equation, (15). T-he accuracy i-13

poorest, of course, near the n- and k- transi:ion lines. The simple

asymptotic estimates, (22) and (23), are not as accurate as (17) ar4

2 2(19' but are tolerably good when '•.ti. They are very poor near or :+

below the n-transition line.

-Wl

4{

a... cAcWh-ons based on (15) do not . co::e -

.:vious r ts c :hdi : . 7 For m=l the ..

. -~e::~ ;.oud. F1or ;.=2 and 2 +-.ere is good a.'rcement for sone -*,z :

- .c prest;ea caIcu : ato:s do not zsixa cerin frequencies .:`a

":, .:a,-rlier0 Separate hand calculations were Pade for each of .he

i7.. ,'ntable frequencies. In no case was a frequency found. ZorCav4:-,

w "e plot these doubtfuI frequencies on iigure 1, we see tha ti Cy L;3

not fit roell With the pattern of the rnainir<.: frequencies. Wc! concl-:,iz,

In closing we ay remark tnat these membrane frequencies are all

-;h cc=:)ared with the inextensioaal fre,.encles. This does not nelec-

.n.ly =nan that they are unimportant, -hwever. Their importance in a

- of :itc-dcpendent loading depends on whether their r-ode ia=kes.

siP:.nficant contribution:to the response of the Thhll. This in turn dopta.,

on the .ud of applied load and its spatial distribution. It is relevant,

"that Lh. modcs5 associated with most of these frequencies involve mnch',

more rot;oa in the stell-surface than uLrmal to it. Thereýfore, il

"wxperim ents on shell vibration are made in 'hich only motion -no-n-1- to

the surface is =easured, it will be very easy to ziss these modes and

conclude (erroneously) that they are uaimportant.

I ,.

/1 ,

J- . P. Wilkinson and A. Katlirs"

" :;,-; , 5 'rici NShells". J. App. Mcch. 32, 525-532 (1965)

"C. P.rasad, "On Vibrations of Spherical Shells", J. Acoust. Soc. AT&:'

34S. 4s9-4 4 (19%4)

11K W. . i+:;, Jr. ," 'N•.turaIl 'F:-qutfccit-i-; a;nd M4huiv Shiaq'c:; f+,.r A\xisymai

Vibration of Deep, Spierical Shells", J. App. ýIech. 32, 553-561 (19o5)

E- . W. Ross, Jr., "Asymptotic Anal~sis of the Axisymmetric V\ibrationn!

0oF Shaells", J. App. Mech. 33, 85-92 (1966)

5..o +ii

- E. W. Ross, Jr., "Approximations in Nonsymmetric Shell Vibratios"% :i.

To appear

- Chintsun 1lwang, "Some Experiments on the Vibration of a llc~ispherical

Sh.A11", J. App. Nech. 33, 817-b-4 (1966)

- P. M. Naghdi and A. Kalnins, "On Vibrations of Elastic. Spherical

Shells", J. App. Mech. 29, 6j-72 (1962)

"- II. Jefreysfoand B. S. Jeffreys, "Nethods of Mathematical Physics"

(C~mbrid-e at thcUniversity Press, 1950) 2nd Ed., p. 653.

9.- A. E. i1. Love, "A Treatise on the Mathematical !::¢ory of Elasticity",

(Dover Publications, Inc., New York, 1944) 4th Ed., pp. 5S3-5S6.

4.

iii+I i'i~iii"+IIIII+IlI~I'IU+]i][lM - ~i.iiHIil'-H|Iili ilII-+[I,

A~i~aV.11 i

.7

,Si .i 87) -;7

7. -,

47 1.47

f-, -S 3.56"2. ••2 2.923.-)" 3.92 "

.. S0 4.31 4.72_5.23 5.20 5.276.43 6.414 6.516.39 6.SS 6 .8is

2. . .2'$ (?)

.922i..21

"s C?)2.31 2.31

3.1S S.26 3.41.3.S4 3.83 3.67s4.6- 4.57 4.65 t

5.66 5.69 5 .7 71S

5.99 5.97 SS8t7.06 7.06 7.13t

m=3 .740 C?).)43 .943

1.20 C?)i.S]4 1.33

2.07 C?)3.13 Z.09 2 . 7 9 t3.,3 :; ,,I4 4.03t

:

4.70 4.75- 4.72s

5.33 5.28 5.27t6.35 6.33 6.51tA,9. 3 6.92 6. s

r.=4 2.--13.S7 3.S6

4.67 4.57 4.65'

5.47 5.5S 5.77s6.15 6.08 5.39"

7.04 7.05 7.13t

S4.5 i.55 5.279S3.2i o.35 65

7 ."6.25ii

7. 09 7. 13

7 4.10 3.9S

5.S2 5.31,7.01 6.9S

S 4.65 4.556.44 6.42

a base on Fq;ation (15)

b b"sed on I.Vations (17) - (19)

c bzused on Equations (22), (23)a associated w-th torsional node

s associated with stretching node

~Iy

I . I9-

9 .9

/ 9.- --

.4 7.-...

' ., . .9,- .9I -, . .99- .9 *,. -

9 I ' 9

4. � - .* -- - , - -* 4

- 9/ -..9'.. I

I - -9,9 9-

-.9- -. A' . ,,

I 9-9 �9

* 1/ A __�9) - � _________

9 9-. .- '9.I '9 I.9-

- *,-,r/49. .- *�* --

-I I

/ - 9, I -___ I 7 '9 ....�NS!TiO2�Li�

9 **� � *�4 �-i"�-T��"'_____________ I .'

'IAp .,..�' --- �'�' I

9/ 1�'�

d9��'a / 9 ,q 'I

49� *9'� / '4.

'9--, '9'I

-9'

- 1: � I '.*

______________________ ____________________

*��-99

9 I

0 . . :A>&iTiCK�iAL FF-�EQUiUZX'* 9 1

* 01 1

I 1 I* () j I I

-- _______________________________________________________________________________________ I _______________________________________________________

49� .9 49� 4-- �.

* *.il�. * I I �4 , , I

I ilk. 1.1-.1

%I I It, I I iC.tt l

DOCUMFNT CONTROL DATA - R&D

(Secursitiv Cl.*13 fcanllu of title tody Of abbtrart nnd indoxik ng wux~iclion must be f t-,ad If- i.-' r. II,

I ORIGINATING ACTIVITY (Corporate author) j2. lkCr-O•.O•-" C -Y C -. A.' CA-

U. S. Army Natick Laboratories Unclassifiedi

Natick, Massachusetts 01760 2b Gko0P

3 REPORT TITLE

ON MEMIANE FREQUENCIES FOR SPHERICAL SHELL VIBRATIONS

4 DESCR.PTIVE NOTES (Type of report and inclusive dtesa)

Technical Report

S AUTHOR(S) (Lost name. list name. snitiial)

Ross, Edward W., Jr.

6. REPORT CATE 7&7a TOTAL NO OP PAGES 7b NO Of REPS

May 1967 . 14 9

a&. CONTRACT OR GRANT NO. f 90. ORIGINATORAS REPORT NUMBER(S)

b,,PROJECT NO. 67-74-OSD

C. Fb. OTHER REPO•4T NO(S) (Any other nu."tberts •hat m.ay Le . .this teport)

d.

10. A V A IL ABILITY/LIMITATION NOTICES

Distribution of this document is unlimited.Release to CFSTI is authorized.

11 SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

U. S. Army Natick Laboratories

Natick, Massachusetts

13. ABSTRACT ,

In this paper the membrane solutions for non-symmetric vibration of aspherical dome are studied. The solutions are written in a convenient form,

and it is shown how these reduce to the familiar membrane and inextensiona.solutions in the static limit. This information enables one to perceiveseveral difficulties in finding inextensional frequencies by nunerical meanlsand to suggest ways around these difficulties. Also membrane frequenciesfound for circumferential wave numbers up through eight by direct calculationand compared with several different approxi.mations. One type of approxima-tion, due to Jeffreys and Jeffreys, gives quite good results. It is seen -ht* Iearlier work agrees well with these results exeIept for a few frequencies whichare not found at all in the present calculations.

FORM1A'D D I JAN Ge. 14 3Unclassified1

Secuity ClassificAncn

VI

KEY WORDS LiNK A LNIK .i L, .K -. - __�_ORDROLE VV wT ,OLL - AT i.O.L .

Analysis 3Continuum mechanics 8 tDynamics 8Memtbrane s 9

Frequency 9Vibration 9Elastic shells 9

Spheres 9

Legendre functions 10

Differential equations 10

Asymptotic expansion 10

Prediction 4Performance 4Tents 4

Parachutes 4

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