Turning point solutions for thin shell vibrations

ao:o-7683;w s3.00+ ,o 1991 Pmgamoa Rnr

.I0 pk

TURNING POINT SOLUTIONS FOR THIN SHELL VIBRATIONS R. Department

of Engineering

ZHANG

J.

Mechanics.

Tongji University,

Shanghai 2ooO92. China

and WEI Tsinghua (Rewired

ZHANG

University.

Beijing 100084. China

8 Auyusr 1989 : in revi.wd,/tirm

I5 Ju&

1990)

Abstract-Asymptotic solutions of the equations representing the free vibration of thin shells of revolution with a first-order turning point are developed. These solutions are uniform and fully approximate. namely satisfy the accuracy of the theory of thin shells. Three categories of the generalized function are delined, Y,, (h = I. 2. 3.4). 1 and ./. in terms of which a singular membrane solution and four bending solutions can be expressed. respectively. In particular. the second cntcgory of the gcncralizcd function .A is first obtained. which is a generalization of the new solution of :hc related equation. This new solution is found from the modification of the Laplacc Transform Method.

NOTATION All the quantities are dimcnsionlcss cxccpt the characteristic length R. the elastic modulus Eand mass density I,. s the first principal coordinate along the longitude the principal radii of curvature H,. R: Latnc’s cocllicicnt. which is equivalent IO the corresponding parallel radius B It the thickness of the shell the thickness paramctcr (small paramctcr) 8:.I( tr, frcq ucncy n Ihc frcqurncy paramctcr Y Poisson’s ratio m the number of waves along the circumference. which is restricted to not being too large. namely. nr _ O( I) I,. I’. it’ Ihe tangential. circumferential and normal displacement Langcr’s variable 2 = cp(.r) stretch Langer’s variable c * :lr Y,(i ; p), a(( ; p). j(1; p) the first, second and third category of the generalized function, where h = I. 2,3.4 ; p=O.&I.+2.... :u,. r,. w:) (i = I, 2.3.4) displacement vcytors corresponding lo the four bending solutions (“yn, L.yl,w.yl} the displascmcnt vector corresponding to a singular solution of the mcmbranc system (p:“‘. y:“‘. r!:“} rhc regular part of (u’,“‘, 11;“‘.tt*‘,“‘i (“;u’, rfll’* ,V]“‘) the displacement vector corresponding to regular solutions of the membrane system, where i = 6 for axisymmetric vibrations ())I = 0) and i = 6.7.X for asymmetric vibrations (nr # 0) the displacement vector corresponding to a singular membrane solution whose leading terms (u,. I’,. w,} , 10, ,101 ,018 arc ,u, . t , , H’, i the displacement vector corresponding to the regular mcmbranc solutions whose leading terms (4. (‘,. w,I are (I~]“‘, 1,:“‘.t~f”‘;. where i = 6 for axisymmctric vibrations (nr = 0) and i = 6,7. X for asymmetric vibrations (no # 0) superscript (0) denotes zero-order or primary approximation.

I. INTRODUCTION

In shell vibrations, there is a turning point whose position depends on the frequency. Ross (1966) defined the turning point as any point at which the asymptotic approximation to four bending solutions is singular. Another definition was given by Gol’denveizer (1980) according to the solvability condition of the membrane system. which is obtained by putting h --+0 in the original system of equations. At the turning point, the membrane system has a singularity. These two definitions arc virtually equivalent to each other. The various types ofvibration with turning pointsin solid mechanics, includingshell vibrations, werediscussed hi Steele (I 976). 1311

IJI:.

J

K

The first-order

ZII\\C. .IllLi N

turning point which occurs in shell \ihrations

that in other titrkis of mechanics. The complexity Spects.

First.

the corresponding

order. which is much higher. stability

corresponds

Sommerf~td

equation of h>Jrod)namic

to a related equation of second-order. which is the t\ell-knoun

However. :I second more important point. As a result.

vibr~it~on~) solutions

is more complicated than

comes largely from the follo\bing tmo

related equation denirta! b> Langer‘s \ariabk ij of tifth In fact. the Orr

equation. and the related equation in toroid turning

Z&l \--.\(I

strength

analysis

Airy

is at mobt of third-order.

aspect is that the membrane system ij singular

one of the t\\o (axis~mmetric

Gkttionsf

of the membrane system is singular

at the

or four (asymmetric

and contains :I logarithmic

term

at the turning

point since the indiciai equation arisin, 17from ;t Forbcnius expansion at the turning point tvill have repeated roots. Thus. the turning pclint of the original s\stcrn is also

the branch point of thr: singular solution the turninp

point in shell

Sornmcrfeld

equation has ;I similar

of the solutions is ditkrcnt.

of the mrmtxtne

vibrations

s>strm.

has ;I singularit\

In this scnsc. WY say that

at the branch point. The

Orr

nature. lvherc the turning point is also the branch point

of the inviscous equation. However. the situation in toroid strength ;inalysis

The ditticulty

arising

~ippro~irnntion.

from this is in tindin, ‘1 ;I sofutiw to :i rclntcd equation. mcnthranc solution tc, the high-order

describe the singular

that can be used to uniformly

Here. the singular

original system whose zero-order

mcrnbrane solution

approximation

is defined as the solution

is the singular solution

01‘ the

of the mcrnbranc

S) stem. The first-order

turning

point in shell vibration5

has hccn invcstigatcd for many yc:tr>.

lictss ( 106hl ohtaincd matching ;I’iyrtlptk>tic st~lutions shells

of revolution,

~intl discussal

the possibility

for lhc auisyrttrtictric 01’ tinding

uriilixxi

cibr.atic>ns 01‘

;rqniptcttrc

rcscrlt~lti~~r~s. Ilc Lixuscci his ~Ittcltti~)rt on the li~~~irit~lr~l~c lll~tt~~~r~~~lc solution. coricludcrf that the log;~rithrnic

mcrnhr;tnc siAution

rloc5 rl~~t ayqxir

rcp-

1Io\rcvcr-. hc

to p~~324 ;I uniform

;I nionogr;~ph in IZussi;ln rcprcscntation in any scnsc. In 1979, Ciol‘dcnvcizcr hi l/l. publi&d ((iol‘dunvfi~tr t’f (I/.. 1079) wllich w3s rc;tlly ;I 5urvcy (>f prcviixr% p;ipcr\ in Russian litcraturc

on the ~i~ynipt(~tic solutions

seen from this monograph rcprcsont;tIiori

lhal

of the singular

of shell vihr~ttittnh.

the authors

nicmtxlnc

;lnrl

kid

solution ofthc Orr

(I%I),

qu;ttion,

whose singularit)

t lowcbcr, this gcncr;tlixd

of the generalized

I\iry functions

to dc;ll with the inner cspansions

Sommcrfcid

li~crilbr~tllc solution, solution

success I0 &lair1

uniform

2

alt hougti progress was m;ltlc in finding

solution,

that of the tour bcrtding stAutions. It is worth mentioning that ;I c;ttcgory of gcncrxlitcd by [)ra/in

inciuding our subject. it can bc

lricd without

Airy

is simil;lr

function

%;I> introduced

of the singular

inviscouh

to that ofour singular

V&IS not rccttgnircri

a~ ;f

Airy equation which is the generalization of the rclxtcd equation

of the Or-r Sommcrfcld equation. This rcwgnition solutions of the rclatcd equation ;trc considcrcd to

is thcorcticaliy qtr;llit2tivcly

have

impartant. since Only the wrtw bch2vior 2s

to express the latter. the soiutions of the original cqu3tions. and can thus bc u4 In the prcscnt paper. three c:ltcgorics of the gcncrali/cd function arc d&in& terms of which ;I singular respcctivelq. in p~lrticlll~ir, ;I gcncr;tlil;ition used to uniformly

membrane solution

and four bcnclin,0 4iitions

in

c;iri bc csprc54.

the second category of the gemx~lizcd function is obtrtincd lirst is hich is

of the new solution

of the rclatcd cqiation

expand the singul;lr

mcmbrant solution. Mefhoci, which ut)uld

modifying the Laplxc Transform :1ppro;tch to tinding the uniform solutions

in sllcll

vibrations,

of the ccjuations with the property rcduccd cquations, obtained by putting the small paramctcr 1:- 0 in the ori$nal

equations. arc singular

at lho turning

point of first order.

2. Tli13 S~Sl-f31 Of’ fX.H~~;\ritP.S After the substitutions

of

34

can bC

This new solution is found by probably provik :III ctrective that their system 01

Turning u,

point solutions

1313

for thin shell vibrations

(5. cp.1) = u(s) cos mcp sin of

u:(s, cp, t) = c(s) sin rnv sin or u,(s. cp, t) = W(S) cos mp sin wt for the surface loads X, and the displacements U, in the equations for thin shells of revolution in terms of displacements. the system of equations representing the free vibration of thin shells of revolution may be denoted as follows :

(lL+p5N)uJ= -(I -V’)RuJ; where the ordinary differential operators (Budiansky and Sanders, 1963) are

d = d/ds.

associated

with Sanders’

( ) = d( )/ds.

(1) theory

of thin shells

R. J

lill

Moreover. contain

j = I. 2.3: i+j

Pi,, (i = 1.2.3;

are neglected

.%A\(, =

6) and the loser-order

because they have no contribution

the different

terms

for

Pd., for

LHAV,

.inJ I4

the diflerrnt

thin

shell

value of Q,,. except P&:1. is neglected in the Gol’denvrirer 1979). The frequent) and thickness paramtxers are

Q = pw’R’/E

and

ditrercntial

to our subject.

theories.

terms of BY, : it is possible

However.

to

For example. e\er>

operators

(Gol’denvrizer

et ‘11..

/L’ = e“ = II’, 12, respectively.

(2)

It is clear that the evuluution of natural frequencies and their corresponding modes is. in fact. an eigenvalur problem of eqns ( I ) under an appropriate boundary condition.

When the frequency

have

eqns (I)

turning

points.

eqns (I) can be rewritten displncemcnt

R is in the frequency

parameter

Actually.

interval

;LS mentioned

as the following

hish-order

by Gol’denvcizer

equation

of’thc sccontl-order

vibration.

dcrivativc

Obviously,

for any shells of revolution.

when R is within

the turning

points

turning

turning

exists.

point

I:or ;I certain enc. parxlkl

II = X for ;isymmc1ric

cnctpl

paranictcr

cylintlric:ll

inlcrval

cqualions

points ofcqns

frcqucncy

has positive,

three

into

zero and negative

( I ) have JifTercnt

behaviors.

valid

in the whole

shells

is very dilticult.

interval

11 W;IS Langcr’s merely

in

and spherical

(3). The /cro

( I). and th

( I ). It is assumctl. C! in interval

parts: values.

Introducing

first-order

shells. /J(X) has Lcro 01’ /J(x) arc dclincti

/x0

in this paper,

(3). it is possible

as

of h(x) arc

to find one, and only

Lhc middle
points

that only ;I lirsl-order

surface of [hc shells

I.s-.s*/

rcspccrivcly,

and the corresponding

S (.Y, ,< s < .sl) and satisfy so far the uniformly

idea that

the uniformly

terms

nonclcmcntary

using qns

points

.s, < s < .s*.

of

of the original

valid

I ;~ncl .Y* < .s 6 5:. f Icrc. h(x)

the accuracy

Lalicl solutions

solutions

functions

equations.

of the so-cxllcd rclatcd equation

transformation

2nd thC cocfticicnt

solutions

ot‘eqns

Hcncc. to lind the solutions of cqns (I) which ;lrc uniformly

Actually,

;ls the solutions

the inkgrals

vihralion.

‘(s,]

.s = .Y+which s;ltisfics fZ = RJ .‘(,s,) and tlividcs

[he longitude

behavior

the frcqucncy

of lhc original

the first-or&r

expressed

the normal

is

h,(s) = h(s) = - ( I - vJ)[f2 -- R:

along

only

11’:

whcrc 11= h for asisymmctric

points

CI (I/. (1979).

including

in the lvholc interval

th:it

The

of the theory

have

of thin

have not been found.

the

nonclcmcntary

which can bc obtained

can be

S;I~C qualitative functions

from

arc

the Langcr

(I) or eqn (4).

the Langcr variables z and ;

(5)

and the new dcpcndcnt

vrlriable

Turning

Fig.

pant

soiutlons

I. Contours L, (k = 0. I.. . . .5)

=

vi

[

in the r-plane.

1

h(s) ’ ?f;:fl)

131.5

for thin shell wbrations

x

)lfS)

into cqn (4) and assuming that the solutions of the resulting equation can bc cxpnndcd as the following

asymptotic scrics

wc obtain ;I scrics ofcquations

3s foilows:

whcru the dilTcrcnlial operator

;rnd r(z), /I(:) and T(Z) are slowly varying cocliicirnts. The following

equation

is referred to as the related equation

corresponding

to the

original eqns (I) :

W’(i)

= 0,

the solutions of which are cniied rehtted functions. indcpcndcnt

of the geometrical

(8) It is interesting

that operator

A is

parameters of the shells. From this it follows that iill the

thin shells of revolution have the same related eqn (8) and the same related functions which describe the general charncteristic

4. I .

Tfic*fin-t id

bchnvior of their free vibration.

third ciitiyrrr~ oj’ the ycvicvwlix~circltrtd fiwtiot

According to the standard Laplace approach WC readily find the solutions of the related cqn (8) in intcgrid form

s

i*exp I.&

{;t-t”/5;

dt

(A-= 0, 1.2 .._,t 5).

where &, represents the contours in the complex t-plane. as shown in Fig. I. In addition, the differentials and integrals of/, can all be expressed by

R. J. ZHAW end U

FtAI;p) =

& jLAf p*exp

ZHA\(;

(jr-f’,51

d!

(k = 0. I.?...

.,5).

(10)

wherep=O,kl.+7,... and FK. respectively, represent the solutionsf, of the related eqn (8) when p = - I. and their differentials or integrals when p = - I. The first category of the generalized function _?,(;:p)(A = I. 2.3.4) is detined by a combination of Fh as follows : = -iF,

Y,

2.7 = -F,+F,

which are identically

9,

= i(--F,+F;)

2,

= i(F, +FO+(l

real functions

It is easily verified

for any real value of [.

that Y,, are the solutions

(A+/‘+ and satisfy the following

-i)F,,.

of the generalized

l)Y,,(
uYP,,(;;p)

formula

combination

= 0

(I 1)

l)P,,(l::/~)

(1’)

(13)

= P,,(<;p --I/)

Y,,(;;p-5)-i’%‘,,(<;p-

a linear

equation

relations

ADY',,(~';~J+ I) = -(p+

The recursion

related

I)+(/‘-

I)P,,(j;p)

= 0.

(14)

( 14) shows that for other values of p. Y,,(< :/I) can bc exprcsscd as for example, ;I’,,(< ; 0). Y,,(< ; I ), . . , S,,(< ; 4) with polynomial

of.

cocllioicnts. The asymptotic method

expressions

for

of steepest dcsccnts as follows

Y,(C;/J)

-

-r,,

c’{cos

Yz(<;p)

-

rl, c’{sin

Y,,(<;/J), :

when

(r+cp,,)+X,,[cos

(r+cp,)-&[cos

< + F CL, can bc obtained

(z+cp,,)+sin (z+fp,,)-sin

by the

(51+~~,,)]) (r+cp,,)ji

(Y’, and .P, arc usclcss).

(13,

and when i > 0 Y?(<;p)

4 -r,[sin

(O+cp,,)--_I’,, ~3s (~~+V,,)l

Y,(<;p)

+ r,, e”(l +_I’,)

Y,(;;P)

+ r,, [cos (O-cp,,)+_~*,,

sin ((I--cp,,)l+F,,(i:~)

(Y, is useless); where

(15):

Turning point ci=

sdutions

for thin shell vibrations

1317

’ [h(s)]’ J ds

s I.

The third category of the generalized related function is defined ns

It is easily verified that ,f({;p)

also satisfy relations (I I) ( 14) with Y,, replnccd by ,p, and

have the expressions :

Only four gcneraiized rclntcd functions

among _Th (11= I, ‘2, 3.4) and ,f are linearly

indupcndcnt because of the conncxion formula

4.2. Tlrr sc~nrrtl ctrtqnr I’ of’ fk gcwrtrli~d ActLli\lly. another &&pry

rc~krtcrl,Jirrrc~tic,,r

of solutions

of related equation (8) exists, which hils the

integral form (9) with the contours L,,. rcplaccd by IK (k = I, 2,. . . , 5). as shown in Fig. 2. when the stund:trd L~I~Ixc approach is used. However. thcsc solutions

cilnnot he genoralizcd in the same manner asl_ [see (9) and ’

(IO)], because the integrals (IO). with contours LK replaced by ix. have no si~njfic~nce at the origin t = 0 when p > 0. WC intend to find a new solution of the related equation (8). For standard Laplnce approach should he modified. Substituting

the following solution

this purpose, the

I?IX

R. J. ZHA%G and

Fig. 2. The deserted contours

./‘(i) =

i (‘

I,.

W.

ZHA’IC;

on which

the origin I = 0 lies.

cxp (
into cqn (8). gives

:;I;. dr = 0.

Obviously,j(;)

is the solution ofeqn

(8) when [r*I~*cxp

(jr).]< = 0

and (I’ -Z)c+cl(IL’)/df which is not identically

zero as in the standard

= r/l(r). Laplace approach,

analytic function in a simply connected region surrounded L represents an auxiliary

but is a single-valued.

by a closed contour C+ L, where

contour and satisfies

I

&r)*exp 1.

(if) dc = 0.

In our case, we choose b(r)

=

I

*exp ( --I ‘/S)

and contour C to be the path in the r-plane which starts at an infinite point coK = co *exp {2(k-

I)ni/5)

(A- = l,2..

. . , j),

encircles the origin once counter-clockwise and returns to its starting point; as well as the auxiliary contour L to be the circle whose center is at the origin and radius equals R + co, as shown in Fig. 3.

Turning

point solutions

1319

for thin shell vibrations

In this way, a new solution of the related equation (8) is found as

t.Inr*exp(jt-t5/5)dt

(k=

(18)

I,?,...,55

and. in a similar manner. the second category of the generalized related function is defined as follows :

.w:p)

= “jph.(i;p) =

&

I0 +I

1 ,...

(p=O._+l,+7

tWp In t-exp

(jf-r”/5)

dt

12

I.2 ,...,

;k=

where the first equals sign is valid because S,(i:p)

5);

is independent

of k (see the Appendix).

This generalized function can be used to describe the singular membrane was not obtained by Gol’denveizer

solution. which

PI al. (1979).

It is not difficult to verify that the second category of the generalized function satisfies the following

relations. which are similar to (I I)-( 14).

(A++r+I).WK;p) = j(<;p) AlIM((;p+

The recursion formula

-(p+

1)4i;p)+#(<;p)

(19)

(21) also shows that .M(<;p) can bc cxprcsscd. for diffcrcnt values

of p. as a linear combination ctiicicnts such as 2,

I) =

of, for example.

.&([;O).

. . . ,W(C;4)

and ,a. The asymptotic expressions for *([;p),

\ / /q / \\\ 1\ / 10

/

1

-2-

Fig. 3. Contour

C’? and auxiliwy

/

_J’ \

contour L in the r-plane.

with polynomial when c -P f co,

coare

i ;:(I

R J

where ;’ = 0.57711566-N.

ZH4Wi

and

Vi

LH:\w;

. . is Euler’s constant and 8 has already been given in formula

(15). Four

of the first

and third

categories of the generalized function.

category are independent. which constitute

the five basic solutions

and the second

of the related equation

(S) when p equ~~ls - 1.

I;. THE GENERAL IS TERMS

ESP-+X30\ OF THREE

OF THE SOLC’TIOSS CATEGORIES

OF THE ORIGISXL

OF THE GENER.-\LIZED

As sholvn in the previous sections. NY found the solutions

EQL;.4TIONS

FL’NCTION

of eyns (7)[. (7):. . . . step

by step. The procedure is actually ~qtli~~l~nt to the expansion of the solutions

in terms of

the generalized functions. 5.1. E.~~{itl.si(~~i of the ~~i~i~Ji~I{lr ~~li~i~~~ri~~lt, .si~l~~tii~tl As can bc seen from espression the singularity

of the singular

(IS). .#‘(; ; p) contains ;t factor, In I. which charxterizes

membrane solution

at the turning

membrane solution can be exup;indcd in terms of 91; ;I’). Letting p = - I in formula (20). recalling formula (16), and eqn (71,. WCconclwle

conlpilring

the

the singular result bvith

th;1t

.I

Substituting

point. Thus.

,/5, I,

=

.xc;:0).

(‘2)

(17) into uiln (7): and recalling the ilitTcrcntial

$,‘I =

;‘,

(:)A(,:

-J)+y,(:w(;:

rclatic~n (30). \vcobtain

I)

Ii’/’ = I in ecln (21 ). Ihc ;Ibovc result c;m finally bc rewritten as f,”

I

nscrtr

=

;‘,

(=,;.~(i;o,+g.,(=).d(;;

ng .ro(0 . y,15) , .I.(?0 . . . . in

cxpitnsion

I)+y,(:)f’(i;

(6) mc;ms the singular

1).

(‘1)

mcmbrnne solution

can

be cnprcsscd as :

However.

as mentioned exlicr.

a Iimxr

combination

formukt

(35) is rcwrittcn

for p 2 5, 2Q‘;;p)

and f(<;r)

can ho expressed 3s

of :HP(<; 0). _ . . . %‘(i ; 4) and ,f(< ; I), . . . , ,f(< ; 4), rcspcctivciy. Thus, 3s

where the second summation

term can bc simplicd using expressions (5) and (17) as

1321

Turnmg point solutions for thm shrll wbrskms

which

is the sIowly varying anatytic function representing the regular part of the singular membrane solution. Thus. the singular membrane solution can finrtlfy be expressed in terms

of the second category of the generalized function as follows :

where the slowly varying coefficients possess the follo~in1 2 asymptotic expansions

5.7. E,vprnsioti

of’Iltc*Jhur

htwriitly

salrrtimis

In the sami way 3s in Section 5. I, the four &ding

solutions

can finully bc expressed

as :

where the slowly varying cocllicicnts arc gitcn in terms 01’ the S;IIIIC notatiw. in cxprcssion (27). bccauso both satisfy tho same dilkfcntial

Once the ~CIICfilI expansions arc knowtl, bccomc tllc rlclcfmin;~tion

of tk

SlOWly Vilfyillg

n,,(:;p).

as

cqiiaticrns.

lintling the solutions

of the original cqns (I)

cocllicionts in lhcif expansions. Ih!G~llSC 01

lhc restriction of the accuracy of the theory of thin shells, dctcrrnination tcfm in the asymptotic expansions (70). (28) and (77) is suOicicnt.

of the primary

where the slowly varying cocfiicicnts &,. 3,. [I,, ;‘, and 0, (i = I , 2. 3) arc alI functions of the original vafiablc s. Substituting (29) into eqns (I). equating the cooliicicnts of .#(<:r) (p = 0, 3_ I, -&?) and constant terms at both sides to each other. WCobtain IX ordinary difkential equations illld

i 8 slowly varying cocfficicnts as unknown.

Solving the cqtrntions for thcsc cocfficicnts

yicfds

a, =a:

=o

a,(s) = D”[B(S)] .- “qPp’(.s)]- 3.2, where D,, is ;I constant that remains to bc detcrmincd later.

(30)

R

I.:?

It is not necessary singular

membrane

(p = 0, 2 I. 22)

which

indicates

sis different

hand.

combinations

It can also ;I manner

solution

where

vcrilicd

;p)

than the error

varying

Through

varying (p =

/j,+cp:,+

-

containing

the combinations solutions,

that the thrco combin;ltions

the

I. 0.

of thin

shells.

we obtain

except r ,. x2 and r ,. appear only

and skillful

mcmbranc

If we re-expand

I ) instead of .@‘P(;:p) ./A(;:p) (p = -3. -2. - I)

calculation.

kcp’tl, (i = I . 2.3) satisfy

icp’5,+

as can be seen.

coetfcients.

- 3. -2.

in the theory

coetficients.

a lengthy

hand. due to the singularity

that

the column

(II’;“. l.‘i”, 11,:“’),

F, G,.

and

vector

must

satisfy

formulae

of the column contain

have the gcncral

(7: and G, arc the constants

Substituting

Thus,

I5 slowly

of .d(;

LH\-.(,

it is possible

the membrane

are analytic.

Thus,

the following

relations

the mcmbranc

as

to verify

equations. between

the

exist

equations

in such

thikt

On the other conclude

order

and the three regular

lx

u’

and note that the terms

that the IS SION~!

combinations.

the other

in terms

of higher

that the combinations On

ZH\-..(. .Ird

to find the other

solution

as before.

are small quantities

J

(32).

eqns (34) can be rewritten

at cp = 0 WC immediately of the singular

form

to bc dctcrmincd.

(33) and (34)

as

vector

;I component

into eqns (31).

we obtain

membrane

1323

Turning point solutions for thin shell vibrations

PPS-P(GI

uy’ =

10’ L’S = &S

-P(G’

b(‘Y”= PRj -HG’+

+ Et In cp)ub”‘--~(G~+E~

E, In cp)u\”

In ~)u\~‘-_cI(G,+

+ El In ~)~~“’ --~lfG~ + E, In ~~)t’$“‘-pfG,

+

E, In cp)r’,O’

Et in ~)~~~‘--/J(G~+ EI In cp)~‘i”-p(G~+E,

In cp)~\“‘--~~~‘p-‘. (35)

Equations

(35) should be valid at the turning

point due to their uniform

Namely. provided we expand eqns (35) asymptotically with the singular solution

validity.

at 9 = 0, the result should coincide

of the mernbr~~n~ equation in the power series at the neigh-

bourhood of the turning point s = s, ; which is shown as :

Finally,

we obtain

E, = --it-‘,

EZ = E, =0

pP, = p:()‘, /IQ> = &“.

pR5 = rl:”

Wld

n,, =

-I1

-‘[lq.s,)j’ qfp’(s*,l’ ?,

which appear in formula (30). Thus.

the sin~ul;~r nIcr!~br~lnc solution

(3 I) is linaliy

fimttl

as

u,(< ; p) = p:o’(.s) -.H(< ; l)Ily’(.S) t*jt; ;p) = ll:(l’(.S)--d#yc; l)L~~“(.s~

r’*s(i;/o = f-l:“(s)- 2(( ; I)w~,o’(.s)+ z ,(.s).W(<;O), where

According to the gcnoral expansion (24). the bending solutions can be assumed as

Only the cocficients in the terms shown should be found. due to the limitation accuracy of the linear theory of thin shck.

of the

Substituting formula (37) into cqns (I). and equating the coefficients of L?“~(<;P) at both sides to each other, gave 15 ordinary difTerential equations and I5 slowly varying coeffkients z,,fll.. . . as the unknowns. which are the same as the equations in the previous section (6. I). From these equations, it is easy to obtain

where C,, is a constant. By using the uniform validity of solutions (37). i.e. in both the subintervalss, < J* and sJ 3 s, kvhich are far away from the turning point se. these solutions should bc identicali\, equal to those which are valid only in the two subintcrvnls. solutions

corrcspvnd

respectively.

tn the ~;1scs of low frequency and high frcyuency.

mcntioncd by Zhang. 1588). The constant C,, is dctermincd as

c’,, =

iI

1‘IB(.r*)]’ “[qT’(.s*)]l2.

(The latter two respectively.

as

Turning

5’0

CC0

I

I

I c2

t

I

c:L& \\ to3

/

. \ \

I

to2

\ \ \

\

\ /

to.

I’

I

tot

I

\ \

// .PCY /

\

\

/

to3

\ \

\ c4 Fig. Al.

t

>

\ ,./

L \ Il’

c2

I

/

?

1325

point sohtttons for thin shell vibrations

to4

\

\

\

\ \

\

\

’

’

C,

C&

Four saddle points and the sttupest descent paths in the r-plane.

cg’ = I.

C{) =9

(A3)

+p(p+J,

and the rcmaindcr arc not used since the addends they appear in [see oqn (AZ)1 are beyond the accuracy of the theory of thin shells. The contours of intc~r~~til~n as gtvcn in (IO) can bc dcformcd into the stcvpcst dcsccnt paths. Then F,(;:p)

whsrc f

=1 ,f(<;/~)

is thr third gcnsrali/cd

rclatsd function as dclincd hcforr.

C’lcarly the values of I,,..,. I/I,, and P, are probably all complex although the variable < is real. To circumvent this dkadvantugc, the tirst gcncralired related function 3,(i:p) (/I = I .2.3.4) is dclincd as the following combinations of F,(<;p) (k = I. 2.. . .5):

3, = -iF, r: = -F:+F, Y,=i(-f,+F,) Y-, = OF, +F,)+(I Thus. aftsr some manipulation. pivcn in cqns (IS)].

the asymptotic

-i)jJ.

rcprcscntations

(Ah)

of Y,,(< :p) as ; -+ + *fi arc linally found [as

It GIII bc SCWI from the definition of /(<:p) that d(c;p) zr 0 if p < 0; otherwise it is a polynomial in < of in the expansion ol’ cnp {{r- :I‘;. The first degree p- I which. by the rcsiduc theorem, is the coctiicicnt of I’ few of these polynomials are given in eqn (I 7).

rk

If P ih an InIcgcr. in i\ inIcresIiny

(tip1.k

- 2

IO IWIL’ Ih;tI r(<:p’

r

J(;;p’

(A’)’

(p = 0. + I. ,2 . . . . ).

Thus. if p is an inrsyer. WC conclude IhaI ,Y(; .p’ = 0 for alI p S 0. olhsrwix!

Ihc vx~cs in (AX) Iurm~naIcs, and

which IS jusI formuLl (16’. Diti?renIiaIion of (h7) wlIh rcspcct IO p (p6C) g~vcs Ihc second goncralircd rcl;lIcd funcllon .I(;:pJ = M,(<;p). as defined before. Obvcrscly, jp([;p) is also indcpcndenr of k. Thus. by ditrercnIiaIing boIh sides of (AX) and isrting p taks an inregsr v;~lus again. the ;~sympIoI~c rcprc~ccnI:cIions of .X(<:p) can be found. when < - + 7., ils follows :

whcrc : = 0.5772156649 is Euler’s consIanI. Only 3(;; - I). .#(;;O) and .a(;; I) arc ugful given in Section 3.

in our

sub~cxl.

;~nd

Ihcir asymptotic

represenLtIions

have been