TECHNISCHE MECHANIK, Band 16, Heft 3, (I996),251»256

Manuskripteingang: 16. April 1996

Free Vibrations of a Non-uniformly Heated Viscoelastic

Cylindrical Shell

M.G. Botogova, G. I. Mikhasev

Low—frequency free vibrations of a viscoelastic cylindrical shell with free edges taking into account a non-

uniform temperature field is investigated. Using Tovstik’s asymptotic method, the solution of the shell

equations are derived in the form offunctions, quickly oscillating and damping near “the coldest” generatrix.

1 Introduction

We will study low-frequency free vibrations of a non—uniformly heated viscoelastic cylindrical shell.

Temperature stresses are supposed to be absent here, and a relaxation kernel is assumed to be some function of

the non—uniform temperature.

The characteristic property of the problem under consideration is the localization of vibration modes in a

vicinity of “the weakest generatrix”, which is “the coldest one” here. For the first time, the localization of

vibration modes of an elastic noncircular cylindrical shell with slanted edges was studied by Tovstik (1983).

Later viscoelastic cylindrical shells were investigated by Mikhasev (1992).

2 Problem Setup

Consider a thin circular cylindrical shell of constant length l and thickness h. We introduce an orthogonal

coordinate system s, (p, so that the first quadratic form of the middle surface takes the form R2 (d s2 + d (p 2) ,

where R is the radius of the middle surface of the shell, and s and (p are the axial and circumferential

coordinates, respectively. The shell material is assumed to be linearly viscoelastic with instantaneous Young's

modulus E and Poisson's ratio v.

Suppose the shell edges are free, and the temperature distribution in the shell is

T = TO + u(1—cosq))

where 0 < u, T0 are constants. Then initial temperature stresses are missing (Podstrigach and Schvets, 1978),

and for analysis of the lowest part of the spectrum of free vibrations, the following basic equations, written in

dimensionless form, can be used:

t

N 82d) 82We4A2W— Kit—LT Wtdt+ + =0[ i< <ch (> as, at,

(1)

a2 [Ns‘tAchn—g—2 W— J K(t—t,T(<p))W(t)dt = 0

S

A=32/8s2+82/8(p2 £8=h2/[12R2(1—v2)]

W = e4R‘1W’* c1) = c1>*/(e4Eh) z: t’/tc t, = pRZ/(84E)

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Here W* ,(I)* are the normal deflection and the stress function, respectively, p is the mass density, a is a natural

small parameter, lat, T((p)) is the relaxation kernel of the shell material, and IL. is the characteristic time.

According to the temperature~time analogy (Ferry, 1963), the relaxation kernel may be represented in the form

Ea,m» = Korma) <2)

where

I

r'(T(<p>> = j

0

d1 t

——=— (3)Wrap» wrap»

is the reduced time and aT(T((p)) is the coefficient of temperature—time reduction. For many polymers in a

large range of temperatures the function aT(T((p)) obeys the William— Landel — Ferry formula (Ferry, 1963).

cramp) — Tg>1 T =—

IM (qm c§+T(<p>—T,

where Tg is the reduction temperature, and the coefficients clg and c5 are derived experimentally, It is

assumed =T0 here.

3 Method of Solution

We shall investigate the lowest part of the spectrum of vibrations, corresponding to the main stress-strain state

of the shell (Tovstik, 1983). Then, for the free edges of the shell, the main boundary conditions will take the

form

82W _ 83W _

852 853

0 for s = o, z (4)

The approximate solution of the boundary value problem (1), (4) can be expressed as (Tovstik, 1983;

Mikhasev, 1992)

W = Wz(s,§)exp{i[§2t +e-“Zpg +äbä2}

(5)

WZ 2281/2wn(s,§) §=e'“2(cp—<p0) Imb>0 Im Q>0

n=0

where (p =(p 0 is the weakest generatrix, wn (xi) are polynomials in E. The function (I) is found in the same

form. The last inequalities guarantee attenuation of the wave amplitudes far from the line (p =(p 0 and damping

of vibrations in the course of time.

The required complex frequency Q and the temperature T((p) can be expanded into the series

Q = QO +8.01 +8292+.„

T((p) = T0 + „(1— coscp0 + el’zg sinch +䀀2 cos<p0+...) (6)

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The substitution of equations (5) and (6) into equations (1) and (4) produces a sequence of boundary value

problems

iijn_,. =0 n=0, 1, 2... (7)

/=0

882:" : 56:2" :0 for s = 0, l (8)

where

Lo=“;#%+tp4a—Co)—szozi A=b%é+§%£—i%—:"%

1 28210 8210 8210 2 . aZLO 292L0 a 18210 . 32=— b —+2b + — b + — ——— lb +— —

LZ 2 ap2 8p8<p0 agpozä l öpz aPaQo gag 28192 y 8&2

i ("HO aL0—— Q28p8cp0+ 1990

+m t j _Q

C:C ;Q = K—————-—e""dt0 0(‘P0 0) El; [QHTWOD

4 Zeroth- , First- and Second-order Approximations

The solution of problem (7), (8) for n = 0 is easily seen to be

Wo = P0(§)ym(s) (9)

under the condition

H<p,<po;£20> E 7»;(1—C0)—näp4 +(1— C0>p8 = 0 (10)

Where P0 is apolynomial in ä,

T (Ä l)

= S Ä —M——T 7»MO) |: K( m5) UKOMZ) K( „10]

SK(x) = %(coshx+cosx)

1 . .

TK(x) = E(s1nhx+s1nx)

1

UK(x) = 3(coshx—cosx)

and km is a root of the transcendental equation

cosh(7»l)cos(Ä1)—1 = 0

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Separating in equation (10) the real and imaginary parts , we obtain

(11)

2

=—m+p4 (12)

where (00 =ReQO, 0L0 =ImS20, B0 =Re C0, A0 =——ImC0. It follows from equations (ll) and (12) that

OLO = g (p,q>0), 030 = f(p,(p0) . The minimization of the last function overp and (p0 yields

w3=nnnf(p,<po)=f(p",<p3) (13>

068=g(p”‚(93)

where

p" = x142 wszo (14>

Thus, the weakest line (pO =0 is the coldest one here.

It is of interest to note that

is the lowest frequency of the elastic shell vibrations corresponding to the eigenvalue km .

Taking into account equations (13) and (14), equation (7) is reduced to the identity LlwO E O. Then the

solution of problem (7), (8) for n = 1 can be represented in the form

W1 = P1(&)ym(s) (15)

where Plfi) is a polynomial in g again.

The compatibility condition of boundary value problem (7) and (8) for n = 2 gives the equation

1 82H” 82H” 2H" 1 182H” 2P_ 2 b2+ 2 §2p0_iba_T _PO+ ü ___7_az_0_

2 3p 3% 3p“ 2 9€ 2 3p“ 5i

(16)

4(21)»; Qg+kfilaC0 PO=O

890

The mark 0 in equation (16) and below means that all functions are calculated at

pail}, (p0=0, w0=wg‚a0=ag.Equation(16)hasthesolution P”(E_,)=aN§” +...+a1§+a0 if

b2: 82H”/8<p%_17

aZHU/apZ ( )

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Then

_ i(N+1/2)b82H”/8p2 (18)

l 2ng (s23 aims/ago)

5 Example

Numerical computations for a shell made from a polymer material , with cf =18,l , cg =45 , Tg =276 K

(Ferry, 1963), and the kernel (thanitsyn, 1968)

0 4 e"

K t =’——— (19)

0 r(o,1)z°~9

were performed. Here F(x) is the gamma function. The calculations were conducted for N =0, m=1 .

30 (03 ' ‘ ' ' ' ' 1 (18’ ' ' ' ' '

20 -

0.5-

10 -

‚ 602 602

O 10 20 30 0 10 20 30

Figure 1. Fundamental frequency 008 and damping decrement a8 vs. 0)”6

Figure 1 shows that the relationships between the parameters 033 ‚ org and the frequency a): of the elastic

vibrations are almost linear.

0.4 I I I I I I

u: 10

O. -2 F5

:0 u

I I I\ I I I())6

0 10 20 30

Figure 2. Parameters (of and 0t? vs. 0):

Figure 2 represents the graphs of the functions (hf zu)? ((1) ”), (xi’=0L{’ for various u. Here 031”e

= R6521, oci7 =Ile. It may be seen that the corrections 8001', eoc‘l’ (see formula (6)) for the fre-

quency m3 and the damping decrement 0t f ‚ respectively, depend on parameter it, which specifies the power

of nonhomogeneity of the temperature field. In the case of thanitsin’s kernel (19), the influence of parameter

u is more pronounced at wzz4 .

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6 Conclusion

The influence of a nonuniform temperature field on low-frequency free vibrations of a viscoelastic cylindrical

shell with free edges has been demonstrated.

Literature

1. Ferry J.: Viscoelastic Properties of Polymers, Inost. Lit, Moscow, (1963), 535 p.

2. Goldenveizer A. L.;Lidsky V. B.;T0vstik P. E.: Free Vibrations of Thin Elastic Shells, Nauka,

Moscow, (1979), 384 p.

3. Mikhasev G. I.: Lowfrequency Free Vibrations of Viscoelastic Cylindrical Shells, Prikl. Mekh.,

Vol. 28, No. 9, (1992), 50-55.

4. Podstrigach Yu. S.; Schvets R. N.: Thermoelasticity of Thin Shells, Kiev, (1978), 343 p.

5. thanitsyn A. R.: Theory of Creeping, Stroyizdat, Moscow, (1968), 416 p.

6. Tovstik P. E.: Some Problems of the Stability of Cylindrical and Conical Shells, Prikl. Mat. Mekh.,

Vol. 47, No. 5, (1983), 815-822.

Addresses: Marina G, Botogova, Research Fellow, Byelorussion State Polytechnic Academy, Block 1,

65 Fr. Skorina Ave, BY - 220027, Minsk; Associate Professor Dr. Gennadi I. Mikhasev, Vitebsk

State University, 33 Moskovsky Ave., BY — 210036 Vitebsk

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