Free Vibrations of a Non-uniformly Heated Viscoelastic Cylindrical Shell

Transcription

1 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 nonuniform 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) 82W e4a2w Kit LT Wtdt+ + =0 [ i< <ch (> as, at, a2 [N s tachn g 2 W J K(t t,t(<p))w(t)dt = 0 S (1) A=32/8s2+82/8(p2 8=h2/[12R2(1 v2)] W = e4r 1W * c1) = c1>*/(e4eh) z: t /tc t, = prz/(84e) 251

2 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 d1 t r'(t(<p>> = j Wrap» = wrap» (3) 0 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). 1 T = IM (qm cramp) Tg> 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 _ 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} WZ 2281/2wn(s, ) =e' 2(cp <p0) Imb>0 Im Q>0 n=0 (5) 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 T((p) = T0 + (1 coscp0 + el zg sinch +ä 2 cos<p0+...) (6) 252

3 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% :"% azlo 292L0 a = b +2b + b + lb + LZ 2 ap2 8p8<p0 agpozä l öpz apaqo gag y 8&2 i ("HO al0 Q 28p8cp m t j _Q C:C ;Q = K - e""dt 0 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 Where P0 is apolynomial in ä, H<p,<po; 20> E 7»;(1 C0) näp4 +(1 C0>p8 = 0 (10) T (Ä l) = S Ä M T 7» MO) : K( m5) UKOMZ) K( 10] SK(x) = %(coshx+cosx) TK(x) 1.. = E(s1nhx+s1nx) 1 UK(x) = 3(coshx cosx) and km is a root of the transcendental equation cosh(7»l)cos(ä1) 1 = 0 253

4 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> where Thus, the weakest line (po =0 is the coldest one here. 068=g(p (93) p" = x142 wszo (14> 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 p 5i 4(21)»; Qg+kfilaC0 PO=O 890 (16) 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 ( ) 254

5 Then l _ i(n+1/2)b82h /8p2 (18) 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 ' ' ' ' ' O 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()) Figure 2. Parameters (of and 0t? vs. 0): Figure 2 represents the graphs of the functions (hf zu)? ((1) ), e (xi =0L{ for various u. Here 031 = R6521, oci7 =Ile. It may be seen that the corrections 8001', eoc l (see formula (6)) for the frequency 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. 255

6 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), 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), Addresses: Marina G, Botogova, Research Fellow, Byelorussion State Polytechnic Academy, Block 1, 65 Fr. Skorina Ave, BY , Minsk; Associate Professor Dr. Gennadi I. Mikhasev, Vitebsk State University, 33 Moskovsky Ave., BY Vitebsk 256