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TECHNISCHE MECHANIK, Band 19, Heft 3, (1999), 187-195
Manuskripteingang: 30. November 1998
Parametric Vibrations of Viscoelastic Cylindrical Shell
under Static and Periodic Axial Loads
S. P. Kuntsevich, G. I. Mikhasev
Low-frequency parametric vibrations of a viscoelastic cylindrical shell subjected to axial static and additional
periodic loads are studied It is assumed that the shell is noncircular and the load is non—uniform in the circum-
ferential direction. It is supposed that for the weak parametric excitation the shell vibrations are localized near
the weakest generatrix on the shell surface. By using Fourier transformations over the circumferential co-
ordinate and the multiple scale method with respect to time, the solutions of the shell equations are constructed
in the form offunctions that decrease quickly outside a small neighbourhood of the weakest line. The region of
instability ofthe shell is determined with regard to the viscosity.
1 Introduction
To the present time a lot of investigations on the parametrical instability of thin cylindrical shells have been
carried out. For example, vibrations of shells subjected to various combinations of static and periodic loads have
been considered by Yao (1963), Wenzke (1963), Vijayaraghavan and Evan-Iwanowski (1967), and Grundman
(1970). However, the majority of the obtained results concern ideal shells with constant parameters. It has been
known that in this case the parametric Vibrations are accompanied by the formation of waves covering the whole
surface of the shell, and the problem of dynamic instability reduces to the Mathieu equation with coefficients
that are functions of a static bifurcation load and a fundamental frequency of a shell.
In this paper we examine the parametric instability of non-circular thin cylindrical shells, which experience static
and additional periodic axial loads. Both load components are inhomogeneous in the circumferential direction,
and the frequency of excitation is close to double the fundamental frequency ofthe shell. The case is considered
when vibrations are characterized by the localization of modes near the weakest (Tovstik, 1995) generatrix on
the shell surface.
This investigation is a continuation of Mikhasev’s article (1997). The main goal of this paper is to study the
special case that cannot be examined by the methods that have been used by Mikhasev (1997). In addition, the
influence of a viscous damping coefficient on the main instability region is studied here.
2 Setting a Problem
We consider the thin viscoelastic non—circular cylindrical shell that is sufficiently thin for the applicability of
both the assumptions of the classical shell theory and the asymptotic methods. The orthogonal co-ordinate sys-
tem (x, (p) is assumed as shown in Figure 1 so that the first quadratic form of the middle surface has the form
R2 (ds2 + dcpz). Here s : x R'1 is a dimensionless longitudinal co-ordinate (0 S s S l = L/R), R is the characteristic
size of the middle surface (it will be defined below), L is the shell length, q) is a circumferential co-ordinate
(cpl S q) 3 (p2). In this case the curvature radius R2 = R x“ ((p) is variable.
Figure 1. The Co-ordinate System
187
Let the shell be subjected to the combined non—uniform axial load (see Figure 1)
w = 83 EhF<<p, r) Fm), r) =F0 (so) + 221i (cp) cos 9* t* (1)
where E is the Young’s modulus, h the shell thickness, 86 = h2 / [12 R2 (1 — v2)] a small parameter, v Poisson’s
ratio, 9* the circular frequency of the additional periodic axial load, and t* the time.
It is supposed that the vibrations are accompanied by the formation of a large number of short waves covering
the shell surface. It is assumed also that F (q), t) is a slowly varying function so that the stress state of the shell
due to the axial stress T1* may be considered as the membrane one. Taking into account these assumptions, for
the analysis of parametric vibrations the semi-membrane shell equations (Bolotin, 1956; Vlasov, 1958; Tovstik,
1995)
62(1) 62w 62w Öw
söAzw—äxap) 852 +83F((P;l)mi‘+—Ö7+SZY57:O
62 (2)
83A2C13+x(<p)a;:=0
written in the dimensionless form may be used. Here A : 82 / 8s2 + 62 / 8902. The parameters 8 and 1F 1 are con-
nected with the corresponding magnitudes introduced in Mikhasev’s paper (1997) by the following relations
8 = M2/3 F‘1 z 8-1/2 13‘]
The dimensionless magnitudes are introduced as follows:
w=w*/R cI>=cI>*/(a3EhR2)
(3)
t=t*/tc Q=Q*tc y=8"/ZtC5/p t,.=R p/E
where w* is the normal deflection, (13* is the stress function, to the characteristic time, ö is the viscous damping
coefficient, p is the mass density. The functions x ((9), F0 ((p), F1 ((p) are supposed to be infinitely differentiable.
Let the shell edges be joint supported so that at the edges s = 0, s = lNavier’s conditions
_62w _ 262(1) 2
“W43 asz 0 (4)
W
are posed. To satisfy boundary conditions (4) the solution of equations (2) is assumed to be ofthe form
’ p„s . nS
w : w„ ((p, t) Sln q) =fn ((1), f) 51" (pas/2)
5
83/21/17.[ ()
p": 1 "21,2,
The cases p" < l and p" > l have been investigated by Mikhasev (1997). Now we consider the case p,, z 1. Sub-
stituting (5) into (2) yields the sequence of equations