Methods for calculation of free vibrations of a cylindrical shell with attached rigid body

Abstract

We consider a mechanical system consisting of a circular cylindrical shell and an absolutely rigid body attached to one of the ends of the shell. Using the principle of possible displacements, we construct a mathematical model for the equilibrium state of the considered system under loads of general form. Using this model, we formulate an eigenvalue boundary-value problem that describes the free vibrations of the "shell-body" system and propose its approximate solution. In the case where the shell is replaced by an equivalent Timoshenko beam, we construct an exact solution of the problem under consideration. We also give an estimate for the effect of the rigid body on the vibrations of the system and investigate the accuracy of the beam approximation of flexural vibrations of the shell.

Dynamic instability of a circular cylindrical shell carrying a top mass under base excitation: Experiments and theory

“…Bondarenko and Telalov (1982) studied experimentally the dynamic instability and nonlinear oscillations of shells; the frequency response was hardening in the region of the main parametric resonance for circumferential wave number n = 2 and softening for n > 2. Trotsenko and Trotsenko (2004), studied vibrations of circular cylindrical shells, with attached rigid bodies, by means of a mixed expansion based on trigonometric functions and Legendre polynomials; they considered only linear vibrations. Pellicano (2005) presented experimental results about violent vibration phenomena appearing in a shell with base harmonic excitation and carrying a rigid mass on the top.…”

Solution of a Problem of Free Vibrations of a Nonclosed Shell of Revolution under Conditions of its Singular Perturbation

“…It should be noted that the regular part of basis (25) coincides with the basis used earlier in the solution of an analogous problem by the Ritz method for average values of the small parameter µ [17].…”

“…A certain progress can be made by approximating the required solutions by Legendre polynomials whose moduli do not exceed 1 for any argument. This enables one to increase the limit value of the number N by a factor of three to four with respect to the power basis and, hence, to extend the range of input parameters of the problem for which computations can be carried out with given accuracy [17].…”