Parametric resonance of arbitrarily layered composite circular cylindrical shells.

Abstract

A composite circular cylindrical shell subjected to harmonic axial loading is studied. The shell is composed of arbitrarily layered fiber reinforced laminae. Lamination theory is used to develop the stiffness properties of the overall shell structure. Coupling between bending, stretching, and twisting in the shell's response can result because of the generality of the layering. The shell is modeled using thin shell theory. An element of the shell in a deformed configuration gives the equations of motion, one of which is nonlinear. Linearization is achieved via a perturbation technique which involves splitting the solution into a pre-instability response and a perturbed response. The equations of motion for the perturbed response of the shell are first partially uncoupled. Fourier series' are then used to suppress spatial dependence and generate a set of Mathieu-type differential equations. The technique allows any type of boundary conditions to be satisfied. Clamped conditions are studied here. The static and dynamic stability, as well as the free vibration natural frequency of a response described by a set of Mathieu equations each involve the numerical calculation of a determinant. This is done by solving the associated eigenvalue problem. The unperturbed response is assumed to be axisymmetric and inertialess; the unperturbed response equations of motion are then solved. Spatial variation in these solutions is allowed, in contrast to the commonly used assumption of a constant membrane state. Results are presented showing that the spatial variation is confined to regions near the shell ends. One difference over isotropy, however, is the appearance of significant shear stress brought upon by material coupling. The effects on parametric resonance of including unperturbed response spatial dependence is studied. It is shown that these effects are largely dependent on the laminate configuration. Specific results are presented for an example in which these effects increase the width of the instability zone by 112% for moderate loading. The variation of the parametric resonance frequency with lamina orientation is studied. It is shown that for a given shell geometry, the parametric resonance frequency and static buckling load do not always attain their respective relative maxima at the same orientations.