Wave-vector analysis of the forced vibrations of cylindrical shells of finite length

Abstract

The forced vibrations of thin shells of finite length are analyzed from a structural wave viewpoint. The proposed theory is restricted to in-vacuo vibrations and to radial harmonic excitation at a surface point on the shell. The theory is based on Donnell’s equations of motion for thin shells. The solution to the forced problem is obtained by solving a boundary value problem that involves a 16×16 system. The 16 equations that are necessary to solve the system are: 4 boundary conditions at each end of the shell, 7 continuity conditions at the excitation point, and 1 discontinuity condition at the excitation point. The method is based in part on Forsberg’s original paper [AIAA J. 2 (12), 2150–2156 (1964)] on the free vibrations of shells of finite length, and, in part, on a general technique for solving one-dimensional forced vibration problems. The theory can be easily implemented on a microcomputer. Some sample axial mode shapes are calculated. To gain physical insight, all the waves that combine into the given mode shape are presented. The theory is also used to perform a wave-vector analysis of the predicted vibration field on the shell surface to separate the contributions from flexural, shear, and longitudinal waves. It is found that a Prony decomposition algorithm in the axial direction of the shell is more useful than the standard fast Fourier transform because it yields a much finer resolution of the helical wave spectrum. The algorithm developed in this study does not include internal losses in the shell. It appears that, with no internal losses, the amplitude of the helical wave spectrum is very sensitive to the physical parameters of the shell, in particular to its thickness, when the excitation frequency is very close to a natural frequency of the shell.