A perturbation analysis of non-linear free flexural vibrations of a circular cylindrical shell

Non-linear free vibrations of a circular cylindrical shell are examined using Donnell's equations. A modal expansion is used for the normal displacement that satisfies the boundary conditions for the normal displacement exactly, but the boundary conditions for the in-plane displacements are satisfied approximately by an averaging technique. Galerkin technique is used to reduce the problem to a system of coupled non-linear ordinary differential equations for the modal amplitudes. These non-linear differential equations are solved for arbitrary initial conditions by using the multiple-time-scaling technique. Explicit values of the coefficients that appear in the forementioned Galerkin system of equations are given, in terms of non-dimensional parameters characterizing the shell geometry and material properties, for a three mode case, for which results for specific initial conditions are presented. A comparison of the results with those obtained in previous studies of the problem is presented and the discrepancies are discussed.