On the Parametric Resonance of a Cylindrical Shell Subjected to a Periodic Axial Load

Abstract

The dynamic stability of thin-walled circular cylindrical shell is studied for the case of periodic axial load. In computing the internal membrane stress induced by the axial pressures, the cylinder is assumed to act as a longitudinal bar. The internal membrane stress is computed by considering the shell to be a longitudinal rod with axial inertia terms included. Hence, the internal axial coordinate as well as with time and to include the resonances of the cylinder acting as a longitudinal rod. The problem under study is to determine the stability of the flexural motions of the shell oscillating about this inextensional mode. The shell motion is represented by Donnell's equations. A study of the solutions of these equations reveals parametric resonance of the well-known type and a second parametric resonance which appears to be new. The latter includes the combination resonance between two modes having the same modal pattern, and also between two modes having different modal patterns. In particular, this includes combination resonance between two transverse modes having a different number of axial half waves. This result is believed to be new and is considered the principal result of the study being described. It is only obtained when axial inertia is included in computing the axial membrane stress. It is found that combination resonance does not exist between modes having a different number of circumferential waves. The two modes must have the same number of axial waves. An estimate is given for the width of the unstable regions, and numerical results are presented. The extension to a shell having the effects of rotatory inertia and shear deformation is briefly discussed.