Vibrations of Plates and Shells

This chapter provides an overview of the vibration of rectangular and circular plates and of cylindrical and shallow shells. It should be recalled that if a structure is subjected to a harmonic applied force, large amplitudes will occur if the frequency of this force is equal, or nearly equal, to one of the natural frequencies, and that the natural frequencies may be required to evaluate response, using an extension of the method. Thus, in some sections only natural frequencies and mode shapes are considered. The necessary relations between stress, strain, and displacement are derived in Cartesian coordinates for use, when considering rectangular plates. The components of direct stress and strain are related by Hooke's law, extended to include Poisson's ratio effects. Homogeneous isotropic elastic solids are considered in this chapter; for these solids there are only two independent elastic constant. The corresponding relations in cylindrical polar coordinates are given for use when considering cylindrical shells.