Generation of waves in an elastic plate by a vertical force and by a moment in the vertical plane

Abstract

An approximate solution is determined for the motion of an infinite elastic plate, excited by a vertical force (normal to the plate) and by a moment in the vertical plane (with the axis of the moment parallel to the plate). The driving force and moment are sinusoidal in time and applied to a small, rigid indenter with a circular base, fixed to the plate. The solution is obtained from a three-dimensional approach but is restricted to low frequencies, where the wavelength of the longitudinal, transverse and bending waves are much larger than the thickness of the plate. The solution contains, for both cases of excitation, three parts that describe a bending wave, a longitudinal wave and a local reaction. The solution for the bending wave is exactly the same as that derived from the usual two-dimensional plate theory. The amplitude of the longitudinal wave will, in most cases in practice, be small compared with the amplitudes of the bending wave or the local reaction. The local reaction is built up of four infinite sums of waves with complex wave numbers. For the case of a thick plate, the amplitude of the local reaction for a constant force or moment increases with increasing thickness of the plate and approaches the value found for a semi-infinite medium. The amplitude decreases exponentially with the distance from the center of the indenter when this distance is larger than the thickness of the plate. In the opposite case the amplitude decreases with distance in approximately the same way as on the surface of a semi-infinite medium. Except for the local reaction at a distance from the indenter that is larger than thickness of the plate, the amplitudes of all waves are found to be independent of the size of the indenter, provided the indenter is small compared with the appropriate wavelength.