Nonlinear Vibrations of a Floating Longitudinally Compressed Elastic Plate in the Interaction of Wave Harmonics of Finite Amplitude

Abstract

The equations for three nonlinear approximations of flexural-gravity vibrations of a longitudinally compressed elastic plate which take into account the nonlinearity of acceleration of vertical plate displacements are obtained using the method of multiple scales. The ice plate unbounded in horizontal directions floats on the surface of a homogeneous ideal fluid. Asymptotic expansions up to the third order of smallness are constructed on the basis of the obtained equations for the plate-fluid surface elevation and the velocity potential of the liquid particles formed in the nonlinear interaction of two harmonics of progressive periodic surface waves. The amplitude-phase characteristics of the formed elevation of the fluid surface (bending of the plate) are analyzed as functions of the basin depth, the parameters of ice plate and interacting harmonics, and the nonlinearity of acceleration of vertical ice displacements.