Mathematical modeling and investigation of nonlinear vibration of perfectly conductive plates in an inclined magnetic field

Abstract

Nonlinear differential equations and boundary conditions describing the vibrations and stability of perfectly conductive, viscoelastic plates in an inclined homogeneous external magnetic field have been obtained. On the basis of obtained equations two various problems of wave propagation in the plate are considered: The problems of stability and propagation of modulated waves. The problem is reduced to the nonlinear equation of Shrodinger type. In the particular case when viscosity of the material is not taken into account, the condition for magnetic field is obtained when the modulated waves are unstable. Problems of nonlinear resonance interaction of magnetoelastic waves. For simplicity three-wave resonance is studied. In this case a system of nonlinear ordinary differential equations for amplitude and phase of interacting waves are obtained. In a particular case, when the viscosity of material is neglected, the obtained system was solved analytically. In the general case it is solved numerically.