Principal parametric resonance of axially accelerating rectangular thin plate in magnetic field

Nonlinear parametric vibration and stability is investigated for an axially accelerating rectangular thin plate subjected to parametric excitations resulting from the axial time-varying tension and axial time-varying speed in the magnetic field. Considering geometric nonlinearity, based on the expressions of total kinetic energy, potential energy, and electromagnetic force, the nonlinear magneto-elastic vibration equations of axially moving rectangular thin plate are derived by using the Hamilton principle. Based on displacement mode hypothesis, by using the Galerkin method, the nonlinear parametric oscillation equation of the axially moving rectangular thin plate with four simply supported edges in the transverse magnetic field is obtained. The nonlinear principal parametric resonance amplitude-frequency equation is further derived by means of the multiple-scale method. The stability of the steady-state solution is also discussed, and the critical condition of stability is determined. As numerical examples for an axially moving rectangular thin plate, the influences of the detuning parameter, axial speed, axial tension, and magnetic induction intensity on the principal parametric resonance behavior are investigated.