The Second-Harmonic Resonance Of An Annular Thin Plate

Nonlinear viberations of plates and shells have drawn widely attentions over a long period of time. With the developing of nonlinear dynamics, some complicate behavior of plates and shells has been discovered. The second-harmonic resonance is the distinctive features of nonlinear oscillation. When the second-harmonic resonance occurs, the oscillation system can be excited into resonance by the external excitation whose frequency is far from the innate frequency of the system.Selecting suitable orthogonal functions and using the Galerkin method, the residuals of the dynamic von Karman partial differential equations are eliminated; thereby the governing equation of an annular thin plate for forced vibration is got. This equation is also called the hard-spring type of Duffing equation. Due to the regulating by the cubnonlinearities, some parametric ranges in which 3-times superharmonic vibration or 1/3-times subharmonic vibration arises exist in the sytem. When the excitation frequency approaching 3-times or 1/3-times resonance frequency, the system response is stable and periodic .This fact shows that regulating action of the cubic nonlinearities on the vibration system is strong. The method of multiple scales was used to solve the Duffing equation, the solutions were used to analyze qualitatively the effect of boundary conditions, ratio of damp, external excitation, and ratio of inner radius to outer radius on the second-harmonic resonance of an annular thin plate. As to strongly nonlinear vibrations of an annular thin plate without damping, we seek the solution of the second-harmonic resonance by MLP method.