Large amplitude vibration and parametric instability of inextensional beams on the elastic foundation

The nonlinear vibration and parametric instability of the inextensional beam on the elastic foundation are investigated. Considering the inextensional condition and second-order moment of the subgrade reaction, the motion equation of the beam on the elastic foundation is obtained via the Hamilton principle. Then the Galerkin method is applied to obtain the discretized model. The periodic motion is examined by means of the continuation method. Whereas, the numerical simulation is performed to study the nonperiodic motion, particular attention is placed on the nonlinear response of two resonant beams and the parametric instability. Finally, the effects of the boundary conditions and foundation models on the nonlinear vibration of the beams are discussed.