Three to One Resonant Responses of Inextensional Beams on the Elastic Foundation

Abstract

The large amplitude vibrations of the inextensional beam resting on the elastic foundation under threetoone internal resonance are investigated. The inextensional condition and multimodal discretization are used to obtain the equation of inplane motion and modulation equations. Due to the secondorder moment of the subgrade reaction, the quadratic nonlinearity is included in the present model. Moreover, this moment destroys the conservative character of the system. The nonlinear response and the associated stability are examined by means of frequency (force)response curves, and the shooting method is applied to investigate the dynamic solutions. Particular attention is placed on the effects of the cutoff frequency and boundary conditions. The results show that the cutoff frequency and boundary conditions do not significantly affect the contribution of nonresonant modes to the nonlinear coefficients. Moreover, the effects of foundation models on the threetoone resonant dynamics of the beam are discussed.