Nonlinear vibration and resonance analysis of a rectangular hyperelastic membrane surrounded by nonlinear elastic foundation

Abstract

In this article, the nonlinear vibration analysis of a rectangular hyperelastic membrane embedded within the nonlinear elastic foundation exposed to uniformly distributed hydrostatic pressure is investigated. The aim is to study the nonlinear vibration behaviors of a rectangular hyperelastic membrane when both primary and 1:1 internal resonances happen in the system. Also, by considering the system embedded within the nonlinear elastic foundation, the model of the structure has been developed close to the real situation. The structure is made of an isotropic, homogeneous, incompressible, and hyperelastic material. To model the system, the neo-Hookean hyperelastic constitutive law is used. The nonlinear elastic foundations include two linear terms Winkler and Pasternak, and a nonlinear cubic term. The governing equations are obtained with Hamilton’s principle of thin hyperelastic membrane and the assumption of finite deformations. Then, using Galerkin’s method, the equation of motion in the transverse direction, which is considered in two modes, is discretized. The resonant case includes the 1:1 internal resonance between two modes and primary resonance, in which the excitation frequency is close to the natural frequency of the first mode. In this analysis, the responses of nonlinear vibration behavior, including primary and 1:1 internal resonances, are obtained with analytical and numerical methods. The analytical and numerical methods are, respectively, the multiple scales method and the fourth-order Runge–Kutta method. Finally, the effects of the elastic foundation parameters and geometrical characteristics on the nonlinear vibration behavior are examined.