Nonlinear vibration and resonance analysis of a rectangular hyperelastic membrane resting on a Winkler–Pasternak elastic medium under hydrostatic pressure

Abstract

In this paper, the nonlinear vibration and resonance analyses of a rectangular hyperelastic membrane embedded within a nonlinear Winkler–Pasternak elastic medium subjected to a uniformly distributed hydrostatic pressure are investigated. The material of the membrane is incompressible, homogeneous, isotropic, and hyperelastic. The constitutive law of neo-Hookean is utilized for modeling the system. The elastic foundation includes two Winkler and Pasternak linear terms and a Winkler term with cubic nonlinearity. Using the theory of thin hyperelastic membrane, Hamilton’s principle, and assuming the finite deformations, the governing equations are obtained. Then, the motion equation in the transverse direction is discretized by applying Galerkin’s method. The transverse nonlinear oscillations of the membrane are considered in two-modes. Then, utilizing the multiple scales method, the secondary resonance for the case of $\mathrm{\Omega }\approx 2{\text{ω}}_2+{\text{ω}}_1$ is analyzed. Also, to analyze the nonlinear vibration behavior, numerical methods including the fourth-order Runge–Kutta methods are utilized. In addition, in analyzing the nonlinear vibration responses of the rectangular hyperelastic membrane, the bifurcation diagram, quasi-period motion responses, and the phase portrait are examined. Finally, the influence of the geometrical characteristics and elastic foundation parameters on the resonance analysis of a rectangular hyperelastic membrane is investigated.