Superharmonic and Subharmonic Resonance Analysis of A Rectangular Hyperelastic Membrane Resting on Nonlinear Elastic Foundation Using The Method of Multiple Scales

In this paper, the nonlinear vibrations of a rectangular hyperelastic membrane resting on a nonlinear elastic Winkler-Pasternak foundation subjected to uniformly distributed hydrostatic pressure are investigated. The membrane is composed of an incompressible, homogeneous, and isotropic material. 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. Also, the kinetic energy, the work of uniform distributed force and pressure, and the effects of damping are determined, according to the strain energy function for neo-Hookean hyperelastic constitutive law. By applying Galerkin’s method, the nonlinear partial differential equation of motion in the transversal direction is transformed to the ordinary differential equations. Then, utilizing the method of multiple scales, the superharmonic and subharmonic resonances including the 1:3 superharmonic and 3:1 subharmonic, 1:5 superharmonic, and 5:1 subharmonic, 1:7 superharmonic, and 7:1 subharmonic are analyzed. Also, the analytical results are compared with those presented by other researchers. Finally, the effect of the Winkler and Pasternak stiffness, the material properties, and various geometrical characteristics on the superharmonic and subharmonic resonances of the vibration behavior of a rectangular hyperelastic membrane is investigated.