The aim of the present work is to investigate the nonlinear vibration response of a pre-stretched rectangular hyperelastic membrane resting on a nonlinear elastic foundation. The membrane is composed of an isotropic, homogeneous and hyperelastic material, which is modeled as a Mooney-Rivlin incompressible material. The elastic foundation is described by a Winkler type nonlinear model with cubic nonlinearity. First the exact solution of the membrane under a biaxial stretch is obtained. Then the equations of motion of the pre-stretched membrane resting on the nonlinear foundation are derived. From the linearized equations, the natural frequencies and mode shapes of the membrane are obtained analytically. Then the natural modes are used to approximate the nonlinear deformation field using the Galerkin method. The results compare well with the results evaluated for the same membrane using a nonlinear finite element formulation. The results show the strong influence of the initial stretching ratio and foundation parameters on the linear and nonlinear oscillations and stability of the membrane.

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