Nonlinear Oscillations of Hyperelastic Annular Membranes With Varying Density

This research presents the mathematical modeling for the nonlinear oscillations analysis of a pre-stretched hyperelastic annular membrane with varying density under finite deformations. The membrane material is assumed to be homogeneous, isotropic, and neo-Hookean and the variation of the membrane density in the radial direction is investigated. The membrane is first subjected to a uniform radial traction along its outer circumference and the stretched membrane is fixed along the outer boundary. Then the equations of motion of the pre-stretched membrane are derived. From the linearized equations of motion, the natural frequencies and mode shapes of the membrane are obtained analytically. The vibration modes are described by hypergeometric functions, which are used to approximate the nonlinear deformation field using the Galerkin method. The results are compared with the results evaluated for the same membrane using a nonlinear finite element formulation. The results show the influence of the stretching ratio and varying density on the linear and nonlinear oscillations of the membrane.