Investigating nonlinear vibrations of higher-order hyper-elastic beams using the Hamiltonian method

This paper presents a higher-order shear deformation beam theory for modeling and nonlinear vibration analysis of hyper-elastic beams made of silicon rubber and unfilled natural rubber. Four models named neo-Hookean, Mooney–Rivlin, Ishihara, and Yeoh models are presented, and their efficacy in nonlinear dynamic modeling of hyper-elastic beams has been explored. Geometric nonlinearity of the hyper-elastic beam is considered based on von-Kármán-type nonlinearity. The hyper-elastic beam is resting on a nonlinearly hardening elastic foundation. It is shown that the Ishihara model is a suitable model for nonlinear vibration analysis of hyper-elastic beams accounting for the shear deformation effect. The nonlinear governing equations based on the presented beam theory are analytically solved via the Hamiltonian method to find nonlinear vibration frequencies. It is shown that the nonlinear vibration behavior of hyper-elastic beams is influenced by rubber-material type and material parameters of the hyper-elastic model.