Analyzing softening and hardening behavior in vibration of a thin incompressible hyperelastic cylindrical shell

Abstract

This study focuses on nonlinear vibration of a thin incompressible hyperelastic cylindrical shell subjected to radial harmonic excitation. Mooney–Rivlin model is used to find the Strain Energy Density and model the material’s nonlinear behavior. The independent bending terms are considered for shape deformation of the middle surface. The nonlinear relevant equations of motion are obtained using Donnell’s nonlinear shallow-shell theory and Lagrange’s equation based on the small strain hypothesis. In this study, the linear and nonlinear models are verified by comparing the present theory’s natural frequency results with corresponding results obtained using a finite element method. Subsequently, the derived equations are solved under undamped and free vibration conditions, leading to an equation of motion in the form of a Duffing equation. This equation is then analyzed using the Lindstedt-Poincaré technique. Multiple Scale Method is used to address the system of coupled nonlinear equations in the presence of harmonic force and damping term. Backbone curves and amplitude response for both asymmetric and axisymmetric modes are presented by considering coupled and, non-coupled governed equations. The effects of the geometrical and material characteristics are studied on the free and forced vibration of shell structure as well as the impact of shape function, which describe the deformation of the middle surface, on both backbone and amplitude-response.