Bifurcation and Chaos of Hyperelastic Spherical Membrane under Structural Damping

This paper investigates the dynamic responses of spherical membrane composed of an incompressible hyperelastic material following the Yeoh model, under the simultaneous influence of periodic disturbance loads and structural damping. Based on the energy variational principle, a strongly nonlinear differential equation is established to provide an approximate description of the radially symmetric motion of a spherical membrane. Through qualitative analysis of the solution, some meaningful conclusions are obtained for the spherical membrane: (1) For constant loading without damping, the influence of constant load on the equilibrium points of the system was analyzed through the equilibrium point curves and potential energy curves, and the critical parameter E0 that determines the motion trajectory of the spherical membranes was obtained. Periodic motion and amplitude jumping phenomena are discussed by analyzing the system's potential wells. (2) For loading of periodic disturbance loads without damping, the quasi-periodic and chaotic motions of the spherical membranes in the secondary steering bifurcation scenario have been detailed. The effects of the periodic disturbance load on the chaotic motions of the membrane have been further analyzed. (3) For combined loading of periodic disturbance loads and structural damping, the effects of factors such as the damping coefficient and periodic disturbance loads on the chaotic motion of the spherical membrane are analyzed primarily using Poincaré sections. Especially, the motion of the membrane under structural damping generates a strange attractor, which provides significant implications for its practical applications in engineering fields.