Soft-material structures have excellent characteristics of infinite degrees of freedom and large deformation, and it has important theoretical significance and application value to perform mathematical modeling and dynamic analysis. This paper studies the large-amplitude oscillation of the cylindrical shell under a harmonic excitation, where the constitutive relationship is described by the Zener rheological model based on the Rivlin-Saunders hyperelastic model. First, the Euler Lagrange equation is used to establish the nonlinear ordinary differential equation describing the radially symmetric motion of the structure, and the viscous evolution equation of the material is derived based on the rheological model, thus obtaining the governing equations of the nonlinear system. Second, based on the zero-viscosity and infinite-viscosity models, the bifurcation behaviors and natural frequency analyses of the nonlinear dynamics of thin-walled structures under constant loads are carried out. Third, based on the small perturbation assumption of the Maxwell unit, an improved Melnikov method suitable for the dynamic analysis of the visco-hyperelastic shells under harmonic excitation is proposed and verified by numerical methods. Finally, the chaos threshold of the system is analyzed based on the improved Melnikov method.