Nonlinear frequency response of breathing modes in hyperelastic cylindrical shells under harmonic excitation

Cylindrical shells can vibrate in three distinct modes: shell modes, beam modes, and breathing modes. Understanding breathing modes is crucial, especially when shells are subjected to internal pressure or explosive loads in spacecraft and pressure vessel applications. This study focuses on the breathing mode of simply supported thin-walled cylindrical hyperelastic shells made from Mooney–Rivlin material, subjected to radial harmonic excitation. By analyzing the shell’s spatial deflections, the coupled equations of motion are derived using the Lagrangian in terms of generalized coordinates and substituting it into the Euler–Lagrange equations. Assuming radial motion is dominant, the system simplifies to a single governing equation, which unlike asymmetric modes, contains both quadratic and cubic nonlinearities. The quadratic nonlinearity poses a softening behavior to the shell for some parametric values. Introducing dimensionless parameters, this equation is solved using the method of multiple scales to examine nonlinear frequency responses and numerically to explore nonlinear dynamics. The frequency response curves reveal that, unlike shell modes, which typically show hardening behavior, breathing modes of hyperelastic shells can display either softening or hardening characteristics depending on the shell’s radius-to-length ratio. Softening for low and hardening for high radius-to-length ratios. Additionally, the presence of damping is shown to transform the shell’s quasi-periodic motion into asymptotically stable.