Axisymmetric 3: 1 internal resonance of thin-walled hyperelastic cylindrical shells under both axial and radial excitations

Nonlinear vibration with axisymmetric 3: 1 internal resonance is investigated for an incompressible neo-Hookean hyperelastic cylindrical shell under both axial and radial harmonic excitations. A full nonlinear strain-displacement relation is derived from the large deflection theory of thin-walled shells. A set of nonlinear differential equations describing the large deflection vibration are formulated by the Lagrange equation and the assumption of small strains. Steady-state responses of the system are predicted via the harmonic balance method with the arc length continuation, and their stabilities are determined via the modified sorting method. The effects of excitations on the steady-state responses are analyzed. The results reveal a crucial role played by the phase difference in the structural response, and the phase difference can effectively control the amplitude of vibration.