Nonlinear analysis of the internal resonance response of an L-shaped beam structure considering quadratic and cubic nonlinearity

The nonlinear behaviors of the L-shaped beam structure based on 1:2 internal resonance are investigated. The governing equations considering quadratic and cubic nonlinearity of the L-shaped beam structure are verified by the transient dynamic analysis of the finite element model. The approximate analytical solution for this system is derived by the method of multiple scales, and verified by the numerical solution. The Jacobian matrix of the modulation equation is employed to determine the equilibrium stability of the vibration response. The effects of the excitation frequency, amplitude and cubic nonlinear terms on the nonlinear responses of the primary resonance of the first and second modes are discussed. The bifurcation diagram, spectrum, phase plane and Poincaré section are applied to investigate the nonlinear vibration response. The results reveal that many nonlinear phenomena are observed, such as double-jump, hysteresis, Hopf bifurcation, modal saturation, multiple solutions, multi-softening and so on. The influence of cubic nonlinear term on internal resonance response should be considered. The broadband characteristics of internal resonance system are broadened by the nonlinear softening and hardening in the frequency response curve. In the primary resonance of the second mode, the hardening region in the frequency response curve becomes the softening region with the increase of excitation amplitude. The double-jump phenomenon is found in this newly formed softening region, and up to five solutions are observed in this region, including three sink points and two saddle-nodes. The broadband characteristics of the system is of great benefit to the application of broadband piezoelectric vibration energy harvesting.