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

Abstract

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.

Nonlinear Dynamic Response of Galfenol Cantilever Energy Harvester Considering Geometric Nonlinear with a Nonlinear Energy Sink

“…It was concluded that with the increase in the excitation amplitude, the geometric nonlinearity will significantly impact the output features. Nie et al [51][52][53][54] introduced the structure's and piezoelectric materials' geometric nonlinearities into the L-shaped piezoelectric energy harvester. The analysis indicated that the energy harvester with the geometric nonlinearities considered could harvest power more efficiently with a reduced vibration displacement over a wider frequency bandwidth.…”

“…As a new type of magnetostrictive material, Galfenol has become a particularly promising transducer material due to its excellent characteristics and is gradually being applied in vibration energy harvesting. In our previous work, the research focus was more on introducing geometric nonlinearity into piezoelectric energy harvesters [51][52][53][54]. In 2023, based on Galfenol materials, we proposed a magneto-electric energy harvester without taking into account the cantilever structure's geometric nonlinearity [15].…”

Aerodynamic Force and Aeroelastic Response Characteristics Analyses for the Galloping of Ice-Covered Four-Split Transmission Lines in Oblique Flows

“…Nino et al [30] and Ferretti et al [31] analyzed the nonlinear aero-elastic behavior of cables undergoing dynamical in-plain galloping. Nie et al [32,33] studied the internal resonance response of an L-shaped beam structure, considering quadratic and cubic nonlinearities. The in-plane and out-of-plane resonance theories can be used in discussing the galloping conditions of iced conductors from the perspective of the whole span, which provides a useful supplement to the vertical and torsional galloping mechanism.…”