Vibration Energy Localization from Nonlinear Quasi-Periodic Coupled Magnets

The present study investigates the modeling of the vibration energy localization from a nonlinear quasi-periodic system. The periodic system consists of n moving magnets held by n elastic structures and coupled by a nonlinear magnetic force. The quasi-periodic system has been obtained by mistuning one of the n elastic structures of the system. The mistuning of the periodic system has been achieved by changing either the linear mechanical stiffness or the mass of the elastic structures. The whole system has been modeled by forced Duffing equations for each degree of freedom. The forced Duffing equations involve the geometric nonlinearity and the mechanical damping of the elastic structures and the magnetic nonlinearity of the magnetic coupling. The governing equations, modelling the quasi-periodic system, have been solved using a numerical method combining the harmonic balance method and the asymptotic numerical method. This numerical technique allows transforming the nonlinearities present in the governing equations into purely polynomial quadratic terms. The obtained results of the stiffness and mass mistuning of the quasi-periodic system have been analyzed and discussed in depth. The obtained results showed that the mistuning and the coupling coefficients have a significant effect on the oscillation amplitude of the perturbed degree of freedom.