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The nonlinear dynamics of a vibration-controlled magnetic system are studied via a three-mode discretization of the governing partial differential equations. The analysis focuses specifically on the effects of modal coupling through the nonlinear terms of the system equation. A bifurcation analysis of the system is performed using sophisticated nonlinear theories, including the center manifold theory and the normal form theorem. The results show that when the first mode and the higher modes are excited simultaneously by the control forces, the three-mode approximation method predicts the existence of a triple zero degeneracy accompanied by complicated bifurcation phenomena. Comparing the dynamics structure predicted by the three-mode approximation model with that obtained from a single-mode approach, it is found that if the higher modes are excited by the control forces, the effects of modal coupling should be taken into consideration since a complicated dynamics structure may exist as a result.

This paper presents a Finite Element Analysis (FEA) of modal energy exchange and harvesting in a simple portal frame structure. We consider both horizontal and vertical support excitations resonant first with the first mode (sway mode) and latter with the second mode (the first symmetrical mode). As 2:1 internal resonance is present between these two modes, the phenomena of mode saturation and energy exchange (modal coupling) may occur. Thus, energy pumped into the system through one of the modes, is partially transferred to the other mode, not directly excited. An evaluation of the energy available for harvesting in each of the considered mode is computed.

Modal coupling is an important characteristic of most nonlinear dynamic systems. Although the existence of modal coupling can make the dynamic responses of a system quite complicated, it can be utilized to provide potential solutions to regulate nonlinear vibration. In this paper, the mechanism of modal coupling is used as a means to suppress vibration of a flexible arm undergoing joint motion. A secondary oscillatory system is attached to the flexible arm to generate appropriate modal coupling. Via this nonlinear coupling, internal resonance can be successfully induced and used to transfer vibration energy from the flexible arm to the vibration absorber. Moreover, the damping enhancement effect is studied, which is found to help suppressing vibrations. The feasibility of the idea presented herein has been demonstrated in the numerical simulations.

Wind-induced response analysis is an important process in the design of large-span roofs. Conventional time-domain methods are computationally more expensive than frequency-domain algorithms; however, the latter are not as accurate because of the ill-treatment of the modal coupling effects. This paper revisited the derivations of the frequency-domain algorithm and proposed a fast algorithm for estimating the dynamic wind-induced response considering duly the modal coupling effects. With the wind load cross-spectra modeled by rational functions, closed-form solutions to the frequency-domain integrals can be calculated by Cauchy’s residue theorem, rather than by numerical integration, thereby reducing the truncation errors and enhancing the efficiency of computation. The algorithm is applied to the analysis of a grandstand roof and a spherical dome. Through comparison with time domain analyses results, the algorithm is proved to be reliable. A criterion of the coupling modal combination was suggested based on the cumulative modal contribution rate of over 70%.