A new modification of homotopy analysis method (HAM) is proposed in this paper. The auxiliary differential operator is specifically chosen so that more than one secular term must be eliminated. The proposed method can capture asymmetric and period-2 solutions with satisfactory accuracy and hence can be used to predict symmetry-breaking and period-doubling bifurcation points. The variation of accuracy is investigated when different number of frequencies are considered.
A new modi¯cation of homotopy analysis method (HAM) is proposed for capturing asymmetric solutions of wire rope isolation systems. Analytical expressions of asymmetric solutions to wire rope isolation systems are obtained. A dynamic system with quadratic polynomial restoring force is investigated speci¯cally. Then the analytical results are applied to a single-degree-of-freedom (SDOF) system with wire rope vibration isolator to investigate the response curve and other dynamic characteristics. The analytical approximations match satisfactorily with the numerical results. The presented analytical approximation is a useful method to derive the response curves and examine limit cycles without resorting to numerical simulations.
Effect of passive vibration isolation heavily depends on force-displacement characteristic of the isolation system. In view of this dependence, this paper investigates the influence of negative stiffness device (NSD) on the effect of vibration isolation system. Detailed evaluation of transmissibility is performed.The critical parameters are identified. It is found that with NSD, significant vibration reduction for both absolute and relative displacement transmissibility is obtained. A modified Lindstedt-Poincaré method (modified L-P method) is used to obtain analytical periodic solutions for the approximated piecewise linear dynamic system. The analytical limit cycles by the modified L-P solution agree satisfactorily with the ones by numerical simulation. The most important finding of this study is that larger damping in a system with NSD helps in reducing transmissibility, thus increasing the efficacy of the isolation system; this is in contrast to other conventional isolation systems, wherein increased structural damping decreases the efficacy of vibration control in the frequency range of interest.
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