Nonlinear vibration of a slightly curved beam with quasi-zero-stiffness isolators

Bending vibration of isolated structures has always been neglected when the vibration isolation was studied. Isolated structures have usually been treated as discrete systems. In this study, dynamics of a slightly curved beam supported by quasi-zero-stiffness systems are firstly presented. In order to achieve quasi-zero-stiffness, a nonlinear isolation system is implemented via three linear springs. A nonlinear dynamic model of the slightly curved beam with nonlinear isolations is established. It includes square nonlinearity, cubic nonlinearity, and nonlinear boundaries. Then, the mode functions and the frequencies of the curved beam with elastic boundaries are derived. The schemes of the finite difference method (FDM) and the Galerkin truncation method (GTM) are, respectively, proposed to obtain nonlinear responses of the curved beam with nonlinear boundaries. Numerical results demonstrate that both the GTM and the FDM yield accurate solutions for the nonlinear dynamics of curved structures with nonsimple boundaries. The multi-mode resonance characteristics of the curved beam affect the vibration isolation efficiency. The quasi-zero-stiffness isolators reduce the transmissibility of modal resonances and provide a promising future for isolating the bending vibration of the flexible structure. However, the initial curvature significantly increases the resonant frequency of the flexible structure, and thus the frequency range of the effective vibration isolation is narrower. Furthermore, the quadratic nonlinear terms in the curved beam make the dynamic phenomenon more complicated. Therefore, it is more challenging and necessary to investigate the isolation of the bending vibration of the initial curved structure.