History dependent analysis of compliant beams for nonlinear vibration isolation

Compliant beams exhibit nonlinear restoring forces at large deformations, and are adopted for nonlinear vibration isolators with quasi-zero stiffness and/or high-static-low-dynamic stiffness. The processing, the fixing, and the assembly of compliant beams induce pre-deformations before they are adopted for the isolation. The neglect of historical factors leads to unexpected experimental phenomena without theoretical explanations. To address the issue, a history dependent method is proposed for compliant beams. The deformation history is divided into sequential deformation processes, and each process is discretized into sequential deformed states following a determined deformation path. A beam constraint model is adopted with the geometric constraints characterizing the historical factors. A path-following method is introduced to solve the model and ensures the continuity of the deformation history. The restoring forces, deformed shapes, and stress are obtained, and the frequency responses are acquired with a subsequent harmonic balance method. A ring-type isolator and a bridgetype isolator are investigated and verified by experiments. The processing of bending initially straight beams to circular/semi-circular rings induces pre-deformations. The deformed shapes of the rings are related to the deformation paths during the processing. The historical factors affect the restoring forces, the stress, and the resonant frequency. The neglect of the history dependence results in a misestimation on the maximum stress of the rings, and the misestimation leads to the unexpected yields in experiments. A slight fixing error induces predeformations in a bridge-type isolator, and leads to multiple force-displacement relations in load and unload processes. The combined history dependent effect of fixing error and asymmetric processing leads to a reduction in the restoring force and an asymmetric deformation of the bridge-type structure. The history dependent effects originate from the nonlinearities in the structure.