Bifurcation and singularity analysis of HSLDS vibration isolation system with elastic base

In order to reveal the bifurcation mechanism and optimize the system design for high-static-low-dynamic-stiffness (HSLDS) vibration isolation system (VIS) with elastic base, the local bifurcation analyses both in unfolding parameter space and physical parameter space were carried out theoretically and numerically. Firstly, the restoring force of the HSLDS-VIS was approximated to linear and cubic stiffness by applying the Maclaurin series expansion and the motion equations of HSLDS-VIS with elastic base were established. Subsequently, the motion equations of HSLDS-VIS with elastic base were formulated to transform the system into a standard form and the averaging method was applied to obtain the single-variable bifurcation equation for the HSLDS-VIS with elastic base in case of primary resonance and 1:2 internal resonance. Furthermore, the transition sets and bifurcation diagrams in the unfolding parameter space were studied by means of singularity theory. Finally, for the engineering application, the transition sets were transferred back to the physical parameter space, thus to obtain the bifurcation diagrams of the amplitude with respect to the external force. The numerical simulation results show that the local bifurcations of HSLDS-VIS with elastic base in case of 1:2 internal resonance are considerable complex and need to be analyzed in six two-parameters spaces, meanwhile, the necessary condition of multiple solutions lies in some physical parameters, which can provide a theoretical basis and reference for design and application of the HSLDS-VIS with elastic base.