Bursting Oscillations and Vibration Isolation in Geometrical Nonlinear Systems with Complex Softening Characteristics

In this paper, the influences of complex softening characteristics from geometrical nonlinear restoring forces on bursting oscillations and vibration isolation for ultra-low frequency excitation are investigated. By adjusting three independent geometrical parameters of the restoring force, various quasi-zero stiffness characteristics with different numbers and degrees of softening intervals can be achieved. The mechanisms through which the complex softening characteristics affect bursting oscillations are revealed from the perspectives of softening interval number and softening degree. The influence on ultra-low frequency transmissibility is analyzed by revealing how variations in softening characteristics govern the distribution and magnitude of transmissibility peaks. A previously unreported phenomenon of transmissibility below one in unconventional isolation frequency bands is identified, and the influence of static characteristics on its occurrence is investigated. The results indicate that the locations of transmissibility peaks are closely related to the restoring force values at zero-stiffness points, while the number of peaks is determined by the parity and the number of the softening intervals in the restoring force curve. Increasing the softening degree of the restoring force curve results in higher transmissibility peaks due to stronger spiking states generated when the system trajectory passes through steeper equilibrium branches. The increase in softening degree also affects the evolution patterns of hysteresis loops in systems exhibiting multiple softening intervals. The steep equilibrium branch near zero-stiffness points away from the origin amplifies sensitivity to excitation amplitude, leading to the observed phenomenon of transmissibility below one in unconventional isolation bands. Enhanced softening characteristics narrow the frequency range of this phenomenon and drive the minimum transmissibility toward one, thus reducing its vibration isolation effectiveness.