Nonlinear dynamics of a bio-inspired 2-DOF low-frequency X-shaped vibration isolator with m-to-n layers driven harmonically under simultaneous primary and 1:1 internal resonances

It is well-established that $X-$shaped bio-inspired structures function effectively as low-frequency vibration isolators when excitation frequencies remain below the structure's natural frequency. However, when subjected to high-frequency excitations, the nonlinear behavior of these structures dominates, leading to the emergence of multistable solutions and strong vibrations, even with small base excitation amplitudes. The primary objective of this work is to mitigate strong vibrations and eliminate solution bifurcations in such isolators under high-frequency excitations while maintaining their high-performance low-frequency isolation capabilities. For the first time, an $m-$layer $X-$shaped vibration absorber has been integrated into an $n-$layer bio-inspired $X-$shaped low-frequency vibration isolator system to enhance the overall vibration isolation performance. The mathematical model governing the oscillatory behavior of the combined isolator-absorber system was derived as a two-degree-of-freedom nonlinear dynamical system using the principles of Lagrangian mechanics. By applying the harmonic balance method, an accurate analytical solution for the isolator-absorber system was obtained. The evolution of steady-state oscillation amplitudes for both the isolator and absorber was then explored as a function of both the base excitation frequencies and the excitation force amplitude. The effects of various $X-$shaped absorber parameters, such as rod length, number of layers, stiffness coefficient, mass, and inclination angle, on the isolator's oscillatory behavior were thoroughly investigated. The key findings confirm that a multilayer, high-stiffness $X-$shaped absorber system can effectively eliminate multistable solutions and mitigate resonant vibrations of the isolator system under high-frequency excitations, without compromising the system's low-frequency vibration isolation performance. However, a poor design of the $X-$shaped absorber parameters may destabilize the motion of the $X-$shaped isolator system, leading to the emergence of a quasi-periodic or chaotic response rather than enhancing its vibration isolation performance.