# Nonlinear Vibration Isolation via a NiTiNOL Wire Rope
## 1 Introduction  
Vibration isolators are widely used in mechanical engineering, civil engineering, and aerospace technology. Nonlinear vibration isolators [1; 2; 3] are superior to linear vibration isolators, with better broadband isolation performance. The stiffness nonlinearity endows the isolator with a low-frequency vibration isolation ability without weakening the load capacity. Damping nonlinearity is beneficial for suppressing the resonance without degrading the high-frequency isolation efficiency. Isolators with both stiffness and damping nonlinearities show promise for presenting compound advantages. Nonlinear stiffness can be accomplished with linear springs with different orientations [4; 5; 6], buckling beams [7; 8; 9], spring-roller-cam mechanisms [10; 11; 12], and magnetics [13; 14; 15]. Nonlinear damping can be induced by geometric constraints [16; 17; 18; 19; 20], frictions [21; 22], materials with intrinsic nonlinearities [23; 24], and eddy current effects [25; 26]. Vibration isolators with both stiffness and damping nonlinearities have been proposed in recent years. [PERSON] et al. [16] proposed an isolator with nonlinear stiffness and damping derived from horizontal linear springs and dampers, respectively. [PERSON] et al. [17] proposeda quasi-zero-stiffness isolator with nonlinear damping, and the stiffness and damping nonlinearities resulted from geometric constraints. [PERSON] et al. [18] investigated a quasi-zero-stiffness isolator with a spring-roller-cam mechanism integrated with nonlinear damping accomplished with two horizontal dampers. [PERSON] et al. [19] studied the nonlinear damping characteristics of the limb-like structured or X-structured isolators. [PERSON] et al. [20] proposed a quasi-zero-stiffness isolator with nonlinear damping using a three-link mechanism. In these isolators, the damping nonlinearity is derived from deliberately designed geometric constraints, and thus, the isolator structures are complicated. In addition, the damping is viscous in most cases, and there is a lack of research regarding the nonlinear stiffness isolator enhanced with non-viscous damping. Generally, the nonlinear damping force is approximated as an explicit function of the velocity and displacement for dynamic analysis. [PERSON] et al. [27] investigated a vibration isolation system with real-power nonlinearities in restoring and damping forces through a multiple scales method. [PERSON] et al. [28] studied a vibration isolator with high-order stiffness and cubic damping via an output frequency response function approach. [PERSON] et al. [29] investigated the effects of the hardening and softening of the damping on a quasi-zero-stiffness vibration isolator with an averaging method. [PERSON] et al. [22] performed a harmonic balance analysis on a quasi-zero-stiffness isolation system with dry friction damping, expressed as a piecewise function. However, some hysteretic damping induced by frictions and phase-change materials is history-dependent and characterized by phenomenological models, such as the Bouc-Wen model [30; 31]. The Bouc-Wen model expression is both piecewise and implicit. There are two major approaches to dealing with the dynamic system with the Bouc-Wen model. The first approach is to take a piecewise fitting on the Bouc-Wen-type hysteretic force, and thus, the implicit expression can be transformed into a piecewise explicit expression. [PERSON] et al. [30] and [PERSON] et al. [31] performed multiple scales analyses on hysteretic systems with piecewise fittings on the Bouc-Wen model, respectively. The second approach is to adopt an alternating frequency/time domain technique [32], and the implicit function is numerically solved in the time domain. This study was focused on a modified Bouc-Wen model. The expression is more complicated than the classic Bouc-Wen model, and it is difficult to perform a piecewise fitting on the hysteretic force. The alternating frequency/time domain technique is more suitable and its effectiveness for the modified Bouc-Wen model remains to be verified.