# Nonlinear Vibration Isolation via a NiTiNOL Wire Rope
## 2 Models  
Figure 1 shows a vibration isolator with a NiTiNOL wire rope. The wire rope is horizontally suspended in tension. Both ends of the wire rope are fixed on the base with two clamping blocks, and the middle of the wire rope is fixed on a guiding block. A guiding rod is assembled on the base with screw threads, and thus, its height can be adjusted. The guiding block and the guiding rod are assembled with a linear bearing, and thus, the guiding block can move only vertically. A linear spring is assembled between the guiding block and the guiding rod. The moving part of the system includes the guiding block, and the middle clamping block and the payload are fixed to it. At the initial state, the gravity of the moving part is balanced by the linear spring, and the wire rope is horizontal. A vertical displacement excitation \(Y_{\text{b}}\) is applied to the base, and the vertical responding displacement of the payload is \(Y_{\text{p}}\). The relative displacement is \(Y=Y_{\text{p}}-Y_{\text{b}}\). The restoring force of the wire rope was modeled with a combined method. The wire rope was cut from the middle and divided into two segments. Firstly, each segment was regarded as a cantilever beam fixed on the base with a constant section, and the inner frictions and the phase transitions of NiTiNOL were ignored. Thus, this part of the restoring force is due to the linear elastic deformation with geometric constraints, and it was modeled with a physical method. The length, the area of the section, the moment of inertia, and Young's modulus of the beam are denoted as \(L\), \(A\), \(I\), and \(E\), respectively. The free end of the beam is attached to the payload and the external loads applied to the free end are denoted as \(F_{\text{wx}},F_{\text{wy}}\), and \(M_{\text{wz}}\), respectively. The elastic deformations of the free end are denoted as \(X_{\text{w}}\), \(Y_{\text{w}}\), and \(\theta_{\text{w}}\), respectively. The free end is fixed on the payload, and thus, the rotation angle \(\theta_{\text{w}}\equiv 0\). At the initial state (\(Y_{\text{w}}=0\)), the beam is horizontal with an axial force \(F_{\text{wx}0}\). The restoring force due to the linear elastic deformation was modeled based on a beam con  
Figure 1: Vibration isolator with a NiTiNOL wire rope. straint model [43]. More details can be found in our previous research [42], and the vertical restoring force is expressed as  
\[F_{\rm wy}=\left(\frac{12 EI}{L^{3}}+\frac{6F_{\rm wx0}}{5L}-\frac{F_{\rm wx0}^{2} L}{700 EI}\right)Y_{\rm w}+\left(\frac{18 EA}{25L^{3}}-\frac{3F_{\rm wx0}A}{1750 IL}\right)Y_{\rm w}{}^{3} \tag{1}\]  
Secondly, the inner frictions and the phase transitions of NiTiNOL were modeled with a phenomenological method. The hysteretic effect was characterized by a Bouc-Wen model, and an exponential function was added to describe the pinching effect [35]. This part of the restoring force is denoted as \(Z_{\rm w}\) and is expressed as  
\[Z^{\prime}{}_{\rm w}=\left\{K_{\rm wd}\Bigg{(}1-\xi{\rm e}^{-\frac{Y_{\rm w}^{ 2}}{Y_{\rm b}}}\Bigg{)}-\big{[}G_{\rm w}+B_{\rm w}{\rm sgn}\big{(}Y^{\prime}{}_ {\rm w}Z_{\rm w}\big{)}\big{]}\big{|}Z_{\rm w}{}^{\rm|}n_{\rm b}\right\}Y^{ \prime}{}_{\rm w} \tag{2}\]  
where \(K_{\rm wd}\), \(G_{\rm w}\), \(B_{\rm w}\), and \(n_{\rm b}\) are the parameters of the Bouc-Wen model, and \(\xi\) and \(Y_{\rm b}\) are the parameters of the pinching effect function. The symbol (\({}^{\prime}\)) denotes the derivative with respect to time. The total restoring force of the wire rope is the summation of the two parts and  
\[F_{\rm r}=2F_{\rm wy}+2Z_{\rm w}=K_{\rm w1}Y_{\rm w}+K_{\rm w3}Y_{\rm w}^{3}+Z \tag{3}\]  
\[Z^{\prime}=\left\{K_{\rm d}\Bigg{(}1-\xi{\rm e}^{-\frac{Y_{\rm w}^{2}}{Y_{\rm b }}}\Bigg{)}-\big{[}G+B{\rm sgn}\big{(}Y^{\prime}{}_{\rm w}Z\big{)}\big{]}|Z|^{ n_{\rm b}}\right\}Y^{\prime}{}_{\rm w} \tag{4}\]  
where  
\[\left\{