# Nonlinear Vibration Isolation via a NiTiNOL Wire Rope
## 3 Methods  
### Harmonic Balance Analysis  
A harmonic balance method was adopted to solve Equations (7) and (8). The restoring force of the wire rope, as shown in Equation (3), is an odd function, and thus, there are only odd harmonics in the responses. Only the first- and third-order harmonics were considered for approximate analysis. The amplitudes of higher-order harmonics are less than \(1/10\) of the first- and third-order harmonics, provided that the responding amplitude is not so high (\(a<3\)). Assume that  
\[y =a_{1}\cos t+b_{1}\sin t+a_{3}\cos 3t+b_{3}\sin 3t \tag{10}\] \[z =c_{1}\cos t+d_{1}\sin t+c_{3}\cos 3t+d_{3}\sin 3t \tag{11}\]  
where \(a_{1}\), \(b_{1}\), \(a_{3}\), \(b_{3}\), \(c_{1}\), \(d_{1}\), \(c_{3}\), and \(d_{3}\) are the coefficients. The Fourier expansion of \(y^{3}\) can be obtained from a recursive method introduced in our previous research [42], and  
\[y^{3}=p_{1}\cos t+q_{1}\sin t+p_{3}\cos 3t+q_{3}\sin 3t \tag{12}\]  
where  
\[\left\{\begin{array}{l}p_{1}=\frac{3}{4}\big{(}a_{1}^{3}+a_{1}^{2}a_{3}+2a_ {1}a_{3}^{2}+a_{1}b_{1}^{2}+2a_{1}b_{3}^{2}-a_{3}b_{1}^{2}+2a_{1}b_{1}b_{3} \big{)}\\ q_{1}=\frac{3}{4}\big{(}b_{1}^{3}-b_{1}^{2}b_{3}+2b_{1}b_{3}^{2}+b_{1}a_{1}^{2 }+2b_{1}a_{3}^{2}+b_{3}a_{1}^{2}-2b_{1}a_{1}a_{3}\big{)}\\ p_{3}=\frac{1}{4}\big{(}a_{1}^{3}+3a_{3}^{3}+6a_{1}^{2}a_{3}-3a_{1}b_{1}^{2}+6a _{3}b_{1}^{2}+3a_{3}b_{3}^{2}\big{)}\\ q_{3}=\frac{1}{4}\big{(}-b_{1}^{3}+3b_{3}^{3}+6b_{1}^{2}b_{3}+3b_{1}a_{1}^{2}+ 6b_{3}a_{1}^{2}+3b_{3}a_{3}^{2}\big{)}\end{array}\right. \tag{13}\]  
According to Equation (7), there are  
\[\left\{\begin{array}{l}c_{1}=\frac{1}{\delta}\big{[}(1+a_{1})\eta^{2}-k_{1 }a_{1}-k_{3}p_{1}\big{]}\\ d_{1}=\frac{1}{\delta}\big{(}b_{1}\eta^{2}-k_{1}b_{1}-k_{3}q_{1}\big{)}\\ c_{3}=\frac{1}{\delta}\big{(}9a_{3}\eta^{2}-k_{1}a_{3}-k_{3}p_{3}\big{)}\\ d_{3}=\frac{1}{\delta}\big{(}9b_{3}\eta^{2}-k_{1}b_{3}-k_{3}q_{3}\big{)}\end{array}\right. \tag{14}\]  
Therefore, when the parameters (\(a_{1}\), \(b_{1}\), \(a_{3}\), \(b_{3}\), \(\eta\)) are determined, \(z\) can be calculated. Define a residual \(r\), according to Equation (8), as  
\[r=\dot{z}-\left\{1-\xi\mathrm{e}^{-\frac{y^{2}}{b_{0}}}-\big{[}\gamma+\beta \mathrm{sgn}\big{(}\dot{y}z\big{)}\big{]}|z|^{n_{b}}\right\}\dot{y} \tag{15}\]  
and its Fourier expansion is expressed as  
\[r=u_{1}\cos t+v_{1}\sin t+u_{3}\cos 3t+v_{3}\sin 3t \tag{16}\]  
An alternating frequency/time domain method [41] was adopted to calculate the parameters \(u_{1}\), \(v_{1}\), \(u_{3}\), and \(v_{3}\). A discrete time sequence from \(t=0\) to \(t=2\pi\) was selected, and \(y(t)\), \(z(t)\), and \(r(t)\) can be calculated as discrete sequences based on Equations (10), (11), and (15), respectively. The parameters (\(u_{1}\), \(v_{1}\), \(u_{3}\), \(v_{3}\)) were obtained by applying the Fourier transform to \(r(t)\). If the determined parameters (\(a_{1}\), \(b_{1}\), \(a_{3}\), \(b_{3}\), \(\eta\)) are the exact solutions of Equations (7) and (8), the parameters (\(u_{1}\), \(v_{1}\), \(u_{3}\), \(v_{3}\)) are all 0. Thus, solving the equations is equivalent to the least-mean-square problem of searching (\(a_{1}\), \(b_{1}\), \(a_{3}\), \(b_{3}\), \(\eta\)) to achieve the minimum value of \(\sqrt{u_{1}^{2}+v_{1}^{2}+u_{3}^{2}+v_{3}^{2}}\). An arc-length continuation method [44] was adopted to solve the problem. The absolute displacement of the payload was calculated as  
\[y_{\mathrm{p}}=(a_{1}+1)\cos t+b_{1}\sin t+a_{3}\cos 3t+b_{3}\sin 3t \tag{17}\]Therefore, the frequency responses of \(a_{\text{p}}\) (\(T_{\text{D}}\)) could be acquired.