## 2 Materials and Methods
### Electrothermal SMA Spring Drive Characteristics  
Figure 10a illustrates the physical model of the electrothermal SMA coil spring, and a simplified model based on the sample is shown in Figure 10b. In the simplified model of the electrothermal SMA spiral spring, the diameter of SMA wire is \(d=2.3\) mm, the helix angle of the spring helix is \(\alpha=6^{\circ}\), the diameter of the spring is \(D=10.6\) mm, the free height of the spring is \(L=21\) mm, and the number of effective spiral turns is \(n_{e}=6\). The upper and lower ends of the spring are closed and ground, the diameter of the copper-core polyurethane enameled wire spirally wound on the SMA wire is \(d^{\prime}=0.4\) mm, and the number of turns of the enameled wire is \(N_{1}=300\). Figure 10: Schematic diagram of electrothermal SMA spring. (**a**) Sample diagram and (**b**) principal diagram. Figure 9: Relations among current, working gap radial distance, and magnetic induction intensity. In this transmission device, the main reliance is on the SMA's restoring force to the MRF for squeeze. The specific material parameters of the electrothermal SMA spring are shown in Table 3[16]. \(M_{\text{S}}\), \(M_{\text{F}}\), \(A_{\text{S}}\), and \(A_{\text{F}}\) are the starting and finishing transformation temperatures of martensite and austenite, respectively, as shown in Table 3. \(G_{\text{M}}\) and \(G_{\text{A}}\) are the shear moduli of martensite and austenite, respectively. The SMA 3D fine-scale intrinsic equation under the pure shear mode can be expressed as [22]:  
\[\gamma_{\text{S}}=\frac{\tau}{G(K)}+\frac{\tau_{0}}{G(K_{0})}+\frac{\sqrt{6}}{ 2}\epsilon_{L}(\beta-\beta_{0})+\gamma_{0} \tag{10}\]  
where all variables without subscripts are current variables, and those with zero subscripts are the initial values. \(K\) is the current temperature of the SMA spring, \(\epsilon_{L}\) is the maximum residual strain, and \(\beta\) is the shape memory factor. During loading of an electrically heated SMA coil spring, the direction of the force is along the axis of the spring, so that the SMA wire is always subject to a pure shear load. The distance from the center of the SMA wire section to the outer edge of the section is denoted by \(x\), and the linear shear strain of the SMA wire section can be expressed as:  
\[\gamma_{\text{S}}=\theta x,x\in(0,r_{\text{S}}) \tag{11}\]  
where \(\theta\) refers to the angle of twist of the SMA spring wire, and \(r_{\text{S}}\) refers to the radius of the SMA spring wire. The deformation of the spring can be expressed as:  
\[f=L-L_{0}=2\pi n_{e}\theta R^{2}{}_{\text{SMA}}=\frac{\pi n_{e}\theta D^{2}} {2} \tag{12}\]  
By combining Equations (10) and (12), the axial deformation of the electrothermal SMA coil spring can be obtained as:  
\[f=\frac{\pi n_{e}D^{2}}{d}\left(\frac{\tau}{G(K)}+\frac{\tau_{0}}{G(K_{0})}+ \frac{\sqrt{6}}{2}\epsilon_{L}(\beta-\beta_{0})+\gamma_{0}\right) \tag{13}\]  
where \(\tau\) refers to the shear stress of the SMA wire section, and the shear stress of the coil spring is:  
\[\tau=k\frac{8 FD}{\pi d^{3}} \tag{14}\]  
where \(F\) refers to the axial load of the coil spring, and \(k\) refers to the stress correction factor, which can be expressed as:  
\[k=\frac{4C_{\text{S}}-1}{4C_{\text{S}}-4}+\frac{0.615}{C_{\text{S}}} \tag{15}\]  
where \(C_{\text{S}}\) refers to the coiling ratio of the coil spring (\(C_{\text{S}}=D/d\)).