## 2 Shell Dynamics
### Estimation of the Characteristic Properties of MR  
This section introduces the IHP model. The FSDT used in this paper assumes small strains, and previous research by [PERSON] et al. [34] has demonstrated that the mechanical properties of MR are approximately linear under such conditions. Therefore, this paper does not consider the effect of strain state on the mechanical properties of MR. The IHP model combines the internal structure of MR reduced to the inclined helix unit with the friction pyramid model, as illustrated in Figure 2.  
The equivalent Young's modulus of MR in the IHP model can be calculated using the following expression [12]:  
\[E_{U/L}==\frac{N_{A}l}{nN_{L}A}(k_{U/L}^{\epsilon}\mu_{U/L}^{\epsilon}) \tag{39}\]  
where the subscript \(U/L\) represents the unloading process and the loading process, respectively. After experimental verification, \(N_{A}\) and \(N_{L}\) satisfy the following expression:  
\[\frac{N_{A}l}{nN_{L}A}=(\rho_{m}-\overline{\rho})\lambda(\epsilon) \tag{40}\]  
where \(\overline{\rho}\approx 0.16\) represents the relative density of the MR blanks; \(n\) represents the number of screws of the equivalent spring unit, and usually \(n=1\). Furthermore, \(\lambda(\epsilon)\) represents the strain influence factor, which has the following expressions:  
\[\lambda(\epsilon)= \begin{cases}a&\dot{\epsilon}>0\\ 1.25\times 10^{4}\bigg{[}b-\frac{b-1}{1+\exp\left(4\left(\epsilon-\frac{\epsilon }{\epsilon_{G}^{2}}\right)/\epsilon_{\alpha}\right)}\bigg{]}&\dot{\epsilon}>0 \end{cases}  \tag{41}\]  
In Equation (39), \(k^{\epsilon}\mu_{U/L}\) represents the equivalent stiffness of the elasto-plastic unit in various contact states, which can be expressed as follows:  
\[ \begin{split} k_{U/L}^{\epsilon}=\int_{0}^{2\pi}&\left[ \begin{array}{c}\frac{\sin\beta+\mu_{U/L}^{\epsilon}\cos\beta}{\cos(\varphi- \beta)+\mu_{U/L}^{\epsilon}\sin(\varphi-\beta)}\frac{\sin\varphi}{K_{U}}+ \cdots\\ &+\frac{\cos\beta-\mu_{U/L}^{\epsilon}\sin\beta}{\cos(\varphi-\beta)+\mu_{ U/L}^{\epsilon}\sin(\varphi-\beta)}\frac{\cos\varphi}{K_{T}}\end{array}\right]^{-1}\\ &\cdot\frac{3}{\overline{\psi}\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\frac {\varphi-\overline{\varphi}}{\overline{\psi}/3}\right)^{2}\right]\text{d} \varphi\end{split}  \tag{42}\]  
Figure 2: Equivalent viscoelastic cell and arrangement of metal coils: (**a**) Cell structure of IHP model; (**b**) arrangement direction of IHP model in MR.  
where \(\beta=\arctan(d/2/D)\) represents the half cone angle of the equivalent elastic unit as shown in Figure 2a. In addition, \(\overline{\varphi}\) represents the mean tilting angle of the equivalent elastic unit, usually given as \(\overline{\varphi}=\arctan(1/r)\), and the molding ratio \(r\) is usually given as \(r=3\). Furthermore, \(K_{V}\) and \(K_{T}\) represent the axial and tangential stiffness of the equivalent elastic unit, respectively, and their expressions are as follows:  
\[ \begin{array}{c}K_{T}=\frac{E_{d}d\cos\alpha}{8\pi c^{3}\left[\frac{4(2+\pi) \mu^{2}\pi^{2}+3\beta/2}{3}\tan^{2}\alpha+1\right]}\\ K_{V}=\frac{E_{md}\cos\alpha}{16\pi c^{3}\left(1+\cos^{2}\alpha\right)}\end{array}  \tag{43}\]  
where \(\alpha=\arctan(d/\pi/D)\) is the guide angle. In addition, in Equation (39), \(\mu^{s}\omega_{U/L}\) represents the friction coefficient between the wires which have different expressions in the loading and unloading processes, and they can be expressed as follows:  
\[\begin{array}{c}\mu^{s}_{L}= \begin{cases}\frac{\xi}{2\epsilon_{cr}^{2}}\mu^ {s}_{cr}&\epsilon<\epsilon^{L}_{cr}\\ \left(1-\frac{\epsilon^{L}_{cr}}{2\epsilon}\right)\mu^{s}_{cr}&\epsilon\geq \epsilon^{L}_{cr}\\ \left(1-\frac{1}{2}\frac{\epsilon^{L}_{cr}-\epsilon_{cr}}{\epsilon-\epsilon_{ z}}\mu^{s}_{cr}&\epsilon>\epsilon^{L}_{cr}\end{cases}