## 3 Mechanical Modeling  
### Modeling for Leaf Spring Group Deformation  
Euler-Bernoulli beam theory is a classical beam theory [33]. The displacement field of this theory can be expressed as:  
\[\left\{ \begin{array}{l}u(x,z)=-z\frac{\mathrm{d}w_{0}}{\mathrm{d}x}\\ w(x,z)=w_{0}(x)\end{array} \right. \tag{1}\]  
where \(w_{0}\) represents the transverse displacement of every point along the neutral axis, synonymous with deflection. The utilization of Euler-Bernoulli beam theory relies on two fundamental assumptions: firstly, the cross-section of the beam remains flat both before and after deformation, and secondly, the cross-section remains perpendicular to the neutral layer post-deformation. The primary strength of this theoretical beam model lies in having a single generalized displacement, typically necessitating only one positional function during solving, significantly simplifying the calculation process [34; 35]. Figure 3 illustrates the state of the Euler-Bernoulli beam pre- and post-deformation. Figure 1b illustrates that within the leaf spring group's structure, a red copper gasket is positioned between the two leaf springs to prevent contact wear between identical materials. During the bending deformation of the leaf spring group, the red copper gasket undergoes bending, ensuring that it only makes contact with the two leaf springs at \(T_{2}\) in Figure 2, facilitating force transmission. Consequently, during model establishment, we segment the function at \(T_{2}\) to more effectively address the impact of the force at this point on the  
Figure 2: Structural parameter model of leaf spring torsional vibration damper. bending moment of the leaf spring. In the coordinate system depicted in Figure 2, the bending moments \(M_{1}(x)\) and \(M_{2}(x)\) of leaf springs 1 and 2 are expressed as follows:  
\[M_{1}(x)=\left\{ \begin{matrix}F(L-x)-F_{\text{c}}(S-x),&x\in[0,S]\\ F(L-x),&x\in(S,L]\end{matrix} \right. \tag{2}\]  
\[M_{2}(x)=\left\{ \begin{matrix}F_{\text{c}}(S-x),&x\in[0,S]\\ 0,&x\in(S,L]\end{matrix} \right. \tag{3}\]  
Based on Euler-Bernoulli beam theory, the bending mechanical model of two leaf springs is:  
\[M_{1}(x)=EI\frac{\text{d}^{2}w_{1}(x)}{\text{d}x^{2}} \tag{4}\]  
\[M2(x)=EI\frac{\text{d}^{2}w_{2}(x)}{\text{d}x^{2}} \tag{5}\]  
where \(w_{1}\) and \(w_{2}\) are the deflections of leaf spring 1 and leaf spring 2, respectively, in m.  
\[\frac{\text{d}^{2}w_{1}(x)}{\text{d}x^{2}}=\left\{ \begin{matrix}\frac{F(L-x)-F _{\text{c}}(S-x)}{EI},&x\in[0,S]\\ \frac{F(L-x)}{EI},&x\in(S,L]\end{matrix} \right. \tag{6}\]  
\[\frac{\text{d}^{2}w_{2}(x)}{\text{d}x^{2}}=\left\{ \begin{matrix}\frac{F_{\text {c}}(S-x)}{EI},&x\in[0,S]\\ 0,&x\in(S,L]\end{matrix} \right. \tag{7}\]  
where \(E\) is the elastic modulus of spring material, in MPa. The sectional moment of inertia \(I\) is:  
\[I(x)=\frac{bh^{3}(x)}{12} \tag{8}\]  
\(h(x)\) is the thickness of any position of the leaf spring,  
\[h(x)=a_{1}+kx \tag{9}\]  
where \(k\) is the thickness change rate of the leaf spring,  
\[k=\frac{(a_{2}-a_{1})}{l} \tag{10}\]  
where \(l\) is the length of the curved portion of the leaf spring. Substitute Formulas (8)-(10) into Formulas (6) and (7) to obtain:  
\[\frac{\text{d}^{2}w_{1}(x)}{\text{d}x^{2}}=\left\{ \begin{matrix}\frac{12F(L-x )-F_{\text{c}}(S-x)}{Eb(a_{1}+kx)^{3}},&x\in[0,S]\\ \frac{12F(L-x)}{Eb(a_{1}+kx)^{3}},&x\in(S,L]\end{matrix} \right.\] (  
where:  
\[ \begin{split}&\varphi_{11}(x)=\int_{0}^{x}\frac{12[F(L-t)-F_{\rm c}( S-t)]}{Eb(a_{1}+kt)^{3}}\mathrm{d}t\\ &=6x(FLkx-F_{\rm c}Skx-2 FLa_{1}-Fa_{1}x-2F_{\rm c}Sa_{1}+F_{\rm c} a_{1}x)/[Eb(kx+a_{1})^{2}a_{1}^{2}]\\ &\varphi_{12}(x)=\int_{0}^{S}\frac{12[F(L-t)-F_{\rm c}(S-t)]}{Eb( a_{1}+kt)^{3}}\mathrm{d}t+\int_{5}^{x}\frac{12F(L-t)}{Eb(a_{1}+kt)^{3}}\mathrm{d}t\\ &=6S(FLSk-F_{\rm c}S^{2}k+2 FLa_{1}-FSa_{1}-F_{\rm c}Sa_{1})/[Eb( Sk+a_{1})^{2}a_{1}^{2}]-\\ &\ 6F(LS^{2}k-Lkx^{2}-2S^{2}kx+2 Skx^{2}+2 LSa_{1}-2 La_{1}x-S^{2} a_{1}+a_{1}x^{2})/\\ &\ [Eb(kx+a_{1})^{2}(Sk+a_{1})^{2}]\\ \end{split}  \tag{15}\]  
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