## 2 Mathematical Model of Multi-Disc Wet Clutch  
### Piston and Friction Components Axial Kinetic Model  
The piston and friction elements of the wet clutch are driven by the control oil pressure, which undergo an axial shift and circumferential rotation. The friction elements are successively numbered and the corresponding axial forces are analyzed in Figure 2.  
The displacement of the \(i\)-th element can be obtained by  
\[x_{i}=(n+1-i)h_{o}+\sum_{j=i}^{n}H_{dj}-\sum_{k=i}^{n}h_{k},\;i=0,1\cdots n \tag{1}\]  
where \(H_{dj}\) is the steel plate or friction plate thickness, \(h_{o}\) is the initial gap of friction pairs, \(h_{k}\) is the friction pair gap, and \(n\) is the number of friction pairs.  
The kinetic model of the friction elements and piston axial was established based on force balance conditions, as shown in Equation (2) [3].  
\[\left[\begin{array}{ccccc}F_{0}+F_{0}+F_{d0}-F_{app}\\ F_{c1}+F_{v1}+F_{s1}+F_{d1}-F_{c0}-F_{v0}\\ F_{c2}+F_{v2}+F_{f2}-F_{c1}-F_{v1}\\ \vdots\\ F_{c(n-1)}+F_{v(n-1)}+F_{s(n-1)}+F_{d(n-1)}-F_{c(n-2)}-F_{v(n-2)}\\ F_{cn}+F_{vn}+F_{fn}-F_{c(n-1)}-F_{v(n-1)}\end{array}\right]+\left[\begin{array} []{ccccc}k_{r}&0&0&\cdots&0&0\\ 0&k_{s}&k_{s}&\cdots&0&0\\ 0&0&0&\cdots&0&0\\ &&&\ddots&\\ 0&\cdots&-k_{s}&-k_{s}&k_{s}&k_{s}\\ 0&\cdots&0&0&0&0\end{array}\right]\left[\begin{array}{c}h_{o}-h_{0}\\ h_{o}-h_{1}\\ h_{o}-h_{2}\\ \vdots\\ h_{o}-h_{n-1}\\ h_{o}-h_{n}\end{array}\right]\]  
Figure 2: Axial force model of the multi-disc wet clutch.  
\[=\left[\begin{array}{ccccc}m_{p}&&&\\ &m_{s}&&&\\ &&m_{f}&&\\ &&&\ddots&\\ &&&m_{s}&\\ &&&m_{f}&\end{array}\right]\left[\begin{array}{c}x_{0}^{\prime\prime}\\ x_{1}^{\prime\prime}\\ x_{2}^{\prime\prime}\\ \vdots\\ x_{n-1}^{\prime\prime}\\ x_{n}^{\prime\prime}\\ \end{array}\right] \tag{2}\]  
where \(m_{p}\) is the piston mass, \(m_{s}\) is the steel plate mass, \(m_{f}\) is the friction plate mass, \(F_{ci}\) is asperities bearing capacity in the friction pair, \(F_{vi}\) is the oil film bearing capacity in the friction pair, \(F_{si}\) is the friction force between the steel plate and the spline, \(F_{fi}\) is the friction force between the friction plate and the spline, \(F_{di}\) is the damping force of the steel plate, \(F_{app}\) is the driven force of the hydraulic oil, and \(k_{r}\) is the stiffness of the return spring. The wave spring stiffness \(k_{s}\) is described as follows [27]:  
\[k_{s}=\frac{16 kEb\delta^{3}N^{4}}{\pi^{3}D_{M}^{3}} \tag{3}\]  
where \(D_{M}\) is the center diameter of the wave spring, \(N\) is the wave number of the wave spring, \(b\) is the wave spring width, and \(\delta\) is the wave spring thickness.  
The damping force between the steel plate and spline is described as follows:  
\[F_{d}=c_{s}x_{i}^{\prime} \tag{4}\]  
The torque is transmitted from the steel plate to the input shaft via the spline and the torque is transmitted from the friction plate to the output shaft via the spline. Therefore, there is friction resistance between the friction components and the spline during the friction element axial shift [20]. The spline resistance forces are deduced as follows:  
\[F_{fi}=\text{sign}(x_{i}^{\prime})\frac{\mu_{s}T_{fi}}{R_{f}\text{cos}(\alpha)} \tag{5}\]  
\[F_{si}=\text{sign}(x_{i}^{\prime})\frac{\mu_{s}T_{fi}}{R_{s}\text{cos}(\alpha)} \tag{6}\]  
where \(\mu_{s}\) is the spline friction coefficients; \(T_{fi}\) is the torque transmitted by the friction pair, \(\alpha\) is the pressure angle of the spline; and \(R_{s}\) and \(R_{f}\) are the spline pitch circle radius on the friction plate and steel plate, respectively.