## 2 Finite Element Model of Pressure Plate Assembly
### Boundary Conditions  
#### 2.2.1 Thermal Boundary Conditions  
In the continuous sliding condition, it is assumed that the relative speed between the clutch pressure plate and the driven plate is invariable and the pressure plate always receives more heat than the friction plate. The power of the shaded area (as shown in Figure 2) is calculated as follows when the pressure plate is under the sliding condition [15],  
\[\mathrm{dp}=\frac{F(t)}{\pi\left(r_{2}^{2}-r_{1}^{2}\right)}.2\pi rdr\text{-} \text{f}\omega_{h} \tag{1}\]  
where p is the sliding power of the shaded area; \(r_{1}\) and \(r_{2}\) inner radius and outer radius of pressure plate, respectively; \(r\) is the radius where the shaded area is located in the friction surface; \(F(t)\) is the clamping load of pressure plate; f is the coefficients of friction; \(\omega_{h}\) is the angular velocity difference between the pressure plate and the friction surface; and \(2\pi rdr\) is the area of the shaded portion.  
In the continuous sliding condition, as the clamping load of pressure plate and the relative angular velocity between the pressure plate and the driven disk remain unchanged, we have  
\[\mathrm{F(t)}=\mathrm{N} \tag{2}\]  
\[\omega_{h}=\omega \tag{3}\]  
where N and \(\omega\) are constant.  
Then, Equation (1) can be rewritten as Equation (4) after substituting Equations (2) and (3) into Equation (1):  
\[\mathrm{dp}=\frac{\mathrm{fN}\omega r}{\pi\left(r_{2}^{2}-r_{1}^{2}\right)}.2 \pi rdr \tag{4}\]  
where \(\frac{\mathrm{fN}}{\pi\left(r_{2}^{2}-r_{1}^{2}\right)}.2\pi rdr\) represents the friction forces of the shaded area, and \(\omega r\) is the linear velocity of the shaded area.  
By definition [16], the heat flux on the friction surface of pressure plate can be obtained as Equation (5):  
\[\mathrm{q}=\frac{\mathrm{dp}}{\mathrm{ds}}=\frac{\frac{\mathrm{fN}\omega r}{ \pi\left(r_{2}^{2}-r_{1}^{2}\right)}.2\pi rdr}{2\pi rdr}=\frac{\mathrm{fN} \omega r}{\pi\left(r_{2}^{2}-r_{1}^{2}\right)} \tag{5}\]  
Figure 2: Friction surface of the pressure plate.  
The total heat flux induced by the relative sliding is absorbed by the pressure plat and the driven plate. The heat distribution coefficient between them follows Equation (6),  
\[\mathrm{k}=\frac{\mathrm{q_{Pressureplate}}}{\mathrm{q_{frictionplate}}}=\left( \frac{\lambda\mathrm{_{Pressureplate}}\mathrm{^{CPressureplate}}\mathrm{^{P Pressureplate}}}{\lambda\mathrm{_{frictionplate}}\mathrm{^{C}}\mathrm{_{frictionplate}}\mathrm{^{Prictionplate}}}\right)^{0.5} \tag{6}\]  
where \(\lambda\) is the thermal conductivity, \(c\) is the thermal capacity, and \(\rho\) is the density.  
Therefore, the heat flux on the friction surface of the pressure plate after the heat distribution coefficient is taken into account can be expressed as  
\[\mathrm{q_{Pressure\ plate}}=\frac{\mathrm{qk}}{1+\mathrm{k}} \tag{7}\]  
In our case, here the parameters needed have been provided by the clutch company, and the heat flux of friction surface of pressure plate is \(\mathrm{q_{Pressure\ plate}}\) = 2,188,000 r J/(m\({}^{2}\)-s).  
According to the condition of actual continuous sliding, the slip time of pressure plate is set to 60 s, and a natural convection cooling process of 3500 s is subsequently applied at room temperature (25 \({}^{\circ}\)C). During the cooling process, the convective heat transfer coefficient on the surface of the pressure plate is estimated as [17]  
\[\mathrm{h}=0.023\frac{\lambda}{d}R_{e}^{0.8}p_{r}^{0.4} \tag{8}\]  
where \(\lambda\) is the air thermal conductivity, \(d\) is the characteristic size, \(R_{e}\) is the Reynolds number, and \(p_{r}\) is the Prandtl number.