# Integrated Thermal and Dynamic Analysis of Dry Automotive Clutch Linings
## 5 Combined Thermal-Dynamic Analysis  
The Equations of motion (1)-(9) for the case of slipping at the clutch interface during clutch engagement and Equations (13)-(20) for the fully clamped clutch state are solved numerically as a system of dynamic equations of motion, together with the thermal differential equations of the clutch, based on Equation (32). The dynamics and thermal equations are coupled together into a single thermal-dynamics model by transient angular velocities, which are an output of the dynamics model and serves as an input to the thermal analysis. The 4 th-order Runga-Kutta step-by-step integration \begin{table}
\begin{tabular}{c c c} \hline \hline
**Disc Temperature** & \
\begin{tabular}{c} **Linear Regressed Equation for** \\ **New Friction Material** \\ \end{tabular} &
\begin{tabular}{c} **Linear Regressed Equation for** \\ **Worn Friction Material** \\ \end{tabular} \\ \hline
**20 °C** & 0.48–0.0210 \(v\) & 0.32–0.00189 \(v\) \\
**40 °C** & 0.49–0.0191 \(v\) & 0.32–0.00452 \(v\) \\
**60 °C** & 0.48–0.0178 \(v\) & 0.31–0.00761 \(v\) \\
**90 °C** & 0.47–0.0167 \(v\) & 0.34–0.00436 \(v\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variation of the coefficient of friction with slip speed, \(v\), for new and worn clutch lining materials [21]. algorithm is used for the solution of equations of motion. The initial engine speed at the instant of clutch actuation is chosen as 100 rad/s or an equivalent engine speed of 960 r/min, which is the idle engine speed for the vehicle under investigation. The duration of clutch engagement is set at 0.5 s, as can be seen in Figure 2 in line with the range reported by [PERSON] et al. [6,7]. Additionally, the initial temperature for all clutch elements under consideration was set to 20 \({}^{\circ}\)C (the average ambient Spring/Summer temperature in the UK). Additionally, the temperature of the engine and the surrounding compartments is considered to be constant at 20 \({}^{\circ}\)C (start-up conditions) for the short duration of the simulation. All the parameters used in the model are listed in Tables 15-16 in Appendix A. Figures 5 and 6 show the variations in displacement, angular velocity and angular acceleration of the clutch friction disc during a single clutch engagement for the new and worn clutch friction linings. The main differences in the characteristic responses for the new and worn lining materials are in the variations of angular velocity during clutch engagement. Close to full engagement, stick-slip oscillations occur in the form of sudden angular velocity fluctuations (Figure 5). Additionally, the stick-slip oscillations can also be noted in the abrupt angular acceleration reversals. Larger negative \(\mu\)-\(v\) characteristic in Table 1 for the new lining material provides good explanations for the observed judder vibrations in Figure 5 for the new lining material. This is in line with the other findings such as [PERSON] et al. [6,7] and [PERSON] et al. [12]. Figure 7a shows the temperature rise for the new friction lining for a single clutch engagement, whilst Figure 7b is the same for the case of the worn lining. Higher interfacial surface temperatures are reached in the case of the new lining, owing to higher generated friction. It should be noted that as the thermal analysis uses a lumped-parameter model, any rise in the temperature should be considered as a bulk temperature rise. The flash temperature rise in the contact is expected to be higher than the rise in the bulk temperature. Nevertheless, any rise in the bulk temperature can be considered as an indication of a rise in the contacting surface temperature where frictional heating takes place. The new friction material reaches a temperature of 22.4 \({}^{\circ}\)C for a single engagement, whilst the temperature of the worn lining reaches 21.3 \({}^{\circ}\)C (Noting that in both cases an initial temperature of 20 \({}^{\circ}\)C is assumed). The temperature of counter face surfaces of the flywheel and pressure plate is almost the same in both cases.