# Modeling of Dry Clutch Wear for a Wide Range of Operating Parameters
## 5 Clutch Wear Model and Its Run-In Submodel  
### Model Structure  
Since in real vehicles, particularly those equipped with manual transmission, the clutch operates in closing cycles, the wear model is designed to be executed in a cycle-wise manner. As the current clutch wear state estimate is of central interest, the cumulative worn volume (\(V_{w}\)) is set to be the model output. To capture inherent wear rate variability, it is assumed that the wear rate for \(k\)th closing cycle is a random variable following a normal distribution (see the experimental evidence in Figure 7b) and having the expectation \(w_{\mu,k}\) and the standard deviation \(w_{\sigma,k}\) dependent on the operating parameters. The related worn volume increment \(\Delta V_{w,k}\) then follows a normal distribution, as well, with the expectation and variance parameters \(\Delta V_{w,\mu_{k}}\) and \(\Delta V_{w,\sigma_{k}^{2}}\) obtained by multiplying the related wear rate parameters with the dissipated energy \(E_{dis,k}\):  
\[\Delta V_{w,k}\sim\mathcal{N}\left(\underbrace{w_{\mu,k}E_{dis,k}}_{\Delta V_ {w,\mu_{k}}}\underbrace{(w_{\sigma,k}E_{dis,k})^{2}}_{\Delta V_{w,\sigma_{k}^{ 2}}}\right). \tag{2}\]  
The total worn volume after a certain number of cycles can be obtained by summing up the corresponding worn volume increments:  
\[V_{w}(k)=\Delta V_{w,1}+\cdots V_{w,2}+\ldots+\Delta V_{w,k}\sim\mathcal{N} \left(\underbrace{\sum_{l=1}^{k}\Delta V_{w,\mu_{l}}}_{V_{w,\mu}(k)}, \underbrace{\sum_{d=1}^{k}\Delta V_{w,\sigma_{k}^{2}}}_{V_{w,\sigma^{2}}(k)} \right). \tag{3}\]  
where the worn volumes of individual cycles are assumed to be statistically independent (see the mathematical background in Appendix A). Taking the square root of variance yields the total worn volume standard deviation \(V_{w,\sigma}\), which can be used for calculating related worn volume 95% confidence interval (CI) as \(V_{w,\mu}\pm 2V_{w,\sigma}\). To execute the wear model in a cycle-wise manner, the four operating parameters (initial slip speed \(\omega_{0}\), clutch torque \(M_{z}\), clutch temperature \(T_{d}\), and closing time \(t_{2}\), updated in each closing cycle; see Section 3 and Figure 6) would represent the model inputs that should be measured or estimated. To reduce the input space dimensionality and, thus, simplify the modeling process, the initial slip speed \(\omega_{0}\) and closing time \(t_{2}\) are merged into a so-called surrogate distance input \(d_{s}=\omega_{0}^{r}t_{2}^{\theta}\), with the exponents \(r\) and \(q\) being optimized for a favorable modeling accuracy (see Appendix B). The dissipated energy is calculated from the measured slip speeds and torques in each (\(i\)th) sampling step of the (\(k\)th) closing cycle and summed up over the total number of sampling steps (\(M\)):  
\[E_{dis}(k)=\sum_{i=1}^{M}M_{z}(i)\omega(i)(t(i)-t(i-1)). \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline
**Correlation of Wear Rate, \(w\), with** & **Correlation Coefficient** \\ \hline Clutch temperature, \(\overline{T}_{d}\) & 0.744 \\ \hline Initial slip speed, \(\overline{w}_{0}\) & 0.295 \\ \hline Torque, \(\overline{M}_{z}\) & 0.210 \\ \hline Closing time, \(\overline{t}_{2}\) & 0.173 \\ \hline \end{tabular}
\end{table} 
Table 1: Correlation indices of wear rate with respect to individual input parameters. The cumulative dissipated energy \(E_{c}\) can then be calculated as:  
\[E_{c}(k)=\sum_{l=1}^{k}E_{dis}(l). \tag{5}\]  
The wear model derived from Equations (2), (3) and (5) is shown in the form of block diagram in Figure 10. It consists of three main submodels: (i) run-in model providing a wear rate weighting function \(g(E_{c})>1\) to account for the increased wear rate during the run-in period, (ii) wear rate expectation model with the output \(w_{\mu,0}\), and (iii) wear rate variability model providing the wear rate standard deviation \(w_{\sigma}\). The expected wear rate \(w_{\mu,0}\) is multiplied by the run-in weighting function \(g(E_{c})\) to obtain the actual/corrected wear rate \(w_{\mu}\).