# Modeling of Dry Clutch Wear for a Wide Range of Operating Parameters
## 7 Wear Rate Variability Model  
### Modeling Approach  
The wear rate variability submodel from Figure 10 has been developed by using the maximum likelihood method [32]. Based on the experimental evidence in Figure 7, the wear rate probability distribution for particular operating parameters is assumed to have normal distribution, where the mean is given by the wear rate expectation model and the standard \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{\
\begin{tabular}{c} **Friction** \\ **Material** \\ \end{tabular} } & \multicolumn{3}{c}{**Modeling Points**} & \multicolumn{3}{c}{**Validation Points**} & \multirow{3}{*}{
\begin{tabular}{c} **Number of** \\ **Model** \\ **Parameters** \\ \end{tabular} } \\  & \multicolumn{3}{c}{**Residuals**} & & & **Residuals** & & **Number of** \\  & \multicolumn{3}{c}{**Standard**} & \multicolumn{3}{c}{**Standard**} & \multicolumn{3}{c}{**Standard**} \\  & \multicolumn{3}{c}{**Deviation (p.u.)**} & & & **Deviation (p.u.)** & & **Parameters** \\ \hline A & 0.154 & 0.875 & 0.841 & 0.148 & 0.411 & 10 \\ \hline B & 0.142 & 0.922 & 0.887 & 0.132 & 0.560 & 14 \\ \hline C & 0.149 & 0.865 & 0.820 & 0.125 & 0.724 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparative performance of wear rate model for three dry clutch friction materials. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{
\begin{tabular}{c} **Model  
deviation is to be provided by the wear rate variability model. The model parameters are determined by maximizing probability of capturing the recorded points used in the model parameterization dataset (characterization and extrapolation points from Figure 3). To this aim, a likelihood is used, which is defined as a value of probability density function (PDF) for a certain argument/input value (e.g., operating parameters used as inputs for the wear rate expectation model). Maximizing the following product of likelihood values of all (\(N\)) modeling points leads to maximization of probability of capturing the whole set of observed data points:  
\[L^{\prime}(\mathbf{\Theta})=f(\mathbf{x}_{1},y_{1},\mathbf{\Theta})\cdot f(\mathbf{x}_{2},y_{2 },\mathbf{\Theta})\cdot\cdot\cdot\cdot f(\mathbf{x}_{N},y_{N},\mathbf{\Theta}), \tag{16}\]  
where \(\mathbf{\Theta}\) is the vector of model parameters, \(y_{j}\) is the observed, \(j\)th output value of a random variable accompanied with an input vector \(\mathbf{x}_{j}\), \(f\) is a PDF providing the likelihood of occurrence of each \(\big{(}\mathbf{x}_{j},y_{j}\big{)}\) pair in dependence of parameters \(\mathbf{\Theta}\). To avoid numerical issues, the multiplication of PDF values in (16) is typically transferred to a sum by taking its logarithm [32]. Furthermore, the minus sign is added to convert the problem from maximization to minimization. This finally results in the negative log likelihood (NLL) cost function to be minimized [29]:  
\[L(\mathbf{\Theta})=\sum_{j=1}^{N}-log\big{(}f\big{(}\mathbf{x}_{j},y_{j},\mathbf{\Theta} \big{)}\big{)}.\] (17a) The normal distribution PDF reads: \[f\big{(}\mathbf{x}_{j},\ y_{j},\mathbf{\theta}\big{)}=\ \frac{1}{\sigma\big{(}\mathbf{x}_{j},\mathbf{\theta} \big{)}\sqrt{2\pi}}e^{-\frac{1}{2}\big{(}\frac{y_{j}-y_{j}(\mathbf{x}_{j})}{e(\mathbf{ x}_{j},\mathbf{\theta})}\big{)}^{2}},\] (17b) where \[\mu\] and \[\sigma\] are its expectation and standard deviation, respectively.