## 3 Improved PIM under Dynamic Conditions
### Real-World Cases  
This section takes motorized spindles, which are used in the machining centers, as an example to confirm the practicability of the method proposed in this paper. According to engineering experience, cutting force, rotational speed, cutting fluid, and number of tool changes are selected as the four WC factors, which may have different influences on FI. They are represented by \(Z_{1}\), \(Z_{2}\), \(Z_{3}\), and \(Z_{4}\) with the corresponding coefficients \(k_{1}\), \(k_{2}\), \(k_{3}\), and \(k_{4}\). There are eight different WCs when the motorized spindles operate. The exact parameters in these WCs are shown in Table 1. In \(Z_{3}\), 1 means using cutting fluid, 0 means not using cutting fluid. In the likelihood ratio test, \(\alpha\) is set as 0.05. When the numbers of covariates are 4 and 3, the test does not meet the criteria. When the number is 2, and the covariables are rotational speed and number of tool changes, respectively, the test meets the standard. Thus, the two covariates are identified as the factors that influence the FI. The coefficients of these two factors are estimated by the partial likelihood estimation, with the results shown in Table 2.  
By using the maximum likelihood estimation, the shape and scale parameters of the Weibull function are estimated, which are \(a=0.0013\) and \(b=0.8829\). For further explanation, a set of dynamic WCs can be built to compare the difference between the traditional and the proposed PIMs. First, it is assumed that the motorized spindles operate in \(e_{5}\) from 0 to 500 h, in \(e_{8}\) from 500 h to 1000 h, in \(e_{6}\) from 1000 h to 1500 h, in \(e_{7}\) from 1500 h to 2000 h, in \(e_{3}\) from 2000 h to 2500 h, in \(e_{2}\) from 2500 h to 3000 h, in \(e_{4}\) 3000 h to 3500 h, and in \(e_{1}\) from 3500 to 4000. Then, the FIs calculated by the traditional PIM and the PIM, which \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Working Conditions** & \(Z_{1}\)**(KN)** & \(Z_{2}\)**(r/min)** & \(Z_{3}\) & \(Z_{4}\) \\ \hline \(e_{1}\) & 0.25 & 2200 & 1 & 10 \\ \(e_{2}\) & 0.23 & 3500 & 0 & 12 \\ \(e_{3}\) & 0.39 & 2800 & 1 & 6 \\ \(e_{4}\) & 0.42 & 1800 & 1 & 6 \\ \(e_{5}\) & 0.26 & 4200 & 0 & 2 \\ \(e_{6}\) & 0.13 & 3500 & 1 & 9 \\ \(e_{7}\) & 0.51 & 3000 & 1 & 6 \\ \(e_{8}\) & 0.60 & 3600 & 1 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The specific value of parameters in the 8 working conditions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Covariate Number**} & \multicolumn{6}{c}{**Working Conditions Covariates**} \\ \cline{2-7}  & \(k_{1}\) & \(k_{2}\) & \(k_{3}\) & \(k_{4}\) & \(H^{2}\) & \(\lambda_{0.05}\)\({}^{2}\) \\ \hline
4 & \(-0.510233\) & 0.000505 & 0.012975 & \(-0.080421\) & 7.603 & 9.488 \\
3 & \(-0.493228\) & 0.000500 & 0 & \(-0.080318\) & 7.602 & 7.815 \\
2 & 0 & 0.000536 & 0 & \(-0.075618\) & 7.503 & 5.991 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The results of coefficients of all WC factors. consider the past effects, are shown in Figure 5. The exact results at specific points in time are shown in Table 3.  
In this case, with \(b=0.8829<1\), the system can be seen as gradually improving. The FI decreases with the increase of service time. This situation usually occurs in the early failure period of electromechanical systems. In other words, at this stage the FI decreases with the increase of cumulative damage. As shown in Figure 5 and Table 3, the results calculated by the proposed PIM are always smaller than those of the traditional PIM, except from 0 to 500 h. The reason for this is that the proposed method considers the influence of past effects. In these built dynamic WCs, which are arranged in descending order of severity, the cumulative damage at a specific moment is always greater than the cumulative damage assumed to always exist in the current state. In the 0 to 500 h period, since no other specific WCs appeared beforehand, the calculation results of the two methods are the same.