## 3 Performance Degradation and Acceleration Model of Motorized Spindle
### Acceleration Model  
According to the results in Section 2.2, it can be concluded that the stress of the motorized spindle is mainly mechanical stress and electrical stress, which is close to the stress described by the inverse power law model [35]. Therefore, the inverse power law model is selected as the acceleration model of the motorized spindle acceleration test, and its standard equation is as follows:  
\[P=CS_{l}^{A} \tag{8}\]  
where \(P\) is the reliability characteristic quantity, \(S_{l}\) is the \(l\)-th stress level of the stress, and C and A are constants independent of the stress level, but related to the failure model and stress type.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Stress** & \(\beta_{Z0}\) & \(\beta_{Z}\) & \(\beta_{0}^{Z}\) & \(\beta^{Z}\) \\ \hline Cutting force \(S_{1}\) & 1 & 0.2 & 3 & 1 \\ Spindle speed \(S_{2}\) & 1 & 0.2 & 1 & 1/3 \\ Tool change times \(S_{3}\) & 2 & 0.4 & 1/9 & 1/27 \\ Cutting power \(S_{4}\) & 5 & 1 & 1 & 1/3 \\ Temperature \(S_{5}\) & 4 & 0.8 & 1 & 1/3 \\ \hline \end{tabular}
\end{table}
Table 8: The influence range coefficient \(\beta_{Z}\) and action coefficient \(\beta^{Z}\).  
\begin{table}
\begin{tabular}{c c} \hline
**Stress** & \(H_{Z}(R|\,\mathcal{D}/10^{-2}\) \\ \hline Cutting force \(S_{1}\) & 0.189 \\ Spindle speed \(S_{2}\) & 0.063 \\ Tool change times \(S_{3}\) & 0.001 \\ Cutting power \(S_{4}\) & 0.004 \\ Temperature \(S_{5}\) & 0.003 \\ \hline \end{tabular}
\end{table}
Table 9: Comprehensive importance of stress.  
The standard inverse power law model can only be used as an acceleration model for a single stress. In order to make it applicable to the case of multiple stresses, the standard inverse power law model is extended [35]:  
\[P=CS_{1l}^{A1}S_{2l}^{A2}\ldots S_{ll}^{AI} \tag{9}\]  
Take a logarithm on both sides of Equation (9), and Equation (9) becomes:  
\[\ln P=A1\ln S_{1l}+A2\ln S_{2l}+\ldots+A1\ln S_{ll}+\ln C \tag{10}\]  
The performance degradation or failure of motorized spindle is usually related to a variety of stresses. In Reference [4], a generalized logarithmic linear model was established by means of stress combination, as shown in Equation (11):  
\[\ln(\mu)=\gamma_{1}\varphi_{1}+\ldots+\gamma_{I}\varphi_{I}+\gamma_{0} \tag{11}\]  
where \(\gamma_{0},\gamma_{1},\ldots,\gamma_{s}\) are the model parameter and \(\varphi_{1},\varphi_{2},\ldots\varphi_{n}\) are the stress function.  
Since \(P\) is the reliability characteristic, then in accelerated degradation test, \(P=\mu\), if \(\ln C=\gamma_{0},A1=\gamma_{1},\ldots,A1=\gamma_{I}\), by combining Equation (10) with Equation (11), a generalized logarithmic acceleration model between the motorized spindle and stress is obtained, as shown in Equation (12):  
\[\ln(\mu)=\gamma_{0}+\gamma_{1}\ln S_{1l}+\ldots+\gamma_{I}\ln S_{ll}+\gamma_{0} \tag{12}\]  
Based on the above analysis and assumption, the accelerated degradation model of the spindle is established, as shown in Equation (13):  
\[\left\{\begin{array}{c}\ln\mu_{k}=\gamma_{1}lnS_{1l}+,\ldots,+\gamma_{I}lnS_ {ll}+\gamma_{0}\\ Y\sim N\big{(}\mu_{k}(\Delta t_{k}),\sigma^{2}(\Delta t_{k})\big{)}\end{array}\right. \tag{13}\]  
where \(\Delta t_{k}=t_{k}-t_{k-1}\).