## 5 Case Study  
### A Case Study of Motorized Spindle  
Taking the motorized spindle produced by Sipu Company in China as an example, the proposed optimization method was applied. The parameters of the spindle are shown in Table 10. The parameters of the accelerated degradation model of the motorized spindle obtained from previous reliability test (December 2016-December 2019) are shown in Table 11.  
Figure 5: Optimization process of accelerated degradation test scheme for the motorized spindle. By substituting the model parameters in Table 11 into Equation (13):  
\[\left\{ \begin{array}{c}\ln\mu_{k}=0.0875\times lnS_{1l}+0.3423\times lnS_{2l}-11. 6428\\ Y\sim N\Big{(}\mu_{k}(\Delta t_{k}),(0.006339)^{2}(\Delta t_{k})\Big{)}\end{array} \right. \tag{25}\]  
As the motorized spindle is a complex electromechanical system, in order to adapt to its complex failure mechanism, the linear Brownian motion of Equation (25) is transformed into nonlinear Brownian motion, and the \(\Delta t_{k}\) in Equation (25) is transformed into \(\Delta t_{k}\)[38], where \(\lambda=1.506\). According to Section 2.2, the cutting force \(S_{1}\) and rotation speed \(S_{2}\) are selected as the test stress. According to the technical parameter information, subsystem fault characterization and engineering experience, the failure threshold of motorized spindle is determined as 30 \(\upmu\)m The maximum cutting stress \(S_{1L}\) is 1000 \(N\), and the maximum spindle speed stress \(S_{2L}\) is 10,000 \(rpm\); in order to ensure the acceleration of the test, the minimum cutting stress \(S_{11}\) is set to 500 \(N\) and the minimum spindle speed stress \(S_{22}\) is set to 5000 \(rpm\) according to the previous working conditions. The surface fitting diagram of accelerated degradation model is shown in Figure 6. The coordinate point \(X\) is the value of speed \(S_{2}\), \(Y\) is the value of cutting force \(S_{1}\), and \(Z\) is the value of \(\mu_{k}\). Figure 7 shows the projection of equation surface fitting Figure 6 on the plane, that is, the stress contour line. According to Figure 7, it can be seen that the influence of cutting force on degradation rate is far greater than that of rotation speed, which is consistent with the conclusion of stress analysis in Section 2.2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & **Rated Power** & **Rated Voltage** & **Rated Current** & **Rated Torque** & **Rated Speed** & **Frequency** \\  & \(P_{e}\)/\(kW\) & \(U_{e}\)/\(V\) & \(I_{e}\)/\(A\) & \(T_{e}\)/\(Nm\) & \(n_{e}\)/\(rpm\) & \(f_{m}\)/\(Hz\) \\ \hline SPS1530-18Z & 7.3 & 350 & 15 & 5.9 & 12000 & 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Technical parameters of the sample motorized spindle. Figure 6: Three-dimensional fitting of the accelerated degradation model of the motorized spindle under two stresses. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\gamma_{1}\) & \(\gamma_{2}\) & \(\gamma_{0}\) & \(\sigma\) \\ \hline
0.0875 & 0.3423 & \(-11.6428\) & 0.006339 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Parameters of accelerated degradation model of the motorized spindle under double stress. According to the \(D_{s}\)-optimality and the actual situation of the calculation example, the cutting force is selected as the main stress concerned (\(M_{m}^{11}(\xi)\) is \(1\times 1\) matrix), and the information matrix \(M_{\xi}\) is  
\[M_{\xi}^{m}=\left[