## 5 Case Study
### A Case Study of Motorized Spindle
\begin{array}{ccc}E\Big{(}-\frac{\partial^{2}\ln p(x|\theta )}{\partial^{2}\gamma_{1}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{ \partial\gamma_{1}\partial\gamma_{2}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p (x|\theta)}{\partial\gamma_{1}\partial\gamma_{2}}\Big{)}&E\Big{(}-\frac{ \partial^{2}\ln p(x|\theta)}{\partial\gamma_{1}\partial\sigma^{2}}\Big{)}\\ E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\sigma^{2}\gamma_{1}} \Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\sigma^{2}\gamma_ {2}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\sigma^{2} \gamma_{2}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\gamma _{2}\sigma\sigma^{2}}\Big{)}\\ E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\gamma_{0}\partial\gamma_ {1}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\gamma_{0} \partial\gamma_{2}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{ \partial\gamma_{0}\partial\sigma^{2}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p (x|\theta)}{\partial\gamma_{0}\partial\sigma^{2}}\Big{)}\\ E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\sigma^{2}\gamma_{1}} \Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\sigma^{2}\gamma _{2}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\sigma^{2} \gamma_{0}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial( \sigma^{2})^{2}}\Big{)}\end{array} \right] \tag{26}\]  
Because \(M_{m}^{11}(\xi)\) is \(1\times 1\) matrix, and according to the principle of matrix partition, \(M_{22}^{m}(\xi)\) can be divided into  
\[M_{22}^{m}(\xi)=\left[ \begin{array}{ccc}E\Big{(}-\frac{\partial^{2}\ln p(x| \theta)}{\partial^{2}\gamma_{2}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x| \theta)}{\partial^{2}\gamma_{2}\partial\gamma_{0}}\Big{)}&E\Big{(}-\frac{ \partial^{2}\ln p(x|\theta)}{\partial\gamma_{2}\partial\sigma^{2}}\Big{)}\\ E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\gamma_{0}\partial\gamma_ {2}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\gamma_{0} \partial\gamma_{0}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{ \partial\gamma_{0}\partial\sigma^{2}}\Big{)}\\ E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\sigma^{2}\gamma_{2}} \Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{\partial\sigma^{2} \partial\gamma_{0}}\Big{)}&E\Big{(}-\frac{\partial^{2}\ln p(x|\theta)}{ \partial(\sigma^{2})^{2}}\Big{)}\end{array} \right] \tag{27}\]  
The objective function of scheme \(\xi\) is obtained as follows  
\[M_{m}^{11}(\xi)=\frac{M_{\xi}^{m}}{M_{22}^{m}(\xi)} \tag{28}\]  
According to engineering experience and test experience, the cost of accelerated degradation test for motorized spindle is shown in Table 12.  
Figure 7: Stress contour of the model. According to Reference [39], for SSADT with two stresses, when the number of stress levels is 4, the evaluation of the test scheme is the highest. Therefore, in order to ensure the accuracy and effectiveness of extrapolation, four stress levels (i.e., \(L=4\)). According to the objective function, Equation (28), all possible scheme sets are solved, and the optimal test scheme under each sample size (\(n=20,n=24,n=28\)) is obtained, as shown in Table 13.  
From the results in Table 13 that under the same constraint conditions, the scheme with the largest objective function of the motorized spindle accelerated degradation test with different sample sizes (\(n=20\), \(n=24\), \(n=28\)) has the same stress level, i.e., cutting force \(S_{1}=\) [800 850 900 950], speed \(S_{2}=\) [6000 6500 7000 7500]. However, due to the constraint of test cost, the inspection times are very different (i.e., the test time is very different), and then the objective function is very different. Therefore, under the same cost constraint condition, the test time should be extended as much as possible, instead of pursuing large samples, when the \(D_{s}\)-optimality is used to optimize the multi stress accelerated degradation test of motorized spindle.