## 5 Stochastic Design Optimization
### Current Performance Prediction of the Design Parameters and Tolerance Designs  
The basic tolerance of the optimum value for the design parameters of the magnetic circuit is shown in Figure 15 and Table 7. Owing to the difficulty in estimating the level of process capability level in the development stage, the probability distribution of the design parameters is defined as a triangular distribution that can assume an uncertain process capability level. For performance prediction, we performed stochastic analyses of the sensitivity of design parameters, performance distribution of electromagnetic force, and magnetic circuit thickness distribution after generating 10,000,000 random numbers for design parameters using the Markov chain Monte Carlo method. We defined the assumption of the probability distribution of the basic tolerance design as a triangular distribution that can assume an uncertain process capability level for the four design parameters, as illustrated in Figure 16. The triangular distribution of each design parameter was assumed to be a probability distribution that averages the minimum and maximum tolerances of the design parameters and the median of the tolerances. The maximum and minimum design ranges were set to the following conditions: magnet thickness (MT) = [8.00\({}^{+0.02}_{-0.04}\)] mm; plate thickness (PT) = [3.50\({}^{0}_{-0.05}\)] mm; yoke position (YP) = [4.80 \(\pm\) 0.03] mm; yoke thickness (YT) = [4.50\({}^{+0.05}_{0}\)] mm. The stochastic performance prediction of the basic tolerance design for the four design parameters is presented in Figures 17 and 18, which plot the results of the stochastic analysis of the electromagnetic force performance distribution and magnetic circuit thickness distribution by generating random numbers within the design range for the upper and lower limits of each design parameter, respectively. As show in Figures 19 and 20, considering the variations in the design parameters of the basic design  
Optimization of Design Parameters Tolerance  
The tolerance of design parameters for the magnetic circuit was optimized by calculating the standard deviation at the 3\(\sigma\) level of design tolerance for each parameter, and stochastic analyses were performed based on the sensitivity of the design parameters, electromagnetic force performance distribution, and thickness variation by the probabilistic analysis method after generating 10,000,000 random numbers. The standard deviation of each design parameter can be obtained from the 3\(\sigma\) level process capability index (Cp = 1) and Equation (15). Figure 17: The performance distribution of electromagnetic force by design tolerance. Figure 18: The performance distribution of thickness minimization by design tolerance. \[C_{\text{P}}=\frac{\text{USL}-\text{LSL}}{6\sigma}=1. \tag{12}\] \[\sigma_{\text{MT}}=\frac{8.02-7.96}{6}=0.01.\] (13) \[\sigma_{\text{PT}}=\frac{3.50-3.45}{6}=0.008\] (14) \[\sigma_{\text{YT}}=\frac{4.55-4.50}{6}=0.008\] (15) \[\sigma_{\text{YP}}=\frac{4.83-4.77}{6}=0.01 \tag{16}\]  
The probability distribution of the design parameters is a normal distribution that averages the standard deviations calculated using Equations (13)-(16), and the median values of the upper and lower limits of the nominal dimension, as defined in Figure 21. The mean and standard deviation of each design parameter are listed in Figure 16.0  
Figure 16.1 Analysis results of sensitivity by performance factor. Figure 16.2 Analysis results of sensitivity by performance factor of magnetic thickness minimization. Figure 16.3 Performance distribution of thickness minimization by 3 sigma level. Figure 16.4 Analysis results of sensitivity by performance factor of 3 sigma level. Figure 16.5 Comparison of reliability of electromagnetic force performance for design tolerance. (**A**) Electromagnetic force performance level within the range of design tolerance. (**B**) Electromagnetic force performance level in the range of design tolerance by the 3 sigma (3\(\sigma\)) level.