## 4 Numerical Results
### Case 3: Study with a Random Friction Coefficient and Disc's Material Properties  
In the final case study, the friction coefficient, Young's modulus and density of the disc are considered to be random. These properties are modeled using random variables as follows:  
\[ \begin{split}\mu&=0.5+0.5\;\zeta_{1}\\ E_{\text{disc}}&=125\times 10^{9}(1+0.2\zeta_{2})\\ \rho_{\text{disc}}&=7200(1+0.2\zeta_{3})\end{split}  \tag{24}\]  
Figure 14: Evolution of the 2-norm and 1-norm error with the number of samples in case 2. Stationary and nonstationary kriging with polynomial order from 0 th to 3 rd are considered. Figure 13: Response surfaces for the real part of three unstable eigenvalue pairs using nonstationary kriging. where \(\zeta_{1}\), \(\zeta_{2}\), and \(\zeta_{3}\) are uniform random variables within the interval \([-1,1]\). By using 200 Sobol sequential samples, the stationary and nonstationary kriging with constant trend function are applied. The hyperplane results of the SVM for three unstable eigenvalue pairs are shown in Figure 15. Based on the obtained metamodels, probability density functions (PDF) for the real part of the three unstable eigenvalues are estimated; the results are presented in Figure 16. In obtaining the reference results, 10,000 samples using Latin hypercube sampling (LHS) are considered. It can be seen from the stationary kriging results that for the first eigenvalue, the estimated PDF is inaccurate. Although the overall shape of the PDF for the second and third eigenvalues matches well with the reference results, the error still exists near the stable region (i.e., Real(\(\lambda\)) = 0). However, the proposed method yields almost identical results to the reference for the second and third eigenvalues, even in the stable region. For the first eigenvalue, although there is still an error, the proposed method shows better performance than stationary kriging. Next, to evaluate the metamodel performance, the Sobol sequential samplings, from 200 to 500 with increments of 25, are applied. Polynomial functions from 0 th to 3 rd order are considered as a trend function. The 2-norm and 1-norm error for the real part of eigenvalue pairs are presented in Figure 17. The obtained results indicate that for 2-norm-based error criteria, stationary and nonstationary kriging show similar performance in the small sample sizes. However, for 1-norm-based error criteria, nonstationary kriging outperforms over the stationary kriging within the all considered the sample sizes and polynomial orders. To further investigate the performance of the metamodels, the propensity of stability is considered. Given \(N\) samples, the \(i\)-th eigenvalue stability can be estimated as follows:  
\[P_{S}^{i}=\text{E}\Big{[}I_{\Omega}^{i}(\zeta)\Big{]}\simeq\frac{1}{N}\sum_{j =1}^{N}I_{\Omega}^{i}\Big{(}\zeta^{j}\Big{)}=\frac{N_{\text{stable}}^{i}}{N} \tag{25}\]  
Figure 16: Estimated PDF for the real part of three unstable eigenvalues using 200 samples. Figure 15: Trained hyperplane results for the three unstable eigenvalue pairs in case 3.  
where \(N_{\text{stable}}^{i}\) is the number of stable samples for \(i\)-th eigenvalue, and \(I_{\Omega}^{i}(\zeta)\) is the indicator function given by  
\[I_{\Omega}^{i}(\zeta)= \begin{cases}1,&\lambda_{i}(\zeta)<\text{tol}\\ 0,&\text{otherwise}\end{cases}  \tag{26}\]  
where tol is the tolerance of the indicator function. Under 10,000 LHS samples and 0.1 tolerance, the reference eigenvalue stability in Equation (25) is computed. The propensity of stability based on the constructed metamodels is computed using the predicted responses at the LHS sample points. The obtained results for the three eigenvalues are presented in Figure 18. As seen from the results, although stationary kriging shows similar performance with nonstationary kriging based on 2-norm error criteria, their stability results are completely different.