# Uncertainty Representation of Natural Frequency for
## 4 Construction of Accurate Kriging Model Based on Prediction Interval Criterion
### The Proposed Improved Kriging Model  
After transforming interval variables and probabilistic variables into PDF, an accurate Kriging model is constructed, where the sampling points are selected based on the maximization criterion of the performance target prediction interval, the balance between modeling efficiency and model robustness is achieved by increasing the sampling points of uncertain variables and reducing the uncertainty of the performance response. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Parameter Value** & \(\boldsymbol{u}\) & **The Range of \(\boldsymbol{x}\)** & **Type** \\ \hline \(q\geq 0\) & & & \(\sqrt[3]{A}+\sqrt[3]{B}-\Theta/3\) & \((-\infty,+\infty)\) & IV \\ \hline \multirow{6}{*}{\(q<0\)} & \multirow{6}{*}{\(a_{4}>0\)} & \multirow{6}{*}{\(M_{3}\geq 0\)} & \(2r\cos(\theta/3)-\Theta/3\) & \(I_{1}^{*}<x<J_{2}^{*}\) & II \\ \cline{3-6}  & & & \(\sqrt[3]{A}+\sqrt[3]{B}-\Theta/3\) & \(x\geq J_{2}^{*}\) & \\ \cline{3-6}  & & & \(\sqrt[3]{A}+\sqrt[3]{B}-\Theta/3\) & \(x\leq J_{1}^{*}\) & III \\ \cline{3-6}  & & & \(-2r\cos[(\theta-\pi)/3]-\Theta/3\) & \(I_{1}^{*}<x<J_{2}^{*}\) & \\ \cline{2-6}  & & \(a_{4}<0\) & & \(-2r\cos[(\theta+\pi)/3]-\Theta/3\) & \(J_{2}^{*}\leq x\leq J_{1}^{*}\) & I \\ \cline{2-6}  & & \(a_{4}=0\) & \(M_{3}\
eq 0\) & \(\frac{-a_{2}+\sqrt{a_{2}^{*}+4a_{3}(a_{3}+x_{*})}}{2a_{3}}\) & \(a_{2}^{2}+4a_{3}(a_{3}+x_{*})\geq 0\) & V \\ \cline{3-6}  & & \(M_{3}=0\) & \(x_{s}\) & \((-\infty,+\infty)\) & VI \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The division criterion of standardized transformation random variable \(u\)According to the mean value \(\mu_{G}(x)\) and mean square error \(\sigma_{G}(x)\) of the Kriging surrogate model at the prediction point \(\tilde{x}_{i}\), the prediction interval of the performance target is determined by the \(3\sigma\) rule as \([\mu_{G}(x)-c\sigma_{G}(x),\mu_{G}(x)+c\sigma_{G}(x)]\). Aiming at the optimization design problem with the minimum of natural frequency, the initial Kriging model is constructed according to the existing sampling points, and the optimal value of the uncertainty variable \(x_{\min}\) is calculated with the objective of minimum of natural frequency. The upper bound of the performance prediction at the optimal value \(x_{\min}\) is \(\mu_{G}(x_{\min})+c\sigma_{G}(x_{\min})\), and the lower bound of the prediction interval based on any other input variable \(x\) is \(\mu_{G}(x)-c\sigma_{G}(x)\). Then a new sampling point is selected based on the criterion of maximizing the prediction interval, the new input variable \(x_{m+1}\) is the corresponding input variable when the following function \(g(x)\) takes the maximum value, where it can be expressed by  
\[g(x)=(\mu_{G}(x_{\min})+c\sigma_{G}(x_{\min}))-(\mu_{G}(x)-c\sigma_{G}(x)), \tag{23}\]  
Step 1: sampling the initial sample points. Assuming that the uncertain input variables and the distribution parameters all obey the random normal distribution, the optimal Latin hypercube method in DOE is used for uniform sampling to obtain the initial set of \(m\) sampling points. Step 2: construction of the initial Kriging surrogate model. Using computer numerical analysis, the output response values corresponding to each sample point are calculated. The initial Kriging surrogate model based on the \(m\) sampling points and response values of uncertain variables \(G_{m}(x)\) is constructed to obtain the predicted mean and predicted variance function of the Kriging model. Step 3: the solution of the optimal input variables based on the Kriging model \(G_{m}(x)\). Taking the minimum natural frequency response as the objective, the corresponding sampling points \(x_{\min}\) in the model are selected as the optimal input variables, and the upper and lower bounds of the prediction interval are obtained. Step 4: the selection of new sampling points.