# Uncertainty Representation of Natural Frequency for
## 3 Interval and Probabilistic Variables Conversion Based on the Four Statistical Moments Method  
In Section 2, the modulus of longitudinal elasticity \(E_{1}\), the transverse modulus of elasticity \(E_{2}\), the main and secondary Poisson ratio \(\mu_{1}\) and \(\mu_{2}\), and in-plane shear modulus \(G\) are considered as the uncertainty variables. According to the available information of input data, these uncertainty variables can be divided into probabilistic variables and interval variables, which need to be solved by different uncertainty methods. Therefore, to unify the uncertainty analysis method, the interval and probabilistic variables conversion method based on the four statistical moments method is proposed [18]. For the probabilistic variables, the PDF value of the sampling points can be easily obtained, so the four statistical moments (mean \(M_{1}\), variance \(M_{2}\), skewness \(M_{3}\), kurtosis \(M_{4}\)) can be directly calculated, as shown in Equation (14). \[\left\{ \begin{array}{l}M_{1}=E(x),\\ M_{2}=E\big{(}x^{2}\big{)}-(E(x))^{2},\\ M_{3}=E\big{(}x^{3}\big{)}-3E\big{(}x^{2}\big{)}E(x)+2(E(x))^{3},\\ M_{4}=E\big{(}x^{4}\big{)}-4E\big{(}x^{3}\big{)}E(x)+6E\big{(}x^{2}\big{)}(E( x))^{2}-3(E(x))^{4},\end{array} \right. \tag{14}\]  
Among them, according to the \(n\)s random discrete sampling points \(\widetilde{x}_{i}\) of the statistical variables and the corresponding PDF value \(p(\widetilde{x}_{i})\), the 1-4 order statistical moments can be directly calculated by \[\left\{ \begin{array}{l}E(x)=\sum\limits_{i=1}^{ns}\widetilde{x}_{i}p(\widetilde{x }_{i}),\\ E(x^{2})=\sum\limits_{i=1}^{ns}\widetilde{x}_{i}^{2}p(\widetilde{x}_{i}),\\ E(x^{3})=\sum\limits_{i=1}^{ns}\widetilde{x}_{i}^{3}p(\widetilde{x}_{i}),\\ E(x^{4})=\sum\limits_{i=1}^{ns}\widetilde{x}_{i}^{4}p(\widetilde{x}_{i}), \end{array} \right. \tag{15}\]  
For the interval variables, the PDF value \(p(\widetilde{x}_{i})\) of these sampling points is not available, and the exact PDF of interval variables is difficult to obtain. In the uncertainty design domain, the number and location of selected sampling points are different, and the calculated value \(E\Big{(}x^{l}\Big{)}\) of the 1-4 order statistical moments will also be variable. Therefore, its four statistical moments are an uncertainty interval \(\big{[}M_{l}^{L},M_{l}^{L}\big{]},l=1,2,3,4\). The upper and lower bounds of the interval \([M1,M4]\) are calculated by using the optimization function of Equation (16). \[M_{l}^{L}/M_{l}^{L}=\underset{p(\widetilde{x}_{i})}{\text{min}}\text{/max}M_{ l}, \tag{16}\]  
First, sample uniformly within the uncertainty interval of interval variables to obtain \(ns\) sampling points \(\widetilde{x}_{i}(i=1,\cdots,ns)\). Then the PDF value \(p(\widetilde{x}_{i})\) of each sampling point is used as the optimization variable, and \(p(\widetilde{x}_{i})\) is optimized within the variable interval of the uncertainty variables to determine the upper and lower bounds of the interval \(\big{[}M_{l}^{L},M_{l}^{L}\big{]}\). For example, if the interval variable \(x\) is an interval variable represented by a single interval method, the PDF value \(p(\widetilde{x}_{i})\) of each sampling point is a random number in the range of (0, 1), and the total PDF value of the random sampling points is 1. According to the four statistical moments \([M_{1},M_{2},M_{3},M_{4}]\) of interval variable \(x\), the standardized variable \(x_{s}\) is transformed into the cubic function of random variable \(u\), as shown in Equation (17). \[\frac{x-\mu_{x}}{\sigma_{x}}=x_{s}=S_{u}(u)=a_{4}u^{3}+a_{3}u^{2}+a_{2}u+a_{1}, \tag{17}\]  
where the mean value is solved by \(\mu_{x}=M_{1}\), standard deviation is solved by \(\sigma_{x}=\sqrt{M_{2}}\), and the coefficients \(a_{1}\sim a_{4}\) of the cubic function \(S_{u}(u)\) are functions of the four statistical moments \([M_{1},M_{2}]\), and the specific calculation details can be found in Ref. [17].