# Uncertainty Representation of Natural Frequency for
## 5 Uncertainty Measurement of Natural Frequency  
The above sections only describe the basic principles of the four statistical moments method and Kriging surrogate model, which are not introduced into engineering examples. The natural frequency analysis process of the laminated composite cylindrical shells based on the four statistical moments method and the Kriging surrogate model with insufficient available data of uncertain input parameters is listed below.  
Step 1. Uncertainty transformation of interval variables. According to the amount and information of data input, the uncertain input parameters can be divided into probabilistic variables and interval variables. Since the data amount of uncertain input parameters of laminated composite cylindrical shells is smaller, the interval variables account for more than half of all uncertain variables, and it is difficult to obtain accurate PDF, so the uncertain transformation of interval variables becomes the research focus.  
Step 2. Uncertainty measurement of probabilistic and interval variables. Probabilistic variables can be directly converted into PDF through sampling data. In the uncertainty interval of interval variables, the _ns_ sampling points are obtained by uniform sampling, and the PDF value \(p(\widetilde{x}_{i})\) of each sampling point is used as the optimization variable. The four statistical moments of interval variables are obtained to convert into a cubic normal distribution function according to the corresponding conversion method. Finally, according to the partition criterion of standardized transformed random variables \(u\), the PDF of interval variables is acquired by Equation (18).  
Step 3. Probabilistic and interval variables sampling, and performance value calculation at sampling points. After the operation of step 2, the uncertain input variables of the laminated composite cylindrical shells all obey the random normal distribution, and the optimal Latin hypercube method is used for uniform sampling to obtain the initial set of \(m\) sampling points.  
Step 4. Kriging surrogate model construction and the optimal input variables solution. Using numerical calculation software, the natural frequency of the laminated composite cylindrical shells corresponding to each sampling point is obtained. Based on the sampling points of the uncertain input variables and the natural frequency calculation values, the initial Kriging surrogate model is constructed to acquire the predicted mean and predicted variance functions. Taking the minimum natural frequency response as the objective, the upper and lower bounds of the prediction interval are acquired.  
Step 5. The adaptive increase in sampling points. Based on the prediction interval maximization criterion, according to Equation (19), the new sampling point and the corresponding natural frequency are obtained to construct the new Kriging surrogate model.  
Step 6. Uncertainty representation of natural frequency based on the Kriging surrogate model. The error \(h(x)\) of the corresponding natural frequency is calculated by Equation (24), and the error mean \(E(h(x))\) is shown in Equation (25). If the error mean is less than the set value \(\varepsilon\), the sampling is stopped, that is, the natural frequency of the laminated composite cylindrical shells with uncertain input variables is accurately represented. Otherwise, step 5 is performed again until the natural frequency is accurately represented.