# Uncertainty Representation of Natural Frequency for
## 7 Conclusions  
To accurately represent the natural frequency of laminated composite cylindrical shells with probabilistic and interval variables, the uncertain representation method based on the four statistical moments and accurate Kriging surrogate models is proposed. The uncertain input parameters obtained in the experiment are converted into probabilistic and interval variables. Based on the four statistic moments method, the probabilistic cubic  
Figure 5: The mean and standard deviation of DNF for the five-layer laminated composite cylindrical shells with clamped–supported boundary conditions. (**a**) mean, (**b**) standard deviation.  
Figure 6: The mean and standard deviation of DNF for the seven-layer laminated composite cylindrical shells with clamped–supported boundary conditions. (**a**) mean, (**b**) standard deviation.  
normal distribution functions are constructed and, according to standardized partition criteria for transforming the random variables, the PDF and CDF of interval variables are calculated. Subsequently, the optimal Latin hyper method is used to uniformly sample the interval variables and obtain the corresponding natural frequency to form the initial Kriging surrogate model. Finally, combined with the prediction interval maximization criterion, the sampling points are adaptively increased to establish to the accurate Kriging surrogate model, which can accurately represent the natural frequency of laminated composite cylindrical shells with uncertain input variables. The proposed method and MCS method are conducted to calculate the uncertain natural frequency of laminated composite cylindrical shells with different boundary conditions and different layers, and these results are highly fitting. In addition, the proposed method can greatly reduce the execution number of calculation functions and improve calculation efficiency.  
Although the proposed method can accurately represent the uncertain natural frequency of laminated composite cylindrical shells, there are some limitations for solving some engineering examples, which can be further studied: (1) the types of uncertain variable should be varied according to the amount and information of available input data. When the amount of input data increases, the interval variables can be transformed into the probabilistic variables, where the probabilistic variables can simplify the algorithm with very little calculation. Therefore, the accurate representation method of natural frequency should be adaptively adjusted when the input number of uncertain parameters changes. (2) The proposed uncertainty representation method is applicable to inherent characteristic of laminated composite cylindrical shells, such as natural frequency. However, there are many external excitations and the proposed method is no longer applicable. Therefore, the further research required is to represent the response characteristics and vibration propagation law of laminated composite cylindrical shells with different boundary conditions under the external excitation of uncertain input variables.  
The individual contributions of authors are specified as follows. Conceptualization, G.C. and X.P.; methodology, X.P.; software, T.W.; validation, T.W., C.L. and Y.Y.; formal analysis, T.W.; investigation, L.L.; resources, X.P.; data curation, Z.Y.; writing--original draft preparation, T.W. and G.C.; writing--review and editing, T.W.; visualization, Y.Y.; supervision, G.C.; project administration, X.P. and C.L.; funding acquisition, C.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (Grant No. 51775501, 51775510, 51875525) and Natural Science Foundation of Zhejiang province (Grant No. LZ21E050003, LY21E050008).  
Data Availability Statement: Data sharing not applicable.  
The authors declare no conflict of interest.