# Stochastic Natural Vibration Analyses of Free-Form Shells
## 4 Results and Discussion
### Shell Model 1: A Negative-Gaussian Free-Form Shell
#### 4.1.2 Stochastic and Sensitivity Analysis  
Three representative shapes generated in the optimization, i.e., the initial shape, the 20 th-step shape, the 400 th-step optimal shape, see Figure 4, are selected for the stochastic analysis of natural frequencies. LHS is adopted to generate samples. To determine the sample size, sampling is repeated on the optimal shape with size of 350, 400, 450, 500, 550, and 600, respectively. The mean values and standard deviations of first three frequencies are solved and checked by Equation (3), with \(\xi_{1}\), \(\xi_{2}\) set up as 0.01 and 0.001, respectively. It is found that, when the sampling size is 400, Equation (3) is satisfied for the  
Figure 4: Shape variation of Model 1 during the optimization process: (**a**) Initial guess; (**b**) 20 th step; and, (**c**) 200 th/400 th step (optimal). Figure 5: Evolution of the first three natural frequencies of Model 1 during the optimization process. Figure 3: The convergence process of Model 1.  
first three frequency of all the selected shapes. For brevity, only the check result of the first natural frequency of the optimal shape is given in Figure 6. Herein, the size of samples is determined as 400 and Figure 7 depicts the samples. The stochastic analysis is implemented on the initial shell model, the 20 th-step shell model and the 400 th-step (optimal) shell model, respectively. The natural frequencies of all samples are solved by FEM using ANSYS. We first focus on the study of first natural frequencies, which are shown in Figure 8. Mean values and standard deviations of the first natural frequency are obtained as 4.56 Hz and 0.15 Hz (initial), 6.13 Hz and 0.20 Hz (20 th step), and 11.76 Hz and 0.36 Hz (optimal shape), respectively. Figure 8: First natural frequency of Model 1. Figure 6: Sampling size test of first frequency every 50 samples. Figure 7: Samples of uncertainty parameters with size of 400: (**a**) Initial geometric imperfection; (**b**) Shell thickness deviation; and, (**c**) Elastic modulus deviation. Via the stochastic analysis, it is found that the first natural frequency follows a normal distribution. The cumulative distributions of first natural frequency of the three models are depicted and fitted in Figure 9 for clear analysis and comparison. As presented in Figure 9a, it can be seen that the value of first natural frequency of the initial guess ranges from 4.05 to 5.00 Hz and the cumulative distributions function (CDF) is fitted as  
\[F(x)=\left\{ \begin{array}{cc}\frac{1}{1+e^{-10.97(x-4.55)}}&4.05\leq x\leq 5.00\\ 0&\textit{else}\end{array} \right. \tag{7}\]  
where \(x\) denotes the first natural frequency. The determination coefficient is 0.9883, which shows a good approximation. It is also observed that the CDFs of 20 th-step shape and optimal shape is similar to Equation (7) and fitted as Equations (8) and (9), respectively  
\[F(x)= \begin{cases}\frac{1}{1+e^{-4.86(x-6.12)}}&5.52\leq x\leq 6.80\\ 0&\textit{else}\end{cases}  \tag{8}\]  
\[F(x)= \begin{cases}\frac{1}{1+e^{-4.88(x-11.75)}}&10.74\leq x\leq 12.78\\ 0&\textit{else}\end{cases}  \tag{9}\]  
Partial correlation coefficients with respect to the first natural frequency are calculated and subsequently normalized to rank the significance of each factor, as shown in Figure 10. It is illustrated  
Figure 9: Cumulative distributions functions (CDFs) of the first natural frequencies of Model 1: (a) Initial guess; (b) 20 th step; and, (c) 400 th step (optimal). that, for initial guess, shell thickness and elastic modulus deviations impose relatively significant effects with similar weights, while the initial geometric imperfection is the least significant factor. In the case of the 20 th-step shell, the significance ranking is similar to the initial guess, while the case of the optimal (400 th-step) shape shows a difference that the effect of elastic modulus is remarkably higher than that of shell thickness deviation.