## 5 Illustrative Examples  
### Vibration Reduction Optimization Design of S-Shaped Curved Stiffened Shell  
As a typical complex thin-walled structure in the aviation field, the S-shaped curved stiffened shell is widely used in fighter planes due to its advantages of infrared stealth and low exploration [36]. With extremely complex service conditions, the dynamic response of the S-shaped curved stiffened shell is very significant, as it is prone to large deformation and vibration, affecting its working stability and even leading to structural failure and destruction. Therefore, it is important to perform the vibration reduction optimization to ensure the strength and safety of the S-shaped curved stiffened shell under service conditions. However, due to the complexity of the structure, the degrees of freedom of mesh model of the S-shaped curved stiffened shell are large, which would affect the efficiency of frequency response analysis and prolong the design period of the structure. In this section, the structure is optimized using the proposed vibration reduction optimization framework based on the global POD ROM. The inner profile of the S-shaped curved stiffened shell is composed of a center line and a series of sections varying along the center line. The section at the inlet is circular and at the outlet is fillet rectangular, as shown in Figure 4. Equations (24) and (25) give the expression of the center line, where \(\Delta_{1}\) and \(\Delta_{2}\), respectively, represent the vertical distance from the center of the left circular section and the right rectangular section to the horizontal reference line. \(L\) is the length of the horizontal reference line, \(L_{1}\) and \(L_{2}\) are the horizontal reference line lengths of the first and second center line expressions, respectively. \[\frac{y_{1}(x)}{\Delta_{1}}=3\bigg{(}\frac{x}{L_{1}}\bigg{)}^{4}-4\bigg{(} \frac{x}{L_{1}}\bigg{)}^{3},\;0\leq x\leq L_{1}, \tag{24}\]  
\[\frac{y_{2}(x)}{\Delta_{2}}=3\bigg{(}\frac{x-L_{1}}{L_{2}}\bigg{)}^{4}-8 \bigg{(}\frac{x-L_{1}}{L_{2}}\bigg{)}+6\bigg{(}\frac{x-L_{1}}{L_{2}}\bigg{)}^{ 2}-\frac{\Delta_{1}}{\Delta_{2}},\;L_{1}\leq x\leq L. \tag{25}\]  
Figure 3: Flowchart of vibration reduction optimization approach accelerated by global POD ROM. The geometric dimensions of the structure are as follows: \(L=4170\) mm, \(L_{1}/L_{2}=0.66\), \(\Delta_{1}/L_{1}=0.28\), \(\Delta_{2}/L_{2}=0.516\). The width and height of the fillet rectangular section on the right are \(W_{ex}=1500\) mm and \(H_{ex}=620\) mm, respectively, and the fillet radius is \(R_{ex}=130\) mm. The radius of the circular section on the left is \(R_{in}=700\) mm. The number of ring stiffeners and axial stiffeners, respectively, are 30 and 6. The height of the stiffeners is 50 mm. The thickness of skin and stiffeners of the initial configuration are 3 mm. Material properties are as follows: [PERSON]'s modulus \(E=167\) GPa, [PERSON]'s ratio \(\mu=0.3\), and density \(\rho=7800\) kg/m\({}^{3}\). The Rayleigh damping coefficient is obtained according to the previous data and set as \(\alpha=25\), \(\beta=1.25\times 10^{-4}\). The finite element model is discretized by S4 element, with 13,470 nodes, 13,680 elements and 80,820 degrees of freedom. The boundary conditions are that the inlet section is clamped, and the degrees of freedom of the outlet section is only relaxed along \(x\) direction. A harmonic load with 1 N amplitude is applied to all nodes in the \(x\) direction of the outlet section on the right. The analysis frequency range is [40 Hz, 120 Hz], and the frequency sweep interval is 0.1 Hz. Using the FOM, the energy-frequency curve is calculated as shown in Figure 5; the calculation takes 3752.8 s. As shown in Figure 5. Energy-frequency response of initial configuration calculated by the FOM of the S-shaped curved stiffened shell, the first resonant frequency is 60.1 Hz, and the second one is 88.0 Hz. The frequency range of interest is [88.0 Hz, 92 Hz], the energy sum of which is 1409.2 db (shown in Figure 5, shadow area).