## 5 Illustrative Examples
### Vibration Reduction Optimization Design of PAF  
Satellites are subjected to a variety of vibration loads, among which the most severe vibration environment is in the launch process, including rocket engine pulse, transient impact load generated by rocket separation, explosive impact, and transverse aerodynamic load [37]. The load is directly transmitted from the satellite and rocket interface to the satellite, and then acts on the sub-stages and instruments of the satellite system, often leading to the exceeding of the satellite's dynamic environment, affecting the life of the satellite, and putting forward more stringent design indexes for the instruments carried on the satellite. As an important part connecting the satellite to the rocket, the PAF plays the role of load transmission, so it is necessary to carry out vibration reduction optimization design to improve the dynamic environment of the satellite during launch. Also, due to the complexity of the structure, the frequency response analysis is inefficient, and the optimization design circle is too long to satisfy the engineering. To solve above problems, in this section, the PAF is optimized using the proposed vibration reduction optimization framework based on the global POD ROM. As shown in Figure 13, the PAF structure in the engineering is thin-walled stiffened structure. The numerical model of the PAF structure is constructed according to the actual structure, and the simplified satellite structure is connected at top, as shown in Figure 14. \(D\) and \(d\) are the diameters of the lower and upper end frames, respectively. \(H\) is the height of the PAF structure. Figure 12: Iterative convergence curve for vibration reduction optimization of the S-shaped curved stiffened shell (FOM). The geometric sizes of the PAF structure are as follows: \(D=1000\) mm, \(d=240\) mm, and \(H=300\) mm. The number of U-shaped stiffeners is eight, and the flange and web are 20 mm and 15 mm, respectively. The number of ring stiffeners is one, the height of the stiffener is 15 mm. The thickness of all skin and stiffeners in the initial configuration of the structure is 3 mm. The shell thickness of the simplified satellite structure is 15 mm and the size is \(500\times 500\times 400\) mm. The satellite is separated by square thin shells at 130 mm and 270 mm relative to the height of itself. The hole is symmetrical at the top of satellite, the diameter is 205 mm, the distance of the center of the two circles is 225 mm, and the circular interface is 20 mm high. The material properties are as follows: [PERSON]'s modulus \(E=70\) GPa, [PERSON]'s ratio \(\mu=0.3\), density \(\rho=2700\) kg/m\({}^{3}\), and the Rayleigh damping coefficient of the whole structure is obtained according to the previous data and set as \(\alpha\)= 12.5, \(\beta=1\times 10^{-4}\). The finite element model is discretized by S4 and S3 elements, with 19,282 nodes, 18,928 elements and 115,692 degrees of freedom. The edge of the lower end frame is fixed, and the harmonic load of amplitude 1 N is applied to all nodes in the \(x\), \(y\) and \(z\) directions of the two circular interface boundaries at the top of the satellite. The analysis frequency range is [20 Hz, 140 Hz], and the frequency sweep interval is 0.1 Hz. The FOM is used to calculate the energy-frequency curve, as shown in Figure 15, which takes 5849.6 s. As can be seen from Figure 15, the first resonance frequency of the structure is 37.6 Hz, the second resonance frequency is 99.3 Hz, and the energy sum in the frequency range of interest [95.0 Hz, 115.0 Hz] is 2117.8 db (shaded area in Figure 15). The design variables for vibration reduction optimization of the PAF structure are, respectively, the thickness of 9 regions (_T1_, _T2_, _T3_, _T4_, _T5_, _T6_, _T7_, _T8_, _T9_) of the lower end frame, upper end frame, skin area A, skin area B, skin area C, skin area D, U-shaped stiffener web and flange, and ring stiffener). The optimization formulation is consistent with that of (26) in Section 5.1.