# Structural Optimization of Combine Harvester Plate-Shell Undergoing Multi-Source Excitation
## 3 Results and Discussion
### Results of Optimization  
Based on the response surface optimization module in Workbench, this paper conducts the structural optimization design of the studied plate-shell structure. Since the engine speed of the combine harvester is 2400 rpm and the frequency reaches 90 Hz, which will cause resonance effects on most of the plate and shell structures on the body. Therefore, it is necessary to raise the fundamental frequency of the plate to avoid the resonance region of the excitation source. In addition, it is necessary to optimize the structure of the plate for its vibration response in a multi-source excitation environment. The specific steps of structural  
Figure 14: Response diagram of plate with variable edge stiffening thickness. (**a**) 2 mm thickness. (**b**) 3 mm thickness. (**c**) 4 mm thickness. (**d**) 5 mm thickness. Figure 13: Response diagram of plate with variable edge stiffener width. (**a**) 50 mm wide. (**b**) 40 mm wide. (**c**) 30 mm wide. (**d**) 0 mm wide. optimization design are DOE experimental design, construction of response surface model, and optimization of algorithm parameters. Figures 15 and 16 are the optimization steps of Workbench and the parametric table in Workbench. The purpose of structural optimization in this paper is to: a. Raise the fundamental frequency of the plate-shell structure to avoid the resonance range of each vibration source on the combine harvester; b. Improve the vibration response of the plate-shell structure to the multi-source excitation vibration environment. In addition, since the structural optimization in engineering design should be based on not increasing the weight of structural materials, the structural optimization design is based on the fact that the overall mass of the plate and shell is less than or equal to the original structural weight. For the plate-shell structure described in the example, the main factors are the width of the edge stiffener, the width of the stiffener, and the thickness of the foundation plate, the overall quality of the plate and shell is the constraint condition, and the first-order natural frequency and the maximum dynamic response are the optimization objectives. The plate-shell structure is structurally optimized. The optimization problem can be formulated as:  
\[\left\{ \begin{array}{ll}min:&F(X)=f(x_{1},x_{2},\ldots,x_{n})\\ &G(X)=g(x_{1},x_{2},\ldots,x_{n})\\ s.t.&g_{j}(X)\leq 8.7596\,j=1,2,\ldots n\\ &X_{i}^{l}\leq X_{i}\leq X_{i}^{u}\,i=1,2,\ldots,n\end{array} \right.\]  
Among them, \(X=\left(x_{1},\ x_{2},\ \ldots,\ x_{n}\right)^{\mathsf{T}}\) is the design variable, \(n\) is the number of design variables; \(F(X)\) is the first-order natural frequency, \(G(X)\) is the maximum dynamic response; \(g_{j}(X)\leq 8.7596\) is the overall quality constraint of the plate-shell structure; \(X_{i}^{l},\ X_{i}^{u}\) is the upper and lower limits of the \(i\)th component of the design variable \(X\). Figure 16: Parametric table. Figure 15: Optimization steps of Workbench. Since the above three influencing factors are all continuous variables, this paper restricts the numerical range of them according to the actual situation. The value range of each parameter is shown in Table 2. In the DOE experimental design, the Latin hypercube experimental design is used to generate 100 sample points, as shown in Table 3, and the finite element analysis of the parametric model is used to obtain the overall mass of the plate and shell corresponding to these sample points, the first-order natural frequency and Dynamic response under the action of two-point concentrated harmonic force. Existing structural optimization categories are divided into response surface optimization and direct optimization. Direct optimization refers to the optimization iteration directly on the finite element model. This method is more accurate but requires a large number of iterations and a lot of time.