## 3 Results
### Tangential Support Plate Optimization Design  
This research chose the overall equivalent static load static optimization of the RSS base, using deformation and stress values as objective functions. The design parameters A, D, B, and L of the support plate were treated as design variables, and multi-objective dimensional optimization was performed using finite element methods. The basic functional form for dynamic optimization is as follows:  
\[\left\{\begin{aligned} &\min\phi_{t}(V_{1},V_{2},V_{3},V_{4}, \breve{x},x),t=1,2\\ & s.t.\left\{\begin{aligned} & M(V)\breve{x}+[K+S](V)x=0,\\ & S_{i}(V_{1},V_{2},V_{3},V_{4},\breve{x},x)\leq 0,i=1,2\ldots,n \end{aligned}\right.,\\ & V_{k}^{L}\leq V_{k}\leq V_{k}^{U},k=1,\ldots,4\end{aligned}\right. \tag{6}\]  
In this context, V\({}_{i}\) represents the design variable, M is the total mass matrix, K is the total stiffness matrix, x is the displacement vector related to force (design variable), and \(\breve{x}\) is the acceleration related to force (design variable). V\({}_{1}\), V\({}_{2}\), V\({}_{3}\), and V\({}_{4}\) correspond to the angle, width, thickness, and length of the support. \(\varphi\) is the objective function, \(\varphi_{1}\) indicates the overall deformation of the base, and \(\varphi_{2}\) represents the stress experienced by the base. g\({}_{i}\) is the constraint function, while V\({}_{K}^{\text{L}}\) and V\({}_{K}^{\text{U}}\) denote the lower and upper bounds of the kth design variable, and m represents the total number of inequality constraints.  
The initial parameters, the range of parameter variations, and the final optimization results for the existing tangential support plate during the optimization process are presented in Table 3.  
The optimized dimensions of the support plate are A = 0.507\({}^{\circ}\), B = 7 mm, D = 1.5 mm, and L = 169.22 mm. Modal analysis of the optimized support plate base model was performed, and the comparison with the original model is shown in Figure 6. From Figure 6, it can be seen that after optimization using the equivalent static load model, the first to sixth modal natural frequencies of the RSS base became 30.34 Hz, 125.33 Hz, 135.75 Hz, 145.89 Hz, 272.17 Hz, and 328.08 Hz, respectively. It is clear that the modal natural frequency values of the optimized support plate are lower than the pre-optimized model, with the first-order natural frequency being 30.34 Hz. This meets the condition that the first-order natural frequency is lower than the operating frequency and well away from the resonance frequency. It confirms the dynamic transformation of the base model from rigidity to flexibility and validates the effectiveness of the optimization using the equivalent static load model.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameters of the** & **Initial Values** & **Range of Variation** & **Optimized Values** \\
**Support Plate** & & & \\ \hline A (\({}^{\circ}\)) & 1.02 & 0–1.5 & 0.507 \\ B (mm) & 11.5 & 0–20 & 7 \\ D (mm) & 3 & 0–5 & 1.5 \\ L (mm) & 150 & 0–300 & 169.22 \\ \hline \end{tabular}
\end{table}
Table 3: The initial values, variation ranges, and optimization outcomes of the parameters set during optimization.