## 5 Conclusions  
Topology optimization based on machining characteristics was applied for structural design improvement and relevant stiffness reinforcement of a machine tool. The machining characteristics such as grinding vibrations and forces obtained from the ultrasonic-assisted grinding experiments of zirconia were utilized as the structure vibration investigations and boundary condition setting in static analysis, respectively. Both equivalent stress distributions obtained from the static analysis and the geometric shape (model) after topological analysis were utilized as important references for further structural design improvement of the column, base, and saddle subsystems. The differences in the effective stress, natural frequency, corresponding modal shape, weight, and stiffness of a machine tool between the original structure and after the design improvement may thus be compared completely. From the above analyses, the following conclusions can thus be drawn:  
1. [leftmargin=*]
2. Machining characteristics were successfully applied to topology optimization in this study. Through experiments, the grinding vibration was detected, and the distinct frequencies obtained were employed for modal shape investigations in numerical modal analysis. The subsystems of a machine tool system with larger displacement may thus be identified, such as the column, base, and saddle for further topology optimization. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Original Structure**} & \multicolumn{3}{c}{**After Design Improvement**} \\ \cline{2-4}  & & **Column + Base + Saddle** & **Variation (\%)** \\ \hline
**Mode** & **Stiffness (N/mm)** & **Mode** & **Stiffness (N/mm)** & \\ \hline \(1\) & \(1.21\times 10^{7}\) & 3 & \(4.73\times 10^{7}\) & 291.00 \\ \hline \(2\) & \(1.90\times 10^{7}\) & 4 & \(8.66\times 10^{7}\) & 355.97 \\ \hline \(3\) & \(4.16\times 10^{7}\) & 5 & \(1.13\times 10^{8}\) & 171.62 \\ \hline \(4\) & \(7.40\times 10^{7}\) & 6 & \(1.52\times 10^{8}\) & 105.34 \\ \hline \(5\) & \(1.01\times 10^{8}\) & 7 & \(2.56\times 10^{8}\) & 152.72 \\ \hline \(6\) & \(1.51\times 10^{8}\) & 8 & \(2.87\times 10^{8}\) & 89.82 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of stiffnesses between the original structure and after design improvement for a machine tool system. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Original Structure (kg)**} & \multicolumn{2}{c}{**After Design Improvement (kg)**} & \multirow{2}{*}{**Variation (\%)**} \\  & **Column + Base + Saddle** & & \\ \hline
2417 & 2492.4 & 3.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of weight between the original structure and after design improvement for a machine tool system. Meanwhile, the grinding forces measured during the experiments can be utilized for boundary condition setting in the static analysis for topological optimization. 2. Through topology optimization analysis, a comparison of a machine tool structural model before and after design improvements revealed that, under the consistency conditions of modal shapes, the sequential mode number was advanced by two modes after design improvement. This means the original machine tool after design improvement may be out of its inherently resonant frequency range. Furthermore, after design improvement, the natural frequency in each mode experienced an increasing phenomenon, with a maximum improvement of 110.28%. Despite only a minor increase in the machine tool weight, the stiffness of the entire system exhibited a significant increase, with a maximum improvement of 355.97%. This confirms the applicability and effectiveness of the proposed method and the accompanying procedures for topology optimization analysis in this study. Conceptualization, supervision, writing, S.-Y.L. ; methodology, analysis, C.-H.C. All authors have read and agreed to the published version of the manuscript.