## 4 Topology Optimization Design
### Variations in Effective Stress, Natural Frequency, and Rigidity  
Both the effective stress distributions resulting from the actions of the grinding forces along different axial directions in the static analysis and the geometric shape (model) after topological analysis are integrated to determine the selection priority for structural design improvement. Thus, the structure areas of a machine tool with larger effective stress distributions may be identified and the relevant subsystem emerge for stiffness reinforcement and improvement design. The geometric shape (model) after topological analysis can be used concurrently as a reference for a lightweight structure design. On this basis, comparison of the locations of maximum effective stress between the original structure and after design improvement for a machine tool system subjected to different direction loadings is shown in Table 3. The maximum effective stresses all occurred at the same locations of the subsystem before and after the design improvement. Moreover, the maximum effective stresses were all reduced after design improvement, and the maximum variation was about 25% along the \(z\)-axis direction. The relationship between the natural frequency, mass, and stiffness is represented by \(\mathrm{k}=\omega^{2}m\), where \(\mathrm{k}\) represents the stiffness (\(\mathrm{N/mm}\)), \(\omega\) is the natural frequency (\(\mathrm{Hz}\)), and \(m\) is the mass (\(\mathrm{kg}\)). The unit of \(\omega\) can also be expressed in terms of angular velocity (\(\mathrm{rad/s}=1/\mathrm{s}\)) and \(m\) can also be derived from the formula \(m=F/a\) and its unit is \(\mathrm{Ns}^{2}/\mathrm{m}\). The comparisons of the natural frequencies before and after the structural design improvement of a whole machine tool system are shown in Table 4. The corresponding modal shapes are depicted in Figure 19. Since the comparisons of the natural frequencies are based on the condition of having quite similar modal shapes, increasing the modal number order not only enhances the overall natural frequencies but also allows the entire machine tool system model to be out of its inherently resonant frequency range. This will avoid the occurrence of structure resonance. Moreover, there was a slight increase in natural frequencies for the six modes (modes from the 3 th to 8 th after the design improvement contrast with those from the 1 st to 6 th before the design enhancement), with the largest increase being 110.28%. Figure 18: Geometrical models before and after design improvements for the saddle subsystem. Left to right: before design improvements, after design improvements. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Original Structure**} & \multicolumn{2}{c}{**After Design Improvement**} & \\ \cline{2-5}
**Loading** & \(\vec{\sigma}_{\text{max}}\) & **Location** & \(\vec{\sigma}_{\text{max}}\) & **Location** & **Variation (\%)** \\
**Direction** & **(MPa)** & **(MPa)** & **(MPa)** & & \\ \hline \(x\) & 2.083 & column & 2.059 & column & \(-\)1.15 \\ \hline \(y\) & 2.795 & saddle & 2.221 & saddle & \(-\)20.53 \\ \hline \(z\) & 28.358 & column & 21.331 & column & \(-\)24.77 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the locations of the maximum effective stress between the original structure and after design improvement for a machine tool system. The weight comparison before and after the structural design improvement of a machine tool system is shown in Table 5, while the stiffness comparison corresponding to each mode is shown in Table 6. It can be observed that the maximum variation in