# Estimation and Adjustment of Interface Stiffnesses for Machine-Tool
## 4 Result and Discussion
### Contact Stiffness Adjustment
The variation, \(\varphi\), of the natural frequency obtained from the FMA is defined as  
\[\varphi=\frac{\text{after adjustment}-\text{before adjustment}}{\text{before adjustment}} \tag{6}\]  
The variation of the natural frequency obtained from FMA is calculated by Equation (6), which is defined as the difference between the simulation result after adjustment and before adjustment, and then this is divided by the simulation result before adjustments. The yellow mark in Figure 9 shows the footing stiffnesses after a multiple of \(10^{1}\) multiplications, which interprets only one footing direction, which is adjusted in one group for each adjustment step of footing stiffness. Figure 10 shows the natural frequencies corresponding \begin{table}
\begin{tabular}{c c c c c} \hline
**EMA Mode** & **EMA (Hz)** & **FMA Mode** & **FMA (Hz)** & **Error (\%)** \\ \hline
1 & 51.57 & 1 & 59.73 & 15.8 \\ \hline
2 & 110.11 & 3 & 76.70 & \(-\)30.3 \\ \hline
3 & 148.89 & 6 & 159.47 & 7.1 \\ \hline
4 & 169.98 & 7 & 166.39 & \(-\)2.1 \\ \hline
5 & 188.11 & 8 & 234.97 & 24.9 \\ \hline
6 & 259.41 & 10 & 307.40 & 18.5 \\ \hline \end{tabular}
\end{table} 
Table 3: Comparison of the natural frequency of the whole-machine-tool between EMA and FMA. to different modes and different footings resulting from FMA after the estimated stiffness is multiplied by \(10^{1}\). The variation shown in Figure 11 is a case of taking this multiple of \(10^{1}\) multiplications. Note that only one estimated footing stiffness is varied for each adjustment step. Figure 10: The natural frequencies corresponding to different modes and different footings resulting from FMA after the estimated stiffness multiplied by \(10^{1}\). Figure 9: One Footing direction is adjusted in one group for each footing stiffness adjustment step. It can be observed that the stiffness values of the footings after \(10^{1}\) multiplication adjustment results in the natural frequencies corresponding to different modes and different footings from FMA. It can be found that the variation of the footing stiffness value at different locations and different directions has no obvious regularity in the influence of the natural frequency of each mode, and the variation is also quite different. Therefore, the display of the analysis data is switched to a specific color presentation that can clearly understand the ranking of the influence level among different footings. The variation ranking corresponding to different modes of the analysis results are shown in Figure 12.  
It can be observed that, when the estimated footing stiffnesses were performed by a multiple of \(10^{1}\) multiplications, the first four ranking stiffness va  
Figure 11: The variation corresponding to different modes and different footing locations after the estimated stiffness multiplied by a multiple of \(10^{1}\). Figure 12: The ranking of the variation expressed by different specific colors corresponding to different modes after the estimated footing stiffnesses were multiplied by \(10^{1}\). It can be observed that, when the adjustment is a \(10^{1}\) multiplication, the natural frequency will be increased due to the adjustment of the footing stiffnesses. However, when the multiples are \(10^{3}\) and \(10^{5}\), the 1 st, 6 th, 8 th, and 10 th modes of the natural frequency will be decreased down due to these multiple magnifications of the footing stiffness adjustments. This indicates that the multiplication tendencies of the footing stiffness adjustments may be different from the variation trends (increase up or decrease down) of the natural frequencies, which are those obtained from FMA. Therefore, these generalized results are helpful for adjusting the interface stiffnesses of the footings. From Table 3, it can be noted that the sign and magnitude of the error between the simulation results and experimental results, and the adjustment guidance of the footing stiffness, are dominated by this error-related information.