# Estimation and Adjustment of Interface Stiffnesses for Machine-Tool
## 4 Result and Discussion
### Contact Stiffness Adjustment  
In the numerical modal analysis module of the analysis software, all contact interfaces are established by the corresponding contact models for each different joint interface, and the initial stiffness values estimated are substituted into these models just mentioned above. The parameterization function of the software should be used for footing interface adjustment. The stiffness values of the footing in all directions are set by parameterization methods. As a result, the parameterized footing stiffness value becomes the input of the simulation system, and the natural frequency analysis result is the output of the simulation system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode**Interface** & **Footing** & **Column-Base Locking** & **Saddle Sliding Set** & **Workable Sliding Set** & **Head Sliding Set** \\ \hline
1 & \(0.8\%\) & \(1.15\%\) & \(0.09\%\) & \(0.14\%\) & \(0.01\%\) \\ \hline
2 & \(-47.0\%\) & \(0.98\%\) & \(0.10\%\) & \(0.05\%\) & \(0.13\%\) \\ \hline
3 & \(-38.3\%\) & \(0.38\%\) & \(0.20\%\) & \(0.21\%\) & \(0.21\%\) \\ \hline
4 & \(-44.3\%\) & \(0.28\%\) & \(0.17\%\) & \(0.04\%\) & \(0.13\%\) \\ \hline
5 & \(-25.4\%\) & \(0.27\%\) & \(0.11\%\) & \(0.03\%\) & \(0.20\%\) \\ \hline
6 & \(-10.7\%\) & \(0.28\%\) & \(0.04\%\) & \(0.01\%\) & \(0.14\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Change of the natural frequency of the whole machine-tool under the individual addition of each different contact interface stiffness. The finite element model is constructed from the design drawings, and there are some differences with the real structure. Moreover, this study does not account for the interface stiffness for the lead screw and transmission shaft. Hence, the corresponding results of the modal shape between FMA and EMA of the whole machine-tool model will be slightly different. Therefore, based on the modal shape results of the experimental modal analysis, the corresponding modal shapes are determined from the first 12 modes of the FMA results, with very similar vibration manner to EMA for this existing difference. The related error deduced between them is defined in Equation (5). Table 3 shows the comparisons of the natural frequency of the whole machine-tool between these two analyses for the six corresponding modes. \[\mathbf{e}=\frac{FMA-EMA}{EMA} \tag{5}\]  
It can be observed that the mode numbers are not exactly the same in terms of correspondence, while the EMA and FMA have the similar modal shapes. The calculation results of FMA do not just fall within the first six modes corresponding to those of EMA exactly in this study, which is presumed to be the reason for the boundary condition setting or some uncertain factors in the simulation. This results in modal shapes obtained from FMA that may not match those of the same mode number of EMA properly due to the setting of the boundary conditions, which cannot completely fit the real situations. Therefore, the corresponding mode numbers shown in Table 3 will be referred for the error comparison between EMA and FMA as the footing interface stiffness is adjusted in the subsequent analysis. The interface stiffness adjustment method applied at this stage may not only improve the shortcomings of the traditional try-and-error method, but it also may be easy in use. The proposed adjustment method may make the adjustment toward the solution processes and make the error converge logically. In this specific methodology, the estimated stiffness of the footings is multiplied by different certain values, which may be used as the input of the simulation system. The natural frequency resulting from FMA under different multiple multiplications are obtained, that is, there are 24 groups of FMA results for each adjustment step of one multiple multiplication, since there are eight footings in this study, and each footing possesses three stiffness values along the X, Y and Z directions, respectively.