# Estimation and Adjustment of Interface Stiffnesses for Machine-Tool
## 5 Conclusions  
In this study, estimation and adjustment of the interface contact stiffness of a three-axis machine-tool is investigated. Firstly, the contact model for each different contact interface is established, and the initial value of the estimated stiffness on each different contact interface is calculated. Then, each different contact interface, or their different combinations of the initially estimated stiffnesses, are prescribed on the interfaces for FMA. The interface that has a great impact on the natural frequency variation of the whole machine-tool is determined subsequently. Finally, the main adjusted object, such as footing, is obtained, and the adjusting method proposed in this study is applied for the stiffness adjustment on the footing interface. The adjusted procedures are performed repeatedly until the natural frequencies of the whole machine-tool obtained from FMA are very close to those of the EMA. From the above analyses, the following conclusions can be drawn. 1. In the convergent analysis of each subsystem three-dimensional geometrical model, as the element size meshed in FMA resulting in the natural frequency falls within the convergent range of 1%, the parameter of mesh-average approximating 0.8 is just accompanied. Hence, the attainment of this parameter (around 0.8) may be noted as the basis for the judgement whether the element size mesh is properly converged or not. 2. From the static analysis results, the reaction force states of the footings of the whole machine-tool indicate that the reaction force on each footing depends on both the self-weights and the positions of the parts stacked above. Hence, if the parameters related to footing interface may be properly accounted and separately assumed, then the interface stiffness distributions on the footings can be identified closer to the real situation accordingly. 3. By substituting each different contact interface or their various combinations of the initially estimated stiffness into the finite element model of the whole machine-tool in this study, it was found that the effects of these interface stiffnesses on the impact of the natural frequency of the whole machine-tool are not more obvious than those of the footing interface stiffness. This indicates that the interface stiffness of the footing is the main factor affecting the natural frequency difference between FMA and EMA. 4. Through the improvement method for footing stiffness adjustment proposed in this study, the FMA results of the footing stiffness under different multiple multiplications were collected and arranged. Thus, the location and direction of the specific footings that affect the natural frequencies corresponding to different modes are summarized by logical reasoning, which is regarded as the adjusting locations of the footing stiffness according to the error sign of the natural frequencies, i.e., it would be adjusted by increasing up or decreasing down to match the EMA results. These adjusted procedures are performed repeatedly until the natural frequencies obtained from FMA are very close to those of EMA. Finally, the error of the natural frequencies of the whole machine-tool between FMA and EMA is less than 5% after being completely  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Location**} & \multirow{2}{*}{**Direction**} & \multirow{2}{*}{**K, (N/mm)**} & \multicolumn{2}{c}{**K\({}_{g}\) (N/mm)**} & \multicolumn{2}{c}{**K\({}_{g}\) (N/mm)**} \\ \cline{2-7}  & **Estimated** & & **After Adjustment** & **Estimated** & **After Adjustment** & **Estimated** & **After Adjustment** \\ \hline A1 & \(3.23\times 10^{3}\) & \(3.23\times 10^{4}\) & \(9.10\times 10^{3}\) & \(9.10\times 10^{3}\) & \(1.30\times 10^{3}\) & \(1.30\times 10^{3}\) \\ \hline A2 & \(1.8  
adjusted, which indicates that the proposed method for adjusting the footing interface stiffnesses has a valuable reference in practical use. Conceptualization, supervision, writing, S.-Y.L. ; methodology, analysis, G.-C.L.