# Modeling a Milling Dynamometer as a 3 DOF Dynamic System by Stiffness Identification
## 3 Static Stiffness Determination
### FEM Ring Stiffness Determination  
The third method used simulated the stiffness using the finite element method [20]. The stiffness based on the FEM was measured considering the tools given by the program ANSYS Mechanical APDL. The 3D geometry of the ring was built using the module for modeling from the pre-processor step (Figure 7a). The rings were meshed with a 3D element defined as the 8-Node Structural Solid element SOLID185. The FE model consists of 1337 nodes and 5620 elements (Figure 7b). For the FEM analysis, the force was applied on the node in two directions, which are similar for the experiments:  
Figure 6: Ring geometry with main dimensions considered in stiffness analytic calculation. Figure 7: Ring models obtained in ANSYS: (**a**) 3D model; (**b**) ring finite element model. The longitudinal direction along the \(Oz\) axis, normal to the hole's surface (Figure 7b);
* The transverse direction, along the \(Ox\) axis, perpendicular to the \(Oz\), and normal to the hole (Figure 7b). As boundary conditions, taking into consideration the structure of the dynamometer, the following were considered:
* For the longitudinal direction of stiffness: the displacement of the nodes from the bottom screw faces is fixed in all three directions, and the nodes from the upper surface of the upper screw are blocked on both directions (\(Ox\) and \(Oy\)) (Figure 7b);
* For the transverse direction of stiffness: the same conditions were held for the bottom screw, and for the upper screw, the nodes were blocked in both directions (\(Oy\) and \(Oz\)) (Figure 7b);  
In the case of the ring's longitudinal direction stiffness, simulations were performed for six values of loads (Table 2). The same values for load forces were used in the experimental set-up. The deformation and stiffness values obtained using FEM, for each force, are presented in the same table. For the stiffness calculation, the values of the force and deformations in the same node were considered. The deformed shape in the case of longitudinal load is presented in Figure 8.  
Considering the values presented in Table 1, the average value for stiffness in the longitudinal direction is \(\mathrm{k_{FEM\_long}}=28.953\times 10^{6}\) [N/m]. In the case of the transverse stiffness determination, it was achieved in the same simulation as in the case of the longitudinal stiffness. The obtained values for deformation and stiffness are presented in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Force F [N]** & **Deformation [mm]** & **Longitudinal Stiffness** \\  & **(\$FEM\_long)** & **(\$\(\mathrm{k_{FEM\_long}}\)) [N/m]** \\ \hline
0 & 0 & 0 \\
98.1 & 0.003388 & \(28.948\times 10^{6}\) \\
196.2 & 0.006777 & \(28.948\times 10^{6}\) \\
293.32 & 0.010133 & \(28.947\times 10^{6}\) \\
390.44 & 0.013487 & \(28.949\times 10^{6}\) \\
487.56 & 0.016842 & \(28.949\times 10^{6}\) \\
585.66 & 0.020208 & \(28.981\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of main quantities for longitudinal stiffness. Figure 8: Ring FEM analysis in longitudinal direction: (**a**) boundary conditions and load; (**b**) longitudinal deformed ring for a load of 585.66 N.  
Based on the values presented in Table 2, the average value for stiffness in the transverse direction is kFEM_tr = 29.041 \(\times\) 10\({}^{6}\) [N/m]. The deformed shape in the case of longitudinal load is presented in Figure 9.  
A synthesis of the obtained values for the ring's stiffness in both the longitudinal and transverse directions is presented in Table 4.  
As can be seen from Table 4, there are three different values for longitudinal and transverse stiffnesses. For the next simulation, their average values were considered: klong = 29.195 \(\times\) 10\({}^{6}\) [N/m], and ktr = 28.894 \(\times\) 10\({}^{6}\) [N/m]. By comparing the values of the average stiffness with the obtained values, one can find small differences in the absolute values of individual errors (Table 5).