## 5 Discussion  
In order to illustrate different applications of the proposed model, we perform an analysis of a CNC structure similar as the structure shown in Figure 1. The frequencies and modes depend solely on the material and geometry parameters. These parameters are sourced from the manufacturer of the CNC _router_ type being modeled. If not available, they can be easily measured. For amplitudes, damping is required. The damping can be adjusted from experimentation. We assume that the structure is built with standard 40 mm \(\times\) 20 mm aluminum profiles as shown in Figure 10, having the following parameters: \(A\) = 3.405 \(\times\) 10\({}^{-4}\) m\({}^{2}\), \(I\) = 5.4113 \(\times\) 10\({}^{-8}\) m\({}^{4}\), \(\rho\) = 2700 kg/m\({}^{3}\), \(E\) = 70 GPa. A damping coefficient of \(\gamma\) = 0.04 was considered. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \multicolumn{2}{c}{\(\boldsymbol{l_{1}}\) **= 0.2 m**} & \(\boldsymbol{l_{1}}\) **= 0.3 m** & \(\boldsymbol{l_{1}}\) **= 0.4 m** & \(\boldsymbol{l_{1}}\) **= 0.5 m** \\ \hline \multirow{3}{*}{\(l_{5}\) = 0.2 m} & exp. & 32.58 & 28.30 & 27.33 & 24.96 \\  & model & 44.8 & 34.0 & 26.8 & 24.4 \\  & error & 0.375 & 0.201 & 0.019 & 0.022 \\ \hline \multirow{3}{*}{\(l_{5}\) = 0.4 m} & exp. & 29.08 & 28.46 & 25.80 & 24.83 \\  & model & 33.4 & 29.4 & 25.4 & 23.6 \\ \cline{1-1}  & error & 0.148 & 0.033 & 0.015 & 0.049 \\ \hline \end{tabular}
\end{table} 
Table 4: First damped resonant frequency in Hertz obtained with the experimental setting (exp.) and our model. The average error is 0.108. Figure 8: Structure for experimental validation (**left**) and detail of the aluminum profile (**right**). Figure 9: Experimental results for the case \(l_{1}\) = 0.4 m, \(l_{5}\) = 0.2 m. The peak acceleration occurs at 27.33 Hz. Considering the structure decomposition as in Figure 1, attached masses are considered at the coupling endpoints of \(e_{1}\) and \(e_{3}\), representing motors NEMA23 with masses \(m_{1}=m_{2}=1.15\) kg and a vertical distance between the mass centers and the neutral axis of \(r_{1}=r_{2}=0.06\) m. Moreover, a third mass is attached at the end of \(e_{5}\), representing the Z-axis displacement mechanism and a spindle, involving a total mass of \(m_{3}=12.5\) kg with \(r_{3}=0.06\) m. In this way, the masses, material, and cross-section parameters are defined. Notice that the length of the beam elements depends on the X-Y spindle position, which changes according to the manufacturing program. The total length of the X-axis profiles is 1 m, and the total length of the Y-axis profiles is 0.8 m, i.e., \(l_{1}+l_{2}=l_{3}+l_{4}=1\) m and \(l_{5}+l_{6}=0.8\) m.  
The first analysis approach is to compute a map of fundamental frequencies, as shown in Figure 11. These maps are computed by building the model for different values of \(l_{1}\) and \(l_{5}\), representing the X-Y position of the spindle, and computing the corresponding fundamental frequencies. As expected, the lower frequencies are found when the spindle is at the center of the CNC range, i.e., when \(m_{3}\) is farthest from the fixed endpoints. On the other hand, the fundamental frequencies are larger when \(l_{1}\) is near 0 m or 1 m, i.e., when the spindle is near the CNC supports, since the structure becomes more rigid at these configurations. These maps could be used by open-loop control strategies to avoid excessive vibrations. For instance, the spindle speed may be shifted away from the fundamental frequencies, depending on the spindle position. Moreover, when working on a small piece, this could be placed in the region where the first frequencies are high (the higher the frequency, the lower the vibration displacement). In a second analysis, a map of vertical displacements is performed, as shown in Figure 12 (left). This map indicates the vertical displacement of the spindle for different X-Y positions (i.e., for different values of \(l_{1}\) and \(l_{5}\)), when a load of \(f(t)=-100N\) is applied on the spindle.