## 4 Results
### Model Validation through Experiments  
Experimental validation was performed using the experimental setting shown in Figure 8. Considering a segmentation as in Figure 1, the total lengths of segments are such that \(l_{1}+l_{2}=1\) m, \(l_{3}+l_{4}=1\) m and \(l_{5}+l_{6}=0.8\) m. The structure is built with standard 20 mm \(\times\) 20 mm aluminum profile with the following properties: \(A=1.6\times 10^{-4}\) m\({}^{2}\), \(I=0.7\times 10^{-8}\) m\({}^{4}\), \(\rho=2700\) kg/m\({}^{3}\), and \(E=70\) GPa. A 30 W vibration BLDC motor with a speed controller was used to induce vibrations. The minimum speed of the controller is 450 RPM. An accelerometer MPU9250 was used as a sensor device, having a bandwidth of 200 Hz. Thus, the experimental setting is able to produce and detect damped resonant frequencies in the range of 7.5 Hz-40 Hz. Consequently, only the first damped resonant frequency was investigated. Second, third, and larger resonance frequencies are not analyzed in this experiment due to practical phenomena. First, the second and larger fundamental frequencies correspond to actuator speeds larger than 4000 RPM, which are not excited in practical CNC router machines. Moreover, magnitudes associated with the second and larger fundamental frequencies are several times smaller than that of the first fundamental frequency. Furthermore, when damping exists (i.e., when considering real structures), second and larger resonance frequencies tend to vanish, thus making it impractical to detect those frequencies in experimental settings. Different simulations were performed, characterized by the combinations of values \(l_{1}\in\{0.2,0.3,0.4,0.5\}\) and \(l_{5}\in\{0.2,0.4\}\). For instance, Figure 9 shows the vertical accelerations obtained (in \(g\) units) for different load frequencies (motor speed). The first damped resonant frequencies are shown in Table 4 (indicated as exp.)  
Simulations based on the proposed model were performed for the same combinations, applying harmonic loads with a frequency in the interval 0 Hz-50 Hz to estimate the first damped resonant frequency. For this, a damping coefficient of \(\gamma=0.3\) was considered. Masses were considered as \(m_{1}=m_{2}=0.45\) kg and \(m_{3}=2.9\) kg, with \(r_{1}=r_{2}=0\) and \(r_{3}=0.035\) m, associated to the motor and other accessories. The obtained first damped resonant frequencies are shown in Table 4, together with the resulting absolute percentage errors. The error of the first frequency is below 0.05 for the cases with \(l_{1}=0.4\) m and \(l_{1}=0.5\) m. Similarly, the error is below 0.15 in all the cases in which \(l_{5}=0.4\) m. The largest error (0.375) was obtained with the configuration (\(l_{1}=0.2\) m, \(l_{5}=0.2\) m). Notice that the error becomes larger when the structure reaches a position with larger rigidity (where the motor is closer to a corner), in which vibration amplitudes are expected to be smaller. The average error is 0.108. In this way, we consider that the experiments validate that the proposed model provides a fair approximation of the first resonance frequency, specially at the less rigid positions in which vibration amplitudes are expected to be more relevant.  
Figure 7: Fourth mode for \(a=0.01\) m, \(M=1\) kg, \(l_{1}=0.5\) m and \(l_{5}=0.5\) m, computed with proposed Model (**left**) and with FEM (**right**).