# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 3 Results and Discussion
### Experimental Data Collection: Time Domain Analysis  
In Table 4, the most significant tests have been reported, with the IC of points P11 and P14 coincident with the single-point laser L2 and L1, respectively, evaluated from the acquired time histories of Figures 7-9.  
The BD test is used to evaluate the first natural frequency of the beam in a clamped-free configuration and to properly update the numerical model, which will be used for computing the dynamic properties in case of clamped-pinned-free beam. Moreover, the BD test is used to calibrate the single-point laser offsets to obtain the beam displacement perfectly symmetric with respect to static equilibrium. Only results of the profilometer are analysed, addressing the paper aim. The acquired time history of the BD test is reported in Figure 7.  
Instead, in Figure 8, the time histories of the MaD test, i.e., the test with magnets in attraction, are shown. From the two details of Figure 8b, it is possible to notice that the points of the beam oscillate around a non-null value, hence a positive drift, due to magnets in attraction and forward the fixed magnets, is present.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test** & **Label** & **L2 (P11) (mm)** & **L1 (P14) (mm)** & **Description** \\ \hline
1 & BD & \(-9.95\) & \(-16.42\) & Only beam dynamics \\
2a & MaD & \(-14.74\) & \(-24.17\) & Beam dynamics + attractive magnets \\
2r & MrD & \(-14.38\) & \(-23.61\) & Beam dynamics + repulsive magnets \\
3 & CD & \(-11.68\) & \(-19.07\) & Beam dynamics + contact (non-holonomic) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Tests carried out during experimental campaign.  
Figure 7: Time history of BD test using profilometer (**a**), with details at 0–2 s and 18–20 s (**b**); red line identifies the detected intermedial point, blue line the point detected at the mass.  
Instead, in case of magnets in repulsion, a negative drift would be seen. Similar behaviour but with more extended semi-oscillations along negative displacements is evinced in the MrD test, i.e., the test with magnets in repulsion, not shown in this work.  
The CD test aims at investigating the nonlinear behaviour in the beam's dynamics, due to the non-holonomic constraint, which modifies the beam's boundary conditions, and, consequently, the beam's natural frequencies.  
By analysing CD test, the acquired point P11 and P14 time histories are reported in Figure 9. The point P11 corresponds to the contact point of the beam with the non-symmetric constraint, while point P14 is the beam's free end, where the lumped mass is attached. As it is possible to notice from Figure 9b, the oscillation frequency changes in time, passing from -4.5 Hz in the first two seconds to -3 Hz in the last two seconds, in which the contact with the non-holonomic constraint is almost no longer occurring. The  
Figure 8: Time history of MaD test using profilometer (**a**), with details at 0–2 s and 18–20 s (**b**); red line identifies the detected intermedial point, blue line the point detected at the mass.  
Figure 9: Time history of CD test using profilometer (**a**), with details at 0–2 s and 18–20 s (**b**); red line identifies the detected intermedial point, blue line the point detected at the mass..  
point P11 displacement is flat when the contact with the constraint occurs, as indicated by the green horizontal line.