# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 3 Results and Discussion
### Post-Processing Methodology
#### 3.2.2 Continuous Wavelet Method
For a deeper analysis, a detail of the time interval of \(1\div 5\) s and in the frequency range of \(0\div 50\) Hz is shown for both P11 (contact) and P14 (mass), as reported below. In Figure 16, there is the evidence of some relevant frequency contents. For sake of clarity, a list of all marked frequencies, also summarised in Table 5, is provided to investigate the system dynamics in frequency domain:  
Figure 16: Detail of CWT time–frequency analysis of CD test for P11 (contact) (**a**) and P14 (mass) (**b**) profilometer points. Figure 14: Time–frequency contents of MaD test using CWT method for P11 (magnets) (**a**) and P14 (mass) (**b**) profilometer points. Figure 15: Time–frequency contents of CD test using CWT method for P11 (contact) (**a**) and P14 (mass) (**b**) profilometer points. * \(\sim\)4.7 Hz identifies a mixed frequency, which is a weighted average of the first natural frequencies of the clamped-free and clamped-pinned-free configurations over the time in each configuration. This is confirmed by the periods corresponding to these bending modes which are larger than the contact time of about 0.032 s, as shown in Figure 17;
* \(\sim\)9.4 Hz is the 2 nd superhamonics of the mixed frequency at -4.7 Hz. It is usual to have superharmonics of even order even if a magnet is not present, indeed this contribution comes from the truncation of the displacement when there is the contact;
* \(\sim\)14.1 Hz is the 3 rd superharmonics of the mixed frequency of \(\sim\)4.7 Hz. It is not visible on the contact since it is stopped by the angular profile which does not allow the nonlinearity to locally develop its effect, since this displacement is not as describable as at the mass point with an expansion mainly characterised by terms of 3 rd order;
* \(\sim\)18.8 Hz is the 4 th superhamonics of the mixed frequency at \(\sim\)4.7 Hz;
* \(\sim\)37.6 Hz is related to the 2 nd bending mode of beam in clamped-free configuration. The time-frequency analysis performed at the mass point in Figure 16b shows a frequency bouncing between the 2 nd mode of the clamped-free (\(\sim\)37.6 Hz) and the 2 nd superharmonics of the mixed frequency (\(\sim\)9.4 Hz). On the other side, the time-frequency analysis performed at the contact point in Figure 16a shows a frequency bouncing between the 2 nd mode of the clamped-free system (\(\sim\)37.6 Hz) and the 4 th superharmonics of the mixed frequency (\(\sim\)18.8 Hz). The lower frequency value of the bouncing changes since the time-frequency analysis of the measurements performed at the contact point does not show the behaviour related at the 3 rd superharmonics of the 1 st mode because of the asymmetry of the contact, hence the closer and higher frequency is the 4 th superharmonics of the first mode at \(\sim\)18.8 Hz. Figure 17 is used to clarify that the contact of the beam with the constraint during the first fully acquired oscillation lasts from 1.366 s to 1.398 s, while the complete oscillation ends at 1.577 s.  
Therefore, the value of \(\sim\)4.7 Hz derives from Equations (7) and (8). Figure 12:0 \begin{table}
\begin{tabular}{c c} \hline \hline
**Frequency (Hz)** & **Description** \\ \hline
4.7 & Mixed frequency \\
9.4 & 2 nd superharmonics of the mixed frequency \\
14.1 & 3 rd superharmonics of the mixed frequency \\
18.8 & 4 th superharmonics of the mixed frequency \\
37.6 & \multicolumn{1}{c}{Clamped-free 2 nd mode frequency} \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Main frequencies. Figure 17: Instants of contact (**a**), and combination with CWT analysis (**b**); red line identifies the detected intermedial point, blue line the point detected at the mass. which is a weighted average of values of the fundamental frequencies of the clamped-free and clamped-pinned-free configurations, i.e., 2.97 Hz and 13.88 Hz, respectively, as reported in Table 6. The \(x\) value corresponds to the time in which the beam is not in contact, and it is computed with the instants marked in Figure 17a and is equal to:  
\[x=\frac{1.577-1.398}{1.577-1.366}=84.83\% \tag{8}\]