# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 4 Conclusions  
Results of the experimental campaign, carried out using Keyence laser profilometer LJ-X8900, have proven the feasibility of conducting high-frequency dynamic analyses with an instrument conventionally adopted in the production line for static detections of dimensions and tolerances in manufactured products. The analyses were performed on experimental data measured in different setups, i.e., the common Euler-Bernoulli cantilever beam alone, the same subject to continuous nonlinearity of attractive or repulsive magnets, and, finally, the cantilever beam subject to non-holonomic contact. The post-processing of the profilometer acquisitions show advantages and drawbacks in the exploited time-frequency analysis methods, which can be summarised in the following final remarks:  
* The FFT allows to globally investigate the system behaviour, mainly concerning the fundamental frequencies and their superharmonics due to non-linearities, since the frequency resolution is defined, and it is maintained constant in the time-frequency analysis, from low to high frequencies. On the other hand, since the FFT frequency resolution is inversely proportional to the time resolution, an appropriate compromise between these two parameters is required. Consequently, the choice of a short time window that allows one to study fast-varying phenomena would result in a low  
Figure 21: Bending mode shapes for clamped-free beam (**a**) and clamped-pinned-free beam (**b**) with colour scale, from blue to red, related to the displacement field. frequency resolution that cannot properly allow one to obtain detailed frequency content information about the system;
* The CWT is used in addition to the latter, since the results of this approach demonstrate that it suits the analysis of dynamics characterised by almost instantaneous phenomena. The main drawback is a worsening of frequency resolution with the increment of frequency, in contrast with FFT analysis;
* FFT and CWT are both useful to study the same system from different perspectives and, together, allow to investigate both nonlinearities with their respective superharmonics and almost instantaneous phenomena. The results achieved with the approaches above described are validated using the Hilbert transform to evaluate the instantaneous lowest frequency of the system, relating it with both time and displacement envelope. In addition to the described time-frequency analysis methods, two linear numerical models are developed to assess the experimental frequency overcomes, based on the averaged superposition of the natural frequencies of the two linear systems. The correlation between the experimental data and the numerical data will be addressed in future research, based on Multi-Phi methodology, to investigate how nonlinear phenomena can be efficiently described by the convolution of linear systems effects. Finally, the correlation between the experimental data and the numerical ones in the case of attractive magnets is highlighted by Figure 22, which confirms the compliance between CWT method and Multi-Phi frequencies (a) and between the CWT approach and the Hilbert mean frequency (b), respectively. In Figure 22a, taking Table 6 into account, the blue lines correspond to the first three frequencies of the clamped free (CF) beam modes; the red lines to the first two ones of clamped pinned free (CPF) system modes; the continuous green line to the mixed frequency obtained as superposition of the first frequencies of CF and CPF systems, calculated in Equation (7); and the dashed green lines to the superharmonics of the latter. Figure 22b shows the correlation between the CWT time-frequency analysis and the mean frequency calculated with the Hilbert transform, instead. **Author Contributions:** Conceptualization, E.B., L.D., S.V. and S.P.C. ; methodology, E.B., L.D., S.V. and S.P.C. ; software, E.B., L.D., S.V. and S.P.C. ; validation, E.B., L.D., S.V. and S.P.C. ; formal analysis, E.B., L.D., S.V. and S.P.C. ; investigation, E.B., L.D., S.V. and S.P.C. ; resources, E.B.