# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 3 Results and Discussion
### Post-Processing Methodology
#### 3.2.2 Continuous Wavelet Method  
An alternative time-frequency analysis approach to FFT is the CWT approach. CWT consists of a set of projections \(W(\tau,s)\) of the signal \(x(t)\) in time \(\tau\) and frequency scale \(s\) onto rescaled and translated versions of an analysing function \(\psi(t)\), also called a \"wavelet\" [43; 44; 45; 46]. The projection is translated in a frequency domain integral in which \(\Psi_{P,\gamma}*(\omega)\) is the complex conjugate of the Fourier transform of the Morse reference wavelet, and \(X(\omega)\) is the Fourier transform of \(x(t)\). \[ \begin{array}{ll}W(\tau,s)&=\frac{1}{s}\int_{-\infty}^{\infty}\psi_{P, \gamma}*\left(\frac{l-\tau}{s}\right)x(t)dt=\\ &=\frac{1}{2\pi}\int_{-\infty}^{\infty}\Psi_{P,\gamma}*(s\omega)X(\omega) \mathrm{e}^{\mathrm{i}\omega t}d\omega\end{array}  \tag{5}\]  
Figure 11: Time–frequency contents of MaD test for P11 (magnet) (**a**) and P14 (mass) (**b**) profilometer points. Figure 12: Time–frequency contents of CD test for P11 (contact) (**a**) and P14 (mass) (**b**) profilometer points. Hence, the Fourier transform of the generalised Morse wavelet is:  
\[\Psi_{P,\gamma}(\omega)=U(\omega)2\bigg{(}\mathrm{e}\frac{\gamma^{2}}{P^{2}} \bigg{)}^{\frac{P^{2}}{\gamma^{2}}}\omega^{\frac{P^{2}}{\gamma}}\mathrm{e}^{- \omega^{\gamma}} \tag{6}\]  
where \(U(\omega)\) is the unit step Fourier transform; \(2\bigg{(}\mathrm{e}\frac{\gamma^{2}}{P^{2}}\bigg{)}^{\frac{P^{2}}{\gamma^{2}}}\) is a normalising constant; \(P^{2}\) is the time-bandwidth product; and \(\gamma\) characterises the symmetry of the Morse wavelet, used in this work for its versatility. By adjusting the time-bandwidth product and symmetry parameters it is possible to obtain wavelets with different behaviours. Therefore, a time-frequency analysis of the evolution of the measured points is performed with CWT based on Morse wavelets. The use of CWT does not provide further details for the BD test, where the beam is analysed in clamped-free conditions. Results reported in Figure 13 are similar to the time-frequency analysis using FFT reported in Figure 10. The two labels at the beginning and at the end of time history confirm the nonlinear effects due to large oscillations are negligible, since the frequency is the same of the first bending mode at \(\sim\)3 Hz. Regarding the MaD test, the softening effect that decreases the starting value of each harmonic when the amplitude of oscillations is higher is more evident respect to the FFT analysis, where overlapping is applied to the time history, with a subsequential averaging of results. From Figure 14a, the 2 nd and the 3 rd superharmonics can be noticed in the first 4 s. This is the confirmation that a nonlinear magnetic force in the equation of motion, if developed up to the 3 rd term, completely describes the action of the magnets. The signal contributions at a near zero frequency at the beginning and end of time history are related to the wavelet cone of influence, which shows where the edge effects of the CWT become significant, hence this region should be excluded from the analysis. Moreover, by changing wavelet parameters to have a better frequency resolution, the cone of influence changes as well. The application of the CWT method to the CD test returns the most interesting results, as shown in Figure 15. From a global point of view, several high spectral contents can be noticed at different frequencies. The markers in Figure 15a, which represent the time-frequency analysis for the contact point, show a frequency contribution at \(\sim\)86 Hz and \(\sim\)114 Hz, which are the 2 nd natural frequency of the system in clamped-pinned-free conditions and the and 3 rd natural frequency of the system in a clamped-free configuration, as stated in Table 5.  
Figure 13: Time–frequency contents of BD test using CWT method for P11 (intermedial) (**a**) and P14 (mass) (**b**) profilometer points.