# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 3 Results and Discussion
### Post-Processing Methodology
#### 3.2.1 Fast Fourier Transform Analysis  
The time-frequency analysis using Fourier transform consists of a series of frequency analyses on windowed time histories with high overlapping percentages. For time-frequency co-domains directing the Fast Fourier Transform (fft) (Equation (2)) and the inverse the Fast Fourier Transform (ifft) (Equation (3)) relationships for the discrete time signal \(x(t)\) and the signal Fourier transform \(X(\omega)\) are dual and commonly used.  
\[X(\omega)=X(x(t),\omega)=\int_{-\infty}^{+\infty}\mathrm{e}^{\mathrm{i}\omega t }x(t)dt=\mathrm{fft}(x(t))\in\mathbb{C} \tag{2}\]  
\[x(t)=\int_{-\infty}^{+\infty}\mathrm{e}^{-\mathrm{i}\omega t}X(x(t),\omega)d \omega=\mathrm{ifft}(X(\omega))=\mathrm{ifft}(\mathrm{fft}(x(t)))\in\mathbb{R} \tag{3}\]  
In the adoptedfft and ifft approaches and amplitudes are normalised with the following scale factor \(Amp\), where \(N\) is the total amount of samples in \(x(t)\), then:  
\[Amp=\frac{2}{N-1} \tag{4}\]  
Hence, in the performed FFT time-frequency analysis, a \(T_{w}\) = 4 s has been chosen as time of single windowed signal, with an overlap of 97% and Hamming window.  
For the BD test shown in Figure 10, a constant harmonic at ~3 Hz is evident, which corresponds to the frequency of the first bending mode in the clamped-free configuration. Instead, the several light horizontal superharmonics are due to leakage phenomena related to Hamming windowing. Moreover, to confirm the softening nonlinearities for large displacements of the clamped-free beam are quite negligible, it is suggested to see the wavelet analysis in the next section.  
By analysing Figure 11, processed for MaD test with magnets in attraction, it is possible to notice two nonlinear effects. The former is the presence of the 2 nd superharmonics of the fundamental frequency, while the latter is the softening effect. In this configuration, the softening behaviour reduces the global stiffness of the cantilever beam and, consequently, decreases the fundamental frequency and its superharmonics at the beginning of the time history with respect to the end, as noticeable from the markers in Figure 11b. Moreover, the  
Figure 10: Time–frequency contents of “BD” for P11 (intermedial) (**a**) and P14 (mass) (**b**) profilometer points.  
softening effect is more evident when the magnetic restoring force variability is higher, due to larger oscillations of the cantilever beam.  
Results of the time-frequency analysis for the CD test are shown in Figure 12. By analysing the mass point P14, reported in Figure 12b, it is evident the 1 st frequency transitions with a decrement in time. The 1 st natural frequency decreases from a value of \(\sim\)5 Hz to \(\sim\)3 Hz when the contact is no longer present. Moreover, the superharmonics of the fundamental natural frequencies are visible in both signals, while, in Figure 12a, it is possible to distinguish the 2 nd resonance frequency corresponding to the point near the contact, appearing at \(\sim\)37 Hz, which agrees with Table 5. This frequency is more evident in Figure 12a since the contact is the main physical cause exciting this second vibrational mode. Hence, the contact corresponds to the point where the beam assumes this behaviour the most.