# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 3 Results and Discussion
### Numerical Modelling
#### 3.3.2 Hilbert Transform  
With reference to the methodology developed by [PERSON] [49], recalling the approach that has already been used in [50], the unforced behaviour of a nonlinear mechanical single degree of freedom (sdof) system can be expressed in the form:  
\[m\ddot{z}(t)+c(t)\dot{z}(t)+k(t)z(t)=0 \tag{12}\]  
where the nonlinear damping coefficient \(c(t)\) and the nonlinear stiffness coefficient \(k(t)\) depend on displacement and velocity. The linear canonical form of the equation of motion is written as follows:  
\[\ddot{z}(t)+2\zeta(t)\omega_{n}(t)\dot{z}(t)+\omega_{n}^{2}(t)z(t)=0 \tag{13}\]  
where the non-dimensional damping factor \(\zeta(t)\) and the natural frequency \(\omega_{n}(t)\) are time dependent. By considering the definition of the complex analytic signal of \(z(t)\), expressed in exponential form, it holds:  
\[Z(t)=z(t)+\mathrm{i}\widetilde{z}(t)=A(t)\mathrm{e}^{\mathrm{i}\overline{ \tau}(t)} \tag{14}\]  
where \(\widetilde{z}(t)\) is the Hilbert transform of the real signal \(z(t)\). The real envelope function \(A(t)\) and the real instantaneous phase function \(\Psi(t)\), which are both time-dependent, result in, respectively:  
\[\begin{split} A(t)&=\sqrt{{z(t)}^{2}+\widetilde{z }(t)^{2}}\\ \Psi(t)&=\tan^{-1}\frac{\widetilde{z}(t)}{\widetilde {z}(t)}\end{split} \tag{15}\]  
where \(\omega(t)=\dot{\Psi}(t)\) is the instantaneous angular frequency. The time-dependent damping \(\zeta(t)\) and natural frequency \(\omega_{n}(t)\) functions are related to the envelope and phase functions through the relations:  
\[\begin{split}\zeta(t)&=-\frac{1}{\omega_{n}} \Big{(}\frac{\dot{A}}{A}+\frac{\dot{\omega}}{2\omega}\Big{)}\\ \omega_{n}(t)&=\sqrt{\omega^{2}+\frac{\dot{A}\dot{ \omega}}{A\omega}+2\frac{\dot{A}^{2}}{A^{2}}-\frac{\dot{A}}{A}}\end{split} \tag{16}\]  
The Hilbert transform is applied to the BD (simple clamped-free beam case), MaD (beam with attraction magnetic configuration), and MrD (beam with repulsive magnetic configuration) tests.  
The outcomes in Figure 19 (left) show the correlation between envelope and damping trends, highlighting how the latter are not characterised by wide variations. A similar behaviour has been observed in as small-scale application i.e. the Levitron (c) device, in which the contact between moving and fixed parts due to constraints is absent [54]. Furthermore, Figure 19 (right) shows the correlation between frequency and envelope trends that confirm the linear, softening, and hardening behaviours, respectively, caused by the presence of magnetic nonlinearities, in similarity with other applications such as Levitron (c).  
Red and blue trajectories correspond to points P11 and P14, respectively, demonstrating the filtering effects of the free mass with respect to points closer to the nonlinear effect. The natural frequency for smaller oscillations, hence for smaller envelopes, converges to the natural frequency of the magnetic configurations that are linearised about the respective equilibrium positions, demonstrating the nature of the nonlinear characteristics, which appears to depend on displacement.