# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 3 Results and Discussion
### Numerical Modelling
#### 3.3.1 Effect of Magnets on a Cantilever Beam: Hardening or Softening Behaviours  
The mechanical and dynamic properties of the cantilever beam resulting from the numerical model are used to develop a single DoF system model that approximates the first bending mode of the numerical model. Thus, the analytical model is used to investigate the nonlinear effect given by the magnets when they are used in attraction or repulsion.  
The elasto-magnetic system represented in Figure 2a can be described by the following nonlinear equation:  
\[m\ddot{z}+k(z-z_{e})+F_{m}=0 \tag{9}\]  
where \(z\) is the position of the beam; \(z_{e}\) is the static equilibrium; \(F_{m}\) is the magnetic force which can be attractive or repulsive, depending on the sign positive or negative, respectively, and it is modelled through the empirical formula of Equation (10); while \(m\) and \(k\) are the equivalent mass and stiffness, given by Equation (11):  
\[F_{m}=\pm\frac{A}{\left(1-\frac{z}{B}\right)^{n}} \tag{10}\]  
\[m=m_{l}+m_{b} \tag{11}\]  
where parameters \(A\), \(B\), and \(n\) may be evaluated through magnetic models based on equivalent currents method [33; 34; 53], while \(m_{l}\) and \(m_{b}\) are the lumped mass and the beam mass participating to the first mode, equal to one third of the total one (without considering the clamped part), which, according to Table 6, has a natural frequency of \(f_{n}=2.97\) Hz.  
Figure 18 shows on the top the forces acting on the system, namely, the elastic force of the beam, the magnetic force with a different sign in case of attraction (a) or repulsion (b), and also the total force because of the superposition of these two effects. Points  
O corresponding to null displacement identify beam static positions in attractive and repulsive magnetic configurations. Since points P correspond to the equilibrium beam position without magnets, offsets between the respective cases with and without magnets can be noticed. The two static magnetic configurations are reached when magnetic and elastic forces reach opposite values that balance each other.  
Secondly, point S of attraction magnet configuration corresponds to the transition between the stable region and unstable region. The attractive magnetic force is high enough to interrupt beam oscillation because magnets attach each other when the analysed beam point reaches this unstable region. Point S corresponds to the balance of elastic and magnetic stiffness reached when the derivative of the total force is null, i.e., when the elastic and magnetic stiffness, shown on the bottom of Figure 18, assume opposite values. Consequently, this unstable region is not present in the repulsion magnetic case since the two stiffnesses never assume opposite values, thus the system is always stable.  
Finally, by analysing total stiffness trends it is possible to identify softening and hardening behaviours caused by the nonlinearities due to magnets: hardening occurs when the total stiffness is higher than the respective linearised value, while, in the opposite case, the behaviour is softening. Both attraction and repulsion cases show two regions of softening and hardening, hence the global dynamic behaviour of the beam is a consequence of which one of these two overcomes the other during one oscillation period. The system with attraction magnets shows softening behaviour for positive displacements and hardening for negative ones.  
The system with repulsion magnets shows softening behaviour for positive displacements and hardening for negative ones. The overall effects of attraction and repulsion magnetic system are, respectively, softening and hardening behaviours, as highlighted by frequency analyses. To confirm the results obtained with both time-frequency analyses about the nonlinear characteristics of the system, the Hilbert transform is used, according to the non-parametric [PERSON]'s method [49; 50].  
Figure 18: Effects on elastic restoring force and stiffness with magnets in attraction (**a**), or in repulsion (**b**).