# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 3 Results and Discussion
### Numerical Modelling
#### 3.3.3 Multi-Phi Approach  
A numerical model of the Euler-Bernoulli cantilever beam, with about 400 DoFs, has been developed using LUPOS [41; 42] for the validation of the experimental campaign. Two linearised models have been developed and tuned to describe the nonlinear dynamics incase of non-holonomic constraints. Mass and material properties, according to Table 1, have been assigned to the different components of the experimental setup. Real modal analysis is used for computing natural frequencies and mode shapes of the system in the two different configurations:  
* beam clamped free;
* beam clamped pinned free. Considering the two configurations as two linear systems, for a generic \(l\) linear system, the equations of motions are described by Equation (17), which is written for active DoFs \(n\), after which the boundary conditions or other reductions are applied:  
\[\mathbf{M}^{(l)}\ddot{\mathbf{x}}^{(l)}+\mathbf{C}^{(l)}\dot{\mathbf{x}}^{(l) }+\mathbf{K}^{(l)}\mathbf{x}^{(l)}=\mathbf{f}(t) \tag{17}\]  
Figure 19: Envelope function \(A(t)\) and damping \(\zeta(t)\) (left), and envelope function \(A(t)\) and instantaneous frequency \(\omega(t)\) (right), of signal \(z(t)\) in case of simple clamped-free beam (**a**), beam with attractive magnet (**b**), and beam with repulsive magnet (**c**) configurations. where \(\mathbf{x}^{(l)},\dot{\mathbf{x}}^{(l)},\ddot{\mathbf{x}}^{(l)}\in\mathbb{R}^{m \times 1}\) are the system displacements, velocities, and accelerations, respectively; \(\mathbf{M}^{(l)},\mathbf{C}^{(l)},\mathbf{K}^{(l)}\) are the mass, damping, and stiffness matrices of the considered \(l\) linear system; and \(\mathbf{f}(t)\) is the time depending external forcing function applied to the system. Neglecting static generalised external forces, and assuming a proportional damping matrix, Equation (17) can be written using the homogeneous expression:  
\[\mathbf{M}^{(l)}\mathbf{x}^{(l)}+\mathbf{K}^{(l)}\mathbf{x}^{(l)}=\mathbf{0} \tag{18}\]  
Hence, modal analysis is performed on each level of the system, i.e., in clamped-free and clamped-pinned-free conditions of the CD test. For each level, to evaluate a non-null stationary oscillating behaviour, a generalised exponential solution is assumed, as in Equation (19):  
\[\left\{ \begin{array}{l}\mathbf{x}=\mathbf{x}_{0}\mathbf{e}^{\mathrm{i}\omega t }\\ \dot{\mathbf{x}}=\mathrm{i}\omega\mathbf{x}_{0}\mathbf{e}^{\mathrm{i}\omega t }\\ \ddot{\mathbf{x}}=-\omega^{2}\mathbf{x}_{0}\mathbf{e}^{\mathrm{i}\omega t} \end{array} \right. \tag{19}\]  
Hence, time is separable and erasable from the equation:  
\[\left(\mathbf{K}-\omega^{2}\mathbf{M}\right)\mathbf{x}_{0}=\mathbf{0} \tag{20}\]  
Apart the trivial solution \(\mathbf{x}_{0}=\mathbf{0}\) that corresponds to the static undeformed condition, Equation (20) presents non-null solutions when the rank is non-full, hence an eigenproblem is definable to evaluate eigenvalues \(\omega_{r}^{2}\) and eigenvectors \(\boldsymbol{\varphi}_{r}\), finding the non-null solutions of the eigenproblem in Equation (21) by the means of Equation (22):  
\[\left(\mathbf{K}-\omega_{r}^{2}\mathbf{M}\right)\boldsymbol{\varphi}_{r}= \mathbf{0} \tag{21}\]  
\[\mathrm{det}\Big{(}\mathbf{K}-\omega^{2}\mathbf{M}\Big{)}=0 \tag{22}\]  
Thus, for each eigenvalue \(\omega_{r}^{2}\), its corresponding eigenvector \(\boldsymbol{\varphi}_{r}\) is computed and then normalised to unitary modal mass \(\boldsymbol{\Phi}_{r}\). Equation (21) can be written as:  
\[\left(\mathbf{K}-\omega_{r}^{2}\mathbf{M}\right)\boldsymbol{\Phi}_{r}= \mathbf{0}\text{ or }\mathbf{K}\boldsymbol{\Phi}=\mathbf{M}\boldsymbol{\Phi}\mathrm{diag} \Big{(}\boldsymbol{\omega}_{r}^{2}\Big{)} \tag{23}\]  
Assuming a proportional damping matrix, the modal superposition of physical coordinates \(\mathbf{x}^{(l)}\) results in:  
\[\left\{