# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 3 Results and Discussion
### Numerical Modelling
#### 3.3.3 Multi-Phi Approach
\begin{array}{l}\mathbf{x}^{(l)}=\boldsymbol{\Phi}^{(l)}\mathbf{ \eta}^{(l)}(t)+\mathbf{x}_{\infty}{}^{(l)}\\ \dot{\mathbf{x}}^{(l)}=\boldsymbol{\Phi}^{(l)}\dot{\mathbf{\eta}}^{(l)}(t)\\ \ddot{\mathbf{x}}^{(l)}=\boldsymbol{\[PERSON]}^{(l)}\ddot{\mathbf{\eta}}^{(l)}(t) \end{array} \right. \tag{24}\]  
where \(\boldsymbol{\[PERSON]}^{(l)}\in\mathbb{R}^{n\times n_{l}}\) and \(\mathbf{\eta}^{(l)}\in\mathbb{R}^{r_{l}\times 1}\). In Equation (24), the addition of \(\mathbf{x}_{\infty}{}^{(l)}\), called asymptotic configuration, is necessary whenever the set of mode shapes is used to simulate the system evolution cannot describe the system state because of non-null boundary conditions. It is considered function of time because it depends on the linearised system considered at each time instant. It is determined, using Guyan reduction [55], by means of Equation (25), where \(\mathbf{x}_{k}\) represents the boundary conditions (known DoFs), while \(\mathbf{x}_{u}\) represents the not constrained DoFs (unknown). \[\left\{ \begin{array}{l}\mathbf{x}_{\infty}=\left[\begin{array}{c}-\Big{(} \mathbf{K}_{uu}^{(l)}\Big{)}^{-1}\mathbf{K}_{uk}^{(l)}\\ \mathbf{I}\end{array}\right]\mathbf{x}_{k}\\ \dot{\mathbf{x}}_{\infty}=\mathbf{0}\end{array} \right. \tag{25}\]The evolution of the full system, during the span of time in which the \(l\)-th configuration dynamics is triggered, depends only by the transition to another level occurring when a certain kinematic condition is respected. The transition is governed by Equation (26), in which the initial conditions of the modal coordinates of the level \(m\) are described by means of the final conditions of modal coordinates of the level \(l\):  
\[\left\{\begin{array}{l}\mathbf{\eta}_{m,0}=\mathbf{\Phi}_{m}^{-1}\big{(} \mathbf{\Phi}_{l}\mathbf{\eta}_{l,end}+\mathbf{x}_{\infty,l}-\mathbf{x}_{ \infty,m}\big{)}\\ \dot{\mathbf{\eta}}_{m,0}=\mathbf{\Phi}_{m}^{-1}\mathbf{\Phi}_{l}\dot{\mathbf{ \eta}}_{l,end}\right. \tag{26}\]  
Figure 20 shows the two linearised configurations available in the CD test. The two configurations are modelled in the LUPOS environment. In the clamped-free configuration, the beam nodes at the clamp have been locked imposing on the BCs for the six DoFs of each node, while the pinned configuration has been realised locking only the \(z\) displacement of the beam in correspondence with the non-symmetric constraint. The modal analysis in the two system configurations returns natural frequency values, which are collected in Table 6, only for the bending modes in xz plane. The frequencies in Table 6 are now used to justify and proper interpret the graphs of the frequency contents of previous section, especially of experimental test CD in the 1 st campaign. The natural frequency at 2.97 \(\div\) 3 Hz is visible in most of the figures displayed in the paper, and it is a validation of the experimental activity. The first four bending mode shapes in the \(xz\) plane for the two linearised systems are illustrated in Figure 21. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & **Clamped-Free (Hz)** & **Clamped-Pinned-Free (Hz)** & **Description** \\ \hline
1 & 2.973 & 13.88 & First bending \(xz\) \\
2 & 38.75 & 85.18 & Second bending \(xz\) \\
3 & 117.9 & 224.6 & Third bending \(xz\) \\
4 & 224.9 & 280.1 & Fourth bending \(xz\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of modal analysis in the \(xz\) eigenspace. Figure 20: LUPOS model for clamped-free beam (**a**) and clamped-pinned-free beam (**b**).