# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 2 Experimental Test Measurements
### Instrumentation  
Experimental data are acquired using two different sets of hardware. The former consists of three point-lasers, produced by Keyence, of type LK-H082 and LK-H152, while the latter includes the Keyence laser profilometer LJ-X8900, which has the possibility to monitor the displacement of 3200 points at the same time. Lasers are connected to a Siemens SCADAS Mobile, and their sensitivity \(S\) is set in the software Siemens Test.Lab in compliance to the margin of error \(\Delta V\) = 20 V of the output voltage, corresponding to \(\pm\)10 V. Therefore, the sensitivity is computed as:  
\[S=\frac{\Delta V}{2d} \tag{1}\]  
The displacements of the three monitored points of the cantilever beam must stay within the range \(d\) given in Table 2, hence their position with respect to the length of the beam has been decided to satisfy this condition, since the greater the distance from the beam the greater the measured displacement will be.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Item** & **Weight** & **Material** & **Density** & **Young** & **Poisson** \\  & **(g)** & & **(kg/m\({}^{3}\))** & **Modulus** & **Ratio** \\  & **(g)** & & **(kg/m\({}^{3}\))** & **(GPa)** & **(-)** \\ \hline Constraint base & 2990.5 & Steel & 7800 & 200 & 0.3 \\ Constraint plate & 193.1 & Steel & 7800 & 200 & 0.3 \\ Screws and washers & 20.4 & Steel & 7800 & 200 & 0.3 \\ Total constraint & 3204.0 & Steel & 7800 & 200 & 0.3 \\ Euler–Bernoulli beam & 144.1 & Aluminium & 2699.2 1 & 69 & 0.33 \\ (600 mm \(\times\) 30 mm \(\times\) 3 mm) & 144.1 & Aluminium & 2699.2 1 & 69 & 0.33 \\ Lumped mass disk & 203.37 & Steel & 8225 & 200 & 0.3 \\ Lumped mass & 9.51 & Steel & 7800 & 200 & 0.3 \\ fasteners & & & & & \\ Beam magnets & 7.1 & Nd\({}_{2}\)Fe\({}_{14}\)B & & 7500 & - & - \\ (\#2 \(\mathcal{O}\)12h4) & 13.3 & Nd\({}_{2}\)Fe\({}_{14}\)B & & 7500 & - & - \\ Constraint magnets & & & & & \\ (\#2 \(\mathcal{O}\)12h4 & 13.3 & Nd\({}_{2}\)Fe\({}_{14}\)B & & 7500 & - & - \\ and \#1 \(\mathcal{O}\)8h8) & & & & & \\ Bosch profiled beams & S 1740 1 & Aluminium & 2700 & 69 & 0.33 \\ R 3040 2 & Aluminium & 2700 & 69 & 0.33 \\ \hline \hline \end{tabular} 1
[FOOTNOTE:1]Footnote 1: The density of “Euler–Bernoulli beam” is computed directly thanks to the dimensions reported in Figure 2 and considering 2 holes \(\mathcal{O}\) 5.5 and 1 hole \(\mathcal{O}\) 10. 2  
\end{table}
Table 1: Weight and property of experimental setup objects.  
Figure 3: A clamp (**a**) and a non-holonomic constraint in non-contact (**b**) and in contact (**c**).  
The lasers measure, according to the reference system shown in Figure 2, the displacement of three points, namely:  
* a point (L1) at the end of the beam on the lumped mass (\(x\) = 545 mm);
* a point (L2) near to the permanent magnets or to the non-holonomic constraint (\(x\) = 390 mm);
* a point (L3) between the clamp and the constraints (\(x\) = 210 mm).  
The laser profilometer LX-J8900 is fixed to a rigid table, using the specific support reported in Figure 4a, and projects a laser beam at 980 mm from the cantilever beam to monitor the displacement of 12 fixed points, as displayed in Figure 4b.  
By analysing Figure 4b, the measured points are labelled from P1 to P14, and their distance in \(x\) direction from the origin of reference system are reported in Table 3.  
The point L1, measured with single-point laser, is coincident with the point P14, acquired with the profilometer. For sake of clarity, the profilometer acquisition system is completely independent and not synchronised with the SCADAS Mobile used for the single-point lasers.