# Optical Analysis of the Impact Transmission in Steel Sheet Arrays with Bolted-Type Points
## 3 Results
### Analytical Discussion  
The methodology declared the use of a pendulum oscillation as a way to generate the impacts, where its typical motion equation is:  
\[x(t)=Asin\bigg{[}\sqrt{\frac{S}{l}}t\bigg{]} \tag{5}\]  
where \(A\), \(l\), and \(g\) are the amplitude, the chord length, and the gravity constant, respectively. From Section 3.1, the analytical waveforms obtained in Figure 6 were:  
\[f_{1}(x)=-0.1*sin(5.8x-0.9)-0.035 \tag{6}\]  
The values for Equation (4) were taken from Table 1, and performing the integral of it:  
\[g_{1}(x)=\frac{cos(5.8x-0.9)}{58}-0.035x+c \tag{7}\]  
Figure 9b shows the graph of Equation (7). It displays two parts: first, the incoming oscillation, followed by how it propagates. It can be said that the analytical displacement keeps the waveform of the pendulum, and as a direct consequence, the rigid body and the structural array declare how the wave travels. Also, in Figure 9a, it is seen that the experimental displacement has a similar behavior. The results after the screw were similar since the solution had the same waveform, just different values were obtained. Therefore, analyzing Section 3.2, introducing the values from Table 2, and following the same procedure as before:  
\[f_{2}(x)=46*sin(5x-1.6)*exp(-1.4x)-0.55 \tag{8}\]  
\[g_{2}(x)=\frac{exp(-1.4x)*[805 sin(5x-1.6)+2875 cos(5x-1.6)]}{337}-0.55x+c \tag{9}\]  
Figure 9d displays \(g_{2}(x)\); a peak is present between \(0<t<1\) s, which can be associated with the impact reaction. For \(t>1\) s, the body, the structure, and the joints decrease the movement. This result is similar to the experimental one. A peak between \(0.5<t<1.5\) s can be observed in Figure 9c, followed by a linear decrement. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Graph** & **A (\(\mu\)m)** & **k (Hz)** & \(\omega\) & \(\beta\) & **b** & **Relative Error** \\ \hline a & 6.2 & 4.7 & \(-0.8\) & \(-0.9\) & \(-0.18\) & 1 \% \\ b & 5.8 & 4.5 & \(+0.2\) & \(-0.9\) & \(-0.4\) & 1.1 \% \\ c & 6 & 4.6 & \(-1.1\) & \(-0.6\) & \(+0.4\) & 1 \% \\ \hline Graph & A (\(\mu\)m) & k (Hz) & \(\omega\) & b & Relative Error & \\ \hline d & 1.6 & 4.8 & \(-2.85\) & 1 & 0.8\% & \\ e & 1.8 & 5 & \(+1.5\) & \(+0.75\) & 1.2\% & \\ f & 1.7 & 5 & \(-0.75\) & \(-0.1\) & 1\% & \\ \hline g & 2 & 4.6 & \(+1.33\) & 0.2 & 1.3\% & \\ h & 2.1 & 4.6 & \(-1.55\) & 0 & 1.2\% & \\ i & 2 & 4.4 & \(-0.5\) & 0.9 & 0.8\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values for harmonic and damped solutions. Section 3.3 shows both waveforms. First, in Figure 8a-c, a damped oscillation as an analytical solution is observed. Then, a harmonic oscillation is present in Figure 8d-i. This result may be associated with the fact that the structure has more weight as well as more rigidity. Inspecting Figure 8, the measurement between screws 1 and 2 is where the damped oscillation occurs; however, between screws 2 and 3 and after the third, the impact propagates as a harmonic oscillation. According to the analytical solutions shown in [7; 8; 9], first, the authors declared a sum of two waves as the first analytical approximation to the motion recorded experimentally. This solution gave them an accuracy of 92%. It is important to mention that the waves were a harmonic oscillation plus a damped one, and the experimental data revealed that the movement travels downwards [7]. Years later, the authors announced another analytical approach by solving a damping mass-spring system, in which a damped wave was obtained as a solution. Again, the experimental results evidenced downward movement [8]. Finally, the authors reported another solution for a damped oscillator, in which they showed a combination of harmonic and linear functions [9]. In comparison with these studies, the present article demonstrates that the analytical results reproduced the experimental motion in every array. It adopts harmonic and damped oscillations. The approaches reached more than 98% of accuracy in the impact time-lapse.