# Dynamic Analysis of an Offshore Knuckle-Boom Crane Under Different Load Applications Laws
## 4 Discussion  
The procedure previously described was applied for both configurations and varying the cited \(\epsilon\) time, already described in Figure 7a. Since during this crane's action the overshoot of the Y displacement is critical for the structure, the trend of the values \(\Delta Y_{4}\%\) and \(\Delta Y_{B}\%\) was further investigated and evaluated as demonstrated in Equation (5). In Figure 10, it is possible to see the trend of points A and B related to the settling value after the application of the load. Both of the points have a power trend line with respect to the \(\epsilon\) time, where the values of point B are always lower due to the damping of the structure. It shows that as the time called \(\epsilon\) (explained in Section 3) grows shorter, the difference between the highest points and the flat part of the Y displacement curves \(\%\Delta Y\) grows bigger, with a determination coefficient over 0.95. The curve was described by four points obtained from four different analyses with different values of \(\epsilon\) and another analysis to check the reliability of the curve with \(\epsilon=0.2\) s. This trend line is in both configurations, so it depends on the structure, not the crane's position. The only difference is the coefficient of determination's variation when the value with \(\epsilon=0.2\,s\) is taken or not. Considering also the value of \(\%\Delta Y\) with \(\epsilon=0.2\) s, as it is possible to see in Table 3, for the configuration C1 the coefficient of determination remains unaffected, but for the configuration C2 it decreased very slightly. Looking at the velocity in Figure 8a,b, we can see that as \(\epsilon\) increases, the highest values of the peaks go down and the peaks shift to the right in both setups. For the acceleration, there is also a change in the highest values, reaching up to about 15 m/s\({}^{2}\). From Figure 9a,b, we can tell that configuration C2 has the highest accelerations. For the acceleration, a change of the absolute values is seen even here, with maximum values around 15 m/s\({}^{2}\). It can be seen from Figure 9a,b that the most critical configuration in terms of higher accelerations is the configuration C2. These high accelerations can have a critical effect on stress and buckling phenomena due to the inertial forces, which can influence the overturning moment (which can be an important cause of accidents, as already seen in other types of cranes [48; 49]) and even the loads acting on the ground, with a possibility of injury regarding the operators [43]. To avoid these possible events, a limit on the maximum \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Configuration** & \(\Delta T\) & \(f_{r}=\frac{1}{\Delta T}\) [Hz] & \(f_{1}\) [Hz] \\ \hline C1 & 0.72 & 1.39 & 1.38 \\ C2 & 0.62 & 1.61 & 1.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average values of \(\Delta T\) for the configuration C1 and C2. Figure 10: Percentage variation of the displacement for points A and B. (**a**) Configuration C1. (**b**) Configuration C2. acceleration must be imposed, considered to be the product between the acceleration of gravity and a parameter \(\Psi\). Here, the value of \(\Psi\) is defined as 1.2 [44; 45], which means that the limit on the maximum acceleration is set to 11.77 m/s\({}^{2}\). Considering the absolute value of the maximum acceleration for each load condition, it is possible to find a value of \(\epsilon\) that can make the accelerations within the limits. As shown in Figure 11, it is possible to understand that the optimal value of \(\epsilon\) exceeds 0.3 s. If the value of \(\epsilon\) (take-up and release time) is about 2.5 s, there are no dynamic effects on the structure in both cases. In contrast, the applied forces match the effective load without any induced effects from the acceleration. To check the results obtained, these numerical computations were compared to a similar structure that presents similar loads and geometry [50].