# Dynamic Analysis of an Offshore Knuckle-Boom Crane Under Different Load Applications Laws
## 3 Results
To understand these considerations, Equation (5) shows how the value of the displacement in A, called \(Y_{A}\), was compared with the value after the application of the load, called \(Y_{s}\):  
\[\Delta Y_{A}=|Y_{A}-Y_{s}|\ \ \rightarrow\Delta Y_{A}\%=\frac{|Y_{A}-Y_{s}|}{Y_ {s}}\cdot 100 \tag{5}\]  
Here, the value called \(\Delta Y_{A}\) is the absolute difference between the first local minimum displacement \(Y_{A}\) along the Y-axis described by point A and the expected static displacement \(Y_{s}\). The quantity \(\Delta Y_{A}\%\) represents the percentage deviation of the dynamic response from the expected static displacement at the first local minimum. The same logic can be applied to the other three points previously cited (B,C,D), and changing the time imposed to reach 100% of the application of the maximum lifttable load (the so-called \(\epsilon\)), it is possible to notice the dynamic response of the structure and predict its behavior with different values. For a clearer understanding of the results, the following example can be considered: Figure 7b shows the displacement of the stick's end along the Y-axis, which is the same direction as the load's application. The value of \(\epsilon\) chosen is equal to 1 s, and the configuration here studied is the C2 position, meaning a value of the load equal to 100,000 N. Table 1 shows the values extrapolated with the methodology explained previously for both configurations; it is already noticeable the similarity of the values between points A and C and between points B and D, which are different only for orders of magnitude lower than \(10^{-2}\). This means that the study of the application and the release of the load are similar. Therefore, to reduce the computational time, only the first load step could be taken into consideration. Furthermore, the obtained values confirm the small deformationshypothesis, considering that the height of the crane reaches 18 m for configuration C1 and 27 m for configuration C2. Confirming even the linear elasticity assumption, the calculated stress remains below the material's yield strength, also due to the fact that the load is intended to be applied cyclically rather than statically. Moreover, the evaluation of the velocity and the acceleration of the end of the stick was considered to also analyze the contribution of the inertia forces. Figure 8 shows the trend of the velocity and Figure 9 the acceleration for both configurations evaluated by the probe, whose position is already described in Figure 3, considering only the first second and observing what happens if the variable \(e\) is changed. Figure 7: Application’s law of the load and the main values of the results. (**a**) Chart of the load application law. (**b**) Example of displacement result graph. Another analyzed aspect is the amount of time described in Figure 7b as \(\Delta T\): this is the period of a complete oscillation. To understand the results, Table 2 is shown. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{Configuration C1} \\ Point & Y [mm] & \(\Delta Y\) [mm] & \%\(\Delta Y\) [\%] \\ \hline A & \(-231.44\) & \(24.73\) & \(11.96\) \\ B & \(-187.44\) & \(18.97\) & \(9.18\) \\ C & \(24.71\) & \(24.71\) & \(11.97\) \\ D & \(-18.96\) & \(18.96\) & \(9.17\) \\ \hline \multicolumn{3}{c}{Configuration C2} \\ Point & Y [mm] & \(\Delta Y\) [mm] & \%\(\Delta Y\) [\%] \\ \hline A & \(-329.11\) & \(30.56\) & \(10.24\) \\ B & \(-274.03\) & \(24.52\) & \(8.21\) \\ C & \(30.56\) & \(30.56\) & \(10.24\) \\ D & \(-24.54\) & \(24.54\) & \(8.22\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the displacement along the Y-axis of the stick’s end with \(\epsilon\) equal to 1 s for the configurations C1 and C2. The value of \(Y_{s}\) is equal to 206.71 mm for the configuration C1 and 298.55 mm for the configuration C2. Figure 8: Trend of velocity of both configuration C1 and C2, taken the first second of lifting and considering \(\epsilon\) as a variable (in the charts it is called “eps”): (**a**) Velocity for configuration C1.