# Dynamic Analysis of an Offshore Knuckle-Boom Crane Under Different Load Applications Laws
## 3 Results  
Many different modal analyses were conducted in order to evaluate the fundamental natural frequencies of the crane. The first goal is to assess the structural response under time-variable actions using the principle of modal superposition. The second one is to study the dynamic behavior of the structure in relation to load conditions. The analyses were performed for both configurations, examining the first four modes of vibration along the Y-axis, as illustrated in Figure 5 for configuration C1 and in Figure 6 for configuration C2. These figures denote the relative displacement at a specific frequency, with the reference for these displacements being the stationary components of the crane. The modal analyses were conducted without considering the mass to be lifted, as this mass does not substantially affect the behavior. The frequency values of the initial vibration modes range from 1 to 5 Hz, which is characteristic of hoisting machines [45, 46]. Figure 4: Geometry simplification process of the translation cart: (**a**) Original translation cart. (**b**) Simplified translation cart. (**c**) Section of the original translation cart. (**d**) Section of the simplified translation cart. Figure 5: First four modes of vibration for the configuration C1 along the Y-axis, which corresponds to the direction of the load force. For each mode, natural frequency and mass participation factor (\(\%mpf\)) are shown: (**a**) \(f_{1}=1.61\) [Hz] \(\%mpf_{1}=3.31\%\). (**b**) \(f_{2}=3.04\) [Hz] \(\%mpf_{2}=5.22\%\). (**c**) \(f_{3}=3.92\) [Hz] \(\%mpf_{3}=62.47\%\). (**d**) \(f_{4}=5.62\) [Hz] \(\%mpf_{4}=1.04\%\). About the application of the load, Figure 7a shows the laws of load applied to the end of the stick component. This pattern was chosen because it has a constant segment that checks how much the accelerations have slowed down and a descending segment that can mean either a load release or, if the time for this segment grows close to zero seconds, a failure of the secure system. Here, we can see two different time periods: \(\epsilon\), which is the time needed to apply the maximum lifttable load completely, and \(T_{trans}\), which is the time set to be longer than the settling time, the time required to reach the final value within a tolerance of \(\pm 5\%\). In particular, \(\epsilon\) is related to the crane's stiffness and the load's movement system. As an example, the presence of a magnet or a grab at the end of the component called stick can modify the law of load applied: a magnet defines the release and the application of the load almost instantaneously, meaning a lower value of \(\epsilon\) and higher accelerations, while the presence of a grab can give a longer time of application and release of the load, therefore lower stress. Figure 7b shows an example of the displacement result which can be obtained by the probe (its position is previously shown in Figure 3). This first result must be observed because it is noted that if the displacement and therefore the relative oscillations are not negligible values, this can induce overstress conditions in the structure that can lead to the collapse of the same [47]. Through this image, it is possible to extrapolate several values, which can be summarized as:  
* point A, which is the lowest peak of the load step;
* point B, which is the highest peak of the application of the load;
* point C, which is the highest peak of the load step;
* point D, which is the lowest peak after the release of the load;
* the settling value after the application of the load \(Y_{s}\). These values can be compared with each other; in particular, points A and B can be compared with the settling value after the application of the load, while points C and D can be compared with the settling value after the release of the load, which is the starting displacement (equal to 0 mm).