# Dynamic Analysis of an Offshore Knuckle-Boom Crane Under Different Load Applications Laws
## 2 Materials and Methods
### Simplified Model  
In this research study, a simplified model was implemented to run the FE analysis, with the aim of reducing computational time. This model is characterized by the same stiffness as the detailed one, as well as a similar mass. In Figure 3, it is possible to see the simplified model and the position of the crane in the different evaluated configurations. Even here, the position of the probe where the displacement is extrapolated for both configurations is shown.  
The FEM was composed of 72,000 quadratic solid elements and 130,000 nodes. This model was the result of an iterative process of numerical simulations, aiming for convergence in the displacements. The analyses were performed on SolidWorks(r) (2024) software through the direct integration method. This methodology can be defined through the Equation (1) [40; 41]:  
\[[M]\{\hat{D}\}_{n}+[C]\{\hat{D}\}_{n}+[K]\{D\}_{n}=\{R^{ext}\}_{n} \tag{1}\]Here, the subscript \(n\) denotes the time instant, \([M]\) represents the mass matrix, \([C]\) signifies the damping matrix, \([K]\) indicates the stiffness matrix, \(\{\dot{D}\}\), \(\{\dot{D}\}\), and \(\{D\}\) correspond to the nodal acceleration, nodal velocity, and nodal displacement vectors, respectively, while \(\{R^{ext}\}\) denotes the external action vector. Specifically, Solidworks(r) employs the Newmark method, a technique that presumes the material properties and structural geometry remain constant throughout the time interval \(\Delta t\), utilizing two parameters (\(\gamma\) and \(\phi\)) to facilitate the solution update at each iteration. The Equations (2) and (3) delineate the updates for the displacement and velocity of the numerical solution:  
\[\{D\}_{n+1}=u_{n}+\Delta t\{D\}_{n}+\frac{\Delta t^{2}}{2}\big{(}(1-2\phi)\{D\} _{n}+2\phi\{D\}_{n+1}\big{)} \tag{2}\]  
\[\{\dot{D}\}_{n+1}=\{\dot{D}\}_{n}+\Delta t\big{(}(1-\gamma)\{\dot{D}\}_{n}+ \gamma\{\dot{D}\}_{n+1}\big{)} \tag{3}\]  
Figure 3: 3D of the simplified model. (**a**) Configuration C1. (**b**) Configuration C2.  
Figure 2: Position of the analyzed configurations: (**a**) Operating range of the crane and values of liftable weight in different configurations; (**b**) Angles of the configuration C1; (**c**) Angles of the configuration C2.  
About the damping matrix, Equation (4) shows how it is obtained in these analyses:  
\[[C]=\alpha[M]+\beta[K] \tag{4}\]  
where \(\alpha\) and \(\beta\) are the Rayleigh damping variables, set to 1.5 and 0.001, respectively, for this structure [42]. These values are the results of numerical evaluation on a step load law: here, the logarithmic decrement was calculated in order to observe the damping of the structure, as already discussed in the literature [43, 44].  
Regarding constraints and loads, Figure 3 also shows the boundary conditions imposed: the model was defined with four fixed supports to simulate its attachment to the ground at the ends of the legs of the portal structure. For the applied load, a force equal to 300,000 N for configuration C1 and 100,000 N for configuration C2 is applied to the end of the component called the \"stick\". The reason behind different loads for different configurations is the result of an analytical evaluation described by Figure 2a.  
To understand the process of modification of the model, Figure 4 shows an example of the simplification process: here it is possible to see that most of the components were removed to lighten the computational effort. The main goal in this phase was to obtain similar stiffness and mass of the whole structure, since our goal is to analyze the dynamic behavior of the whole structure and not just the single component. This process was already implemented in previous research works [39, 45].