# Dynamic Analysis of an Offshore Knuckle-Boom Crane Under Different Load Applications Laws
## 1 Introduction  
Hositing machines are structures subjected to various loads, including wind forces [1, 2], seismic activity [3, 4], and the weight being lifted [5, 6]. These forces can produce vibrations, leading to overloads that, if not assessed, may compromise the structure due to excessive stress, buckling, and fatigue. High-performance cranes have been a primary focus for increasing efficiency, emphasizing compact designs and large capacities. Nonetheless, due to reduced production volumes and limited opportunities for prototype development, designers must develop and use simulation models that cover all possible real-world scenarios. During crane design, consider standards (e.g., [7, 8, 9]) that take into account various load conditions, such as self-weight, payload effects, and wind effects, treating them as constant forces acting on the crane structures. Nonetheless, these behaviors are not temporally consistent. Several studies have examined the dynamic analysis of load conditions and their temporal variations [10], abrupt release of a portion of the hoist load [11], and the impact of load motion [12, 13]. Other aspects taken into consideration are the crane's fatigue behavior [14, 15], buckling phenomena [16, 17], or particular components of the crane [18, 19, 20]. These factors must be considered when designing high-performance cranes. The knuckle-boom crane, also known as an articulating crane, is an example of a hoisting machine that cannot beeconomically prototyped, necessitating a thorough examination of its design. These cranes have two booms connected by a knuckle, which increases versatility and functionality. The literature has already examined the dynamic behavior of these structures: [PERSON] and [PERSON] [21] introduced a model of a knuckle-boom crane as a two-dimensional multi-body system incorporating structural flexibility and damping. They created a reliable model using the finite segment method to check flexibility and a method to measure structural damping. These methods are not optimal for modelling flexibility; nonetheless, they are more effective in minimizing the workload required for modelling and computation. Utilizing experimental data, they successfully validate the proposed model in three phases: verifying the steady-state specifications of the mechanical system model, validating the steady-state characteristics of the motion control system model, and assessing the dynamic behavior of the crane model. [PERSON] et al. [22] established a three-dimensional mathematical model of a knuckle-boom crane, which was validated by many dynamic analyses to assess the dynamic coefficient and compare it with the values specified by the standards. The primary benefit of this model is the capability to incorporate lumped masses and stresses at any location on rigid parts, excluding the joints. Testing several crane configurations with specified parameters significantly reduces the time needed for intricate construction analysis. Regarding the methodology for modelling the dynamics of these systems, there is an increasing amount of literature. [PERSON] et al. [23] employed a digital twin methodology for the condition monitoring of a small-scale knuckle-boom crane; the utilization of a non-linear finite-element program facilitated real-time condition assessment, hence improving predictive maintenance and safety. [PERSON] et al. [24] devised a control methodology to mitigate payload swing in offshore knuckle-boom cranes utilizing an iterative learning algorithm while also demonstrating adaptation and experimental implementation via a manipulator to encapsulate the fundamental dynamics of a conventional offshore knuckle-boom crane. It was found that although simulated movements often show inconsistencies, adding a round joint between the crane's end and the wire is expected to provide nearly the best anti-swing performance for real loads, too. [PERSON] et al. [25] modelled the dynamics of the crane, specifically the luffing motion driven by hydraulic cylinders, as a planar multi-body system.