Nonlinear complementarity framework for sliding cable analysis with elastic catenary equation considering frictional nonsmoothness

In this paper, a model is presented for the elasto-static problem of planar point mass robots suspended by m-cables . In particular, each cable configuration is described by an elastic catenary and static equations and compatibility conditions for the system are given, thus the 2m force reaction unknowns can be evaluated. The proposed formulation has been used to solve the direct problem and it is suitable for investigating the influence of elastic catenary on the end-effector exact positioning. The model allows evaluating the relation between end-effector position and the involvement of each cable in sustaining the payload.

In order to simulate the process of ship mooring more accurately, catenary equations are used to model mooring lines. Use different catenary models to simulate different phases of ship mooring operation. Establish a basic elastic catenary equation of mooring line by using the micro mechanical equilibriums theory. Then, a double-catenary model of heaving line and cable with a point load, imitating the effect of an eye splice, is developed based on existing catenary models in the literature. By calculating the mooring line tension, analyze whether the tension exceeds the safety load. The simulator is verified through simulations. A set of case studies is used to verify the effectiveness and practicability of the model. Integrate and realize the three-dimensional visualization of mooring line operation in ship mooring operation. The established model can guide mooring operation of ships.

Topological derivatives for a class of quasilinear elliptic equations

A general framework for sliding cable analysis with elastic catenary equation

The interaction between free-surface waves and marine structures plays a critical role in underwater radiated noise (URN) generation and structural integrity, particularly for ships and submarines. Hydroelastic vibrations of hull panels under wave-induced dynamic loading represent a major URN source, with implications for marine ecosystems and acoustic stealth. This study investigates the hydroelastic response of partially submerged hull panels subjected to free-surface wave interactions, focusing on deformation dynamics, modal characteristics, and the impact of panel stiffening. A representative thin square panel, immersed to varying depths, is analyzed using a nonlinear finite element framework coupled with a phase-field method for free-surface modeling. Structural motion is resolved through hydroelastic eigenmodes derived from linear elasticity equations. Results quantify the influence of the steepness of the wave and the immersion ratio, revealing an exponential decrease in deformation with increasing steepness of the wave and a shift toward suction-dominated deformation modes at greater immersions. Replacing the idealized panel with a stiffened configuration demonstrates a 94% reduction in deformation amplitudes, highlighting the efficacy of structural reinforcement. These findings advance the understanding of hydroelastic vibration mechanics and provide actionable insights for noise-mitigating marine design.

Because of the huge workspace of the cable-supporting system for the large radio telescope,sag of cables needs to be considered in the stiffness analysis. Therefore,the relationship between cable end tension and cable length is deduced by using elastic catenary equation. The static model of the cable-supporting system is constructed with this relationship. Based on the nonlinear relationship between the cable end tension and the cabin displacement,the incremental expression of the resultant force on the cabin exerted by cables is formulated in terms of the cabin displacement. Then the analytical expression of the static stiffness matrix is obtained. Numerical examples are presented to validate the method. Stiffness matrix is reported at a point and the stiffness distribution is depicted within the workspace. The results show that the rotational stiffness about the cabin central axis is relatively weak.

Most methods used to solve overhead line sag-tension problems assume that the conductor is inextensible. This assumption leads to the classical catenary curve. In this article, the conductor is assumed to be extensible, which leads to the more accurate elastic catenary curve. First, an original mathematical approach of establishing the equations of this curve is proposed: it uses the slope to connect the abscissa and the ordinate of each point on the curve. Then, it is used to solve sag-tension problems for perfectly linear elastic conductor or composite conductor with nonlinear inelastic behaviour. In each case, all steps of the algorithms proposed are explained. The numerical examples included in this paper show that the number of iterations needed to solve a sag-tension problem is quite small. Comparisons with previous studies are also made to assess the accuracy of the proposed method.

Time-domain simulation of nonlinear motions of moored floating body in irregular waves using rankine source method

In this article, we analyze the linear stability of tandem offloading systems in wind, current, and waves. The wind and current forces are evaluated with the help of published experimental data, while the hydrodynamic coefficients and wave drift forces are rigorously estimated by using a three-dimensional singularity distribution method based on potential theory. The bow hawser and mooring lines are described quasistatically by elastic catenary equations. In order to examine the linear static and dynamic stability of the system, the equations for surge, sway, and yaw are linearized. The effect of design parameters such as turret position, mooring stiffness, and hawser length and stiffness on stability is investigated based on linearized model. The stability analysis clarifies the mechanism of the limit cycle for tandem offloading systems, which is known as fishtailing motion. The theoretical results of the shape and amplitude of the limit cycle are found to be in good agreement with those of simulations and experiments.

Most methods used to solve overhead line sag-tension problems assume that the conductor is inextensible. This assumption leads to the classical catenary curve. In this article, the conductor is assumed to be extensible, which leads to the more accurate elastic catenary curve. First, an original mathematical approach of establishing the equations of this curve is proposed: it uses the slope to connect the abscissa and the ordinate of each point on the curve. Then, it is used to solve sag-tension problems for perfectly linear elastic conductor or composite conductor with nonlinear inelastic behaviour. In each case, all steps of the algorithms proposed are explained. The numerical examples included in this paper show that the number of iterations needed to solve a sag-tension problem is quite small. Comparisons with previous studies are also made to assess the accuracy of the proposed method.

Floating wind turbines (FWTs) with shared mooring systems can be one of the most cost- effective solutions in reducing mooring costs. First, the static configuration of a shared line is estimated using the elastic catenary equation. The present study investigates the global responses of two FWT with a shared mooring system. Two shared mooring configurations with different horizontal distances between the FWTs are considered. In the first configuration, the FWTs are placed 750m apart; and in the second configuration, they are placed 1000m apart. Two different environmental conditions (ECs) are used to simulate the global responses of the system in time domain. The shared mooring line results in higher extreme motions in surge and sway (degree of freedoms) DOFs due to the reduction of mooring restoring stiffness. The lower mooring restoring stiffness can be attributed to the reduction of one seabed anchoring point for each FWT as compared to a single FWT with three anchors installed. In the rotational DOFs, the shared mooring line configurations result in slight mean offset in each direction and significant increase in the motion standard deviations. This is caused by the reduced mooring stiffness associated with the change in platform orientation.

The process of multi-span tension stringing in transmission line is analysed based on elastic catenary equation. From the tower condition of transmission line, the conservation equations of span spacing and elevation difference are built for conductor, and the tension balance equations of conductor between different spans are got considering the rolling friction of the pulley. Added the external tension condition, the overall nonlinear equations of multi-span tension stringing are set up which can be solved by Newton iteration method. The iterative process is listed for the calculation of the conductor tension and sags in engineering problems. Compared the results with the numerical example, this method are proved the reliability. The calculation amount, efficiency and accuracy of this method can meet the needs of the practical engineering.

: The U.S. Navy Civil Engineering Laboratory is conducting a series of dynamic cable experiments in order to evaluate computer models of cable systems used in the ocean. The results of an experiment using 60 foot cables are compared to two computer simulations in this report. Other experiments at scales of six feet and 2,500 feet have been performed. Three cases from the experiment conducted in the hydroballistics tank of the Naval Surface Weapons Center in 1976 are compared to the SNAPLOAD and SEADYN computer models. Two of the runs simulate the anchor-last deployment of a mooring; the third shows the relaxation of a mooring displaced laterally, then released. The quality of the experimental data is evaluated by comparing each case to the static, elastic catenary equations at the start and finish of each run. The measured positions of points along the static catenaries are found typically to agree with the catenary calculations within 1 to 2 percent of the cable length. Tension measured at the fixed end typically agrees with the calculated value within about 12 percent.

Model testing of deepwater offshore structures often requires the use of statically-equivalent deepwater mooring systems. The need for such equivalent systems arises due to the spatial limitations of wave basins in accommodating the dimensions of the direct-scaled mooring system. With the equivalent mooring system in place and connected to the model floater, the static global restoring forces and global stiffness of the prototype floating structure can be matched (to within some tolerance) by those of the model for specified offsets in the required degrees of freedom. A match in relevant static properties of the system provides the basis for comparisons of dynamic responses of the model and prototype floaters. Although some commercial programs are capable of designing equivalent mooring systems, the physics applied in these programs are protected by intellectual property, and their methodologies are generally inflexible. This paper illustrates a concise approach to the design of statically-equivalent deepwater mooring systems. With this approach, either manual or advanced optimization techniques can be applied as needed based on the complexity of the equivalent system to be designed. A simple iterative scheme is applied in solving the elastic catenary equations for the optimal static configuration of each mooring line. Discussions cover the approach as applied in developing a fit-for-purpose tool called STAMOORSYS, its validation, and its application to the design of an equivalent mooring system for a spar platform in deepwater. The spar model parameters are representative of a structure which could be tested in the Offshore Technology Research Center, College Station, Texas, USA. Results show that the method is capable of producing good design solutions using manual optimization and a genetic algorithm.

Existence of competitive equilibria via Smith's non-linear complementarity result