Catenary Theory Research Articles

Numerical Modeling and Analysis of Pendant Installation Method Dynamics Using Absolute Nodal Coordinate Formulation

Accurately simulating the deployment process of coupled systems in deep-sea environments remains a significant challenge. This study employs the Absolute Nodal Coordinate Formulation (ANCF) to dynamically model and analyze multi-body systems based on the Pendant Installation Method (PIM). Utilizing the principle of energy conversion, this study calculates the stiffness, generalized elastic forces, mass matrices, and Morison equation, formulating a motion equation for the dynamic coupling of nonlinear time-domain forces in cables during pendulum deployment, which is numerically solved using the implicit generalized-α method. By comparing the simulation results of this model with those from the catenary theory model, the advanced modeling capabilities of this model are validated. Lastly, the sensitivity of the multi-body system under various boundary conditions is analyzed. The results indicate that deployment operations are more effective in environments with strong ocean currents. Furthermore, upon comparing the impacts of structural mass and deployment depth on the system, it was found that deployment depth has a more significant effect. Consequently, the findings of this study provide a scientific basis for formulating subsequent optimization strategies.

Design and analysis of grid mooring system for gravity net cage with a comprehensive optimization method

The design of gravity net cage mooring systems is a complex endeavor, which is critical for preventing major accidents caused by mooring system failures. A comprehensive optimization method is proposed to enhance the reliability and performance of the net cage mooring system by integrating static and dynamic phases. The proposed method initiates static simulations based on the catenary theory to refine the initial range of parameters. This preparatory step ensures the subsequent dynamic analysis is carried out based on feasible initial conditions with the lumped mass method. The dynamic simulation captures the complex interactions and behavior of the mooring systems under various environmental conditions. The multi-objective optimization provides feasible solutions with simultaneous consideration of both processes. Three different scenarios are optimized with the forces and motions compared after the time domain analysis, resulting in a satisfactory set of solutions. The results show that variations in the angle of the mooring line of the net cage primarily impact the net cage's movement and volume in different environments.

Remote estimation of cable tension using catenary theory and point clouds obtained by terrestrial laser scanning

This study addressed the structural health monitoring of civil engineering cable structures, in which single cable tension is the key parameter influencing safe and intended behaviour. Traditional methods for monitoring the tension force in a cable are labour-intensive and require direct interaction with the cable. In contrast, the proposed approach provides a remote and rapid assessment of the tension force in the cables of engineering structures using only the shape of the cable. First, a terrestrial laser scanner was used to acquire a point cloud of the cable geometry from a distance. Catenary theory was subsequently applied to the point cloud data to calculate the theoretical tensile force in the observed cable. When the results were compared with those obtained through conventional direct measurements using a dynamometer, the differences were negligible. The proposed approach was subsequently demonstrated on the actual in-service overhead contact lines of a tram system. The results indicated that the precision of the proposed approach is highly dependent on the accuracy of the data describing the physical properties of the cable (cable linear mass). Thus, the proposed approach was proven to be precise, reliable, and rapid, and directions for future research were discussed accordingly.

Automated geometric reconstruction and cable force inference for cable-net structures using 3D point clouds

Laser scanning provides an efficient solution to digital twin construction in civil engineering. The complexity and redundancy of large-scale point clouds substantially prolong the labor-intensive model reconstruction process. This paper presents an automated and high-precision geometric reconstruction approach for cable-net structures with a complete workflow from raw points to the extraction of cable shapes and forces. The strategy involves the extraction of coordinates of key cable regions through the characterization of cable geometry and local registration methods, followed by the computation of cable forces and shapes using segmented catenary theory and nonlinear optimization. The approach is validated with a single-curvature cable-net structure, with the accuracy of cable shapes within ±5 mm and errors in cable forces <5%. The method contributes to health monitoring and rapid reconstruction of cable-net structures. The precise geometry and forces will facilitate the creation of the mechanical models reflecting the physical authenticity of structures.

Analyzing the Effect of Tethered Cable on the Stability of Tethered UAVs Based on Lyapunov Exponents

In the working process of the tethered unmanned aerial vehicle (UAV), there is interference from the tethered cable, which can easily lead to the instability of the UAV. To solve the above problems, a method based on the Lyapunov exponent is proposed to analyze the stability of tethered cables for tethered UAVs. The dynamics equation of the UAV platform is established using the Euler–Poincare equation. The tension formula of the tethered cable is derived from the catenary theory and the principle of micro-segment equilibrium. Based on the Lyapunov exponential method, the stability changes of the tethered UAV in the takeoff, hovering, and landing stages are simulated and analyzed in a MATLAB environment. Prototype tests are carried out to prove the correctness of the simulation model and calculation conclusions. The results show that with an increase in the density of the tethered cable, the stability of the tethered UAV tends to decrease. At the same time, stability is affected by the density of the tethered cable more often during takeoff than during landing.

Dynamics modeling and simulation of steel cord conveyor belt based on dynamic elastic modulus

Steel cord conveyor belts are widely used in long-distance transportation of various materials. The steel cord belt is composed of rubber and steel cord core, exhibiting excellent viscoelasticity and complex dynamic characteristics during operation. The existing static elastic modulus cannot accurately reflect the real dynamic properties of the conveyor belt. In this paper, based on the catenary theory, the formula for the dynamic elastic modulus of the conveyor belt is derived by introducing the sag and elastic deformation of the steel cord conveyor belt on the basis of the static elastic modulus. The dynamic modeling of the steel cord conveyor belt is carried out using rigid belt block element + flexible beam method. Then, the RecurDyn software is used to establish the dynamic simulation system of the belt conveyor and conduct simulations, which are validated through a test bench. The results show that dynamics simulation based on the dynamic elastic modulus is closer to the real dynamic characteristics of the belt conveyor. Finally, the dynamic characteristics of belt conveyors under different start-up times, different start-up speeds, and idler spacing are analyzed using this modeling and simulation method. The results demonstrate that the dynamic elastic modulus can accurately reflect the real dynamic properties of the conveyor belt, providing a more accurate dynamic modeling and simulation method for the design of belt conveyors.

Surrogate models for the stiffened catenaries: applications in subsea pipe laying

In this paper, the governing nonlinear ODE of the suspended stiffened catenary is reinvestigated. It is shown that strong nonlinearity arises from stiffened catenary length, which should be checked by an iterative numerical solution. The two concepts of stiffened catenary (guessed length) and natural catenary (known length) geometries of the suspended pipe, are compared and critically commented upon. In applying the theory to subsea pipeline installation, it is shown that natural catenary assumption, underestimates the installation stresses, particularly in shallow water and low laying depth. However, the true values of the stresses can be computed via stiffened catenary theory, in which the bending stiffness of the suspended pipe is not ignored. Thereafter, substantial iterative numerical solution of the governing nonlinear differential equation, in each load case is carried out. From these batch simulations, a surrogate expression is developed via optimization techniques. This model provides a correction factor by which, the accurate installation stress can be found. Moreover, the accuracy of results is verified by FEM analysis. It is concluded that for the initial estimation of the stresses, the simple natural catenary assumption, which is currently practiced can be used. However, the results should be corrected by the new surrogate expression, that has been produced in this paper. This can eliminate the underestimation of the installation stresses when a simple computational procedure is used.

Fully coupled aero-hydrodynamic modelling of floating offshore wind turbines in nonlinear waves using a direct time-domain approach

In standard operating conditions, a floating offshore wind turbine (FOWT) suffers from the combined action of nonlinear waves and wind. It is important to explore the effect of higher-order hydrodynamic loads induced by the nonlinear waves on its motion response. In this work, a fully coupled aero-hydrodynamic analysis method for FOWT is developed. The hydrodynamic part is based on the nonlinear potential flow theory and the perturbation approach, using a higher-order boundary element method in the time domain. Blade element momentum theory is applied to evaluate the aerodynamic load on the turbine rotor. The mooring system is modelled by the catenary theory. The developed method is verified against the published data of the NREL's 5-MW baseline wind turbine supported by the DeepCwind semi-submersible platform. The verifications are conducted stepwise in the following ways, including hydrodynamic wave excitation load, free-decay response, the load-displacement relationship of the mooring line, the steady-state response of the wind turbine, and fully coupled aero-hydrodynamic interaction. It is found that the nonlinear effect mainly influences the surge motion. The coupled configuration has little effect on the linear wave force and platform displacement but notably impacts the nonlinear wave force and displacement. The FOWT motion is found to be critical in changing the wave field, the maximum wave elevation position around the platform, and the nonlinear wave force acting on the floating platform. The results illustrate that the traditional indirect time-domain method using the Quadratic Transfer Function (QTF) based on assumed response cannot appropriately evaluate the nonlinear wave force of a FOWT in coupled motions. The results also reveal that nonlinear wave hydrodynamics is necessary to understand the motion response of a fully coupled FOWT.

An analytical method for adjusting dead load allocation between side and central cables in a cable-stayed bridge with three cable planes

Mechanical performance of cable-stayed bridges, especially those with super-wide main beams, can be significantly improved by adopting the three cable planes' scheme. However, broad application prospects of the novel bridge scheme require a proper adjustment of the ratio of the main beam's dead load allocated to the side and central cables. Otherwise, the central cables will bear an excessive dead load, deteriorating the overall bridge appearance and torsional stiffness. To this end, a simple analytical method was proposed for dead load allocation, reducing the share of the main beam's dead load borne by the central cable and re-allocating it to the side and central cables in any ratio by adjusting the unstressed lengths of the stay cables. This method takes full advantage of the convenience of adjusting the unstressed lengths of the stay cables and involves calculating the stay cables based on the catenary theory, which ensures high calculation accuracy. The proposed method first converts the spatial model of cable-stayed bridges with three cable planes into a planar model, employing the principle of virtual work to determine the elevation differences between the two anchor points of the three stay cables. Then, the unstressed lengths of stay cables are calculated based on the catenary theory. Finally, the proposed algorithm's feasibility is proved by applying the finite element method to a floating-type cable-stayed bridge with three cable planes with a main span of 1120 m and a bridge deck width of 75 m.

Reinforced Concrete Beam Design Using Catenary Theory

Reinforced Concrete Beam Design Using Catenary Theory

Study on the hydrodynamics of a twin floating breakwater by using SPH method

Numerical and experimental studies are performed on the hydrodynamics including wave-attenuation performances measured by the transmission and reflection coefficients and dynamic responses of a twin floating breakwater. In the present work, the twin floating breakwater consists of two identical single pontoon-type floating breakwaters with independent identical mooring systems. Numerical simulations are performed with the δ -SPH, a Smoothed Particle Hydrodynamics model, after introducing a mooring line computation based on the catenary theory. The experimental tests are carried out in a wave flume at the Ocean University of China. The displacements of the twin floating breakwater and the water surface elevation are recorded in the experiment. Comparison of the numerical and experimental results demonstrates the reasonable accuracy of the present numerical model. Based on the numerical results, the wave-attenuation performances of the twin floating breakwater and a single pontoon floating breakwater are compared. Furthermore, the effects of the wave period and the horizontal spacing between the two pontoons on the hydrodynamics of the twin floating breakwater are studied, and in this process, Bragg resonance reflection of water waves by the twin floating breakwater is discussed. The results reveal that the twin floating breakwater is superior to the single pontoon floating breakwater on wave-attenuation performances. Moreover, comparing with the horizontal spacing between two pontoons, the wave period has a more significant effect on the hydrodynamics of the twin floating breakwater. Furthermore, there could be Bragg resonance reflection between waves and the twin floating breakwater. However, this phenomenon can be significantly weakened by the movement of the twin floating breakwater. • The hydrodynamics of a twin floating breakwater are numerically studied using δ -SPH method after introducing a mooring line computation based on the catenary theory. • Comparison of the numerical and experimental results demonstrates the reasonable accuracy of the present numerical model. • The hydrodynamic performance of the twin floating breakwater is studied numerically. Moreover, Bragg resonant reflection is discussed.

Design Of the Tension Mechanism Of a Knotter And Optimization Of the Motion Parameters

Abstract Currently, the knotting of chemical fiber filament packages in factories primarily comprises manual operations. It also has a high labor intensity and is subject to artificial subjective factors, due to which it is difficult to guarantee the quality of the finished fiber products. Furthermore, it also significantly restricts the intelligent construction of chemical fiber production. This work employs partially oriented yarn (POY) and full draw yarn (FDY) to develop an automatic knotting device. The catenary theory was used to confirm the state of the yarn and to determine the relevant parameters of the tension mechanism. The results showed that the developed automatic knotting device satisfactorily meets the conditions required for manual knotting. The curve equation that was calculated using the catenary conformations of the yarn state verified that the actual yarn state was consistent with the theory. Moreover, the work provides a theoretical foundation for subsequent mechanism optimizations.

Nonlinear dynamic modeling of a tether-net system for space debris capture

In this paper, a flexible tether-net system is applied to capture the space debris and a numerical framework is established to explore its nonlinear dynamic behaviors, which comprises four principal phases: folding, spreading, contacting, and closing. Based on the discretization of the whole structure into multiple nodes and connected edges, elastic force vectors and associated Jacobian matrix are derived analytically to solve a series of equations of motion. With a fully implicit method applied to analyze the nonlinear dynamics of a slender rod network, the involved mechanical responses are investigated numerically accounting for the interactions. Contact between the deformable net and a rigid body is handled implicitly through a cost-effective modified mass algorithm while the catenary theory is utilized to guide the folding process (from planar configuration to origami-like pattern). The dragging and spreading actions for the folded hexagon net could be realized by shooting six corner mass toward a specific direction; next, the six corners would be controlled to move along a prescribed path producing a closing gesture, when touch between the flying net and the target body is detected, so that for the space debris could be captured and removed successfully. We think the established discrete model could provide a novel insight in the design of active debris removal (ADR) techniques and promote further development of the model-based control of tether tugging systems.

Reasonable completed state evaluation for hybrid cable-stayed suspension bridges: An analytical algorithm

Hybrid cable-stayed suspension bridges have broad application prospects due to their superior spanning ability, excellent overall stiffness, and economical efficiency, which outperform other long-span bridges. An analytical approach is proposed to determine the reasonable completed bridge state of hybrid cable-stayed suspension bridges in the paper. The specific steps are as follows: Firstly, the vertical supports offered by some stay cables and hangers in the main span to the main beam are estimated by assuming a continuous beam on multi-point rigid supports. For regions where both stay cables and hangers are used, the ratio of hanger force to the vertical force in the stay cable is assigned manually. Next, according to the assumption that the horizontal component force exerted by the main cables on both sides of the bridge tower and each pair of stay cables on the bridge tower are equal, the configuration and the internal force of the main cables and the stay cables are determined using the multi-segment catenary theory. The vertical forces exerted by the main and stay cables on the towers are summed up to estimate the axial force and strain acting at each position of the tower. In the side spans, the difference between the vertical support reaction in the continuous beam on multi-point rigid supports and the vertical force in the stay cable is used to calculate the weight of ballast of the main beam. The weight of ballast is apportioned as a uniformly distributed load acting on no more than three segments. The horizontal forces exerted by the stay cables on the main beam are summed up to estimate the axial force and strain acting at each position of the main beam. Finally, the analytical data of the full bridge are obtained as inputs of the finite element model. These data include the node coordinates and internal forces of the main cables, stay cables, towers, and the main beam. The calculation results obtained by the analytical method can be used for the preliminary design of hybrid cable-stayed suspension bridges. The proposed analytical algorithm can quickly solve the reasonable completed bridge state without complex adjustment of cable forces. It has the benefits of simplicity, high accuracy, and a clearly defined physical meaning. Finally, we verify the reasonability and effectiveness of the proposed analytical method in an asymmetric hybrid cable-stayed suspension bridge with a main span of 1400 m.

Automatic and inverse design of broadband terahertz absorber based on optimization of genetic algorithm for dual metasurfaces.

In this study, we introduce a genetic algorithm (GA) into the catenary theory model to achieve automatic and inverse design for terahertz (THz) metasurface absorbers. The GA method was employed by seeking optimal dispersion distributions to achieve broadband impedance matching. A THz dual-metasurface absorber was designed using the proposed approach. The designed metasurface absorber exhibits an absorbance exceeding 88% at 0.21-5 THz. Compared to the traditional design method, the proposed method can reduce time consumption and find the optimal result to achieve high performance. The investigations provide important guidance and a promising approach for designing metasurface-based devices for practical applications.

Static analysis of elastic cable structures under mechanical load using discrete catenary theory

In this paper, the nonlinear mechanical response of elastic cable structures under mechanical load is studied based on the discrete catenary theory. A cable net is discretized into multiple nodes and edges in our numerical approach, which is followed by an analytical formulation of the elastic energy and the associated Hessian matrix to realize the dynamic simulation. A fully implicit framework is proposed based on the discrete differential geometry (DDG) theory. The equilibrium configuration of a target object is derived by adding damping force into the system, known as the dynamic relaxation method. The mechanical response of a single suspended cable is investigated and compared with the analytical solution for cross-validation. A more intricate scenario is further discussed in detail, where a structure consisting of multiple slender cables is connected through joints. Utilizing the robustness and efficiency of our discrete numerical framework, a systematic parameter sweep is performed to quantify the force displacement relationships of nets with the different number of cables and different directions of fibers. Finally, an empirical scaling law is provided to account for the rigidity of elastic cable net in terms of its geometric properties, material characteristics, component numbers, and cable orientations. Our results would provide new insight in revealing the connections between flexible structures and tensegrity structures, and could motivate innovative designs in both mechanical and civil engineered equipment.

Study on Array Floating Platform for Wind Energy and Marine Space Optimization

The concept of multiline anchor, whose application is mainly considered in water depths beyond 100 m and analyzed only by numerical simulation, has been discussed for half a decade, yet previous studies have not conducted the wave basin experiment. Thus, this paper set this concept firstly with a shallow water mooring system designed for a Taiwan offshore water area, where the suitable water depth for floating offshore wind turbine is located from 50 to 100 m, and then conducted a 1:144 scaled model wave basin experiment to validate the results from numerical simulation. In this paper, the numerical model simulated and analyzed three identical DeepCwind OC4 semi-submersible platforms equipped with NREL 5MW wind turbines in OrcaFlex and the experiment carried out by using three 1:144 scaled semi-submersible platforms with equivalent disks which simulated different operations of wind thrusts. To consider the possible influence of the wake effect, the minimum turbines spacing was set at 750 m in a full scaled model and the length of mooring lines was redesigned according to the catenary theory. This paper utilized OrcaWave to calculate hydrodynamic parameters and input it into OrcaFlex to simulate the line tension and the three degrees of freedom (surge, heave, and pitch) of the platforms under regular and irregular wave tests, and coordinate with scaled model tests carried out in Tainan Hydraulics Laboratory (THL). In addition to the reduction in the number of anchors, the concept of multiline anchor was also discussed in this study for the spatial configuration of offshore wind farms. It shows that the wind farm composed of three floating wind turbines can reduce the ocean space by roughly 24% compared to that with a single-line anchor. According to the comparison of numerical and experimental results, this study finally optimized the mooring lines by changing the diameter to increase the stability and the threshold of Minimum Breaking Load (MBL) and proposed a multiline anchor configuration for shallow offshore water area in Taiwan based on the results obtained.

Research on path generation and stability control method for UAV‐based intelligent spray painting of ships

AbstractThis paper presents an intelligent path planning method with multiple constraints to solve the problem of instability of unmanned aerial vehicle (UAV) intelligent spray painting (UAV‐ISP) systems for spray painting the outer panels of ships. Spray‐painting path planning is realized using the multiobjective particle swarm algorithm and ant colony algorithm. In addition, a stability control method for the UAV based on the linear active disturbance rejection control (LADRC) algorithm is proposed and controlled by Mission Planner ground station software to solve the stability control problem of UAVs during spray painting. Through the above two key technologies, the painting quality of the UAV‐ISP system is guaranteed. First, a three‐dimensional (3D) model of the ship was obtained in a 3D modeling environment. Then the paint paths were optimized using the multiobjective particle swarm algorithm. The optimized path planning results were then combined using the ant colony algorithm to obtain the final spray painting path. Force analysis of the UAV during spray painting was carried out using the Newton–Euler equation and catenary theory, and a mathematical model of the UAV spray‐painting system with multiload coupling was established. Finally, a stability control system for the spray‐painting UAV based on LADRC was designed. The experimental and simulation results show that the proposed painting path is accurate, the spray‐painting UAV has good stability, and the spray‐painting quality of the UAV‐ISP system meets the standards for the outer panels of ships.

Dynamic Behavior of the Deepwater Flexible Pipeline during Pipe Laying Process

The dynamic behavior of the flexible pipeline during deepwater Flex-lay directly determines the structures of laying facilities and the actual installation process. A coupled dynamic model considering the effects of different factors was established in this paper. Based on the model, the initial attitude of the flexible pipeline during the laying process was determined by using the natural catenary theory and Morison equation. The hydrodynamic analysis of the HYSY201 pipelaying vessel was carried out by using the finite element software AQWA. Under the specific sea condition, a flexible pipeline with outer-diameter of 352.42 mm being laid onto the 3000 m deep seabed was simulated by using the software OrcaFlex to study the pipeline dynamic behaviors including axial tension, bending moment and stress-strain in the laying process, and the factors affecting the dynamic behavior of the pipeline were analyzed. The results show a significant correlation between the marine loads, vessel motion and the dynamic response of the pipeline. Compared with the static state case, the maximum axial tension, bending moment and stress-strain of the pipeline under the interaction of the marine loads and the vessel motion increased by 42.7%, 220%, 52% and 18.7%, separately. Among the marine loads, the surface wave had the most significant effect on the dynamic performance of the pipeline. When the wave direction acts on the width of the ship, the wave height is greater than 2 m and the spectrum period is eight seconds, the wave has the greatest influence on the dynamic response of the pipeline.

A Study on Shallow Water Mooring System of Floating Offshore Wind Turbine with Sinker

The purpose of this study is to investigate the feasibility of catenary chain mooring with sinker in the shallow waters of 50 to 100 meters depth facing the open ocean. Although the effect of the sinker can be expected to reduce the hardening nonlinearity of the catenary characteristics statically, it is necessary to consider the dynamic effects of sinkers ascending or sitting off the sea floor. As the procedure, we set the environmental design conditions assuming the coastal area of Japan and the semi-submersible type floating offshore wind turbine model, and perform static analysis to investigate the effect of parameters on the mooring characteristics, and then dynamic analysis to judge the strength feasibility of the mooring chains in storm conditions. Catenary theory is applied to the static analysis, and OrcaFlex is used for dynamic analysis to consider the interaction between the floating body, mooring system and sea floor in time domain. The pros and cons of using sinker in “Catenary chain with intermediate sinker type” and “Catenary with Hinged Arm Sinker Mooring type” are discussed to clarify the design considerations.