Catenary Configuration Research Articles

Cantilever modelling in the railway catenary shape-finding problem

Catenary designers need accurate numerical simulation tools to improve the system performance, increase operational speed, minimize wear and prevent arcing. To obtain reliable results, the catenary numerical model must avoid over-simplifying the hypotheses, which significantly affect the outcome, even if this involves increasing the model’s complexity and computational cost.While the cantilevers that support the catenary wires are frequently omitted or over-simplified in many published studies, we incorporate the cantilevers into the catenary model to examine their impact on the static and dynamic behaviour of the catenary, and to develop a shape-finding procedure to include the cantilevers by a substructuring approach.For this we compared three scenarios: (i) a conventional catenary model without cantilevers, plus two other more realistic models that include the cantilevers: (ii) considering cantilever deformation in the geometric design, and (iii) disregarding cantilever deformation in the geometric design.The findings suggest that including the cantilevers reduces the stiffness of the system and produces a similar contact force. Although the two cantilever design options analysed result in slightly different initial catenary configurations, they also yield a very similar contact force. These findings therefore validate using simplified supports, provided they are appropriately calibrated, for which the proposed comprehensive cantilever model can be applied.

An experimental study of oscillations induced by two-phase slug flows of water and air on a flexible catenary shaped submerged pipe

In this paper, an apparatus is described to conduct an experimental study on a small-scale laboratory model of a submerged suspended flexible pipe, in a catenary layout, oscillating in response to the intermittent loads caused by the internal two-phase slug flow of water and air.The slug flows were obtained by injecting water and air into the pipe with sixteen combinations of flow rates. The flexible pipe was set in a catenary configuration inside a water tank with a depth of 2 m. The oscillations of the pipe were recorded by cameras registering the position of fifty-five targets attached to points along the body of the pipe, providing a time-history of the movement of the pipe.The flow rates of the liquid and gas phases, the speed of the slugs and their frequencies were measured, leading to an evaluation of slugs‘ lengths and hold-ups, and to the assessment of the fluid loads applied to the pipe due to gravity and curvature. The video images of the pipe movement were processed to provide the amplitudes and frequencies of oscillation of the pipe.The methodology developed in this work may be used for various geometries and flow regimes, and it could be employed to the validation of physical models and numerical simulations of pipes’ oscillations excited by fluid interactions through the comparison between the experimental results and the computer calculations.The application of the experimental results to offshore pipes in operations, carrying oil and gas from petroleum reservoirs submarine wells to floating facilities in the ocean, may be obtained by the proper scaling up in conjunction with computer simulations.

Catenary configuration and geometric stiffness matrix of inextensible cables: Analytical high-order asymptotic solutions for parametric design

The analytical and geometric study of catenary curves is a classic matter of applied mathematical modeling. The paper systematizes a methodological strategy to achieve fully analytical solutions for the mechanical problem of determining the equilibrium configuration assumed by inextensible inclined cables under gravitational loads. By developing a two-step perturbation scheme, the statically indeterminate equilibrium problem is asymptotically formulated, and asymptotic solutions are obtained in the form of convergent polynomial series of terms with increasing orders of smallness. As a major achievement, the asymptotic analytical solutions are explicit functions of the governing parameters and automatically satisfy the integral compatibility condition, which traditionally requires a numerical solution to assess the hyperstatic unknowns. The mathematical conditions for the existence of admissible solutions and asymptotic consistency of the perturbation scheme are provided, while the characteristic properties of the asymptotic series are recognized or demonstrated. Parametric analyses successfully verify the high approximation accuracy achievable by high-order asymptotic series, truncated to a large but finite number of terms. With the aim of extending the methodology to the largest possible variety of cable structure applications, a fully analytical, although asymptotic, expression of the geometric stiffness matrix of inextensible inclined cables is determined. Parametric analyses confirm that high-order asymptotic expressions can accurately approximate the exact stiffness matrix assessed numerically. This achievement opens up viable opportunities for analytically studying and parametrically designing complex cable structures (including collaborations with other structural elements), within the framework of the direct stiffness method. Finally, the feasibility and effectiveness of the asymptotic direct stiffness method are successfully verified by solving in a fully-analytical way a paradigmatic static problem for the catenary cable-stayed beam.

Numerical Investigations on Suspended Power Cable Configurations for Floating Offshore Wind Turbines in Deep Water Powering an FPSO

Abstract Floating offshore wind turbines (FOWTs) are an opportunity for floating production storage and offloading units (FPSOs) to reduce emissions. To avoid long connecting power cables with long transmission distances between a FOWT and an FPSO, the novel concept of a suspended power cable in a large water depth of 1000 m is investigated in this study. The power cable is kept floating between the sea surface and the seabed without touching either of them. A catenary configuration and two configurations with subsea buoys attached at different locations along the cable are investigated. The OC3-Hywind 5 MW reference FOWT is set up with a deepwater mooring system, and a spread-moored FPSO is modeled with characteristics similar to existing FPSOs. Steady-state and dynamic simulations are carried out in the numerical software OrcaFlex. The different configurations are first evaluated in steady-state analyses. The largest tensions are observed for the catenary configuration, whereas it shows the lowest horizontal cable excursions. Buoys attached along the center section of the cable lift it into regions with strong currents. This results in a large horizontal excursion of the cable and large tensions. The suspended configuration with buoys attached evenly over the cable results in significantly lower tensions than the other two configurations. It is studied further with dynamic analyses. The tensions at the floater hang-offs increase by a maximum of 24% compared to steady-state results indicating that dynamic analysis is crucial for the design of suspended cable configurations.

Tether-Based Localization for Cooperative Ground and Aerial Vehicles

Considering a multi-robot system composed of a ground vehicle and a multi-copter connected each other with a tether, it is crucial for them to know their relative positions in order to carry out any operating task. Localization is usually accomplished by using GPS in outdoor environments. However, such a system does not work in indoor and more generally in so-called GPS denied environments. We investigated a solution which exploits the catenary configuration assumed by tether and does not require any computationally demanding resource or complex cable tension sensors. Different approaches have been experimentally tested, compared and validated with classical localization systems in both indoor and outdoor environments.

Cable structures: An exact geometric analysis using catenary curve and considering the material nonlinearity and temperature effect

Structures formed by cables constitute structural systems of great practical application in engineering, such as suspension bridges, cable-stayed bridges, cable cars, transmission lines in which case they are used as supporting elements that conduct electricity, overcoming the spans between the towers or in stays for the so-called cable-stayed towers, among other applications. Cable structures have a highly non-linear behavior both from a geometric point of view, arising from their initial equilibrium configuration, and from a material point of view, since the constitutive laws of the cables present an elastoplastic behavior. The objective of this work is to present a geometrically exact formulation for static analysis of cable structures considering the catenary configuration. For this, both geometric and material nonlinearities are included in the development of the the catenary cable element formulation. The effect of temperature variation is also considered. A rigorous updated Lagrangian description combined with the corotational approach is employed to obtain the consistent tangent stiffness matrix of the exact cable element. The proposed methodology was implemented in the software program called Advanced Structural Analysis System (ASTRAS), which is an academic software based on the Finite Element Method (FEM). Finally, numerical examples are analyzed using the present procedure and the obtained results are compared with those found in the literature and given by commercial softwares in order to demonstrate the consistency and precision of the formulation.

Experimental investigation on responses characteristics of a mooring line with imposed top-end motions

A series of experiments were performed to investigate the dynamic response of a mooring line based on the catenary configuration with imposed surge and heave motions in a water flume with a considered model scale factor of 1:70. The surge and heave motions with different amplitudes and periods were driven by a programmable electrical motor at the fairlead position, respectively. The dynamic tensions and motion trajectories of the mooring line were measured to illustrate the effects of pretension and motion period on the predominant harmonic components in the dynamic response. The results show that the pretension and motion period have significant effects on the dynamic tensions and nonlinear motion trajectories of the mooring line. Multiple harmonic components of the dynamic tension are observed at the fairlead and the predominant harmonic tensions are found to be changed with variations of the pretension and motion period. A sharp increase of the maximum tension force is observed at large pretension for the mooring line with the surge and heave imposed motions. Such sharp tension increase may result in floating platform and mooring system breakage. The results of this paper revel the foundation of the further studies on the transforming mechanism of the slack and taut state of the mooring line as well as for optimism of the engineering designs in floating offshore devices mooring installations.

Effects of blade configuration and solidity on starting torque of Darrieus wind turbine

This study investigates the effects of blade configuration and solidity of Darrieus wind turbine on the starting torque characteristics. Generally, the configuration of Darrieus wind turbine is divided into Troposkien, parabola, Catenary, Sandia, modified-parabola and straight types. A numerical analysis has been carried out using Multiple Stream Tube (MST) method to investigate the effect of blade configuration and solidity of Darrieus wind turbine on the starting torque under the initial low range of rotational speed. The simulation results show that the starting torque of Darrieus wind turbine varies considerably depending on the blade configuration. The initial starting torque was larger with Troposkien, Parabola, Catenary, and Sandia configurations than with modified parabola or straight types. The increase in solidity with increasing number of blades raised the starting torque and improved the dynamic stability during the initial operational speed of Darrieus wind turbine. Additionally, these torque results represent basic data for fluid-structure interaction (FSI) simulation of the steady-dynamic operation of the turbine.

Approximate calculation of the static analysis of a lifted stay cable in super-long span cable-stayed bridges

The sag effect of long stay cables is one of the key factors restricting further increase in the span of cable-stayed bridges. Based on the formerly proposed concept of long stay cables lifted by an auxiliary suspension cable in cross-strait cable-stayed bridges, corresponding static approximate calculations and analytical theory based on catenary and parabolic cable configurations are established. Taking a main span 1400 m cable-stayed bridge as the research object, three typical lifting conditions and the whole process of auxiliary cable lifting are analyzed and discussed. The results show that the sag effect is effectively reduced. The support efficiency is only improved when the cables are lifted above the original cable chord. Reduction of the horizontal component force of the cable is limited. The equivalent elastic modulus and the vertical support stiffness of the lifted cables are significantly increased with increased horizontal projection length and not sensitive to the change of the lifting point position. The scheme of lifting the cable to the chord midpoint is more economical because of the less steel required for the auxiliary suspension cable, but its effect on improving the vertical support efficiency is limited. The support efficiency is better when the cable is lifted to the cable end tangential to the original cable chord, but the lifting force and the cross-sectional area of the auxiliary suspension cable are doubled. The approximate calculation results of the lifted cables are very close to the numerical analysis results, which verifies the applicability of the approximation method proposed in this study. The results of parabolic approximation calculations are approximately equal to that of catenary cable geometry. As the parabolic approximation analysis theory of lifted cables is more convenient in mathematical processing, it is feasible to use parabolic approximation analysis theory as the analytical method for the conceptual design of lifted cables of super-long span cable-stayed bridges.

Analysis of the overlap section in a high-speed railway catenary by means of numerical simulations

The transition between catenary tensioning sections is accomplished smoothly by overlapping a number of spans in each catenary section. This work presents an analysis of the overlap section in a high-speed railway catenary based on numerical simulations. The paper studies the influence on the system’s dynamic behaviour of features such as double cantilevers and tensioning devices efficiency. Four and five-span overlaps are compared and the effect of train speed and overlap contact wire geometry are also analysed. Finally, an entire catenary section is optimised by Bayesian Optimisation techniques, leading to a catenary configuration with an interaction force with a standard deviation notably lower than that provided by the nominal catenary design.

Numerical Calculation of Two-Dimensional Subsea Cable Tension Problem Using Minimization Approach

Subsea cable laying is a risky and challenging operation faced by engineers, due to many uncertainties arise during the operation. In order to ensure that subsea cables are laid out diligently, the analysis of subsea cable tension during the laying operation is crucial. This study focuses on the fatigue failure of cables that will cause large hang-off loads based on catenary configuration after laying operation. The presented problem was addressed using mathematical modelling with consideration for a number of defining parameters, which include external forces such as current velocity and design parameters such as cable diameter. There were two types of subsea cable tension analyses studied: tensional analysis of catenary configurations and tensional analysis of lazy wave configurations. The latter involved a buoyancy module that was incorporated in the current catenary configuration that reduced subsea cable tension and enhanced subsea cable lifespan. Both analyses were solved using minimization through the gradient-based approach concerning on the tensional analysis of the subsea cable in differentconfigurations. Lazy wave configurations were shown to successfully reduce cable tension,especially at the hang-off section.

The behavior of anchor lines embedded in layered soils

The behavior of deepwater anchors in the seabed is significantly affected by the anchor lines embedded in soils, whether in anchor installation or in mooring. Unlike the anchor lines submerged in seawater, the ones embedded in soils exhibit special and complex behaviors due to soil resistance in either tension transmitting or profile properties of the lines. This problem becomes more complicated when encountering a complex seabed condition, such as the layered soils with both cohesive and cohesionless layers. Full knowledge of the anchor lines embedded in layered soils is imperative for anchor installation and safe operation of mooring systems. The literature review indicates that the relevant studies apply only to a single layer of either cohesive or cohesionless soil. This highlights the importance of the present study and also the challenge to deal with the problem in layered soils composed of both cohesive and cohesionless layers. In the present work, a mechanical model as well as analytical procedures is proposed to analyze the tension transmitting and profile properties of anchor lines embedded in layered soils. This model can well capture the complexity of seabed conditions, applying to not only the layered soils with both cohesive and cohesionless layers, but also the cohesive or cohesionless layers with different soil strengths. To gain further knowledge of the behavior of anchor lines embedded in layered soils, numerous cases of layered soils are designed and analyzed by the present model. The results indicate that, the configuration of the embedded anchor line presents complex and various reverse catenary shapes due to different strengths of layered soils; companying with increasing curvature of the reverse catenary configuration, both the tension loss and the tension angle at the shackle of the anchor increase with increasing proportion of cohesionless layers. These phenomena exhibit the complexity and variety of the behavior of anchor lines embedded in layered soils. A specific case of such problems can be efficiently solved by the present model to gain quantitative analytical results.

Iterative Nonlinear Cable Shape and Force Finding Technique of Suspension Bridges Using Elastic Catenary Configuration

AbstractIt is important from the safety and economical perspectives that a suspension bridge should have minimal deformation under the self-weight of the bridge while the tension forces of the main...

Experimental modelling of mooring systems for floating marine energy concepts

New floating devices are emerging thanks to a highly active marine renewable energy sector. These concepts, featuring floating offshore wind installations as well as other projects in the wave energy sector require innovative mooring systems. Therefore, a complete analysis of this element is required to achieve a successful design.This paper investigates the behaviour of mooring systems based on catenary configuration thanks to a detailed experimental program. Tests to characterize the properties of the mooring lines, and geotechnical and hydrodynamic trials were also performed on a 1:40 laboratory scale. Mooring truncation problem is normally faced by the researchers working on model testing of mooring lines in deep water, because the depth of the lab is typically small. However, it does not seem to be a problem for offshore floating renewable energy devices which are usually in relatively shallow water. The influence of catenary weight, imposed displacements at the fairlead, wave-current loads and different types of sea bottom friction on the mooring systems were analysed in terms of tensions, movements and shape of force-displacement curve. However, scaling of the sea bottom and the mooring line interaction is a challenge. Although the damping effect was investigated and compared under different movement conditions, the conclusions achieved can be upscale to real scale but it must be noticed that there are limitations due to scale effect issues. A total of 111 test were performed. The influence of the hydrodynamic inertia and drag is negligible on long wave and slow motions, while they have more importance during shorter waves and quick motions. The results reveal the importance of acceleration in the movement being possible to distinguish dynamic and quasi-static performance depending on the acceleration of the mooring line.

Pipeline walking and anchoring considerations in the presence of riser motion and inclined seabed

Steel Catenary Risers (SCRs), are increasingly becoming an attractive option for many deepwater field developments. SCRs are typically used to transport fluids between floating production vessels and pipelines. Typical uses may also involve the transport of produced fluids from a subsea production system to a floating production vessel or the transport of gas or water for re-injection into the producing reservoirs. The floating production vessel on which the steel catenary riser is supported will be subject to motions caused by environmental loads, and influenced by the mooring system and other risers. Horizontal movement of the vessel causes changes in the riser catenary configuration in near, mean, and far positions. On the seabed, the riser is connected to a pipeline that extends for some distance from the riser touchdown point, to its tie-in point on a pipeline or other facility. Effective tension at the touchdown point is necessary to maintain the riser configuration which may cause the pipeline to walk in the axial direction. The development of axial walking is in part due to the pull experienced on the pipeline at the touchdown point from the SCR tension. In this paper, the results of the effective axial force and the pipeline end expansion using a finite element study are presented to highlight the effect that the changing SCR tension, combined with the thermal transients and a global seabed slope along the pipeline length, has on the pipeline walking. Additionally, the paper provides some guidance in regards to the selection of the optimum location for the hold-back anchors, to ensure that pipeline walking does not compromise the integrity of both the SCR and the pipeline system.In general, the results show that SCR bottom tension provides the dominant walking mechanism and can exceed the other walking mechanisms associated with thermal transients and seabed slope. For a straight short pipeline, in the range of 2–3 km, where there is no lateral buckling, it is recommended to install the anchor towards the PLET (Pipeline End Termination) and away from the SCR transition point.

Analysis of guyed masts by the stability functions based on the Timoshenko beam-column

A new formulation for the analysis of guyed towers in which the mast is modeled as a continuous equivalent beam-column on non-linear elastic supports, is proposed. Because second-order effects must not be neglected and given the importance of consider the shear deformation in the analysis of this type of structures, the method proposes the calculation of the second-order deformation using the stability functions based on the Timoshenko beam column. The equivalent beam-column properties of the mast are calculated depending on the pattern construction of the tower and the guys are replaced by non-linear elastic supports. Based on the catenary configuration of the cable, the spring constant is obtained from the secant modulus of the cable. In order to validate the proposed method, a comparative study is carried out analyzing a guyed mast using the proposed method and a numerical model through the use of finite elements. As a main conclusions we will mention that, the proposed method are sufficiently accurate compared to the finite element method, confirming the validity of the hypotheses adopted in the development of the method and; the reduction of the computational effort, since the structure does not need to be discretized into a large number of elements for the convergence.

Analysis of Large-Sag Extensible Catenary with Free Horizontal Sliding at One End by Variational Approach

A variational approach for the static equilibrium configuration analysis of a large-sag extensible catenary with free horizontal sliding at one end is developed, considering the virtual work done by the catenary self-weight and the horizontal tension at the free sliding end support. The virtual work of the cable system is expressed in terms of the vertical coordinate, which is a function of the unstrained arc length of the cable. The stationary condition is applied to the virtual work functional to obtain the equilibrium configuration of the catenary. The validity of the variational approach is ensured by Euler’s equation, which is identical to the one derived by the force equilibrium consideration of a catenary segment. The finite element method is used to evaluate the numerical results. A parametric study is performed to demonstrate the effect of axial deformation on the equilibrium configuration of a large-sag extensible free horizontal sliding catenary. The results indicate that as the axial deformation increases, the total arc-length and horizontal span length of the cable also increase. Particularly, the very large-sag catenary configurations are studied and exhibited in this study.

Static form-finding analysis of a railway catenary using a dynamic equilibrium method based on flexible multibody system formulation with absolute nodal coordinates and controls

This paper proposes a dynamic equilibrium method for finding the initial equilibrium configuration of a railway catenary. In the proposed method, the catenary composed of flexible wires is modeled using two-node cable elements with absolute nodal coordinates based on a flexible multibody system formulation. Dynamic conditions that characterize the initial equilibrium configuration of the catenary are given and addressed as control processes in the form-finding procedure. The key feature of the proposed method is that the catenary configuration is continually evolved by dynamic simulation until characterization conditions are attained and an equivalent configuration of the centenary at static equilibrium is thus computed. It is validated using two examples and applied to the form-finding analysis of a two dimensional simple railway catenary. The obtained results are analyzed and discussed. It is general and can be applied to catenaries with complex configurations.

PACDIN statement of methods

PAntograph–Catenary Dynamic Interaction (PACDIN) is a code developed by the vehicle technology research centre (CITV) of the Universitat Politècnica de València in collaboration with the railway company Talgo S.L. The model of the catenary is a finite element model using absolute nodal coordinates. It is based on a general formulation that can be applied for analysing a wide range of catenary configurations, including stitch wire, transitions or non-straight path tracks. The formulation is fully non-linear and includes large deformations, dropper slackening and contact interaction. The model is linearised when deformations are small, as in the case of the benchmark dynamic analysis. The results of the PACDIN code show a good agreement with the average results of other benchmark codes.

A 3D absolute nodal coordinate finite element model to compute the initial configuration of a railway catenary

In this paper we propose a method of finding the initial equilibrium configuration of cable structures discretized by finite elements applied to the shape-finding of the railway overhead system. Absolute nodal coordinate formulation finite elements, which allow for axial and bending deformation, are used for the contact and messenger wires. The other parts of the overhead system are discretized with non-linear bars or equivalent springs. The proposed method considers the constraints introduced during the assembly of the catenary, such as the position of droppers, cable tension, and height of the contact wire. The formulation is general and can be applied to different catenary configurations or transitions both in 2D and 3D with straight or curved track paths. A comparison of the results obtained for reference catenaries in the bibliography is also included.