Comparisons between Catenary and Dynamic cables

This benchmark is dedicated to comparisons between the two cable models in FRyDoM : quasi-static catenary line and dynamic cable.

Two benchmarks are reported here : the first is for a taut line case and the second a slack line case. The starting and ending nodes are fixed in the world reference frame, at respectively \(\mathbf{n1} = (-10,0,-10)\) and \(\mathbf{n2} = (10,0,10)\), with a distance \(D\) between them.

The unstretched length varies from \(0.85 D\) for the taut lines, to \(1.5 D\) for the slack lines.

In both cases, we compare the stretched length given by the two models, as well as the tensions at the starting and ending nodes of the lines.

The relative error on the stretched length, \(L_s\) is computed by

\[\epsilon_{Length} = 100 \dfrac{L_s(Dynamic) - L_s(Catenary)}{L_s(Catenary)}\]

The relative error for the tension in the line, \(\mathbf{t}\), is integrated on the line

\[\epsilon_{Tension} = 100 \sum_{i = 0}^{N_{elements}} \dfrac{ | \mathbf{t}_{Dynamic}(s_i) - \mathbf{t}_{Catenary}(s_i) | }{N_{elements} |\mathbf{t}_{Catenary}(s_i)|}\]

The lines properties are given in the following table :

Parameters Values
Radius 0.05 m
Linear density 616.538 kg/m
Section area 0.007854 m²
Young modulus 636620000 Pa
Rayleigh damping 0
Number of elements 50

Taut case

The following figures show the stabilization of the stretched length and tensions of the dynamic cable to the values estimated by the quasi-static catenary model.

Relative errors for the stretched length and tension in the lines are given in the following table

Relative Errors Values
Stretched length 6.15955e-05 %
Tension in the lines 0.1686 %
Taut Irr

Fig. 43 Line profiles given by the dynamic and catenary models (superposed)

Taut Length

Fig. 44 Stretched length given by the dynamic (orange) and catenary (blue) models

Taut Tension_Starting_X

Fig. 45 Horizontal tension in the starting node given by the dynamic (orange) and catenary (blue) models

Taut Tension_Starting_Z

Fig. 46 Vertical tension in the starting node given by the dynamic (orange) and catenary (blue) models

Taut Tension_Ending_X

Fig. 47 Horizontal tension in the ending node given by the dynamic (orange) and catenary (blue) models

Taut Tension_Ending_Z

Fig. 48 Vertical tension in the ending node given by the dynamic (orange) and catenary (blue) models

Slack case

The following figures show the stabilization of the stretched length and tensions of the dynamic cable to the values estimated by the quasi-static catenary model.

Relative errors for the stretched length and tension in the lines are given in the following table. Discrepancies on the tension results given by the two models are larger than in the taut case, since bending is larger in the slack case.

Relative Errors Values
Stretched length 0.00138824 %
Tension in the lines 4.13197 %
Slack Irr

Fig. 49 Line profiles given by the dynamic and catenary models (superposed)

Slack Length

Fig. 50 Stretched length given by the dynamic (orange) and catenary (blue) models

Slack Tension_Starting_X

Fig. 51 Horizontal tension in the starting node given by the dynamic (orange) and catenary (blue) models

Slack Tension_Starting_Z

Fig. 52 Vertical tension in the starting node given by the dynamic (orange) and catenary (blue) models

Slack Tension_Ending_X

Fig. 53 Horizontal tension in the ending node given by the dynamic (orange) and catenary (blue) models

Slack Tension_Ending_Z

Fig. 54 Vertical tension in the ending node given by the dynamic (orange) and catenary (blue) models