A mass-lumped cable

Spring-mass cable SimObject with Hydrodynamic loads and bottom contact; winchable from both ends.

Collaboration diagram for A mass-lumped cable:

Simulation model of a discrete mass point cable, connected with linear springs, in three degrees of freedom. The cable is numElements + 1 equal segments; each segment's mass and loads are lumped half on each of its end nodes, so a free node carries one segment's mass and the end points (the connected objects) the outer half-segments (owner ruling R96, BASE-0070).

• Reynolds dependent hydrodynamic load model by the independence principle (owner ruling 2026-10-04, BASE-0076): the normal drag Cd |v_n| v_n on 1/2 rho D L across the segment, v_n the relative velocity normal to it, plus the axial skin friction along it. A normal coefficient Cd(Re_n) at the cross-flow Reynolds number Re_n = |v_n| D / nu (owner ruling 2026-10-04, fhsim_environment ENV-0104) and a tangential (skin-friction) coefficient Ct(Re) = 0.015 + 0.7535 / (Re^0.9 + 0.1) at Re = |v| D / nu on 1/2 rho D L q|q|, q the axial relative speed, the law of fhsim_marine_elements' TrawlCable (R68, BASE-0060).
• Buoyancy of the fraction of each segment's displaced volume that is below the wave surface: a circular cylinder of the nominal diameter between the two nodes, under the surface taken flat at the wave elevation above the segment midpoint (owner ruling R66, BASE-0058); the drag and the bottom-proximity lift act on the same wetted fraction. A horizontal segment at the surface thus changes its buoyancy smoothly over one diameter, a vertical one over its length. Where a segment lies deeper than the environment's MaxWaveElevation bound it is fully submerged without evaluating the wave elevation.
• Froude-Krylov and added-mass wave excitation on each wetted segment, the Morison inertia term (1 + Ca) rho A a_n with A the displacementCrossSectionArea, a_n the water particle acceleration at the midpoint normal to the segment (owner ruling R67, BASE-0059), and the matching added mass in the node inertia: each free node's mass is its own plus Ca rho A L times the wetted fractions of its two half-segments, isotropic as fhsim_marine_elements' CableElBasics (R93, BASE-0068). Ca = 1, the potential-flow value for a circle (tabulated in DNV-RP-C205, Appendix A; the section and table numbers are not verified against the standard). None of the excitation acts along the axis; the added mass does.
• Contact forces with seabed: the soil load of each segment (two half-length cylinders, one at each node) comes from the environment's SeafloorForces shared resource (CableElement); the cable holds no soil parameter. The load is the soil's alone, the cable keeps its full water load.
• Cable can be winched in/out at both ends.

Author

Jorgen Haavind Jensen

Date

11.04.2011 JHJ: Initial version.

Example configuration excerpt

<Lib
  LibName     = "fhsim_base"
  SimObject   = "Cable"
  Name        = "C"
  cableLength = "10"
  numElements = "50"
  diameter    = "0.01"
  cableWeight = "100"
/>
<Lib
  LibName     = "fhsim_environment"
  SimObject   = "Environment"
  Name        = "Env"
  Waves.Model="Component"
  Waves.NumWaves="0"
  Bathymetry.Depth    = "100"
/>
...
<Connection
  C.PositionA        = "0,0,-10"
  C.PositionB        = "0,0,-0.1"
  C.VelocityA        = "0,0,0"
  C.VelocityB        = "0,0,0"
  C.RetractedLengthA = "0"
  C.RetractedLengthB = "0"
  C.RetractedSpeedA  = "0"
  C.RetractedSpeedB  = "0"
/>

Note

The simulation must include an Environment

Input ports

Name	Width	Description
PositionA	3	The position of end A.
PositionB	3	The position of end B.
VelocityA	3	The velocity of end A.
VelocityB	3	The velocity of end B.
RetractedLengthA	1	Retracted cable length at endpoint A
RetractedLengthB	1	Retracted cable length at endpoint B
RetractedSpeedA	1	Rate of retraction at endpoint A
RetractedSpeedB	1	Rate of retraction at endpoint B

Output ports

Name	Width	Description
ForceA	3	The spring force from end A.
ForceB	3	The spring force from end B.

Configuration parameters

Name	Width	Description
cableLength	1	Total cable length [m]
numElements	1	Number of mass nodes in cable [#]
diameter	1	nominal cross sectional diameter [m]
tensileCrossSectionArea	1	cross sectional area of load bearing material [m^2]; Default: 75% of nominal area
displacementCrossSectionArea	1	fluid displacement cross section area [m^2]; Default: 90% of nominal area
E-modulus	1	Young's modulus of load bearing material [Pa],[N/m^2]; Default: 10^11 (100GPa)
cableWeight	1	Cable weight per meter [kg/m]; Default: tensileCrossSectionArea*7900kg/m^3
hydrodynamicDiameter	1	effective hydrodynamic diameter [m]; Default: +5% of nominal diameter
Rho	1	Water density [kg/m^3]; Default: 1025. Sets the buoyancy, the drag and the added mass of the wetted cable.
internalDampingCoefficient	1	structural axial damping factor. A value of 1 gives critical damping. Default: 0.25
ExternalWakeUpdate	1	How the other objects' wakes reach the segment midpoints; listed in fhsim_environment's doc/user/wake-parameters.md. (Default: Step)

Initial conditions

By specifying no initial conditions, the end points are being used to calculate the catenary. Note that this will fail if the distance between the end points is greater than the cable length, or if the points have the same north and east components (hanging straight down). Otherwise, it is possible to specify each and every point's position and velocity:

Name	Width	Description
element0_position .. element<numElements-1>_position	3	The position of element NUM, 0-based (element0_position is the first mass node).
element0_velocity .. element<numElements-1>_velocity	3	The velocity of element NUM, 0-based.

Jacobian

OdeJacobian divides each node's force derivatives by the node's mass and differentiates that mass's dependence on the node positions through the wetted fractions (R93). It differentiates the axial tension and damping and the hydrodynamic drag, including its Reynolds-dependent coefficients, with respect to both node velocities and, through the segment direction, both node positions. It does not differentiate:

• the current, the water density, the wave elevation and the wave particle acceleration, which are taken as given at each segment midpoint (the wetted fraction's dependence on the node positions is differentiated, with the elevation held fixed);
• the bottom-proximity lift and the drag ramp near the seabed;
• the seabed contact forces: while any segment is in contact (at the last evaluation) HasJacobians is false and the engine differences the cable;
• the virtual endpoint's dependence on the first free node while cable is retracted.

The tension is not differentiable where a segment is exactly at its rest length, and the drag coefficients jump between Reynolds-number regimes. The default initial condition puts the two outer segments exactly at their rest length; give the element positions explicitly where the Jacobian at the initial state matters.

Full example file

<Contents>
  <OBJECTS>
    <Lib
        LibName="base" 
        SimObject="Cable"  
        Name="C"
        cableLength="10"
        numElements="50"
        diameter="0.01"
        cableWeight="100"
    />
  
    <Lib
        LibName="environment"
        SimObject="Environment"
        Name="Env"
        Waves.Model="Component"
        Waves.NumWaves="0"
        Bathymetry.Depth="100"
    />
   
  </OBJECTS>
  
  <INTERCONNECTIONS>
    <Connection
        C.PositionA="0,0,-10"
        C.PositionB="0,0,-0.1"
        C.VelocityA="0,0,0"
        C.VelocityB="0,0,0"
        C.RetractedLengthA="0"
        C.RetractedLengthB="0"
        C.RetractedSpeedA="0"
        C.RetractedSpeedB="0"
    />
  </INTERCONNECTIONS>
 
  <INITIALIZATION>
    <InitialCondition
    />
  </INITIALIZATION>
 
  <SIMULATION>
    <Timing TStart="0" TEnd="31"/>
    <Integrator Method="RK45_i" NumCores="1" LogStates="1" UseRSSNorm="0">
      <StepControl StepMax="0.002" StepMin="0.00000001" AbsTol="1e-3" RelTol="1e-3"/>
    </Integrator>
    
  </SIMULATION>
 
<OBSERVERS>
    <FileOutput TOutput="0,0:1:30,31" Select="union(object.FineCable:port.ForceA,object.FineCable:port.ForceB)"/>
  </OBSERVERS>
</Contents>

This SimObject is referred to as Cable