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Three dimensional transient analysis of the installation of marine cables

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Summary

A numerical solution for the three dimensional transient motion of a marine cable during installation is presented for the case of a cable laying vessel arbitrarily chaning speed and direction while paying out cable with seabed slack. The cable transient behaviour is governed by the numerical solution of a set of non-linear partial differential equations with the solution methodology incorporating both spatial and temporal integration. The space integration is carried out by dividing the cable inton straight elements with equilibrium relationships and geometric compatibility equations satisfied for each element. The position of each element is described by its elevation and azimuth angles and, therefore, a system of 2n non-linear ordinary differential equations is established. The time integration of this set of equations is performed using a high order Runge-Kutta technique. Results are presented fro the cable tension and element elevation and azimuth angles as functions of time and for transient cable geometries when the cable ship executes horizontal planar manoeuvres.

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Abbreviations

A t, An, Ab :

tangential, normal and binormal components of cable element acceleration vector

A c (s, t):

cable element acceleration vector

C D ,C f :

normal and tangential friction drag coefficients

C m :

added mass coefficient

D n n,D b b,D t t :

normal, binormal and tangential drag forces

d :

cable diameter

G i (t):

constant of integration

g :

gravitational acceleration

H :

hydrodynamic constant of the cable

I, J, K :

inertial system unit vectors

i, j, k :

tow vessel system unit vectors

L i :

length of the cable element

N, B, T :

normal, binormal and tangential node forces

n :

number of straight elements

2N i /L i , 2B i /L i :

geometric stiffness terms

OXYZ:

inertial system

oxyz:

system attached to the tow vessel

o′x′y′z′:

local system

R c (s, t):

vector position of a cable element

r o(t),r c (s, t):

ship and cable element position vectors

\(\frac{\partial }{{\partial t}}r_c \left( {s, t} \right)\) :

cable element velocity vector

\(\frac{{\partial ^2 }}{{\partial t^2 }}r_c \left( {s, t} \right)\) :

cable element acceleration vector

\(\frac{d}{{dt}}r_0 \left( t \right) = - \frac{d}{{dt}}x_0 \left( t \right)i\) :

ship speed

\(\frac{{d^2 }}{{dt^2 }}r_0 \left( t \right) = - \frac{{d^2 }}{{dt^2 }}x_0 \left( t \right)i\) :

ship acceleration

s :

unstretched distance along the cable

T t :

effective tension

T i (s, t):

axial tension in cable elementi

T n, 0 = −L n H(θ n ):

axial tension in the beginning of elementn

T ship(t):

cable top force

0054xx s, t :

“true” axial force

t, n, b :

tangent, normal and binormal unit vectors

V p 0 :

cable pay out rate

V 0 :

initial ship speed

V t ,V n ,V b :

tangential, normal and binormal components of cable element velocity vector

V c (s, t),A c (s, t):

velocity and acceleration vectors of a cable element

w :

cable weight in sea water per unit length

w j :

self weight

X(s, t), Y(s, t) Z(s, t) :

functions describing cable geometric configuration

β:

final ship azimuth direction

ϱ:

sea water density

ϱ c :

cable physical mass per unit length

ϱ c A c :

d'Alembert force

\(\varrho C_m \frac{{\pi d^2 }}{4}\left( {A_n n + A_b b} \right)\) :

added inertia

τ:

time taken for the ship to change heading

ψ = ψ(s, t), θ = θ(s, t):

azimuth and elevation angles

ψ i , θ i :

azimuth angle and elevation angle

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Author notes

  1. M. A. Vaz

    Present address: Laboratory for Submarine Technology, Ocean Engineering Program, COPPE, Federal University of Rio de Janeiro, 21941-590, Rio de Janeiro, Brazil

  2. J. A. Witz & M. H. Patel

    Present address: Santa Fe Laboratory for Offshore Engineering, Department of Mechanical Engineering, University College London, London, UK

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    1. M. A. Vaz
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    2. J. A. Witz
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    3. M. H. Patel
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    Vaz, M.A., Witz, J.A. & Patel, M.H. Three dimensional transient analysis of the installation of marine cables. Acta Mechanica 124, 1–26 (1997). https://doi.org/10.1007/BF01213015

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