22 SEP. 1982
Lab.
v.
thepsbouwkin
ARCHIEF
Technische Hogesch[
I
Deift
Ocean Engng. Vo!. 5, pp. 209-223. © Pergamon Press Limited, 1978. Printed in Great Britain.
0029-8018/78/0601-0209 502.00,0
STEADY-STATE ANALYSIS OF UNDERSEA CABLES
A. P. K. DE ZOYSA
National Engineering Laboratory, East Kilbride, Glasgow, Scotland
AbstractA numerical solution and computer program developed to analyse undersea
flexible cable problems is presented. The cable is subject to hydrodynamic loads from a sur-rounding fluid medium. The object is to determine the cable profile and tensile forces developed. The cable equations and boundary conditions result in a classical two-point boundary value problem. An iteration 'shooting method' is used for solution. The program has been written
to analyse a class of tosíng and anchoring problems. However the analysis is general and
applicable to most steady'state cable problems.
NOTATION
a, b, il, e, g, h coefficients depending on force model
C'1 tangential drag coefficient C. normal drag coefficient e chord length of cable aerofoil
F,, hydrodynamic force per unit length normal to the cable F hydrodynamic force per unit length tangential to the cable
f
Pode's constantk, r number of boundary conditions
i, coordinate vector along the increasing s direction
io coordinate vector along the increasing O direction
coordinate vector along the increasing p direction J("p) Jacobian of r("p)
I total length of cable
p vector of guessed initial conditions
q computed boundary condition R ship-to-buggy distance along seabed R, hydrodynamic force component along i,
R0 hydrodynamic force component along i0
R, hydrodynamic force component along ¡,
s distance along the cable
T cable tension
T, cable tension at s = O
T T, T
components of cable tension in .v, i', z directionst diameter or effective thickness
V velocity of flow
w apparent weight per unit length of cable in water
X, y,z Cartesian coordinates
- body trail behind ship
CL, boundary conditions
vector of computed boundary errors
A daming factor used in the Newton iterations
friction coefficient form drag coefficient
E, vector containing the variables x, y, :, 0.qL. T
p surrounding fluid density
w angle between the flow velocit and the cable tangent angle between current and ship-to-buggy line
O angle made with the x axis by a vertical plane passing through the tangent at .4 (see Fig. 1)
e, valueofOats = O
Resolv where R, and w is fu nct ions The di 210 A. P. K. DE ZOYSA
angle made by the tangent at if to the horizontal plane (see Fig. 1)
(p aIue ofç, at s = 0.
I. INTRODUCTION
THE ANALYSIS of undersea cables has been of interest due to its many engineering
appli-cations. Examples are found in cceanoraphic research, hydroraphic surveying, salvage,
fishing. towing and offshore technology. These applications require the ability to accurately
predict the static and dynamic behaviour of cables. The static case has been considered in
this work. In cable analysis the designer is faced with essentially two problems. His first is
to obtain a realistic model of the hvdrodynamic forces and this depends very much on the physical conditions prevailing. The second is to devise a reliable and efficient solution technique to obtain the cable profile and tensions.
A considerable amount of research has been done on hydrodynamic forces acting on
cables (Caserella and Parsons. 1970). Pode (1950) developed a force model which included
tangential and normal drag. He numerically integrated the cable equations to obtain
non-dimensional 'cable functions and tabulated them for difterent cases. His solutions are still used extensively in design. However, the solutions are only valid for a two-dimensional
case with a constant velocity field. Whicker (1958) modified Pode's force model to make it
applicable to faired cabies. Eames (1968) suggested a model which accounted for viscous and form drag on the cable. He also provided an extensive discussion on the relationship
between the hydrodynamic force model and the physical conditions of the problem. Wilson (1964) modified Pode's model to include a varying tangential drag. In the program developed here, four of the commonly used force models are included so that the designer may choose
one which is most suited to his problem.
The problem for the steady-state cable when formulated reduces to a system of six, first-order, coupled, differential equations. These equations may be numerically integrated for a given set of initial conditions. However, in the real case it is usual to know boundary conditions at both ends of the cable. The resulting two-point boundary value problem may be solved for the two-dimensional case simply by searching a family of initial value solutions. This is essentially what Pode's cable functions help to do. This method is,
how-ever, cumbersome for the three-dimensional case and almost impossible for a non-uniform
velocity field. Use of the 'shooting method' for solution, and starting with two guessed values for the initial conditions was first suggested by Bedendender (1970). However, his method did not give convergent solutions for a number of difficult cases, especially when the initial guess was poor. Improvements have been made to the method to ensure
con-vergence for any practically feasible solution.
The computer program developed operates in an interactive form, hence the input and
output is self-explanatory. The program has been used to analyse a power cable, connecting
a ship to an undersea buggy. The buggy is to be used for inspection of offshore structures.
2. FORMULATION OF TUE PROBLEM
The general problem of the undersea cable may be formulated with regard to Fig. I in which s is the distance along the cable, O is the angle made with the x-axis by a vertical plane passing through the tangent at A, p is the angle made by the tangent at A to the horizontal plane, and i, i, i are local coordinate vectors along the s, 0, p directions.
Steady-state analysis of undersea cables 211
X
Fio. 1. Coordinaie system for general problem.
Resolving forces along the ¡, i. i5, directions, we have
dT = w sin R (x, i, z, O, p) ds dO
(Tcos p) = - R0 (x, y, z, O, p)
dsTf?
= w cos p - R5, (x, y, z, O, p) dswhere R5, R0 and R5, are hydrodynamic force components in the three vector directions
and w is the weight per unit length of cable. The hydrodynamic forces are, in general, functions of position (x, y, :) and cable angle (O, p).
The directional derivatives of the cable element ds are
dx =z p ds dy = sin O COS ds dz
- = sin p.
dsç
212 Â. P. K. DE ZoYsA
Equations (l-6) form a set of first-order, coupled. differential equations, to be solved for
boundary conditions specified at the two ends of the cable. The boundary conditions in general could be a non-linear function of the independent variables.
Examples are
(x. r, z) at s = O and s = / are known.
(T cos ? x. i', z) at s = O and (z, r) at s = /are known.
The problem is now reduced to a classical two-point boundary value problem and has
been solved using a 'shooting methodS, which is described in Section 4. A detailed derivation
of equations (l-6) is given in Appendix L.
3. HYDRODYNAMIC FORCE MODELS
Cables in relative motion within a fluid experience lift and drag forces. Lift forces are present in stranded or faired cables with unsymmetrically placed aerofoil sections. Lift
forces have not been considered in the present analysis: however, they could be incorporated
into the theory without any difficulty. Drag forces are of two types, pressure and friction drag. Pressure drag is duc to the normal component of flow. Friction drag contributes to
both normal and tangential drag forces. Earnes has pointed out the possibility of form drag
on a cable. He introduced a form drag coefficient. 'i, and assumed the force decreased linearly with the effective horizontal thickness ratio as the cable angle decreased. Form drag acts in the resultant direction of flow. However, insufficient data are available at
present. on form drag coefficients to make reliable computations.
A generai expression for the drag forces, which includes four hydrodynamic force
models, was given by Bedendender. This is written as
F
pt,V2
(a sin sin+ b sin ').
(7)F, PtnV2 cos j + e cos
±
g COSJ ± h cos '4 sin j] (8)2
cosy
where F is normal force,
F, is tangential force,
C is drag coefficient for normal flow, V is velocity of flow,
p is density of the fluid,
t is diameter or effective thickness, and
is angle between flow velocity and the cable tangent. Coefficients a, b, d, e, g and I, depend on the force model.
3. I Pode model
In the model developed by Pode a = 1, g f (f is Pode's constant) and the other
coefficients are zero, giving
ptCV2 . F -- 2 y sin y j, (9) Note that P. 0.0 1-0.03 fo 3.2 J Vilsom, ii In Wilso, where CJiS and Pode in 3.3 W/,icker For Whi giving where i/c is ti 3.4 Eames' For Eani The form dr
Steady-slate anIyss of undersea cables
213
j:' ptC,,V V
>(f
2Note that Pode's tangential drag force is independent of . The coefficient f ranges from 0.01-0.03 for round cables.
3.2 Wilso,z model
In Wilson's model a = 1, d = ltCf/C,,, and the other coefficients
are zero, giving
F,, - Csin
sin (11)F
-
p)TtCV2cos4 ¡cos
(12)
2
where Cris the tangential dragcoefficient, the value of d ranges from 0.015-0.04. Wilson
and Pode models are applicable only to round cables.
3.3 W/,icker model
For Whicker's model
a = i/c,
d = -0.055
- 0.02 I/c, g = Ob
I - I/c,
e = 0.386 - 0.303 i/c, h = Ogiving
ptC,j'2
2 {t/c sin sin t
+ (1 - i/c) sin
},F
=P1CV{(_Q055
-0.02 1/c)coscos 'j + (0.386 - 0.303 i/c) cos '} (14)
2
where tIc is the thickness
to chord length ratio for a faired cable; i/c = I for roundcables.
3.4 Ea,nes' model
For Eamcs model,
a=1
-The forni drag term with coefficient 'i is included in this model, giving
r,, PICnV2{(1
- t)sin
sin 4+
sin 4m),2 (15)
(10)
214 A. P. K. Ds Zo'sÂ
F piC,,V ( cos '4.' -f- V COS ) Sifl 'i.' )
2
where p. is friction coefficient. and i., ')have a ranre between 0.025-0.05.
It is seen that, for ic = i and p. = 0. all four models agree on the normal drag term.
There is however no such agreement on the tangential drag term, which is very much dependent on the nature of the problem and cable shape. Typical values for C, taken.from
Bedendender, are given in Table 1.
4. SOLUTION TECHNIQUE
The equations (l-6) could be numerically integrated for a given set of initial conditions.
It is however more common to know boundary conditions at both ends of the cable. A solution for this case may be obtained iteratively using the 'shooting method'. A
mathe-matical discussion of the shooting method' is given by Hall and Watt (1976). The appli-cation of the method for the cable problem is explained here.
The six differential equations may be written in the form,
= f (s,
,...
E6) (i = 1, 6)where vector consists of the variables x, y, :, O, ? arid T; ' is the differential of with respect to s.
in general r boundary conditions may be known at s = O and k boundary conditions
at s = / (with r ± k = 6). Let the boundary conditions be of the form, g {(0)} = ec
(i = 1, r)
and /t {(1)}
= f,
(i = 1, k).Initially the k unknown conditions at s = O are guessed. The problem is now treated as an initial value problem and the equation set (17) integrated to obtain a solution. In general
there will be a discfepancy between the computed and actual boundary conditions at
s = i;
a Newton Raphson iteration is then performed on the error to obtain a better initial guess. This process is repeated until the boundary conditions at s = / are satisfied to within a
given accuracy.
TABLE. I. VALUES OF C,, FOR VARIOUS TYPES OF CABLE
(16) (I 7) Let E, ditions at The error A Newton would be, where J('p is a dampi There method, in points take In this where p1 is grations fo iteration. A fourt used is, 4. I Diverge. To avoi the solution of 'IX. The f Conside exist here w tension and and it lies complication conditions. I The correct region T0 > Cable type C,
Round cable, smooth 1.05-1.15 Round cable, stranded 1.15-1.25
Hair fairing 0.6 -0.8
Trailing fairing 0.3 -0.45 Enclosed fairing 0.03-0.15
Steady-sttc analysis of undersea cables 215 Let (0) = p he thekguessed values at s == O. Let q1 be the computed boundary con-ditions at s == 1. The error is then
-- q1 = c (i 1, k).
The error a is a function of the guessed values p. It is desired to have
= o.
A Newton Raphson iteration on the error function to obtain an improved guess for p would be,
= "/3 .- "). [J(flp)}_1 E("p) (18)
where J("p) is the Jacobian of a(p) given by n is the iteration number and "X
is a damping factor such that O < "X 1.
There are at least three ways of computing(Eal/pJ)k.k. Bedender used a mean slope
method, in which is computed for two different p1 values and the mean slope through the
points taken as equai to akp1.
In this work,
pj 6p1
r1(np1 ± Spi) c("p1)
(19)
where p1is a small change in "p1. Usually Sp1 1O X "p1. To compute the Jacobian,
inte-grations fork and ± S
initial conditions are required, resulting in k + I integrations periteration.
A fourth-order RungeKutta integration scheme is used. The convergence criterion
used is,
k
< tolerance. ¡=1
4. 1 Dirergcni solutions
To avoid divergent and sometimes, incorrect solutions, it was found necessary to bound
the solution to lie within possible solution regions. This was done by adjusting the value
of "X The following example is used to illustrate the technique.
Consider a cable held nearly straight between two points A and B (Fig. 2). Two solutions
exist here which satisfy the boundary conditions. En one solution the cable hasa positive tension and lies below the line AB. The other solution gives a negative tension in the cable
and it lies above the line AB. The first solution is obviously the correct one. A further complication results because the fully stretched cable also closely satisfies the boundary conditions. Divergent or sometimes incorrect solutions may be obtained for such a case.
The correct solution is obtained if the results at each iteration are constrained to lie in the
216 A. P. K. DE Zosi Let If then and FtG. 2. Stretched cable.
Let (&p*) and '(6T) he the computed corrections for "p0 and "T0 with "X1 = 1.0 fl
equation (18). * n n
'10 - Po
(8p*)> 0
nr-I = (np0 + cç,) (fl-t-lpo-
flpO - "Po otherwise "X1 = 1.0. Let .jT0* = "T0 - "X1 n(T0*). If n-.iTo* < O then n1-11=
"1 and 'lx =10
2("T0 fl4 'T 8(s"X is substituted in equation (18) to compute the "'p values.
Bounding the solution in this manner also acts as a solution search coupled to the
Newton iteration. A value of X > I could sometimes be used to accelerate the convergence
if runaway solutions are constrained by bounding the solution.
The con Appendix I 5.1 Underse A cable and tension sought for -- ho
T - X I
Ty - Y- z c
Uniform Cu dimension s otherwise "X = "X1.Scjdv-statc analysis of undersea cables 217
Current ft
- .___ ._.._. ., Shp -. .-_ drect'on
FIG. 3. Undersea buggy and ship.
The computer program written for the solution of the cable problem is explained in
Appendix IL.
5. EXAMPLES 5.1 Undersea buggy
A cable is used to supply power to a temotely controlled sea buggy. The cable profile and tension at the buggy end are required (see Figs. 3-5). Solutions of the following are
sought for varying depths and current directions:
- horizontal angle made by current with ship-to-buggy line (Fig. 5). T - x component of tension at the buggy end.
T - y component of tension at the buggy end, arid T2 - z component of tension at the buggy end.
Uniform current with a surface speed of 2 knots (3.7 km/hi) was assumed and the given
dimensions are
B Ship end (s()
Buggy
4
Buggy end (sO) FIG. 4. Coordinate system for undersea buggy.
218 A. P. K. DE Z0YsA
FIG. 5. PIan view of ship and buggy.
Height of buggy = 3 ft (0.91 m)
Ocean depths = 800 and 600 ft (244 and 183 rn) Ship-to-buggy distance along seabed (R) = 550 and 750 ft (168 and 229 ni)
Length of cable = 1000 ft (305 m)
Nett weight of cable per unit length = 0.841 lbjift (12.3 N/rn)
Cable diameter = 0.9 in (23 mm).
The Wilson hydrodynamic force model, equations (li) and (12), was used for the solution with C,. = 1.0 and C,- = 0.005. Boundary conditions were
x.v,
= O at s = Ox = R cos
, y =
R sin and z = ocean depth at s = i.Variations in T,., T'y, T, with the current direction angle, , are plotted in Fig. 6. 5.2 Tow problem
A ship is towing a submersible body (see Fig. 7) at a depth of 500 It (152 m) and a speed of 15 knots (28 km/hr). A surface current of 3 knots (5.5 km/hr) moves in the direction of the
ship's motion. The current velocity profile with depth is assumed to be linear, and is zero
at a depth of 500 ft (152 m).
The body drag is given by T = kV2 where k .0, Vis the relative velocity in knots, T, = 900 lbf(4 kN) for V = 15 knots (28 km/hr).
rt is desired to know:
a - the body trail
, behind the ship,b - the vertical force T: necessary to maintain the body at a depth of 500 it (152 ni), and
e - cable tensions at the ship's end.
Boundary conditions are
x = 0, z = 0, Tcos
= 900 lbf(4 kN)ats = O
= 500 ft (152 n) ats =1.
2000 800 600 400 200 000 305,< 400 300 00 o o
Staady-stte analysis of undersea cabks 219
-300 200 '80 20 00 80 60 40 20 I I I X
(a) x-Tens,or, at buggy end
b) y-Tenson at buggy end
Fio. 6. Variation of tension with current direction. -1 '000 200 -'000 -'00 - 200
-
Depth D,stance 0244m 800ff 49m 550ft x183m 600ft 229m 750ff -300 -400 - 2000 30 60 90 120 degrees¿C) z-Ten5on al buggy end
220 A. P. K. DE ZOYSA
¡
FIG. 7. Tow problem.
Two cases are considered. In Case 1. the cable has a round cross-section of diameter
=
0.6 in (15 mm). ¡n Case 2, a faired cable with thickness 0.85 in (21.6 mm) and chord to thickness ratio = 0.2 is used. The Wilson force model with C,, = i and Ci = 0.005 is used
for the round cable solution. The Whicker force model, equations (13) and (14), with = 0.2 is used for the faired model solution.
Results are shown in Table 2. The tensions developed for the round cable are con-siderably higher than those for the faired cable. Fairing is essential for this cableto reduce the very high tensions developed.
TABLE 2. Tow PROBLEM RESULTS
T, Trail = . Tensions at
hf (N) ft (m) ship end
lhf(N)
AcknowledgementsI wish to thank the Oxford University Mathematics Department staff for thevery
useful suggestions and comments made on this problem. This paper is published by permission of the
Director, National Engineering Laboratory, I)epartrnent of Industry. lt is Crown copyright.
REFERENCES
BEDENDENDER, J. W. 970. Three-dimensional boundary value problems for flexible cables. OTC 1281.
2nd Ann. Offshore Technology Conf.. Houston. Texas.
CASARELLA, M. J. and PARSOHS. M. l90. A survey of investigations in the configuration and motion of cable systems under hydrodynamic loading. Mariì,e Technol. Soc. J. 4, 27-44.
EAMES, M. C. 1968. Steady-state theory of towing cables. Trous. R. Juts/n.oar. Arclzit. 110, 185-206.
HALL. G. and WAy r. J. M. 1976..umerical Methods Jór Ordinary D'rential Equations.pp. 216-238.
Clarendon Press, Oxford.
Pone, L. 1950. An experimental investigation of the hydrodynamic forces on stranded cables. Rept No-7 13. David Taylor Model Basin, Washington, D.C.
WHICKER, L Rept No WILSON, B. A & M I)cruvat ion u' The local i, is t is, S t Ì S t R. R0.Ra w ist Resoling fc' 1?, ds Neglecting a Similarly for
and for equi
Resolving ib
Round cable 4785 (21 285) 829 (253) 5526 (24 580)
-s
2-.
Steady-state analysis of undersea cables 221
WJIICKER. L. F. 1958. Theoretical analysis of the effect of ship motion on mooring cabes n deep water. Rept No 1221. David Taylor Model Basin, Washington. I).C.
WILSON, B. W. 1964. Characteristics of deep sea anchor cables in strong ocean currents. Rept NO 204-3 A & M Research Foundation. Texas.
APPENDIX I
Dcripat ion of tite diffi'renr ial equations
The local coordinate system for an element of cable length da is shown n Fig. 8 in which i, is the unir, tangent sector to cable element
-i is the normal to cable in the ',-ertical plane
i is the normal to i, i, vectors and lies in the horizontal plane.
R, R0,R are hvdrodynamic force per unit length components in the i,. i, i directions respectively, and w is the nett force per unit length on cable.
Resolving forces for equilibrium along the i, vector.
/ dTds\ /1 dip \ /1 dO R, da w sin p d.c -j- ,T -- ---jcos - th) cos
2 - da
/
dTds\ ¡I dT \ ji i/O\
(T __Jcosl__ds)cos(_ _d.iJ =0.
\ da 2 / \2 da . \2 ds /Neglecting all second order terms sve have
dT R,ds - ; sin ip d.c = - --- ds.
ds
Similarly for equilibrium in the i direction,
dO R0 S-- Tcosrp
--ds
and for equilibrium along ¡Q direction
- w cos (p
Resolving the element length da along the x, y and a directions dx = d.c cos p cos O ¿
= T.
(1.4) da da 5 + A/
z
Fio. 8. Loca! coordinate system.
y
(1.2)
222 A. P. K. DE ZOYSA
Hence have the equation set:
dy = ds COSQ SiflÛ ds sin Q. /dT (dv ds f dy k ds fd: k ds w SIn
)
-- R0 = WCOS( - R1, = COSO COS Q Sin OCOS (P = sintp.These equations are the same as equations (I-X) in the mails text.
Description o A Comput written to oç expensive on to a main fra: The progri is sufficient1 flowchart of t
Stcadv-statc analycis of undersea cables 223
APPENDIX Il
Description of computer progra,n
A computer program svas ritten in standard Fortran to do the analysis described in Section 4. It was
written to operate interacti'.elv so that thc Input and Ouipitt is sell-explanatory. The program is not
expensive on resources or time and could sithout difficulty he executed on line from a terminal linked tip to a main frame computer.
The orogram analyses a certain class of tosing and mooring problems. However the core of the program
is sufficiently general to include other types of problems with minor changes in the coding. An overall
flowchart of the computing procedure is given in Fig. 9.
input Current flow doto / Cable dimensions Tow conditions, Mooring position Fluid force model [ampute coefficients
for fluid force model
nitiol solution search to
obtain first oDproximatibn
for iterations Input Boundary conditions No of iterations Accuracy Solution bounds lnteqrote equilibrium equations to obtoiri boundary conditions Is the solution sufficiently No Occurate7 Yes Output Tensions
Conf uratian of cable
Flot cable _J Correction opplied to initial conditions Max irnuri iterations reached?
Fio. 9. Overall flowchart.
