- Email: [email protected]
Transient behaviour of towed marine cables in two dimensions
Applied Ocean Research 17 (1995) 143% 153 0 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0141-1187/95/$09.50
0141-1187(95)00012-7
ELSEVIER
Transient behaviour of towed marine cables in two dimensions M. A. Vaz* & M. H. Pate1 Santa Fe Laboratory for Ofshore
Engineering, Department of Mechanical Engineering, London WCIE 7JE, UK
University College London, Torrington Place,
(Received 4 May 1995; accepted 8 August 1995)
This paper presents a numerical solution for the transient motion of marine cables being towed from a cable ship which is changing speed. The cable ship is assumed to move rectilinearly, hence the cable configuration is two-dimensional. The solution methodology consists of dividing the cable into n straight elements with equilibrium relationships and geometric compatibility equations satisfied in each element. A system of n non-linear ordinary differential equations is derived from this and then solved by fourth- and fifth-order Runge-Kutta formulations with the dynamic axial tension calculated iteratively because it is itself dependent on the solution. Results are presented for the cable-top tension and element angles as functions of time and for transient cable geometries when the towing velocity is linearly or parabolically increased (or decreased). It is shown that the results from this analysis compare reasonably well with full-scale experimental data from Hopland (Proc. Int. Wire and Cable Symp., 1993, pp. 734-39).
towards the objective of developing a reliable prediction model for the transient behaviour of cables when towed or laid. Since there is some full-scale data available’ for transient behaviour of a towed cable, the first phase of this work has been to develop a general analysis for towed cables and for the cable laying process and to verify the towed cable aspect of the work by comparing its predictions with full-scale data from Hopland. This is the objective of this paper. The study of towed marine cables has been an area of interest to engineering research for the past three decades. Schram and Reyle2 investigated the threedimensional motion of a cable-body towing system. They employed the method of characteristics to solve the governing equations of a towed system using numerical interpolation. A transfer function, defined as the ratio of the resultant towed body’s amplitude of motion to the amplitude of the ship motion, was developed for various towing speeds and towline lengths, which decreased as the speed and length increased. However, their analysis did not include the effort of hydrodynamic mass. Sanders3 took Schram and Reyle’s solution for a three-dimensional towed system and developed a new formulation which could readily handle non-uniform properties, and could be easily implemented. Sanders disregarded inertia and bending terms and removed the requirement to have a first-order truncation error in the integration scheme. He employed
1 INTRODUCTION Marine cables, towed by ships or submarines, have been used in a variety of civil and military tasks. Civilian applications include acoustic surveying for sub-seabed hydrocarbon reservoirs and for geophysical mapping of ocean basins. Military applications are principally concerned with the detection of submarines. Figure 1 shows a schematic diagram of the towing arrangement. There is also a parallel activity, undertaken for many years now-the laying of undersea communication cables across virtually all of the world’s oceans. The installation of this world-wide communication infrastructure is increasingly becoming commercially competitive, with technical issues such as the speed and accuracy of cable laying yielding significant commercial benefits. In this climate, the feasibility of high-speed cable laying becomes important, as does an understanding of the transient behaviour of the cable laying process as the cable ship changes speed. Prediction of the transient behaviour of the cable as it is being laid requires a non-linear time-domain analysis-and once the analysis is available, its verification poses a difficult problem. The work described in this paper is the first phase *Now at Ocean Engineering Programme, Coppe, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. 143
144
M. A. Vaz, M. H. Pate1 xdt)
L-_
I
/LO
/
/
I
h
Fig. 1. Schematic diagram of the element model
a physical finite-difference formulation to solve the three-dimensional equations for cable dynamics. Delmer et al4 presented a numerical method for the dynamic simulation of towed cables. The system was constructed from a set of generic elements incorporating cable strands, knots, kinks, cable ends and winches. These elements were then organised into a general and flexible analysis procedure allowing consideration of complex systems. From here it was possible to solve the pre-defined towed system under a variety of load cases using a finite-element method. Ablow and Schechter’ devised a robust finite-difference approximation to the differential equations governing the behaviour of towed cables. The code was fully three-dimensional and yielded very good results when compared to actual measured data. Triantafyllou6 on the other hand, studied the dynamics of a translating catenary. The static and linearised dynamic governing equations were derived along the local tangential and normal directions. It was shown that two simpler equations could be derived and solved asymptotically for both small and large, horizontal or inclined cable sags. Triantafyllou also deduced that the most important parameters for travelling cables are the nondimensional speed of translation, the ratio of the elastic-to-catenary stiffness, the inclination angle, the cable’s weight-to-tension ratio and the added-mass-tophysical-mass ratio. this work up by Triantafyllou et a1.7 followed producing a detailed report on mooring dynamics for offshore applications, jointly funded by the Sea Grant Program and a consortium of companies. Investigations centred around the effect of principal non-linearities on predictions from a linear analysis of cable dynamics. The study confirmed the existing perception that the principal non-linearity for vibrating cables in water is fluid drag. The problem of the effect of drag force linearisation and of cable-sea bed interactions was addressed and a solution procedure was derived. A novel approach towards the problem of snap loading for a curved cable was also developed.
Milinazzo et al.* presented an efficient algorithm which simulated the three-dimensional behaviour of a towed cable. The formulation was based on an implicit, second-order finite-difference approximation of the equations of motion. The results obtained correlated well with the results of Ablow and Schechter’ and with experimental data. Triantafyllou and Chyrssostomidis’ developed a procedure for calculating the response of a towed array to a harmonic excitation applied at the upstream end. The fluid forces on the array were modelled using a slender body approximation and the cross-flow principle with the quadratic damping due to cross-flow separation replaced by an equivalent linear damping. This was determined via an iterative procedure. It was discovered that when separation drag is included, the array exhibits the behaviour of an over-damped system, responding only to low-frequency excitations. Triantafyllou and Hover” investigated the dynamic behaviour of cables employed as tethers for underwater vehicles. The dynamics of the vehicle are significantly affected by the presence of the tether, particularly in deep water when the mass and drag of the cable are comparable to, or much larger than, the drag and inertia of the vehicle. A numerical scheme was presented to analyse the dynamics of the tether. Although all of the above work has fully examined towed cable behaviour under a variety of ‘quasi-static’ and steady static dynamic conditions, the work has not addressed the verification of predictions from the theory sufficiently and the transient problem of the cable and its response to a change in velocity of the towing ship has not been given the attention it deserves. This issue is addressed in this paper in association with the work of Hopland,’ who measured the full-scale behaviour of submarine cables by studying the shape of towed cable lengths. It was found that a straight line configuration of the towed cable was a good approximation to steady-state towing, and experimentally determined hydrodynamic constants were presented for this steady state. Hopland also studied transient towed cable geometry when the ship’s speed is abruptly changed. His measurements are compared with predictions from the analysis formulated in this paper. This analysis resembles that of Sanders3 but is aimed at predicting the transient motions of and the tension in a towed cable when the surface ship changes speed. The analysis is restricted to two dimensions, but incorporates self weight, normal and tangential drag forces, axial tension, geometric stiffness and physical and added inertia. The transient cable configuration and axial tension are determined for the case of the towing ship accelerating or decelerating from one steady velocity to another through any arbitrary variation with time between the two states.
145
Towed marine cables in two dimensions
where ro(t) = -xo(t)i and rC(s, t) = X(s, t)i + Y(s, t)j. The ship and cable element position vectors-respectively ro(t) and rc(s, t)-are defined relative to the system oxy. The functions X(s, t) and Y(s, t) describe the cable configuration as seen from the ship. The velocity and acceleration vectors of a cable element-respectively V,.(s, t) and A,($, t)-are obtained by VA Fig. 2. Reference axis system. 2 FORMULATION EQUATIONS
4 = $
A&, 4 =
OF THE GOVERNING
The analysis is formulated on the basis that the towed cable is continuous, inextensible and flexible, that is, with negligible bending stiffness. For most applications, cable extensibility is negligible because of the relatively high axial stiffness and low axial tension. The set of equations governing the transient behaviour of towed marine cables comprises equilibrium and geometric compatibility relationships. These equations are derived in Sections 2.2 and 2.3, respectively, although prior to this, the kinematic characteristics of marine cables are derived in Section 2.1.
[r0(4
+
$ h(t) + rds, 41
where the ship’s speed and acceleration are respectively given by a,(t)/& = -6’x,(t)i/& and a2ro(t)/dt2 = -$xo(t)i/dt2. Furthermore, the cable element velocity ac(s, t)/dt and acceleration #r,(s, t)/dt2 vectors are given by gr,(+
t) = $X(S,
t)i + $ Y(s, t)j
and
grc(s, t) = $x(S,
t)i + $
Three Cartesian systems of reference, shown in Fig. 2, are adopted: OXY (the inertial system), oxy (the noninertial system which moves with the cable ship) and o’x’y’ (the non-inertial system, local axes). To begin with, the transformation rules between these three frames of reference are defined. Referring to Fig. 2, the following relations apply:
The cable element velocity vector VC(s, t) is obtained by substituting eqn (3a) into eqn (2b) and using eqn (1) to produce
Vt(s, f) =
coscqs, t) --gX,(l, +$s,
1
1
+ sine(s, t) g Y(s, t) [
(4b)
and
+
:x0(r) [
cose(s, t) $
(2a)
t)
[
(1)
where t and n are tangential and normal unit vectors referred to the local system o’x’y’; i, j and I, J are the unit vectors referred to oxy and OXY, respectively; 6’= 13(s, t) is the angle between i and t and s is the unstretched distance along the cable. Equation (1) allows vector parameters to be expressed in any one of the three systems of reference axes. The local system o’x’y’ changes with time and space since it is ‘attached’ to a cable element. The vector position of a cable element RC(s, t), when referred to the inertial system OXY, may be decomposed into (see Fig. 2)
(44
where
V,(s, t) = sine(s, t)
4 = ro(4 + rc(s, 4
(3b)
Y(s, t)j
VC(s, t) = Vt(s, t)t + Vn(s, t)n
2.1 The kinematics of marine cables
R&
WI
41
rds,
[
- $X(S,
Y(~, t)
1
1
t)
where V, and V,, are, respectively, the tangential and normal components of VC(s, t). The cable element acceleration vector A,(s, t) is given by substituting eqn (3b) into eqn (2~) and using eqn (1) to produce A,(s, t) = A,(s, t)t + A,($, t)n
(54
where
A,(~, t) = cose(s, t) -$~~(t)
[ + sin@,
t)
a2
+ dr2x(s, t) 1
$ Y(s, t)
(5b)
146
M. A. Vaz, M. H. Pate1
and A,($, t)sinQ(s, t) $x0(t)
+ cosO(s,t)
- -$X(s,
t)
1
$ Y (s, t)
where A, and A, are, respectively, normal components of AC(s, t).
the tangential
and
2.2 The equilibrium relationships These are obtained from the equilibrium of forces acting on the infinitesimal element in Fig. 2. Newton’s second law must be written in an inertial frame of reference (e.g. OXY). However, it is more convenient to transform the resulting equation into the systems oxy or o’x’y’. The marine cable being considered is taken to be fully immersed in water, i.e. no portion of cable above the sea water level (‘dry cable’) is considered in this analysis. Inertia forces, self weight, axial tension, hydrodynamic added inertia (due to normal acceleration only) and normal and tangential drag forces are considered. Furthermore, the cable is assumed to be inextensible and continuous and bending stiffness is neglected. In the absence of flexural stiffness, the axial tension cannot be negative, otherwise a singularity occurs and local buckling (loop formation) may take place. The forces (per unit length, except T) involved are: self weight (~9); normal drag (L&n); tangential drag (D,t); ‘effective’ tension (Tt); inertia (d’Alembert) force (p,A,) and added inertia [pC,(rd2/4)A,n], where pc is the cable’s physical mass per unit length, $21is the cable’s weight in sea water per unit length, p is the sea water’s density, C, is the added mass coefficient and d is the cable diameter. For conciseness, the parameters s and t will be omitted from the independent variables except when necessary to make the text clearer, i.e. differentiation with respect to s and t will be denoted by the inverted comma and overdot respectively. Note that the ‘true’ axial force T(s, t) is given by T(s, t) = T(s; t) - pg(-ird2/4) Y(s, t), where g is the acceleration due to gravity. This takes account of the effect of external hydrostatic pressure in producing a tension ‘look-alike’ term that acts in an analogous way to physical tension in the governing equation. The physical tension and hydrostatic-pressure-induced tension-like term are combined to define an ‘effective tension’-Young and Fowler” and Seyed and Patel” give detailed formulations of this term. The normal and tangential drag forces, per unit length, can be written as D, =
T’ + n,sinQ + ;pCprdV;
and (6b)
= pc [(x - ,Y,)cosB + YsinQ] (7a)
To’-
~pCDdC/,IV,l+wcosQ
= p[Ycos0+(&-%)sinB] (7b)
where
These equations are valid for any ship motion. Equations (7a) and (7b) reduce to Zajac’s equations for the particular case of a towed cable in a stationary configuration. There are apparently four unknown variables, T(,Y, t), Q(s, t), X(s, t) and Y(s, t). However, the last three variables must satisfy geometric conditions of compatibility. 2.3 Equations for geometric compatibility The cable configuration is governed by the interrelation of forces [eqns (7a) and (7b)] and by geometric compatibility which constrains the displacements. These latter compatibility equations are derived in Appendix A. At the cable’s upper point, i.e. at s= 0, the cable element velocity and acceleration vectors are given by V,(O, t) = -&(t)i and A,(O, t) = -&(t)i, hence f(t) = g(t) = 0 and df(t)/dt = dg(t)/dt = 0. In summary it may be written that s k = fi sin0 d< @a) .I0 J? =
1 (d)‘cos0 .I[
I; =
?cos0d,C 10
+ (9sin01 d<
(gb)
(8c)
and Y=
-;&dv”I V”l
D, = ; pCprdVf
where d is the cable diameter, and Co and Cr are the normal and friction drag coefficients, respectively. Both Cn and Cr are obtained experimentally. Zajac’” provides experimentally measured coefficient values for some typical marine cables. The normal drag coefficient Co is fairly constant for typical ranges of velocities. However, the friction drag coefficient Cr may vary substantially with towing speed and cable length, i.e. Cr may be a function of Reynolds’ number. Summing forces parallel to the tangential and normal axes, respectively, results in
.Y o [-(b)2sinQ .r
+ ri’cosG]d<
(gd)
To solve eqns (7a) and (7b) and determine T(s, t) and 8(s, t), it is necessary to know k, .?, Y and Y which are themselves complicated integral functions of 0(&s,t) as given by eqns @a)-(8d).
147
Towed marine cables in two dimensions A finite-difference method could solve the above nonlinear equations. However, a very large computational effort would be required because the solution involves integration in time and space and is iterative in nature. The technique would require a solution to be guessed or estimated for a future time step before the governing equation could be solved iteratively until a convergence criterion was satisfied. This process would need to be repeated for subsequent time steps. An approximate method (for example, Galerkin’s technique) could also be used to solve this set of nonlinear equations-requiring the definition of shape functions (which need to satisfy the geometric and ‘dynamic’ boundary conditions), substituting them into the governing equations and integrating them over the cable length to calculate the time-dependent coefficients.
3 SOLUTION ELEMENTS
USING
MULTIPLE
CABLE
and
i-1
li; =
-
C
[SfCOSQj
tijSiIlOj] Lj
+
-
(4fCOS8i
+ 8iSiIlOi)
2
j=l i-l
fi = C
[-$sinBj
+
djC0St9j]
Lj
j=l + ( -@sinei
+
8iCOS8i)
2
(9b)
where 0 and Li (i= 1,2, . . ., n) are the angle and length of the the cable and -Ci(*j) and ‘I;i( Pi) are, respectively, horizontal and vertical components of the velocity and acceleration at the element mid-point. For i= 1, the summation terms are zero. Hence the normal and tangential mid-point velocities can be obtained by substituting eqn (9a) into eqns (4b) and (4~) to give i-l
An approximate solution for the governing equation is obtained, instead, by subdividing the total cable length into n straight elements connected by pin-nodes-as shown in Figure 1. As a result, eqns (7a) and (7b) must be satisfied for each cable element, the integral operators in eqns (8a)-(8d) then reduce to summations of n terms and hence a system of non-linear ordinary differential equations must be solved. This spatial approximation reduces the partial differential eqns (7a) and (7b) to a set of coupled ordinary differential equations hence simplifying the problem. If a large number of elements is used (this is actually impractical for numerical reasons), the model of multiple elements reduces to an assemblage of chains. The cable element angle is a function of time only and the axial force within an element is a function of time and position. Note that the condition of continuous cable can be relaxed when using such an approximate solution, hence each cable element may be permitted its own geometric dimensions and material properties. This implies, for instance, that a variation of drag coefficient along the cable can be easily considered. 3.1 Geometric compatibility
for multiple cable elements
The cable elements and element nodes are numbered from 1 (top-most element) to n (bottom-most element) and from 0 (top-most node) to n (bottom-most node), respectively. Equations (8a)-(8d) can be rewritten for each cable element as
j=I 1 Fi=di 1
Xi = -
ej 2 sinQi + 2
ijsinejLj
[
$COSBi
[
+ 2
j=l
PjCOS*j&j
(94
V,,i = -.+OSOi
+ C
djLjSiIl(Qi
-
ej)
(104
j=l
and i-1 V”,i
=
&sinQj +
C4jLjCOS(Bi
-
dj)
+ di+f
(lob)
i=l
where i= 1, 2, . . ., n, and again the summation terms are zero for i= 1. These mid-point velocities actually represent average values over the cable element i, hence the normal and friction drag forces are not represented in this case by their average values. The error ensuing from this averaging tends to diminish as more elements are used.
3.2 Equations for multiple cable elements The cable elements are connected by ‘pins’, hence the bending moments are zero at the nodes. The nodal forces can be resolved into normal and tangential components (see Fig. 3). The normal components represent the geometric stiffnesses (‘spring’ forces) which tend to straighten the cable and reduce localised points of high curvature. The geometric stiffness term for the continuous cable-see eqn (7b)-is slightly modified to accommodate the finite-angle variation between adjacent elements. Hence the governing equations are rewritten by substituting eqns (9a) and (9b) into eqns (7a) and (7b) to give, for the tangential and normal terms, respectively,
T; = fqei)
(114
,
148
M. A. Vaz, M. H. Pate1
T,.,,C
node i-2
is calculated iteratively because it itself depends on the solution: first the Njs are ‘guessed’ [eqn (B3b)] by assuming that the angular accelerations are zero, then the t9,s are calculated [eqn (12)] and finally the Nis and &s are recalculated. Actually, for the examples studied, convergence is very quick and the overall results do not vary significantly with the number of iterations. In the standard cases, though, three iterations are used. Two initial conditions (i.e. the angles and angular velocities) are required for each cable element. In the case of a uniform cable, these are obtained from the stationary configuration at the initial ship speed, i.e. I \
/
node i+l
Fig. 3. Geometrical stiffness model.
e;(o) =
and 2Ni
COS-l
1 +;[H/VO]~
-;[H/V,I’
,
1 i
PCD,idi
Vn,i
I Vn,;I + WiCOS~i
i= 1,2,...,n
Li
= Pi
i-l
C [$Sin(B; -
fj)
and +
tijCOS(Bi - Sj)] Lj
&(O) = 0,
j=l
.. L;
(1lb)
+ Qi- + .Y0sin8, 2 where
g
[BjCOS(B;
-
Oj)
-
HjSin(f3; -
1,2,...,n
COS-l
t9j)] Lj
(13b)
{ /~-;vr:li12}
where V, is the final (constant) . . ‘> n.
i-l
PC,;
i=
where His the hydrodynamic constant of the cable and V. is the initial ship speed. Also note that the cable element angles will all converge, after a ‘very’ long time r, to e,(r) = 0 and
eicT) =
H(ei) = - ~ pCf,i~di V~i - wisinO,
-
(13a)
(13~) ship speed and i= 1, 2,
{ 4 RESULTS L
I
4.1 Input data
Note that Co,;, Cr,i, di, Wi, pc,i and pi (i= 1, 2, . . ., n) may be separately defined for each cable element. This is one merit of the technique which partly balances the drawback of it being a pin-jointed approximation to a continuous cable. The calculation of the cable-top force 7’shir(t) is given in Appendix B. 3.3 Solution methodology Equation (1 lb) is rewritten to allow numerical by a Runge-Kutta solver using the computer Matlab.14 Then di = $
I
+ 2
- &pCo,idi PI
1 I
i=
- fiJ Lj
Vn,iI V”,il + Tcos8; 1
[$Sill(Bi
-
0,)
+
solution package
- jlosinf?i
djcOs(ei - ej)]
j=l
1,2 ,...,a
(12)
Equation (12) represents a system of n non-linear ordinary differential equations. The geometric stiffness term 2NJL;
AND DISCUSSION
,
Results are presented for a light armoured cable (LA) and a heavy armoured cable (HA), with the properties described in Table 1. These cables contain, respectively, one and two layers of armour with a smooth outer surface. They were used in this analysis so as to enable a comparison with the experimental data presented by Hopland.’ Hopland’ obtained the hydrodynamic constants H of these cables by towing them at constant speeds and curve fitting the results using a least-squares method. He obtained 0.6173 radm/s and 0.7956 radm/s, respectively, for the LA and HA cables, giving measured normal drag coefficients of 2.054 and 1.649 respectively, using the expression H = (2w/pCDd)‘r2 and p= 1025 kg/m3. The LA and HA cables were tested for two conditions. 1. The LA cable was towed with the tow ship accelerating from O-9 to 2.5 knots (O-463 to 1.286 m/s) and decelerating from 2.4 to 0.9 knots (1.235 to 0,463 m/s).
149
Towed marine cables in two dimensions Table 1. Cable characteristics Cable type
Weight in air (N/m)
Weight in sea (N/m)
16.09 26.49
10.59 17.76
LA HA
Cable diameter
Cable length (m)
(m)
0.0264 0.0332
and the parabolic
360 300
profiles
are
&,(t) = V, + 2( I’, - V&/T
- (V, - V,,)(t/T)2
j;_,(t) = 2[1 - t/T](Vf - VO)/T
where V, and Vr are the ship’s initial and final speeds, respectively, and T is the elapsed time taken to reach the final speed. In both cases, tangential drag forces are neglected and the geometrical and material properties of the cable elements are assumed constant with time and along the length of the cable.
Ship Decelerating
Ship Accelerating
Fig. 4. Velocity
profiles.
2. The
HA cable was towed with the tow ship accelerating from 1.1 to 2.4 knots (0.566 to 1.235 m/s) and decelerating from 2.5 to 1.0 knots (l-286 to 0.514 m/s).
4.2 Sensitivity
= V, + (V, - V&/T
jr,(t) = (V, - VO)/T
studies
Initially, the robustness and performance of the governing equation and the solution technique are examined from three points of view-the effect of the number of elements used, the relative influence of the cable length and the likely errors induced by the precise ship acceleration and deceleration profiles used in the analysis. Numerical tests were carried out to investigate solution accuracy as a function of the number of elements used. They demonstrated that convergence was achieved with a relatively small number of elements. To illustrate this, Fig. 5 shows the evolution of the cable configuration for the LA tow cable with the ship accelerating linearly. The results using five (omitted from Fig. 5) and six elements virtually coincided and even two elements produced reasonable results. Hence, the cable discretization can be performed with five or six elements. The use of more elements does not necessarily imply longer computation times as convergence can deteriorate with the use of fewer elements. Another factor investigated was the importance of the cable length on cable dynamics. Figure 6 presents the evolution of the cable configuration for the towed LA cable as the towing ship is decelerating linearly. It can be
Table 2 shows the expected initial (ao) and final (of) cable angles, based on eqns (13a) and (13~). It also shows the cable-top tensions [Tship(O)=LOwsina and Tship(7)=Low sinar] for these conditions. The results converged to the initial and final stationary values, given by Table 2, for all simulations performed in this paper. The precise variation of the ship’s speed with time is not given by Hopland. The acceleration is said to be ‘abrupt’ and the ship’s final speed was reached within 45-60 s. Physical arguments would suggest that the acceleration (or deceleration) is higher at the beginning due to the power law variation of the ship’s resistance with speed. Therefore, in the absence of a detailed description of the ship’s motion, parabolic and linear velocity profiles are used for comparison (see Fig. 4). The analysis presented here can deal with any ship motion profile, although for the purposes of comparison with Hopland’s measurements, the tow ship’s linear velocity and acceleration profiles used in this simulation are as follows io(t)
(14a)
Table 2. Stationary cable angles Cable type
LA
wh
HA
b-ad)
Ship accelerating Ship decelerating TshqdO)/Tship(T)
1*105/0.470 0.489/l-105
l-140/0-620 O-597/1.200
3406/l 727 1791/3406
484113096 299714966
0’0
Ship accelerating Ship decelerating
(14b)
150
M. A. Vuz, M. H. Pate1
LA Cable
Ship Accelerates
from 0.463 to 1 266 m/s in 30 s
(a) HA Cable
- Ship
Accelerates
from 0.566 to 1 235 m/s in 120 s
I
0
50
100
150 2w Layback (m)
250
3w
:.
.parabolic
400
500
profile
350
im
0
Fig. 5. Cable configuration.
(b)
seen that the short cable reaches the final state more quickly. In order to assess the error potential arising from uncertainty in the ship’s velocity variation with time, the relative behaviour of the towed HA cable is considered for the linear and parabolic velocity profiles given by eqns (14a) and (14b). Figure 7(a)-(c) presents these results for a ship transient time of 120 s. The ship’s velocity profile influences the first 480 s, and after that, both configurations converge as the effect of the past history of the ship’s velocity decays away. Note that the parabolic ship velocity profile has a higher initial deceleration, hence the cable angles and cable-top tensions [Fig. (a) and (b)] have steeper initial gradients in this case. Figure 7(c) illustrates how the ship’s motion affects the evolution of the cable configuration. On the other hand, if the cable ship experiences a lower initial deceleration, the opposite behaviour is expected. This emphasises the importance of knowing the ship’s as this does modify the cable velocity profile, configuration.
HA Cable
MO
300 Time (s)
Ship Accelerates
600
from 0.566 to 1.235 m/s In 120 s
5000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .._ i.. .perebolic profile
30001 im
0
2m
400
300 Time (s)
I 600
500
(c) HA Cable 0
- Ship
Accelerates
from 0.566 to 1.235 m/s In 1M s
-50
hj
:
i linear profile
-250
-:---.
0
50
im
150
200
250
3w
: i0
Layback
Fig. 7. (a) Cable element angles; (b) cable-top configuration. 150 200 Layback (m)
250
3M)
Fig. 6. Cable Configuration.
350
tension;
(c) cable
Towed marine cables in two dimensions LA Cable - Ship Accelerates
:
from 0.463 to 1.286 m/s In 60 s
j
:
*
151
HA Cable - Ship Accelerates
horn 0.566 to 1.235 m/s in 60 s
experiment
___ 0
50
100
150 MO Layback (m)
250
300
I 350
0
with experimental
150
MO
250
300
:
Fig. 10. Cable configuration.
data
Figures 8-l 1 show the evolution of the cable configuration for the LA and HA cables when the tow ship accelerates and decelerates at a constant rate. As explained before, the main difficulty in quantitatively comparing the experimental and theoretical results is the uncertainty regarding the ship’s velocity profile. Furthermore, it should be noted that the experimental results do not always converge to the straight line configurations predicted by the stationary model. This may suggest the presence of surface currents, or be simply due to systematic experimental errors. It is also believed that the constant normal drag coefficients used in the theoretical model may not represent the actual dissipative forces. This issue should be considered further. By examining Figs S-11, it is seen that the cable transient is longer when the ship decelerates and also that the results compare better with the measurements LA Cable - Ship Decelerates
lM3
Layback
Fig. 8. Cable configuration.
4.3 Comparisons
50
from 1 235 to 0 463 m/s in 60 s
for this case. This may be because the normal drag forces tend to ‘lift’ the cable when the tow ship accelerates, but when it decelerates, the cable falls through the water under its own weight with a longer transient governed by its free-fall velocity in water.
5 CONCLUSIONS The governing equations for the transient behaviour of towed cables have been formulated. In the interest of computational efficiency, a discretised linked-element model has been used for solution of the governing equation. This approach has been selected partly because it leads to very rapid solution times using a modern mathematical manipulation package such as Matlab. l4 Sensitivity studies on the technique show that it requires use of a relatively small number of elements, but that the geometric stiffness term in the formulation HA Cable 0
- Ship
Decelerates
from 1 286 to 0 514 m/s in 60 s
-50
j Time iiterval =j 60 s
_&
j 0
xl
j 100
i
:
150 MO Layback (m)
,;
250
Fig. 9. Cable configuration.
; I
300
350
-350
0
50
100
150 200 Layback
250
Fig. 11. Cable configuration.
300
350
152
M. A. Vuz, M. H. Pate1
and the numerical values of the drag coefficient are important for accuracy. Despite the fact that precise ship velocity profile data were not available, the fullscale test data of Hopland’ were compared with predictions from the analysis and showed good agreement. This work suggests that the more advanced threedimensional steady-state and transient analysis which is increasingly required for cable towing and laying can be provided, in a computationally efficient manner, by a linked-element solution of the governing equation as presented here.
12. Seyed F. B. and Pate1 M. H., Mathematics of flexible risers including pressure and internal flow effects. Marine Strut. 5 (1992) 121-50. 13. Zajac, E. E., Dynamics and kinematics of the laying and recovery of submarine cable. Bell System Technical J., September (1957). 14. The Math Works Inc., Pro-matlab fbr VAX/VMS Computers. The Math Works Inc., 1991
APPENDIX A: EQUATIONS COMPATIBILITY The following
ACKNOWLEDGEMENTS This work was carried out with the support of the facilities of the Santa Fe Laboratory for Offshore Engineering, University College London-the support of Santa Fe Exploration (UK) Ltd is gratefully acknowledged. M. A. V. acknowledges the support of the Brazilian Council of Research (CNPq) for this work. Both authors would like to thank Dr J. A. Witz, Dr W. T. Pinto and Dr J. Feng for their invaluable suggestions and comments. The assistance of Mr T. Wilne during preparation of the paper is also acknowledged.
relationships
OF GEOMETRIC
are derived
from Figure
1.
X(s, t) = cosqs, t)
(Al4
g Y(s, t) = sinQ(s, t)
@lb)
; and
Therefore eqns (Ala)
(aY/&)* + (8X/&)* = 1. Differentiating and (Alb) with respect to t results in
&Xc,,
t) = -sino(s,
t) $S(.s, t)
(A24
and
REFERENCES 1. Hopland, S., Investigation of cable behaviour in water during laying of fiberoptic submarine cables. Proc. Znt. Wire and Cable Symp., 1993, pp. 734-39. 2. Schram, J. W. & Reyle, S. P., A three-dimensional dynamic analysis of a towed system. J. Hydronautics, 2
&
cosfqs, t) ;
Y(s, t) =
Now, integrating produces ~X(S,
5. Ablow, C. M. & Schechter, S., Numerical simulation of undersea cable dynamics. Ocean Engng, 10 (1983) 443-57. 6. Triantafyllou, M. S., The dynamics of translating cables. Report No. MITSG 85-385, MIT, Cambridge, Massachusetts, 1985. 7. Triantafyllou, M. S., Bliek, A., Burgess, J. & Shin, H., Mooring dynamics for offshore applications, Parts 1 and 2. Report No. MITSG 86-l/2, MIT, Cambridge, Massachusetts, 1986. 8. Milinazzo, F., Wilkie, M. & Latchman, S. A., An efficient algorithm for simulating the dynamics of towed cable systems. Ocean Engng, 14 (1987) 513-26. 9. Triantafyllou, M. S. & Chyrssostomidis, C., The dynamics of towed arrays. Report No. MITSG 89-325, Sea Grant College Program, MIT, Cambridge, Massachusetts, 1989. 10. Triantafyllou, M. S. & Hover, F., Cable dynamics for tethered underwater vehicles. Report No. MITSG 90-4, Sea Grant College Program, MIT, Cambridge, Massachusetts, 1990. 11. Young R. D. & Fowler 0. O., Mathematics of marine risers. The Energy Technology Conference and Exhibition, Houston, Texas, November 1978.
(A2b)
eqns (A2a) and (A2b) with respect to s
t) = - r sinQ(<, t) gO(<, t)dE +f(t) 0
(1968) 213-20. 3. Sanders, J. V., A three-dimensional dynamic analysis of a towed system. Ocean Engng, 9 (1982) 483-99. 4. Delmer, T. N., Stephens, T. C. & Coe, J. M., Numerical simulation of towed cables. Ocean Engng, 10 (1983) 119-32.
@, t)
(A34
and s
;
Y(s, t) =
Differentiating results in
.Icos~(~,t) &‘K.WE+ g(t) 0
eqns (A2a) and (A2b) with respect
Wb) to t
sine(s, t) -$t9C,, t)
(A44
and
&
coskys,t)
Y(s, t) =
$H(s, t)
i Wb)
Towed marine cables in two dimensions Integrating results in
eqns
(A4a)
and
(A4b)
with
respect
to s
153
T,,f= 0 = NJ. Substituting eqn (Bl) results in
this boundary
condition
L,_t i s < L,
T,(s, t) = (S - L,)H(&),
into
(B2)
The axial tension at the end of element n is T,,O= -L,Jf(B,). The equation of continuity of force at the node i-l is (see Fig. 3)
Ti_l,f = Ti,ocos(Bi - 8,_1) + Nisin(Bi - Bi_t),
Wa)
i=2,3,...,n
(B34
and
Ni-1 = Ti,osin(Bi - Bi_r) - Nicos(Bi - 8,-t),
2
$ Y(s, t) =
i = 2,3, . . , n
+-$(t) +cos(E, t)-$e(t, t)dE (A5b)
However,
substituting
Ti-1 (~,t) =
B:
CALCULATING
Applying
i= 1,2,...,n
Li-t
Ti,oCOS(Bi- Bi_l)
NiSiIl(Bi- f3_1),
Li-2 < S 5 Li-1,
i = 2,3,. . . , n
THE AXIAL
The axial tension Ti(s, t) in cable element i is calculated by integrating eqn (1 la) with respect to s to produce Ti(s, t) = H(8i)s + G,(t),
eqn (B3a) into eqn (Bl) produces
(s-L~~~)H(O~_,)+
+ APPENDIX TENSION
(B3b)
eqn (B4a) to element
Ti,o(s, t) =
- LiH(ei)
+ Ti+t,ocos(6’i+t - 0,)
+ Ni+tsin(ei+r
5 S < Lij
i= (Bl)
where &, = 0, Li = c&r Lj, Gi(t) is a constant of integration, and i= 1, 2, . . ., n. The constant G,(t) can be determined from the boundary condition at the bottom end of the cable by assuming that the force on node n is equal to zero (i.e.
Pa) i at s = Li_t produces
1,2,...,n-
- ei), 1
(B4b)
The nodal forces are obtained from the recurrence formulae given by eqns (B3b) and (B4b) and the cabletop force is given by Tship(t) = Jm
(B9
0141-1187(95)00012-7
ELSEVIER
Transient behaviour of towed marine cables in two dimensions M. A. Vaz* & M. H. Pate1 Santa Fe Laboratory for Ofshore
Engineering, Department of Mechanical Engineering, London WCIE 7JE, UK
University College London, Torrington Place,
(Received 4 May 1995; accepted 8 August 1995)
This paper presents a numerical solution for the transient motion of marine cables being towed from a cable ship which is changing speed. The cable ship is assumed to move rectilinearly, hence the cable configuration is two-dimensional. The solution methodology consists of dividing the cable into n straight elements with equilibrium relationships and geometric compatibility equations satisfied in each element. A system of n non-linear ordinary differential equations is derived from this and then solved by fourth- and fifth-order Runge-Kutta formulations with the dynamic axial tension calculated iteratively because it is itself dependent on the solution. Results are presented for the cable-top tension and element angles as functions of time and for transient cable geometries when the towing velocity is linearly or parabolically increased (or decreased). It is shown that the results from this analysis compare reasonably well with full-scale experimental data from Hopland (Proc. Int. Wire and Cable Symp., 1993, pp. 734-39).
towards the objective of developing a reliable prediction model for the transient behaviour of cables when towed or laid. Since there is some full-scale data available’ for transient behaviour of a towed cable, the first phase of this work has been to develop a general analysis for towed cables and for the cable laying process and to verify the towed cable aspect of the work by comparing its predictions with full-scale data from Hopland. This is the objective of this paper. The study of towed marine cables has been an area of interest to engineering research for the past three decades. Schram and Reyle2 investigated the threedimensional motion of a cable-body towing system. They employed the method of characteristics to solve the governing equations of a towed system using numerical interpolation. A transfer function, defined as the ratio of the resultant towed body’s amplitude of motion to the amplitude of the ship motion, was developed for various towing speeds and towline lengths, which decreased as the speed and length increased. However, their analysis did not include the effort of hydrodynamic mass. Sanders3 took Schram and Reyle’s solution for a three-dimensional towed system and developed a new formulation which could readily handle non-uniform properties, and could be easily implemented. Sanders disregarded inertia and bending terms and removed the requirement to have a first-order truncation error in the integration scheme. He employed
1 INTRODUCTION Marine cables, towed by ships or submarines, have been used in a variety of civil and military tasks. Civilian applications include acoustic surveying for sub-seabed hydrocarbon reservoirs and for geophysical mapping of ocean basins. Military applications are principally concerned with the detection of submarines. Figure 1 shows a schematic diagram of the towing arrangement. There is also a parallel activity, undertaken for many years now-the laying of undersea communication cables across virtually all of the world’s oceans. The installation of this world-wide communication infrastructure is increasingly becoming commercially competitive, with technical issues such as the speed and accuracy of cable laying yielding significant commercial benefits. In this climate, the feasibility of high-speed cable laying becomes important, as does an understanding of the transient behaviour of the cable laying process as the cable ship changes speed. Prediction of the transient behaviour of the cable as it is being laid requires a non-linear time-domain analysis-and once the analysis is available, its verification poses a difficult problem. The work described in this paper is the first phase *Now at Ocean Engineering Programme, Coppe, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. 143
144
M. A. Vaz, M. H. Pate1 xdt)
L-_
I
/LO
/
/
I
h
Fig. 1. Schematic diagram of the element model
a physical finite-difference formulation to solve the three-dimensional equations for cable dynamics. Delmer et al4 presented a numerical method for the dynamic simulation of towed cables. The system was constructed from a set of generic elements incorporating cable strands, knots, kinks, cable ends and winches. These elements were then organised into a general and flexible analysis procedure allowing consideration of complex systems. From here it was possible to solve the pre-defined towed system under a variety of load cases using a finite-element method. Ablow and Schechter’ devised a robust finite-difference approximation to the differential equations governing the behaviour of towed cables. The code was fully three-dimensional and yielded very good results when compared to actual measured data. Triantafyllou6 on the other hand, studied the dynamics of a translating catenary. The static and linearised dynamic governing equations were derived along the local tangential and normal directions. It was shown that two simpler equations could be derived and solved asymptotically for both small and large, horizontal or inclined cable sags. Triantafyllou also deduced that the most important parameters for travelling cables are the nondimensional speed of translation, the ratio of the elastic-to-catenary stiffness, the inclination angle, the cable’s weight-to-tension ratio and the added-mass-tophysical-mass ratio. this work up by Triantafyllou et a1.7 followed producing a detailed report on mooring dynamics for offshore applications, jointly funded by the Sea Grant Program and a consortium of companies. Investigations centred around the effect of principal non-linearities on predictions from a linear analysis of cable dynamics. The study confirmed the existing perception that the principal non-linearity for vibrating cables in water is fluid drag. The problem of the effect of drag force linearisation and of cable-sea bed interactions was addressed and a solution procedure was derived. A novel approach towards the problem of snap loading for a curved cable was also developed.
Milinazzo et al.* presented an efficient algorithm which simulated the three-dimensional behaviour of a towed cable. The formulation was based on an implicit, second-order finite-difference approximation of the equations of motion. The results obtained correlated well with the results of Ablow and Schechter’ and with experimental data. Triantafyllou and Chyrssostomidis’ developed a procedure for calculating the response of a towed array to a harmonic excitation applied at the upstream end. The fluid forces on the array were modelled using a slender body approximation and the cross-flow principle with the quadratic damping due to cross-flow separation replaced by an equivalent linear damping. This was determined via an iterative procedure. It was discovered that when separation drag is included, the array exhibits the behaviour of an over-damped system, responding only to low-frequency excitations. Triantafyllou and Hover” investigated the dynamic behaviour of cables employed as tethers for underwater vehicles. The dynamics of the vehicle are significantly affected by the presence of the tether, particularly in deep water when the mass and drag of the cable are comparable to, or much larger than, the drag and inertia of the vehicle. A numerical scheme was presented to analyse the dynamics of the tether. Although all of the above work has fully examined towed cable behaviour under a variety of ‘quasi-static’ and steady static dynamic conditions, the work has not addressed the verification of predictions from the theory sufficiently and the transient problem of the cable and its response to a change in velocity of the towing ship has not been given the attention it deserves. This issue is addressed in this paper in association with the work of Hopland,’ who measured the full-scale behaviour of submarine cables by studying the shape of towed cable lengths. It was found that a straight line configuration of the towed cable was a good approximation to steady-state towing, and experimentally determined hydrodynamic constants were presented for this steady state. Hopland also studied transient towed cable geometry when the ship’s speed is abruptly changed. His measurements are compared with predictions from the analysis formulated in this paper. This analysis resembles that of Sanders3 but is aimed at predicting the transient motions of and the tension in a towed cable when the surface ship changes speed. The analysis is restricted to two dimensions, but incorporates self weight, normal and tangential drag forces, axial tension, geometric stiffness and physical and added inertia. The transient cable configuration and axial tension are determined for the case of the towing ship accelerating or decelerating from one steady velocity to another through any arbitrary variation with time between the two states.
145
Towed marine cables in two dimensions
where ro(t) = -xo(t)i and rC(s, t) = X(s, t)i + Y(s, t)j. The ship and cable element position vectors-respectively ro(t) and rc(s, t)-are defined relative to the system oxy. The functions X(s, t) and Y(s, t) describe the cable configuration as seen from the ship. The velocity and acceleration vectors of a cable element-respectively V,.(s, t) and A,($, t)-are obtained by VA Fig. 2. Reference axis system. 2 FORMULATION EQUATIONS
4 = $
A&, 4 =
OF THE GOVERNING
The analysis is formulated on the basis that the towed cable is continuous, inextensible and flexible, that is, with negligible bending stiffness. For most applications, cable extensibility is negligible because of the relatively high axial stiffness and low axial tension. The set of equations governing the transient behaviour of towed marine cables comprises equilibrium and geometric compatibility relationships. These equations are derived in Sections 2.2 and 2.3, respectively, although prior to this, the kinematic characteristics of marine cables are derived in Section 2.1.
[r0(4
+
$ h(t) + rds, 41
where the ship’s speed and acceleration are respectively given by a,(t)/& = -6’x,(t)i/& and a2ro(t)/dt2 = -$xo(t)i/dt2. Furthermore, the cable element velocity ac(s, t)/dt and acceleration #r,(s, t)/dt2 vectors are given by gr,(+
t) = $X(S,
t)i + $ Y(s, t)j
and
grc(s, t) = $x(S,
t)i + $
Three Cartesian systems of reference, shown in Fig. 2, are adopted: OXY (the inertial system), oxy (the noninertial system which moves with the cable ship) and o’x’y’ (the non-inertial system, local axes). To begin with, the transformation rules between these three frames of reference are defined. Referring to Fig. 2, the following relations apply:
The cable element velocity vector VC(s, t) is obtained by substituting eqn (3a) into eqn (2b) and using eqn (1) to produce
Vt(s, f) =
coscqs, t) --gX,(l, +$s,
1
1
+ sine(s, t) g Y(s, t) [
(4b)
and
+
:x0(r) [
cose(s, t) $
(2a)
t)
[
(1)
where t and n are tangential and normal unit vectors referred to the local system o’x’y’; i, j and I, J are the unit vectors referred to oxy and OXY, respectively; 6’= 13(s, t) is the angle between i and t and s is the unstretched distance along the cable. Equation (1) allows vector parameters to be expressed in any one of the three systems of reference axes. The local system o’x’y’ changes with time and space since it is ‘attached’ to a cable element. The vector position of a cable element RC(s, t), when referred to the inertial system OXY, may be decomposed into (see Fig. 2)
(44
where
V,(s, t) = sine(s, t)
4 = ro(4 + rc(s, 4
(3b)
Y(s, t)j
VC(s, t) = Vt(s, t)t + Vn(s, t)n
2.1 The kinematics of marine cables
R&
WI
41
rds,
[
- $X(S,
Y(~, t)
1
1
t)
where V, and V,, are, respectively, the tangential and normal components of VC(s, t). The cable element acceleration vector A,(s, t) is given by substituting eqn (3b) into eqn (2~) and using eqn (1) to produce A,(s, t) = A,(s, t)t + A,($, t)n
(54
where
A,(~, t) = cose(s, t) -$~~(t)
[ + sin@,
t)
a2
+ dr2x(s, t) 1
$ Y(s, t)
(5b)
146
M. A. Vaz, M. H. Pate1
and A,($, t)sinQ(s, t) $x0(t)
+ cosO(s,t)
- -$X(s,
t)
1
$ Y (s, t)
where A, and A, are, respectively, normal components of AC(s, t).
the tangential
and
2.2 The equilibrium relationships These are obtained from the equilibrium of forces acting on the infinitesimal element in Fig. 2. Newton’s second law must be written in an inertial frame of reference (e.g. OXY). However, it is more convenient to transform the resulting equation into the systems oxy or o’x’y’. The marine cable being considered is taken to be fully immersed in water, i.e. no portion of cable above the sea water level (‘dry cable’) is considered in this analysis. Inertia forces, self weight, axial tension, hydrodynamic added inertia (due to normal acceleration only) and normal and tangential drag forces are considered. Furthermore, the cable is assumed to be inextensible and continuous and bending stiffness is neglected. In the absence of flexural stiffness, the axial tension cannot be negative, otherwise a singularity occurs and local buckling (loop formation) may take place. The forces (per unit length, except T) involved are: self weight (~9); normal drag (L&n); tangential drag (D,t); ‘effective’ tension (Tt); inertia (d’Alembert) force (p,A,) and added inertia [pC,(rd2/4)A,n], where pc is the cable’s physical mass per unit length, $21is the cable’s weight in sea water per unit length, p is the sea water’s density, C, is the added mass coefficient and d is the cable diameter. For conciseness, the parameters s and t will be omitted from the independent variables except when necessary to make the text clearer, i.e. differentiation with respect to s and t will be denoted by the inverted comma and overdot respectively. Note that the ‘true’ axial force T(s, t) is given by T(s, t) = T(s; t) - pg(-ird2/4) Y(s, t), where g is the acceleration due to gravity. This takes account of the effect of external hydrostatic pressure in producing a tension ‘look-alike’ term that acts in an analogous way to physical tension in the governing equation. The physical tension and hydrostatic-pressure-induced tension-like term are combined to define an ‘effective tension’-Young and Fowler” and Seyed and Patel” give detailed formulations of this term. The normal and tangential drag forces, per unit length, can be written as D, =
T’ + n,sinQ + ;pCprdV;
and (6b)
= pc [(x - ,Y,)cosB + YsinQ] (7a)
To’-
~pCDdC/,IV,l+wcosQ
= p[Ycos0+(&-%)sinB] (7b)
where
These equations are valid for any ship motion. Equations (7a) and (7b) reduce to Zajac’s equations for the particular case of a towed cable in a stationary configuration. There are apparently four unknown variables, T(,Y, t), Q(s, t), X(s, t) and Y(s, t). However, the last three variables must satisfy geometric conditions of compatibility. 2.3 Equations for geometric compatibility The cable configuration is governed by the interrelation of forces [eqns (7a) and (7b)] and by geometric compatibility which constrains the displacements. These latter compatibility equations are derived in Appendix A. At the cable’s upper point, i.e. at s= 0, the cable element velocity and acceleration vectors are given by V,(O, t) = -&(t)i and A,(O, t) = -&(t)i, hence f(t) = g(t) = 0 and df(t)/dt = dg(t)/dt = 0. In summary it may be written that s k = fi sin0 d< @a) .I0 J? =
1 (d)‘cos0 .I[
I; =
?cos0d,C 10
+ (9sin01 d<
(gb)
(8c)
and Y=
-;&dv”I V”l
D, = ; pCprdVf
where d is the cable diameter, and Co and Cr are the normal and friction drag coefficients, respectively. Both Cn and Cr are obtained experimentally. Zajac’” provides experimentally measured coefficient values for some typical marine cables. The normal drag coefficient Co is fairly constant for typical ranges of velocities. However, the friction drag coefficient Cr may vary substantially with towing speed and cable length, i.e. Cr may be a function of Reynolds’ number. Summing forces parallel to the tangential and normal axes, respectively, results in
.Y o [-(b)2sinQ .r
+ ri’cosG]d<
(gd)
To solve eqns (7a) and (7b) and determine T(s, t) and 8(s, t), it is necessary to know k, .?, Y and Y which are themselves complicated integral functions of 0(&s,t) as given by eqns @a)-(8d).
147
Towed marine cables in two dimensions A finite-difference method could solve the above nonlinear equations. However, a very large computational effort would be required because the solution involves integration in time and space and is iterative in nature. The technique would require a solution to be guessed or estimated for a future time step before the governing equation could be solved iteratively until a convergence criterion was satisfied. This process would need to be repeated for subsequent time steps. An approximate method (for example, Galerkin’s technique) could also be used to solve this set of nonlinear equations-requiring the definition of shape functions (which need to satisfy the geometric and ‘dynamic’ boundary conditions), substituting them into the governing equations and integrating them over the cable length to calculate the time-dependent coefficients.
3 SOLUTION ELEMENTS
USING
MULTIPLE
CABLE
and
i-1
li; =
-
C
[SfCOSQj
tijSiIlOj] Lj
+
-
(4fCOS8i
+ 8iSiIlOi)
2
j=l i-l
fi = C
[-$sinBj
+
djC0St9j]
Lj
j=l + ( -@sinei
+
8iCOS8i)
2
(9b)
where 0 and Li (i= 1,2, . . ., n) are the angle and length of the the cable and -Ci(*j) and ‘I;i( Pi) are, respectively, horizontal and vertical components of the velocity and acceleration at the element mid-point. For i= 1, the summation terms are zero. Hence the normal and tangential mid-point velocities can be obtained by substituting eqn (9a) into eqns (4b) and (4~) to give i-l
An approximate solution for the governing equation is obtained, instead, by subdividing the total cable length into n straight elements connected by pin-nodes-as shown in Figure 1. As a result, eqns (7a) and (7b) must be satisfied for each cable element, the integral operators in eqns (8a)-(8d) then reduce to summations of n terms and hence a system of non-linear ordinary differential equations must be solved. This spatial approximation reduces the partial differential eqns (7a) and (7b) to a set of coupled ordinary differential equations hence simplifying the problem. If a large number of elements is used (this is actually impractical for numerical reasons), the model of multiple elements reduces to an assemblage of chains. The cable element angle is a function of time only and the axial force within an element is a function of time and position. Note that the condition of continuous cable can be relaxed when using such an approximate solution, hence each cable element may be permitted its own geometric dimensions and material properties. This implies, for instance, that a variation of drag coefficient along the cable can be easily considered. 3.1 Geometric compatibility
for multiple cable elements
The cable elements and element nodes are numbered from 1 (top-most element) to n (bottom-most element) and from 0 (top-most node) to n (bottom-most node), respectively. Equations (8a)-(8d) can be rewritten for each cable element as
j=I 1 Fi=di 1
Xi = -
ej 2 sinQi + 2
ijsinejLj
[
$COSBi
[
+ 2
j=l
PjCOS*j&j
(94
V,,i = -.+OSOi
+ C
djLjSiIl(Qi
-
ej)
(104
j=l
and i-1 V”,i
=
&sinQj +
C4jLjCOS(Bi
-
dj)
+ di+f
(lob)
i=l
where i= 1, 2, . . ., n, and again the summation terms are zero for i= 1. These mid-point velocities actually represent average values over the cable element i, hence the normal and friction drag forces are not represented in this case by their average values. The error ensuing from this averaging tends to diminish as more elements are used.
3.2 Equations for multiple cable elements The cable elements are connected by ‘pins’, hence the bending moments are zero at the nodes. The nodal forces can be resolved into normal and tangential components (see Fig. 3). The normal components represent the geometric stiffnesses (‘spring’ forces) which tend to straighten the cable and reduce localised points of high curvature. The geometric stiffness term for the continuous cable-see eqn (7b)-is slightly modified to accommodate the finite-angle variation between adjacent elements. Hence the governing equations are rewritten by substituting eqns (9a) and (9b) into eqns (7a) and (7b) to give, for the tangential and normal terms, respectively,
T; = fqei)
(114
,
148
M. A. Vaz, M. H. Pate1
T,.,,C
node i-2
is calculated iteratively because it itself depends on the solution: first the Njs are ‘guessed’ [eqn (B3b)] by assuming that the angular accelerations are zero, then the t9,s are calculated [eqn (12)] and finally the Nis and &s are recalculated. Actually, for the examples studied, convergence is very quick and the overall results do not vary significantly with the number of iterations. In the standard cases, though, three iterations are used. Two initial conditions (i.e. the angles and angular velocities) are required for each cable element. In the case of a uniform cable, these are obtained from the stationary configuration at the initial ship speed, i.e. I \
/
node i+l
Fig. 3. Geometrical stiffness model.
e;(o) =
and 2Ni
COS-l
1 +;[H/VO]~
-;[H/V,I’
,
1 i
PCD,idi
Vn,i
I Vn,;I + WiCOS~i
i= 1,2,...,n
Li
= Pi
i-l
C [$Sin(B; -
fj)
and +
tijCOS(Bi - Sj)] Lj
&(O) = 0,
j=l
.. L;
(1lb)
+ Qi- + .Y0sin8, 2 where
g
[BjCOS(B;
-
Oj)
-
HjSin(f3; -
1,2,...,n
COS-l
t9j)] Lj
(13b)
{ /~-;vr:li12}
where V, is the final (constant) . . ‘> n.
i-l
PC,;
i=
where His the hydrodynamic constant of the cable and V. is the initial ship speed. Also note that the cable element angles will all converge, after a ‘very’ long time r, to e,(r) = 0 and
eicT) =
H(ei) = - ~ pCf,i~di V~i - wisinO,
-
(13a)
(13~) ship speed and i= 1, 2,
{ 4 RESULTS L
I
4.1 Input data
Note that Co,;, Cr,i, di, Wi, pc,i and pi (i= 1, 2, . . ., n) may be separately defined for each cable element. This is one merit of the technique which partly balances the drawback of it being a pin-jointed approximation to a continuous cable. The calculation of the cable-top force 7’shir(t) is given in Appendix B. 3.3 Solution methodology Equation (1 lb) is rewritten to allow numerical by a Runge-Kutta solver using the computer Matlab.14 Then di = $
I
+ 2
- &pCo,idi PI
1 I
i=
- fiJ Lj
Vn,iI V”,il + Tcos8; 1
[$Sill(Bi
-
0,)
+
solution package
- jlosinf?i
djcOs(ei - ej)]
j=l
1,2 ,...,a
(12)
Equation (12) represents a system of n non-linear ordinary differential equations. The geometric stiffness term 2NJL;
AND DISCUSSION
,
Results are presented for a light armoured cable (LA) and a heavy armoured cable (HA), with the properties described in Table 1. These cables contain, respectively, one and two layers of armour with a smooth outer surface. They were used in this analysis so as to enable a comparison with the experimental data presented by Hopland.’ Hopland’ obtained the hydrodynamic constants H of these cables by towing them at constant speeds and curve fitting the results using a least-squares method. He obtained 0.6173 radm/s and 0.7956 radm/s, respectively, for the LA and HA cables, giving measured normal drag coefficients of 2.054 and 1.649 respectively, using the expression H = (2w/pCDd)‘r2 and p= 1025 kg/m3. The LA and HA cables were tested for two conditions. 1. The LA cable was towed with the tow ship accelerating from O-9 to 2.5 knots (O-463 to 1.286 m/s) and decelerating from 2.4 to 0.9 knots (1.235 to 0,463 m/s).
149
Towed marine cables in two dimensions Table 1. Cable characteristics Cable type
Weight in air (N/m)
Weight in sea (N/m)
16.09 26.49
10.59 17.76
LA HA
Cable diameter
Cable length (m)
(m)
0.0264 0.0332
and the parabolic
360 300
profiles
are
&,(t) = V, + 2( I’, - V&/T
- (V, - V,,)(t/T)2
j;_,(t) = 2[1 - t/T](Vf - VO)/T
where V, and Vr are the ship’s initial and final speeds, respectively, and T is the elapsed time taken to reach the final speed. In both cases, tangential drag forces are neglected and the geometrical and material properties of the cable elements are assumed constant with time and along the length of the cable.
Ship Decelerating
Ship Accelerating
Fig. 4. Velocity
profiles.
2. The
HA cable was towed with the tow ship accelerating from 1.1 to 2.4 knots (0.566 to 1.235 m/s) and decelerating from 2.5 to 1.0 knots (l-286 to 0.514 m/s).
4.2 Sensitivity
= V, + (V, - V&/T
jr,(t) = (V, - VO)/T
studies
Initially, the robustness and performance of the governing equation and the solution technique are examined from three points of view-the effect of the number of elements used, the relative influence of the cable length and the likely errors induced by the precise ship acceleration and deceleration profiles used in the analysis. Numerical tests were carried out to investigate solution accuracy as a function of the number of elements used. They demonstrated that convergence was achieved with a relatively small number of elements. To illustrate this, Fig. 5 shows the evolution of the cable configuration for the LA tow cable with the ship accelerating linearly. The results using five (omitted from Fig. 5) and six elements virtually coincided and even two elements produced reasonable results. Hence, the cable discretization can be performed with five or six elements. The use of more elements does not necessarily imply longer computation times as convergence can deteriorate with the use of fewer elements. Another factor investigated was the importance of the cable length on cable dynamics. Figure 6 presents the evolution of the cable configuration for the towed LA cable as the towing ship is decelerating linearly. It can be
Table 2 shows the expected initial (ao) and final (of) cable angles, based on eqns (13a) and (13~). It also shows the cable-top tensions [Tship(O)=LOwsina and Tship(7)=Low sinar] for these conditions. The results converged to the initial and final stationary values, given by Table 2, for all simulations performed in this paper. The precise variation of the ship’s speed with time is not given by Hopland. The acceleration is said to be ‘abrupt’ and the ship’s final speed was reached within 45-60 s. Physical arguments would suggest that the acceleration (or deceleration) is higher at the beginning due to the power law variation of the ship’s resistance with speed. Therefore, in the absence of a detailed description of the ship’s motion, parabolic and linear velocity profiles are used for comparison (see Fig. 4). The analysis presented here can deal with any ship motion profile, although for the purposes of comparison with Hopland’s measurements, the tow ship’s linear velocity and acceleration profiles used in this simulation are as follows io(t)
(14a)
Table 2. Stationary cable angles Cable type
LA
wh
HA
b-ad)
Ship accelerating Ship decelerating TshqdO)/Tship(T)
1*105/0.470 0.489/l-105
l-140/0-620 O-597/1.200
3406/l 727 1791/3406
484113096 299714966
0’0
Ship accelerating Ship decelerating
(14b)
150
M. A. Vuz, M. H. Pate1
LA Cable
Ship Accelerates
from 0.463 to 1 266 m/s in 30 s
(a) HA Cable
- Ship
Accelerates
from 0.566 to 1 235 m/s in 120 s
I
0
50
100
150 2w Layback (m)
250
3w
:.
.parabolic
400
500
profile
350
im
0
Fig. 5. Cable configuration.
(b)
seen that the short cable reaches the final state more quickly. In order to assess the error potential arising from uncertainty in the ship’s velocity variation with time, the relative behaviour of the towed HA cable is considered for the linear and parabolic velocity profiles given by eqns (14a) and (14b). Figure 7(a)-(c) presents these results for a ship transient time of 120 s. The ship’s velocity profile influences the first 480 s, and after that, both configurations converge as the effect of the past history of the ship’s velocity decays away. Note that the parabolic ship velocity profile has a higher initial deceleration, hence the cable angles and cable-top tensions [Fig. (a) and (b)] have steeper initial gradients in this case. Figure 7(c) illustrates how the ship’s motion affects the evolution of the cable configuration. On the other hand, if the cable ship experiences a lower initial deceleration, the opposite behaviour is expected. This emphasises the importance of knowing the ship’s as this does modify the cable velocity profile, configuration.
HA Cable
MO
300 Time (s)
Ship Accelerates
600
from 0.566 to 1.235 m/s In 120 s
5000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .._ i.. .perebolic profile
30001 im
0
2m
400
300 Time (s)
I 600
500
(c) HA Cable 0
- Ship
Accelerates
from 0.566 to 1.235 m/s In 1M s
-50
hj
:
i linear profile
-250
-:---.
0
50
im
150
200
250
3w
: i0
Layback
Fig. 7. (a) Cable element angles; (b) cable-top configuration. 150 200 Layback (m)
250
3M)
Fig. 6. Cable Configuration.
350
tension;
(c) cable
Towed marine cables in two dimensions LA Cable - Ship Accelerates
:
from 0.463 to 1.286 m/s In 60 s
j
:
*
151
HA Cable - Ship Accelerates
horn 0.566 to 1.235 m/s in 60 s
experiment
___ 0
50
100
150 MO Layback (m)
250
300
I 350
0
with experimental
150
MO
250
300
:
Fig. 10. Cable configuration.
data
Figures 8-l 1 show the evolution of the cable configuration for the LA and HA cables when the tow ship accelerates and decelerates at a constant rate. As explained before, the main difficulty in quantitatively comparing the experimental and theoretical results is the uncertainty regarding the ship’s velocity profile. Furthermore, it should be noted that the experimental results do not always converge to the straight line configurations predicted by the stationary model. This may suggest the presence of surface currents, or be simply due to systematic experimental errors. It is also believed that the constant normal drag coefficients used in the theoretical model may not represent the actual dissipative forces. This issue should be considered further. By examining Figs S-11, it is seen that the cable transient is longer when the ship decelerates and also that the results compare better with the measurements LA Cable - Ship Decelerates
lM3
Layback
Fig. 8. Cable configuration.
4.3 Comparisons
50
from 1 235 to 0 463 m/s in 60 s
for this case. This may be because the normal drag forces tend to ‘lift’ the cable when the tow ship accelerates, but when it decelerates, the cable falls through the water under its own weight with a longer transient governed by its free-fall velocity in water.
5 CONCLUSIONS The governing equations for the transient behaviour of towed cables have been formulated. In the interest of computational efficiency, a discretised linked-element model has been used for solution of the governing equation. This approach has been selected partly because it leads to very rapid solution times using a modern mathematical manipulation package such as Matlab. l4 Sensitivity studies on the technique show that it requires use of a relatively small number of elements, but that the geometric stiffness term in the formulation HA Cable 0
- Ship
Decelerates
from 1 286 to 0 514 m/s in 60 s
-50
j Time iiterval =j 60 s
_&
j 0
xl
j 100
i
:
150 MO Layback (m)
,;
250
Fig. 9. Cable configuration.
; I
300
350
-350
0
50
100
150 200 Layback
250
Fig. 11. Cable configuration.
300
350
152
M. A. Vuz, M. H. Pate1
and the numerical values of the drag coefficient are important for accuracy. Despite the fact that precise ship velocity profile data were not available, the fullscale test data of Hopland’ were compared with predictions from the analysis and showed good agreement. This work suggests that the more advanced threedimensional steady-state and transient analysis which is increasingly required for cable towing and laying can be provided, in a computationally efficient manner, by a linked-element solution of the governing equation as presented here.
12. Seyed F. B. and Pate1 M. H., Mathematics of flexible risers including pressure and internal flow effects. Marine Strut. 5 (1992) 121-50. 13. Zajac, E. E., Dynamics and kinematics of the laying and recovery of submarine cable. Bell System Technical J., September (1957). 14. The Math Works Inc., Pro-matlab fbr VAX/VMS Computers. The Math Works Inc., 1991
APPENDIX A: EQUATIONS COMPATIBILITY The following
ACKNOWLEDGEMENTS This work was carried out with the support of the facilities of the Santa Fe Laboratory for Offshore Engineering, University College London-the support of Santa Fe Exploration (UK) Ltd is gratefully acknowledged. M. A. V. acknowledges the support of the Brazilian Council of Research (CNPq) for this work. Both authors would like to thank Dr J. A. Witz, Dr W. T. Pinto and Dr J. Feng for their invaluable suggestions and comments. The assistance of Mr T. Wilne during preparation of the paper is also acknowledged.
relationships
OF GEOMETRIC
are derived
from Figure
1.
X(s, t) = cosqs, t)
(Al4
g Y(s, t) = sinQ(s, t)
@lb)
; and
Therefore eqns (Ala)
(aY/&)* + (8X/&)* = 1. Differentiating and (Alb) with respect to t results in
&Xc,,
t) = -sino(s,
t) $S(.s, t)
(A24
and
REFERENCES 1. Hopland, S., Investigation of cable behaviour in water during laying of fiberoptic submarine cables. Proc. Znt. Wire and Cable Symp., 1993, pp. 734-39. 2. Schram, J. W. & Reyle, S. P., A three-dimensional dynamic analysis of a towed system. J. Hydronautics, 2
&
cosfqs, t) ;
Y(s, t) =
Now, integrating produces ~X(S,
5. Ablow, C. M. & Schechter, S., Numerical simulation of undersea cable dynamics. Ocean Engng, 10 (1983) 443-57. 6. Triantafyllou, M. S., The dynamics of translating cables. Report No. MITSG 85-385, MIT, Cambridge, Massachusetts, 1985. 7. Triantafyllou, M. S., Bliek, A., Burgess, J. & Shin, H., Mooring dynamics for offshore applications, Parts 1 and 2. Report No. MITSG 86-l/2, MIT, Cambridge, Massachusetts, 1986. 8. Milinazzo, F., Wilkie, M. & Latchman, S. A., An efficient algorithm for simulating the dynamics of towed cable systems. Ocean Engng, 14 (1987) 513-26. 9. Triantafyllou, M. S. & Chyrssostomidis, C., The dynamics of towed arrays. Report No. MITSG 89-325, Sea Grant College Program, MIT, Cambridge, Massachusetts, 1989. 10. Triantafyllou, M. S. & Hover, F., Cable dynamics for tethered underwater vehicles. Report No. MITSG 90-4, Sea Grant College Program, MIT, Cambridge, Massachusetts, 1990. 11. Young R. D. & Fowler 0. O., Mathematics of marine risers. The Energy Technology Conference and Exhibition, Houston, Texas, November 1978.
(A2b)
eqns (A2a) and (A2b) with respect to s
t) = - r sinQ(<, t) gO(<, t)dE +f(t) 0
(1968) 213-20. 3. Sanders, J. V., A three-dimensional dynamic analysis of a towed system. Ocean Engng, 9 (1982) 483-99. 4. Delmer, T. N., Stephens, T. C. & Coe, J. M., Numerical simulation of towed cables. Ocean Engng, 10 (1983) 119-32.
@, t)
(A34
and s
;
Y(s, t) =
Differentiating results in
.Icos~(~,t) &‘K.WE+ g(t) 0
eqns (A2a) and (A2b) with respect
Wb) to t
sine(s, t) -$t9C,, t)
(A44
and
&
coskys,t)
Y(s, t) =
$H(s, t)
i Wb)
Towed marine cables in two dimensions Integrating results in
eqns
(A4a)
and
(A4b)
with
respect
to s
153
T,,f= 0 = NJ. Substituting eqn (Bl) results in
this boundary
condition
L,_t i s < L,
T,(s, t) = (S - L,)H(&),
into
(B2)
The axial tension at the end of element n is T,,O= -L,Jf(B,). The equation of continuity of force at the node i-l is (see Fig. 3)
Ti_l,f = Ti,ocos(Bi - 8,_1) + Nisin(Bi - Bi_t),
Wa)
i=2,3,...,n
(B34
and
Ni-1 = Ti,osin(Bi - Bi_r) - Nicos(Bi - 8,-t),
2
$ Y(s, t) =
i = 2,3, . . , n
+-$(t) +cos(E, t)-$e(t, t)dE (A5b)
However,
substituting
Ti-1 (~,t) =
B:
CALCULATING
Applying
i= 1,2,...,n
Li-t
Ti,oCOS(Bi- Bi_l)
NiSiIl(Bi- f3_1),
Li-2 < S 5 Li-1,
i = 2,3,. . . , n
THE AXIAL
The axial tension Ti(s, t) in cable element i is calculated by integrating eqn (1 la) with respect to s to produce Ti(s, t) = H(8i)s + G,(t),
eqn (B3a) into eqn (Bl) produces
(s-L~~~)H(O~_,)+
+ APPENDIX TENSION
(B3b)
eqn (B4a) to element
Ti,o(s, t) =
- LiH(ei)
+ Ti+t,ocos(6’i+t - 0,)
+ Ni+tsin(ei+r
5 S < Lij
i= (Bl)
where &, = 0, Li = c&r Lj, Gi(t) is a constant of integration, and i= 1, 2, . . ., n. The constant G,(t) can be determined from the boundary condition at the bottom end of the cable by assuming that the force on node n is equal to zero (i.e.
Pa) i at s = Li_t produces
1,2,...,n-
- ei), 1
(B4b)
The nodal forces are obtained from the recurrence formulae given by eqns (B3b) and (B4b) and the cabletop force is given by Tship(t) = Jm
(B9