Dynamic motion and tension of marine cables being laid during velocity change of mother vessels

China Ocean Eng.,Vol. 27, No. 5, pp. 629 – 644 © 2013 Chinese Ocean Engineering Society and Springer-Verlang Berlin Heidelberg DOI 10.1007/s13344-013-0053-5, ISSN 0890-5487

Dynamic Motion and Tension of Marine Cables Being Laid During Velocity

Change of Mother Vessels*

XU Xue-song (徐雪松)1, WANG Sheng-wei (王盛炜) and LIAN Lian (连琏)

School of Naval Architecture, Ocean and Civil Engineering, Shanghai JiaoTong University,

Shanghai 200240, China

(Received 4 June 2012; received revised form 11 October 2012; accepted 25 January 2013)

ABSTRACT

Flexible segment model (FSM) is adopted for the dynamics calculation of marine cable being laid. In FSM, the cable

is divided into a number of flexible segments, and nonlinear governing equations are listed according to the moment

equilibriums of the segments. Linearization iteration scheme is employed to obtain the numerical solution for the

governing equations. For the cable being laid, the payout rate is calculated from the velocities of all segments. The

numerical results are shown of the dynamic motion and tension of marine cables being laid during velocity change of the

mother vessels.

Key words: dynamics calculation; discretization method; marine cable; cable laying; cable payout rate

1. Introduction

In the current information age, more and more marine telecommunication cables have been laid to

connect two offshore areas or islands. Laying cable is a common task in ocean engineering. In the

cable laying operation, the velocity of mother vessel may change according to the task requirement.

The most typical examples of velocity change of the mother vessel are: (1) the mother vessel starts

from the initial static status; (2) the mother vessel decelerates from a forwarding velocity to zero

velocity. During the Velocity Change of the mother vessel, the cable configuration and tension may

vary all the time. It will take a long time for the cable to transform in different statuses. In the

meanwhile, the payout rate should be set to the value by which the lower end of cable freely drops to

the seafloor without tension. In the past decades, there have been a lot of researches for the dynamics calculation of marine

cables (Patel, 1995; Howell et al., 1992; Park et al., 2003; Grosenbaugh, 2007). Howell et al. (1992)

formulated the three-dimensional nonlinear equations of motion for a submerged cable. The derivation

of the equations can be viewed as a general formulation in that all forces and moments are equated for

an incremental cable segment. The numerical techniques had been provided to analyze the nonlinear

dynamics of low-tension cables and chains. Park et al. (2003) also made a numerical and experimental

study on the dynamics of a towed low tension cable. An implicit finite difference algorithm is

* The research was jointly supported by the National Natural Science Foundation of China (Grant Nos. 51009092 and 51279107),

Doctoral Foundation of Education Ministry of China (Grant No. 20090073120013) and Scientific Research Foundation of State Education Ministry for the Returned Overseas Chinese Scholars.

1 Corresponding author. E-mail: [email protected]

XU Xue-song et al./China Ocean Eng., 27(5), 2013, 629 – 644

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employed for solving the three-dimensional cable equations. Fluid and geometric nonlinearities are

solved by NewtonRaphson iteration.

Patel et al. (1995) presented a numerical model for the transient behavior of marine cables during

laying operation. The solution methodology consists of dividing the cable into a number of straight

elements with equilibrium relationships and geometric compatibility equations satisfied for each

element. Vaz et al. (2000) presented a 3D steady-state formulation for elastic segmented marine cables

during installation in sheared currents. The time independent set of first-order nonlinear ordinary

differential equations are solved by a Runge-Kutta integrator.

In the laying operation, the cable tension may vary in a large range if the cable is set at the

negative slack (Yoshizawa et al., 1983). The payout rate should be set at a desirable value to avoid

negative slack cable. Jung et al. (2001) calculated the slack from a comparison of the cable payout rate

and the ship ground speed in accordance with laying conditions, and the speed controller of the cable

engine based on an H∞ servo control is designed for adjusting the cable engine in order to lay a desired

amount of slack.

In this paper, a flexible segment model (FSM) based the method of Xu et al. (2012) is employed

to calculate the dynamic motion and tension of marine laying cables during Velocity Change of mother

vessels. In the FSM-based method, a cable is discretized into a series of flexible segments. For each

flexible segment, its deflection feature and external forces are analyzed independently. For the whole

cable, the nonlinear governing equations are listed according to the moment equilibrium at nodes. To

solve the nonlinear equations, a linearization iteration scheme is provided. The linearization iteration

scheme can avoid the widely used NewtonRapson iteration scheme in which the calculation stability

is influenced by the initial points.

The remaining sections will be organized as follows: Section 2 details the FSM of cable being laid,

and analyzes the deformation, displacement, velocity and acceleration of segment based on the FSM;

Section 3 lists the governing equations of cable motion; Section 4 presents the linearization iteration

scheme for the nonlinear governing equations; Section 5 comes up the equation to calculate the payout

rate for slack cable laying; Section 6 and Section 7 show the numerical results of dynamic motion and

tension in Velocity Change of mother vessel; Section 8 sums up the paper.

2. Flexible Segment Model of Cable Being Laid

As shown in Fig. 1a, the marine cable is laid out from the mother vessel, and its lower end drops

to seafloor accordingly. FSM of the cable being laid is shown in Figs. 1b and 1c. In Fig. 1b, the cable is

discretized into n flexible segments, marked as S1, S2, …, Sn. At the ends of these flexible segments,

there are n+1 nodes N1, … , Nn, Nn+1. N1 is the upper end, connected with the mother vessel. Nn+1 is the

cable lower end, linked with the seafloor, i.e. touch-down point (TDP). Between Ni and Ni+1, there is a

midpoint Ci on Si.

The cable being laid deforms under external loads. The deformation includes lateral deflection

and axial elongation respectively caused by bending moment and axial tension. As the axial elongation

is much smaller than the lateral deflection, axial elongation is neglected as Patel and Vaz did in 1995


631

(Patel and Vaz, 1995).

Fig. 1. Flexible segment model of the cable being laid: (a) marine cable being laid; (b) segments and nodes of the model;

(c) deformation of Si.

To illustrate the deformation, the segment Si is magnified, as shown in Fig. 1c, although the real

deformation of each segment is very small. There are three hypotheses for each segment:

(1) The bending moment and internal tension are uniform in each segment;

(2) The external forces lump at the midpoint of each segment;

(3) The deformation of each segment is very small, although the deformation of the whole cable

may be large.

The 2-D global and local coordinate systems are shown in Figs. 1b and 1c. The global coordinate

system XY is set as X axis parallel to the horizontal plane, Y axis pointing to the sea floor, and its origin

is set at the initial position of the cable upper end. If the upper end moves, it will leave the origin.

At each node Ni, there is a local moving coordinate system niti. It is set as axis ti axial, axis ni

perpendicular to ti, the origin always at Ni. In the same way, at the midpoint Ci, there is a local

coordinate system nc,itc,i.

2.1 Deformation Caused by Bending Moment

To illuminate the deflection feature of segments, the following parameters are denominated, as

shown in Fig. 1c:

Segment deflecting angle i is the angle from the tangent at Ni to the tangent at Ni+1.

Segment deflection in is the deflection from the tangent at Ni to the point Ni+1.

Segment angle i is the angle from Y axis to the line NiNi+1;

Assuming that Mi is the bending moment of Segment Si, the segment deflection angle i

between the tangents at Ni and Ni+1, as shown in Fig. 1c, can be calculated by

ii i

Ml

EI , (1)


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where li is the length of Si, E is Young’s modulus, and I is inertia moment of the cross-section area.

The segment deflection δni from the tangent at Ni to the point Ni+1, as shown in Fig. 1c, can be

calculated by

2

2 2i i

i i i

Mn l l

EI

. (2)

The deflecting angle from the tangent at Ni to the tangent at Ci can be calculated by

, 2 2i i i

c i

M l

EI

. (3)

The deflection from the tangent at Ni to the point Ci can be calculated by 2

, 2 2 4 2i i i i

c i

M l ln

EI

. (4)

The segment angle i from Y axis to the line NiNi+1 can be expressed by

1

01 2

ii

i jj

, (5)

where 0 is the angle between Y axis and the tangent at N1.

2.2 Displacement Calculation

The rotational transform from niti to ni+1ti+1 can be denoted by:

1 1

cos sinˆ ˆ ˆˆ ˆ ˆ

sin cosi i

i i i i i i i

i i

n t n t P n t , (6)

where iP is the variable matrix about i , ˆin and it are unit vectors of axes ni, ti, 1

ˆin and

1it are unit vectors of axes ni+1, ti+1.

The coordinate transformation between Ni and Ni+1 can be denoted as:

1

1

ˆ2

i i ii i

i i

x xl

y y

P t . (7)

From Eq. (3), the rotational transform between nc,itc,i and niti can be calculated by

, ,ˆ ˆˆ ˆ

2i

c i c i i i

n t P n t . (8)

From Eq. (4), the coordinate transformation between Ci and Ni can be calculated by

,

,

ˆ2 4

c i i i ii

c i i

x x ly y

P t . (9)

The first and second derivatives of ( )iP can be written as:

cos sin sin cos

sin cos cos sini i i i i i

i

i i i i i i

P

; (10)


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2 2

2 2

1 2

sin cos

cos sin

sin coscos sin

cos sin sin cos

,

i i i i

i

i i i i

i ii i i i

i

i ii i i i

i i i

P

P P

(11)

where 1 iP and 2 iP are variable matrixes about i . In the remaining part, P matrixes are all

variable matrixes also.

2.3 Velocity Calculation

From Eq. (7), the relationship between velocities of Ni and Ni+1 can be expressed by

1

1

ˆ ˆ ˆ2 2 2

i i i i ii i i i i i

i i

x xl l l

y y

P t P t P t , (12)

where il is the time derivative of il . 1l

is the payout rate of cable, while for 2i , 0il .

i

t is the time derivative of it , and can be obtained from 1it :

1 1 1 1 1 1ˆ ˆ ˆ ˆ( ) ( ) ( )i i i i i i i

t P t P t P t . (13)

But for N1,

0 01 0

0 0

0 cosˆ ( )1 sin

t P . (14)

In the same way, from Eq. (9), the relationship between velocities of Ci and Ni+1 can be expressed

by

,

,

ˆ ˆ ˆ2 4 2 4 2 4

c i i i i i i i ii i i

c i i

x x l l ly y

P t P t P t

. (15)

2.4 Acceleration Calculation

Based on Eq. (15), the second derivative of 1t can be calculated by

2 200 0 0 0 0 0 0 0

1 02 200 0 0 0 0 0 0 0

coscos sin cos sinˆsinsin cos sin cos

t

. (16)

Set

T

0 1 n Θ , (17)

Eq. (16) can be simplified by

1 1,1 2,1ˆ t K K Θ , (18)

where 1,1K and 2,1K are variable matrixes about Θ . In the remaining part, the following K matrixes

are all variable matrixes also.


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From Eq. (14), the second derivative of 1it can be obtained by

1

1 2

3 4

1, 1 2, 1

ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ, , ,

i i i i i i i

i i i i i i i i i

i i i i i i

i i

i

t P t P t P t

P t P t P t P t

P t P t Θ P t

K K Θ

(19)

Based on Eq. (13), the acceleration of Ni+1 can be expressed by

1

1

ˆ ˆ ˆ2 2 2

ˆ ˆ ˆ ˆ ˆ2 2 22 2 2 2 2

i i i i ii i i i i i

i i

i i i i i ii i i i i i i i i i

i

x xl l l

y y

xl l l l l

y

P t P t P t

P t P t P t P t P t

1 4 1, 2,

3, 1 4, 1

ˆ2

ˆ ˆ ˆ ˆ2 2 22 2 2 2

ˆˆ , ,

2 2 2 2

2

ii i

i i i i ii i i i i i i i

i

i i i ii i i i i i

j

i i

l

xl l l l

y

l l i l

P t

P t P t P t P t

tP t P Θ P K K Θ

K K Θ P

1

ˆ .i

j jj

l

t

(20)

The acceleration of Ci can be expressed by

,

,

1 ˆ ˆ ˆ2 4 4 4

ˆ ˆ ˆ ˆ ˆ4 4 4 2 4 2 4

c i i i i ii i i i i i

c i i

i i i i i i i ii i i i i i i i

i

x xl l l

y y

x l ll l l

y

P t P t P t

P t P t P t P t P t

1 4 1, 2,

5, 6,

ˆ2 4

ˆ ˆ ˆ ˆ4 4 4 2 4

ˆˆ , ,

2 4 2 4 4 2 4 2 4

i ii

i i i i i ii i i i i i i

i

i i i i i i i i ii i i

i i

l

x ll l l

y

l l l li

P t

P t P t P t P t

tP t P Θ P K P K Θ

K K

1

1

ˆˆ .

2 4 2

ij i i

j j ij

l l

tΘ P t P

(21)

3. Governing Equation

The governing equations are derived from the principle of moment equilibrium at the segment

midpoints. The external forces working on Segment Si include:

(1) Hydrodynamic force 1,iF ;


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(2) Net gravity 2,iF ;

(3) Inertia force 3,iF .

External forces on each segment here are assumed lumped at its midpoint. In the non- inertial

coordinate system, these external forces must reach moment equilibrium at all midpoints C1, C2, …, Cn.

Equations of moment equilibrium at all midpoints will be listed as governing equations. The external

forces acting on segments can be calculated as follows.

3.1 Hydrodynamic Force

To simplify the calculation of hydrodynamic force, the segments of cable are all viewed as

oblique cylinders. Herein the semi-empirical Morrison approach is adopted for modeling

hydrodynamic force (Pao et al., 2000). The hydrodynamic force can be decomposed into tangent,

normal components. Thus, the hydrodynamic force of Si at Ci can be calculated by

1, 1, 1,i n i t i F F F , (22)

where 1,n iF and 1,t iF are the normal and tangent components of the hydrodynamic force, calculated

by

1, d w , , , , M w ,

M w ,

1( )

2 ( 1) ;

n i i i wn i n i wn i n i i i wn i

i i n i

c d l c Al

c Al

F v v v v v

v

(23)

1, f w , , , ,

1( )

2t i i i wt i t i wt i t ic d l F v v v v , (24)

in which, cd, cM are drag and inertia coefficients, di is the diameter of Si, Ai is the outer area of

cross-section of Si, ρw is the water density, and cf is the frictional coefficient.

,n iv is the normal component of velocity vector ,c iv of Ci, calculated by

2

,, , ,, , ,, , , , , 7,2

,, , ,, , ,

1( )

1cx ic i c i c icx i cx i cy i

n i cx i c i cy i c i i

cy ic i c i c icx i cy i cy i

tx x xt t tt x t y

ty y yt t t

v K

, (25)

where ,cx it and ,cy it are components of cit , denoted by

T

, ,ci cx i cy it t t . (26)

Similarly, ,n iv is the normal component of acceleration vector ,c iv at Ci, calculated by

,, ,

, , , , , 7,

,, ,

( ) cx ic i c i

n i cx i c i cy i c i i

cy ic i c i

tx xt x t y

ty y

v K , (27)

where ,wn iv is the normal component of current velocity vector ,w iv at Ci, ,wn iv is the normal

component of current acceleration vector ,w iv at Ci, and ,t iv is the tangent component of current

velocity vector ,c iv at Ci, expressed by

2, , ,, , ,

, , , , , 8, 8, ,2, , ,, , ,

( ) cx i c i c icx i cx i cy i

t i cx i c i cy i c i i i c i

cy i c i c icx i cy i cy i

t x xt t tt x t y

t y yt t t

v K K v . (28)


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,wt iv is the tangent component of current velocity vector ,w iv at Ci.

The hydrodynamic force of Si at Ci can be summed as:

1, d w , , , , M w , M ,

f w , , , ,

4, 5,

1( ) ( 1)

21

( )2

,

i i i wn i n i wn i n i i i wn i i i w n i

i i wt i t i wt i t i

i i

c d l c Al c Al

c d l

F v v v v v v

v v v v

F F

(29)

where

4, d w , , , , M w ,

f w , , , ,

1( )

21

( ) ;2

i i i wn i n i wn i n i i i wn i

i i wt i t i wt i t i

c d l c Al

c d l

F v v v v v

v v v v

(30)

5, M w , M w 7, ,( 1) ( 1)i i i n i i i i c ic Al c Al K F v v . (31)

3.2 Net Gravity

As the cross-section area of each segment is much smaller than the whole surface area of segment,

the segment is viewed approximately enclosed by water. Therefore, Archimede’s principle can be applied.

The net gravity at Ci is calculated by

2, wi i i rA l F g , (32)

where iA is the cross-section area of Si, g is the gravity vector, and rw is the cable density in

water.

3.3 Inertia Force

According to D'Alembert’s principle, the inertia force at Ci is calculated by

3, r ,i i i c iA l F r , (33)

where r is the cable density in air.

3.4 Sum-up of External Force

Sum up the above F1,i, F2,i and F3,i, the external forces working on Si at Ci can be summed up:

,

1, 2, 3, 4, 2, 5, 3,

,

4, 2, M w 7, , r ,

4, 2, M w 7, r ,

9, 10,

( 1)

( 1)

.

x i

i i i i i i i i

y i

i i i i i c i i i c i

i i i i i i i c i

i i

F

F

c Al Al

c Al Al

F F F F F F F F

F F K r r

F F K I r

K K Θ

(34)

3.5 Tension of Segment

For each segment Si, the axial tension is viewed uniform, and can be obtained by

T T T T

, , 9, 10, , 9, , 10,ˆ ˆ ˆ ˆ

n n n n

i c i j c i j j c i j c i jj i j i j i j i

T t F t K K Θ t K t K Θ . (35)


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3.6 Moment Equilibrium

The moment at Ci caused by the external forces can be calculated by

, , , , , ,1

,

, , , ,1 ,

, , , , 9, 10,1

11, 12,

( ) ( )

( ) ( )

( ) ( ) ( )

n

i x j c j c i y j c j c ij i

nx j

c j c i c j c ij i y j

n

c j c i c j c i j jj i

i i

M F y y F x x

Fy y x x

F

y y x x

K K Θ

K K Θ

(36)

By combining Eqs. (1) and (36), the following equation can be obtained:

11, 12,i i i i

i

EIM

l K K . (37)

3.7 Boundary Conditions

For the deep-sea cable, the upper end can be viewed as freely connected. The moment equilibrium

at N1 can be expressed by

0 , , , , ,1

,

, , , ,1 ,

( ) ( )

( ) ( )

0.

n

x j c j i y j c j c ij

nx j

c j c i c j c ij i y j

M F y y F x x

Fy y x x

F

(38)

Eqs. (37) and (38) are governing equations of the marine cable motion while being laid.

4. Numerical Solution

In the dynamics calculation, Θ as an unknown variable matrix needs to be calculated from Eqs.

(37) and (38) as governing equations. Since Eqs. (37) and (38) are highly nonlinear about Θ , it is

difficult to get the analytical solution to the equations directly, so we resort to the numerical solution.

Here we employ a linearization iteration scheme to get the numerical solution.

There is a special and important feature about the governing equations: if the current Θ and Θ

are known, they become linear equations about the current Θ . By solving the linear equations, the

current Θ can be obtained directly. This is the key point in the linearization iteration scheme.

From the current Θ , the next Θ and Θ can be calculated by the following iteration: ( 1) ( ) ( )i i i t Θ Θ Θ ; (39)

( 1) ( ) ( ) ( 1)1( )

2i i i i t Θ Θ Θ Θ , (40)

where i in the superscript bracket means the iteration times (e.g. ( )iΘ is the value of Θ in the i-th

iteration), and t is the time interval of iteration. By the iteration, the time history of Θ can be


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obtained from the initial Θ and Θ .

In the above numerical solution, 1l , 1l and 1l

are calculated beforehand. 1l is the cable payout

rate, can be determined by Eq. (51) in the next section.

With the current ( )

1

il , the length of S1 in the next iteration will be extended by

( 1) ( ) ( )

1 1 1

i i il l l t . (41)

The length of Sn in the next iteration will be subtracted by the length dropping to the sea floor, as Eq.

(43) in the next section.

1l in the next iteration can be obtained by

( 1) ( )( 1) 1 1

1

i ii l l

lt

. (42)

5. Payout Rate Calculation

If the cable is negative slack, the tension will increase rapidly, as presented by Yoshizawa et al.

(1983). The payout rate should be set appropriate to lay the cable to the seafloor slackly.

To ensure the lower end of the cable is laid down to the seafloor without tension and waste, the

payout rate should be set to a value by which the length of the lower end freely dropping to the sea is

equal to the length of lower end going through the seafloor.

As shown in Fig. 2, the mother vessel goes leftward, and the marine cable is laid down to the

seafloor accordingly. The TDP in the i-th iteration is Nn+1(i), denoted by A. And then the TDP in the

(i+1)-th iteration is Nn+1(i+1), denoted by B. If there is no seafloor, A would freely drop to C.

Fig. 2. Payout rate analysis:

AV is the velocity vector of Nn+1(i), parallel to the displacement vector from A to C.

Assuming that BCl is the length between points B and C below the seafloor, i.e. the length of the

lower end freely drops to the seafloor, the length of Sn(i) is shortened by

( 1) ( )i i

n n BCl l l . (43)

Assume ABl is the distance between the TDPs of the i-th and (i+1)-th iterations, i.e. the forwarding


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distance of the cable lower end.

To ensure the cable is freely dropped to the seafloor with no waste, the payout rate should be set

to a value by which the freely-dropping length is equal to the forwarding length: BC ABl l , and the

triangle ABC is isosceles.

Assuming that the seafloor is horrizonal, is the angle between AB and AC,

π2 π

2 n

; (44)

π

4 2n . (45)

If the seafloor is oblique, should be adjusted accordingly.

The velocity vector of Nn+1(i) is parallel to the displacement vector from A to C. The velocity

vector of Nn+1 is T

1 1n nx y , thus

1 1cos sinn ny x ; (46)

1

1

sin cos 0n

n

x

y

. (47)

Set

5 sin cos P , (48)

the equation can be denoted by

1

5

1

0n

n

x

y

P

. (49)

From Eq. (12), the velocity of Nn+1 can be calculated by

1

1

1 11 1

11

11 1 6

ˆ ˆ ˆ2 2 2

ˆ ˆ ˆ2 2 2

ˆ ,2

n n n n nn n n n n n

n n

ni i

i i i ii

x xl l l

y y

xl l l

y

l

P t P t P t

P t P t P t

P t P Θ

(50)

where 6P Θ is a variable matrix.

Input Eq. (49) into Eq. (48), the following can be obtained by

11 5 1 5 6

ˆ 02

l

P P t P P Θ , (51)

so the payout rate can be calculated by

5 6

1

15 12

l

P P Θ

P P t

. (52)


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6. Dynamic Motion and Tension During Acceleration of Mother Vessel

Velocity Change of the mother vessel often occurs in cable laying operation. During the Velocity

Change of the mother vessel, the cable configuration and tension may change. To simulate the

changing process of the dynamic motion and tension of marine cables being laid, the numerical

calculation was carried out. Section 6 presents the numerical results of dynamic motion and tension

during the acceleration of the mother vessel, while Section 7 presents the numerical results of dynamic

motion and tension during the deceleration of the mother vessel.

Assuming the velocity of the mother vessel is set as

1 o f o( ) /x V V V t T , (53)

where oV and fV are the initial and final vessel velocities, respectively; and T is the accelerating

time taken for the vessel to reach the final velocity.

Assuming that the cable is armored, its features are shown in Table 1.

Table 1 Cable features

Weight in air Weight in sea Cable diameter Young’s modulusDrag

coefficient Inertia

coefficient Frictional coefficient

26.49 N/m 17.76 N/m 0.0332 m 1.4393e+10 N/m2 1.649 2.0 0.017

Here we used the above FSM based method to do the numerical calculation. The numerical

calculation was implemented by Matlab programming. In the acceleration of the mother vessel,

o 0V m/s, f 2V m/s and 100T s. The initial cable state is vertically hanged in water.

The numerical results are shown in Figs. 3–6. Fig. 3 shows the configurations of (a) 120-m-depth

and (b) 1200-m-depth cables. We can see that it will need about 300 s for the 120-m-depth cable to

reach the final steady configuration, while it will need about 2700 s for the 1200-m-depth cable to

reach the final steady configuration.

The cable tensions are shown in Fig. 4. The tensions vary in small ranges: 2.13–2.14 kN for the

120-m-depth cable; 21.2–21.4 kN for the 1200-m-depth cable.

The payout rates are shown in Fig. 5. For the 120-m-depth cable, it takes about 350 s to reach the

final payout rate (2 m/s). And for the 1200-m-depth cable, it takes about 3000 s to reach the final

payout rate (2 m/s).

As the accelerating time T will influence the tension varying range, the comparison of different

accelerating times is done. The results are shown in Fig. 6, and the time histories of cable tensions at

10T s, 50T s and 200T s are plotted with dash-dot, dashed and solid lines, respectively. The

influence of the accelerating time T on the cable tension can be concluded by

(1) As T goes smaller, i.e. acceleration is larger, the tension becomes larger;

(2) As the depth goes larger, the influence of T on the cable tension becomes smaller;

(3) Even though the acceleration is large, the varying range of cable tension is relatively small by

the calculated payout rate;

(4) The initial and final tensions are approximate.


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Fig. 3. Cable configurations of (a) 120-m-depth and (b) 1200-m-depth cables during acceleration with 100T s.

Fig. 4. Cable tensions of (a) 120-m-depth and (b) 1200-m-depth cables during acceleration with 100T s.

Fig. 5. Cable payout rates of (a) 120-m-depth and (b) 1200-m-depth cables during acceleration with 100T s.


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Fig. 6. Cable tensions of (a) 120-m-depth and (b) 1200-m-depth cables during acceleration with

10T s, 50T s and 200T s.

7. Dynamic Motion and Tension During Deceleration of the Mother Vessel

This section will present the numerical results of dynamic motion and tension of marine cable

being laid during the deceleration of the mother vessel. The parameters of Eq. (53) are set as

o 2V m/s, f 0V m/s and 100T s. The initial cables are in the 2m/s steady forwarding state.

The numerical results are shown in Figs. 7–9. Fig. 7 shows the configuration of (a) 120-m- and (b)

1200-m-depth cables during the deceleration. We can see that it will need about 350 s for the 120-m-

depth cable to reach the steady static configuration, while it will need about 3000 s for the 1200-m-

depth cable to reach the steady static configuration.

Fig. 7. Cable configurations of (a) 120-m- and (b) 1200-m-depth cables during deceleration with 100T s.

The cable tensions are shown in Fig. 8. The tensions vary in small ranges: about 2.11–2.13 kN for

the 120-m-depth cable; about 21.1–21.3 kN for the 1200-m-depth cable.


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Fig. 8. Cable tensions of (a) 120-m- and (b) 1200-m-depth cables during deceleration with 100T s.

The payout rate is shown in Fig. 9. For the 120-m-depth cable, it takes about 350 s to reach the

final payout rate (0 m/s). And for 1200-m-depth cable, it takes about 2800 s to reach the final payout

rate (0 m/s).

Fig. 9. Cable payout rates of (a) 120-m- and (b) 1200-m-depth cables during deceleration with 100T s.

8. Conclusions

FSM-based method is employed to do dynamics calculation of marine riser being laid. In FSM,

the cable is divided into a number of flexible segments. The whole cable deformation is composed of a

series of small deformation of these flexible segments. Moment equilibrium equations on these

segments are listed as governing equations. A linearization iteration scheme is presented to obtain

numerical solution to the nonlinear governing equations. The iteration scheme can avoid the selection

of initial values which is important for NewtonRapson iteration. Therefore, the stability of the

dynamics calculation by the present linearization iteration scheme is improved.


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To lay the cable to the seafloor slackly, the equations to calculate the payout rate are presented.

With the calculated payout rate, the cable lower end can freely drop to the seafloor without tension and

waste.

The numerical results of the dynamic motion and tension of marine cables during Velocity

Change of mother vessels are presented. From the results, we can see that with the calculated payout

rate, the tension of the cable is not much influenced by the acceleration and deceleration of the mother

vessel.

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Dynamic motion and tension of marine cables being laid during velocity change of mother vessels