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Dynamic behaviour of deep water mooring lines with submerged buoys
0
Pergamon
Compurers & Srrucrures Vol. 64, No. 14, pp. 819-835, 1997 1997 Civil-Comp Ltd and Elsevier Science Ltd. All rights reserved Printed in Great Britain 0045-7949/97 $17.00 + 0.00
PII: 80045-7949(%)00169-l
DYNAMIC
BEHAVIOUR OF DEEP WATER MOORING LINES WITH SUBMERGED BUOYS S. A. Mavrakos and J. Chatjigeorgiou
Department of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens 15773, Greece Abstract-The paper deals with the dynamic behaviour of wire mooring lines in deep water. Numerical methods in the time- and the frequency-domain are presented to evaluate the dynamic tensions and the mooring line motions. Special attention has been paid to the assessment of the impact that submerged buoys have on the dynamic behaviour of mooring lines on which they are attached. The modification of the mooring line dynamic performance in dependence on the buoys’ characteristics (number, size and location of a.ttachment) has demonstrated through a parametric numerical study. 0 1997 Civil-Comp Ltd and Elsevier Science Ltd.
1. INTRODUCTION Mooring systems are used extensively in the offshore field of applictions to position floating drilling or production systems and other marine vehicles. Special attention has been paid in the last few years
in deep water applications, where dynamic considerations should be incorporated in a rational design procedure. This has been pointed out by several investigators [l-3] and led to a modification of the design rules and guidelines of several organizations [4, S]. The quasi-static approach, suitable for the design of catenary mooring systems in shallow water (depth below approximately 100 m) [5], is generally considered inadequate for deep water applications. In such cases, the maximum attainable value of the dynamic tension a.mplification, being approximately equal to the elastic stiffness of the line for nonsnapping conditions, is shifted within the wave frequency range and, thus, large dynamic tensions due to the vessel’s first-order motions can be excited [3]. It is therefore evident that for such cases the principal parameter affecting the dynamic response of the line is its elastic stiffness and, hence, its reduction would improve the dynamic performance of th.e line. On the other hand, when a snapping condition has occurred, i.e. when the dynamic tension exceeds the static one in certain parts of the cable, hence causing the cable to become slack for part of the cycle and then to tighten suddenly, the principal parameter controlling this phenomenon is the free-falling velocity of the ca.ble. The latter can be calculated assuming that the cable’s weight is supported by the drag force when it becomes slack [6, 71. It was found that by increasing the free-falling velocity (for
example, by making the line heavier), a considerable reduction of the peak dynamic tension during snapping can be achieved [68]. Considering nonsnapping cables, it has recently been demonstrated numerically [9] and confirmed experimentally in a series of small- and large-scale experiments [lo, 111, that the beneficial reduction of the cable’s elastic stiffness can be achieved through the dynamic modification of the system, using properly designed buoys attached to the mooring lines. The buoy-cable system behaves like an inverted pendulum, with the buoyancy acting as a gravity force. Thus, the proper selection of the number, size and location of attachment of these buoys can manipulate the natural frequencies of the buoy-cable system to the benefit of the mooring line’s dynamic tension reduction, thus forming the basis of an optimization algorithm. A first attempt towards the development of an optimization algorithm for deep water mooring applications has recently been presented by exploiting the dynamic features of a double pendulum in air [ 121. In the present contribution, extensive numerical results of a parametric study are presented dealing with the systematic evaluation of the effect that the buoys’ characteristics have on the dynamic behaviour of deep water mooring lines. Both frequency- and time-domain analysis methods have been applied to obtain the numerical results. The former involves an equivalent linearization technique to accommodate the effect of the quadratic drag on the cable and the attached buoys. The latter, which additionally accounts for geometric nonlinearities due to large cable motions, makes use of a simplified formulation of the nonlinear governing equations, which is suitable for mooring applications with wire cables. The present contribution extends previous related 819
S. A. Mavrakos and J. Chatjigeorgiou
820
work [9, 121 by including in the formulation of the problem the buoys’ rotational motion around their attachment point.
2. GOVERNING
infinite long cylinder with the same cross-section, and cp denotes the angle between the horizontal and the local tangential direction of the cable in the moving coordinate frame. It possesses one static, ‘pa, and one dynamic component, cp,. The former is the angle between the horizontal and the static tangential direction of the cable, b, whereas the latter denotes its dynamic deviation from the static equilibrium condition. Finally, a constitutive tension-strain relation has to be added, which in the present contribution is assumed to be a linear one
EQUATIONS
The cable is modelled as a slender rod with negligible bending stiffness. Let s denote the unstretched Lagrangian coordinate measured from the lower cable end up to a material point of the cable, v (u, v) its velocity vector, m the mass per unit unstretched length, T the tension vector (pointing along the local tangential direction, characterized by its unit vector t), F (fi, F.) the vector of the distributed external force per unit stretched length and e the local strain. Then the equations of the in-plane motion of a cable having a two-dimensional static shape can be expressed along the local tangential, t, and normal, n, directions of its moving configuration (moving dynamic reference) as [3,91
m{~-v~}=g+fi(l+e)
T = EAe.
The external forces F, and Fn, include the submerged weight of the cable per unit stretched length, W, and the fluid drag forces which are obtained on the basis of the so-called separation principle, i.e. relating the large normal force to the relative fluid-cable velocity in the normal direction, and the tangential force to the corresponding relative velocity in the tangential direction
(1)
F, = - w sin 4 + 0.5rrp,crd
x(Ucos$~-u)]Ucosd-u]
(m+a)$+muz= T$+F.(l de
au
acp
z=as-“-i%
+e)
(2)
(4) where c1is the two-dimensional added mass, which is assumed to be equal to its potential-flow value for an
(6)
F, = - w cos 4 - O.Sp,cdd
(Usinf$+v)(Usinr#r+v]. (3)
(5)
(7)
Here, cf, cd are the friction and normal drag coefficients, respectively, pWis the water density, d the stretched cable diameter and U the local current velocity (assumed horizontal). Equations (l)-(5) form the complete set of the nonlinear, two-dimensional dynamic equations of the mooring line motion, For cables with two-dimensional static configuration, the
ZA
Current
Fig. I. Buoy attached on the mooring line.
Deep water mooring lines with submerged buoys
out-of-plane motion is first order decoupled from the in-plane one [3]. In the case of buoys attached to the cable, eqns (l)(4) must be supplemented with the corresponding ones of the buoys’ motion. Let M and M, be the mass and the added mass of the buoy, respectivedly, J, and J,, its mass- and added-moment of inertia about C, B0 its net buoyancy in water, A, its projected surface and cM the drag coefficient. Then, the equations of motions for a buoy located at the junction of two successive segments of the cable (i and i + l), and undergoing small translational and rotational motions in the X-Z plane with its centre of gravity coinciding with the centre of the displaced water volume, are (see Fig. 1)
Table 1. Cable data for Figs 2-12 Length Mass per unit length Added mass per unit length Diameter Submerged weight per unit length Water depth EA Normal drag coefficient Tangential drag coefficient
rp in
(l)-(4)
eqns
- T; cos qbi+ D,
ap
u = at cos
(8)
(M + Ma)&, = T., , sin &+,
- T sin #ii + 0; + B. (A4 + M.)R&
(9)
-t- (Jc + Jac)e’ =: - BoRd + (D, - D,O)Rb.
Lj,
v, = ,,/(u
=
&CM
(ib
+
Rbd))
A, Kib
- (ii, + Rd)’
invoking
fpl
+
a4an.
5
-aps. at
the
-
following
ql
(14)
a4cos CPI
(1%
ln CPI + -g
which relate the velocity, (u, v), and the displacement, (p, q), components in the moving and the static reference coordinate systems, a set of simplified equations, correct to second order, can be obtained after lengthy mathematical manipulations for the in-plane cable’s motion:
(10)
(16)
Here dots represent time derivatives, x0, zb are the motions of the buoy’s attachment point in the horizontal and vertical directions, 6 the buoy’s angle of rotation about C, Rb its radius, Ti+, , (pi+, and Ti, ‘pi are the total tension and angle in the segments i + 1 and i, respectively, at the buoy’s attachment point and D, = ;,&CW’&v,(u -
and
3000m 94.5 kg m-i 16.0 kgm-’ 0.141 m 770 N m-’ 375 m 1*109N 1.2 0.05
expressions:
(j=
= 7;+,cos&+,
821
aT,
hh
+F.I +~‘PI
+ T,as
g-q!$+&=e,
(11)
(17)
of9
(12) + ii.
$+p%=~,(l
(13)
The nonlinear governing equations for the cable’s and buoy’s dynamics, i.e. eqns (l)-(4) and (8x10), respectively, can be simplified, so that without losing on important nonlinear contributions, effective numerical schemes can be applied for their treatment. In this simplifiction procedure the assumption is made that the dynamic tensions and the cable’s motions are small around its static configuration. The latter defines a new reference coordinate system. By expanding the set of nonlinear eqns (I)-(4) about this static frame, the equations of motions for the cable can be expressed in terms of its tangential, p and normal, q, displacements. Let T,, cpI be the components of the dynamic tension and the dynamic angle, respectively, and To, cpo their static values, i.e. we set T = To + T, and cp = cp0+ cpl. Substituting the expressions for T and
+e0),
(19)
where fiI = (fi - W,)(l
+ e) -
F,0(1
+
e0)
(20)
Table 2. Buoys’ characteristics
Buoy Type Type Type Type Type Type Type Type Tvue
tyPe I II III IV V VI VII VIII IX
Mass (kg) 29,651 33,508 19,991 8218 25,878 43,538 27,840 39,614 51,387
Added mass (kg)
Net buoyancy (%l
Diameter (m)
73,694 57,963 33,543 33,543 33,543 33,543 33,543 33,543 33,543
50.00 35.00 20.00 25.00 17.50 10.00 16.67 11.67 6.67
6.50 6.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
S. A. Mavrakos and J. Chatjigeorgiou
822
Static :oniiauration 4 Cable 3COG r-77 ct 2100 m irark
;vith one cnchor
buoy
j--i
7--
1000.0 horizontal
I.0
3000.0 2000.0 distance(m)
Fig. 2.
F.1 = (F,CPL+ F”)(l + e) - F&l + eo).
-
(21)
Here, F, and F,, are given by eqns (6) and (7), respectively, whereas Fto and F,o denote their static values, obtained by inserting in eqns (6) and (7), u = u = 0. In eqns (16x19) subscripts 0 and 1 are used to distinguish static from first-order dynamic quantities. Besides the nonlinear drag forces involved in eqns (16)-(19), only the last two terms in the right hand side of eqn (17) represent nonlinear contributions. The first of them accounts for large dynamic tension variation along the cable, whereas the second one for geometric nonlinear effects due to large transverse in-plane motion of the cable. Neglecting these two nonlinearities, the well-known two-dimensional, linearized equations of motion of a cable can be obtained, accounting for the effects of quadratic drag only [3]. By applying a similar linearization procedure to the equations of the buoy’s motion, i.e. eqns (8X10), its dynamic equilibrium along the tangential and normal directions of the static cable configuration can be expressed as
To,~w
sin$ =
Ti,i+I + Dp,i cos $
- DqVisin $
(22)
(M + M&h sin $ + (M + M&j, cos I(/ + TI,~sin $ +
TO,FPI,~
cos
$
=
To,i+ I(pl.i+I
+ Dp,i sin $ + Dq,i cos t,b
(M + Ma)]&& cos +(J+
J&e;
=
x RdD,., cos
~0,~ -
RG& sin ~p~.~]
BoR&
-
(~0.~ -
(23)
+
D,,; sin cpo,,).
(24)
Here, Dp,i = (Ox - D&OS cpo,i+ D, sin ~p~.~, Dq,i = - (Ox - D&in ~0.~ + D, cos cpoei, with D, and D, being given by eqns (11) and (12), respectively, and D,, is the static force on the buoy in the horizontal
direction for & = ib = B = 0; $ = qo,i - ~p~,~+ ,. In deriving the above equations, the following kinematic compatibility relations at the buoy-cable junction have been taken into account: xb
=
pi
cos
p0.i
-
qi
sin
f$O,i
-ql+lsin~o,i+,
=P/+ICOS~O,~+~
(25)
(M + Ma)pi COS* - (M + Ma)41sin $ tb
+W + Ma)Rtae;+ T,,
cos +
=
pi
sin
40,i =
+
qi cos
4O.i
p,+~sin~o.i+I-qi+Icos~oo,i+,.
(26)
823
Deep water mooring lines with submerged buoys
The second-order governing equations (16x19) can be further simplified in cases where the excitation frequencies are assumed to be well below the first elastic frequency of the cable. This assumption, which is a reasonable one for most applications with wire cables, implies almost uniformly distributed dynamic tension along the cable and smaller tangential than normal displacements. As a result, the second-order term in eqn (17), which includes differentiation of the dynamic tension with respect to the cable’s length, can be neglected. Moreover, in eqn (19), the dynamic angle due to the axial motion of the cable can be considered small with respect to the angle induced by dynamic lateral motions. Using these assumptions, together with the hypothesis of large static tension relative to the total weight of the cable, eqn (16) becomes a trivial one, whereas eqn (17) can be expressed as follows [9]:
(m + a) 3
==(To + T,) !$ + T, dJ$ - Fd (27)
where
I 1 - u;
4.00
,
(28)
UN = U sin I#J z=A{pi-p,-[q$+/.
(29) (30)
In eqn (30), p2 and p, denote the upper and lower end displacements, respectively, L is the suspended cable length and A is the equivalent stiffness of the line. The latter accounts for additional elasticity provided to the system by the axial and transverse motions of the lower, on the bottom lying, part of the cable, if any. Let 1, be its length, EC its modulus of elasticity and A, its cross-sectional area, then A is given by 1 1 1 x = (E,A,J1,) + (EA/L) .
(31)
The second term in the parenthesis on the right hand side of eqn (27) accounts for geometric nonlinear effects due to large transverse motions of the cable, whereas the third term describes the cable’s ability to straighten under a dynamic tension. The simplified eqns (27) and (30) can be used to study the dynamics of a snapping cable as well. This has been demonstrated in Ref. [7] where reference is made for more details.
Cable 3000 m one buoy, at from anchor. Tension at the
2450 top
.______ buoy’s N.B. SOW’*,\ - - buoy’s N.B. 35% ‘\\ buoy’s N.B. 20% / A*&&& cable witout ,,,(,,, buoy , 0.00 -j1!L111111,111111111 0.00 0.50 1 .oo 1 20 w (rad/sec) Fig. 3.
m
t.
2.
S. A. Mavrakos and J. Chatjigeorgiou
824 3. SOLVING
THE GOVERNING
EQUATIONS
on the buoys in eqns (22)-(24) is concerned, this is achieved by having them expressed in Fourier series using the commercial software package MATLAB [14] and then retaining the mean and the harmonically oscillating terms only. Having the linear damping force determined in this way, the system of the linearized eqns (16~(19) is reduced to a set of four ordinary differential equations with two boundary conditions at each end, which is solved in the Fourier domain using an explicit centred difference scheme. The linearized dynamic equations of cable’s motion can be written in matrix form as
For cables excited sinusoidally at their top end, the drag forces on them and the attached buoys can be equivalently linearized. Thus, in cases where the last two nonlinear terms on the right hand side of eqn (17) can be omitted, the system of eqns (16)-(19) can be numerically treated in the frequency domain. The proposed linearization procedure, which is similar to the one already presented by the authors [12] relies on a decomposition of the drag forces into a mean, plus an equivalent linear damping force by minimizing the quadratic error between the equivalent representation and the actual nonlinear drag force averaged over one period. The mean force can be included in the static analysis, whereas the linear damping force is introduced in the linearized dynamic equations to solve them in the frequency domain. The procedure is similar to the one proposed by Krolikowski and Gay [ 131.As far as the linearization of the drag forces
1 dqo
-
Y’(S) = (TI&dpaqa)
(33)
and 0
-mm2 ---
To ds
A(s) =
(32)
where
0
---
2 =A(s).Y(s),
1 dTo
0
To ds
1
0
EA
[(m + c()o.?+ ibw]
-
TO
(34)
dqo ds
0
1
0
4.00
m cne L”cble X00 cable’s weight.Tension -,:---
buov, N.5. 20% a’t the top
3.00
u-l W
6 2.00 >
A* * AA cable witout buov buoy at 2100 n - - buoy at 245 .____-- buoy at 28:
0.00
I 0.00
I
I
/
I
I
I
1
I,,
0.50
I,,
I
I
I
I
I
I,,
,
,
1 .oo w (rad/sec) Fig. 4.
,
,
,
,
,
I,,
1.50
,
(
(
,
,
,
,
,
2.1
825
Deep water mooring lines with submerged buoys
Here, the subscript a denotes the amplitude of the corresponding dynamic quantity and b is the equivalent damping coefficient for the cable. This coefficient, which is a motion-dependent quantity, is used together with a difference scheme to determine the unknown local amplitude of oscillations by an iterative procedure. The difference scheme has the form
-I 1
Yj+I =
I-?FA(Sj+,)
[
x
and by
can be obtained discretizing the
I
A(S)) yj
using centred cable into
yb,i+
(-
w’kfb
sin II, -.
_7’o.icos $ T0.1-f I
(-
02hfb
(36)
Ab’yb.i,
(35) & = (T,,,~rp,,,,p,,,q,,,~e,)
differences n nodes
- TO,]sin $
I =
where yb,i+, and yb,j are vectors calculated at the buoy’s attachment point in the segments i + 1 and i, respectively. They are defined by
1
-w
cos 3 TOJf & =
[
0’ = 1,2, . . . , n - 1). The procedure gives acceptable results with an overall error 0(As2), As being the grid length. The equations of the buoy’s motion, i.e. eqns (22)(24), can accordingly be written in the following matrix form:
The transformation
+
iob,)cos $
+
iob,)sin rj
(02kfb
(-
+
k = 4 i + 1. (37)
matrix Ab is given by
- W2M&
iwb,)sin If+
-
w2hfb
for
iob,)cos $
To,;+ I
To,;, I
0
0
0
cos *
- sin $
0
0
0
sin $
cos $
0
0
0
4.00
(-w2Mb
+
COS q%,i
i&)Rb
(w2hfb
W2Jb- B,,Rb
Cable 3000 m cable’s weight.
- iob,)Rb sin (pO,/ d&a - &Rb
one buoy, N.E. 50% Tension at the top
I
-I
A A A A A cable - - buoy ______ buoy buoy
without at 2100 at 2450 at 2850
w
b&v \ m ‘+ m
(rad/sec)
Fig. 5.
p
I
0
826
S.
A. Mavrakos and J. Chatjigeorgiou
Here, Mb = M + M, and & = J + J,,, whereas the damping coefficients b, and 6, are defined by Dr., = - &,.A and D,,, = - bq.gr, respectively. In cases where, in addition to the quadratic drag forces, nonlinearities arising from large transverse in-plane cable- and buoy-motions have to be taken into account, then eqn (27) together with eqns (8)-( 10) can be solved using a time-domain simulation scheme. According to this scheme, the solution of the partial differential equation (27) is expanded in each cable’s segment into an infinite sum of sinusoids, plus additional terms to accommodate the boundary conditions at its end, i.e.
inserting the solution for q, i.e. eqn (38), into the relation (30)
T&T! ’
x
L
p2 -pl
- I,qz - (14 - I&, -
“= I
Q. sin(nax),
&+
b,(m,Qnz =;(-@I
+ ii2(-1)“‘) + c
n-1
nc “I Q.I,(n, m) + (bdm)Q,
-2
&
+ h(m))T,
sin(mnx)dx,
(39)
(40)
(41)
=
2
’ 1 so m+a
sin(nnx)sin(mnx)dx
(42)
d% . sm(mnx)dx. ds
(43)
Moreover, Fd is given by eqn (28), whereas the dynamic tension, T,, involved in eqn (39) can be expressed in terms of the temporal coordinates Qn by
‘x&dx ds
0
14= L
I&)
, (44
(45)
(46)
s
’ dq, sm(nrrx)dx. .
= L oz
(47)
Each component of the temporal coordinates, Qm, defined by the set (38) is then integrated using Newmark’s method [ 151. By incorporating the coupled equations of motion for the buoy, i.e. eqns (8X10), the proposed time domain simulation scheme extends the one presented in Ref. [9] by including buoy rotations around its attachment point on the cable. 4. NUMERICAL
where
h(m)
s
(38)
where x = s/L, s being Lagrangian coordinate and L the segment’s length. The latter is defined as the distance between buoys, or between a buoy and one cable’s end or between ends. The temporal coordinates Q. involved in eqn (38) are the principal unknown of the problem. They are derived from a set of ordinary differential equations which are obtained by implementing Galerkin’s method. To this end, eqn (38) is first substituted for q in the governing equation (27). Both sides of the resulting expression are then multiplied by sin(mnx) and integrated in their definition interval, yielding
QJ&z,
where
I 3 CL
q(x, t) = q2x + q,(l - x) + 1
2
II= I
RESULTS
AND DISCUSSION
Using the frequency- and the time-domain analysis methods presented in the previous section, a systematic study was conducted to assess the impact that one or more buoys have on the dynamic behaviour of the lines to which they are attached. The data for the selected all-wire mooring line are shown in Table 1. The water depth is assumed to be 375 m, whereas the horizontal force at the top is equal to 7.8 x IO6N. The line is excited tangentially at its top with an amplitude of 1.0 m. The basic mooring line configuration does not have any buoy attached to it. To study the effect that one or several buoys have on the dynamic behaviour of the line to which they are attached, one, two and three spherical buoys are considered to be attached to the mooring line having characteristics summarized in Table 2. A total of nine types is used, each one giving by itself or in combination with the others net buoyancy equal to 20, 35 or 50% of the submerged cable’s weight. The buoys’ drag coefficient is assumed to be equal to 0.8. The static configurations of the mooring line without buoys and with a buoy attached to it at a distance 2100 m from the anchor are shown in Fig. 2. The buoy size is varied, so that its net buoyancy
827
Deep water mooring lines with submerged buoys Ccile cable’s
(a) 4.00
3000 m weight,
two buoys, Tension at
N.8. 20% the top
*PA&A cable witout buoys
a
-A-
d
1
buoys at buoys ct buoys at
-a---
!
2100 2450 Z:CC
and and end
I
2850 2850 2150
m m rr
3.00
ul W
6 2.00 >
1 .oo
0.00
11 0.00
0.50
1.00 w
(b)
2.00
Cable 3000 m two buoys, N.8. 2055 cable’s wei ht. Location of the buoys: 2100 and 4 850 m from anchor
Tension
at various
4.00 A -
__
locations ‘A&A*& at at
A
0.00
1.50
(rad/sec)
A
f
--
cz --&_
-
top tap before before
without with
the
1st buoy
the
2nd
;~~~~~~~~~,~~~~~,,,,,,,,,,,,~,,~,~,,,,,,~ 0.00 0.50 1 .oo 1.50 w
(rad/sec)
Fig. 6.
buoys
buoys buov
2.00
S. A. Mavrakos and J. Chatjigeorgiou
828
Cable cable’s
(a)
,
4.00
3000 m weight.
two buoys, Tension at
~~hba .a
N.B. 35% the top
cable wiiout buoys buoys at 2100 and buoys at 2450 and buoys at 21CO and
& -o --P
2850 2850 2350
r r r
3.00
u-l
I
W 6 2.00 >
1 .clo A
0.00
1
0.00
0.50
1.50
1 .oo
2.1
w (rad/sec) 09 4 o.
Cable 3000 m two buoys, N.B. cable’s wei ht. Location of the 2100 and s 850 m from anchor Tension at various locations oa ~AA at top A A ..
r-r
_ --
d /
N
=
ot
top
before before
c--
3% buoys:
without buoy: with buoys the 1st buoy the 2nd buoy
. d,
‘1-F
A
A A
,,,(,,~,,1,,1,,~,,1,,,~
0.00 0.00 ;11,,,,,11,,,,,, 0.50
8 1.00 w (rad/sec) Fig. 7.
1.50
2.i 0
829
Deep water mooring lines with submerged buoys
(4 4.00
Cable cable’s _I
3000 m weight.
two buoys, N.E. 50~ Tension at the top &AA&~ cable witout buoys buoys at 2100 and 2850 buoys at 2450 buoys at 2100
and and
2850 2450
_ w (rad/sec)
(b)
Cable 3000 m two buoys, N.B. cable’s wei ht. Location of the 2100 and s 850 m from anchor Tension at various locations
4.00
ibid. A a
4 T-4---
3.00
u-l I W 6
2.00 d
>
1 .oo
0.00
_ --
b at
tap
at
top
before before
50~ buoys:
without
buoys with buoys the 1st buoy the 2nd buoy
Cable Buoys Total Fnsion
(a) 4.00
SOCO m.with three buoys ar 2100, 2450, ‘2850 m net buoyancy 20% cable’s at various locations
from anchor weight
,
4
h
1 .oo
---
before _____ before
0.00
4.00
-
~~,,,,l,ll,,,,,,,,,,,l,,,,,,,,,,,,i,,,,, 0.00
(b)
the 2nd buo the 3rd buoy
1.00 w (rad/sec)
0.50
1.50
Cable 3000 m.with three buoys Buoys at 2100, 2450, 2850 m Total net buoyancy 50% cable’s Tension at various locations
2.t 0
from anchor weight
h A p-, /
A
--.-_
3.00
U-I I W i
2.00
>
1 .oo
h
0.00
0.00
0.50
1 .oo
w (rad/sec) Fig. 9. 830
1 so
2.1 0
Deep water mooring lines with submerged buoys
831
Cable 3000 m, three buoys, N.B. 50% cable’s weight. Buoys at 2100, 2450 and 2850 m from anchor
(4 400000
2 frequency: I ,344 r-ad/s
$
-200000~
5: Cl
-400000
1 _
-600000
a&A&A nobuay case _ -___ ____-
, a,, 0.00
tension tension tension tension
at the before before before
top the 1st buoy from anchor the 2nd buoy from oncho r the 3rd buoy from anchal
, , v, t,, ,a, 1 t 1, I,, 50.00 100.00 Time(sec)
Cable 3000 m, three buo ;b~.Bi4~% cable’s weight. Buoys at is , and 2850 m from anchor
W 8200000
frequent : .344 ro dy/s
8000000
1
7800000
-
A\
z
b’
z Q 9 w, I-
, , , , b,,
-+
z; 7600000 Cl I-
100.00
50.00
Time(sec) Fig. IO.
, 15c 1. 00
832
S.
A. Mavrakos and J. Chatjigeorgiou
ranges from 20 to 50% of the submerged cable’s weight. In Fig. 3 the dynamic tension at the top of the cable having one buoy attached to it at 2450 m from the anchor is plotted against the frequency of excitation, with parametric dependence on the net buoyancy values. A considerable modification of the dynamic behaviour of the cable is obtained entirely due to the buoy. Its presence causes the minimum value of the dynamic tension to appear in all cases at lower frequencies than the one corresponding to the first elastic mode of the cable. The substantial difference between the two minima lies in the fact that in the former case the low values of the dynamic tension appear along the whole segment of the cable above the buoy, whereas in the latter the minimization occurs only at the cable’s top [9]. It is apparent from Fig. 3 that, as the net buoyancy of the buoy increases, the minimization frequency attains lower values, whereas the minimum value of the tension remains almost constant. As far as the maximum values of the dynamic tension are concerned, it seems that they are greatly affected by the amount of net buoyancy offered to the system. The higher the buoyancy is, the lower the maximum attainable dynamic tension. Figures 4 and 5 show results concerning the dynamic tension variation at the top of a cable having
one buoy attached to it at various locations and offering to the system net buoyancies equal to 20 and 50% of the submerged cable’s weight. It can be seen that the frequency of the dynamic tension minimization is almost insensitive to the buoy’s location, depending primarily on the buoy’s size. Numerical results for a two-buoy configuration with total net buoyancy equal to 20, 35 and 50% of the submerged cable’s weight are presented in Figs 6a, b, 7a, b and 8a, b, respectively. In each arrangement the two buoys have the same diameter and are of Type IV for the 20% case, of Type V for the 35% case and of Type IV for the 50% case (see Table 2). Figures 6a, 7a and 8a show the dynamic tension variation at the top for various locations of the buoys. A considerable reduction of the maximum dynamic tension is observed for the 50% case, as compared with the 20 and 35% cases, as well as with the no-buoy case. The dynamic tensions exhibit two peaks which are more pronounced for the buoys’ locations at 2100 and 2450 m from the anchor. In order to investigate the variation of dynamic tension along the cable’s length, Figs 6b, 7b and 8b are given. Here, the dynamic amplification per unit imposed upper end cable motion is plotted against the excitation frequency. It can be seen that significant tension amplifications may occur at points
Ccble 3000 m wiih~iwo buoys Buoys at 2100 and 2850 m from anchor Total net buoyancy 35% cable’s weight 4.00 Lines Symbols
* frequency domain ‘: time domain
S - 3.00 IfI & Y l
5 E 2.00 E 0 Z S & 1 .oo
_ -
_ -._____ ***** AAAAA 000~0
tension tension tension tension tension tension
ot the before before at the before before
top the the top the the
1st buoy 2nd buoy
\
1st buoy 2nd buoy
0.00 0.40
0.80
w(rad/s) Fig. 11.
1.
Deep water mooring lines with submerged buoys directly below the locations of buoys, a phenomenon discussed in some detail in Ref. [9]. Figures 9a and 9b refer to three-buoy configurations with various buoyancies. The buoys have the same diameter of 5.0 m and are of Type IX for the 20% case and of Type VII for the 50% case. It is evident from Fig. 9a, b, that as the net buoyancy increases, the maximum dynamic tension attains lower values along the line. From the three-buoy cases considered here, it appears that the 50% buoyancy case gives the most favourable results. Both peaks are less pronounced, whereas the dynamic tension amplificauon has an almost constant value in the whole frequency range of interest. The 50% case of the three-buoy configuration has also been treatedi using the time-domain analysis code. A representative sample is shown in Fig. lOa, b, where the dynamic and the total tension at the cable’s top and at the buoys’ attachment points are depicted for an excitation frequency equal to 0.344 rad SK’. For comparison purposes, Fig. lOa, b also shows the corresponding results for the no-buoy case. Comparisons between the time- and frequencydomain analysis results are given in Fig. 11 for a two-buoy configuration. Here, the buoyancy ratio equals the 35% of the submerged cable’s weight. The results compare favourably with each other for lower
833
frequencies. For higher frequencies, however, although the depicted trend of the time- and frequency-domain results remains the same, discrepancies are present. They can be traced back to the additional nonlinearities, which are incorporated into the time-domain solution scheme. Figure 12 is given in order to investigate the effect that the large cable’s upper end motion has on both its dynamic behaviour and the validity of the linearizations. Here, the dynamic tension variation at the top of a line having one submerged buoy attached to it at 2100 m from the anchor and offering to the system a net buoyancy of 50% of the submerged cable’s weight, is plotted against the excitation frequency for top tangential motions equal to 10 and 1 m. It is noted that the 1 m amplitude case has already been examined (see Fig. 5) and it is reproduced here for reasons of comparisons only. The frequency-domain results are supplemented by the corresponding ones obtained with the more accurate time-domain analysis. It is evident that an increase of the cable’s upper end motion affects the tension amplifications for both the buoy- and the no-buoy cases. In particular, it can be seen that the tension amplification is altered in the whole investigated frequency range for the no-buoy case, and in its lower part only for the case of the line with one buoy attached to it. Considering, moreover, the
Cable 3000 m one buoy, N.B. 50% cable’s weight. Tension at the top Buoy at 2100 m from anchor
a=lO.m
l_<:z? +++++
a= 1.m Nobuoy cosa a=lO.m Nobuoy case
1”“““‘1”““‘1’1”“1”“l1”““‘r
0.00
With buoy
0.50
1.oo w (rad/s) Fig. 12.
1.50
2. 0
834
S.
A. Mavrakos and J. Chatjigeorgiou
beneficial effect that buoys have towards the reduction of the dynamic tension build-up in the line, this feature is also still present in the case of the large cable’s top motions, though less efficient in comparison to the case of smaller motion excitations. Finally, we find that the frequency- and time-domain analysis results compare better between each other in the 1 m amplitude case than in the 10 m one. This can be traced back to the fact that in the case of 1 m motion amplitude, the nonlinear terms involved in the time-domain scheme to account for the large transverse cable’s motion do not essentially affect the results, which are primarily dominated by the nonlinear drag term. On the contrary, for the 10 m amplitude the influence of these terms seems to be more pronounced in obtaining reliable results, thus restricting the validity of the linearization in the frequency-domain analysis. Finally, numerical results from the time-domain analysis are given in Fig. 13 for the three-buoy configuration investigated in Fig. 9b. Here, the cable’s top is subjected to a bichromatic excitation. The two sinusoids have frequencies 0.312 rad s-’ and 0.772 rad s-l, whereas the corresponding amplitudes of motions in the tangential and transverse directions at the top are 0.486 and 0.49 m for the first- and 0.276 m and 0.175 m for the second-frequency,
400000
respectively. The time histories of the dynamic tension for the no buoy case are also given for comparison purposes.
5. CONCLUSIONS
Frequency- and time-domain analysis methods have been used to investigate the dynamic behaviour of wire mooring lines in deep water. The presented numerical schemes account for the dynamic modification caused to the system due to the insertion of submerged buoys attached on the mooring line. Besides the horizontal and vertical translations, the angular motion of the buoy around its attachment point has been taken into account in the formulation of the problem. It was shown that a considerable reduction of the dynamic tension build-up can be obtained through the insertion of buoys, even in the case of the large cable’s upper end motion, provided that their principal characteristics (number, size and location of attachment) are properly selected. The latter can form the basis of an optimization algorithm, as shown by the authors [9]. This possibility, however, should be further investigated before it becomes an intrinsic part of a rational design procedure.
Cable 3000 m, three buoys, N.S. 50% cable’s weight. Buoys ct 2100, 2450 and 2850 m from anchor _I frequencies:
.312
_ --
and
.772
tension at the nobuoy case
rad/s
top
-400000
-600000
I 1 r r ai I aI I I I I I I 0I I I I I I I I I I I I I I I 50.00 100.00 15c 0.00 Time(sec) Fig. 13.
.oo
Deep water mooring lines with submerged buoys REFERENCES
1. I. J. Fylling, Ci. 0. Ottera and F. Gottliebsen, Optimization and safety considerations in the design of stationkeeping systems. In: Proc. 3rd Int. Symp. on Practical Design
ofShipsand
Mobile Units, PRADS’87,
Trondheim, Norway, pp. 445461 (1987). 2. C. T. Kwan, Design practice for mooring of floating production systems. In: Proc. Offshore Station Keeping Symp., SNAME (1990). 3. M. S. Triantafyl‘lou, A. Bliek and H. Shin, Dynamic
analysis as a tool for open sea mooring system design. Trans. SNAME ‘93, 303-324 (1985). 4. DnV, Rules for Classification of Mobile Offshore Units. Position Mooring (POSMOOR), Part 6, Chap. 2. Det
Norske Veritas, Oslo (1989). 5. A.P.I., Recommended practice for design, analysis and maintenance of mooring for floating production systems (RP 2FPl). American Petroleum Institute (1991). 6. J. H. Milgram, M. S. Triantafyllou, F. C. Frimm and G. Anagnostou, Seakeeping and extreme tensions in offshore towing. Trans. SNAME 96, 35-72 (1988). 7. V. J. Papazoglou, S. A. Mavrakos and M. S. Triantafyllou, Non-linear cable response and model testing in water. J. Sound Vibr. 140, 103-I 15 (1990). 8. H. Shin, Analysis of extreme tensions in a snapping cable. In: Proc. of the 1st Int. Offshore and Polar Engineering Conf., ISOPE’91, Edinburgh, Vol. II, pp. 216221 (1991).
835
9. S. A. Mavrakos, L. Neos, V. J. Papazoglou and M. S. Triantafyllou, Systematic evaluation of the effect of submerged buoys’ size and location on deep water mooring dynamics. In: Proc. of the 4th Int. Symp. on Practical Design of Ships and Mobile Units, PRADS’89,
Varna, Bulgaria, Vol. 3, pp. 105.1-105.8 (1989). 10. S. A. Mavrakos, V. J. Papazoglou and M. S. Triantafyllou, An investigation into the feasibility of deep water anchoring systems. In: Proc. 8th Int. Offshore Mechanics and Arctic Engineering Conf., OMAE’89, Vol. 1, The Hague, The Netherlands,
pp. 683-689 (1989). 11. S. A. Mavrakos, V. J. Papazoglou, M. S. Triantafyllou and P. Brando, Experimental and numerical study on the effect of buoys on deep water mooring dynamics. In: Proc. 1st Int. Offshore and Polar Engineering Co&. ISOPE’91, Vol. Ii, Edinburgh, pp. 243-251 (1991). ”
12. S. A. Mavrakos, J. Chatjigeorgiou and V. J. Papazoglou, Use of buoys for dynamic tension reduction in deep water mooring applications. In: Proc. 7th Int. Conf. on the Behaviour of Offshore Structures, BOSS’94, Vol. 2, pp. 417426. MIT, Cambridge, MA
(1994). 13. L. P. Krolikowski and T. A. Gay, An improved linearization technique for frequency domain riser analysis. In: Proc. Offshore Technology Co&.. OTC 3777, Houston, TX, pp. 341-353 (1980). ” Guide. The Math Works (1993) 14. MATLAB-Users 15. R. W. Clough and J. Penzien, Dynamics of Structures. McGraw-Hill, New York (1993).
Pergamon
Compurers & Srrucrures Vol. 64, No. 14, pp. 819-835, 1997 1997 Civil-Comp Ltd and Elsevier Science Ltd. All rights reserved Printed in Great Britain 0045-7949/97 $17.00 + 0.00
PII: 80045-7949(%)00169-l
DYNAMIC
BEHAVIOUR OF DEEP WATER MOORING LINES WITH SUBMERGED BUOYS S. A. Mavrakos and J. Chatjigeorgiou
Department of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens 15773, Greece Abstract-The paper deals with the dynamic behaviour of wire mooring lines in deep water. Numerical methods in the time- and the frequency-domain are presented to evaluate the dynamic tensions and the mooring line motions. Special attention has been paid to the assessment of the impact that submerged buoys have on the dynamic behaviour of mooring lines on which they are attached. The modification of the mooring line dynamic performance in dependence on the buoys’ characteristics (number, size and location of a.ttachment) has demonstrated through a parametric numerical study. 0 1997 Civil-Comp Ltd and Elsevier Science Ltd.
1. INTRODUCTION Mooring systems are used extensively in the offshore field of applictions to position floating drilling or production systems and other marine vehicles. Special attention has been paid in the last few years
in deep water applications, where dynamic considerations should be incorporated in a rational design procedure. This has been pointed out by several investigators [l-3] and led to a modification of the design rules and guidelines of several organizations [4, S]. The quasi-static approach, suitable for the design of catenary mooring systems in shallow water (depth below approximately 100 m) [5], is generally considered inadequate for deep water applications. In such cases, the maximum attainable value of the dynamic tension a.mplification, being approximately equal to the elastic stiffness of the line for nonsnapping conditions, is shifted within the wave frequency range and, thus, large dynamic tensions due to the vessel’s first-order motions can be excited [3]. It is therefore evident that for such cases the principal parameter affecting the dynamic response of the line is its elastic stiffness and, hence, its reduction would improve the dynamic performance of th.e line. On the other hand, when a snapping condition has occurred, i.e. when the dynamic tension exceeds the static one in certain parts of the cable, hence causing the cable to become slack for part of the cycle and then to tighten suddenly, the principal parameter controlling this phenomenon is the free-falling velocity of the ca.ble. The latter can be calculated assuming that the cable’s weight is supported by the drag force when it becomes slack [6, 71. It was found that by increasing the free-falling velocity (for
example, by making the line heavier), a considerable reduction of the peak dynamic tension during snapping can be achieved [68]. Considering nonsnapping cables, it has recently been demonstrated numerically [9] and confirmed experimentally in a series of small- and large-scale experiments [lo, 111, that the beneficial reduction of the cable’s elastic stiffness can be achieved through the dynamic modification of the system, using properly designed buoys attached to the mooring lines. The buoy-cable system behaves like an inverted pendulum, with the buoyancy acting as a gravity force. Thus, the proper selection of the number, size and location of attachment of these buoys can manipulate the natural frequencies of the buoy-cable system to the benefit of the mooring line’s dynamic tension reduction, thus forming the basis of an optimization algorithm. A first attempt towards the development of an optimization algorithm for deep water mooring applications has recently been presented by exploiting the dynamic features of a double pendulum in air [ 121. In the present contribution, extensive numerical results of a parametric study are presented dealing with the systematic evaluation of the effect that the buoys’ characteristics have on the dynamic behaviour of deep water mooring lines. Both frequency- and time-domain analysis methods have been applied to obtain the numerical results. The former involves an equivalent linearization technique to accommodate the effect of the quadratic drag on the cable and the attached buoys. The latter, which additionally accounts for geometric nonlinearities due to large cable motions, makes use of a simplified formulation of the nonlinear governing equations, which is suitable for mooring applications with wire cables. The present contribution extends previous related 819
S. A. Mavrakos and J. Chatjigeorgiou
820
work [9, 121 by including in the formulation of the problem the buoys’ rotational motion around their attachment point.
2. GOVERNING
infinite long cylinder with the same cross-section, and cp denotes the angle between the horizontal and the local tangential direction of the cable in the moving coordinate frame. It possesses one static, ‘pa, and one dynamic component, cp,. The former is the angle between the horizontal and the static tangential direction of the cable, b, whereas the latter denotes its dynamic deviation from the static equilibrium condition. Finally, a constitutive tension-strain relation has to be added, which in the present contribution is assumed to be a linear one
EQUATIONS
The cable is modelled as a slender rod with negligible bending stiffness. Let s denote the unstretched Lagrangian coordinate measured from the lower cable end up to a material point of the cable, v (u, v) its velocity vector, m the mass per unit unstretched length, T the tension vector (pointing along the local tangential direction, characterized by its unit vector t), F (fi, F.) the vector of the distributed external force per unit stretched length and e the local strain. Then the equations of the in-plane motion of a cable having a two-dimensional static shape can be expressed along the local tangential, t, and normal, n, directions of its moving configuration (moving dynamic reference) as [3,91
m{~-v~}=g+fi(l+e)
T = EAe.
The external forces F, and Fn, include the submerged weight of the cable per unit stretched length, W, and the fluid drag forces which are obtained on the basis of the so-called separation principle, i.e. relating the large normal force to the relative fluid-cable velocity in the normal direction, and the tangential force to the corresponding relative velocity in the tangential direction
(1)
F, = - w sin 4 + 0.5rrp,crd
x(Ucos$~-u)]Ucosd-u]
(m+a)$+muz= T$+F.(l de
au
acp
z=as-“-i%
+e)
(2)
(4) where c1is the two-dimensional added mass, which is assumed to be equal to its potential-flow value for an
(6)
F, = - w cos 4 - O.Sp,cdd
(Usinf$+v)(Usinr#r+v]. (3)
(5)
(7)
Here, cf, cd are the friction and normal drag coefficients, respectively, pWis the water density, d the stretched cable diameter and U the local current velocity (assumed horizontal). Equations (l)-(5) form the complete set of the nonlinear, two-dimensional dynamic equations of the mooring line motion, For cables with two-dimensional static configuration, the
ZA
Current
Fig. I. Buoy attached on the mooring line.
Deep water mooring lines with submerged buoys
out-of-plane motion is first order decoupled from the in-plane one [3]. In the case of buoys attached to the cable, eqns (l)(4) must be supplemented with the corresponding ones of the buoys’ motion. Let M and M, be the mass and the added mass of the buoy, respectivedly, J, and J,, its mass- and added-moment of inertia about C, B0 its net buoyancy in water, A, its projected surface and cM the drag coefficient. Then, the equations of motions for a buoy located at the junction of two successive segments of the cable (i and i + l), and undergoing small translational and rotational motions in the X-Z plane with its centre of gravity coinciding with the centre of the displaced water volume, are (see Fig. 1)
Table 1. Cable data for Figs 2-12 Length Mass per unit length Added mass per unit length Diameter Submerged weight per unit length Water depth EA Normal drag coefficient Tangential drag coefficient
rp in
(l)-(4)
eqns
- T; cos qbi+ D,
ap
u = at cos
(8)
(M + Ma)&, = T., , sin &+,
- T sin #ii + 0; + B. (A4 + M.)R&
(9)
-t- (Jc + Jac)e’ =: - BoRd + (D, - D,O)Rb.
Lj,
v, = ,,/(u
=
&CM
(ib
+
Rbd))
A, Kib
- (ii, + Rd)’
invoking
fpl
+
a4an.
5
-aps. at
the
-
following
ql
(14)
a4cos CPI
(1%
ln CPI + -g
which relate the velocity, (u, v), and the displacement, (p, q), components in the moving and the static reference coordinate systems, a set of simplified equations, correct to second order, can be obtained after lengthy mathematical manipulations for the in-plane cable’s motion:
(10)
(16)
Here dots represent time derivatives, x0, zb are the motions of the buoy’s attachment point in the horizontal and vertical directions, 6 the buoy’s angle of rotation about C, Rb its radius, Ti+, , (pi+, and Ti, ‘pi are the total tension and angle in the segments i + 1 and i, respectively, at the buoy’s attachment point and D, = ;,&CW’&v,(u -
and
3000m 94.5 kg m-i 16.0 kgm-’ 0.141 m 770 N m-’ 375 m 1*109N 1.2 0.05
expressions:
(j=
= 7;+,cos&+,
821
aT,
hh
+F.I +~‘PI
+ T,as
g-q!$+&=e,
(11)
(17)
of9
(12) + ii.
$+p%=~,(l
(13)
The nonlinear governing equations for the cable’s and buoy’s dynamics, i.e. eqns (l)-(4) and (8x10), respectively, can be simplified, so that without losing on important nonlinear contributions, effective numerical schemes can be applied for their treatment. In this simplifiction procedure the assumption is made that the dynamic tensions and the cable’s motions are small around its static configuration. The latter defines a new reference coordinate system. By expanding the set of nonlinear eqns (I)-(4) about this static frame, the equations of motions for the cable can be expressed in terms of its tangential, p and normal, q, displacements. Let T,, cpI be the components of the dynamic tension and the dynamic angle, respectively, and To, cpo their static values, i.e. we set T = To + T, and cp = cp0+ cpl. Substituting the expressions for T and
+e0),
(19)
where fiI = (fi - W,)(l
+ e) -
F,0(1
+
e0)
(20)
Table 2. Buoys’ characteristics
Buoy Type Type Type Type Type Type Type Type Tvue
tyPe I II III IV V VI VII VIII IX
Mass (kg) 29,651 33,508 19,991 8218 25,878 43,538 27,840 39,614 51,387
Added mass (kg)
Net buoyancy (%l
Diameter (m)
73,694 57,963 33,543 33,543 33,543 33,543 33,543 33,543 33,543
50.00 35.00 20.00 25.00 17.50 10.00 16.67 11.67 6.67
6.50 6.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
S. A. Mavrakos and J. Chatjigeorgiou
822
Static :oniiauration 4 Cable 3COG r-77 ct 2100 m irark
;vith one cnchor
buoy
j--i
7--
1000.0 horizontal
I.0
3000.0 2000.0 distance(m)
Fig. 2.
F.1 = (F,CPL+ F”)(l + e) - F&l + eo).
-
(21)
Here, F, and F,, are given by eqns (6) and (7), respectively, whereas Fto and F,o denote their static values, obtained by inserting in eqns (6) and (7), u = u = 0. In eqns (16x19) subscripts 0 and 1 are used to distinguish static from first-order dynamic quantities. Besides the nonlinear drag forces involved in eqns (16)-(19), only the last two terms in the right hand side of eqn (17) represent nonlinear contributions. The first of them accounts for large dynamic tension variation along the cable, whereas the second one for geometric nonlinear effects due to large transverse in-plane motion of the cable. Neglecting these two nonlinearities, the well-known two-dimensional, linearized equations of motion of a cable can be obtained, accounting for the effects of quadratic drag only [3]. By applying a similar linearization procedure to the equations of the buoy’s motion, i.e. eqns (8X10), its dynamic equilibrium along the tangential and normal directions of the static cable configuration can be expressed as
To,~w
sin$ =
Ti,i+I + Dp,i cos $
- DqVisin $
(22)
(M + M&h sin $ + (M + M&j, cos I(/ + TI,~sin $ +
TO,FPI,~
cos
$
=
To,i+ I(pl.i+I
+ Dp,i sin $ + Dq,i cos t,b
(M + Ma)]&& cos +(J+
J&e;
=
x RdD,., cos
~0,~ -
RG& sin ~p~.~]
BoR&
-
(~0.~ -
(23)
+
D,,; sin cpo,,).
(24)
Here, Dp,i = (Ox - D&OS cpo,i+ D, sin ~p~.~, Dq,i = - (Ox - D&in ~0.~ + D, cos cpoei, with D, and D, being given by eqns (11) and (12), respectively, and D,, is the static force on the buoy in the horizontal
direction for & = ib = B = 0; $ = qo,i - ~p~,~+ ,. In deriving the above equations, the following kinematic compatibility relations at the buoy-cable junction have been taken into account: xb
=
pi
cos
p0.i
-
qi
sin
f$O,i
-ql+lsin~o,i+,
=P/+ICOS~O,~+~
(25)
(M + Ma)pi COS* - (M + Ma)41sin $ tb
+W + Ma)Rtae;+ T,,
cos +
=
pi
sin
40,i =
+
qi cos
4O.i
p,+~sin~o.i+I-qi+Icos~oo,i+,.
(26)
823
Deep water mooring lines with submerged buoys
The second-order governing equations (16x19) can be further simplified in cases where the excitation frequencies are assumed to be well below the first elastic frequency of the cable. This assumption, which is a reasonable one for most applications with wire cables, implies almost uniformly distributed dynamic tension along the cable and smaller tangential than normal displacements. As a result, the second-order term in eqn (17), which includes differentiation of the dynamic tension with respect to the cable’s length, can be neglected. Moreover, in eqn (19), the dynamic angle due to the axial motion of the cable can be considered small with respect to the angle induced by dynamic lateral motions. Using these assumptions, together with the hypothesis of large static tension relative to the total weight of the cable, eqn (16) becomes a trivial one, whereas eqn (17) can be expressed as follows [9]:
(m + a) 3
==(To + T,) !$ + T, dJ$ - Fd (27)
where
I 1 - u;
4.00
,
(28)
UN = U sin I#J z=A{pi-p,-[q$+/.
(29) (30)
In eqn (30), p2 and p, denote the upper and lower end displacements, respectively, L is the suspended cable length and A is the equivalent stiffness of the line. The latter accounts for additional elasticity provided to the system by the axial and transverse motions of the lower, on the bottom lying, part of the cable, if any. Let 1, be its length, EC its modulus of elasticity and A, its cross-sectional area, then A is given by 1 1 1 x = (E,A,J1,) + (EA/L) .
(31)
The second term in the parenthesis on the right hand side of eqn (27) accounts for geometric nonlinear effects due to large transverse motions of the cable, whereas the third term describes the cable’s ability to straighten under a dynamic tension. The simplified eqns (27) and (30) can be used to study the dynamics of a snapping cable as well. This has been demonstrated in Ref. [7] where reference is made for more details.
Cable 3000 m one buoy, at from anchor. Tension at the
2450 top
.______ buoy’s N.B. SOW’*,\ - - buoy’s N.B. 35% ‘\\ buoy’s N.B. 20% / A*&&& cable witout ,,,(,,, buoy , 0.00 -j1!L111111,111111111 0.00 0.50 1 .oo 1 20 w (rad/sec) Fig. 3.
m
t.
2.
S. A. Mavrakos and J. Chatjigeorgiou
824 3. SOLVING
THE GOVERNING
EQUATIONS
on the buoys in eqns (22)-(24) is concerned, this is achieved by having them expressed in Fourier series using the commercial software package MATLAB [14] and then retaining the mean and the harmonically oscillating terms only. Having the linear damping force determined in this way, the system of the linearized eqns (16~(19) is reduced to a set of four ordinary differential equations with two boundary conditions at each end, which is solved in the Fourier domain using an explicit centred difference scheme. The linearized dynamic equations of cable’s motion can be written in matrix form as
For cables excited sinusoidally at their top end, the drag forces on them and the attached buoys can be equivalently linearized. Thus, in cases where the last two nonlinear terms on the right hand side of eqn (17) can be omitted, the system of eqns (16)-(19) can be numerically treated in the frequency domain. The proposed linearization procedure, which is similar to the one already presented by the authors [12] relies on a decomposition of the drag forces into a mean, plus an equivalent linear damping force by minimizing the quadratic error between the equivalent representation and the actual nonlinear drag force averaged over one period. The mean force can be included in the static analysis, whereas the linear damping force is introduced in the linearized dynamic equations to solve them in the frequency domain. The procedure is similar to the one proposed by Krolikowski and Gay [ 131.As far as the linearization of the drag forces
1 dqo
-
Y’(S) = (TI&dpaqa)
(33)
and 0
-mm2 ---
To ds
A(s) =
(32)
where
0
---
2 =A(s).Y(s),
1 dTo
0
To ds
1
0
EA
[(m + c()o.?+ ibw]
-
TO
(34)
dqo ds
0
1
0
4.00
m cne L”cble X00 cable’s weight.Tension -,:---
buov, N.5. 20% a’t the top
3.00
u-l W
6 2.00 >
A* * AA cable witout buov buoy at 2100 n - - buoy at 245 .____-- buoy at 28:
0.00
I 0.00
I
I
/
I
I
I
1
I,,
0.50
I,,
I
I
I
I
I
I,,
,
,
1 .oo w (rad/sec) Fig. 4.
,
,
,
,
,
I,,
1.50
,
(
(
,
,
,
,
,
2.1
825
Deep water mooring lines with submerged buoys
Here, the subscript a denotes the amplitude of the corresponding dynamic quantity and b is the equivalent damping coefficient for the cable. This coefficient, which is a motion-dependent quantity, is used together with a difference scheme to determine the unknown local amplitude of oscillations by an iterative procedure. The difference scheme has the form
-I 1
Yj+I =
I-?FA(Sj+,)
[
x
and by
can be obtained discretizing the
I
A(S)) yj
using centred cable into
yb,i+
(-
w’kfb
sin II, -.
_7’o.icos $ T0.1-f I
(-
02hfb
(36)
Ab’yb.i,
(35) & = (T,,,~rp,,,,p,,,q,,,~e,)
differences n nodes
- TO,]sin $
I =
where yb,i+, and yb,j are vectors calculated at the buoy’s attachment point in the segments i + 1 and i, respectively. They are defined by
1
-w
cos 3 TOJf & =
[
0’ = 1,2, . . . , n - 1). The procedure gives acceptable results with an overall error 0(As2), As being the grid length. The equations of the buoy’s motion, i.e. eqns (22)(24), can accordingly be written in the following matrix form:
The transformation
+
iob,)cos $
+
iob,)sin rj
(02kfb
(-
+
k = 4 i + 1. (37)
matrix Ab is given by
- W2M&
iwb,)sin If+
-
w2hfb
for
iob,)cos $
To,;+ I
To,;, I
0
0
0
cos *
- sin $
0
0
0
sin $
cos $
0
0
0
4.00
(-w2Mb
+
COS q%,i
i&)Rb
(w2hfb
W2Jb- B,,Rb
Cable 3000 m cable’s weight.
- iob,)Rb sin (pO,/ d&a - &Rb
one buoy, N.E. 50% Tension at the top
I
-I
A A A A A cable - - buoy ______ buoy buoy
without at 2100 at 2450 at 2850
w
b&v \ m ‘+ m
(rad/sec)
Fig. 5.
p
I
0
826
S.
A. Mavrakos and J. Chatjigeorgiou
Here, Mb = M + M, and & = J + J,,, whereas the damping coefficients b, and 6, are defined by Dr., = - &,.A and D,,, = - bq.gr, respectively. In cases where, in addition to the quadratic drag forces, nonlinearities arising from large transverse in-plane cable- and buoy-motions have to be taken into account, then eqn (27) together with eqns (8)-( 10) can be solved using a time-domain simulation scheme. According to this scheme, the solution of the partial differential equation (27) is expanded in each cable’s segment into an infinite sum of sinusoids, plus additional terms to accommodate the boundary conditions at its end, i.e.
inserting the solution for q, i.e. eqn (38), into the relation (30)
T&T! ’
x
L
p2 -pl
- I,qz - (14 - I&, -
“= I
Q. sin(nax),
&+
b,(m,Qnz =;(-@I
+ ii2(-1)“‘) + c
n-1
nc “I Q.I,(n, m) + (bdm)Q,
-2
&
+ h(m))T,
sin(mnx)dx,
(39)
(40)
(41)
=
2
’ 1 so m+a
sin(nnx)sin(mnx)dx
(42)
d% . sm(mnx)dx. ds
(43)
Moreover, Fd is given by eqn (28), whereas the dynamic tension, T,, involved in eqn (39) can be expressed in terms of the temporal coordinates Qn by
‘x&dx ds
0
14= L
I&)
, (44
(45)
(46)
s
’ dq, sm(nrrx)dx. .
= L oz
(47)
Each component of the temporal coordinates, Qm, defined by the set (38) is then integrated using Newmark’s method [ 151. By incorporating the coupled equations of motion for the buoy, i.e. eqns (8X10), the proposed time domain simulation scheme extends the one presented in Ref. [9] by including buoy rotations around its attachment point on the cable. 4. NUMERICAL
where
h(m)
s
(38)
where x = s/L, s being Lagrangian coordinate and L the segment’s length. The latter is defined as the distance between buoys, or between a buoy and one cable’s end or between ends. The temporal coordinates Q. involved in eqn (38) are the principal unknown of the problem. They are derived from a set of ordinary differential equations which are obtained by implementing Galerkin’s method. To this end, eqn (38) is first substituted for q in the governing equation (27). Both sides of the resulting expression are then multiplied by sin(mnx) and integrated in their definition interval, yielding
QJ&z,
where
I 3 CL
q(x, t) = q2x + q,(l - x) + 1
2
II= I
RESULTS
AND DISCUSSION
Using the frequency- and the time-domain analysis methods presented in the previous section, a systematic study was conducted to assess the impact that one or more buoys have on the dynamic behaviour of the lines to which they are attached. The data for the selected all-wire mooring line are shown in Table 1. The water depth is assumed to be 375 m, whereas the horizontal force at the top is equal to 7.8 x IO6N. The line is excited tangentially at its top with an amplitude of 1.0 m. The basic mooring line configuration does not have any buoy attached to it. To study the effect that one or several buoys have on the dynamic behaviour of the line to which they are attached, one, two and three spherical buoys are considered to be attached to the mooring line having characteristics summarized in Table 2. A total of nine types is used, each one giving by itself or in combination with the others net buoyancy equal to 20, 35 or 50% of the submerged cable’s weight. The buoys’ drag coefficient is assumed to be equal to 0.8. The static configurations of the mooring line without buoys and with a buoy attached to it at a distance 2100 m from the anchor are shown in Fig. 2. The buoy size is varied, so that its net buoyancy
827
Deep water mooring lines with submerged buoys Ccile cable’s
(a) 4.00
3000 m weight,
two buoys, Tension at
N.8. 20% the top
*PA&A cable witout buoys
a
-A-
d
1
buoys at buoys ct buoys at
-a---
!
2100 2450 Z:CC
and and end
I
2850 2850 2150
m m rr
3.00
ul W
6 2.00 >
1 .oo
0.00
11 0.00
0.50
1.00 w
(b)
2.00
Cable 3000 m two buoys, N.8. 2055 cable’s wei ht. Location of the buoys: 2100 and 4 850 m from anchor
Tension
at various
4.00 A -
__
locations ‘A&A*& at at
A
0.00
1.50
(rad/sec)
A
f
--
cz --&_
-
top tap before before
without with
the
1st buoy
the
2nd
;~~~~~~~~~,~~~~~,,,,,,,,,,,,~,,~,~,,,,,,~ 0.00 0.50 1 .oo 1.50 w
(rad/sec)
Fig. 6.
buoys
buoys buov
2.00
S. A. Mavrakos and J. Chatjigeorgiou
828
Cable cable’s
(a)
,
4.00
3000 m weight.
two buoys, Tension at
~~hba .a
N.B. 35% the top
cable wiiout buoys buoys at 2100 and buoys at 2450 and buoys at 21CO and
& -o --P
2850 2850 2350
r r r
3.00
u-l
I
W 6 2.00 >
1 .clo A
0.00
1
0.00
0.50
1.50
1 .oo
2.1
w (rad/sec) 09 4 o.
Cable 3000 m two buoys, N.B. cable’s wei ht. Location of the 2100 and s 850 m from anchor Tension at various locations oa ~AA at top A A ..
r-r
_ --
d /
N
=
ot
top
before before
c--
3% buoys:
without buoy: with buoys the 1st buoy the 2nd buoy
. d,
‘1-F
A
A A
,,,(,,~,,1,,1,,~,,1,,,~
0.00 0.00 ;11,,,,,11,,,,,, 0.50
8 1.00 w (rad/sec) Fig. 7.
1.50
2.i 0
829
Deep water mooring lines with submerged buoys
(4 4.00
Cable cable’s _I
3000 m weight.
two buoys, N.E. 50~ Tension at the top &AA&~ cable witout buoys buoys at 2100 and 2850 buoys at 2450 buoys at 2100
and and
2850 2450
_ w (rad/sec)
(b)
Cable 3000 m two buoys, N.B. cable’s wei ht. Location of the 2100 and s 850 m from anchor Tension at various locations
4.00
ibid. A a
4 T-4---
3.00
u-l I W 6
2.00 d
>
1 .oo
0.00
_ --
b at
tap
at
top
before before
50~ buoys:
without
buoys with buoys the 1st buoy the 2nd buoy
Cable Buoys Total Fnsion
(a) 4.00
SOCO m.with three buoys ar 2100, 2450, ‘2850 m net buoyancy 20% cable’s at various locations
from anchor weight
,
4
h
1 .oo
---
before _____ before
0.00
4.00
-
~~,,,,l,ll,,,,,,,,,,,l,,,,,,,,,,,,i,,,,, 0.00
(b)
the 2nd buo the 3rd buoy
1.00 w (rad/sec)
0.50
1.50
Cable 3000 m.with three buoys Buoys at 2100, 2450, 2850 m Total net buoyancy 50% cable’s Tension at various locations
2.t 0
from anchor weight
h A p-, /
A
--.-_
3.00
U-I I W i
2.00
>
1 .oo
h
0.00
0.00
0.50
1 .oo
w (rad/sec) Fig. 9. 830
1 so
2.1 0
Deep water mooring lines with submerged buoys
831
Cable 3000 m, three buoys, N.B. 50% cable’s weight. Buoys at 2100, 2450 and 2850 m from anchor
(4 400000
2 frequency: I ,344 r-ad/s
$
-200000~
5: Cl
-400000
1 _
-600000
a&A&A nobuay case _ -___ ____-
, a,, 0.00
tension tension tension tension
at the before before before
top the 1st buoy from anchor the 2nd buoy from oncho r the 3rd buoy from anchal
, , v, t,, ,a, 1 t 1, I,, 50.00 100.00 Time(sec)
Cable 3000 m, three buo ;b~.Bi4~% cable’s weight. Buoys at is , and 2850 m from anchor
W 8200000
frequent : .344 ro dy/s
8000000
1
7800000
-
A\
z
b’
z Q 9 w, I-
, , , , b,,
-+
z; 7600000 Cl I-
100.00
50.00
Time(sec) Fig. IO.
, 15c 1. 00
832
S.
A. Mavrakos and J. Chatjigeorgiou
ranges from 20 to 50% of the submerged cable’s weight. In Fig. 3 the dynamic tension at the top of the cable having one buoy attached to it at 2450 m from the anchor is plotted against the frequency of excitation, with parametric dependence on the net buoyancy values. A considerable modification of the dynamic behaviour of the cable is obtained entirely due to the buoy. Its presence causes the minimum value of the dynamic tension to appear in all cases at lower frequencies than the one corresponding to the first elastic mode of the cable. The substantial difference between the two minima lies in the fact that in the former case the low values of the dynamic tension appear along the whole segment of the cable above the buoy, whereas in the latter the minimization occurs only at the cable’s top [9]. It is apparent from Fig. 3 that, as the net buoyancy of the buoy increases, the minimization frequency attains lower values, whereas the minimum value of the tension remains almost constant. As far as the maximum values of the dynamic tension are concerned, it seems that they are greatly affected by the amount of net buoyancy offered to the system. The higher the buoyancy is, the lower the maximum attainable dynamic tension. Figures 4 and 5 show results concerning the dynamic tension variation at the top of a cable having
one buoy attached to it at various locations and offering to the system net buoyancies equal to 20 and 50% of the submerged cable’s weight. It can be seen that the frequency of the dynamic tension minimization is almost insensitive to the buoy’s location, depending primarily on the buoy’s size. Numerical results for a two-buoy configuration with total net buoyancy equal to 20, 35 and 50% of the submerged cable’s weight are presented in Figs 6a, b, 7a, b and 8a, b, respectively. In each arrangement the two buoys have the same diameter and are of Type IV for the 20% case, of Type V for the 35% case and of Type IV for the 50% case (see Table 2). Figures 6a, 7a and 8a show the dynamic tension variation at the top for various locations of the buoys. A considerable reduction of the maximum dynamic tension is observed for the 50% case, as compared with the 20 and 35% cases, as well as with the no-buoy case. The dynamic tensions exhibit two peaks which are more pronounced for the buoys’ locations at 2100 and 2450 m from the anchor. In order to investigate the variation of dynamic tension along the cable’s length, Figs 6b, 7b and 8b are given. Here, the dynamic amplification per unit imposed upper end cable motion is plotted against the excitation frequency. It can be seen that significant tension amplifications may occur at points
Ccble 3000 m wiih~iwo buoys Buoys at 2100 and 2850 m from anchor Total net buoyancy 35% cable’s weight 4.00 Lines Symbols
* frequency domain ‘: time domain
S - 3.00 IfI & Y l
5 E 2.00 E 0 Z S & 1 .oo
_ -
_ -._____ ***** AAAAA 000~0
tension tension tension tension tension tension
ot the before before at the before before
top the the top the the
1st buoy 2nd buoy
\
1st buoy 2nd buoy
0.00 0.40
0.80
w(rad/s) Fig. 11.
1.
Deep water mooring lines with submerged buoys directly below the locations of buoys, a phenomenon discussed in some detail in Ref. [9]. Figures 9a and 9b refer to three-buoy configurations with various buoyancies. The buoys have the same diameter of 5.0 m and are of Type IX for the 20% case and of Type VII for the 50% case. It is evident from Fig. 9a, b, that as the net buoyancy increases, the maximum dynamic tension attains lower values along the line. From the three-buoy cases considered here, it appears that the 50% buoyancy case gives the most favourable results. Both peaks are less pronounced, whereas the dynamic tension amplificauon has an almost constant value in the whole frequency range of interest. The 50% case of the three-buoy configuration has also been treatedi using the time-domain analysis code. A representative sample is shown in Fig. lOa, b, where the dynamic and the total tension at the cable’s top and at the buoys’ attachment points are depicted for an excitation frequency equal to 0.344 rad SK’. For comparison purposes, Fig. lOa, b also shows the corresponding results for the no-buoy case. Comparisons between the time- and frequencydomain analysis results are given in Fig. 11 for a two-buoy configuration. Here, the buoyancy ratio equals the 35% of the submerged cable’s weight. The results compare favourably with each other for lower
833
frequencies. For higher frequencies, however, although the depicted trend of the time- and frequency-domain results remains the same, discrepancies are present. They can be traced back to the additional nonlinearities, which are incorporated into the time-domain solution scheme. Figure 12 is given in order to investigate the effect that the large cable’s upper end motion has on both its dynamic behaviour and the validity of the linearizations. Here, the dynamic tension variation at the top of a line having one submerged buoy attached to it at 2100 m from the anchor and offering to the system a net buoyancy of 50% of the submerged cable’s weight, is plotted against the excitation frequency for top tangential motions equal to 10 and 1 m. It is noted that the 1 m amplitude case has already been examined (see Fig. 5) and it is reproduced here for reasons of comparisons only. The frequency-domain results are supplemented by the corresponding ones obtained with the more accurate time-domain analysis. It is evident that an increase of the cable’s upper end motion affects the tension amplifications for both the buoy- and the no-buoy cases. In particular, it can be seen that the tension amplification is altered in the whole investigated frequency range for the no-buoy case, and in its lower part only for the case of the line with one buoy attached to it. Considering, moreover, the
Cable 3000 m one buoy, N.B. 50% cable’s weight. Tension at the top Buoy at 2100 m from anchor
a=lO.m
l_<:z? +++++
a= 1.m Nobuoy cosa a=lO.m Nobuoy case
1”“““‘1”““‘1’1”“1”“l1”““‘r
0.00
With buoy
0.50
1.oo w (rad/s) Fig. 12.
1.50
2. 0
834
S.
A. Mavrakos and J. Chatjigeorgiou
beneficial effect that buoys have towards the reduction of the dynamic tension build-up in the line, this feature is also still present in the case of the large cable’s top motions, though less efficient in comparison to the case of smaller motion excitations. Finally, we find that the frequency- and time-domain analysis results compare better between each other in the 1 m amplitude case than in the 10 m one. This can be traced back to the fact that in the case of 1 m motion amplitude, the nonlinear terms involved in the time-domain scheme to account for the large transverse cable’s motion do not essentially affect the results, which are primarily dominated by the nonlinear drag term. On the contrary, for the 10 m amplitude the influence of these terms seems to be more pronounced in obtaining reliable results, thus restricting the validity of the linearization in the frequency-domain analysis. Finally, numerical results from the time-domain analysis are given in Fig. 13 for the three-buoy configuration investigated in Fig. 9b. Here, the cable’s top is subjected to a bichromatic excitation. The two sinusoids have frequencies 0.312 rad s-’ and 0.772 rad s-l, whereas the corresponding amplitudes of motions in the tangential and transverse directions at the top are 0.486 and 0.49 m for the first- and 0.276 m and 0.175 m for the second-frequency,
400000
respectively. The time histories of the dynamic tension for the no buoy case are also given for comparison purposes.
5. CONCLUSIONS
Frequency- and time-domain analysis methods have been used to investigate the dynamic behaviour of wire mooring lines in deep water. The presented numerical schemes account for the dynamic modification caused to the system due to the insertion of submerged buoys attached on the mooring line. Besides the horizontal and vertical translations, the angular motion of the buoy around its attachment point has been taken into account in the formulation of the problem. It was shown that a considerable reduction of the dynamic tension build-up can be obtained through the insertion of buoys, even in the case of the large cable’s upper end motion, provided that their principal characteristics (number, size and location of attachment) are properly selected. The latter can form the basis of an optimization algorithm, as shown by the authors [9]. This possibility, however, should be further investigated before it becomes an intrinsic part of a rational design procedure.
Cable 3000 m, three buoys, N.S. 50% cable’s weight. Buoys ct 2100, 2450 and 2850 m from anchor _I frequencies:
.312
_ --
and
.772
tension at the nobuoy case
rad/s
top
-400000
-600000
I 1 r r ai I aI I I I I I I 0I I I I I I I I I I I I I I I 50.00 100.00 15c 0.00 Time(sec) Fig. 13.
.oo
Deep water mooring lines with submerged buoys REFERENCES
1. I. J. Fylling, Ci. 0. Ottera and F. Gottliebsen, Optimization and safety considerations in the design of stationkeeping systems. In: Proc. 3rd Int. Symp. on Practical Design
ofShipsand
Mobile Units, PRADS’87,
Trondheim, Norway, pp. 445461 (1987). 2. C. T. Kwan, Design practice for mooring of floating production systems. In: Proc. Offshore Station Keeping Symp., SNAME (1990). 3. M. S. Triantafyl‘lou, A. Bliek and H. Shin, Dynamic
analysis as a tool for open sea mooring system design. Trans. SNAME ‘93, 303-324 (1985). 4. DnV, Rules for Classification of Mobile Offshore Units. Position Mooring (POSMOOR), Part 6, Chap. 2. Det
Norske Veritas, Oslo (1989). 5. A.P.I., Recommended practice for design, analysis and maintenance of mooring for floating production systems (RP 2FPl). American Petroleum Institute (1991). 6. J. H. Milgram, M. S. Triantafyllou, F. C. Frimm and G. Anagnostou, Seakeeping and extreme tensions in offshore towing. Trans. SNAME 96, 35-72 (1988). 7. V. J. Papazoglou, S. A. Mavrakos and M. S. Triantafyllou, Non-linear cable response and model testing in water. J. Sound Vibr. 140, 103-I 15 (1990). 8. H. Shin, Analysis of extreme tensions in a snapping cable. In: Proc. of the 1st Int. Offshore and Polar Engineering Conf., ISOPE’91, Edinburgh, Vol. II, pp. 216221 (1991).
835
9. S. A. Mavrakos, L. Neos, V. J. Papazoglou and M. S. Triantafyllou, Systematic evaluation of the effect of submerged buoys’ size and location on deep water mooring dynamics. In: Proc. of the 4th Int. Symp. on Practical Design of Ships and Mobile Units, PRADS’89,
Varna, Bulgaria, Vol. 3, pp. 105.1-105.8 (1989). 10. S. A. Mavrakos, V. J. Papazoglou and M. S. Triantafyllou, An investigation into the feasibility of deep water anchoring systems. In: Proc. 8th Int. Offshore Mechanics and Arctic Engineering Conf., OMAE’89, Vol. 1, The Hague, The Netherlands,
pp. 683-689 (1989). 11. S. A. Mavrakos, V. J. Papazoglou, M. S. Triantafyllou and P. Brando, Experimental and numerical study on the effect of buoys on deep water mooring dynamics. In: Proc. 1st Int. Offshore and Polar Engineering Co&. ISOPE’91, Vol. Ii, Edinburgh, pp. 243-251 (1991). ”
12. S. A. Mavrakos, J. Chatjigeorgiou and V. J. Papazoglou, Use of buoys for dynamic tension reduction in deep water mooring applications. In: Proc. 7th Int. Conf. on the Behaviour of Offshore Structures, BOSS’94, Vol. 2, pp. 417426. MIT, Cambridge, MA
(1994). 13. L. P. Krolikowski and T. A. Gay, An improved linearization technique for frequency domain riser analysis. In: Proc. Offshore Technology Co&.. OTC 3777, Houston, TX, pp. 341-353 (1980). ” Guide. The Math Works (1993) 14. MATLAB-Users 15. R. W. Clough and J. Penzien, Dynamics of Structures. McGraw-Hill, New York (1993).