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Stability of single point mooring systems
Stability of single point mooring systems M I C H A E L M. B E R N I T S A S and F O T I S A. P A P O U L I A S
Department o f Naval Architecture and Marine Engineering, The UniversiO, of Michigan, Ann Arbor, M148109-2145, USA A mathematical model for the horizontal plane slow motions of ships connected to a mooring terminal through a single elastic line is developed based on the ship manoeuvring equations and a nonlinear elastic mooring line model. Slowly varying excitations from current, wind and drift forces are included. Local perturbation analysis about the critical points of the corresponding autonomous system reveals the asymptotic stability characteristics of the mooring system. Time simulations using a tanker and a barge confirm the theoretically predicted long term behaviour and show the short term system behaviour. The mooring system may exhibit instability in sway, yaw and surge Or reach a limit cycle depending on the properties of the critical points, vessel type, propeller, initial conditions, current and wind. Key Words: single point mooring, stability, asymptotic stability, short time behaviour, instabilities in surge sway and yaw, limit cycle, local perturbation, simulation, current, tanker, barge, propeller, initial condition
INTRODUCTION The horizontal plane motions of vessels connected to a mooring terminal by a single mooring line may exhibit short term instabilities in sway (galloping),yaw (fish-tailing), and surge, or in the long term reach a limit cycle. The mooring system dynamics are investigated in this work by local and global stability analysis and simulation of a mathematical model comprised of the third order non: linear sway, yaw and surge equations of motion, and a model for the nonlinear elastic mooring line behaviour. Slow motion excitations due to current, wind and drift forces are included. Model tests, 17measurements in the North Sea,ll theoretical analysis 9'12'ts and stability analysis and simulations is have been performed in the past in order to study the behaviour of single point mooring (SPM) systems. Problems related to the SPM dynamics are those of ship manoeuvring and towing and have been studied extensively in the past in references 1 , 2 , 4 , 5, 7 and 8. In this work the mathematical model is derived in Section I. In Section II the asymptotic stability features of the mooring system are analyzed by local perturbation analysis in the vicinity of the critical points, and the global system behaviour is assessed. Finally, in Section III, simulations of six different SPM systems with a barge or a tanker are used to verify the theoretical stability conclusions and investigate the effects of current, initial conditions, cable characteristics, vessel type and propeller on the short term system response. I. MATHEMATICAL FORMULATION The mathematical model used in this work consists of the equations of the ship motions in the horizontal plane, that is sway, yaw and surge, the mooring line dynamics model Accepted June 1985. Discussion closes March 1986. 0141-1187[86/010049-10 $2.00 9 1986 Computational Mechanics Publications
and a set of appropriate initial conditions. In thebody fixed reference frame (X, Y, Z) with its origin at the centre of gravity of the moored vessel, where (X, Y) is the horizontal plane and (X,Z) is the centre-plane of symmetry (Fig. 1), the nonlinear equations of motion in sway, yaw and surge are given by (1), (2) and (3) respectively:
m ( d U - - r v ) = Fx
(I)
\dt
dr
I= .--= mz ot
(3)
where the tight hand sides represent the total excitation forces in the X and Y directions, and the moment about the Z axis respectively, and the rest of the symbols are explained in the nomenclature. At the connection of the mooring line and the mooring terminal two orthogonal systems of co-ordinates are introduced in the horizontal plane, namely an earth fixed system (x',y') and ( x , y ) where x represents the direction of the current. In the (x,y) system the equations of motion take the following form as explained in Appendix A:
(m - - X t i ) f i = X + TCOSt.O+ F x W I N D + F X w A V E
(4)
m -- Yfl ..
COS
y -- Y~f = Y - - T sin t.o + FyWIND + FyWAVE -
-
(m -- YT3)(-- ti tan $ + vr tan $ -- ur)
(5)
cos ~/ fi + (Izz --Nk ) i" = N - - xpT sin w +MZwlN n +3LWAVi ~ " : +N~(--t'l tan ~ + vr tall $ --ur)
Applied Ocean Research; 1986, Vol. 8, No. 1
(6)
49
Stability o f sblgle point moorblg systems: 3/. M. Bernitsas and F. A. Papoulias where T is the mooring line tension, X , Y , N are functions of the rudder angle, propeller and relative velocities, u, v, r of the moored vessel with respect to the water. Nonlinear third order expressions for X, Y , N derived by Taylor expansion are given in Appendix A. From the geometry of the system in Fig. 1 we get: 4' = 4 + fl
(7)
co = v + 4
(8)
sin 7 =
y + xp sin 4
(9)
l
12 = (x + xp cos 4) 2 + (y + Xp sin 4) =
(10)
where l = mooring line.length. Further, the kinematics of the system yield the following equations:
r= ~ v=
(1 l) 5'
sin3 --utan~'--Uc-cos 4' cos 4'
dy
.9 = - - .
dt (14)
where Uc = absolute current velocity and 3 = angle between the x and x ' axes. Finally, when the tension in the mooring line is positive (ew > 0), it is given by equations (15)-(17): r = p(ew) q
(15)
T r = -Sb
(16)
ew = (1 -- lw)flw
(17)
where r = specific tension, Sb = average breaking strength, p and q are empirically determined constants and lw = working length of the mooring line) 4 The excitation due to current, wind and slowly varying wave forces can be computed as follows. The current force is evaluated by using in the expression for X, Y, and N the relative velocitiesof the moored vessel with respect to the water. This approach is correct to the third order for time independent current since the expressions for X, Y, and N in Appendix A are used. The wind forces are computed using equations (18)-(20): FxWIN D 9
=
[P~Cx WIN D A r V =
F y W I N D = 1Pa CyWIND A L 1''2
'
_= t
M'zWIND -- ~ Pa CzWINDA L L
V2
(18)
(19) (20)
where Pa = air density, V = relative velocity of wind, A T and AL are the transverse and lateral projected areas of the moored vessel above the waterplane and the three coefficients are functions of the ship type and the wind angle of attack) 3 The slowly varying wave forces can be computed by using the simple method described in reference 12 or the more accurate method developed by Faltinsen and Loken) ~ The mathematical model described in this section is nonlinear of third order and can be solved numerically by time simulation for a given set of initial conditions. Results are presented and discussed in Section III.
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Applied Ocean Research, 1986, Vol. 8, No. 1
In this section a method is developed for analysis of the stability of the model described by equations (4)-(17). The method is demonstrated by analysing the effect of current on the asymptotic and finite time interval stability of the system. The method is general and can be applied for studying the effects of wind and waves as well) s The three steps used in the stability analysis are the following. First the critical points of the autonomous system are found. Then local perturbation analysis about these critical points shows the local behaviour of the system in the phase space. Finally the global behaviour of the system is revealed based on the local stability properties of the system. The mathematical model of the mooring system (4)-(17) is composed of the six differential equations (4)-(6), (11), (12) and (14), and eight auxiliary equations. By selecting as state variables the zi's, i --- 1. . . . . 6 given by equations (21)(26): zl = y
(21)
Z2 = . y
(22)
za= 4
(23)
z4=r
(24)
Zs=X
(25)
(12)
(13)
2 = u cos 4 ' - - v sin 4 ' + Uc cos3
II. STABILITY ANALYSIS
and (26)
Z 6 ~- U
we can rewrite the six differential equations as six first order nonlinear coupled state equations. Since the system is autonomous we can derive the critical points by setting all time derivatives equal to zero. In general these equations produce three critical points which have the form: (Zlo =Yi, Z2o= O, zao = 4i, Z4o= O, Zso=Xt, Zro= ui) (27) for i = 1, 2, 3. These three points satisfy equations (28)-(38): vi = Uc sin 4i
(28)
ui = -- Uc cos 4t
(29)
Xi = Ri -- ~(Xw -- X w , Uc cos 4i) U~ sin 2 4i + ~ Xn n 82 --~X~u62Ue cos t~i+ Xv~rU c sin 4i --Xv~uU~c6 sin 4i cos 4i (30) Yi = Yo + (YvuUe sin ~ t - You) Ue cos 4 i - - YvUc sin 4i _ z6YvvvUacsin3 ~:i + U~c(Y~u + ~YvuuUc sin 4l) 1 x cos24i + Y~8 + ~Y~n~fa + ~Y~vvU~c8 sin24i 1 + ~Yv~sUe82 sin ~:i-- Y~uUc8 cos ~i 1 + ~Y~uuU~c8 cos2 4i (31) rt tan coi = - Xi
(32)
& Ti = - -
(33)
COS r i ')'i = r
_(
-- 4i
(34)
T, ~'/q
ew~-\Sb "p/
(35)
1l = lw(1 + ewi)
(36)
Yt = li sin 7i -- xp sin ffi
(37)
xt = - li cosTi--xp cos 4t
(38)
Stability o f single pohlt mooring systems: 3/. M. Bernitsas and F. A. Papoulias where $i, i = 1, 2, 3, are the three roots of (39) assuming that -- 1r/2 < tpi < 7r]2:
decayed. This principle is used in the numerical applications of the next section to predict local behaviour in the vicinity of critical points for six SPM systems involving a barge or a tanker. Further, features of the global behaviour o f the system can be derived on the basis of local properties. Both the local and global behaviour properties are verified by simulations.
as sin 3 ~t + a4 sin 2 $i + a3 sin ~i + aa + (a2 + a6 sin ~t) cos ~i = 0
(39)
al = ao--Xp Yo + ~(N8 -- Xp Ys) + 61-~3(Ns/~tl --Xp Y5~8) + U2[(Nouu--xpYouu) + ~6(N~uu--xpY~uu)] (40)
a2 = V c [ ( % Y o . - N o u ) + 8(xpY~.--N,,,)]
(41)
III. NUMERICAL APPLICATIONS
a3 = Vc[(xpYv--Nv) + ~8:(xpYv~, "Nvs,)l +t
3 ~ U~(xp Yvut, -- N~,,u)
In this section six different mooring configurations are used to investigate the effects o f several system features and environmental conditions on the horizontal plane motions and stability of the SPM. The six configurations used are described in the first column o f Table 1. Five of those involve a barge with or without propeller and/or directionally stabilising skegs moored by nylon or polyester rope, while the sixth is that of a tanker with nylon rope. The tools used in this investigation are first, the derivation o f the critical points as defined by equations (28)-(38) and listed in Table 1; second, the computation o f eigenvalues for each critical point as given by equation (47) and listed in Table 2; and third the simulations depicted in Figs. 2-12. The simulations are performed in the ( x , y ) system of co-ordinates, where x is the direction o f the current, as shown in Fig. 1. The program simulates the system modeled by equations (4)-(17) and the input consists o f the initial values of the state variables, the direction and velocity of the current, the mooring line working length (lw = 58.5 m) and constants, which are for tile nylon wetted rope
(42)
aa = U2 [(Xp Youu --Nouu) + ~5(Xp Y*uu --Nnuu)]
+ 89
oo)
(43)
as = ~u3c(xprvvv--N~ro) + ~U3c(Nvuu--xpYvu,,)
(44)
a~ = U~(N~, -- xp Vv~)
(45)
Local perturbation analysis about these critical points can reveal significant properties of the asymptotic and finite time interval stability of the mooring system. Assuming that the state variables can be written as:
gi = Zio + OZil + O(02) for i = 1. . . . . 6
(46)
where (Zio, i = 1. . . . 6) is a critical point, tr = small perturbation parameter and (zil,i = 1 . . . . . 6) are the first order correction terms of the state variables near the critical point, we get an equivalent linearised system valid in the vicinity o f each critical point. In order to study the stability of this system we must find the six eigenvalues by solving the following eigenvalue problem: A z l = XBzt
(47)
where zl =(Zll,Z21,z31,z41,ZSl,Z61) T and A and B are 6 x 6 matrices whose non-zero terms are given in Appendix B. The values o f ;ki, i = 1. . . . . 6 for each critical point, reveal significant properties o f the behaviour o f the system in the vicinity of the critical point. This theory is well developed for first and second order autonomous systems. For higher order systems like the one derived here it is still a subject of current research in mathematics? We can infer, however, several properties Of the system behaviour, on the basis of second order systems. For the sixth order system we are working with, this will limit the validity o f these properties to the close vicinity o f the critical points where the effects associated with four of the eigenvalues have
p = 9.78
(48)
q = 1.93
(49)
and S b = 680 625 lbf
(50
and for the polyester rope p = 176
(51)
q = 1.86
(52)
and S o = 607 500 lbf
(53)
It should be mentioned that all quantities shown in Figs. 2-12 are dimensionless. All linear dimensions, e.g. the
Table 1. Criticalpoblts for bargeand tanker SPM systems Variables Case 1. Barge, propeller, nylon rope 2. Barge, propeller, polyester 3. Barge, nylon rope 4. Barge, skegs, propeller, nylon 5. Barge, skegs, propeller, polyester 6. Tanker, propeller, nylon
Critical point
y
r
x
1
T
1st 2nd 3rd Ist 2nd 3rd 1st 2nd, 3rd 1st 2nd, 3rd 1st 2nd, 3rd 1st 2nd, 3rd
0.2436 0.7099 -0.7132 0.2415 0.6995 -0.7014 0.0 • 0.7131 0.0603
0.0369 0.2300 -0.2681 0.0369 0.2300 -0.2681 0.0 • 0.2511 0.0051
1.0101 1.0159 1.0175 1.0019 1.0031 1.0034 1.0099 1.0168 1.0099
0.0014 0.0035 0.0042 0.0014 0.0035 0.0042 0.0014 0.0039 0.0020
0.0598
0.0051
1.0019
0.0020
-0.0710
-0.0103
-1.4801 - 1.0845 -1.0509 -1.4722 -1.0770 - 1.0431 -1.5149 -1.0642 -1.5129 Complex co~ugate -1.5049 Complex conjugate -0.6380 Real, atinfinity
0.1859
0.00064
Applied Ocean Research, 1986, Vol. 8, No. 1
51
Stability o f single point mooring systems: M. 3/. Benfftsas and F. A. Papoulias Table 2. Eigenvaluesfor the barge and tanker SPM systems at the criticalpoints Eigenvalues Case
Critical point
ht
k2
1
1st 2nd 3rd Ist 2nd 3rd 1st 2nd, 3rd 1st 1st Ist
-0.0902 -0.2230 -0.2510 -0.0968 -0.2330 -0.2610 -0.0755 -0.2381 -0.1084 -0.1089 -0.1743
• i4.1906 • i7.9074 • i 8.4206 • i9.5244 -+i 17.8825 • i 18.9997 • i3.8452 • i 8.1978 • i4.5715 • i0.3311 • i 3.5840
2 3 4 5 6
h3 -1.5636 -0.2458 -0.1004 • i 0.5370 -1.5486 -0.2476 -0.0993 -+i0.5414 -1.6983 -0.3630 -1.6547 -1.6528 -2.5767
h4
ks
0.3632 -0.7515
0.0822 -0.0839 -0.5250 0.0831 -0.0829 -0.516i 0.0725 -0.0940 -0.0245 -0.0240 0.0354
0.3596 -0.7318 0.4986 -0.6600 -0.5410 -0.5426 -0.1109
h6 • i0.2108 • i0.4997 • i 0.1102 _*i0.2122 • i0.5031 • i0.1259 • i0.1889 _*i 0.5206 • i0.6194 • i0.6211 • i 0.4774
The asymptotic behaviour of trajectories initiated in the vicinity of a critical point can be assessed on the basis of the eigenvalues in Table 2. Specifically: q \ LY
,~.,,~ _
/
V = drift ~ngle
~
Figure 1.
N
Geometry o f mooring system
moored vessel lateral offset y and the distance x, are nondimensionalized with respect to the ship length L. Dimensionless expressions for time t and tension T are given in Table 3. Finally, the drift angle ff is in radians. Regarding the critical points of the autonomous SPM System we can draw the following conclusions: (1) The barge in the first three cases shown in Table 1 has three real critical points because equation (39) has three real roots. The first critical point is not at zero offset and drift angle and the second and third are not symmetric in the first two cases because of the asymmetry caused by the propeller. (2) In cases four and five the directionally stabilising skegs change drastically the stability characteristics o f the barge. More specifically, the first critical point is closer to the zero offset and zero drift angle point while the second and third points are complex conjugate. Skegs are vertically placed fins at the ship stern, like fixed rudders, which change the slow motion derivatives Iv, Yr,Nv and Nr, as can be seen from Table 3 by comparison of the third and fourth columns. They also increase the barge towing resista n c e by 40%.4 Such skegs are used in practice to increase the directional stability of barges at the expense of increased resistance. (3) In case six, the tanker has one critical point with negative offset and drift angle because al given by equation (40) is negative. It also has two critical points at infinity because a4 and as given by equations (43) and (44) respectively, are zero.
52
Applied Ocean Research, 1986, Vol. 8, No. 1
(4) The SPM system in case 1 (SPM # 1 ) i s unstable about the first critical point because L~> 0 and R e ( k s , X 6 ) > 0 . This implies that the point has features of unstable node due to X4 and unstable spiral point due to ks and k6. Accordingly Figs. 2 - 4 and 6 show that t h e trajectories initiated near the first critical point diverge from that point. (5) The second critical point of SPM #1 has characteristics of a stable node because X3, X4 < 0 and a stable spiral point because Re(hi, X2), R e ( k s , X 6 ) < 0 . Thus the trajectory in Fig. 2 is attracted by this point and exlfibits two oscillatory components, a fast one due to the cable dynamics which decays very fast and a slow one. a Similar is the behaviour shown in the simulation of the distance x (see Fig. 4) between the mooring point and the centre of gravity of the moored vessel, as shown in Fig. 1, and the simulation of the drift angle ff in Fig. 6. The fast oscillations due to the cable dynamics affect mostly the drift angle, have some effect on the offs e t y and hardly affect the distance x. (6) The third critical point o f SPM #1 is a stable spiral point because it has three pairs of complex conju. gate eigenvalues with negative real parts. A trajectory attracted by this point will in general have three oscillatory components. In the offset and distance simulations in Figs. 3 and 4 respectively, two of the oscillatory components are apparent, while the component due to the cable dynamics is relatively insignificant. (7) SFM # 2 exhibits the same properties as those described above for SPM #1. The effect of the cable stiffness is obvious in k l and k2 since the frequency of the corresponding component is higher due to the increased stiffness of the polyester in comparison to the nylon rope. The effect of this stiffness on the other eigenvalues is minor. (8) SPM #3 has similar properties with SPM #1 with the exception of having symmetric critical points due to the absence of propeller. (9) SPM # 4 and SPM #5 have properties of a stable node due to Xa and X4 and a stable spiral point due to kl, X2, Xs and X6. (lO) SPM # 6 has one unstable finite critical point which has properties of an unstable spiral point due to
Stability o f single point mooring systems: ill M. Bemitsas and F. A. Papoulias Table 3.
Sumnzary of moored vessel particulars and slow motion derivatives used hz simulations Vessel Non-dimensional expressions
Quantity
Barge with skegs~
Barge~
L ft
Super tanker ~
191.56
xplL
1066.3 0.4683 0.1796
0.505
lwlL t T Ibs
tUc]L T](O.5pL 2U~)
1.0
Xt} m Y~,
Xfi/(O.5pL 3) m/(O .5pL 3) Y~/(O.5pL3)
-0.01383
Yv
Yd(O.5oC'Ue)
-0.01153
YF Yr Yo N~ Nv (Izz -- N i) Nr NO Yvvr Yorr
Yi/(O.5pL') Yr/(O.5pL3Ue) YJ(O.5pL'U~) N~/(O.S pr , ) Nv/(O.5pL3Uc) " (Izz -- NF)/(O.SpL3) Nr/(O .SOL"V c) No/(O.5pL ~U~) Yvvr/(O.5pL3/Uc) Yvrr/(O.5pL'[Uc)
0.0 0.00238 0.0000746 0.0 -0.007285 0.00188 - 0.0 0128 0.00001509
Yvo~l(o.spcVuc)
-0.3005
Yrrr/(O.5pLS/Ue) Nvvr/(O.5pL ~/Ue) Nvrrl(O.SpL SlUc) Nvvvl (O.5o L ~lUe) Nrrrl(O.5pL~lUc)
0.0 - 0.0111 0.01025 - 0.009669 0.0
Y~vv Yrrr Nvvr Nvrr Nvvv Nrrr
-0.00136 0.017 0
-0.0009 0.0181 0.0171 -0.0261 0.0 0.00365 -0.00056 0.0 -0.0105 0.00222 -0.0048 -0.00028
-0.0259 0.002815
-
-0.0026425 -0.00265
0.0193l
-0.045
- 0.01025
0.00611
-
e*
'3_. I . ~ I I ! :I~,I~. N~.I)I ROPe,PROPEI.LER_ 2. ~'I~3 : B.~R6F.,NYLONROPE
?
r
~_o o.
Lt.
Lt-~
I--4 GO Z~
%
Zm LUr I. SPtl~I :SA.~E-~,NYLONROPs163 Z .~2:8.~P.GE, Pt;L~STER RC~E,PRCPELLER_
*o
O.
10.
20.
30.
40.
DIMENSIONLESS
S". TIME
60.
713.
O.
t
Figure 2. Offset for SPM barge system. Mooring line stiffness effect
;ks and ;k6- Thus all trajectories originated in the vicinity o f the first critical point diverge while all those originated near infinity also diverge so that the cable length tension achieve finite values. Therefore there must be a limit cycle in between, to which all trajectories converge3 This is shown in Figs. 10-12. Further, we can assess the properties of the finite time interval behaviour o f SPM systems and the effects o f current direction, propeller, rope stiffness and skegs by studying the trajectories in the simulation graphs 2-12. In these figures, time is non-dimensionalised by L / U e where U c =
Figure 3.
10.
20.
30. 40. OIMENSIONLESS
50,
TIME t
60.
70.
Offset f o r SPM barge system. Propeller effect
2.5 knots = 1.286 m/s. Therefore the dimensionless time of 75 used in the figures corresponds to 57 rain o f real time for the barge and 5 h and 16 min for the tanker. Specifically: (1 I) Figure 2 shows that SPM #1, with the nylon rope, converges to the second critical point. SPM #2, with the polyester rope, exhibits similar behaviour. This is in agreement with the stability analysis results shown in Tables 1 and 2. (12) Figure 3 shows the effect o f the propeller by comparing the behaviour of the barge system with a propeller (SPM #1) and without one (SPM #3). It
Applied Ocean Research, 1986, Vol. 8, No. 1
53
Stability of s#zgle pohzt mooring systems: 3/. M. Bemitsas and F. A. Papoulias oa i
.. x
/~'%
~
(~)
t
/
tl = 180~
~
//I
7 t//'///)t ; vv 0.
In.
20.
30.
/ 'tJ
40.
50.
DIMENSIONLESS
TIME
60.
(13)
(14)
(15)
(16)
(17)
0.
70.
ft.
20.
appears that the convergence of the system with the propeller to the third critical point is slower than that of the system without the propeller. The effect of the combined action of the current and the propeller on the distance x and the offsety is shown for SPM #1 in Figs. 4 and 5 respectively. In case the'direction of the current is ( 1 8 0 ~ 15 ~) degrees the barge is attracted by the second critical point while in case the direction is (180 ~ 15~) degrees the moored barge is attracted by the third critical point at a slower rate. This shows that the asymmetry of the flow past the barge, created by the propeller, affects the finite time interval stability of the mooring system. In the long run one of the two stable critical points will attract the barge provided that the environmental conditions do not change. High stiffness of the mooring line ma3) have another undesirable effect on the drift angle as shown in Fig. 6. When a stable critical point attracts the mooring system quickly tile action of the fast cable dynamics becomes evident. Thus in Fig. 6, for SPM # I , the high frequency oscillations in the drift angle simulation are attributed to the fast cable extensional dynamics shown in Fig. 7.4 Several offshore barges which are unstable in forward motion, may exhibit improved stability behaviour in towing or mooring when equipped with skegs. SPM # 4 and SPM # 5 have one stable critical point. As shown in Fig. 8 this point attracts the barge moored by either rope. Further, it should be noticed, as shown in Fig. 9, that SPM # 4 and SPM #5 are galloping over a significant period of time before they are eventually attracted by the critical point. The tanker system (SPM #6) which has one finiie unstable critical point exhibits, as explained in (10) above, periodic asymptotic behaviour. This is shown in Figs. 10 and 11. It exhibits, however, a large amplitude motion in surge and sway over a 5 h period before reaching the limit cycle. The tension in the rope for SPM # 6 initially exhibits large amplitude oscillations as shown in Fig. 12. After the transient effects have decayed, T has a quasiperiodic response and asymptotically becomes a periodic function.
Applied Ocean Research, 1986, Vol. 8, No. 1
30.
40.
DIMENSIONLESS
t
Figure 4.. Distancefor SPM barge system. Current effect
54
I .SPill I : BARGE,NYLON ROPE,PROiOELLER II= 180"*IS'. 2 SPl'leI : BARGE,NYLON ROPE,PROPELLER,
I..~tl#1:0ARf~.NYI.0NROPE,PROPEDJER, = I~i~162o. ~ 1 1 1 :BARC-T..I'Ctl.0NROPs a = le~p, iso.
/ k i
50.
S0.
TIME
70.
t
Figure 5. Offset for SPM barge system. Current effect
I
r
$M"/ 2 : BARGE. POLYs
ROPE, PROPfl.I.~
g 3. ~
EE
w l t27
"~
T
0.
10.
2o.
30.
~0.
OIMENSIONLESS
so. TIME t
s0.
70,
8
Figure 6. Drift anglefor SPx~barge system. Polyester rope
I
O
Spile2 : BAR~E,F~L~ESTE~ ROPE,PROPEILt~
H LU~
LUoa
d ~. LO LLI~
O.
,I II,
IIIIlilllll l,,, = 10.
20.
40.
50.
DIMENSIONLESS
30.
TIME
60.
70.
8
t
Figure 7. Tension in polyester rope for SPM barge system
CLOSING REMARKS Nonlinear stability analysis and time simulations have been used to study the mathematical model developed in this work for the horizontal plane slow motions of ships moored to a terminal through a single non-linear elastic line. The
Stability o f shtgle pohtt mooring systems: M. M. Bernitsas and F. A. Papoulias
I S f ~ ' 4 : BARE.NYlON ROPE.I:'ROPUJJER.~ 6 5 . : B.kP.r.~oPOLYESTERROPE.PROPELLER.$~:r.,s
2 ~
SPt'l#6 : TANKER,NYLON R ~ , PR0P[Lt[R
>.. o
o,
"'"
= R~O-IS O. :LIMIT CYCLE
Gq
LL~ C3 .":
~
o7
1
iJJ~
w~ U~ Zt
//XX//\\ z \//
o z~
O'),..,i
z~
s ,.o
? ,:g O.
20.
I0.
30.
40.
DIMENSIONLESS
Figure 8.
?
so. TIME
ca.
7o.
-0.70
t
-0.62
-0.58
>I
Figure 11. subspace
-0.46
-0.42
I--
) &~'1#6 : T ~ ,
NYLONR ~ . PROPlRLER,
e - t~~
~
tOm
/
o
-0,50
DISTANCE x
Tanker traiectory in the ofset-distance phase
______-- J
037 Z~.
-0.54
DIMENSIONLESS
Offset for SPM barge system. Skeg effect
LL~
-0.66
tJ~ Zr I. ~11~ 4 : BARC4[,EiS.CI*,I R ~ ,
.
3>
PR(/~[XLER,~ G S : O.&RGdE,POtY[ST[R ROPE,PROPB.LER,$1(EGS.
2. ~
,:? -1.60
-I.52
-1.44
-I.36
-I.52
(POtYESTeRROPE) DIMENSIONLESS
-I.44
-1.36
DISTANCE
x
O.
ca
I L $P~#6 : TANI~R,N~I.0N ROPE.PROPELLER,
a - taO~ ~ NYLON R~E,I~01>BIER,
~.l~-is o
>.
0 Illllllti
-I.28
(m~o~ Ropt)
Figure 9. Barge trajectory ht the offset-distance phase subspace. Skeg effect
2 $i~'~6:TJ~R
to
I
I0.
20.
30.
4~.
DIMENSIONLESS
Figure 12.
5~.
TIME
eo.
70.
t
Tension hz nylon rope for SPM tanker system
limit cycle. Further, combinations of the above effects may result in undesirable system response with large changes in the tension of the mooring line. Such conditions may make the asymptotic and the finite time interval behaviour of the mooring system hazardous.
| ACKNOWLEDGEMENTS
O
to~
This work is supported by the University Research Program of the Department of Transportation under contract #DTRS5683-C-00005.
(-o~i z-~
O,
Figure 10.
10.
/
///
kj
L 9I 20.
/1 // \J
30. 43. OIMENSZONLESS
k.J SO. TIME
60.
' 70
t
NOMENCLATURE a, b, c
Offset for SPM tanker system. Current effect Fx, Fy Iz=
effects of vessel type, propeller, direction of current and mooring line stiffness, on the system short and long term behaviour have been studied. It was shown that the system may exhibit instability in sway, yaw and surge, or reach a
l
lw L
dummy independent variables representing u, v, r and 6 in functions X, Y and N total excitation forces in the X and Y directions respectively mass moment of inertia of moored vessel about the z-axis length of mooring line working length of mooring line length of moored vessel
Applied Ocean Research, 1986, Vol. 8, No. 1
55
Stability o f single pofllt mooring systems: M. M. Bernitsas and F. A. Papoulias m
M~ N
No N~b Nabc r
R Re t
T u
v~ V. X Xp
X
xo x~ x~b Xabc Y Y
Yo Ya Yob Yabc
mass o f moored vessel total excitation m o m e n t about the Z axis yaw m o m e n t yaw moment due to propeller derivative o f yaw m o m e n t with respect to a derivative o f yaw m o m e n t with respect to a and b derivative o f yaw moment with respect to a, b and e yaw angular velocity moored vessel resistance real part o f complex number time mooring line tension moored vessel forward velocity with respect to the water velocity o f current moored vessel lateral velocity distance between mooring point and vessel's centre o f gravity in the current direction distance between the centre o f gravity and the point o f connection o f the mooring line for the moored vessel surge force surge force due to propeller derivative o f surge force with respect to a derivative o f surge force with respect to a and b derivative o f surge force with respect to a, b and c m o o r e d vessel lateral offset sway force sway force due to propeller derivative o f sway force with respect to a derivative o f sway force with respect to a and b derivative o f sway force with respect to a, b, and c
Greek symbols
~' p
rudder angle drift angle between current direction and vessel longitudinal axis in radians yaw angle water density
REFERENCES Abkowitz, M. A. Stability and Motion Control o f Ocean Vehicles, MIT Press, Cambridge, 1972 2. Abkowitz, M. A. Measurement of hydrodynamic characteristics from ship maneuvering trials by system identification, Transactions, SNAME 1981,89, 283 3 Bender, C. M. and Orszag, S. A. Adranced Mathematical Methods for Scientists and Engineers, McGraw-ttill Book Company, 1978 1
4 5 6 7 8 9
10
56
Bernitsas, M. M. and Kekridis, N. S. Simulation and stability of ship towing, h~ternational Shipbuilding Progress 1985, 32 (368) Comstock, J. P. (ed.) Principles of Natal Architecture, The Society of Naval Architects and Marine Engineers, 1967 Cox, J. V. Statmoor - A single point mooring static analysis program, Naval Civil Engineering Laboratory, Report No. AD-AII9 979, June 1982 Eda, H. Low-speed controllability of ships in wind, Journalof Ship Research 1968, 12 (3) 181 Eda, H. Course stability, turning performance, and connection force of barge systems in coastal seaway, Transactions, SNAME 1972, 80 Faltinsen, O. M., Kjaerland, O., Liapis, N. and Naldcrhaug, 1t. Hydrodynamic analysis of tanker at single-point nmoring systems, Proceedings of Second hzternational Conference on Beharior of Offshore Structures, London, August 1979 Faltinsen, O. M. and Loken, A. E. Drift forces and slowly varying forces on ships and offshorestructures in waves, Norwegian Marithne Research, 1978
Applied Ocean Research, 1986, Vol. 8, No. 1
11 12
Hating, R. E. Single-point tanker mooring measurements in the North Sea, Proceedings of Offshore Technology Conference, ltouston, paper no. 2711, 1976 Hsu, F. It. and Blenkarn, K. A. Analysis of peak mooring forces caused by slow vessel drift oscillations in random seas,
Proceedhzgs of Offshore Technology Conference, Houston, 13 14
paper no. 1159, 1970 Isherwood, R. M. Wind resistance of merchant ships, Transactions, RINA 1972, 114, 327 MeKenna, H. A. and Wong, R. K. Synthetic fiber rope, properties and calculations relating to mooring systems, in Deepwater
Mooring and Drilling, A SME, OED 1979, 7 15
Oortmersen, G. van and Remery, G. F. M. The mean wave, wind and current forces on offshore structures and their role in the design of mooring systems, Proceedings of Offshore Technology Conference, Houston, paper no. 1741, 1973 16 Parsons, M. G. and Greenblatt, J. E. 'SttIPSIM]OPTSIM: Simulation program for stationary liner optimal stochastic control systems, Department of Naval Architecture and Marine Engineering, The University of Michigan, Report #188, June 1977 17 Pinkster, J. A. and Remery, G. F. M. The role of model tests in the design of single-point mooring system, Proceedhlgs of Offshore Technology Conference, Houston, paper no. 2212, 1975 18 Sorhekn, tt. R. 'Analysis of motion in single-point mooring systems, Modeling, Identification and Control 1980, 1 (3)
APPENDIX A Tile equations o f motion in the ( x , y ) system, that is (4)-(6), are derived in this Appendix from the basic equations ( 1 ) (3). The total excitation forces F x and F y in the X and Y directions respectively and the moment M z about the Z axis are comprised o f four groups o f terms, namely hydrodynamic terms due to the relative motion o f the water, terms due to the tension in the mooring line, and forces and moments and second order drift forces due to waves. Expressions for the surge and sway forces X and Y, and yaw moment N can be derived by Taylor expansion in terms o f the independent variables, the rudder angle 6 and the relative velocities u, o, r o f the vessel with respect to the waterJ 's In non-linear analysis, terms up t o the third order are used. Terms beyond the third order and second and higher order acceleration terms may be neglected since there is no significant interaction between viscous and inertia properties o f the fluid and acceleration and inertia forces calculated from potential t h e o r y when a p p l i e d t o 2 X ~ b ,2 submerged bodies give linear terms that is Xfifit't, 9 . "2 .. .a Xkrr , Xhuh u . . . . . = 03 The inclusion o f the non-linear terms used here, is necessary because the linear model fails to predict accurately the maneuvers o f unstable ships,s The significant terms, as explained in reference 2, are those included in equations (54)-(57). The surge force X is X = Xo + X u u + 2XuuU I 2 + ~XuuuU 1 3 + 21X w v 2 + 1Xrrr2
+ ~x~ ~ + ~X~v2u + 89
+ lX~.*2U
+ (Xvr + m) vr + Xv~v8 + Xr~r5 + Xrvurvu + Xv~,,v~tt + Xrrurru
(54)
where the first four terms are equal to the negative o f the hull resistance in forward m o t i o n at velocity u
- - R ~- Xo + X . u + ~X.,,u 2 + ~x~..u i 3
(55)
The sway force is
Y = Yo + Youu + Youu u2 + Yo v + 16YvvvV3+ 12YvrrVr2 + ~2Yv~6v52 + YvuvU + ~YvuuVU2+ ( Y r - mu) r + 61Yrrrr 3 + ~1 YrvvrV2 + ~ y r ~ r 8 2 + Yru+ 12YruurU2 + yr5+~y~53+
~y~vv~ 2 + ~ Y , rrSr2+ Y~u~U
+ ~Y, uu 6u2 + Yvr~ vr6
(56)
Stability of single point mooring systems: M. M. Bernitsas attd F. A. Papoulias and the yaw moment is
A ( 1 , 5 ) = Tosinz3ocoszaOlo + cos 3'0 (6"T - - / ~ )
N = No + Nouu + Nouuu2 + Nvv + gNvvVl 3 + ~Nvrrv 2 _Ft~Nv~v6 2 + NvuvU + ~ N ~ v u + Nrr + ~N.r? + ~Nrrvro2 + ~Nr6~r6~ + NrurU + 89 2 + N~ 6 + gt N ~ 6 3 + ~2N6vv6V2 + ~N~rr6r2 + N~u6U + ~N~uu6U~+ Nvr~ vr6 (57) In the above equations the index '0' indicates propeller dependent terms. In the numerical applications with the barge and the tanker only the terms shown in Table 3 are used since these are the only significant terms for the above vessels. Table 3 gives values for the non-dimensionalised slow motion derivatives.4 Combining the above non-linear Taylor expansions for X, Y and N with the dynamic response terms of the equations of motion (1)-(3),the non-linear equations of motion inthe horizontal plane become (m
-
-
X,~) ti = X + T cos co + FxWIND + FXWAVE
X cosz3o sin 6oo A (1,6) =
• (You + 2YouuZ6o + Yvuvo+ YvuuVoZ6o
--mz4o+ Yruz4o+ YruuZ4oZ6o + Ysu6 + Y&,u6Z6o) A(2, l)=[sin'rO(~o--CT)COSZ3osinw o
To cos~z~o-I
io
+ l~lZwlND + MZWAVE (60) We can identify the added mass terms Xi~, Yi, Yi, Ni., Ni, the mooring line forces and moment and those due to wind FXWlND, F~,V~ND, MZWlNO and waves FxWAVE, F~XVhVE, M5~,AaVnEdThe X, Y and N terms, given by equations (54), (57), represent mainly damping terms due to viscous effects and energy dissipation due to waves. Equations (58)-(60) may be further developed by taking the time derivative of the lateral velocity v, using equation (12). + or tan ~ -- ti tan ~ -- ur
+ (Vo tan z3o--Z6o) sin Z3o+ Vocos Z3o] + (Vo tan z 3o-- z6o) Cc, cos z3o-- Co sin z3o
A ( 2, 4 ) = (cos z 3o) [NvrvoZao+Nr + t~NrrrZ4u2 +1
A (I, 3) --- To sin Z3osin 6o0
+ vo sin Z3o) A (2, 5) = [ L
/o
To + cosTo(CT----~o)COSZ3osin Wo]xv 1
A(2, 6) =--Nfiz4o - -
CC, sinz3o
COSZ30
Toxp cosz3o
+ N~.uVoZeo + N.uz4o + N...z~oz~o
to
+ N~.6 + N,.,,SZ6o) To A (3, 1) = -- - - cos 3'0 sin 6oo + CT cos COosin 3'0 1o
To coszao cos COo+ (Vo tan Z3o--Z~) C~,
I
[Xwvo + (Xvr + m) Z,o + Xv~ 6
A (3, 2) =
• cosz3o--Bosinz3o+ (m -- Y;~)
COSZ30
x Z4o[--z6o sin z3o--(vo tan Z3o--z~) sin zzo
+ X.~.z4o zGo+ Xo~,,6z6o] A (3, 3) = (vo tan Z~o-- Z6o)[Xvvvo+ (Xvr + m) Z4o
- - Uo COS Z30]
A (1,4) = (cos z3o) [YvrrVoZ~+ (Yr--mzc,o) + ~YrrrZ42o 2
To sinz3o COSZ30
+ cos z3o(Nou + 2NouuZ6o + Nvuvo
--Xp sin2wo ( C T - - / ~ ) cosz3o
+l
t 2Nrwvo2 + ~gr56 62 + Nruz6o + igruuZ6o 1 z
+ N6rrZ4o6 +Nvr66Vo] +Nr3 (--Z6o coszs0
A (1,2) = C~, -- (m -- Yfi)z4o tanz3o
-
To cosz3o cos cooJxp+N~z4o [Z6osinz3o
--
To cos 2Z3o 1o
Io
2 [ To~ --x v sin C o o | C r - - - - l c o s z 3 o \ !o ]
(61)
Upon substitution of (61) into (50) and (60), we get the equations of motion in the (x,y) system of co-ordinates as given by equations (4)-(6).
To% COS Z30
A (2, 3) = [ To sin Zao sin COo L
COS
A ( I , I ) = sinTo(T~ \ 1o
I x"
A (2, 2) = Cr +N;)z4otanzao
--N~ b + (Izz - N~) i" = N -- Txp sin 60
APPENDIX B
CB, sin Z3o+ (cos Zzo)
COS Z30
(58)
(m -- Yt;) b -- Yfi"= Y-- T sin co + FyWIND+ FyWAVE (59)
b=
(m -- Y~)z4o
1
2
I
+ Xv86 + XrvuZ4o z6o + Xv~u6Z6o]
2
~YrvvVo + ~Yr566 + Yruz6o + 2Yrm, Zc,o --
+ Y, rrZ4o6 + Yvr~6Vo] + ( m - Yfi) x (Z6ocosZ3o-- Vosin Z3o)
To sin~o(,xpc~176 t-1 ) lo
+ CTXp
COS 6O0 sin
6oo
Applied Ocean Research, 1986, Vol. 8, No. I
57
Stability o f single point mooring systems: M. M. Bentitsas and F. A. Papoulias A (3, 4) = Xrrz4o + Xwuzao'z~o + X m , z4ozoa
B(4, 1 ) = 1
+ (Xvr + m) Vo + Xr~8+XrvuVoZ6o
B ( 5 , 3) = 1
+ X~,,8 zoo
B(6, 5) = 1
A ( 3 , 5 ) = ~ ~Xvvu V2o " F ~~tXrruZ~o 2 + ~~ X ~ . 8 2 + X~,,VoZ4o
To
+ Xv~uSVo + Xr6uSZ4o -- -~o sin 3'0 sin too
and Bo = Yo + Youz6o + YouuZ~ + Yvvo + ~1Ywv Voa +i
-- C T COS r
+1
2 3 iYvuuVoZ6o + (Yr -- mz6o) Z4o + 61 Yrrrz4o
A (3, 6) = (-- tan Z3o) [Xwvo + (Xvr + m) z4o + Xv~ 8
+ Xm,z~oz6o + Xv~.Sz6o]
+ ~1.. : + ~yr~Z4o82 + I rvvZ 4OVO
+ (xi, + x,,.z~o + ~ x . . . z ~ )
+1
A(5,4) = I -
-
1
r r u z 40z 60
~YvuuZaoZ6o + Y88 + g Y ~ d i 3 +
1
2
CB = Yv+ ~YvvoVo l : + 1Yvrr Z 4o 2 "q- ~Yv~68 1 2 "4- Yvuz6o tanzao Vo
A (6, 3) =
COSZ30
A (6, 6)
2
1 2 + ~Y~rrSZ4o+ Y~uSZ6o + 2~-Ya,uSZ60 2 + Yw~ VoZ408
A(4,2) = 1
A (6, 2) =
i 2 YvrrVoZ24o + ~Yv~vo~ "1- YvuvoZ6o
COS "f0
1
2 + 12Y~uZ6o+ Ym, voz4o + Y~vSVo + Yvr~SZ4o
Co = No + Nouzoa + NouuZoo 2 + Nvvo + gNwvvo i 3 +~ 2 1 2 ~NvrrVoZ4o + ~Nv6~ Vo5 + N v u v o Z 6 o 1 2 1 3 "1 2 + 2 N ~ , u V o Z 6 o + N r z 4 o + g N r r r Z 4 o q- ~NrrvZ4oVo
-
COS Z30
..~1
1 2 ~N,.~z4o8 2 + Nruz4oZ6o + ~NruuZ4oZ6o + N88
B ( 1 , 2 ) = m -- Y~; B(1, 4) = -- Y/. coszzo
q_ 16N~6683 q_ ~ I N~SVo 2
B(1,6) =-
.t_1~N~,#,Sz6o 2 +
(m-
Y~) sin zao
~(2, 2) = - N ~
1
Applied Ocean Research, 1986, Vol. 8, No. 1
2
+ 1~NvuuZ6o 2 + NrvvVoZ4o + NbvvSVo + Nvr85z4o
B(2, 6) =Nil sinz3o
58
Nv~6 Voz~o8
1 2 Cc, = N~ + ~ N ~ v o + ~Nv~:4o + ~Nv~ 8 2 + No,,z~o
B(2, 4) = (Izz--Nk) cosz3o
B(3, 6) = m -- X~
1 2 + N~,,6z6o + ~N~.6z4o
qTo CT =
-
eWolw
Department o f Naval Architecture and Marine Engineering, The UniversiO, of Michigan, Ann Arbor, M148109-2145, USA A mathematical model for the horizontal plane slow motions of ships connected to a mooring terminal through a single elastic line is developed based on the ship manoeuvring equations and a nonlinear elastic mooring line model. Slowly varying excitations from current, wind and drift forces are included. Local perturbation analysis about the critical points of the corresponding autonomous system reveals the asymptotic stability characteristics of the mooring system. Time simulations using a tanker and a barge confirm the theoretically predicted long term behaviour and show the short term system behaviour. The mooring system may exhibit instability in sway, yaw and surge Or reach a limit cycle depending on the properties of the critical points, vessel type, propeller, initial conditions, current and wind. Key Words: single point mooring, stability, asymptotic stability, short time behaviour, instabilities in surge sway and yaw, limit cycle, local perturbation, simulation, current, tanker, barge, propeller, initial condition
INTRODUCTION The horizontal plane motions of vessels connected to a mooring terminal by a single mooring line may exhibit short term instabilities in sway (galloping),yaw (fish-tailing), and surge, or in the long term reach a limit cycle. The mooring system dynamics are investigated in this work by local and global stability analysis and simulation of a mathematical model comprised of the third order non: linear sway, yaw and surge equations of motion, and a model for the nonlinear elastic mooring line behaviour. Slow motion excitations due to current, wind and drift forces are included. Model tests, 17measurements in the North Sea,ll theoretical analysis 9'12'ts and stability analysis and simulations is have been performed in the past in order to study the behaviour of single point mooring (SPM) systems. Problems related to the SPM dynamics are those of ship manoeuvring and towing and have been studied extensively in the past in references 1 , 2 , 4 , 5, 7 and 8. In this work the mathematical model is derived in Section I. In Section II the asymptotic stability features of the mooring system are analyzed by local perturbation analysis in the vicinity of the critical points, and the global system behaviour is assessed. Finally, in Section III, simulations of six different SPM systems with a barge or a tanker are used to verify the theoretical stability conclusions and investigate the effects of current, initial conditions, cable characteristics, vessel type and propeller on the short term system response. I. MATHEMATICAL FORMULATION The mathematical model used in this work consists of the equations of the ship motions in the horizontal plane, that is sway, yaw and surge, the mooring line dynamics model Accepted June 1985. Discussion closes March 1986. 0141-1187[86/010049-10 $2.00 9 1986 Computational Mechanics Publications
and a set of appropriate initial conditions. In thebody fixed reference frame (X, Y, Z) with its origin at the centre of gravity of the moored vessel, where (X, Y) is the horizontal plane and (X,Z) is the centre-plane of symmetry (Fig. 1), the nonlinear equations of motion in sway, yaw and surge are given by (1), (2) and (3) respectively:
m ( d U - - r v ) = Fx
(I)
\dt
dr
I= .--= mz ot
(3)
where the tight hand sides represent the total excitation forces in the X and Y directions, and the moment about the Z axis respectively, and the rest of the symbols are explained in the nomenclature. At the connection of the mooring line and the mooring terminal two orthogonal systems of co-ordinates are introduced in the horizontal plane, namely an earth fixed system (x',y') and ( x , y ) where x represents the direction of the current. In the (x,y) system the equations of motion take the following form as explained in Appendix A:
(m - - X t i ) f i = X + TCOSt.O+ F x W I N D + F X w A V E
(4)
m -- Yfl ..
COS
y -- Y~f = Y - - T sin t.o + FyWIND + FyWAVE -
-
(m -- YT3)(-- ti tan $ + vr tan $ -- ur)
(5)
cos ~/ fi + (Izz --Nk ) i" = N - - xpT sin w +MZwlN n +3LWAVi ~ " : +N~(--t'l tan ~ + vr tall $ --ur)
Applied Ocean Research; 1986, Vol. 8, No. 1
(6)
49
Stability o f sblgle point moorblg systems: 3/. M. Bernitsas and F. A. Papoulias where T is the mooring line tension, X , Y , N are functions of the rudder angle, propeller and relative velocities, u, v, r of the moored vessel with respect to the water. Nonlinear third order expressions for X, Y , N derived by Taylor expansion are given in Appendix A. From the geometry of the system in Fig. 1 we get: 4' = 4 + fl
(7)
co = v + 4
(8)
sin 7 =
y + xp sin 4
(9)
l
12 = (x + xp cos 4) 2 + (y + Xp sin 4) =
(10)
where l = mooring line.length. Further, the kinematics of the system yield the following equations:
r= ~ v=
(1 l) 5'
sin3 --utan~'--Uc-cos 4' cos 4'
dy
.9 = - - .
dt (14)
where Uc = absolute current velocity and 3 = angle between the x and x ' axes. Finally, when the tension in the mooring line is positive (ew > 0), it is given by equations (15)-(17): r = p(ew) q
(15)
T r = -Sb
(16)
ew = (1 -- lw)flw
(17)
where r = specific tension, Sb = average breaking strength, p and q are empirically determined constants and lw = working length of the mooring line) 4 The excitation due to current, wind and slowly varying wave forces can be computed as follows. The current force is evaluated by using in the expression for X, Y, and N the relative velocitiesof the moored vessel with respect to the water. This approach is correct to the third order for time independent current since the expressions for X, Y, and N in Appendix A are used. The wind forces are computed using equations (18)-(20): FxWIN D 9
=
[P~Cx WIN D A r V =
F y W I N D = 1Pa CyWIND A L 1''2
'
_= t
M'zWIND -- ~ Pa CzWINDA L L
V2
(18)
(19) (20)
where Pa = air density, V = relative velocity of wind, A T and AL are the transverse and lateral projected areas of the moored vessel above the waterplane and the three coefficients are functions of the ship type and the wind angle of attack) 3 The slowly varying wave forces can be computed by using the simple method described in reference 12 or the more accurate method developed by Faltinsen and Loken) ~ The mathematical model described in this section is nonlinear of third order and can be solved numerically by time simulation for a given set of initial conditions. Results are presented and discussed in Section III.
50
Applied Ocean Research, 1986, Vol. 8, No. 1
In this section a method is developed for analysis of the stability of the model described by equations (4)-(17). The method is demonstrated by analysing the effect of current on the asymptotic and finite time interval stability of the system. The method is general and can be applied for studying the effects of wind and waves as well) s The three steps used in the stability analysis are the following. First the critical points of the autonomous system are found. Then local perturbation analysis about these critical points shows the local behaviour of the system in the phase space. Finally the global behaviour of the system is revealed based on the local stability properties of the system. The mathematical model of the mooring system (4)-(17) is composed of the six differential equations (4)-(6), (11), (12) and (14), and eight auxiliary equations. By selecting as state variables the zi's, i --- 1. . . . . 6 given by equations (21)(26): zl = y
(21)
Z2 = . y
(22)
za= 4
(23)
z4=r
(24)
Zs=X
(25)
(12)
(13)
2 = u cos 4 ' - - v sin 4 ' + Uc cos3
II. STABILITY ANALYSIS
and (26)
Z 6 ~- U
we can rewrite the six differential equations as six first order nonlinear coupled state equations. Since the system is autonomous we can derive the critical points by setting all time derivatives equal to zero. In general these equations produce three critical points which have the form: (Zlo =Yi, Z2o= O, zao = 4i, Z4o= O, Zso=Xt, Zro= ui) (27) for i = 1, 2, 3. These three points satisfy equations (28)-(38): vi = Uc sin 4i
(28)
ui = -- Uc cos 4t
(29)
Xi = Ri -- ~(Xw -- X w , Uc cos 4i) U~ sin 2 4i + ~ Xn n 82 --~X~u62Ue cos t~i+ Xv~rU c sin 4i --Xv~uU~c6 sin 4i cos 4i (30) Yi = Yo + (YvuUe sin ~ t - You) Ue cos 4 i - - YvUc sin 4i _ z6YvvvUacsin3 ~:i + U~c(Y~u + ~YvuuUc sin 4l) 1 x cos24i + Y~8 + ~Y~n~fa + ~Y~vvU~c8 sin24i 1 + ~Yv~sUe82 sin ~:i-- Y~uUc8 cos ~i 1 + ~Y~uuU~c8 cos2 4i (31) rt tan coi = - Xi
(32)
& Ti = - -
(33)
COS r i ')'i = r
_(
-- 4i
(34)
T, ~'/q
ew~-\Sb "p/
(35)
1l = lw(1 + ewi)
(36)
Yt = li sin 7i -- xp sin ffi
(37)
xt = - li cosTi--xp cos 4t
(38)
Stability o f single pohlt mooring systems: 3/. M. Bernitsas and F. A. Papoulias where $i, i = 1, 2, 3, are the three roots of (39) assuming that -- 1r/2 < tpi < 7r]2:
decayed. This principle is used in the numerical applications of the next section to predict local behaviour in the vicinity of critical points for six SPM systems involving a barge or a tanker. Further, features of the global behaviour o f the system can be derived on the basis of local properties. Both the local and global behaviour properties are verified by simulations.
as sin 3 ~t + a4 sin 2 $i + a3 sin ~i + aa + (a2 + a6 sin ~t) cos ~i = 0
(39)
al = ao--Xp Yo + ~(N8 -- Xp Ys) + 61-~3(Ns/~tl --Xp Y5~8) + U2[(Nouu--xpYouu) + ~6(N~uu--xpY~uu)] (40)
a2 = V c [ ( % Y o . - N o u ) + 8(xpY~.--N,,,)]
(41)
III. NUMERICAL APPLICATIONS
a3 = Vc[(xpYv--Nv) + ~8:(xpYv~, "Nvs,)l +t
3 ~ U~(xp Yvut, -- N~,,u)
In this section six different mooring configurations are used to investigate the effects o f several system features and environmental conditions on the horizontal plane motions and stability of the SPM. The six configurations used are described in the first column o f Table 1. Five of those involve a barge with or without propeller and/or directionally stabilising skegs moored by nylon or polyester rope, while the sixth is that of a tanker with nylon rope. The tools used in this investigation are first, the derivation o f the critical points as defined by equations (28)-(38) and listed in Table 1; second, the computation o f eigenvalues for each critical point as given by equation (47) and listed in Table 2; and third the simulations depicted in Figs. 2-12. The simulations are performed in the ( x , y ) system of co-ordinates, where x is the direction o f the current, as shown in Fig. 1. The program simulates the system modeled by equations (4)-(17) and the input consists o f the initial values of the state variables, the direction and velocity of the current, the mooring line working length (lw = 58.5 m) and constants, which are for tile nylon wetted rope
(42)
aa = U2 [(Xp Youu --Nouu) + ~5(Xp Y*uu --Nnuu)]
+ 89
oo)
(43)
as = ~u3c(xprvvv--N~ro) + ~U3c(Nvuu--xpYvu,,)
(44)
a~ = U~(N~, -- xp Vv~)
(45)
Local perturbation analysis about these critical points can reveal significant properties of the asymptotic and finite time interval stability of the mooring system. Assuming that the state variables can be written as:
gi = Zio + OZil + O(02) for i = 1. . . . . 6
(46)
where (Zio, i = 1. . . . 6) is a critical point, tr = small perturbation parameter and (zil,i = 1 . . . . . 6) are the first order correction terms of the state variables near the critical point, we get an equivalent linearised system valid in the vicinity o f each critical point. In order to study the stability of this system we must find the six eigenvalues by solving the following eigenvalue problem: A z l = XBzt
(47)
where zl =(Zll,Z21,z31,z41,ZSl,Z61) T and A and B are 6 x 6 matrices whose non-zero terms are given in Appendix B. The values o f ;ki, i = 1. . . . . 6 for each critical point, reveal significant properties o f the behaviour o f the system in the vicinity of the critical point. This theory is well developed for first and second order autonomous systems. For higher order systems like the one derived here it is still a subject of current research in mathematics? We can infer, however, several properties Of the system behaviour, on the basis of second order systems. For the sixth order system we are working with, this will limit the validity o f these properties to the close vicinity o f the critical points where the effects associated with four of the eigenvalues have
p = 9.78
(48)
q = 1.93
(49)
and S b = 680 625 lbf
(50
and for the polyester rope p = 176
(51)
q = 1.86
(52)
and S o = 607 500 lbf
(53)
It should be mentioned that all quantities shown in Figs. 2-12 are dimensionless. All linear dimensions, e.g. the
Table 1. Criticalpoblts for bargeand tanker SPM systems Variables Case 1. Barge, propeller, nylon rope 2. Barge, propeller, polyester 3. Barge, nylon rope 4. Barge, skegs, propeller, nylon 5. Barge, skegs, propeller, polyester 6. Tanker, propeller, nylon
Critical point
y
r
x
1
T
1st 2nd 3rd Ist 2nd 3rd 1st 2nd, 3rd 1st 2nd, 3rd 1st 2nd, 3rd 1st 2nd, 3rd
0.2436 0.7099 -0.7132 0.2415 0.6995 -0.7014 0.0 • 0.7131 0.0603
0.0369 0.2300 -0.2681 0.0369 0.2300 -0.2681 0.0 • 0.2511 0.0051
1.0101 1.0159 1.0175 1.0019 1.0031 1.0034 1.0099 1.0168 1.0099
0.0014 0.0035 0.0042 0.0014 0.0035 0.0042 0.0014 0.0039 0.0020
0.0598
0.0051
1.0019
0.0020
-0.0710
-0.0103
-1.4801 - 1.0845 -1.0509 -1.4722 -1.0770 - 1.0431 -1.5149 -1.0642 -1.5129 Complex co~ugate -1.5049 Complex conjugate -0.6380 Real, atinfinity
0.1859
0.00064
Applied Ocean Research, 1986, Vol. 8, No. 1
51
Stability o f single point mooring systems: M. 3/. Benfftsas and F. A. Papoulias Table 2. Eigenvaluesfor the barge and tanker SPM systems at the criticalpoints Eigenvalues Case
Critical point
ht
k2
1
1st 2nd 3rd Ist 2nd 3rd 1st 2nd, 3rd 1st 1st Ist
-0.0902 -0.2230 -0.2510 -0.0968 -0.2330 -0.2610 -0.0755 -0.2381 -0.1084 -0.1089 -0.1743
• i4.1906 • i7.9074 • i 8.4206 • i9.5244 -+i 17.8825 • i 18.9997 • i3.8452 • i 8.1978 • i4.5715 • i0.3311 • i 3.5840
2 3 4 5 6
h3 -1.5636 -0.2458 -0.1004 • i 0.5370 -1.5486 -0.2476 -0.0993 -+i0.5414 -1.6983 -0.3630 -1.6547 -1.6528 -2.5767
h4
ks
0.3632 -0.7515
0.0822 -0.0839 -0.5250 0.0831 -0.0829 -0.516i 0.0725 -0.0940 -0.0245 -0.0240 0.0354
0.3596 -0.7318 0.4986 -0.6600 -0.5410 -0.5426 -0.1109
h6 • i0.2108 • i0.4997 • i 0.1102 _*i0.2122 • i0.5031 • i0.1259 • i0.1889 _*i 0.5206 • i0.6194 • i0.6211 • i 0.4774
The asymptotic behaviour of trajectories initiated in the vicinity of a critical point can be assessed on the basis of the eigenvalues in Table 2. Specifically: q \ LY
,~.,,~ _
/
V = drift ~ngle
~
Figure 1.
N
Geometry o f mooring system
moored vessel lateral offset y and the distance x, are nondimensionalized with respect to the ship length L. Dimensionless expressions for time t and tension T are given in Table 3. Finally, the drift angle ff is in radians. Regarding the critical points of the autonomous SPM System we can draw the following conclusions: (1) The barge in the first three cases shown in Table 1 has three real critical points because equation (39) has three real roots. The first critical point is not at zero offset and drift angle and the second and third are not symmetric in the first two cases because of the asymmetry caused by the propeller. (2) In cases four and five the directionally stabilising skegs change drastically the stability characteristics o f the barge. More specifically, the first critical point is closer to the zero offset and zero drift angle point while the second and third points are complex conjugate. Skegs are vertically placed fins at the ship stern, like fixed rudders, which change the slow motion derivatives Iv, Yr,Nv and Nr, as can be seen from Table 3 by comparison of the third and fourth columns. They also increase the barge towing resista n c e by 40%.4 Such skegs are used in practice to increase the directional stability of barges at the expense of increased resistance. (3) In case six, the tanker has one critical point with negative offset and drift angle because al given by equation (40) is negative. It also has two critical points at infinity because a4 and as given by equations (43) and (44) respectively, are zero.
52
Applied Ocean Research, 1986, Vol. 8, No. 1
(4) The SPM system in case 1 (SPM # 1 ) i s unstable about the first critical point because L~> 0 and R e ( k s , X 6 ) > 0 . This implies that the point has features of unstable node due to X4 and unstable spiral point due to ks and k6. Accordingly Figs. 2 - 4 and 6 show that t h e trajectories initiated near the first critical point diverge from that point. (5) The second critical point of SPM #1 has characteristics of a stable node because X3, X4 < 0 and a stable spiral point because Re(hi, X2), R e ( k s , X 6 ) < 0 . Thus the trajectory in Fig. 2 is attracted by this point and exlfibits two oscillatory components, a fast one due to the cable dynamics which decays very fast and a slow one. a Similar is the behaviour shown in the simulation of the distance x (see Fig. 4) between the mooring point and the centre of gravity of the moored vessel, as shown in Fig. 1, and the simulation of the drift angle ff in Fig. 6. The fast oscillations due to the cable dynamics affect mostly the drift angle, have some effect on the offs e t y and hardly affect the distance x. (6) The third critical point o f SPM #1 is a stable spiral point because it has three pairs of complex conju. gate eigenvalues with negative real parts. A trajectory attracted by this point will in general have three oscillatory components. In the offset and distance simulations in Figs. 3 and 4 respectively, two of the oscillatory components are apparent, while the component due to the cable dynamics is relatively insignificant. (7) SFM # 2 exhibits the same properties as those described above for SPM #1. The effect of the cable stiffness is obvious in k l and k2 since the frequency of the corresponding component is higher due to the increased stiffness of the polyester in comparison to the nylon rope. The effect of this stiffness on the other eigenvalues is minor. (8) SPM #3 has similar properties with SPM #1 with the exception of having symmetric critical points due to the absence of propeller. (9) SPM # 4 and SPM #5 have properties of a stable node due to Xa and X4 and a stable spiral point due to kl, X2, Xs and X6. (lO) SPM # 6 has one unstable finite critical point which has properties of an unstable spiral point due to
Stability o f single point mooring systems: ill M. Bemitsas and F. A. Papoulias Table 3.
Sumnzary of moored vessel particulars and slow motion derivatives used hz simulations Vessel Non-dimensional expressions
Quantity
Barge with skegs~
Barge~
L ft
Super tanker ~
191.56
xplL
1066.3 0.4683 0.1796
0.505
lwlL t T Ibs
tUc]L T](O.5pL 2U~)
1.0
Xt} m Y~,
Xfi/(O.5pL 3) m/(O .5pL 3) Y~/(O.5pL3)
-0.01383
Yv
Yd(O.5oC'Ue)
-0.01153
YF Yr Yo N~ Nv (Izz -- N i) Nr NO Yvvr Yorr
Yi/(O.5pL') Yr/(O.5pL3Ue) YJ(O.5pL'U~) N~/(O.S pr , ) Nv/(O.5pL3Uc) " (Izz -- NF)/(O.SpL3) Nr/(O .SOL"V c) No/(O.5pL ~U~) Yvvr/(O.5pL3/Uc) Yvrr/(O.5pL'[Uc)
0.0 0.00238 0.0000746 0.0 -0.007285 0.00188 - 0.0 0128 0.00001509
Yvo~l(o.spcVuc)
-0.3005
Yrrr/(O.5pLS/Ue) Nvvr/(O.5pL ~/Ue) Nvrrl(O.SpL SlUc) Nvvvl (O.5o L ~lUe) Nrrrl(O.5pL~lUc)
0.0 - 0.0111 0.01025 - 0.009669 0.0
Y~vv Yrrr Nvvr Nvrr Nvvv Nrrr
-0.00136 0.017 0
-0.0009 0.0181 0.0171 -0.0261 0.0 0.00365 -0.00056 0.0 -0.0105 0.00222 -0.0048 -0.00028
-0.0259 0.002815
-
-0.0026425 -0.00265
0.0193l
-0.045
- 0.01025
0.00611
-
e*
'3_. I . ~ I I ! :I~,I~. N~.I)I ROPe,PROPEI.LER_ 2. ~'I~3 : B.~R6F.,NYLONROPE
?
r
~_o o.
Lt.
Lt-~
I--4 GO Z~
%
Zm LUr I. SPtl~I :SA.~E-~,NYLONROPs163 Z .~2:8.~P.GE, Pt;L~STER RC~E,PRCPELLER_
*o
O.
10.
20.
30.
40.
DIMENSIONLESS
S". TIME
60.
713.
O.
t
Figure 2. Offset for SPM barge system. Mooring line stiffness effect
;ks and ;k6- Thus all trajectories originated in the vicinity o f the first critical point diverge while all those originated near infinity also diverge so that the cable length tension achieve finite values. Therefore there must be a limit cycle in between, to which all trajectories converge3 This is shown in Figs. 10-12. Further, we can assess the properties of the finite time interval behaviour o f SPM systems and the effects o f current direction, propeller, rope stiffness and skegs by studying the trajectories in the simulation graphs 2-12. In these figures, time is non-dimensionalised by L / U e where U c =
Figure 3.
10.
20.
30. 40. OIMENSIONLESS
50,
TIME t
60.
70.
Offset f o r SPM barge system. Propeller effect
2.5 knots = 1.286 m/s. Therefore the dimensionless time of 75 used in the figures corresponds to 57 rain o f real time for the barge and 5 h and 16 min for the tanker. Specifically: (1 I) Figure 2 shows that SPM #1, with the nylon rope, converges to the second critical point. SPM #2, with the polyester rope, exhibits similar behaviour. This is in agreement with the stability analysis results shown in Tables 1 and 2. (12) Figure 3 shows the effect o f the propeller by comparing the behaviour of the barge system with a propeller (SPM #1) and without one (SPM #3). It
Applied Ocean Research, 1986, Vol. 8, No. 1
53
Stability of s#zgle pohzt mooring systems: 3/. M. Bemitsas and F. A. Papoulias oa i
.. x
/~'%
~
(~)
t
/
tl = 180~
~
//I
7 t//'///)t ; vv 0.
In.
20.
30.
/ 'tJ
40.
50.
DIMENSIONLESS
TIME
60.
(13)
(14)
(15)
(16)
(17)
0.
70.
ft.
20.
appears that the convergence of the system with the propeller to the third critical point is slower than that of the system without the propeller. The effect of the combined action of the current and the propeller on the distance x and the offsety is shown for SPM #1 in Figs. 4 and 5 respectively. In case the'direction of the current is ( 1 8 0 ~ 15 ~) degrees the barge is attracted by the second critical point while in case the direction is (180 ~ 15~) degrees the moored barge is attracted by the third critical point at a slower rate. This shows that the asymmetry of the flow past the barge, created by the propeller, affects the finite time interval stability of the mooring system. In the long run one of the two stable critical points will attract the barge provided that the environmental conditions do not change. High stiffness of the mooring line ma3) have another undesirable effect on the drift angle as shown in Fig. 6. When a stable critical point attracts the mooring system quickly tile action of the fast cable dynamics becomes evident. Thus in Fig. 6, for SPM # I , the high frequency oscillations in the drift angle simulation are attributed to the fast cable extensional dynamics shown in Fig. 7.4 Several offshore barges which are unstable in forward motion, may exhibit improved stability behaviour in towing or mooring when equipped with skegs. SPM # 4 and SPM # 5 have one stable critical point. As shown in Fig. 8 this point attracts the barge moored by either rope. Further, it should be noticed, as shown in Fig. 9, that SPM # 4 and SPM #5 are galloping over a significant period of time before they are eventually attracted by the critical point. The tanker system (SPM #6) which has one finiie unstable critical point exhibits, as explained in (10) above, periodic asymptotic behaviour. This is shown in Figs. 10 and 11. It exhibits, however, a large amplitude motion in surge and sway over a 5 h period before reaching the limit cycle. The tension in the rope for SPM # 6 initially exhibits large amplitude oscillations as shown in Fig. 12. After the transient effects have decayed, T has a quasiperiodic response and asymptotically becomes a periodic function.
Applied Ocean Research, 1986, Vol. 8, No. 1
30.
40.
DIMENSIONLESS
t
Figure 4.. Distancefor SPM barge system. Current effect
54
I .SPill I : BARGE,NYLON ROPE,PROiOELLER II= 180"*IS'. 2 SPl'leI : BARGE,NYLON ROPE,PROPELLER,
I..~tl#1:0ARf~.NYI.0NROPE,PROPEDJER, = I~i~162o. ~ 1 1 1 :BARC-T..I'Ctl.0NROPs a = le~p, iso.
/ k i
50.
S0.
TIME
70.
t
Figure 5. Offset for SPM barge system. Current effect
I
r
$M"/ 2 : BARGE. POLYs
ROPE, PROPfl.I.~
g 3. ~
EE
w l t27
"~
T
0.
10.
2o.
30.
~0.
OIMENSIONLESS
so. TIME t
s0.
70,
8
Figure 6. Drift anglefor SPx~barge system. Polyester rope
I
O
Spile2 : BAR~E,F~L~ESTE~ ROPE,PROPEILt~
H LU~
LUoa
d ~. LO LLI~
O.
,I II,
IIIIlilllll l,,, = 10.
20.
40.
50.
DIMENSIONLESS
30.
TIME
60.
70.
8
t
Figure 7. Tension in polyester rope for SPM barge system
CLOSING REMARKS Nonlinear stability analysis and time simulations have been used to study the mathematical model developed in this work for the horizontal plane slow motions of ships moored to a terminal through a single non-linear elastic line. The
Stability o f shtgle pohtt mooring systems: M. M. Bernitsas and F. A. Papoulias
I S f ~ ' 4 : BARE.NYlON ROPE.I:'ROPUJJER.~ 6 5 . : B.kP.r.~oPOLYESTERROPE.PROPELLER.$~:r.,s
2 ~
SPt'l#6 : TANKER,NYLON R ~ , PR0P[Lt[R
>.. o
o,
"'"
= R~O-IS O. :LIMIT CYCLE
Gq
LL~ C3 .":
~
o7
1
iJJ~
w~ U~ Zt
//XX//\\ z \//
o z~
O'),..,i
z~
s ,.o
? ,:g O.
20.
I0.
30.
40.
DIMENSIONLESS
Figure 8.
?
so. TIME
ca.
7o.
-0.70
t
-0.62
-0.58
>I
Figure 11. subspace
-0.46
-0.42
I--
) &~'1#6 : T ~ ,
NYLONR ~ . PROPlRLER,
e - t~~
~
tOm
/
o
-0,50
DISTANCE x
Tanker traiectory in the ofset-distance phase
______-- J
037 Z~.
-0.54
DIMENSIONLESS
Offset for SPM barge system. Skeg effect
LL~
-0.66
tJ~ Zr I. ~11~ 4 : BARC4[,EiS.CI*,I R ~ ,
.
3>
PR(/~[XLER,~ G S : O.&RGdE,POtY[ST[R ROPE,PROPB.LER,$1(EGS.
2. ~
,:? -1.60
-I.52
-1.44
-I.36
-I.52
(POtYESTeRROPE) DIMENSIONLESS
-I.44
-1.36
DISTANCE
x
O.
ca
I L $P~#6 : TANI~R,N~I.0N ROPE.PROPELLER,
a - taO~ ~ NYLON R~E,I~01>BIER,
~.l~-is o
>.
0 Illllllti
-I.28
(m~o~ Ropt)
Figure 9. Barge trajectory ht the offset-distance phase subspace. Skeg effect
2 $i~'~6:TJ~R
to
I
I0.
20.
30.
4~.
DIMENSIONLESS
Figure 12.
5~.
TIME
eo.
70.
t
Tension hz nylon rope for SPM tanker system
limit cycle. Further, combinations of the above effects may result in undesirable system response with large changes in the tension of the mooring line. Such conditions may make the asymptotic and the finite time interval behaviour of the mooring system hazardous.
| ACKNOWLEDGEMENTS
O
to~
This work is supported by the University Research Program of the Department of Transportation under contract #DTRS5683-C-00005.
(-o~i z-~
O,
Figure 10.
10.
/
///
kj
L 9I 20.
/1 // \J
30. 43. OIMENSZONLESS
k.J SO. TIME
60.
' 70
t
NOMENCLATURE a, b, c
Offset for SPM tanker system. Current effect Fx, Fy Iz=
effects of vessel type, propeller, direction of current and mooring line stiffness, on the system short and long term behaviour have been studied. It was shown that the system may exhibit instability in sway, yaw and surge, or reach a
l
lw L
dummy independent variables representing u, v, r and 6 in functions X, Y and N total excitation forces in the X and Y directions respectively mass moment of inertia of moored vessel about the z-axis length of mooring line working length of mooring line length of moored vessel
Applied Ocean Research, 1986, Vol. 8, No. 1
55
Stability o f single pofllt mooring systems: M. M. Bernitsas and F. A. Papoulias m
M~ N
No N~b Nabc r
R Re t
T u
v~ V. X Xp
X
xo x~ x~b Xabc Y Y
Yo Ya Yob Yabc
mass o f moored vessel total excitation m o m e n t about the Z axis yaw m o m e n t yaw moment due to propeller derivative o f yaw m o m e n t with respect to a derivative o f yaw m o m e n t with respect to a and b derivative o f yaw moment with respect to a, b and e yaw angular velocity moored vessel resistance real part o f complex number time mooring line tension moored vessel forward velocity with respect to the water velocity o f current moored vessel lateral velocity distance between mooring point and vessel's centre o f gravity in the current direction distance between the centre o f gravity and the point o f connection o f the mooring line for the moored vessel surge force surge force due to propeller derivative o f surge force with respect to a derivative o f surge force with respect to a and b derivative o f surge force with respect to a, b and c m o o r e d vessel lateral offset sway force sway force due to propeller derivative o f sway force with respect to a derivative o f sway force with respect to a and b derivative o f sway force with respect to a, b, and c
Greek symbols
~' p
rudder angle drift angle between current direction and vessel longitudinal axis in radians yaw angle water density
REFERENCES Abkowitz, M. A. Stability and Motion Control o f Ocean Vehicles, MIT Press, Cambridge, 1972 2. Abkowitz, M. A. Measurement of hydrodynamic characteristics from ship maneuvering trials by system identification, Transactions, SNAME 1981,89, 283 3 Bender, C. M. and Orszag, S. A. Adranced Mathematical Methods for Scientists and Engineers, McGraw-ttill Book Company, 1978 1
4 5 6 7 8 9
10
56
Bernitsas, M. M. and Kekridis, N. S. Simulation and stability of ship towing, h~ternational Shipbuilding Progress 1985, 32 (368) Comstock, J. P. (ed.) Principles of Natal Architecture, The Society of Naval Architects and Marine Engineers, 1967 Cox, J. V. Statmoor - A single point mooring static analysis program, Naval Civil Engineering Laboratory, Report No. AD-AII9 979, June 1982 Eda, H. Low-speed controllability of ships in wind, Journalof Ship Research 1968, 12 (3) 181 Eda, H. Course stability, turning performance, and connection force of barge systems in coastal seaway, Transactions, SNAME 1972, 80 Faltinsen, O. M., Kjaerland, O., Liapis, N. and Naldcrhaug, 1t. Hydrodynamic analysis of tanker at single-point nmoring systems, Proceedings of Second hzternational Conference on Beharior of Offshore Structures, London, August 1979 Faltinsen, O. M. and Loken, A. E. Drift forces and slowly varying forces on ships and offshorestructures in waves, Norwegian Marithne Research, 1978
Applied Ocean Research, 1986, Vol. 8, No. 1
11 12
Hating, R. E. Single-point tanker mooring measurements in the North Sea, Proceedings of Offshore Technology Conference, ltouston, paper no. 2711, 1976 Hsu, F. It. and Blenkarn, K. A. Analysis of peak mooring forces caused by slow vessel drift oscillations in random seas,
Proceedhzgs of Offshore Technology Conference, Houston, 13 14
paper no. 1159, 1970 Isherwood, R. M. Wind resistance of merchant ships, Transactions, RINA 1972, 114, 327 MeKenna, H. A. and Wong, R. K. Synthetic fiber rope, properties and calculations relating to mooring systems, in Deepwater
Mooring and Drilling, A SME, OED 1979, 7 15
Oortmersen, G. van and Remery, G. F. M. The mean wave, wind and current forces on offshore structures and their role in the design of mooring systems, Proceedings of Offshore Technology Conference, Houston, paper no. 1741, 1973 16 Parsons, M. G. and Greenblatt, J. E. 'SttIPSIM]OPTSIM: Simulation program for stationary liner optimal stochastic control systems, Department of Naval Architecture and Marine Engineering, The University of Michigan, Report #188, June 1977 17 Pinkster, J. A. and Remery, G. F. M. The role of model tests in the design of single-point mooring system, Proceedhlgs of Offshore Technology Conference, Houston, paper no. 2212, 1975 18 Sorhekn, tt. R. 'Analysis of motion in single-point mooring systems, Modeling, Identification and Control 1980, 1 (3)
APPENDIX A Tile equations o f motion in the ( x , y ) system, that is (4)-(6), are derived in this Appendix from the basic equations ( 1 ) (3). The total excitation forces F x and F y in the X and Y directions respectively and the moment M z about the Z axis are comprised o f four groups o f terms, namely hydrodynamic terms due to the relative motion o f the water, terms due to the tension in the mooring line, and forces and moments and second order drift forces due to waves. Expressions for the surge and sway forces X and Y, and yaw moment N can be derived by Taylor expansion in terms o f the independent variables, the rudder angle 6 and the relative velocities u, o, r o f the vessel with respect to the waterJ 's In non-linear analysis, terms up t o the third order are used. Terms beyond the third order and second and higher order acceleration terms may be neglected since there is no significant interaction between viscous and inertia properties o f the fluid and acceleration and inertia forces calculated from potential t h e o r y when a p p l i e d t o 2 X ~ b ,2 submerged bodies give linear terms that is Xfifit't, 9 . "2 .. .a Xkrr , Xhuh u . . . . . = 03 The inclusion o f the non-linear terms used here, is necessary because the linear model fails to predict accurately the maneuvers o f unstable ships,s The significant terms, as explained in reference 2, are those included in equations (54)-(57). The surge force X is X = Xo + X u u + 2XuuU I 2 + ~XuuuU 1 3 + 21X w v 2 + 1Xrrr2
+ ~x~ ~ + ~X~v2u + 89
+ lX~.*2U
+ (Xvr + m) vr + Xv~v8 + Xr~r5 + Xrvurvu + Xv~,,v~tt + Xrrurru
(54)
where the first four terms are equal to the negative o f the hull resistance in forward m o t i o n at velocity u
- - R ~- Xo + X . u + ~X.,,u 2 + ~x~..u i 3
(55)
The sway force is
Y = Yo + Youu + Youu u2 + Yo v + 16YvvvV3+ 12YvrrVr2 + ~2Yv~6v52 + YvuvU + ~YvuuVU2+ ( Y r - mu) r + 61Yrrrr 3 + ~1 YrvvrV2 + ~ y r ~ r 8 2 + Yru+ 12YruurU2 + yr5+~y~53+
~y~vv~ 2 + ~ Y , rrSr2+ Y~u~U
+ ~Y, uu 6u2 + Yvr~ vr6
(56)
Stability of single point mooring systems: M. M. Bernitsas attd F. A. Papoulias and the yaw moment is
A ( 1 , 5 ) = Tosinz3ocoszaOlo + cos 3'0 (6"T - - / ~ )
N = No + Nouu + Nouuu2 + Nvv + gNvvVl 3 + ~Nvrrv 2 _Ft~Nv~v6 2 + NvuvU + ~ N ~ v u + Nrr + ~N.r? + ~Nrrvro2 + ~Nr6~r6~ + NrurU + 89 2 + N~ 6 + gt N ~ 6 3 + ~2N6vv6V2 + ~N~rr6r2 + N~u6U + ~N~uu6U~+ Nvr~ vr6 (57) In the above equations the index '0' indicates propeller dependent terms. In the numerical applications with the barge and the tanker only the terms shown in Table 3 are used since these are the only significant terms for the above vessels. Table 3 gives values for the non-dimensionalised slow motion derivatives.4 Combining the above non-linear Taylor expansions for X, Y and N with the dynamic response terms of the equations of motion (1)-(3),the non-linear equations of motion inthe horizontal plane become (m
-
-
X,~) ti = X + T cos co + FxWIND + FXWAVE
X cosz3o sin 6oo A (1,6) =
• (You + 2YouuZ6o + Yvuvo+ YvuuVoZ6o
--mz4o+ Yruz4o+ YruuZ4oZ6o + Ysu6 + Y&,u6Z6o) A(2, l)=[sin'rO(~o--CT)COSZ3osinw o
To cos~z~o-I
io
+ l~lZwlND + MZWAVE (60) We can identify the added mass terms Xi~, Yi, Yi, Ni., Ni, the mooring line forces and moment and those due to wind FXWlND, F~,V~ND, MZWlNO and waves FxWAVE, F~XVhVE, M5~,AaVnEdThe X, Y and N terms, given by equations (54), (57), represent mainly damping terms due to viscous effects and energy dissipation due to waves. Equations (58)-(60) may be further developed by taking the time derivative of the lateral velocity v, using equation (12). + or tan ~ -- ti tan ~ -- ur
+ (Vo tan z3o--Z6o) sin Z3o+ Vocos Z3o] + (Vo tan z 3o-- z6o) Cc, cos z3o-- Co sin z3o
A ( 2, 4 ) = (cos z 3o) [NvrvoZao+Nr + t~NrrrZ4u2 +1
A (I, 3) --- To sin Z3osin 6o0
+ vo sin Z3o) A (2, 5) = [ L
/o
To + cosTo(CT----~o)COSZ3osin Wo]xv 1
A(2, 6) =--Nfiz4o - -
CC, sinz3o
COSZ30
Toxp cosz3o
+ N~.uVoZeo + N.uz4o + N...z~oz~o
to
+ N~.6 + N,.,,SZ6o) To A (3, 1) = -- - - cos 3'0 sin 6oo + CT cos COosin 3'0 1o
To coszao cos COo+ (Vo tan Z3o--Z~) C~,
I
[Xwvo + (Xvr + m) Z,o + Xv~ 6
A (3, 2) =
• cosz3o--Bosinz3o+ (m -- Y;~)
COSZ30
x Z4o[--z6o sin z3o--(vo tan Z3o--z~) sin zzo
+ X.~.z4o zGo+ Xo~,,6z6o] A (3, 3) = (vo tan Z~o-- Z6o)[Xvvvo+ (Xvr + m) Z4o
- - Uo COS Z30]
A (1,4) = (cos z3o) [YvrrVoZ~+ (Yr--mzc,o) + ~YrrrZ42o 2
To sinz3o COSZ30
+ cos z3o(Nou + 2NouuZ6o + Nvuvo
--Xp sin2wo ( C T - - / ~ ) cosz3o
+l
t 2Nrwvo2 + ~gr56 62 + Nruz6o + igruuZ6o 1 z
+ N6rrZ4o6 +Nvr66Vo] +Nr3 (--Z6o coszs0
A (1,2) = C~, -- (m -- Yfi)z4o tanz3o
-
To cosz3o cos cooJxp+N~z4o [Z6osinz3o
--
To cos 2Z3o 1o
Io
2 [ To~ --x v sin C o o | C r - - - - l c o s z 3 o \ !o ]
(61)
Upon substitution of (61) into (50) and (60), we get the equations of motion in the (x,y) system of co-ordinates as given by equations (4)-(6).
To% COS Z30
A (2, 3) = [ To sin Zao sin COo L
COS
A ( I , I ) = sinTo(T~ \ 1o
I x"
A (2, 2) = Cr +N;)z4otanzao
--N~ b + (Izz - N~) i" = N -- Txp sin 60
APPENDIX B
CB, sin Z3o+ (cos Zzo)
COS Z30
(58)
(m -- Yt;) b -- Yfi"= Y-- T sin co + FyWIND+ FyWAVE (59)
b=
(m -- Y~)z4o
1
2
I
+ Xv86 + XrvuZ4o z6o + Xv~u6Z6o]
2
~YrvvVo + ~Yr566 + Yruz6o + 2Yrm, Zc,o --
+ Y, rrZ4o6 + Yvr~6Vo] + ( m - Yfi) x (Z6ocosZ3o-- Vosin Z3o)
To sin~o(,xpc~176 t-1 ) lo
+ CTXp
COS 6O0 sin
6oo
Applied Ocean Research, 1986, Vol. 8, No. I
57
Stability o f single point mooring systems: M. M. Bentitsas and F. A. Papoulias A (3, 4) = Xrrz4o + Xwuzao'z~o + X m , z4ozoa
B(4, 1 ) = 1
+ (Xvr + m) Vo + Xr~8+XrvuVoZ6o
B ( 5 , 3) = 1
+ X~,,8 zoo
B(6, 5) = 1
A ( 3 , 5 ) = ~ ~Xvvu V2o " F ~~tXrruZ~o 2 + ~~ X ~ . 8 2 + X~,,VoZ4o
To
+ Xv~uSVo + Xr6uSZ4o -- -~o sin 3'0 sin too
and Bo = Yo + Youz6o + YouuZ~ + Yvvo + ~1Ywv Voa +i
-- C T COS r
+1
2 3 iYvuuVoZ6o + (Yr -- mz6o) Z4o + 61 Yrrrz4o
A (3, 6) = (-- tan Z3o) [Xwvo + (Xvr + m) z4o + Xv~ 8
+ Xm,z~oz6o + Xv~.Sz6o]
+ ~1.. : + ~yr~Z4o82 + I rvvZ 4OVO
+ (xi, + x,,.z~o + ~ x . . . z ~ )
+1
A(5,4) = I -
-
1
r r u z 40z 60
~YvuuZaoZ6o + Y88 + g Y ~ d i 3 +
1
2
CB = Yv+ ~YvvoVo l : + 1Yvrr Z 4o 2 "q- ~Yv~68 1 2 "4- Yvuz6o tanzao Vo
A (6, 3) =
COSZ30
A (6, 6)
2
1 2 + ~Y~rrSZ4o+ Y~uSZ6o + 2~-Ya,uSZ60 2 + Yw~ VoZ408
A(4,2) = 1
A (6, 2) =
i 2 YvrrVoZ24o + ~Yv~vo~ "1- YvuvoZ6o
COS "f0
1
2 + 12Y~uZ6o+ Ym, voz4o + Y~vSVo + Yvr~SZ4o
Co = No + Nouzoa + NouuZoo 2 + Nvvo + gNwvvo i 3 +~ 2 1 2 ~NvrrVoZ4o + ~Nv6~ Vo5 + N v u v o Z 6 o 1 2 1 3 "1 2 + 2 N ~ , u V o Z 6 o + N r z 4 o + g N r r r Z 4 o q- ~NrrvZ4oVo
-
COS Z30
..~1
1 2 ~N,.~z4o8 2 + Nruz4oZ6o + ~NruuZ4oZ6o + N88
B ( 1 , 2 ) = m -- Y~; B(1, 4) = -- Y/. coszzo
q_ 16N~6683 q_ ~ I N~SVo 2
B(1,6) =-
.t_1~N~,#,Sz6o 2 +
(m-
Y~) sin zao
~(2, 2) = - N ~
1
Applied Ocean Research, 1986, Vol. 8, No. 1
2
+ 1~NvuuZ6o 2 + NrvvVoZ4o + NbvvSVo + Nvr85z4o
B(2, 6) =Nil sinz3o
58
Nv~6 Voz~o8
1 2 Cc, = N~ + ~ N ~ v o + ~Nv~:4o + ~Nv~ 8 2 + No,,z~o
B(2, 4) = (Izz--Nk) cosz3o
B(3, 6) = m -- X~
1 2 + N~,,6z6o + ~N~.6z4o
qTo CT =
-
eWolw