RM-3368 -PR
JANUARY 1963ANALYSIS OF
THE RESPONSE OF MOORED SURFACE
AND SUBSURFACE VESSELS TO
OCEAN WAVES
J. J. Leendertse
This research is sponsored by the United States Air Force under Project RAND
Con-tract No. AF 49 (638)-700 monitored by the Directorate of Development Planning, Deputy Chief of Staff, Research and Technology, Hq USAF. Views or conclusions con-tained in this Memorandum should not be interpreted as representing the official opinion
or policy of the United States Air Force. Permission to quote from or reproduce portions of this Memorandum must be obtained from The RAND Corporation.
7d.
R-11 n
This Memorandum is part of a RAND study of water basing of weap-on systems. For this study it was necessary to obtain information about the behavior of moored surface and subsurface vessels in wave-disturbed water and the induced force fluctuations in the mooring lines. This Memorandum presents an analysis of this problem.
The analysis should be of interest to defense agencies and con-tractors concerned with the mooring of vessels as well as to designers of moorings for various vessels including those used in the exploration and exploitation of the offshore sea bottom.
SUMMARY AND CONCLUSIONS
This study analyzes the response of a moored ship or submersible craft, in uniform or irregular waves, and also the forces in the moor-ing lines induced by ship responses.
Considerations are restricted mainly to waves approaching the vessel head-on, and only the surge, pitch, heave, and the fore and aft mooring-line forces are examined. Moorings with and without buoys are considered. The influence of the assumed linearization on the
mooring-line
force-displacement diagram for ships in irregular waves is discussed, and a method is given for estimating errors introduced by the linearization.The response in heave, surge, and pitch of a moored ship or sub-mersible craft with known hydrodynamic characteristics to waves
ap-proaching head-on can be calculated by use of the method outlined in this Memorandum.
Computed values of the response of a moored 880-ton vessel (simu-lated by a rectangular block of equivalent displacement), moored in uniform waves, are found to compare favorably with results of model tests.
High mooring-line forces are caused principally by the surge motion. These high forces can be expected when waves of significant heignt are present with a frequency equal to the natural frequency of the ship.
The calculation method can be expanded for the other modes of movement (sway, roll, and yaw) and for waves with arbitrary heading if the modes of movement of the unmoored ship are essentially un-coupled.
CONTENTS
PREFACE
SUMMARY AND CONCLUSIONS
LIST OF SYMBOLS ix
Section
INTRODUCTION 1
MOTIONS OF AN UNRESTRAINED VESSEL IN HARMONIC WAVES 2
MOORING-LINE CHARACTERISTICS 4
SPREAD-MOORED SHIP 8
SHIP MOORED BY BUOYS WITH UNIFORM WAVES HEAD-ON 20
SUBMERGED CRAFT WITH UNIFORM WAVES HEAD-ON 23
SPREAD-MOORED SHIP AND SUBMERSTRTY, IN LONG-CRESTED
IRREGULAR WAVES 26
ErraT
OF THE NONLINEAR MOORING-LINE FORCES 28DISCUSSION 33
Appendix
MOORING-LINE CHARACTERISTICS 37
CALCULATION OF RESPONSES 41
A = wave amplitude
A = horizontal cross-sectional area of a ship at water surface
A
cross-sectional area of a buoy atWater surface
X = complex value of the movement in surge for a
Unit height
d = coefficients in linearized mooring-line equations
B beam
li= complex value of the movement in pitch for a wave with unit height
= complex value of the movement in heave for a wave with unit height
d1 = coefficient in linearized mooring-line equations
E = expectation value
F - resultant horizontal component of the restoring forces of
h
the mooring cables
F = resultant vertical component of the restoring forces of
the mooring cables
complex value of the exciting force or moment in the s ex
mode of movement for a wave of unit height
f() function
-s
fex complex value of the exciting force or moment per unit
mass
G = spectral energy of the response for e = o g = acceleration of gravity
g(x) = odd-single valued power function of x
= horizontal force at the holding point (o,o) ko,of
the still
the still
H(w) ratio of response in a particular variable to wave amplitude (complex frequency factor)
[Hkwj]2 = square of the absolute value of the complex frequency factor
vertical distance between the holding point of a mooring line and the sea bottom
I = imaginary axis
I
(0
= random (force) function, derived from the wave spectrumI inertia mass moment of the ship solution around the
8
y axis
I = random (moment) function, derived from the wave spectrum
I" = added inertia mass moment
8
J inertia moment of the horizontal cross-sectional area
of a buoy around the y axis
Keq equivalent linear stiffness coefficient
K stiffness coefficient in force equation of the s mode
ST
for the movement in the T mode
kst = stiffness coefficient half-length of a ship
I = horizontal distance between the point where a mooring line touches the sea bottom and the holding point Mh = total moment of the horizontal components of the bow
and stern lines
M
total moment due to the vertical forces in the mooringlines perpendicular to the long axis of the ship
M
virtual mass or mass inertia moment in the force onST
moment equation of the s mode for the movement in the T mode
Ma = expected number of maxima of the response per unit time exceeding the value of the response R(t) = a
MI = virtual mass in the x Movement
xx
14 = virtual mass in the x movement
xx
lv = virtual mass in the x movement
'xx
MP = added mass'
= linearized damping term in the force equation of the s
ST
mode for movement in the T mode
n = horizontal distance between the point where a mooring line touches the sea bottom and the anchor
p = vertical distance between the holding points of a mooring line and the mass center of the ship
R = average response amplitude. av
Rt periodic force due to other Modes of. movement
R(t) = response amplitude
R = restoring moment in pitch of a hovering submersible
Bliz = average response amplitude of the 1/3 highest responses
" (i.e., of the highest third of all amplitudes)
S = total length of a mooring line
Sr(w) = spectral density of the response in a particular variable
S (w) = spectral density of the waves
= length of a mooring line lifted from the bottom s = general indication for mode of movement
T = total force in a mooring line
Tos
period of free oscillation in the s mode
t = time
V, = vertical force in a mooring line at the holding point o
kolo)
V, = vertical component of the force in the mooring lines
01o)p
perpendicular to the long axis of the vesselV(x ) = vertical force in a mooring line at the holding point (x,
w = net weight of a mooring line per unit length xly,z = Cartesian co-ordinate axes
Y = random variable with zero mean value
= -w2M
+ NN
+ K (impedance)ss ss zz zz
zB = vertical distance between center of gravity and center of
buoyancy
1 2:ss
a m:
wo ss
a = a value of the response
-eq
df(K )
y = coefficient =
eq,
8 = water depth
e = small parameter modifying the nonlinear function = pitch angle
p = density of water a = root mean square
a root mean square of the response of the assumed linear
system in surge
= root mean square of the response of the nonlinear system in surge
T =
general indication for mode of movement (used only as asubscript) p = angle of roll
To = angle of a mooring line at the holding point with the horizontal
* = angle of yaw
w wave frequency
wos
natural frequency in the s mode
I. INTRODUCTION
A vessel moored at sea will experience a complicated series of
translational and rotational oscillations due to sea waves. These
motions can be considered as the summation of six components, three translational and three rotational.
In the presently available analyses of motions of unmoored ships and submerged crafts, differential equations can be written for each mode of movement. Unfortunately, motions in one of these modes are
coupled to motions of other modes, and the analysis becomes rather complicated. Generally, the problem is simplified by neglecting some of the coupling effects and by specifying the position of the vessel in the wave system.
This study develops and analyzes a model for a moored ship or submerged craft restrained by mooring lines, using the presently avail-able mathematical models for the free ship or hovering submerged craft and the force-displacement relationship of the cable-holding points on the ship.
The coupled movement (three degrees of freedom) in a vertical plane through the longitudinal axis of the vessel and the generated
mooring-line forces are considered in detail. The general case of
six degrees of freedom in arbitrary heading is discussed briefly in general terms in Section IX.
MOTIONS OF AN UNRESTRAINED VESSELINMARMONIC WAVES
-
(1)
Referring to the analyses by Weinblum and St. Denis, the Move
ment of a Vessel unrestrained by mooring lines in harmonic waves may be expressed with Certain approximations by the second-order linear
differential equation ds -s jwt + N cis + K s + R
= AF
e dt2 ss dt ss t ex Where A = wave amplitude=. wave force for a wave
of
Unit amplitudeex
stiffness coefficient
ss
.. virtual pass or mass inertia moment of the vessel
ss
N = linearized damping coefficient
ss
R = periodic force due to the other modes of movement
s = considered translational or rotational displacement
W. = wave frequency
The first subscript of the mass, damping, and stiffness coeffi-cients refers to the considered force or moment equation; the second subscript, to the mode of movement to which the coefficient belongs.
The first term on the left in Eq. (1) represents the inertia force; the second term represents the damping force; the third term is the restoring force; and the fourth term is the force due to other modes of movement. The term on the right expresses the periodic force of the waves.
Extensive literature is available concerning the calculations of the mass and damping coefficients for a ship of particular dimensions and the periodic wave force. Weinblum and St. Denis,(1) and Korvin-Kroukovsky,(2) particularly, present readily applicable data for
cal-culating these coefficients and the wave forces.
Information about the coupling, of the different modes of move-ment is limited, and only a few incidental cases have been
investi-gated; fOr example, the coupled heave and pitch
by
KOrvin-KrotkoVsicyand Jacobs. and St. Denis neglect the coupling in their analyses of Ship motion, and in this study, the coupling term will also.bo.neglected initially.
For the unrestrained Ship, the restoring forces and moments
in
the different modes are caused by the displacement of the Ship from
the position of rest; if the Ship is moored, the forces
in
the mooringIII.
IvIOORING-LIDIE cHARAcTERisricsThe forces exerted on a ship or vessel by mooring it with a long single chain or cable that has an embedded anchor at its other end are
functions
of the weight of the chain or line and the location of theholding point in
the ship
relative to the anchor. If it is assumedthat the cable is lying partly on a flat bottom as in (a) of Fig. 1,
then the horizontal and the vertical forces on the
ship
are nonlinearfunctions of the horizontal and vertical displacement. Based on the
analyses of single mooring lines presented in Appendix Al (b) of Fig. 1
presents the total tension and its horizontal and vertical components
as a function of the displacement in nondinensional parameters.
In a particular condition of the mooring chain, for example, as presented in (a) of Fig. 1, a rectangular-coordinate system is fixed
to this point, with the x-axis horizontal and z-axis vertical. For
small displacements around the holding point (0,0), the horizontal and the vertical components of the force in the chain at this point may be assumed to be linear with the displacement and may be expressed by H
=H
(o,o) + ax + bz (xlz) V(x,z) = V.(o o) + cx + dz where= horizontal force at the holding point o H(0
vertical force at the holding point o
17(0,o)
H(x,z) horizontal force at (x,z)
V(z) = vertical force at
(xlz)The coefficients a, b, c, and d can be Obtained directly from Fig. 2, which Is based upon the mooring-line analyses presented in Appendix A. It will be noted that b < a and c < d.
500 100 50 0.5
0.l
0 (b) 0102
03
04
Displacement, (S-L)/ h05
Fig. I
Nondimensional representation of mooring- line
forces as a function of displacements
5000 1000 500 100 50 RD 5 01
02
03
04
05
06
Displacement, (S - L) /hFig. 2
Linear coefficients for small
If a Chain with a sinker is used, the forces can again be ex-pressed by Eqs. (2) and (3), but the calculation of the coefficients becomes cumbersome.
IV. SPREAD-MOORED SEEP
Spread-mooring is used presently in the oil indUAtgy for mooring tender-barges near offshore drilling platforms. The layout of the mooring is represented in Fig. 3. It is assumed that the waves ap-proach the ship head-on. Initially, it is assumed that the ship is subjected to uniform waves; later on, the effect of irregular waves will be introduced.
The Ship's motions in the plane considered involve surging, heaving, and pitching. For the unrestrained (free-floating) Ship, surge does not have important effects on the heave and pitch and consequently PaY be considered uncoupled. In the ease of the moored ship, however,
coupling will enter into the system due to the mooring lines. For
ex-ample, the position of the bow, 'which is determined by heave and pitch, influences the horizontal component of the mooring-line force, and hence the surge.
Referring to Eq. (1), Weinblum and St. Deis, Wilson,(4)
the linearized equation of motion in surge for the center of gravity of the unrestrained ship, compared to a fixed coordinate system taken in the center of the Ship in still water, takes the form
M
+I
=AC
eiwtex
where
M + M" = virtual mass of the ship in x direction xx
M = mass of ship
le
= added mass in x direction= damping coefficient in surge xx
Generally, the drag is small and may be neglected. However, in
some cases, moored crafts may be built specially for mooring purposes, and in that case, no emphasis may be placed on towing or propulsion
characteristics. Then, Nxx is not necessarily small, and estimates of values may be obtained from the propulsion characteristics and a
lineari-Top view
Side view
Note: The ship's principal axes are shown coincident with the fixed axis.
Fig.3
Spreadmoored ship
ration process as developed by Havelock (described in Ref. 1) for the heaving motion. For the time being, the drag term will be maintained,
being important even When small in eases of a resonance condition.
In addition to the inertia and drag forces, a restraining force exists for the moored vessel, and the equatiOn of motion becomes
M
5E +N *+F
= I1dab
XX h ex
where F is the resultant horizontal component of the restoring force of the mooring cables. With reference to Eqs. (2) and (3), taking the direction of the x-axis tqward the left in the direction of wave pro-pagation, the horizontal force of the left cable is
H(xlz) rn
(5 )
-
a(x - p0) + b(z + LO)(6)
where
p = vertical distance between the holding points of a mooring line and the mass center of the Ship
L = half-length of the Ship
and for the cable on the upstream side
H(x/z)bow = H(°'°/bow
a(x - p0) - b(z - LS)
(7)
The other four mooring lines have no significant component in
the x
direction. Consequently, the total restoring -force is
- F - 2ax + 2(bL + ap)0
Thus, Eq.
(5)
becomes'14302 + Nxxi +2ax - 2(bL + ap)G = eiwt
ex
-2(bL + ap)
xe
Eq.;(9) becoMes Jut M -2 + N k.+ K + Ke
'Arc
e-xx xx -xxxe
. ex (5)In this analysis, following the presentation by Kriloff,
Wein-(1) (4)
blum and St. Denis, and Wilson, the coupling effects as induced on the free-floating ship are neglected. Tests on ship models and com-putation of coupled and uncoupled motions indicated that neglecting
the coupling terms is of minor
significance
for the pitching motionbut is more important for the heaving motions. As will appear, since
the effects of heave on the mooring-line forces are relatively minor compared with those of pitch, neglecting these coupling terms in the motion equation of the free-floating ship seems justified and simpli-fies the analysis significantly.
For the moored ship, the heaving motion is influenced by the re-storing force of the chains. The restoring force Fv for the bow and stern chains can be calculated from the vertical mooring-line force
V'(x,z)stern = -V.(o,o).tern
+ c(x - 1)0) - d(z + Le) (13)
V(x,z)bow = - c(x +
pie) -
d(z - Le) (14)kolo)bow
Addition of Eq. (13) and Eq. (14) gives
Fv =
-1/(010
'stern
-
V(opoibow- 2dz
l
(15)
(u)
The constant forces
V-6,o)stern.
+ :V6,o)b0; act downward-.,on- the ship'
and increase its displacement. ,Generally, this increase is very small and may be neglected. Consequently, the stiffness Coefficient
in
heave for the moored Ship becomesK = (pgAs
zz (16)
where As is the horizontal cross-sectional area of a ship at the still-water surface. The first term on the right side of Eq. (16) repre-sents the vertical force due to the displaced volume of water; the second term, the force due to the mooring lines on the bow and stern of the vessel.
The coefficient d appears to be very small compared with pgAs and consequently the bay and stern mooring lines have an insignificant effect on the heaving motion. Likewise, the other mooring lines al-ready neglected in Eq. (16) have no effect on the heaving motion.
Following Eq. (1), the equation of motion in pitch of a free-floating vessel may be written, if coupling with other modes of move-ment is neglected, as
e + Ne
+ Kee0
AF eex
jWt
(17)Where
mee virtual inertia moment of the ship
Nee damping coefficient in pitch
Kee = 138j'y
J = longitudinal moment of inertia of the water plane
= exciting moment in pitch due to waves with unit height ex
The restoring moment
(Keee)
of the free-floating ship is increasedThe total moment of the vertical components of the bow and stern line is M - -
Pe)
V(x,z)stern(L + Pe)
v(xyz)bow
= 2cLx - 2cpL6 -
2dL2e
+ pe
Cr(00)
stern V(°'° ow 2dzThe moments due to the horizontal forces In the stern and bow lines are
=
- (p + L8) H(x,
z)sternp - Le)
bow(2o)
+ 2apx - 2ap2
e - 2isioLe
(21)
- Le (B(°'°)stern +11(0,0)bow)
-2bzLe
The moments due to vertical forces in the mooring lines perpendicular to the long axis of the ship are
H
4
(d1L2 +no,op
+e
(22)
where
V(
o, o vertical component of force in mooring line
)p perpendicular to the long axis of the vessel
di coefficient determining the influence of the
vertical movement
Neglecting the higher-order terms, the resultant moment due to all mooring-line forces is
+ (}1 (o,o)stern + H(o)o)b + 4
V(0,0
Vf°2°)bow + 2dL + ild-1/1) L (v(02°)stern (23)Consequently, the total restoring moment is a function of x and. 0, and. the equation of motion may be written
Mee
+N00 +K00
9+K x =
Ox AFeex ei(ut (210ee
= PgJy +
[2ap2 + 2(b + c )1g" +(0,
o)stern + H(0, Obeli
+ 2dL
+
4d1L) L - Off +V( 0, o)bow + 4
`0101stern /13)111
(25)
-2(cL + ap) (26)
Thus, the three equations of motion are
+
Ni+Kx+KO=A1Vce
iti5E xx xx , xx x9 ex (ut (27)M
+N09 +K88 0+KOx x
Aiexdot
(28) -s jwtM 1+N i+K
z
AF e zz zz zz ex '(9)and.
It will be noticed that Eqs. (27) and (28) are coupled.
Antici-pating a solution
x =
Aejwt
=
Beiwt
Where X and. B are complex quantities, then
2 =
ejwt= jug
eiwt=
-w B e
2Introducing these complex quantities in place of the real quanti-ties in Eqs. (27) and (28) gives
C-w2Mxx + jwN
QCx0 =
+ ExyA
+ KB
Aex
KOxX C-w2mee + Kee) T3
= AFex
We now introduce the impedances
ZXX =
+JWNXX +XX
ZGO=
-w2MGO + jwNee+ xee
(39)
(30 ) (
3 )
( 32 )(33)
( 3 4 )(35)
( 3 8 )and
-Exx-A- Kxer3
Anx
K + 13' = ATA
ex
exSolving for A and
T3 gives
2. ex ee -6
'F
ex
ex
A
(42)
1-3- =(43)
-23CXK
xe K9 2eeby which amplitudes and phase lags with the exciting periodic
waves
can
be calculated.For the vertical motion, a complex
solution is
anticipateddejwt
Thus
where -d is a complex quantity. Following the method for x and
e,
weobtain = 42zz ex where z = -w2m
+
iwN+K
zz zz zz zz FexA
Z_The fluctuations in the mooring cables may now be determined. For
example, rewriting Eq.
(6)
H(x'z)stern = 11(0,o)stern
- ax + (op +bL)19 + bz
If we introduce the following.expression for the force fluctuation in the mooring cable, Which is a function of wave amplitude and frequency
Hstern(A2W
+ Cap
+ bL)e + bz (48)then
Hstern(A'w) = Re[-EA + (ap- +
bL)IT + bU] Ad at
(14.9)
In many instandes, the term K9 in Eq. (40) is very smtll com-pared to 27,., and Kex in Eq. (41) is very small comcom-pared to ;61. Then
the pitch and surge of the moored ship are essentially uncoupled, and r C
ex A
(50) 1-xx
The coupling is important, however, for the resonance movement in surge, lahich is generally not significantly damped, and in that ease Eqs. (42) and (43) have to be used.
If coupled motion for the free-floating Ship in pitch and heave are important--for example, for a ship with the center of mass not ap-proximately in the middle of the Ship as described by
Korvin-Kroukov-sXy(2)--the equation of motion of this vessel when moored becomes
ItcX2 + Nxxi + Kxxx + Kx09
= Alex e (52)
ex
KOx x +M00.6 + N009 + K090 + MOz2 + NOzi + Kezz AF°ex eiwt (53)
\
Mz9.6 +Nez
e
+Kze
9 +M 2 +N i +K z
zz zz= Ar
ex eitut (54)or using the mechanical impedances Z Zze, similarly Z in
Eq. (38) the equations of motion may be expressed:
7
X + =K0T3
(55)
xx xAnx
Kex
I + 2
99 13 + EI3z b-=
09
ex (56)+ 2
-a.=
Alis-z(57)
zte
zz
ex
This Set of linear equations may be :solved
by
using Cramerl s rule, - writing for the determinant of the system-2-
K
6
1,:xxe
A =
K
-2-7
ex
ee
ez
02te 2tt
(51) (58)The unique solutions are
given by
A
(59)
A
=
e
(6o)
(61)where 6s:, Ael Az are the determinant forms obtained by replacing the elements of the first, second, or third columns, respectively, of
Eq.
(58)
byax
V. SHIP MOORED BY BUOYS WITH UNIFORM WAVES BEADrON
The equations of motion for a ship using mooring buoys canbe
de-rived in a fashion
similpr
to that for the ship using mooring cablesonly. In this case, the motions of the buoys have to be considered
in addition to the motions of the ship.
Considering a mooring configuration in Fig. 4, it will be noted that the relative vertical motions between the buoys and the ship will induce Fmpil horizontal displacements between the buoys and the ship, thus relatively small force fluctuations in the lines between ship and buoy. Consequently, the heaving and pitching motions of the ship are
considered as of no importance to the forces in these lines. This is
naturally not the case for the heaving motions of the buoys.
Assuming again a linear relationship between forces and movements,
and neglecting the pitching of the buoys, the equations of motion of
the system neglecting damping in surge become
M22 + Ki
(x2 -
xl)+ K2 (
- x3)
2 ejwt .(63) xx -ex /4.4 R +K (X3 - x2) + a x3 + b z3 = ejwt
(610
'xx 3ex 141 + [pig (2 LiBi) + di z1 - cx1 -sFl(65)
ex zz zzM
2 + N +[og (2
L3B3)+
d]
z3 + cx3 2eiat'
(66)
3zz 3
3zz
exIn many Instances, in mooring with buoys, the connection between the ship and bilOY is made with a cable that is relatively light in
com-parison with the heavy chains used on the
buoys.
If these Cables areplaced
in
high tension, the horizontal movements of the buoys and theShip are practically the same, and it may be assumed that Xi = X2
x3.
Wave propagation
Fig. 4
Buoy mooring
M3
Then Eqs. (62) through
(66)
reduce to' (Mi M2 1413 xx xx lvi(
+ 1 exV`
+ 172c eiwt 2 -3ex)(67)
exmi
Y1 + N1 z1 + [pg(An) + d] z1 - cx = -sF 1 .eiwt
(68)
- . xx Z Z ex --z jwt +N3 z3
+ [pg(Arr) + d] z3 + cx F e(69)
3zz
zz
2exIn Eq. (67), the virtual masses of the buoys are small compared
with the mass of the ship, and also the horizontal wave forces are
small compared with the wave force acting on the ship; consequently,
the effects of the buoys in this horizontal movement of the
ship
maybe neglected.
Generally, the natural frequency in heave of the buoys is higher
than the frequencies of the waves, thus the terns ME and Ni are small compared with the term [Pg(2LB) + d] and may be neglected in our ini-tial investigation of the Ship's movement.
Disregarding the above-mentioned terms, introduction of Eqs.
(68)
and
(69)
into Eq.(67)
givesThis result is important, since in principle it enables the de-sign of a mooring in 'which, at the resonance frequency
gArr + d 2 2bc XX Sc + 2ax bz + bz3 (71)
\
M2xx 1+ENlxx +N2xx/
+ a 2bc b -zF (pgArT d) - (pgArt + d) lex -s F2 ex .-x
+ 2 , ex jwt e (70)(PgA+
d) TI athe excitation term on the right side of Eq. (70) becomes zero by proper placement of the buoys.
:ee
2ap2 + 2(h + c)pL + (2dL +
H(p,o)stern
VI. SUMMED CRAFT WITH UNIFORM WAVES HEAD-ON
Equations (27) through (29) developed in Section IV are also
applicable to submerged craft having the configuration shown in Fig.
5.
Here the vessel is assumed to be moored with two lines, one at the bow and one at the stern, and to be situated in water of intermediate
depth.
Since displacements in heave do not induce changes in the volumes of water displaced by the vessel, as is the ease with surface ships, the restoring forces in the equation for heave (Eq. 16) are determined only by the effect of the Mooring lines. Thus
zz
= 2dz
The restoring moment in pitch (Re) of the hovering submersible is caused by the buqyance
Re
= pg7z0
(73)where
zB = distance between center of buoyancy and center of gravity
V =
displacementConsequently, the stiffness coefficient in pitch for the moored submersible is
Goodman and Sargent,(.77)tudying the response of a submerged
hovering submarine, neglect the damping term in the equation of motion. 0)stern
+(0)°)/Do)
P Cg7zBFig. 5
Moored submersible
Wave
Considering a Submarine with approximately synmetrical fore and aft end, they obtained in the nomenclature used ihere
-M
=As
zz FexeNe
+ cgg . AF e - exLe
jutFor the moored submersible the equations become
M
+ K. x + K.e
xx xxxe
-Mee +Ke
ee+ K x
Ox = exeiwt -s jwt AF e zz zz ex.Neglecting the damping terms is justified here if no motions With fre-quency close to one of the natural frequencies are considered;
VII. _SPREAD-MOORED SHIP AND SUBMERSIBLE IN LONG-CREW= IRREGULAR WAVES
It appears that the actual wave condition in the ocean can best
be represented
by
use of the model of a random process as derived byNeumann and described by Pierson, Neumann, and James.(8) Statistical
values such as average wave height are given, not Wailes of the enr. vironment as a. function of time. The sea-is taken as a sUMmatiOn Of
a large number (or as an Integral
or
an infinite number) of uniformWive trains, each with different amplitudes and directions tOperIm
posed
in
random phase. The profiles of the individual waves areEarned to be sine curves according to Airy't Theory.(9)
Techniques are available to predict the amplitudes of the waves and their distribution over the frequency range from wind velocity, wind duration, and the fetch. Generally, the result can be presented in the form of a wave spectrum, which is the distribution of the mean squares of the wave amplitudes in a given increment of the frequency (spectral density) over the wave frequencies.
In the following analysis, it is assumed that the waves are
uni-directional and meet the ship or submerged vessel head on. This case is
realistic, as it represents the crafts moored in swell.
Following the work by St. Denis and Pierson,(10) the relation
be-tween the spectral density of wave and ship responses is given by
S(W)
= Sw(411(w)12 (80)where
S (w) spectral density of the response in a particular variable (displacement, strain, etc.)
Sw(w) spectral density of the wave
H(w) = ratio of response in a particular variable to wave
amplitude (complex frequency factor)
If the spectrum of the waves is given, the spectrum of the re-sponse can be calculated by Eq. (70). The mean square of the response is then given by
-a2 =
Sr(w)dw =
I
Svt(w)[H(w)]2dw(11)
It has been Shown by Longuet-Higgins, that for a relatively nar-row band of wave frequencies, such as is the case with swells being assumed here, the probability distribution of the wave amplitudes
tends to be Gaussian if the frequency factor
has
nonzero values inthe range of wave frequencies. Consequently, it mgy be expected that the probability distribution of the response amplitudes is also Gaussian.
Longuet-Higgins calculated important statistical relationships for the narrow-frequency spectrum,
which
were consequently tabulated by Pierson, Neumann, and James; (8) for examplewhere
E[R
co]
R = 0.88 a av
R113
= 1.416 a
In many instances, the response spectrum may not be considered to be narrow, and the expected number (Ma) of maxima of the response
per unit time exceeding the value of the response R(t) = a can be
ex-(12)
pressed after Bendat as
Ma =
(2c
01
)1/2( ct2)
a2
w S [H(w)]2dw
(Average response. amplitude)
(Average response amplitude of. the 1/3 highest responses)
Thus, this presentation introduces the
probability
concept into thecalculation of movements and cable stresses.
VIII. EFFECT OF THE NONLINEAR MOORING-LINE FORCES
In the analyses of the response, it has been assumed that the re-storing forces of the cables are linear with the displacement by use of Eqs. (2) and (3). This assumption will introduce certain errors in the calculated response and the mooring-line forces.
Considering the spread-moored ship, it has been seen that the pitch and surge are coupled because of the bow and stern mooring lines.
If the total horizontal restoring force of a system is plotted
as a function of the horizontal displacement for different pitch angles,
a graph of the type presented in Fig.
6
will be obtained. In thisgraph the linearization calculated by Eqs. (2) and (3) is also plotted. The nonlinearity of the total restoring force is much smaller than that of the individual cables.
It will be seen from such graphs that force-displacement curves for different pitch angles are essent1P11,y parallel for equal distances over the expected range of pitch angles.
It is assumed that movements in surge extend into the nonlinear
range. The horizontal restoring force may now be written, following
(13)
the procedures of Crandall, by extending the linear Eq.
(8)
Fh = 2a[x + eg(x)] - 2(bL + ap)9 (82)
where
e = small parameter modifying the nonlinear function
g(x) = odd single-valued power function of x
The values e and g(x) are chosen in such a manner that for zero pitch angle, Eq. (82) is identical with the force-displacement curve obtained by use of Appendix A.
The coupled equations of motion in surge and pitch for the ship in irregular waves can now be written by introducing the nonlinearity in Eq. (27).
Total horizontal
restoring force
Displacement in surge Actual force Linearized forceFig. 6
Typical plot for the horizontal restoring force versus
Where I
(0
and 19(t) are random functions, both derived from the wave xspectrum.'
:Equation
(83)
may be rewritten by introducing the equivalent lin-ear stiffness coefficient K+ 20CC
+K x+k0
e
Ix(t) +E eq x where. = e - w2x)0 X CW20 g(x) X = 24/14xxAssuming that is zero, the mean square response of the ystem
to an irregular sea with a particular spectrum is found by Eq. (81)
stX 30C X m. xx Cc2, =
SS
Di(
0[H(w)]2
ex exe
Ox79
2(88)
The spectral density S(w) is given from the assumed sea condition, and the square of the absolute value of the complex-frequency factor
[H(w)]2
is obtained from Eq. (42)(89)
A 2a [ 1 ke = lx(t)(83)
x + egkx) xx xxKO
Ox(84)
M60 mee m00ee
= Ie(t) ( 87 )= -(02,14
+ iwN
+14K
XX 30C xx xx eq.
Introducing Eqs.
(89) and (90) into Eq. (88)
-2ax
= G f
(1(eq)for small variation of Keq from w2 .2 Eq.
(91)
may be expressedoX
a 11I
fl
+ y
-
m2-2
; w
-2
where G cro = spectral energy of the response for e = o
d f
eqJ w2
d K at Keg.
ox
eq.
In the analysis with Eqs.
(88)
through(92)
It wasassumed
thatthe remainder function E equals Zero, which is naturally
not
the case;17- is again a stationary random process just like Ix(t)
and depends
on the value of the .equivalent stiffness 'coefficient.A measure of
itsvalue is its expected mean square
E[E].
The mean square of the remainder function = can be expressed by use of Eq.
(86)
qrfx2]
oweq w2wix2+
exg(x)]e
`-w4,F
fxcg(x)j.21
t
FE=2This will be a minimum for fixing Keg when
d
(E[E21
= 0
dKeq(95)
(90)which results in
E[xg(x)].
Keqw2
ox
E[x]
Inserting Eq.
(96)
into Eq.(92)
results-2 2 E[xg(x)] ax 1 + ycooxe E[x2] -2 co
The probability.density of a random variable Y with zero mean value is
f(y) 1 eY2/202
(1%/577
Where a =. standard deviation.
The expectation value E[xg(x)].in Eq.
(97) is
for the nonlineargystemwhiChyould:require knowing the response of the nonlinear sys-tem.. Fortunately, the term in Eq.
(97) is
to be _multiplied by thespall
parameter e, and.,the expectation value E[xgx] of the linearsys-tem instead of the nonlinear syssys-tem
will
induce errors of the secondorder. Consequently 2 2 ax
y606.
.= 1 +
2-3
ao -*/-T7 ao -= -x2/2ac2) xg(x)e dx(99)
by which the effect of the linearization can be investigated. The
IX. DISCUSSION
The mathematical models presented here have Shortcomings. The most important one is the assumed linear relationship between the
re-storing forces and the displacement of the ship. The effect of the. nonlinearity of the mooring lines. in the surge motion, which is parti-cularly affected by the nonlinearity, was investigated in detail. in Section VIII, and a method was presented for calculating the ratio of the mean square of the nonlinear response and the linear response.
Naturally, the methods of analyzing the response of the moored ship has the limitations that are imposed on the analysis of a free-floating.
vessel, and the direct force-displacement relationship established in Section II limits the method to mooring in a few hundred feet depth.
Unfortunately, no experimental data are available in the
liter-(i4)
ature to Check the analysis in detail.
APaper
describing model tests performed at the University of California presents no detailed information concerning the important Characteristics of the vessel and its moorings, but by selecting a mooring. with about the sane Charac-teristics in surge, one can obtain good agreement between experimentaland calculated values of the response of an 880-ton vessel
(Fig. 7).
The calcUIatiOn of the responses at a.particular'frequency is given in Appendix B.
The design of moorings by using the formulas of this Memorandum can be expedited considerably by graphical representation of the
eX-citing forces andthe impedances; Such a procedure is illustrated in
Appendix B.
, In practically all cases, the surge response of the vessel is the
main contributor to high forces in the mooring lines. This is' caused
by the fact that very United damping is available in this mode of movement.
In principle,,
a
reduction of the surge response is possible by two methods: namely, by increasing the damping or by mismatching the natural frequency in surge with the main range of frequencies of Wave excitations.dif-.1.0 0 1.2 1.0 > 0:8 o 0.6 g
04
0.2 040
3530
25 20 15 10 5 - Natural frequency _ in surge 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Wave frequency (rad/sec)
1111
115 10 9 8
Wove period (sec)
Fig. 7
Experimental and calculated values
ficult. In an incidental case, surge movements have been limited by the introduction of damping devices in the mooring lines.
Since wind and currents, whose effects are not discussed here, im-pose certain requirements on the mooring-line tensions, the applica-bility of the second method is often also limited.
In the previous sections of this Memorandum, a few relatively
simple but realistic cases where the ship was moored in the
longitu-dinal plane of symmetry were considered.
Unfortunately, the equations of motions in six
degrees
of freedom- become very complicated, as coupling among many modes exists, and many
of the hydrodynamic force coefficients cannot be calculated. Complete solutions of the motions are not yet realized.
Weinblum and St. Denis,(1) in their now classical paper, Piasented
a method for calculating the uncoupled motions of an unrestrained Ship
in its six degrees of freedom in regular waves with arbitrary heading. This work has been expanded by Pierson and St. Denisao) for the move-
-went in irregular waves with a directional spectrum.
If the motions of the free-floating ship are considered uncoupled,_ the same ship in a moored condition will have coupled motions due to the mooring lines. For an arbitrary mooring, for example, the line-arized equation of motion in surge becomes
N
N X +K x + Kxyy + K.z + K 6+K
+ K cr. = Aejultxx xx xx xz x0 xt xrpr' ex .
(100)
The equations of motion in the other modes are similar. The
coeffi-cients K' K
etc., depend on the mooring lines and can becalcu-xy
xz'lated in a manner similar to that for the spread-moored Ship in waves head-on (e.g., Eqs.
(6)
through (8)).For a submerged hovering submarine, a solution has been found by
Goodman and Sargent.(7) It appeared that for an essentisil,y
symmetri-cal fore and aft body, the motions in the six degrees are uncoupled, with exception of the roll. The latter is coupled with yaw and sway.
For the moored vessel, the surge and pitch become coupled in addition
=
gw/H
ds
-Appendix A
MOORING-LINE CHARACTERISTICS
With reference to the symbols indicated in Fig.
8,
thehorizon-tal component (H) of the force in the line is constant, and the
ver-tical component (V) is equal to the net weight of the section that is lifted off the bottom.
At an arbitrary point A, at distance s along the cable from the point where the cable touches the bottom
(101)
(102)
(w/H)2
Integration over the length of the cable gives
clE H F -1
16]
Yo 1,7 LsinhR-N/1
+(gw/H)2 1i h = , ;will d H L +(1.)2 w 1 (104)1/1
+ (gw/H)2 From Fig.8
it can be established thatS - n .(105)
(103)
mak
Anchor point
w = weight per unit length of chain in water
S = total length of the cable
= length of chain lifted from the bottom
Fig 8
Schematic of mooring line and force components
Tw/H
thus From Eq. (104) consequently V wh wh and. S - L = - A = w
H
-1 Sw] 1 Tw/Hv
H H(107)
(1o9)(no)
(in)
Equations (108) through (112) establish the relationship between the
dimensionless parameters .1T/H1 H/Wh, V/wh, T/Whl (S - L)/h, and 00.
5- L
T6/H
-
sinh-1 TW/H(108)
41!
-
1 'It will be noted in Eq. (108) that for a given value of (S - 1)/h, Zw
can only be solved by trial and error.
Solutions are presented in (b) of Fig. 1. From these solutions
the force variations: when the holding point is moved over a small
distance in the horizontal or vertical direction according to Eqs. (2)
CALCULATION OF RESPONSES
The calculations of the responses, e.g., Eqs. (42), (43), and
(47),
areconsiderably
simplified by introducing- (HI) w os Fex
s
fex ss KST STM
SS (113)(115)
(116)
The value z and its inverse l/z are functions of the damping
ss ss
term
KS
S and the ratio of the exciting frequency and the naturalfre-quency
in
the particular mode and are presented in Figs. 9 and 10. The excitation term?ex can also be plotted in the complex plane (for example, Fig. 11).
Here f,
, andr
are plotted for aves-ex ex ex
sel of 880 tons with a length of 80 ft, draft of 5 ft, and beam of
33.6
ft, and a block coefficient bk = 1.This vessel has the same displacement, draft, and beam of a ves-(14)
sel studied by Wiegel. The complex values of the excitation
forces are calculated as outlined by Wilson,(4) which is a particular case
SS
W M
(1110CI EIS
ss
ss 2
M
160° 1800
50
140° 120° Phase angle 2 100° 80° 2.0 10 0 1.0 18 16 I 4 1 21.0 080.6 0.402
60°
2.0Fig. 9
Polar plot of the impedance
3.0 4.0 40°
4.0
30
w/wo 20
180°
-160° 140°
Inverse impedance (I/2)
3.0 2.0 1.0 1.6 1.8 2.0 Phase angle
g 10
Polar plot of the inverse impedance 1/2
1.0 0.4 2.0 3.0 -120° 780°
-60°
Imaginary 0.7 0.8 0.3 0.6 Cl no ** J damping Real Imaginary Imaginary 0.95 0.9 0:8
* *IN/(Iy
lyy Phase a le With Wave95
*pg AzzET(T) / (m+mz) 1°)0.80.7
as .4**!.+_ M
g sin kL 0.9 0.3 ui k L 1.0 f t/sec2 Note:lags with wove
* Nomenclature Ref 1 (To be corrected for
shallow water effects)
**Nomenclature Ref 4
Fig. II Complex representation of forcing functions in surge, pitch
and heave as a function of frequency Cu-) for an
880 ton vessel
Real 5: F71 x 0.01ct
0.3 0.4 0.6 -J tJ ID 0:7 ForcingF x. ixOltForcing function in pitch
Te
Fes
f ax
Forcing function in heave
[
_ex Mu tell] function in surge xx[
N99 1.0 Real 1.1 I 0.01 rod/sec2 1.2M = 60,400 slugs"
9,000 slugsJr
mx e 250 x lo6 slugs-ft2 I" 500 x 106 slugs,ft2 er 204,000 slugs = 264,400 slugs zzfor the shallow-water effects.
The characteristics of the vessel are
then
chosen.M
= 69,400 slugs xx Mee = 750 x 106 slugs-ft2 Kee J= 1/12
pgB(2L)3 = 1040 x 106 lb-ft/rad Kee =0.4
zz =0.4
K
= pg(2LB) = 390,000 lb/ft zz 5 = 195 ft h = 200 ftInitial tension in bow and stern 1-1/2 in. die-lock Chain (w = 20 lb/ft under water) T = 90,000 lb. Thus T/Wh = 22.5, and from (b) of Fig. 1,
(S - L)/13. = 0.1. (Note T At; H.)
The linear coefficients are obtained from Fig. 2.
a = 9000 lb/ft .b = 900 lb/ft = 1400 lb/ft = 140 lb/ft a/W '= 450 b/W
=
45 c/W = 70 d/W = 7Ke
= -2(bL + ap) = - 250,000 lb/h,(Eq, 11)- x
Ox = -2(eL + ap) = -340,000 lb/rad (Eq. 26)
She natural frequencies are
wox = 0.51 rad/sec (T0 = 07/0.51 = 12 sec)
xx xx
woe eeImee . 1.18 rad/sec (Toe = 2r1/1.18 = 5.3 sec)w
=v/K /M
= 1.21 rad/sec (T = 2u/1.21 = 5.2 sec)oz zz zz oz
The uncoupled heave can be Obtained
directly
from Figs. 10 and11 and Eq. (47). For example, at w = 0.6
10.961 L 15 deg
ex
1 = 1.30 L - 13deg
-s Fex =zz
=,0.87 A L
2 deg0.96 x 1.3
AAL 15 deg - 13 deg
1.44
For surge from Eq. (42)
where
ex
= 0.3 L 90 deg
Yeex
= 10.5 x
10-3 L -74
deg
=-250,000/69,400
=-3.61
x0
kex
=-340,000/750 x 106
=-0.455 x 10-3
o.6
woe1.18 rad/sec,
=1:18=
0.508
woezee
= 0.76 L 14 deg2
-k70 fex
38.0 x 10-3
L
-74 deg
2-cio_we.
zou =
1.06 L 14 deg
Graphical addition
re-sults in 0.280
L
104 deg
and
woxx
ex
= = o =zz
o.6
1.18
0.51 rad/sec,
2
L
180 deg,
w z o0.318 L 104 deg
ee
=0.51
0.1 1.. 180 deg
ex
ex
k
2-
z
oe
007x
exe
. w2ee
- k
x0
ex
A=
2 zoe
xx
x0
wo zxx wo ze
.e
- k.
x0
k
ex
k
ex
2eee
zFor the pitch 2 -w z ex 0x xx -0 kex fex 2 -w z o xx xe x 2 7-kex woezee
-
2 wo2 zxx woezee 0.106 L 194 degGraphical addition re-sults in 0..12 L
193
deg 2.33 AL -89
deg -0 2 ?ex woxzxx = 1.05 xL 106
deg ex = + 0.136 x 10-3 L 90 degTx
k ex ox xx ex Ox-
2-w2zwz -kk
ox xx oe 00 xe ex Graphical addition results in1.15 x
10-3 L 104 deg-1.15 x
103L 1C4 de
&0.95
x 10-2 A radL
-89
deg- 0.12
L 193 deg1 0.54
A degL
-89
degThe force variation in the Stern chain can be calculated
Tstern(A/0'6)
= Hstern(A10.6) = [-aA + (op + bL)E + 11;51A= [(-21,000
L -89
(leg) + (1200L-89 (leg)
+ (780L
2 deg)1A -k k xeex
=
=
-1.64 x
10-3 0.280 L 104 degA
= 0.12L
193
degwhich, by solving graphically, results in
Tstern(Al0.6) = [20,000 L
89
deglet
eIt will be noticed that force variation is caused mainly by the surge. The results of the responses in heave, surge, and pitch, as to phase angle and amplitude, are in good agreement with the results of
a model test (Fig. 12) in waves with comparable frequency. The
charac-teristics of the cables could not be determined from the paper, but the natural frequency in surge was approximately the same
(wmodelx
= 0.4 - 0.5; wanal = 0.51).
Figure 6 presents also the calculated responses in heave, pitCh, and surge and the tension in the stern cable for a range of frequencies. The results are in good agreement With experimental Values.
Water-surface elevation at
center of gravity
(ft)
Heave (ft)
Pitch (rod)
+8
8 +16 Surge (ft) 0-16
+0.1 -0.1 Assumed A = 5.6ft
raa
WAM11141111111111111111MMIltrWRIIIMIIIIMIII
11111k5IMIIIIIKU11111111111=1111/1111111.1111W"
liblENEWAIMMINEMSIN
Calculated results for
A = 5.6 ft,
u.) = 0.6 rod/sec
Experimental results by
Wiegel (Fig. 13 of Ref. 14)
Time (sec)
r-
L CG Surge CG Wave directionFig. 12
Comparison between calculated and measured responses of a moored ship in uniform waves
Approximated ship dimensions
used in calcul-
ation0
10
20
Heave
REFERENCES
Weiriblum Georg, and Manley St. Denis, "On the Motions of Ships at
Sea, Trans. SHAME, Vol.
58, 1950, pp. 184-231.
Korvin-Kroukovsky, B. V., Theory of Seakeeping, Society of Naval
Architectt and Marine Engineers, New York, 1961..
Korvin-KroUkovsky, B. V., and Winnifred R. Jacobs, "Pitching and
Heaving Motions of a Ship in Regular Waves," Trans. $NAME,
65, 1957, pp. 590-632.
4..
Wilson, Dr. Basil Wel "The Energy Problem in the Mooring of ShipsExposed to Waves," Perm. Int. Assoc. of Nay. Congresses Bull. No. 50,
1959.
Ntiloff., A., "A General Theory of the Oscillations of a Ship on Waves," and "On Stresses Experienced by a Ship in a Seaway," INA, Vol. 40,
1898, pp. 135-212.
Weinblum, Georg, "Progress of Theoretical Investigations of Ship
Motions in a Seaway," Ships and Waves,
1954, pp. 129-159.
Goodman, Theodore R., and Theodore P. Sargent, "Launching of Airborne Missiles Underwater " Part IX, Allied Research Assoc.,
- Inc., Boston, Mass., (ASTIA AD
251
620),1961.
Pierson, Williard J., Gerhard Neumann, and Richard W. James, Observing and Forecasting Ocean Waves, Hydrographic Office Pub.
603,
Reptinted1960.
Wiegel, R. L., and J. W. Johnson, "Elements of Wave Theory," Proc. of the First Conf. on Coastal Engineering, Council on Wave Research,
Berkeley, California,
1951.
St. Denis, Manley, and Willard J. Pierson, Jr., "On the Motions of
Ships in Confused Seas," Trans. SNAME, Vol.
61, 1953, pp. 280-357.
U. Longuet-Higgins, M. S., "On the Statistical Distribution of the
Heights of Sea Waves," j. Mar. Res., Vol. 11,
1952, pp. 245-266.
Bendat, Julius S., Principles and Applications of Random Noise
Theory, John Wiley and Sons, New York,
1958, pp. 130-133.
Crandall, Stephen, Random Vibrations of Systems with Non-Linear Restoring Forces, Massachusetts Institute of Technology, AFOSR
708
(ASTIA No. AD259693),
June1961.
Wiegel, R. L., "Model Studies of the Dynamics of an 1311 Moored in Waves," Proc. of Sixth Conf. on Coastal Engineering, Council on