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OCR for page 41
Numerical Solutions for Larg  Amplitude
Ship Motions In the Time Domain
W.M. Lin (Science Applications International Corporation, USA)
D. Yue (Massachusetts Institute of Technology, USA)
ABSTRACT
A threedimensional time domain approach is used
to study the largeamplitude motions and loads
of a ship in a seaway. In this approach, the ex
act body boundary condition is satisfied on the
instantaneous wetted surface of the moving body
while the freesurface boundary conditions are lin
earized. The problem is solved using a transient
freesurface Green function source distribution on
the submerged hull.
Extensive results are presented which validate and
demonstrate the efficacy of the method. These re
sults include linear and larg~amplitude motion co
efficients and diffraction forces with and without
forward speed, calmwater resistance and added
resistance with waves and motions, the large
amplitude motion history of a ship advancing in an
irregular seaway, as well as load distributions on
the changing submerged hull. Most of the large
amplitude results we obtained are new and illus
trate the importance of nonlinear effects associated
with the changing wetted hull. Of special signifi
cance are the dramatic changes of the added mass,
the steady resistance, and sinkage and trim forces
as the motion amplitudes increase.
The present method is a major step forward in
the development of design and prediction tools for
ship motions and loads, and represents a signifi
cant milestone towards a full~nonlinear capability
in the forseeable future.
1 INTRODUCTION
The accurate prediction of waveinduced motions
and hydrodynamic loads is of crucial importance
in ship design. In addition to concerns such as
4
efficiency and comfort, severe motions can limit
operability and affect safety, while extreme loads
may lead to structural failure. Thus the general
problem of a moving body interacting with waves
has been pursued actively since at least the time
of Froude (1868) and Michell (1898~.
Traditionally, the problem is linearized and formu
lated in the frequency domain, by assuming the
motions to be small and time harmonic, and the
resulting boundaryvalue problem is solved using a
singularity distribution on the mean body bound
ary. For zero speed problems, this approach is
quite successful and has become a standard tool
for the design of large offshore structures (e.g., Ko
rsmeyer, et al, 1988~. In the presence of forward
speed, the socalled NeumannKelvin problem is
significantly more difficult due primarily to the
complexity of the corresponding Green function.
Thus, despite several earlier attempts (e.g., Chang,
1977; Inglis & Price, 1981; Guevel & Bougis, 1982),
a truly satisfactory numerical solution is as yet
unavailable. A promising variation due to Gadd
(1976) and Dawson (1977) is the use of Rankine
sources on the body surface as well as a portion
of the free surface on which more general quasi
linearized freesurface conditions can be specified.
Such approaches have been developed actively in
the past 5 or 6 years (e.g., Chang & Dean, 1986;
Xia, 1986; Larsson, 1987; Boppe, et al, 1987;
Jensen, et al, 1988; Letcher, et al, 1989; Bertram,
1990; Nakos & Sclavounos, 1990), with increasingly
encouraging results. In all of these methods, how
ever, the free surface and body geometry remain
fixed in the undisturbed positions, and geometric
nonlinearities are not included.
An alternative to the frequencydomain approach
is to formulate the timedomain initialvalue prob
lem (cf., Finkelstein, 1957; Cummins, 1962~. The
OCR for page 41
requisite timedependent Green function which
satisfies the linearized freesurface boundary condi
tion is simpler than the corresponding ones in the
frequency domain, yet is capable of describing ar
bitrary (largeamplitude) motions when the proper
freesurface memory effects are included. While
linearized and even fully nonlinear timedomain re
sults have been available for problems in two di
mensions (or with vertical axisymmetry) for some
time, developments for threedimensional problems
have been relatively recent. Such work include Ko
rsmeyer (1988) for the linearized radiation problem
without forward speed, and Liapis (1986), Beck &
Liapis (1987), King (1987), King et al. (1988) for
the general linearized problem with constant for
ward speed. For submerged bodies, results for lin
earized free surface but large body motions have
been obtained by Ferrant (1988) and recently by
Beck & Magee (1990~. We remark that for lin
earized (smallamplitude) motions with zero or
constant forward speed, these timedomain solu
tions are formally related to the frequencydomain
results via Fourier transforms.
In this paper, we extend the timedomain approach
to arbitrary largeamplitude motions of a surface
piercing body in a seaway. The exact body bound
ary condition is applied on the instantaneous sub
merged hull surface while a linearized freesurface
condition is used. This approximation can be jus
tified in principle upon the assumptions of small
incident wave slopes and slenderness of the body
geometry in the directions of the (largeamplitude)
motions. The practical utility of this approach
must, in the final analysis, be demonstrated by the
validity and accuracy of its predictions. This is the
focus of much of the present work.
In a boundaryelement approach, the submerged
body surface at each time step is divided into a
number of panels over which linearized transient
freesurface sources are distributed. In contrast
to earlier work, the problem is formulated in a
coordinate system fixed in space. This is clearly
necessary for the case of arbitrary largeamplitude
motions and excursions which is the primary ob
jective of the present code. Under this formula
tion, a general and concise waterline integral term
can be derived to account for arbitrary transla
tions and distortions of the body waterplane, and
the diffraction problem can be included straight
forwardly by adding the incident wave contribu
tion to the body boundary condition. For gen
eral nonlinear calculations, the position and ori
entation of the body is updated (by solving the
equations of motion or as prescribed) and the un
derwater body surface is repanelized at each time
step. Since the body boundary condition is satis
fied on the exact instantaneous hull, the so called
"mterm" effects associated with forward speed
(Ogilvie & Tuck, 1969) are automatically and ez
actly included. For the special case of constant
forward speed and small oscillatory motions (the
linearized seakeeping problem), the traditional lin
earization (and decoupling) of the latter is, how
ever, less explicit in the earthfixed (timedomain)
formulation. For these linear forwardspeed cal
culations, the quadratic terms are included in the
force calculations to account for the forwardspeed
couplings but the mterms are otherwise neglected
in the body boundary conditions.
Linear and largeamplitude computational results
are presented for a floating sphere, two Wigley
hulls, and the Series 60 (CB = 0.7) hull under
going free or captive motions and with or with
out forward speed or incident seas. The program
is applicable for general sixdegreeoffreedom mo
tions (without lift) but we restrict ourselves to
vertical plane motions in head seas in this paper.
For the linear problems without forward speed,
(time) impulse response functions are computed
from which the requisite motion coefficients are
obtained via Fourier transforms. For the nonlin
ear cases and for problems with forward speed,
the bodies are started from rest and computations
typically continued until steady states (limit cy
cles) are achieved. Where available, the results are
compared with other time and frequencydomain
calculations and experimental data. In all cases,
the importance of the nonlinear effects due to ge
ometry variation is identified.
2 MATHEMATICAL FORMULATION
We consider a general
threedimensional body
floating on a tree surface and undergoing arbitrary
sixdegreeoffreedom motion in the presence of in
cident waves. An earthfixed Cartesian coordinate
system is chosen with the xy plane coincident with
the quiescent free surface, and z is positive upward.
The fluid is assumed to be homogeneous, incom
pressible, inviscid and its motion irrotational. Sur
face tension is not included and the water depth is
infinite.
42
OCR for page 41
The fluid motions can be described by a velocity Note that G° = 0, so that
potential
4iT(X,t) = ~(x,t) + 4~(x,t), (1)
where ~! is the incident wave potential, 4} = AT
4~r the total disturbance potential, t is time, and x
is the position vector. In the fluid domain V(t),
satisfies Laplace's equation
V2~ = 0 . (2)
On the mean free surface Butt, we impose the lin
earized condition
~` +g4iz = 0 on 7(t), t ~ 0, (3)
where 9 is the acceleration due to gravity. On the
instantaneous body boundary B(t), we require no
normal flux:
~ = Vn  ~, ~ on B(t), t ~ 0, (4)
where the unit normal vector to the body n is pos
itive out of the fluid and Vn is the instantaneous
body velocity in the normal direction. For finite
time, the conditions at infinity, SOO, are
A, 4~' ~ O onSOO, t > 0, (5)
and the initial conditions at t = 0 are
4} = 4~` = 0 on {(t), t = 0. (6)
We introduce the transient freesurface Green func
tion for a stepfunction source below the free sur
face (see, e.g., Stoker, 1957~:
G(P,t;Q,T)=G°+Gf=~_ :1+
2 / [1  cos(~(tTall ek(2+~) Jo~kR) dk
o
for P 76 Q. t > T. (7)
where P = (a, y, z) and Q = (6, 71, <) are the source
and held points, r = IP Ql, r' = IP Q'l' Q' =
((, 9, a), R2 = (x _ ()2 + (y  71)2, G° = 1/r  1/r'
is the Rankine part of the Green function, Gf =
GG° is the freesurface memory part, and JO
Bessel function of order zero. Eq. (7) satisfies the
following initiallo oundaryvalue problem:
V2G = 0
G`` + gGz = 0
G. G' ~ 0
G = G'
in V(t), t > T
on{(t), t > T
for t > ~
= 0 onT(t), t = ~ .
GO = Gi =
2 / ~sin(~(tT))ek(Z+~) Jo~kR) dk . (8)
o
To obtain a boundary integral formulation for
A, we apply Green's identity to IRAQ, T) and
Gr(P,t;Q,T) in a fluid domain V(T) bounded by
~F(T), B(\T), SOO, and a small surface SP excluding
point P:
///V(~)~.VQG,  G7VQ.) dV
= // ~ AgentG~nQ`) dS, (9)
sco +~(~+~(~)+SP
where V(~T), ~F(T), I3(~T) are respectively V(~),
jF(T), I3(T) with the exclusion of possible point P.
The lefthand side of (9) and the integrals over SOO
and SP on the righthand side are all zero. In
tegrating the resulting equation with respect to T
from 0 to t, we obtain
/° //:~(T)+~3(r)(~?G7nQG~r~nQ~) dS = 0.
(10)
To eliminate the integral over F(\T), it is neces
sary to exchange the time and surface integrals
involving ~F(T). It should be noted that (To iS
timedependent in the earthfixed coordinate sys
tem. Applying the linearized freesurface condition
(3) and the transport theorem we obtain finally
// (.G,nQ GT4in~ ~ dS
9 { Br //3F(T) T T) dS
 / (.G??G7.T\)VN dL} , (11)
where (To iS the instantaneous intersection of
the body with the free surface (excluding possi
ble point P), N the unit normal to P(T) on the
free surface, positive out of the fluid domain, and
VN the normal velocity of [(T) in the N direction.
Combining (10) and (11) and applying the free sur
face condition on G and the initial conditions on
4i, we obtain
//() ~ S + /0 dT //~(T)(.G'nQG,tnQ ~ dS
43
OCR for page 41
+  do (.G,,  G?~^,)VN dL = 0. (12)
The integral over the free surface /(T) can now be
eliminated by applying Green's identity again to
4~(Q,t) and G° in F(t) and combining the result
with (12). Finally, we have for P on B(t):
2~r4~(P,t)+/t (EGG4inQG ) dS
/o {//~`T'(qiG7nQSnags) dS
+ 9 /rt ,(.G774~'G])VN dL}. (13)
The overhead bars on the integration surfaces are
dropped since the integration over SO on these sur
faces gives no contribution. The waterline memory
integral over [(T) in (13) is the general form for ar
bitrary large motions in the earthfixed coordinate
system. For a submerged body, this term vanishes.
For the special case of horizontal planar motions
only, this integral reduces to a similar term formu
lated in a bodyfixed coordinate system given by
Liapis (1986~.
For largeamplitude problems, the evaluation of
the tangential velocities on the body is of critical
importance, and, in the absence of lift, a source
formulation is preferred. To obtain the equation
for the source distribution, we formulate the inte
rior problem governed by an equation similar to
(13), which upon combining with (13) yields
'T {/Jrs(~( /°
[// ~ S 9 /r(T) aGf VNVndL] } (14)
where a(Q,T) is the source strength at point Q
at time T. and VN and Vn are related by VN =
V /N
Finally, we apply the body boundary condition for
P on B(t) to obtain:
= Vn(P, t)VAMP, t) · no
47r {//~(t) aGnp dS + / do
+ [// dS9 /r~ ~aGnp`VNVn dL] ~
(15)
Eq.~15) can be solved for the unknown UP, t)
given B(t), Vntt), Arty, and a(P,~) and 8(T) for
O < T < t. Once the source strength is found, ~ is
evaluated by (14), and the velocity on the body,
Vat, is obtained using a vector form of (15~.
The total pressure is given by Bernoulli's equation,
P = Pi {~! + 2~V.T~2 + go), (16)
and the force on the body is obtained by inte
grating (16) over the instantaneous submerged hull
B(t).
The formulation for the general arbitrary motions
problem is thus complete. For linear (small mo
tions) problems without forward speed, the body
boundary condition is linearized by applying (15)
on the mean body position BO (the waterline term
is absent) and the quadratic term is neglected in
integrating (16) on BO for the forces. For the lin
earized problem with forward speed (the linear sea
keeping problem) in the present earthfixed coor
dinate system, the quadratic contributions are in
cluded in the pressure integration (Eqs. 16 and
19 ~ to account for forwardspeed effects, but the
body boundary condition (15) on BO is otherwise
not further expanded. The formulation is thus nei
ther strictly linearized (in terms of the motion am
plitudes) nor consistent (in that the associated m
terms are not included). This is a consequence of
the fact that a formal decomposition of the prob
lem in terms of the steady forwardspeed distur
bance and the (small) oscillatory disturbance.is
no longer straightforward in the earthfixed time
domain system. (In this system, the forward speed
is in fact a largeamplitude motion.) A more con
venient formulation for the linearized constant for
ward speed problem is, of course, to adopt a coor
dinate system translating with the ship (c/., Liapis
& Beck, 1985~.
3 NUMERICAL METHOD
3.1 Solution of the Integral Equation
A panel method is used for the solution of the in
tegral equation (15~. The body surface B(t) is di
vided into No) quadrilateral elements over which
the source strength is assumed constant. Non
planar quadrilaterals are mapped to planar ele
ments by fitting to the corner points in a least
squares sense (Hess & Smith, 1964). Similarly, the
44
OCR for page 41
body waterline I`(t) is divided into NW(t) straight
line segments on which the source strength c,* is
assumed to be the same as that of the adjacent
body panel. The equation is satisfied at collocation
points corresponding to null points of the panels. A
forward Euler scheme with constant time step, At,
is used to integrate forward in time and the con
volution integral is computed using a trapezoidal
rule.
The discretized form of (15) is given by:
NM
~ aM 1l Go p (Pi, Q j; O) d S
i=l s
=Or [V(Pi, t)V.I(Pi, t)] · np
M1 _N,m
At ~ Em ~ aJ  G/pt(Pi, pi; tT) dS
m=0 i=1 S
Nw
~ aim  G/pt(Pi, Qk; tT)VNkVnkdL
k=1 r.
for i = 1,2,...,NM. (17)
In the above, M and m are the indices for t and
T respectively where t = MAt and T = mat; i,
j are respectively the panel indices for collocation
points P and field points Q (k is the index for the
waterline field points); and so = 1/2 and em = 1
for m > 0. Note that the convolution summation
ends at m = M  1 due to a property of Gf.
Eq. (17) is in the form of a system of linear equa
tions:
NM
~ AijaM = Bi i = 1,2,...,NM, (18)
j=1
which can be solved for the unknown panel source
strengths, aM, by standard means.
3.2 Evaluation of the Free Surface Tran
sient Green Function
The integrals involving the Rankine part of the
Green function G° are evaluated using a method
similar to Hess & Smith (1964~. Special efforts
have been made to completely vectorize these eval
uations, resulting typically in several factors of sav
ings on vector processors.
The numerically more time consuming task is the
evaluation of the memory term Gf and its deriva
tives which, because of the convolution integrals,
must be evaluated a large number of times. There
has been much effort in recent years to develop
efficient and accurate numerical methods for cal
culating Gf and its derivatives, including Newman
(1985, 1990), Liapis & Beck (1985), and Magee &
Beck (1989~. The present approach is an improve
ment upon the method given in Newman (1985~.
The domain for Gf is divided into a number of
regions wherein, depending on the arguments, as
cending series, asymptotic series or a combination
of these and twodimensional economized (Cheby
shev) polynomial approximations are utilized. The
final results are maintained to a minimal accuracy
of 6D for Gf and 5D for its first and second deriva
tives. The entire procedure is completely vector
ized and the average computing time required for
the evaluation of Gf and its two first and two sec
ond derivatives is 0~13) microseconds on a Cray
Y/MP depending on the relative frequency of eval
uations in the different regions. Details can be
found in Lin & Yue (1990~.
The numerical integration of Gf and its deriva
tives over the quadrilateral panels is performed
using twodimensional GaussLegendre quadrature
after mapping the general quadrilaterals into unit
squares. A similar onedimensional quadrature is
used for the waterline elements.
3.3 Implementation & Force Evaluation
The present method is general for arbitrary large
amplitude sixdegreeoffreedom motions which in
general requires the modelling of a changing un
derwater body geometry. Depending on the type
and amplitude of the motions, different numerical
treatments are required:
1. Linear motions with zero or constant forward
speed In this case, the underwater geome
try does not change with time and the matrix
Aij in (18), which represents the Rankine in
fluences and depends only on the relative dis
tances between Pi and Qj, is constant. Thus,
A'j needs to be evaluated and inverted only
once at the beginning of the computations.
Furthermore, since Gf and its derivatives de
pend only on Pi  Qj~ and t  T. for zero or
constant horizontal velocity, the previous val
ues of Gf can be reused at each time step.
4s
OCR for page 41
Thus only one new set of evaluations for Gf
and its derivatives is necessary at a new time
step, and the rest can be obtained from pre
vious calculations. The main differences be
tween zero and constant forward speed prob
lems are the absence of the waterline contri
bution in the former, and the need to include
quadratic terms in the force evaluation to ac
count for forwardspeed crosscoupling contri
butions in the latter.
" Linear motions with arbitrary horizontal ex
cursions The body moves with variable
speed on a straight course or perform arbi
trary excursions in the horizontal plane. In
this case, the underwater body geometry still
does not change in time and Aij needs to
be evaluated and inverted only once. How
ever, the fixed relationships of PiQj~ and
tT between time steps no longer exist and
evaluations of Gf and its derivatives have to
be done for different T,S at every time step.
This increases the computational burden sig
nificantly.
" Arbitrary largeamplitude motionsIn this
case, the underwater body geometry changes
with time, and both Aij and all values of Gf
and its derivatives must be reevaluated at each
time step.
For largeamplitude motions, a robust geometry
processing capability is essential and an automatic
repanelizer was developed for this task. The orig
inal input geometry must now include the above
meanwaterline portion. At each time step, the un
derwater geometry is represented by a number of
vertical strips with each strip having a fixed num
ber of panels along its girth. As the body moves,
its new location and orientation is updated in the
global coordinate system and the new waterline is
found from the intersection with the mean free sur
face. The underwater portion of each strip is then
repanelized using a spline curve fitting. For sim
plicity, the number of panels in each strip is kept
the same but any particular vertical strip may be
eliminated completely if it is out of the water.
The force evaluation follows from integration of
(16) over the instantaneous or mean submerged
hull for the nonlinear and linearized problems re
spectively. The quadratic terms are included in the
nonlinear and forwardspeed calculations and are
evaluated directly given the normal and tangential
velocities on the body. The calculation of It
is separated into two parts: B.~/~1t is given from
the incident waves, and B~/0t is evaluated by
/ By! Hi m Am _ Am~
~ fit J i At Vim · V.m , ( 19)
where Vim, the 'grid' velocity at point P., is equal
to ~ Am _ pSm~! )/At for the nonlinear problem, and
is simply the forward speed, U. in the linearized
forwardspeed problem. We remark that because
of repanelization for the nonlinear problems, the
control points Pi in general do not represent the
same global points or the same body points. When
the number of panels changes between time steps,
(19) can not be used in a straightforward manner
and special care must be taken. This is not con
sidered in the present code.
4 RESULTS
Numerical computations were performed for lin
earized radiation and diffraction problems; large
amplitude forced motions and free motions of a
floating body with and without forward speed and
the presence of ambient waves; as well as wave re
sistance and added resistance problems. For sim
plicity we limit ourselves to heave/pitch motions
in head seas although the present code is capable
of general six degreeoffreedom motions (without
lift). A sphere, two Wigley hull forms, and the
Series 60 (CB=0.7) hull were used for this study
The first Wigley form was used by Gerritsma
( 1988) in his seakeeping experiments and has
a beamtolength ratio 2b/L=O.1, and draftto
length ratio H/L=0.0625. This is designated as
the ``WSK', hull hereafter. The half beam y is given
by:
Yb = (1  X)~1  Z)~1 + 0.2X) + Z(1  Z4~1  X)4
where X(2:~/L)2 and Z(z/H)2. The other
Wigley hull is commonly used for wave resistance
studies which we designate the "WRT" hull. This
hull has 2b/L=O.1, and 11/L=0.0625 with half
beam given by y/b=~1 X)~1 Z).
For convenience, unless otherwise noted, all quan
tities in the following are nondimensionalized by
fluid density p, gravitational acceleration 9, and
body length L (or radius a for the sphere). The
panel numbers, N. indicated are always for half
46
OCR for page 41
Hulme (1982)
SAMP N=18
SAMP N=150
. . .
0.0 5.0 10.0 15.0 20.0
told
Fig. 1: Impulse response function for the heave
force on a hemisphere, V _ 2~ra3/3.
Of the (submerged) body. The present compu
tational results are referred to as SAMP (Small
Amplitude Motion Program) when the linear op
tion (fixed underwater geometry) is used, and as
LAMP (kargeAmplitude Motion Program) when
the largeamplitude capability with changing sub
merged geometry is employed. Where available,
numerical calculations from the linearized time
domain method developed at the University of
Michigan (Magee & Beck, 1988) are included and
denoted as "Michigan". Strip theory results are
based on Salvesen, Tuck & Faltinsen (1970~. Those
used in Secs. 4.1 and 4.2 are taken from Magee &
Beck (1988~.
4.1 Linear Radiation Problem
Fig. (1) shows the impulseresponse function, L(t),
for the heave force on a hemisphere, radius a, as
a function of time, obtained using SAMP with
At=0.05. Here L(~) is defined as
~ ~ P.//l30 (20)
for an impulse (delta function) acceleration at
t=0. The curve for Hulme (1982) is obtained from
Fourier transform of his analytic frequencydomain
results. The timedomain results show the charac
teristic oscillations at larger times which are caused
by the presence of irregular frequencies of the in
terior problem (Adachi & Ohmatsu, 1982~. Note
that the amplitudes of these spurious oscillations
decrease as the number of panels is increased.
0 r I
D ~
~ o
jo
Do °


~.
o
A
1
Hulme (1982)
· · · SAMP N=18
SAMP N=150
\~_ ~ ___
(~ 1
it,, 1
Tier a/g 
1
A33/PV
0.0 2.0 4.0 6.0 8.0
w2a/9
Fig. 2: Heave addedmass and damping coefficients
for a hemisphere.
The added mass and damping coefficients are re
lated to the cosine and sine transforms respectively
of the impulseresponse function. These are shown
in Fig. (2) as compared to the frequencydomain
result of Hulme (1982~. For illustration the loca
tion of the lowest irregular frequency (w,2r~a/g
2.56, Hulme, 1983) is also indicated. It is seen
that the oscillations in Fig. (1) are directed related
to the irregular behavior near Micra although the
singular nature is not fully captured because of the
finitetime truncation of L(t) (at t=20 in this case).
We note again the smaller and more confined os
cillations for larger N.
For radiation problems with forward speed, a
forced oscillatory motion is superimposed on a
stepfunction jump of the forward velocity to the
prescribed value. The force coefficients are then
obtained from Fourier transforms of the SAMP
force time histories after steady state is reached.
In the presence of forward speed, quadratic con
tributions must be retained in the present space
fixed formulation and the impulseresponse func
tion is not generally useful. This computational
inefficiency for the special case of linearized sea
keeping is the main (in fact only) disadvantage of
the spacedfixed formulation.
Figs. (3,4) show the heave and pitch diagonal
and offdiagonal addedmass and damping coeffi
cients of the WSK hull at Froude number En 
U/~/~=0.2. The SAMP calculations use N=120
and At=0.1. These results are considered to have
converged in that selected calculations (not shown)
47
OCR for page 41
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using N=200 and At=0.05 show changes of less
than 1% in the force coefficients. As pointed out
earlier, the SAMP calculations do not contain the
contributions of the mterms while the general
LAMP computations do. To gauge the importance
of these effects, we include in the figures the small
amplitude limits of the largeamplitude LAMP re
sults (see Sec.4.5~. These values are obtained from
extrapolations of the small finiteamplitude LAMP
results to zero amplitude and are denoted hereafter
as LAMPo. From Figs.~18), it is seen that LAMP
approaches the LAMPo limit very smoothly so that
the estimates are reliable.
For the diagonal coefficients (Fig.3), the LAMPo
and SAMP results are both reasonably satisfactory
although the mterm effects appear to play an ap
preciable role as the relative frequency, ToU~/9,
decreases and especially for the pitch coefficients.
On the other hand, the Michigan results (which
include mterms) using the same N show unex
plained deviations for intermediate values of To
Not surprisingly, strip theory performs better for
larger To but appears to be affected significantly
only for Bss at lower frequencies.
The results are much more interesting for the off
diagonal coefficients. For the addedmass terms
A3s and As3, the SAMP calculations compare well
to measured data. When effects of the mterm are
included, the deviations of LAMPo from SAMP
are primarily at lower frequencies but the com
parison to experiments is otherwise not improved.
For the damping terms B3s and Bs3, the SAMP
results are inadequate, showing a large somewhat
constant shift, and indicate that the mterm effects
may play an important role for these coefficients.
Indeed, when the mterm effects are included, the
LAMPo and Michigan results are both reasonably
close to experimental measurements.
In summary, we see that the SAMP results are
satisfactory for the offdiagonal added mass coef
ficients, and generally acceptable for the diagonal
coefficients. Taken together, the LAMPo predic
tions provide overall good agreement with exper
iments and are clearly superior. By comparing
SAMP and LAMPo results, the effect of the m
terms is quantified and shown to be important for
pitchrelated coefficients at lower frequencies and
especially for the offdiagonal damping terms. In
is noteworthy that since LAMP uses a linearized
freesurface condition, the present results suggest
that nonlinear freesurface terms such as those that
may be included in Dawsontype codes (e.g., Nakos
& Sclavounos, 1990) may otherwise not be as im
portant in the prediction of the motion coefficients.
A wellknown theoretical difficulty of the linear
seakeeping problem is the singularity at the critical
relative frequency To =1/4 (e.g., Dagan & Miloh,
1980~. Physically, this may be explained by the
simple fact that at To = 1/4 the group velocity
of the waves generated by the oscillatory motion
matches the ship speed U and resonance occurs. In
the frequency domain, the linearized result is un
bounded while in the timedomain the resonance
is manifest as a slow unbounded growth of the so
lution with time.
While actual predictions at the critical frequency
may not be contemplated in the context of lin
earized theory, it is useful to be able to avoid
the undesirable growing solution in geneml time
dependent simulations. An obvious idea (also sug
gested by Beck & Magee, 1990) is to simply trun
cate the duration of the memory effect convolution
terms in (15) by requiring t  T < TCU`, say. This is
illustrated in Fig. (5) which shows the time history
of the vertical force on a WSK hull starting from
rest with a heaving frequency of w=1.25 and a for
ward speed corresponding to To = 1/4 (Fn=0~2~.
When the memory effect is not limited, the force
grows slowly without reaching steadystate. When
we set TCUI=1O' however, steadystate is quickly
reached resembling the results at noncritical To,S.
We point out that because of the relatively slow
growth and the effective truncation of TCU' by the
total duration of the simulation, the problem of
the critical frequency may not pose a severe limi
tation in practice for general (not critically forced)
timedomain applications. The motion coefficients
obtained from SAMP calculations using different
values of TCU! are presented as separate symbols
on the To = 1/4 vertical in Figs. (3,4~. We cau
tion that these results should not be considered as
'predictions' at the critical frequency despite the
fact that they are now finite and appear to give a
roughly continuous trend.
4.2 Linear Diffraction Problem
The presence of incident waves, ~I(x,t), can be
included into the governing equation (15) in a
straightforward manner. In practice, this is intro
duced instantaneously at t=0 and the diffraction
ejects evaluated after the transients vanish typi
so
OCR for page 41
o
o
o
to 
o
0
to
to
o
 SAMP IV= 120
unlimited memory effect
memory effect truncated for tT ~ TO
0.0 20.0 40.0 60.0
t~l~
Fig. 5: Peaks of the leave force history for a lose
hull at Fn=0.2, TO=1/4.
cally in a very short time. In the case of no forward
speed, the linear diffraction problem is validated
for a floating hemisphere. The SAMP results are
indistinguishable from the earlier computations of
Cohen (1986) and King (1987) and the compar
isons are not shown here.
Figs. (6) shows the amplitude and phase of the
wave exciting forces on a WSK hull moving with
forward speed, Fn=0.2, in head seas. For the
SAMP results, N=120 and 30 time steps per inci
dent wave period is used. The comparisons to the
timedomain results of Michigan, strip theory and
experiments are overall satisfactory. The Michi
gan calculations underpredict the secondary hump
at high frequency although both of the 3D time
domain calculations show better correlations to the
experiments than strip theory.
For a more realistic and complicated geometry, we
consider the Series 60 (CB=0.7) hull form for which
extensive experimental data are available. Figs. (7)
show a complete set of results for Fn=0.2 in head
seas. The SAMP results with N=180 and 250 have
clearly converged (30 time steps per incident wave
period are found to be more than adequate) and
compare very well with experimental data. Not
surprisingly, strip theory results are generally ac
ceptable compared to SAMP predictions at higher
frequencies. On the other hand, the Michigan re
sults do not indicate convergence and appear to un
derpredict the amplitudes especially at the higher
frequency second hump.
4.3 LargeAmplitude Motions at Zero
Speed
As a first example, we study the largeamplitude
heaving of a (complete) sphere. The sphere is ini
tially semisubmerged with its center at the ori
gin and a forced heaving motion with zest) =
Ahsin~t is imposed at t=0. We set w=1 and
use lV=150 and wAt = 2,r/30. As pointed out
earlier, at each time step, the submerged portion
of the sphere is repanelized, the Rankine influence
matrix reevaluated and solved, and the memory
terms recalculated from T=0 to ~ = t.
Fig. (8) shows the different components of the ver
tical force on the sphere as a function of time for
the case of Ah/a=0.5. Steadystate (limit cycle) is
rapidly reached (within one period) for all the com
ponents. For this geometry, the hydrostatic force
is a large part of the total. The inertia (~/0t)
term shows distinct higherharmonic components
while the quadratic (~V.~2/2) component is pri
marily at the second harmonic, as expected. Sur
prisingly, these higher harmonic contributions ap
pear to cancel closely so that the total force oscil
lates primarily at forcing frequency together with
a negative (suction) steady component.
To validate the largeamplitude capability of
LAMP, systematic convergence tests were per
formed for this case. A typical example is
Fig. (9) which shows the hydrodynamic force his
tory (FHD) for varying N and w/`t for Ah/a=0.5.
The results are indistinguishable indicating that
IV=150 and w/`t = 2,r/30 are more than adequate.
Fig. (10) shows the time history of the vertical hy
drodynamic forces on the sphere for different Ah/a.
As expected, the nonlinear curves approach the
linear one as Ah/a decreases. For larger heaving
amplitudes, the peaks become sharper and higher,
while the troughs become shallower. The nonlin
ear effect, however, is relatively small even when
the heaving amplitude is 50% of the radius. This
was found also in earlier full~nonlinear simulations
for heaving axisymmetric bodies (Lin, et al, 1984;
Dommermuth & Yue, 1986~.
A detailed analysis of the frequency components of
the force can be obtained from Fourier transform
of the steadystate (limitcycle) time history. For
convenience, we define the frequency components
of the vertical force:
F (`t) = OR (Lfo + feint+ f2ei2W'+
51
OCR for page 41
o
en ~
to
*
~ o ly
~ m
1
lo
11 ~ .
1 .
to .
to  .
1
o
o
1
SAMP N=168
SLAW (Letcher et al, 1989)
xs~ Experiment (ITTC, 1985)
0.0 t.0 2.0 3.0 to 5.0
t/To
Fig. 12: Horizontal force time history for a WRT
hull moving at Fn=0~3.
to
o
o
1
cut
to
*
(a) 0 \
~ 1
 1 ~1
[q up
_i
11 1
~4
~ 0
C`2
SAMP N=168
SLAW (Letcher et al, 1989)
i 1
0.0 1.0 2.0 3.0 4.0 5.0
t/To
Fig. 13: Vertical force time history for a WRT hull
moving at Fn=0~3.
to
*
,~ 0
~ ~
~ 1
:a
~ 0
I'q C`2 I
lo
11 o_
1
0 l
4,
SAMP N=168
SLAW (Letcher at al 10~\
. . . . .
0.0 1.0 2.0 3.0 4.0 5.0
t/To
Fig. 14: Pitch moment time history for a WRT hull
moving at Fn=0~3.
the pitch moment is very good while the two re
sults differ by about 15% for the steady vertical
force.
To evaluate the dependence of SAMP predictions
on panel size and time steps, systematic conver
gence tests were performed for different numbers
of panels and time steps for Fn=0~3 and 0.4. These
results are summarized in Tables 2 and 3. Three
different grid distributionsuniform, cosine and
'geometric' spacingare considered in the lon
gitudinal direction while the girthwise grid sizes
are kept constant. The socalled 'geometric' spac
ing is based on the criterion that the projection of
each panel on the yz plane be constant. For the
parabolic WRT hull, this corresponds to a square
root grid distribution in the longitudinal direction.
From the tables, some indication of convergence is
observed, although a definite trend is still difficult
to obtain. This suggests that possibly larger num
bers of panels may be required for more accurate
predictions at these Froude numbers. Comparing
the three grid distributions, the geometric grids
seem to give the best and most consistent results.
For the calculations in this and later sections, a
geometric grid with N=168 (28x6) and ~t=2'r/40
was used.
To obtain a complete resistance curve, the calcu
lation of Fig. (12) was repeated for Froude num
bers ranging from 0.2 to 0.45. The final results
are shown in Fig. (15) together with the ITTC ex
perimental data. The overall comparison is quite
satisfactory.
Finally, we evaluate the possible effect of the start
up velocity profile on the steadystate force predic
tions. For illustration, three velocity profiles were
considered with V(t)=0 for t < 0, V(t) = U for
t > To, and for O < t < To: (i) impulsive start,
V(t) = U; (ii) ramp start, V(t) = Ut/T~; and (iii)
cosinefunction start, V(t) = Uf1cos(,rt/2T')~/2.
Fig. (16) shows the horizontal force time histories
for these three cases with T~/To=0.6. The initial
transients and the phases of the later oscillations
differ appreciably between (i) and (ii),tiii) but the
same steadystate value is reached. As expected,
the smoother ramp and cosinefunction profiles
produce smaller transient and To oscillations (asso
ciated with milder startup disturbances) and are
preferred for practical resistance predictions using
SAMP.
56
OCR for page 41
Table 2: Wave resistance coefflaents Cat x 103 of the WRT hull at Fn=0~3
an a function of grid size anu distribution.
 At/2,r 1 uniform grid I cosine grid I geometric grid ~ 
1 1 20x6 1 28x6 1 36x6 1l 20x6 1 28x6 1 36x6 11 20x6 1 28x6 ~ 36x6
. l .. ,
1/20 2.53 2.84 1.70 l 2.82 1.99 1.48 1.20 1.48 2.50
1/30 2.50 1.37 1.58 l 1.82 1.87 0.94 1.57 1.62 1.60
1/40 1.68 2.79 1.26 1.22 1.84 1.78 . 1.29 1.64 1.75
Table 3: Wave resistance coefficients Cc x 103 of the WRT hull at Fn=0~4
as a function of grid size and distribution.
 l~tj2,r 1 uniform grid I cosine grid I geomeh
36x6
2.07
1.94
2.09
lo
._
*
20x6 1 28x6 1 36x6 1l 20x6 128x6 1 36x6 1l 20x6 1 28x6
1/20 1.32 1.56 1.53 ~ 1.43 1.62 1.71 1.42 1.68
lt30 0.33 2.50 1.96 ~ 1.68 1.98 2.12 2.24 2.07
1/40 4.22 1.24 2.24 1.61 2.33 2.10 2.05 2.15
0 0
cr) . ~O
· SAMP N=168
Experiment(ITTC, 1985)
lo
N
0.15 0.20 0.25 0.30 0.35 0.40 0.45 050
Fn
Fig. 15: Wave resistance coefficient for the WRT
hull as a function of Froude number.
4.5 LargeAmplitude Motions with For
ward Speed
A main objective of this work is to evaluate and
quantify the importance of largeamplitude mo
tions on seakeeping characteristics. While the
change of such relevant quantities as added mass
(and hence natural frequencies and response ampli
Impulsivestart
Ramp start
Cosine function start.
rem I I r  l
00 10 20 3.0 40 5.0
t /Lo
Fig. 16: Horizontal force time history for a TORT
hull moving at Fn=0~3 using different startup ve
locity profiles (T1 /To=0.6 ).
tudes) due to largeamplitude motions can be ex
pected on physical grounds, and has been reported
in experiments (e.g., O'Dea & Troesch, 1987), the
precise magnitudes or even dependencies are as yet
not well known.
To address this problem, the LAMP program
was applied to the largeamplitude forced heav
57
OCR for page 41
A
°1
.
A
A l
*
t:, A
1
to
o
1
00 0.1 02 0.3 04 ()5
Ah/0
Figs. 17: Steady surge, heave and pitch forces and
moment on a heaving (frequency w=1.0, amplitude
Ah) WRT hull moving with constant forward speed
(Fn=0.2~. .: LAMP (N= 160~;    : 'calmwater'
(Ah=O) value obtained with SAMP.
ing of a WRT hull moving with constant forward
speed. Specifically we chose a Froude number
Fn=0.2, a heaving frequency w=1.0, and consid
ered a range of heaving amplitudes correspond
ing to Ah/B ~7.5  45% where H is the draft.
Figs.~17) show the added resistance, added sinkage
and added trim forces as a function of the heave
amplitude. For contrast, the 'calmwater' values
(cf. Sec.4.4) are also indicated. For small Ah/H,
the results are close to the calmwater values as
expected and the approach to this limit is very
smooth. As the amplitude increases, however, the
A
c ~
.
co
.
o
.
D
~o
~ ,
D

· ~"added mass"
.
0 0 0
"damping"
added m ~ ~,
.
0.0 0.1 0.2 0.3 0 4 0
Ah/0
Fig. 18: Excitation frequency components of the
limitcycle forces and moment on a heaving ~
w=1.0, amplitude Ah ~ WRT hull moving with con
stant forward speed (Fn=0.2~. ~ in phase with
acceleration ~ "addedmass" ); O outofphase with
acceleration ~ "damping" ~ coefficients using LAMP
(N=160). Also indicated are I, \7 : small
amplitude limits (LAMPo); and ·, As: SAMP
added mass and damping coefficients.
added resistance increases rapidly and is as much
as 0~5) or more times the calmwater value for the
larger amplitudes considered. The picture is sim
ilar for the mean vertical force and moments but
somewhat less dramatic.
By analysing the limitcycle force and moment his
tories for the frequency components at the forc
ing frequency that are in phase and out of phase
respectively with the acceleration, the socalled
largeamplitude "addedmass" and "damping" co
efficients with forward speed can again be identi
fied (cf. Sec.4.3~. These diagonal and offdiagonal
s8
OCR for page 41
seakeeping coefficients are shown in Figs.~18) as
a function of Ah/H. For small amplitudes, the
approaches to the linear (zero amplitude) limits
are very smooth allowing accurate extrapolation
for the linearized LAMPo estimates. As noted
in Figs.~3,4), this smallamplitude limit of LAMP
differs from the linearized SAMP prediction (also A,
shown in Figs.18) because of the absence of the
effect of the mterms in the latter.
As in the case for the heaving sphere with no for
ward speed (Figs.ll), the largeamplitude added
mass shows a clear dependence on the amplitude
and decreases appreciably as Ah increases. As an
illustration, when the heave amplitude is ~40% of
the draft, A33 has decreased to about 60~o of the
smallamplitude value. For the present WRT hull,
the waterplane area changes very little with heave
amplitude and consequently the relevant heave
natural frequency would increase by ~20% com
pared to the linear value in this case. This may
explain some of the "unexpected" dependencies of
the response amplitude functions on forcing ampli
tude observed in experiments. As with the sphere
with no forward speed, the LAMP damping coef
ficients here are less sensitive to heave amplitude
probably due to the fact that nonlinear radiation
mechanisms are absent in the present approxima
tion.
4.6 Motions of a Floating Body and Un
steady Loads
With the ability to determine the relevant hydro
dynamic forces at any given instant, the complete
six degreeoffreedom motion history of the body
can be obtained by a direct integration of the dy
namical equations. Thus LAMP can be employed
to study general timedependent motions such as
in the case of a ship advancing in an irregular sea
way. Indeed, one of the important applications is
the study of episodic events involving large loads
and motions in the time domain.
To validate the motions solver, we consider first
the decaying motions of a freelyfloating sphere re
leased from rest with a given initial vertical dis
placement. As a test, we integrate the problem di
rectly in time, although for the linearized problem
the solution can be obtained by a Laplace trans
form involving the force impulseresponse function
of Fig. (1) (Liapis, 1986~. Consistent with the open
time quadrature formula of (17), we employ an ex
59
SAMP
LAMP Ah/a=0.50
0.0 8.0 16.0 24.0 32 0
told
Fig. 19: Vertical displacement history of a float
ing hemisphere released from rest with an initial
height Ah.
plicit multistep scheme for the motion time inte
gration.
First, SAMP computations were performed and
compared to the linearized calculations and small
amplitude experiments of Liapis (1986~. The re
sults are indistinguishable and very close to the
measurements and are not shown. Fig. ( 19)
shows the SAMP and LAMP (for initial ampli
tude Ah/a=0.5) results with N=150 and At=0.1.
The difference in the (normalized) amplitudes is
small but it is of interest to note that the LAMP
oscillation periods are longer than the linearized
values by about 5%. This is caused by the com
peting effects of addedmass decrease due to large
amplitude motions (cf. Fig. (11~) which tends to
decrease the period, and the reduction in the av
erage waterplane area (and hence the hydrostatic
force) which tends to increase the period of oscil
lation. In this case, the latter effect is greater and
consequently an increase in natural period is ob
served. This again underscores the importance of
nonlinear geometry effects.
An interesting application of the motion program
is to reexamine the calmwater resistance problem
of Sec.4.4 but this time allowing the body to freely
sink and trim until steadystate values are reached.
Starting from rest, the ship is held to a constant
forward speed but is otherwise free to heave and
pitch. In the largeamplitude simulation, the un
derwater geometry is allowed to change as steady
OCR for page 41
o
,(
o
lo
o GO
*
o
C\2
lo
lo 
Pree Body:
· LAMP N = 196
:~ Experiments
(ITTC, 1985; Bai, 1979,
Fixed Body:
O SAMP N=168
Experiments
(ITTC, 1985)
, . . . . . .
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Fn
Fig. 20: Wave resistance coefficient for the WRT
hull as a function of Froude number.

_`
~ °
+ f
_'
CO
lo. .~
C~2
lo
lo
~ S;nkage · LAMP JV=196
;~ Experiment
(ITTC, 1985)
4~1
trim
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Fr'
Fig. 21: Sinkage (a) and trim (b) of the WRT hull
as a function of Froude number. [Adraft at AP,
[Fdraft at FP. Positive trim means bow up.
state is attained by an equilibrium between hydro
static forces of the final geometry and the steady
vertical force and pitch moment.
Fig. (20) shows the free sinkage and trim wave
resistance of the WRT hull as compared to cor
responding experimental measurements as well as
the fixed body resistance results of Fig. (15~. The
present LAMP calculations use At=21r/40 and
N196 (28x 7) where an extra horizontal row of
panels is added to the grid of Fig. (15) to model
the extra draft due to linkage. From the figure,
the increase in wave resistance due to sinkage and
trim is clearly predicted. Comparisons between the
LAMP results and experiments are excellent.
The comparison for sinkage and trim is shown in
Figs. (21~. The LAMP predictions are again sat
isfactory both in the magnitude and the forward
speed dependence. The results are somewhat bet
ter for the sinkage and show a slight underpredic
tion of the trim at higher speeds. Since sinkage
is the main reason for the resistance increase, the
underprediction of trim at high En does not affect
the resistance prediction significantly.
Finally, we applied LAMP to study free vertical
plane motions of a ship advancing in irregular head
seas. A time record of the free surface was con
structed using a twoparameter PiersonMoskowitz
spectrum with a significant wave height H~/3=10
feet and a modal period Tm=12 seconds with low
and high frequency cutoffs at 0.1 and 4.25 rad/sec
respectively.
A 400 foot long WRT hull with constant forward
speed corresponding to Fn=0.2 was first chosen.
Figure (22) shows the time history of the inci
dent wave elevation at the ship center of gravity,
along with the pitch and heave displacements from
LAMP calculations (N=160, At=0.3526 seconds).
For contrast, strip theory predictions obtained by
superposition of the linear frequencydomain re
sponses at the component incident frequencies are
also included. To be consistent with the strip the
ory program used here, the LAMP calculations in
clude the hydrostatic (and FroudeKrylov) forces
only up to the mean waterline. To remove the ef
fects of starting transients, the LAMP simulation
has been started from rest at t=10 seconds.
As can be seen in Fig. (22), the differences between
LAMP and strip theory are small for the pitch dis
placement, with the exception of apparent mem
ory (transient) effects around 10 c t < 60 and
60
OCR for page 41
to
cO
T
o
co
LAMP
Strin theory
Do
0
_
to
1
o
CD
1
0
0
0_
1 \ ~ 1/
I O ~BY'
(J ~
INJI
J ~I\
.' ~l:~` I' ~
1 1~\ " :t ~7 :1: ~
,1
v V V \`,1
V1
o
1
_ 1
30.0 60.0 ~ 90.0
t (see)
Fig. 22: Incident wave elevation (~) at the ship's center of gravity, pitch angle All, and heave displacement
(z) of a WRT hull at Fn=0.2 in irregular head seas.
6
OCR for page 41
lo ~
o
~
1
o
r\'

~o
~ o
~  .
1
o 
ll
600
90.0
Fig. 23: Incident wave elevation (~) at the ship's center of gravity, pitch angle All, and heave displacement
(z) of a Series 60 (CB=0.7O) hull at Fn=0.2 in irregular head seas.
62
OCR for page 41
 ~ t=0.9305 second
v 1 0.2 ~ _ _ 1 4 _    1 V
1 Z~.
t=4.4564 second
vim
Fig. 24: Nondimensional dynamic pressure distribution on a WRT hull at two time instances.
80 < ~ < 90. The same is true for the heave dis
placement except for a nearly constant downward
shift of the nonlinear result. This is caused by the
steady sinkage force associated with forward speed
which is present in LAMP but absent in the strip
theory (from Fig. 21, this is estimated at about
0.53 feet for a 400 foot hull). Overall, the rela
tive good performance of the strip theory is not
unexpected for the mathematical WRT hull (with
relatively small geometry changes near the water
line) and for head seas.
For a more realistic geometry, we show in Fig. (23)
the corresponding case for a 400 foot long Series 60
(CB=O.7) hull at Fn=0~2. In this case, strip the
ory overpredicts the pitch motion significantly, in
some instances by over 50% (e.g., near t=75 see).
For the heave displacement, there are appreciable
underpredictions of the troughs in addition to the
absence of the steady downward linkage. These de
ficiencies of strip theory for the more complicated
Series 60 hull are consistent with the observations
of Frank & Salvesen (1970~.
As a final illustration, we display the pressure
distribution on the hull as output from LAMP.
Fig. (24) shows the dynamic pressure distribution
on the WRT hull at t=0.9305 and 4.4564 seconds
respectively corresponding to the case in Fig. (22~.
Note that the pressures are given on the actual sub
merged surface. The complete unsteady loads on
the ship hull can now be obtained from the pres
sure distribution and motion histories.
5 CONCLUSIONS
A timedomain method, LAMP (kargeAmplitude
Motion Program), was developed for the general
largeamplitude motions of a threedimensional
surfacepiercing body in a seaway. The body
boundary condition is satisfied exactly on the in
stantaneous underwater body surface while a lin
earized freesurface condition is used.
To validate the approach and evaluate its accu
racy, the method was applied extensively to obtain
linearized motion coefficients for a number of dif
ferent geometries with or without forward speed.
The results include addedmass and damping coef
ficients, wave exciting forces and steady wave resis
tance, sinkage and trim forces and moments. These
are compared to experimental measurements and
existing linear time and frequencydomain calcu
lations. The comparisons are overall satisfactory
for all the results and show that LAMP is equal
or superior to any of the existing computational
methods in terms of accuracy.
The main feature and purpose of LAMP, however,
is for general nonlinear largeamplitude motions.
To illustrate its effectiveness, we apply LAMP
to study the largeamplitude forced heaving of a
(complete) floating sphere; the large verticalplane
motions and the free sinkage and trim of a Wigley
hull moving with constant forward speed; and the
general timedependent largeamplitude motions of
a Series60 ship advancing in an irregular seaway.
63
OCR for page 41
Some of the main findings are the importance
of (added) steady (and higherharmonic) compo
nents, and the modifications of the firstharmonic
(excitation frequency) motion coefficients. In the
first case, the presence of largeamplitude heaving
motions is shown to result in significant increases
of the wave resistance and steady sinkage and trim
forces. In the latter, the nonlinear "addedmass" is
found to decrease markedly with increasing heave
amplitude. The consequent reduction of inertia
and increase in natural frequency may have impor
tant implications to the motion dynamics of the
ship. This may also explain some of the exper
imentally observed dependencies on amplitude of
normalized motion response functions. When ap
plied to general timedependent motions in irregu
lar waves, LAMP demonstrates the importance of
transient (memory) and nonlinear geometry effects
especially for realistic ship geometries where strip
theory is found to be inadequate. The road is now
laid for nonlinear simulations of extreme episodic
events and complete load and motion predictions.
The current version of LAMP is fully vectorized
for highspeed vector processors. For a nonlinear
(largeamplitude) simulation using 0~150200) un
knowns on the body and a similar number of time
steps, the typical CPU time on a single Cray YMP
processor is 0~12) hours. Further code optimiza
tion may reduce this requirement by a small factor.
For applications involving significantly larger num
ber of unknowns and time steps, the time domain
formulation may be particularly suited for parallel
algorithms on multiple processors.
Acknowledgement
This research was sponsored by the Office of Naval
Research, the U.S. Coast Guard, and the De
fense Advanced Research Projects Agency. We are
grateful to Cray Research, Inc., for the use of their
Cray YMP/832 supercomputer. Some computa
tions were also performed on the NSF Pittsburgh
Supercomputer Center Cray YMP. We thank M.
Meinhold and K. Weems for valuable technical and
graphical help.
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DISCUSSION
Pierre Ferrant
Sirehna S.A., France
I would like to make a comment on your results about the heaving
sphere. I do not completely agree with your analysis of the
behaviour of the damping coefficient which you show to remain
constant when amplitude is varied. My own results show that at least
for a submerged body, the nonlinear phenomena associated to the
body boundary condition are very frequencysensitive. In fact, when
dealing with oscillatory motions, even in the time domain, one cannot
ignore the importance of frequency and this parameter must be varied
before drawing conclusions. I would therefore be very interested if
you could give result for the surfacepiercing heaving sphere at lower
frequencies, say about wet R/g=0.4.
AUTHORS' REPLY
We have preliminary results for the heaving (surfacepiercing) sphere
for normalized frequencies ranging from w ~ 0.4 to ~ 3 and
amplitudes rangin, from Ah/a ~ O to ~ 0.5 or higher. In contrast
to your results for the submerged sphere, the damping coefficient
remains relatively independent of amplitude for lower frequencies (w
< ~ 1) and shows some sensitivity only for intermediate frequencies.
The precise mechanisms for these dependencies (an amplitude and
frequency) are as yet not completely understood.
The following table lists the results for the frequency ~ = 0.4 you
su~gest~l. For comparison, the data for Fig. (1) at w = 1.0, as well
as w = 1.5 and 3.0 are also included. Again, we note that these
results are only preliminary.
Normalized damping coefficients for a (surfacepiercinz) heaving
~here.
Ah/a
0.125
0.250
0.375
0.500
w=0.4 w= 1.0 w= 1.5 w=3.0
0.036 0.195 0.151 0.314
0.036 0. 195 0.154 0.3 12
0.035 0.195 0.170 0.312
0.035 0.194 0.185 0.316
66