CALCULATION OF THE SECOND ORDER MEAN FORCE
ON A SHIP IN OBLIQUE SEAS
by
PAUL ROSS ERB
S.B., Massachusetts Institute of Technology
(1976)
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREES OF
MASTER OF SCIENCE
IN NAVAL ARCHITECTURE AND MARINE ENGINEERING
AND
MASTER OF SCIENCE
IN MECHANICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1977
Signature of Author .
Departmentsof Ocean Engineering
,May 12, 1977
Certified by .
/
Thesis Supervisor
Department of Ocean Engineering
Certified by
.............
Department Reader
Department of Mechanical Engineering
Accepted by .
......
Chairman,
Departmentl Committee on Graduate Students
r~ives
JUN 22 1977
-2CALCULATION OF THE SECOND ORDER MEAN FORCE
ON A SHIP IN OBLIQUE SEAS
by
PAUL ROSS ERB
Submitted to the Department of Ocean Engineering and the
Department of Mechanical Engineering on May 12, 1977 in
partial fulfillment of the requirements for the Degree of
Master of Science in Naval Architecture and Marine Engineering
and the Degree of Master of Science in Mechanical Engineering.
ABSTRACT
A design tool is developed for use by engineers for
the calculation of the mean added resistance or drift force
on an elongated body such as a ship in a seaway. Only
forces arising from wave-ship motion interaction or wave
reflection are considered and developed in a form suitable
for a computer program. This procedure allows computation
of the mean second order force from first order quantities
already known in strip theory of ship motions. Regular
wave computer results generated by the MIT 5-D motions
program for the Mariner cargo vessel are presented and
compared with experiment, including a set of new beam seas
experiments. Comparison is also made with published results from two other programs. The extension of the
regular wave theory to an irregular long-crested seaway
and then to a short-crested seaway is outlined. Finally,
six representative sea spectra are used in a brief design
analysis for the Mariner at service speed.
Thesis Supervisor:
Title:
Chryssostomos Chryssostomidis
Associate Professor of Naval Architecture
-3ACKNOWLEDGEMENTS
The author wishes to extend his sincere appreciation
for the help and guidance given to him by his advisor,
Professor C. Chryssostomidis, not just on this thesis, but
throughout his undergraduate and graduate years at M.I.T.
This work would have been very difficult to complete
if not for the efforts of several individuals.
The experi-
mental work was done with the help of Ysabel Meija. Professors Owen Oakley and Martin Abkowitz generously offered
advice and opinions at various stages.
Fraternity brothers,
Stephen K.Gourley and David B. S. Smith gave invaluable
assistance by producing many of the final graphs and
figures.
Barbara Belt read portions of the draft and
offered valuable suggestions.
Special thanks are also
extended to Elaine Govoni, who prepared the final manuscript.
All computations were performed at the M.I.T.
Information Processing Center with a grant from the
National Science Foundation.
In closing, the author
wishes to express his gratitude to the Society of Naval
Architects and Marine Engineers for their generous
graduate scholarship and to his family for their unfailing
support.
-4TABLE OF CONTENTS
Title Page . . . .
Abstract
....
. .
. . .
. .
. . ....
Acknowledgements . . .
Table of Contents
. .
. .
. . .
. .
. . .
List of Figures and Tables . .
List of Symbols
. . .
.
.
.
&
.
.
.
.
.
.
...
. . .
. . .
. .
. . .
. .
. .
. .
. . .
. .
.
CHAPTER I
Introduction
. . . . . . . . . .
12
CHAPTER II
Historical Background . . . . . . . . . .
16
CHAPTER III
Theoretical Development . . . . .
24
CHAPTER IV
Verification of Theory
.
.
. S . .
60
A.
Comparison with Experimental Results.
64
B.
Comparison with Other Theoretical
Results . . . . . . . . . . . . . . .
75
CHAPTER V
Design Analysis.
CHAPTER VI
Condlusion and Recommendations. . . .
References
. .
Appendices:
A.
.
............
93
.
. 110
. . . . . . . . . . . . . . . . .. 114
Changes to M.I.T. 5-D Seakeeping
Program User's Manual . . . . . .
. . 117
B.
Sample Output . ...........
C.
Subroutines Modified
D.
Listing of Mariner Example Data . .
. .
. . .
119
.
. . 128
. 169
-5LIST OF FIGURES AND TABLES
Figures
No.
Page
Title
1
Coordinate System . ...........
.
25
2
Mariner Added Resistance Components
(Terms in Eq. (86)) . . . . . . . . . . . .
62
3
Added Resistance/Head Waves . . .
66
4
Mariner Model; U=0; A=900 .. . ..........
.
71
5
Mariner Model; U=0;
3=1050
.
.
72
6
Mariner Model; U=0;
3=750 . ........
73
7
Mariner - Added Resistance Head Waves
(p=1800 ) Various Speeds
8
.
.
.
. .
.
11
12
..
.
.
79
.
.
80
Mariner - Drift Force
(B=1200) Various Speeds ..........
82
Mariner - Drift Force
(Speed - 15 knots) Various Headings . ...
84
Mariner - Drift Force
.
.
.
85
. .
.
.
87
(Speed - 15 knots) Various Headings . .
.
.
88
.
98
.
Mariner - Negative Added Resistance
(Speed - 15 knots) Various Headings
14
. .
Mariner - Added Resistance
(Speed - 15 knots) Various Headings
13
77
Mariner - Added Resistance
(Speed - 15 knots) Various Headings
10
.
. ........
(Speed - 15 knots) Various Headings
9
.
. . .
Mariner - Drift Force
............
. .. . .
. . .
15
Sea Spectra ..
16
. 100
Mariner Irregular Seas Results .. ...
(Mean Added Resistance; Pierson-Moskowitz)
-6-
Figures (Continued
No.
17
18
19
Title
Page
Mariner Irregular Seas Results
(Mean Drift Force; Pierson-Moskowitz) .
.
Mariner Irregular Seas Results
(Mean Added Resistance; Bretschneider)
Mariner Irregular Seas Results
(Mean Drift Force; Bretschneider)
.
101
104
. . 0. 105
Tables
Added Resistance/Drift Force in Pou:nds
for Various Sea States . . . . . . S . .
.
107
-7LIST OF SYMBOLS
a(x) or ajk(X)
- Sectional added mass coefficient
AX
x
- Sectional area
Ajk
- Added mass matrix of the ship
b
- Sectional beam
b(x) or bjk(x)
- Sectional damping coefficient
b* (x)
- Gerritsma and Beukelman sectional
quantity (3a)
B
- Ship beam
Bjk
- Damping matrix of the ship
CA
- After cross-section
Cjk
- Hydrostatic restoring force matrix of
the ship
d
- Sectional draft
E
- Radiated energy from ship
E(w)
- Energy spectrum of the sea based on full
wave amplitude
f
- Functional relation
f (x)
- Sectional Froude-Kriloff force
F(B)
- Mean second order force in long-crested
irregular seas
- Mean second order force in short-crested
irregular seas
F.
- Complex exciting force vector
Fj
- Diffraction portion of exciting force
-8FjI
- Froude-Kriloff portion of exciting force
F
- Unsteady hydrodynamic force vector
- Mean value of F
C4-
Magnitude of 3
'D
- Component of I related to the diffraction potential
jx
- Added resistance component of second
order force
0y
- Drift force component of second order
force
j·I
- Component of c related to the FroudeKriloff excitation
C4D
- Component otf 'analogous
fraction excitation
to the dif-
g
- Gravitational acceleration
Gj
- Added mass and damping part of H.
hD(x)
- Sectional quantity, integrand of53D (81)
hj(x)
- Sectional diffraction force
'h1/ 3
hj(x)
- Significant wave height
H.
- Total hydrodynamic force vector
i
- /-1
ID
- Integral used to develop'D expressicn
(82)
Ijk
- Moment or product of inertia
j
- Subscript (1...6; surge, sway, heave,
roll, pitch, yaw)
- Sectional quantity, integrand of*.D (78d)
similar
------------
1-l
-i
J
-9K
- Wave number
L
- Ship length (L.B.P.)
mk
- Related to ship normal, nk
Mjk
- Mass matrix of the ship
nk
- Ship hull normal (generalized)
Nj
- Two-dimensional normal
p
- Pressure in fluid
Q
- Represents either A or B
S
- Ship surface
SF
- Free surface
So
- Control surface at infinity
tjk
- Sectional integrand of Tjk
Tjk
- Complex quantity giving the Ajk and Bjk
elements
U
- Forward velocity of ship
V
- Volume within S, SF, So
Vza(X)
- Relative sectional velocity
x
- Longitudinal axis of the ship
xb
- Distance from center of gravity to
center of buoyancy
y
- Transverse axis of the ship
z
- Vertical axis of the ship
Zc
- Coordinate of the center of gravity
a-
Wave amplitude (1/2 height)
- Heading angle of the ship
-10Ck
- Phase angle between motion and excitation
- Free surface position
Sk
- Complex amplitude of ship motion
p*
- Corrected free surface velocity
6
- Principal wind direction for shortcrested seas
X
- Wave length
- Difference between heading angle ý and
wind direction 8
p
- Water density
- Sectional area coefficient (Ax/bd)
•B
- Full body potential amplitude
PD
- Diffraction potential amplitude
4 DC
- Non-zero mean higher order potential
ýI
- Incident wave potential amplitude
ck
- Motion potential amplitude
•ko
- Speed independent motion potential
kU
- Speed dependent motion potential
ýS
- Steady state potential
ýT
- Time dependent potential amplitude
- Overall velocity potential
•B
- Full body potential
- Full incident wave potential
§+(W)
- Sea spectrum based on half the wave
amplitude
-11Tk
- Two-dimensional (sectional potential
W
- Encounter frequency (rad/sec)
Wo
- Incident wave frequency (rad/sec)
Wp
- Spectral peak frequency (rad/sec)
Ws
- Spectral frequency ordinate
(rad/sec)
-12I.
INTRODUCTION
The forces that act on a ship in a seaway vary widely
both in their nature and their relative importance.
In
general, they represent random processes which can only be
quantified meaningfully through statistical analysis.
Be-
cause of this complexity, naval architects have traditionally
been forced to make many design decisions by combining
judgement and experience with the results of comparatively
simple, idealized analysis
or experiments.
One example of
this process is the use of the ship-beam analogy and the
quasistatic trochoidal wave profile for ship structural
design.
Another illustration can be found in the procedure
used to determine ship power requirements.
Common practice
has been to obtain the results of calm water resistance and
self-propulsion model tests, then, to account for the actual
operating conditions by applyinga power increase of fifteen
to thirty percent.
This service margin must be appropriate
if the ship is to fulfill her owner's requirements consistently, efficiently, and economically in the real ocean
environment.
Recently, great progress has been made in the effort
to include rigorously in the design process some of the
factors that influence a ship at sea.
The first order
-13strip theory of ship motions, due originally to KorvinKroukovsky and Jacobs
[1 31
, has been extended to the point
where motions can be calculated acceptably for a wide variety
of ship types, sea states, and heading angles.
This makes
possible the calculation of dynamic loadings imposed on the
ship by a seaway (Salvesen, et. al
[2 7
, and others).
Statis-
tical methods developed for systems subject to random
excitation now provide useful probabilistic statements
concerning significant design events (e.g., the chance of
slamming or the highest likely bending moment).
Due in part to these advances, the problem of environmental influence on ship power requirements and the related
problem of the sideways drift of a ship in a seaway can now
be much more fully addressed.
The forces involved can be
traced to a wide variety of sources, for example:
1.
The motions of the ship interact with the ocean
waves to produce a net drift or added resistance.
2.
The ocean waves reflect off the ship hull causing
a net force.
3.
Wind present at sea acts on the superstructure to
cause a drift force and/or an extra resistance.
4.
Marine fouling causes an increase in resistance due
to surface roughness.
5.
Involved interactions will also be present (e.g.,
the propeller may operate less efficiently due to
-14the ship motion, rudder motions necessary for
coursekeeping might cause induced drag and side
force, or the side slip of the whole vessel might
similarly cause drag).
The complexity of the problem
complete analysis.
outlined still precludes
However, it can be appreciated that even
a partial solution, which reduces the significance of purely
judgemental factors like the service margin, will greatly
increase the confidence and capability of the navalarchitect.
This is particularly true for design decisions that break
new ground for the profession such as powering for ultralarge tankers, vessels on new trade routes, or even dynamically positioned drilling ships.
In order to developa useful, flexible design tool of
this nature, the following report addresses the portion of
the drift force/added resistance problem included in the
first two points above, namely theship motion-wave interaction and the reflection.
From this point, 'drift force'
and 'added resistance' refer strictly to mean forces averaged
during one wave period of encounter and caused by wave-ship
hull interaction.
'Added resistance' will be the term for
the mean force component parallel to the longitudinal, ship
axis, positive toward the stern.
'Drift force' will signify
the mean force component perpendicular to the longitudinal
ship axis, positive when directed toward the same half-
-15plane as the wave propagation.
To place the method to be presented in a historical
perspective, the paper begins with a discussion of past
analytical efforts on the problem.
Then a brief description
of first order strip theory of ship motions ispresented,
and the method of calculation of second order mean wave
force developed by Salvesen [2 8 ] is detailed.
As will be
seen, it is the mean second order wave force which is the
drift force/added resistance referred to above.
After an
investigation of the characteristics of the final Salvesen
formulation, an outline of past experimental work done on
the problem is given, and an experimental effort devised to
test certain areas of applicability is described.
Following
presentation of the experimental results, some numerical
predictions for regular waves are compared with the results
[1 6 ] .
shown by Salvesen S28e and Loukakis [162
The theoretical
basis for a probabalistic extension of the regular wave
theory to irregular ocean conditions is described, and a
brief design analysis for several sea states is given.
Finally, some general conclusions and recommendations are
offered.
The appendices contain documentation for the
computer program developed in the analysis.
-16II.
HISTORICAL BACKGROUND
was one of the earliest researchers (1939)
Kreitner
to investigate the problem of wave forces on a ship hull.
He concluded that the force was caused primarily by the
reflection of the incident waves from the hull.
However,
the real pioneer in the second order force problem was
Havelock '9
who proposed a theory in 1942 for a ship with
no forward speed in regular head seas that was allowed to
He found that Kreitner's reflection force
pitch and heave.
was unrealistic for a pitching and heaving ship in waves of
reasonable length.
Instead it became clear that the phase
difference between the ship motion and wave excitation was
the primary source of added resistance.
Following this
line of thought, Havelock proposed the following:
-
where F
I
2-
(1)
is the exciting force or moment due to the wave
(assuming that the ship does not affect the wave flow field);
this force (moment) is referred to as the Froude-Kriloff
excitation force (moment).
5 to pitch.
Subscript 3 refers to heave,
fj is the motion amplitude, Ej is the phase
-17angle between the motion and the excitation, and K is the
wave number (Lo/g).
Hanaoka
[6]
examined the case of a moving ship in calm
water that was externally forced to heave and pitch.
He
gave an expression for the wave resistance of the ship
which predicted a considerable increase over the unforced
case.
This increase was related to the damping in the
radiated waves produced by the moving hull.
The damping
could, in turn, be associated with the phase lag of Havelock,
so the two theories were definitely supportive.
During the time when Hanaoka was working on his formulation (1953), Haskind [ 8
employed a potential flow method
to combine the efforts of Havelock and Kreitner.
He pro-
posed that the net wave force was the sum of two parts, one
wave reflection and the other wave-ship motion interaction.
The integral equation that resulted was complicated,-involving Kochin H-functions (surface ingegrals dependent
on frequency and form).
The next big step in the field came with Maruo's [17
potential flow solution, which was presented in 1957.
divided the velocity potential into three parts:
He
the in-
cident waves, the steady-state body potential, and the
time-dependent (oscillatory) body motion potentials.
For
practical purposes, the body potential was evaluated for
-18each section of the ship separately in the conventional
strip formulation.
The end result was a solution for the
added resistance consisting of a sum of six terms, one each
for heave, pitch, and reflection and one each for their interactions.
Maruo's work was very important in that it com-
bined and refined the ideas of Kreitner and Havelock,
added a consideration of forward velocity and considered
interactions between the motion related and reflection related
resistance that had been earlier ignored.
Further informa-
tion on Maruo's theory for head seas can be found in Ref. 7.
The extension of Maruo's work to seas approaching the ship
from any angle (oblique seas) was accomplished recently by
Hosoda 1
0 ],
but it is extremely complex since it involves
twenty-five components for all five degrees of freedom.
Joosen [ ll] offered a new result for the case of head
seas in 1966.
Joosen's equation resembled Havelock's
original work, but included heave and pitch interaction.
Newman presented an oblique seas theory in 1967
[2 0 ]
which
was abstract in that it utilized pure slender body theory
and a long wave approximation.
Boese
[2 9 ]
derived a method
for head seas similar to Havelock's in 1970.
It was based
on the determination of the pressure distribution around
the hull in the wave.
Before describing the most recent work done on the
-19second order wave force problem, some conclusions regarding
the nature of the drift force/added resistance can be
drawn.
These general principles can be inferred from and
are supported by the theoretical and experimental work that
has been done on the problem.
1.
The source of most of the drift force/added
resistance (except for very short waves) is
the phase lag between ship motions and wave
excitations. [9,
17,
29]
A phase lag exists only in the presence of damping and,
for ship motions, damping is almost exclusively by radiated
waves.
Furthermore, the energy loss through damping is
directly related to the work necessary to maintain constant
phasing.
2.
Therefore, the following can be concluded:
The drift force/added resistance problem can be
formulated in terms of the waves radiated from
the hull
3.
[5 ]
The second order force is a wave energy phenomenon
which must be proportional in magnitude to the incident wave amplitude squared
[1 7 '
29]
The case for the last point above is particularly strong
in that experimental data supports it well (See, for example,
Ref. 29).
This assertion is also crucial for the applica-
tion of statistical methods.
Two more statements of
-20importance can also be made:
4.
Since the source of the second order force is the
seaway, and the associated ship motions, added resistance will be independent of calm water resistance.
If some variation affects both, then it
will do so through two distinct mechanisms.[17, 29]
5.
Being totally a surface wave phenomenon, the drift
force/added resistance can be measured in experiments using Froude scaling and fairly small models.
(Ref. 29).
Gerritsma and Beukelman presented a theory in 1972
[5 ]
based on the idea that the added resistance in head waves
could be calculated from the radiated energy contained in
the outgoing damping and reflection waves around the ship.
Their result continues to be extremely useful, due to its
reliability and the fact that it is much easier to combine
with a strip theory ship motions computer program than many
of the other methods mentioned.
In their development, they
postulate that the energy radiated during one wave encounter
period is,
(2)
Tr
0
-21where w is the encounter frequency and b*(x) and Vza(x) are
a sectional coefficient and a relative sectional velocity
defined as follows:
b(x) - U[ Cl(X)
b(X)-
x
V
(3a)
+(3b) 27
+ U S.5-
where b(x) is the sectional damping coefficient for an
oscillating two-dimensional cylinder shaped like the section
and a(x) is the sectional added mass for the same problem.
U is the forward velocity of the ship, xb is measured from
the center of gravity and ý* represents the velocity of the
free surface.
Referring to the work of Hanaoka et al.[7]
it can be shown that the proportionality constant relating
radiated energy and added resistance is the wave length,
X(E =XgXx).
Applying this fact together with (3) in Equa-
tion (2) gives:
L
A
-22-
where b 3 3 (x) and a 3 3 (x) are now strictly interpreted as the
sectional heave damping and added mass coefficients.
Through the use of energy flux consideration, Loukakis
and Sclavounos [1 6 1 1977, have succeeded in extending Gerritsmas's theory to computation of drift force and added resistance (and yaw moment) in oblique seas.
This represents
one of the three newest general methods for computation of
second order mean force in oblique seas that seem well
suited to inclusion in modern ship motions calculation
schemes (the current goal being to obtain good second order
mean force estimations from quantities known in the first
order motion calculation, thereby minimizing computer time).
The second method that seems to hold promise in this
It
sense is a potential flow theory due to Ankudinov
gives the second order force in a form similar to Havelock
except that the total first order exciting force appears
instead of just the Froude-Kriloff portion.
It would be
very easy to incorporate in a motions program, but it
neglects wave reflection.
Furthermore, there is some
question about the validity of the final results presented
(Salvesen[2 8]).
J. N. Newman was the originator of the third theory
which will be derived and employed in this paper.
He has
worked on various aspects of the second order force problem
-23(Ref. 18, 19, 20, 22) but the particular paper that is
most relevant to this report is Ref. 21.
The method
presented there can best be described as a potential flow
formulation which determines a net pressure on the hull of
the vessel due to higher order wave effects.
Newman's equa-
tion was in surface integral form, and strict validity was
assumed only for a submerged body beneath waves.
Salvesen
(Ref. 28) extended the analysis to surface vessels and applied the methods of strip theory to make numerical
solution for a given ship within the context of a first
order ship motions program possible.
-24III.
THEORETICAL DEVELOPMENT
In order to generate a formula for the prediction of
the mean second order force on a ship in oblique waves, it
will first be necessary to develop the general potential
flow problem that applies.
Consider, then, a ship moving at
speed 1 oriented arbitrarily to regular sinusoidal waves, as
shown in Figure 1.
The oscillatory motions that result
will be assumed to be linear and harmonic.
system is illustrated in Figure 1.
The coordinate
It is a right-handed
orthogonal (x,y,z) system with the x-y plane coinciding with
the plane of the undisturbed free surface, x along the
longitudinal centerline, positive in the direction of forward
motion, y positive to port and z positive upward through the
**
ship center of gravity.-
The waves are shown to have their
direction of propagation at an angle, 8, to the ship x-axis
(8 = 1800, head waves).
It is presumed that the ship oscillates as a rigid body
in six degrees of freedom with complex motion amplitudes,
Sk,
k=I..6, which refer to surge, sway, heave, roll, pitch,
and yaw, respectively, as shown in Figure 1.
Disregarding
'The computer program has been generalized so the user can
choose the longitudinal position of the origin arbitrarily.
The choice of origin at the center of gravity is for the
simplicity of this derivation only.
-25-
HEAVEt (3)
SWAY
$f
y(2)
YAW
PITCH
(5)
IURGE
X
(1)
ROLL
(4)
Wave
Crests
n
FIGURE 1:
ýCOORDINATE SYSTEM
-26viscosity and assuming irrotationality, potential flow
theory can be applied.
Thus, there is an overall velocity
potential for the problem, j (x,y,z,t).
This potential must
satisfy Laplace's equation and the following exact boundary
conditions:
On the ship surface (;(x,y,z)=0):
Equation (5) expresses the condition that there must be zero
flow velocity normal to the hull at the hull position.
On the free surface (C=r(x,y,t) the following condition
must apply:
'
L-l(6)
-
which expresses a pressure and gravitational force equilibrium at the boundary.
Also, a radiation condition must be
applied at B = -o guaranteeing that the motion will be
zero (Vr+0).
The first step toward a solution is to separate the
velocity potential into its time independent part due to
the steady forward motion of the ship and its time dependent
part associated with the incident waves and the harmonic
-27motions,
[-Ux +
,(
++,
y)])
twt
(7)
where w is the frequency of encounter, related to the incident wave frequency, wo, by.
C
Uos
(8)
It should be understood that only the real part is to be
taken in expressions involving e i Wt that appear in this
derivation.
At this point it is necessary to linearize the problem
by assuming that
,s,the steady perturbation associated with
wavemaking in calm water, is small and so are its derivatives.
The potential 'T and its derivatives must also be assumed
small.
In this way, higher order terms and cross products
of the potential components can be neglected.
Decomposing
the complex amplitude of the time dependent potential gives,
where
CPI
is the incident wave potential, ýD is the diffrac-
tion potential, and
4k
the kth mode of motion.
is the potential contribution from
It can be seen that the problem has
-28been transformed into the solution and superposition of
several simpler potentials.
ýI is the well-known potential
for an infinite free surface of plane progressive gravity
waves.
#b and the ýk's must be solved, since the first
deals with the wave reflecting properties of the motionless
ship and the rest, of course, represent the properties of
the motion.
An application of a Taylor series expansion about the
mean ship position in connection with the above simplifications will show that the boundary conditions can now be
linearized as follows:
U +
Y
(-(-U(
-0
••e
,ull'(10)
which represents the time independent portion of the body
boundary condition applied at the mean hull position
which represents the time independent part. of the free surface
boundary condition applied at the undistrubed free surface.
-F = 0
Lhull'
O the
(12)
which represents the incident and diffraction portion of the
-29-
time dependent part of the body boundary condition on the
mean hull location.
(CW-U-),+9 C] j:"0"=0
(13)
Oh 'S =
which gives the free-surface condition on 4I and
C
D." In the
above equations, n is the outward normal to the hull.
The oscillatory motion potentials must satisfy:
an
iLWnk +Tmk
oh the •hul'
(14)
and
(15)
In these equations, nk is a generalized normal defined by,
(n.r,,r)
0
a (nnM,n )=
rxn
(16)
where n is the ship hull normal and r is a position vector.
Also, mk is defined as follows,
k
M0
or
4
,5
n.3
rn =i
-2
(17)
-30Further simplification of the hull boundary condition on
the body :motion potentials can be achieved by separating
the potentials into two parts,
Rk0/Pk
(18)
L/-'U
+
where ý 0 will be assumed speed-independent.
Substituting
(18) into (14) gives two hull conditions,
n
k
LWrf
and
(19)
must satisfy all the same conditions and
Since 4 U and 4
k
k
the Laplace equation, it follows from the relations (17)
and (19) that:
O
;
4or k=l.
s
(20)
Thus, the oscillatory motion potential components can be
given from (18) in a form which includes speed only as a
simple factor.
(This 'speed-independence' is an involved
assumption which will be clarified later.)
o
for
o
k=..
3=0k
W -3
(21a)
(21b)
-31-
with the body boundary condition from (19) being
aLu =
kik
(22)
and the free surface condition becoming from (15)
8)±X
k
(23)
This completes the synthesis of the relevant conditions
governing this problem.
To summarize, the general potential
is separated into several terms as per Equations (7) and (9).
These linearized potentials must satisfy Laplace's equation,
the linearized boundary conditions ohnthe calm water surface.
(11),
(13) and (23), and the body boundary conditions (10),
(12), and (22) at the hull position as well as the radiation
conditions at infinity.
Having formulated the potential flow problem for a
ship moving on an arbitrary heading in regular waves, the
equations of motion of the vessel will now be developed.
In this way specific quantities that must be extracted from
the potential flow solution can be identified.
The ship is considered as a rigid body with six degrees
-32of freedom.
Under the assumption of linear harmonic motions
already stated, the governing equations can be written in
matirx form in the frequency domain,
[-w2 (MN
+icA
+ . e+C3f
F,
(24)
where Mjk is the generalized mass matrix of the ship (25),
Ajk and Bjk are the added-mass and damping coefficients (26),
the
Cjk 's are the hydrostatic restoring coefficients, and
the Fj's are the complex exciting forces and moments.
Note
that the ship is idealized as a coupled spring-mass-dashpot
system with harmonic forcing functions.
Assuming that the
ship possesses lateral symmetry, then the matrices above can
be given as follows:
o M o -MC Co
I1
V
M
o
o
M
(25)
o0o
o -
0o I 4
0O
0o -L 1 o
0
-1Iq
where the center of gravity is located at (0 ,0,'&c),
M is the
-33ship mass, and the I
's are mass moments of inertia, when
j=k and products of inertia for jAk.
Q,,
I or
o
Q,,
0
Q,, 0
(26)
O0 C
o
Q0o
Q44
QL o Q44o OQ(
Il
where Q stands for A or B.
These terms (26) are hydrodynamic in nature and must
be developed from the potential flow problem outlined.
For
the hydrostatic coefficients the only nonzero terms are
C33, C4 4 , C 5 5 , and C 3 5 . All these are quantities well
known to: the naval architect from hydrostatics.
An examination of the matrices in Equations (25),
(26),
and the C-matrix will reveal that for the laterally symmetric
ship there are two sets of three equations, one for surge,
heave, and pitch, and another for sway, roll, and yaw.
These two sets are decoupled.
Furthermore, under the
assumption that the hull form is slender , it has been
shown that the forces associated with surge are small, and
-34this mode can be disregarded leaving five degrees of freedom.
To solve these equations, it will be necessary to
develop the added mass and damping matrix elements and the
five exciting forces from the potential flow solution
already outlined.
These are all obtainable from the pressure
in the fluid on the hull.
By Bernoulli's equation,
jIV
a=t
(27)
)
If this pressure is expanded in a Taylor series about the
mean hull position and linearized consistently, it follows
that the unsteady pressure is
p
~--U
;,)
Lut
0, e W-
L\wt
y-
X, e
(28)
The last term is just the hydrostatic restoring force which
was already separately included in the equations of motion
(see above).
Thus, the hydrodynamic force and moment
acting on the ship will be given by
A (LW -U
J
I=z,,G
-!)
,
--
(29)
-35where 0 is the hull surface at its mean position.
The use
of Equation (9) will allow the division of (29) into two
parts, the exciting force and moment due to the wave
(including the diffracting effects of the hull):
-J
Y
v,(LLO-Ui-
+ )is
(30)
and the force and moment due to the body motions (which
physically represent both the added mass and damping).
r) (liw-u
j:Tk
(31)
(32)
where
Equation (24) shows that the real part of (33) will be the
added mass while the imaginary part will be the damping.
-36Therefore,
(34)
The evaluation of Equation (33) is not currently possible
because the motion potentials involved are three-dimensional
in nature and associated with an arbitrary hull shape.
Strip
theory allows the solution of the problem through a simple
lengthwise integration of two-dimensional potentials associated with each section.
This is developed using a variant
of Stokes Theorem derived in Ref. 27.
U
P=
U
r
~ ads -U
S0
na Ocd
CA(35)
Applying this to (33), and assuming that the ship has zero
after cross-section (CA)** yields
k
-fW
fn
00tS +UPI S~4k As
(36)
** In the original MIT motions program, this after section
term was included in the calculation. For consistency with
the second order force subroutine to follow, either all
these terms must be removed from the motions calculation or
a "zero" after section must be included in the input.
-37Using (36), the Tjk's can be written for any desired jk
combination by substituting from (21).
Next, conventional
strip theory approximations can be applied by assuming that
the ship hull is long and slender.
This allows the following
transformation of the surface integral of (36) for the
speed-independent potentials
i -kw:~
-t
L
dX
(37)
LL
As a consequence of the slender-hull assumption, it can be
seen that, along the hull, a/ax<<a/ay or 3/aS
n3 .
and nl<<n 2 or
The last of these allows substitution of a two-
dimensional normal, Nj, (noting that now n 5 =-xN 3 and
n 6 = xN 2 ).
In view of the above assumptions, it is possible to
quantify the limitations inherent in making the motion
potentials 'speed independent' (See (21)).
The free surface
condition (15) gives, upon reduction, that w>>U2/2x for
the speed-independence to be workable.
This is equivalent
to a wave-length on the order of the ship beam.
Fortunately,
this theoretical restriction on strip theory does not
preclude very reasonable answers for fairly long waves,
since the hydrostatic terms grow in importance for the
-38smaller frequencies.
The above observations can now be used to infer that
the speed-independent, three-dimensional motion potentials,
4
o,
can be replaced as follows with sectional potentials
Sor
kzz,3, -(38a)
6•
(38b)
(38c)
where the YK's are the potentials for the two-dimensional
problems of an infinite cylinder with the shape of the
section oscillating in sway, heave, or roll.
The problem
of a cylinder oscillating in any of these modes is a classic
one in hydrodynamics, and several techniques exist for
mapping the solution to an arbitrary section shape.
These
methods include the Frank close-fit source-distribution
Tsai-Porter close-fit mapping method, the Demanche bulb-form,
and the Lewis form.
The last two are used by the M.I.T.
5-D program.
In summary, the sectional potentials can be computed
by known numerical methods, then combined with Equation (37)
-39to give
SLO
fC
Jk
LwbJA
(39)
where ajk and bjk are the sectional added mass and damping
inferred from (34).
This makes possible the computation
of all the speed-independent Tk 's which can be used in
jk
(36) in conjunction with (21) to give the full Tjk 's.
The
real and imaginary parts of the Tjk's then correspond (34)
to the desired damping and added mass.
The process is
complicated in an algebraic sense, but further details,
including results, are available in Reference 27.
The final
answers are given in terms of the sectional added mass and
damping for the two-dimensional problem, for example,
A•C53
-X
X
+
(40a)
or
(40b)
where
-40-
)=
I
CL24
CtX
The B 4 4 damping term developed in this way does not
give realistic answers when used in Equation (24).
This is
due to the fact that ship rolling is governed by viscous
effects.
The resulting non-linearity can be handled by
defining a quasi-linear damping augmentation based on roll
velocity and using an iterative scheme (See Ref. [27]).
The development of the exciting force and moment from
(30) is also crucial to the understanding of the secondorder force theory to be derived.
Salvesen, et. al
[27 ]
show
that the forces and moments can be expressed in terms of
the sectional potentials discussed above.
This method due
to Haskin circumvents the solution of the diffraction
potential, ýD, a very important simplification.
To proceed, the exciting force (30) is separated into
two parts,
-- E i+
(41)
F.D
where
IIj
-
L)
UCI
(42)
-41and
F
n,(LO
c
DCtS
(43)
The potential for the incident waves is well known from
classic linear gravity wave theory:
Se-p
(where
[-
K (xcos-ysi)+
CK=Gve LmpIitU-dLe, K=W
(44)
wave number)
This quantity can be substituted into (42), giving,
n_tiC'
FPL
where Wo is written as a consequence of (8).
(45)
Equation (45)
is the Froude-Kriloff exciting force which is easily computed without knowledge of the sectional potentials.
Continuing with Equation (43), the diffraction part of
the exciting force, application of the special version of
Stokes theorem used earlier (35) gives, for a ship with zero
after cross-section,
F-
iw1 U r
Z
-
(46)
-42Use of the hull boundary condition (19) yields:
fSf
(P)
Uj
-
)
cl-
(47)
A theorem of vector calculus known as 'Green's Second
Identity' can be used with two functions (*andP f ) satisfying the same Laplace equation, free-surface condition,
radiation condition, and bottom condition to yield the
following identity
n
(48)
Applying this to (47) gives the result
Now the use of the hull boundary condition (12) leads to
I
ff
(
T
(50)
It is clear from (50) that the diffraction potential has
-43been extracted from the problem.
The normal derivative of
the incident wave potential (44) becomes, after use of strip
theory assumptions:
ann).= (Ln,sin,8 +n,)K 4,
(51)
Using (51) and the motion potential relations (20) in
Equation (50) and invoking the standard strip theory
assumptions to transform the surface integral gives,
F;
-
-
x co•8
LKY~sir1,8 X2
-x
e
r vL ( Ln-nsAn/6)4t
'.
.-
L
[ (n
"UO[- L08WsO)
5
"
,3
'•
co ax
"A
(52)
Making use of the two-dimensional sectional normal, and
the concept of a sectional force, allows the following
t
simplifications:
From (45) :
FI3= pT{f.
FcX
L
(X)CX
2,3,
(53a)
(53b)
-44-
K'
FK
=
x FZ)
Ax
(53c)
where
S9e
-ý (-A)
- LXcos/
ce
-
LKY"'in/3 d
(53d)
j = aj3j (4
from (52) :
FD
=
FD =
P0
iLkj(x)
+- -
F
f 0(
v
x
(54a)
(x) x
x+U
L U.) ) k(%x)
(54b)
tx
(54c)
e
d
(54d)
where
;(x)
-
e
XCOS/2
C'X
3-N o,•)
... e
••.t
]
-45The strip theory formulation of the linearized ship
motions problem has now been completed.
Once the sectional,
two-dimensional potentials for sway, heave, and roll are
determined by one of the methods mentioned (See Ref. [27]),
then these can be used in (39) to produce the various
sectional damping and added mass coefficients.
Equation
(40) and other similar relationships detailed in Ref. [27]
can then be applied to find the added mass and damping
matrices for the whole ship.
of (24) is known.
from (41),
This means that the left side
The exciting force is then determined
(53), and (54) since the sectional potentials
are known.
After this (24) is easily solved for the complex
motion amplitudes,fk.
These complex quantities contain
both the real motion amplitudes and the phasing.
It should
be reemphasized that these motions are, by assumption, linear
and harmonic in the frequency of encounter.
In a nonlinear analysis of the problem, higher order
terms in the incident wave potential would tend to interact
with the other potentials, producing periodic higher order
forces with non-zero means.
Of course, these forces tend to
be small in comparison with the simple harmonic first order
excitations.
They are, in fact, negligible in the pitch,
roll, and heave modes because of the strong hydrostatic
restoring forces that are acting.
However, they can act to
produce large displacements over time in the horizontal
-46modes.
It can be seen, then, that in order to force the
ship to perform the assumed motion at constant speed, U,
and constant heading angle, B, in the 'real' ocean, an
extra time varying force will have to be applied both
longitudinally and transversely (also a moment will be
necessary).
In the absence of these the ship will drift
and its speed will vary from that expected in calm water.
A method by Newman, already mentioned in Section II,
makes possible the calculation of the mean of the second
order force which produces the speed loss and drift.
As
will be shown, this can be done using only quantities known
from the first order analysis.
The unsteady hydrodynamic force on a body in an inviscid
medium with a free surface is given by Equation (29).
This
can be expressed in a more general form as follows:
FS5
where $ is the wetted surface of the body.
(55)
Applying
Gauss' theorem and utlizing the fact that p= 0 on the free
surface,
Scis
-V
(56)
-47where S is a control surface in the far field and V is the
volume enclosed between S, Sc
, and the free surface.
Ber-
noulli's equation gives the pressure in terms of the total
velocity potential:
+v
F=
v
4 -1171
s (57)
The transport theorem [2 3 ] shows that
aVIv
tdv -
where SFis the free surface.
+
-*
the fact that V-n=0 on S, and V.n=is the surface velocity).
on S and SF
(where V
Using (58) in (57) yields
_V(59)
Pf
(58)
The truth of (58) depends on
-+
+÷
Cs
I VS (I5i'-AV
+
+
ct5
-48Again, Gauss' theorem can be employed to change the first
volume integral to a surface integral, with the result:
at~
_fC-t
C)
(1'
ncs
i
11CLS
ICI~
;40
- - %
-CJ
~
.
(60)
When this is included in (59) the result is
D 45P·d
cis
+
(61)
Ip 7 V, V,1 7-v\
O-V
Once more invoking Gauss
2-•1 v
v
l
theorem,
n V7cids
4+ 4fF
Putting (62) in (61) gives as a final result,
(62)
-49-
F
fen
Is
v-
Y IJ1Ic
(63)
The expression (63) provides the force on a body in a
fluid with a free surface exactly (within the limits of the
theory of potential flow).
Using a first order potential
in (63), would yield (after linearization) the oscillatory
hydrodynamic force components (30) and (31).
The simplest
higher order potential that could be used in the problem
would be of the form
O)c) 11
.tt
(a)
.
(
(64)
(64)
where the numbers in paraentheses indicate the order of the
terms and the 'D.C.' is responsible for the non-zero mean
of the oscillatory potential.
Putting (64) in
(63)
'shows
that there will be no net contribution from the first
integral because the DC-potential has no time derivative.
The second integral will give a steady state contribution
-50which can be developed by writing % = I + @B, substituting
in (63), and performing the vector calculus,
,=
a+SS
!)C
(65)
Newman applied a 'weak scatterer' assumption at this
point (gB<<«I).
This is easily justified for a body sub-
merged beneath the surface.
Salvesen reasoned that this
might also be true for a slender body like a ship, an
assumption that will be more accurate for head than beam
seas.
The result is
F=:Bn
FJ
Writing
gB = s
4
Isa
Beiwt
Be
and
(66)
iwt
I
I = I
ee
, taking the mean value
of (66), and using Green's theorem to change the integration
surface from the far-field to the body, gives the following:
IT
ds3
(67)
-51where ( )* refers to the complex conjugate and
5 represents
the mean force vector for one regular wave period.
Since only the horizontal component of 3 is of interest,
,go,* can be written,
)A
')
s
(68)
Substitution of (68) in (67) allows the magnitude of (67)
to be given as,
n
___b'
a n
cis
r(69)
In the ship coordinate system, the beam and lengthwise
components of this mean force will be,
TX = ITCcos/'T'
= 75L78
(70)
which follows because o'is in the direction of wave propagation.
It should be noted that (69) is a form of the
Kochin function, first developed in connection with this
problem by Haskind.
As a consequence of the fact that the mean higher order
-52force contribution has been reexpressed in terms of the full
body potential, substitution of the first order body potential of strip theory will not give an approximation of the
net force contributed by the higher order DC terms in the
full potential (64).
The extra mean force necessary to
sustain the prescribed body motion in waves is given by
substituting
=K
Z
x fjl
f
+ 0D in ('s).
SS~p%3'c=I2
a) 4
3
-n
an
(71)
This can be rewritten for computational purposes as
-+ D
(72a)
where
O
0
X
dn
(72b)
-53-
psf~l~s
±Tds
(72c)
and after application of the body boundary condition (12)
03
T.jKsD
J±ct5
(72d)
The next problem is to express Equations (72) in strip
theory terms.
Examining (72b) first, a substitution of
the boundary condition (14) gives
+ ULwricts
(73)
Applying the variant of Stokes theorem (35) and substituting
for
oo from (8):
Comparing this with (45) reveals that:
-54-
f(75)
a
This can easily be shown by trigonometric identity to be
the same as the formula developed by Havelock (1).
Continuing now to
theory assumption, nl<<n
(72c), it follows from the strip
2
or n 3 , and (44) that
<BlV(76)
Including (76) in (72c) and replacing
Pj
using (21)
n.5
D
YB3(77a)
((0 Z3,3Z
22.-
(77b)
......(-r + Ls
i Ac{s
-55The next step is to replace the unknown 3-D potentials with
the sectional potentials according to (38) and transform the
surface integral,
CTD-=
SK
j{=x)
2,3,L
,)
(78a)
d-A
D
LCU
(78b)
A
A
where a sectional quantity h(x) has been defined as
j
A()Ip
A.
(-N
J'f
ý=
+ iA/sAz)
fAt
(78c)
;1)3JLI
:Zj 3j 4L
>poc
WCe- X51'/
co)..
+A!x
f
(78d)
;cKYsir1,~Kx~Nf
de
3)LIL
-56The similarity between (78) and (54) can be seen; however,
the two are not algebraically related and must be computed
separately.
For simplicity, let:
5
Id
(79b)
L
^D
(79a)
il'a,3,c]
5(·
iU
L
A~~d
giving
D
D
_-II
~^D
~,-iKF,.
(79c)
Finally, consider the third contributor (72d).
Substituting
(76) and rewriting the expression:
a(x)
Lx
S2SD
Cx
+
(80a)
0)1
(80b)
-57Including the expression for @
:
YxOs/ =W,(-n
r
. Ln
sg
(81)
In order to simplify computation, it is consistent with the
strip theory assumptions already made to replace e
e +Kdand e-iKysin
with
,iK(+b)sin
where d = sectional
with e
draft, a E sectional area coefficient (•x/bd), and
b Esectional beam.
The first of these is conventional in
strip theory and the second should be legitimate if X>>»b.
All this gives:
=e
Lx06
e_ -x-
ihis if x cosA
-Li1(±b)US
e
0.. ...
OD(-n3 + ZnzSen3
The integral in (82) can be rewritten by substituting for
the hull normal from (22) and using Green's second identity
(48) in two dimensions.
-58-
I
=- w
0
3.
w3
3 .l-
sLn )
Lf.J
(83)
Using the now familiar hull condition (12), writing out the
wave potential (44) and applying the same assumptions about
e-KZ and eiKy sin
OTC
outlined above in route to (82) gives:
e -.
ŽMi'
4T(C
0
1
ýe o
(84)
x
where the symmetric section assumption has been used to
neglect two cross products involving the potentials and
hull normals.
Examination of (84) will reveal that when the
two dimensional normals and sectional potentials are introduced (39) will be directly applicable.
This will allow the
reexpression of (84) in terms of the sectional added mass
and damping already known.
-59Where (85)
represents just the real part of (84) because
the imaginary part is not needed in (72a).
(The reader is
reminded, "...only the real part is to be taken in expressions involving eiwt that appear in this derivation.")
In summary, the mean second order 'DC' force on a
ship in regular waves is available from the real part of
the following expression,
L
where the motions are available from the first order
computation already outlined, (F
)* is also available
A
from that process, F P can be developed from (79) using
strictly quantities known from strip theory, andWJD comes
from (80) in combination with (85).
Then the added
resistance and drift force can be given by simply applying
(70).
The M.I.T. 5-D motions program has been modified
to properly extract and recombine these quantities (See
Appendices for details).
-60IV.
VERIFICATION OF THEORY
In Chapter III it was shown how the added resistance
and drift force could be computed for any ship using
Equations (86) and (70).
This capability was incorporated
in the M.I.T. 5-D motions program in the form of two subroutines (ADDRES and RESIST).
The first of these is pri-
marily an output organizer; the second does the actual
calculation outlined earlier.
As mentioned before, all the
input necessary is directly available from the first-order
motions calculation except the sectional quantity given by
(78d).
This quantity is generated from the sectional po-
tentials within subroutine INTRPL.
A user's manual for the
program, briefly describing these routines, is presented in
the appendices, together with input and output samples.
In order to maximize the possibilities for comparison
with existing second-order force results, the Mariner-type
fast cargo vessel was used in all the examples that follow.
The Mariner was designed about 1950 at the Bethlehem Steel
yard, in Quincy, MA
[2 5 ] .
She has a length-between-perpen-
diculars of 528 feet, a beam of 76 feet, and a service speed
of 20 knots.
For this study, a full load draft of 29.75
feet was chosen giving a displacement of 21,000 tons.
The
pitch radius of gyration was set at about 25% of the L.B.P.
-61Further details on the ship can be obtained from Appendix D.
which gives a copy of the input used for this study in the
M.I.T. 5-D program.
As a first step in a consideration of the theory,
Equation (86) will be examined in detail for one particular
oblique regular seas case (8 =
1 5 00,
From the
U=15 knots).
form of the equation, it is clear that the second-order
force prediction is generated as the sum of eleven components:
Froude-Kriloff and Diffraction terms for each of the
five modes of motion and the wave reflection term.
These
components are all plotted separately in Figure 2.
The
non-dimensionalized added resistance component (defined by
2
GAR = Added Resistance/pga2 B /L, where Bis the ship beam, L
is the L.B.P., and
,a is the wave amplitude) is given as
a function of wave-length to ship length ratio.
,
case, the Froude-Kriloff pitch term (5I
For this
Eq. (75), j =5)
makes a large contribution as does the Froude-Kriloff heave
term ('31, Eq. (75), j =3).
The wave reflection term (
in Eq. (80)) also makes a contribution, particularly for
very short waves, where it is totally dominant.
and pitch 'wave diffraction' terms (13D
,
5D,
The heave
Eq. (77),
j = 3,5) act to reduce the added resistance considerably,
while all the terms involving roll, sway, and yaw are practically insignificant.
These relative magnitudes persist
(62) FIGURE 2:
N
MARINER ADDED RESISTANCE COMPONENTS (Terms in Eq. (86))
0-l
E-1
z
z
0
z
0
U
P
U)
H
U)
H
0
I
L-
-63in general in all near-head sea conditions, but in near
beam seas, the pitch terms,* 5I
and " 5 D, play a much less
important role while the sway, yaw, and reflection terms,
(*2 D, 3 2I,
tudes.
6 D,
,
,61
and
3D
)
increase their relative magni-
In following waves, the contributions of the various
components are heavily dependent on encounter frequency and
the associated accuracy of the strip theory, a question
which will be discussed later at greater length.
It can be seen that all the significant motion-related
second-order force components peak at one place (generally
near the heave or pitch resonance, respectively) producing
a sharp peak in the total force (markedZ).
This sharp
peak is present for all heading angles, although its location varies, depending primarily on the location of the
heave and pitch peaks in bow and beam waves.
One final
observation that can be made is that, for near head seas,
the original formula of Havelock((2), marked
applied with some success.
F-k) can be
Since all the force components
peak in the same area, and Havelock's formula contains two
of the most significant positive terms, it will predict
the peak location for the second-order
force.
Several
investigators have found the magnitude of its predictions
to be within a factor of two in most cases
[ 5' 2 8 ] .
This
has engineering significance because the Froude-Kriloff
-b4-
force (See (75) and (45)) can be computed easily without
knowledge of the sectional potentials.
In fact, the cal-
culations can be done by hand if a first order motions
printout giving sectional Froude-Kriloff force is available.
Having generated a working second-order force computational scheme based on (86), the next step is to ascertain
its validity.
This will be done by comparison with experi-
mental results and analytical predictions generated by other
methods.
A.
Compa.rison With Experimental Results
Experimental efforts on the second-order. force problem
are historically very complex and not too repeatable.
The
main reason for most of the difficulties can be easily
identified.
The periodic forces involved are extremely
small,making measuremnt very difficult, especially in the
presence of friction in the mechanical equipment, vibration
in the towing carriage, and electronic noise.
Most of the
towing tank work that has been done has been directed
toward finding the added resistance in regular head seas.
There are two methods for carrying out these measurements:
constant velocity (where the model is free only to heave
and pitch) and constant thrust (where a self-propelled
model is attached to a movable sub-carriage and allowed to
-65surge, as well).
All the experimental work presented in
this report was obtained using the constant velocity method
This allows a more effective comparison with the computer
program, which neglects surge.
As mentioned before, the second-order force is primarily
a wave phenomenon and can be scaled by Froude number so that
the force on the real ship will be proportional to the model
force times the cube of the scale ratio.
Pure Froude
scaling also implies that the use of small models is justifiable,: but this must be tempered with the realization that
the forces involved must remain measurable.
It is also
important that the wave height be kept fairly small to be
consistent with the linear strip theory.
The paper by Strom-Tejsen, et. al.[29] contains a
detailed description of an experimental program carried out
at NSRDC to determine the added resistance in head seas of
a range of Series 60 models, a destroyer, and a high-speed
form.
Other extensive head sea. experiments have been done
by Gerritsma and Beukelman
[5
and the University of Osaka
Beck and Wang at M.I.T.[2,31]
1
. Sibul has also done a
great deal of work with Series 60 models at the. University
of California [28], and some of his results are shown in
Figure 3.
It is intended that this figure will be representative
(66)
9
s)
ccq
H
En
W
kD_
[0
H
U)
01
3
2
I
0
1
2.0
3.0
t-- SLonger Waves
FIGURE 3:
4.0
5.0
6.0
Encounter Frequency,w/ L
Ig
ADDED RESISTANCE/HEAD WAVES
-67of the fact that Equation (86) provides a very good correlation with experiments for the familiar case of head seas.
The plot shows experimental results for a Series 60 model
very similar to the Mariner at nearly the same Froude number.. The agreement with (86) is well within experimental
error throughout most of the range of practical wavelengths.
However, acceptable theoretical prediction methods for added
resistance in head seas have been available for several
years, and this report is primarily intended to address the
more general problem of second-order force in oblique seas.
As soon as the head seas restriction is lifted, the
experimental difficulties involved become almost insurmountable.
A large basin must be available to achieve the
desired range of heading angles, furthermore the model must
now ideally be allowed complete freedom of motion, making
measurement very difficult.
Many other complexities might
be enumerated, but it is perhaps sufficient to say that
very little oblique seas second-order force data is presently
available.
Spens - and Lalangas have done some work on a
Series 60 model at Stevens, measuring drift force and
yawing moment
[2 8 ].
a container ship
Also, Hosoda presents a few results for
Journee
12 ]
did work at Delft with
another container ship in following seas.
The work done at Stevens has already been employed by
-682
Salvesen for comparison in his paper [28]
.
Since comparison
with Salvesen's calculations will later be presented, the
Stevens results are not reproduced here.
Furthermore, ship
characteristics were not readily available for the other
cases.
Consequently, it was decided that an oblique seas
experiment should be carried out in order to provide a
basis for theoretical comparison in this report.
The only
work of this nature that could be done in the M.I.T. ship
model towing tank involved near beam sea cases at zero
speed.
The zero speed restriction is obvious since the
carriage and the generated waves must run parallel to the
long axis of the conventionally shaped tank.
The beam seas
restriction was intended to minimize interference between
the outgoing damping waves of the ship and the tank walls.
An existing fiberglass Mariner model (L.B.P. =5.42')
was employed for the experiment. It was mounted beamwise
in the tank and connected to the carriage by a heave staff
and a roll bearing.
restrained.
All other modes of motion were thus
The M.I.T. tank is 108' long, 8-1/2' wide, and
4' deep, and the model was positioned at approximately the
halfway point.
Waves were generated by a pivoted hydrauli-
cally driven paddle.
Only regular sinusoidal waves of
varying length were used.
Instrumentation consisted of a dynamometer-force block
-69fixed transversely in the model to measure drift force, a
transducer on the heave rod, a transducer on the roll
bearing, and an electrical resistance wave probe for
measuring incident wave height.
All the data was taken to
a Sanborn multi-channel chart recorder which provided a
real-time visualization of the signals.
Data was taken at
regular wavelengths ranging from about one-half to about
four times the ship length.
Smaller waves could not be
generated, and longer ones were pointless because of the
'deep water wave'
Newman states
[2 3 1
assumption inherent to the theory.
that if the depth over wavelength ratio
becomes less than 1/2, the 'deep water wave' assumption
begins to break down.
This affects the incident wave
potential. formulation (44).
For a wave in the tank four
times the ship length, the ratio is less than 1/4.
Throughout the experiment wave height was kept constant at about 1.25 inches.
This represented a compromise
between the desires for small wave heights for linearity
and large wave heights for measurable forces.
At the most,
the mean forces measured represented a few tenths of a
pound, and the motions were attimes unavoidably large.
The experimental procedure consisted of sending a regular
wave train toward the model and taking data until a steady
state was obtained in all the responses.
The data record
-70was stopped when the incident regular wave train became
contaminated (e.g. by reflected waves from the towing tank
beach.)
The mean drift force was obtained by graphical
measurement using the output from the chart recorder.
The
wave amplitude (necessary for nondimensionalization) was
similarly estimated.
The resulting experimental points are
shown as circled-dots on Figures 4, 5, and 6 for the three
heading angles investigated (8=90*, 1050, and 750, respectively).
Two drift force computations are also shown on each of
the graphs.
The solid line corresponds to the full five
degrees of freedom prediction, (Eq. (86), j=2,..6)while the
starred line represents a calculation in which the computer
program was artificially restrained to represent a rollheavetWo-degree of freedom, system. (Eq. (86), j =3,4). The
computer was run using a higher metacentric height (Scale,
8') and a smaller roll radius of gyration (Scale, 11') than
the real ship because it was very impractical to adjust the
measured model characteristics.
At any rate, this made
little difference in the predictions, because these changes
impacted most heavily on roll response, and roll is always
a small contributor to the second-order force calculated
in (86).
Examination of the three graphs will show that the
expected level of agreement is obtained for all wavelengths
*,see Ape'exwx c.
Il'.'
O( /
(rI/,qofd)/aDO,,oa
m
LdO
Idiu
'qa o
c,, r-'
72)
72)
.-L
00
6)
r,
cc)
f-i
L.€)
r-I
r-I
0,
m
,--
(a/,9,7obd)/.guo.
iiaau Iqcl
CN
r-
73)
O
O
O
O
(I/,9,70bd)/aD'daoa
'LA
acD
-74-
between 0.5 and about 1.4 times the ship length.
particularly true for the 8
=
This is
1050 case (Figure 5), where
the points fall nicely on the line predicted by the computer
using the two-degree of freedom assumption.
It can also be
seen that for wavelengths greater than about 3.5 times the
ship length, the mean forces go to zero as the computer
predicts.
However, a serious surprise is contained in the data
for the wavelengths between about 1.5 and 3.5 times the ship
length.
In this area on all three graphs, the computer
predicts near zero mean force, but the measured results
'blow-up' as the wavelength increases.
A solution or a full
explanation for this theoretical discrepancy is, as yet,
unavailable.
It can be noted that the model attains roll resonance
in the area of X/L =1.7 and sustains a very large roll angle
(no less than plus or minus 15 degrees).
tudes persist up to X/L= 3.0.
Large roll ampli-
Furthermore, when the roll
natural period of the ship was shifted by using outrigger
weights, the location of the problem area shifted with it.
Currently, it is felt that the observed behavior does not
represent a problem with the measurement equipment.
Rather,
it is supposed that this represents a nonlinear interaction
associated with the large roll angle.
This would explain
-75the lack of correlation with any linear, small motion
theory.
An appreciation for the experimental difficulties
involved in these measurements can be obtained, when it is
realized that the measurements described above as 'blowingup' represent a few tenths of a pound on model scale.
In
closing, it should be emphasized that good agreement was
obtained over a sizable portion of the range where linear
theory would be expected to apply.
B.
Comparison With Other Theoretical Results
In the preceding portion of this chapter, Salvesen's
second-order force was examined in some detail in order to
gain insight into the behavior of its various components.
Then it was compared with some existing and some new experimental results.
The outcome was generally quite favorable
in the area where linear theory would be expected towork,
but some serious questions were also raised concerning
applications in the area of roll resonance.
The next available step is to compare the results of
the theory in the M.I.T. 5-D program with the results from
other programs and linear theories.
To date, the only
second order force computational results published for
oblique seas have been offered by Salvesen[ 2 8 1 and Loukakis
(Ref. [16]).
Salvesen also programmed Equation (86), but he did so
-76within the framework of the Naval Ship Research and Development Center Ship Motion and Sea Load Program.
This program
uses a different scheme for creating the two-dimensional
sectional potentials.
It also incorporates several other
differences in numerical techniques, so his results (called
'Salvesen'and labeled NAV) can be used to assess the impact
of computational procedure on the theoretical predictions.
Loukakis presents results balled 'Loukakis' and labeled
(NTUA) wnicn are based on a different theory. His method was
briefly mentioned in Chapter II.
It is based on the
original work of Gerritsma and Beukelman, and it represents
a totally different "radiated energy" formulation of the
problem.
However, this new theory was implemented on a
version of the M.I.T. 5-D motions program, so the two
theories can be compared without undue concern for the
impact of numerical techniques.
Admittedly, there will be
discrepancies because Loukakis made his computation using
only vertical motions in order to avoid a roll resonance
problem.
To summarize, it will be possible to make com-
parisons with the same theory in a different motions program
(M.I.T.-NAV) and a different theory in the same motions
program (M.I.T.-NTUA).
The ship for these comparisons is
the Mariner-type fast cargo vessel already described.
Figure 7 shows the predicted impact of forward speed
on added resistance in head waves.
Results are plotted for
-77-
c'
N0a
0-)
H
0
z
0.4
0.8
1.2
1.6
2.0
X/L
FIGURE 7:
MARINER- ADDED RESISTANCE HEAD WAVES
(ý= 180 ) VARIOUS SPEEDS
-78both the M.I.T. program (MIT) and the Salvesen program (NAV).
The magnitude of the peak appears to increase with the
speed, except at speeds near zero.
The higher speeds also
tend to reach their associated peak values at longer wavelengths coinciding with the heave and pitch peaks.
This is
very important when the graph is considered as a response
operator for use in the spectral analysis of irregular sea
states because most of the ocean energy is in relatively
long waves.
The two computational methods compare well
for the shorter wavelengths before the peaks; however, some
discrepancies can be seen in the medium wavelength area
beyond the peak.
No explanation can be offered for this
since the (NAV) program details are unavailable to the
author.
There is also some difference in the actual peak
value predicted, but it is very difficult to obtain enough
data points to truly characterize the peak.
Of course, for
long waves, all the results go to zero.
Figures 8 and 9 show the general effect of a heading
change in bow waves on the added resistance component of
the second order force.
Figure 8 shows Salvesen and M.I.T.
calculations, and Figure 9 gives Loukakis and M.I.T.
Both
Figures 8 and 9 were run at a speed of fifteen knots.
The
graphs show how the second order force peak comes in much
shorter waves as Beta goes to ninety degrees, following
(79
1
00u
C)
o_
H
H
0.2
0.6
FIGURE 8:
1.0
1.4
X/L
1.8
MARINER - ADDED RESISTANCE (SPEED - 15 knots)
Various Headings
-80-
cq
?5
Uri
0-
H
0
0.4
0.8
FIGURE 9:
1.2
1.6
2.0
MARINER - ADDED RESISTANCE
(SPEED - 15 knots); Various Headings
-81closely the movement of the heave motion peak.
The M.I.T.-
Salvesen correltaion is much the same as in Figure 7:
good
correspondence for low wavelengths (except for 8 = 1050) ,
peak magnitude differences and M.I.T. overpredicting NAV
results for medium length waves.
The Loukakis program
predicts much higher peaks than the M.I.T. results for near
head seas, but lower peaks for near beam seas.
It also
gives slightly different peak locations, and noticeable
differences in the medium wavelength area.
It is interesting
to note that the character of the three solutions is
generally the same, but they are by no means the same.
Also, the M.I.T. program predicts more resistance at 8 = 1500
than in head waves (presumably due to contributions from the
three extra degrees of freedom).
Figure 110 illustrates the effect of forward speed on
drift force in oblique bow waves (a = 1200).
Nondimensional
drift force (defined by aDF = drift force/pgg 2B2 /L) is
shown as a function of wavelength over ship length.
sults are shown for the M.I.T. and Salvesen schemes.
ReHigher
forward speed again increases the peak drift force, as was
observed for the added resistance in Figure 7.
Increasing
speed also shifts the peak value to longer wavelengths.
Furthermore, (from the figures in Reference 28),
there is
an indication that the Salvesen curves turn sharply up in
-82-
•n
c'J
ca
N
U
0
H
I-I
a2
4
2
0
X/L
FIGURE 10:
MARINER - DRIFT FORCE
= 1200; Various Speeds
-83the shorter waves; this does not occur in the M.I.T. calculation.
fied.
The reason for this cannot be explained or justiThe usual differences between the two results in
medium wavelengths show clearly.
In general, it could be
said that the correlation is improved by a speed increase.
It would perhaps be useful, at this point, to interject
a quantitative idea concerning the real force magnitudes involved in these nondimensional numbers.
For example, the
peak added resistance at twenty knots (about '12' in Figure
7) in 5-foot amplitude waves, corresponds to a force of
about 200,000 pounds on the real ship, a figure roughly
on the order of the calm water resistance at the same speed.
The peak drift force in Figure 10 at fifteen knots
(about
'17') would translate into almost 300,00 pounds in 5-foot
amplitude waves.
Of course, these numbers are for one
particular highly tuned regular wave frequency.
The spec-
tral analysis to be outlined in Chapter V will lead to much
lower mean values but the significance of the numbers involved can surely be appreciated.
Figures 11 and 12 present the impact on drift force
of a heading angle change in bow waves at fifteen knots.
The first of the two presents M.I.T. and Salvesen (NAV)
results while the second gives M.I.T. and NTUA predictions.
As expected from the added resistance curves of Figures 8
(84
20
18
N
Na
00
H
R4
b
•
0.2
0.4
0.6
0.8
1.0
1.2
1.4
V/L
FIGURE 11:
MARINER - DRIFT FORCE (SPEED - 15 knots)
Various Headings
nt)VrosHaig
SED-1
FOC
DRIF
IE
FIUE1:
N
C4
o5
En
u
C0
rz4
H
P4
0.2
FIGURE 12:
0.4
0.6
0.8
1.0
1.2
X/L
1.4
MARINER - DRIFT FORCE (SPEED-15 knots) Various Headings
-86and 9, the peak values occur at larger wavelengths and have
decreasing magnitudes as the waves come on the bow.
The
M.I.T.-Salvesen comparison becomes better as the heading
The usual peak and medium wavelength dif-
angle increases.
ferences appear except at P =1500 where the Salvesen peak
is higher.
The NTUA results illustrate much the same ten-
dencies, and generally come closer to the Salvesen computationthan the M.I.T. one.
The final two graphs (Figures 13 and 14) delve into
an area that is the subject of considerable controversy in
ship motion theory -- following seas.
Salvesen did not give
any results for heading angles less than 1050 in his 1974
paper, so the only available comparison is provided by
Loukakis.
Figure 13 shows negative added resistance for a range
of headings at fifteen knots.
Both sets of results are
fairly consistent in the longer wavelengths, showing a
tendency to decrease the prediction with increasing heading
angle.
However, the M.I.T. results tend to wander around
the axis for
= 75
°
and 8 = 600.
be attributed to numerics.
by the two methods differ
This can most likely
The peak magnitudes generated
considerably, but anyway, the
existence of any peak, at all, has not as yet been established
experimentally.
The container ship experiments of Journee
FIGI
18
16
e.
ao
N
H
CI
-88-
00
r-I
l
r-I
Nl
r-I
0
I
'-q
(a/,,9,,od)/aD~doa
~
iaia
laci
-89(Ref. 12) show this quite clearly since he obtained
positive added resistance throughout the entire wave range
in following seas (0 = 0).
Loukakis attempted a computa-
tion based on the container ship and received results
similar to those of Figure 13 [ 16].
Further doubt can be
placed on these peak predictions by virtue of the fact
that they occur very near zero encounter frequency for the
lower heading angles.
As the ship passes into the negative
encounter frequency region, the curves all take a large
'jump' to a positive added resistance.
This jump does seem
to occur in a somewhat more orderly fashion in Loukakis'
work.
Additional insight into the problem can be obtained
by considering Figure 14 which shows positive drift force
at fifteen knots for the same following seas cases and for
beam seas.
The results show a large amount of inconsistency
in the magnitudes.
There is no clear-cut decrease of
force with decreasing heading angle for either set of data.
All the same problems mentioned for Figure 13 are, of
course, present for all the following sea headings.
In
addition, the beam seas case gives a rather disappointing
comparison between the two theories.
The M.I.T. and NTUA
results do agree well in beam seas down to a wavelength/
ship length ratio of 0.5; then they diverge rapidly.
The
experiments of Figure 4 might be used to illuminate this
-90matter, but unfortunately they could not be extended to
smaller waves.
In summary, the three theories presented compare very
well with each other over the full speed range of the Mariner
for heading angles greater than 1050 (bow waves).
In most
of these cases, the M.I.T. results show a larger secondorder force in medium wavelengths than the other two methods.
The head seas results of Figure 3 show experimental points
that fall slightly above the M.I.T. predictions in the long
wave range, so this may well be an asset in the M.I.T.
version of the theory.
However, experimental data for any
oblique bow wave case is noticeably absent, and final conclusions cannot yet be drawn for this region.
Agreement between the theories becomes noticeably
poorer in the beam seas regime.
Loukakis' decision to ignore
the lateral motions contribution may be a factor.
'weak-scatterer' potential assumption (See Eqs.
Also, the
(65) and
(66)) incorporated in the Salvesen theory is expected to
cause problems in this area, but, again, the lack of
experimental work prevents a conclusion.
The beam seas
experimental work that was done (Figs. 4,5,6) raised
serious questions concerning the validity of any linear
theory in the presence of large roll amplitudes.
The
Salvesen theory, in particular, is quite insensitive to the
roll mode of motion.
This mode makes little contribution
-91to the seond-order force, even when its predicted amplitude
is large.
On the subject of numerical computation problems, the
M.I.T. version of the Salvesen theory exhibits some oscillations at fifteen knots in quartering seas (Figs. 13,14).
Some of the Salvesen (NAV) results indicate a sharp upward
turn in the second-order force for short bow waves (e.g.,
Fig. 8 or 10).
This is noticeably absent in the M.I.T.
computation.
In following seas, both the NTUA and M.I.T. calculations
exhibit difficulties.
Neither shows any real consistency,
and the agreement between either theory and experiment is
poor.
This problem is discussed in some detail by Loukakis
(Ref. 16).
He shows how the total added mass and damping tend to
zero, while some of the sectional coefficients go to
infinity, as the encounter frequency decreases.
This decay
will take on one of several forms, depending on the order
of magnitude relation between the encounter frequency and
the incident wave frequency.
All these tendencies are
artificially introduced by the strip theory assumptions.
Nevertheless, the equations of motion produce 'reasonable'
(but not necessarily 'accurate') predictions.
well be due to the hydrostatic terms dominating
This may
-92the equations, making the damping and added mass insignificant.
The linear second-order force predictions are a different
matter, though.
They are dependent to a large extent on
the same sectional potentials that cause the problems in
the motions computation.
Very near zero encounter frequency, it has been observed
that several of the components of the Salvesen theory grow
very large; and it is only the fact that they have opposite
signs that keeps the prediction bounded.
In conclusion, it
is obvious that much work remains to be done on the following
seas problem.
-93V.
DESIGN ANALYSIS
An ocean seaway represents a true random process.
The
wave patterns are continually changing in time and space,
and their complexity defies any deterministic analysis.
In
view of this, the results of Chapter IV for regular sinusoidal waves must be considered as only a first step in
a procedure that will lead ultimately to a statistical
formulation of second order wave force in the real ocean.
Conceptually, the method to be used involves obtaining
a Fourier integral representation of the ocean.
The actual
sea surface is represented by superimposing many infinite,
in theory, regular waves of different periods, random phase
and infinitesimal amplitude.
The end result is an amplitude
or energy spectrum for the sea, depciting the energy in
the ocean as a function of frequency.
For a linear system,
the principle of superposition can be invoked
so the re-
sponse to a number of input regular waves is the sum of the
responses to each, individually.
Therefore, if the response
of the system can be obtained as a function of frequency,
then this can be combined with the sea spectrum to yield
a response spectrum.
Under certain other probabilistic
assumptions, predictions regarding, for example, the
highest likely response or the probability of an event, can
-94then be made using the response spectrum.
This theory of random excitations for linear systems
has been used to good advantage to provide a design tool
in the area of ship motions, and many good references on
the technique are available
[4 ] .
However, it is not immedi-
ately obvious that this technique can be used for the second
[30]
order force. Vassilopoulos
proved it rigorously using
nonlinear system theory
[3 0
, but Maruo first gave an
intuitive derivation [7 ] that is repeated below.
If the sea can be described by a long-crested (unidirectional) energy spectrum, E(w), the following will be
true for each infinitesimal frequency component, where do
is the infinitesimal wave amplitude.
E(u) ct ý (cc)2
(87)
Futhermore, if the second order force is purely proportional
to wave amplitude squared for any frequency, as originally
stated in the Introduction, then it follows that:
AF/(d oo-
(w)
where F represents the second order force.
(88)
(Note that
f(W)/(pgb2/L) is the quantity plotted as a function of
-95wavelength (frequency) in all the results of Chapter IV.)
Combining (87) and (88) for a frequency increment, gives:
&S(wE(u) cw.
5
sF
(89)
It follows from probability theory that the integral of
the second order force spectral component (89) over all
possible frequencies will represent the expected mean
second order force for a long period of time in the longcrested seaway,
0O
5(4E)(u.) CLLs
F
(90)
(
The result (90) was developed by Maruo using a spectrum
E(u2),
based on the full wave amplitude (See (87)).
The
spectrum definition which is most commonly used in practice
is based on one-half the amplitude squared.
spectrum
•
(u
),
the formula (90)
Calling this
becomes,
cO
F= a
(Luo
(o (4dw.5
(91)
This is the formula given in Strom-Tejsen, et. al. [29]
Should a more rigorous derivation be desired, Vassilopoulos
-96should be consulted. [30]
If the long-crested unidirectional
seas assumption (made before (87)) is relaxed, Maruo showed
(and Newman
22 ]
recently rederived rigorously) that the
mean second order force in short-crested seas can be approximated in this manner,
AT
,8)co
pcs
'jd
(92)
(h)/3-e)
In this expression, @ is the principal wind direction or
primary directional source of waves (defined the same as
8 in Chapter III.
F(B) is a mean force for a particular
ship heading angle, developed from (91).
The well-known
'cosine-squared' spreading function has been employed to
distribute the effective energy in the sea placing it
primarily in the waves coming from the principal wind
direction.
Equation (92) represents a double integral
over frequency and propagation direction, but it is still,
conceptually, a superposition of many small sine waves.
Returning, for now, to long-crested seas, (91) was
found to be well suited to inclusion in the M.I.T. 5-D
program (which already calculated many statistical quantities associated with ship motions).
Response operators in
the form of (88) were easily developed from the results of
-97regular wave calculations.
Then these were combined with
the various sea spectra and numerically integrated in subroutine STATIS according to (91).
A separate auxiliary program that performed motions
calculations in short-crested seas was already in existence.
This was modified so that it could receive the second
order (long-crested) mean force output from the 5-D program
(according to (91).
forth in
(92),
It
yielding,
then performs a calculation set
finally, a predicted long-time mean
added resistance and drift force in a short-crested, irregular seaway.
In order to test the feasibility of employing (91) and
(92) in the design process, six wave spectra were chosen.
The six are shown in Figure 15.
Spectra 1, 2, 3, and 6 are Peirson-Moskowitz fullydeveloped one-parameter representations from the formula:
e
_
22
where A= .0081g 2 and B=33.56/h/3.
(93)
The quantity, hl/
represents the significant wave height (the average height
of the one-third highest observed waves).
Spectra 4 and 5
were developed by use of the Bretschneider, two-parameter
representation:
98)
0
r-I
3
-I
D
--
-I
H
FZ4
0
O
0
m
DOS-
N
,T. ' (sm) ;•
r-
-99-
(94)
where Wp is the desired location of the spectral peak.
For
spectrum 4, the peak was chosen at a much lower frequency
than that. generated by the Pierson-Moskowitz formula for
the same hl/.
This corresponded to a decaying seaway,where
the energy is found in longer waves.
The peak for spectrum
5 was located at a much higher frequency, thereby simulating
a developing sea, taking energy from the wind in the short
wave region.
These six spectra were used in (91) in combination with
response operators generated by the theory of Chapter III
for regular waves.
The operators were calculated for a full
range of headings at the service speed of twenty knots.
The resulting long-crested seas predictions were then used
in (92) to provide information on second order forces in
the corresponding short-crested seaway.
The results obtained from spectra 1, 2, 3, and 6 are
shown in Figures 16 and 17 for added resistance and drift
force, respectively.
In these plots, the force is non-
dimensionalized using significant wave height squared.
Figure 16 shows mean added resistance as a function
of significant wave height for the Pierson-Moskowitz for-
(100
1.0
0.8
0.6
e'J
0.4
C''(*
H
N~
0.2
U
Hr
0
-0.2
-0.4
FIGURE 16:
MARINER IRREGULAR SEAS RESULTS
(0o1)
1.2
1.0
0.8
c'4
N
oa
i·-4
~I
H
0.6
0i
0.4
0.2
FIGURE 17:
MARINER IRREGULAR SEAS RESULTS
-102mulation.
The first important point revealed is that the
added resistance in long-crested seas, for 8 =150* exceeds
that of 8 =1800 (head wave) throughout the entire wave
height range.
The reason for this must be found in the
relative locations of the two response peaks and the spectral peak.
In smaller waves, the 8
=
120 0 curve outreaches
both the higher beta angle predictions.
The negative pre-
dictions for the following waves are created by the negative
response operators (Shown in Fig. 13 for 15 knots).
Since
these negative response operators have not, as yet, been
confirmed by experiment, any of these results for irregular
following seas should be used very carefully.
be noted that the plot for B =
1 80*
It should
can be compared with
the Series 60, CB =0. 6 0 experimental and regression analysis
results for irregular head seas given by Strom-Tejsen
[2 9 ]
The short-crested seas calculation (92) for the 1800 'wind'
direction tends to be greater than the 1800 long-crested
prediction.
This is a result of extra energy being channeled
into near head-sea headings which also have very large
added resistance operators.
The 1500 'wind' direction curve
falls considerably below the long-crested 1500 plot because,
in this case, the energy is being shifted to headings with
lower added resistance responses.
Figure 17 gives the mean drift force for the same
-103heading range.
The beam seas case produces the biggest
drift in smaller waves, but, again as a consequence of peak
positions, the 8=1200 curve dominates in higher seas.
relative amount of drift produced at hl 20by the 8 =
The
1 500
case is surprising since it almost matches the beam seas
case.
The short-crested seas calculation for a 'wind'
direction of 1500 overshadows its long-crested companion
in smaller waves, then falls beneath in higher waves.
This
can be predicted from a line of reasoning similar to that
given above.
In all these plots, the reader is cautioned to regard
the results for beam and following seas with some suspicion
in view of the results of Chapter IV.
For example, the
sharp dip and peak in the near beam seas experimental data
(Figs. 4, 5, 6),
was measured directly in the way of most
of these spectral peaks.
This underscores even more heavily
the need for further research in this area.
Finally, Figures 18 and 19 show the effect of spectral
peak location on the added resistance and drift force.
Spectra 3, 4, and 5 were used as being representative of
fully developed, decaying, and developing seas.
For the
bow waves;, the decaying and fully developed seas tend to
produce the most drift force or added resistance.
The A=1500
and B =1200 long-crested added resistance cuves plot higher
(104)
1.0
0.8
0.6
c'J
0.4
H
U]
0.2
C12
H
0
-0.2
-0.4
FIGURE 18::
MARINER IRREGULAR SEAS RESULTS
(105)
1.2
1.0
0.8
pq
c4
0.6
p
a.
o_
rX4
N
Um
0
Hr
H
0.4
[Ii
0.2
FIGURE 19:
MARINER IRREGULAR SEAS RESULTS
-106than the head waves case through most of the range.
The
short-crested calculations have the same general character as
their long-crested counterparts, but not the same magnitude.
For the Mariner at service speed, the force maximum for bow
waves seem to occur near the fully developed seas.
The drift force is dominated by the B =
1 200
case for
decaying and developed seas, but the beam seas case is very
large in the developing sea.
The short-crested seas calcu-
lation follows quite closely the comparable long-crested
case.
To provide an assessment of the actual forces in the
different sea states without dealing with the significant
wave height, the predicted forces, in pounds, for all the
points plotted in Figures 16 through 19 are given in Table I.
The maximum added resistance occurs at a heading of
1500, in 20-foot seas and its magnitude represents around
three quarters of the calm water resistance (inferred from
the Mariner shaft horsepower requirements
200,000 pounds).
margins in
[2 5 ]
to be about
Before the requirements for service
ship propulsion are rewritten, it should be
pointed out that these high waves put the linearity assumptions of the theory to a severe test.
Also, there may be
other factors (e.g.structural integrity, motions, etc.)
that prevent operation of a 500-foot ship at service speed
-107-
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-108in such a sea state.
In a more modest sea (say h1 /3= 10')
the maximum added resistance occurs at a heading of 1200
and it represents a more modest twenty percent of the calm
water resistance.
The drift force reaches its maximum in
the developing beam seas.
A similar value is obtained in a
20-foot fully developed sea state at a heading of 1200.
However, the drift force at speeds near zero is more
likely to be of interest to the naval architect.
It has been demonstrated in this section that the
techniques of spectral analysis can be used in combination
with a regular wave theory to gain quantitative insight
into the second order force acting on ships at sea.
On the
basis of this brief study, it is already possible to conclude
that the a prior assumption that irregular head seas will
represent the worst case for added resistance is unjustified.
It is encouraging for the simplicity of calculation that,
at least for Mariner-like ships, fully developed seas
appear to yield the maximum mean added resistance responses.
The problems associated with regular following seas were
seen to carry naturally into the spectral analysis.
These
difficulties and the doubts raised by the beam seas
experimental work should make designers wary of the full
application of the short-crested sea analysis (92).
However,
the author is optimistic that (92) will be very useful
-109in connection with the Salvesen second order force theory
if added resistance predictions are desired for primarily
head winds (e 180o).
Finally, the extreme sensitivity
of the calculated mean force to the input ocean spectrum
can be easily seen.
-110VI.
CONCLUSION AND RECOMMENDATIONS
This report represents an attempt to develop a design
tool for use by the naval architect in calculating the
second order wave force in areal seaway on a proposed
vessel.
This force impacts heavily on the design in that
it is a major source of extra resistance and/or side slip
that must be counteracted by the power plant in a seaway.
The main thrust of the effort has been to develop a reliable
computational method requiring only basic information about
the ship (See Appendix) as an input.
Second order force
predictions were desired for any heading angle of the ship
at any speed in regular waves of any frequency.
The present
study starts by introducing and defining the problem in the
context of the general ship design process used by naval
architects.
Then the previous theoretical work on second
order forces was outlined, and it was shown how several
methods of prediction have been recently developed by hydrodynamicists for use in oblique seas.
One of these theories,
due to Salvesen, was derived in the context of modern strip
theory of ship motions.
This theory was implemented within the M.I.T. 5-D ship
motions program.
Calculations were performed using the
Mariner-type cargo ship in regular waves, and a brief
-111investigation of the character of the final equation was
conducted.
The next step involved comparison with existing
regular head sea experimental data.
Due to the lack of
oblique seas experimental data, an experiment using a
Mariner model for net drift force measurements in near beam
seas was conducted in the M.I.T. Towing Tank.
When this
data was compared with the computer predictions, it was
partially supportive, but it also raised some important
linearity questions.
A theoretical comparison was then made using results
from the same theory in a different motions program and a
different theory in the same motions program.
The results
were encouraging for heading angles between 1800 and 900,
but the predictions were shown to be subject to question
in the following waves.
Next, the theoretical extension
of the regular wave computer results to an arbitrary longcrested or short-crested irregular seaway was outlined.
This extension is crucial to the usefulness of the method
as a design tool.
The resulting spectral analysis tech-
nique was implemented in the M.I.T. 5-D program, and a
brief design analysis of the Mariner was performed using
six representative ocean spectra and a range of headings
at the service speed.
The following important points can be presented as a
-112result of this study:
1.
The linear estimate of second order force used
in this study represents a very useful design
tool in combination with the spectral analysis.
2.
It is very important that a full range: of bow
wave headings be considered in the spectral analysis in order to define the true maximum added
resistance.
3.
Further experimental work in oblique bow waves
is urgently needed to provide comparisons with
the available theories.
4.
The problems encountered in obtaining reasonable
calculated results in following seas indicate
the necessity for further theoretical developments
in this area.
Further experimental work would also
be very helpful.
5.
Every effort must be made to gain an understanding
of the source of the phenomenon observed in
regular beam wave experiments -- particularly in
view of the fact that this second order force
'jump' occurs very near most spectral peaks.
6.
As a result of point 4 above, the short-crested
seaway equation would be most usefully applied, at
this time, for waves propagating in a direction
generally opposite to the ship (head winds).
-1137.
The second order yaw moment, which can also be
calculated in an extension of the theory presented
(Ref. 28), should be developed fcr the M.I.T. 5-D
program.
This should not be a high priority effort,
however, since there are very few theoretical or
experimental comparisons to make.
-114REFERENCES
1.
Ankudinov, V. K., "The Added Resistance of a Moving
Ship in Waves," International Shipbuilding Progress,
Vol. 19, 1972.
2.
Beck, R. F., "The Added Resistance of Ships in Waves,"
Massachusetts Institute of Technology, Department of
Naval Architecture and Marine Engineering, Report 67-9,
1967.
3.
Chryssostomidis, C., Loukakis, T., Steen, A.,
"The
Seakeeping Performance of a Ship in a Seaway," to be
published by the U.S. Maritime Administration.
4.
Comstock, J. P., Principles of Naval Architecture,
Society of Naval Architects and Marine Engineers, 1967.
5.
Gerritsma, J. and Beukelman, W., "Analysis of the Resistance Increase in Waves of a Fast Cargo Ship,"
International Shipbuilding Progress, Vol. 19, 1972.
6.
Hanaoka. T., "Non-Uniform Wave Resistance," Journal of
Zosen KiQokai,.Vol. 94, 1954.
7.
Hanaoka, T., et. al., "Researches on the Seakeeping
Qualities of Ships in Japan", Chapter 5: Resistance
in Waves, Society of Naval Architects of Japan, 60th
Anniversary Series, Vol. 8, 1963.
8.
Haskind, M. D., "Two Papers on the Hydrodynamic Theory
of Heaving and Pitching of a Ship," S.N.A.M.E. Technical
and Research Bulletin, 1-12.
9.
Havelock, T. H., "The Drifting Force on a Ship Among
Waves," Philosophical Magqazine, Vol. 33, Series 7, 1942.
10.
Hosoda, R., "The Added Resistance of Ships in Regular
Oblique Waves," Selected Papers from the Journal of the
Society of Naval Architects of Japan, Vol. 12, 1974.
11.
Joosen, W. P. A., "Added Resistance of Ships in Waves",
Proceedings of the 6th Symposium on Naval Hydrodynamics,
Washington, D.C0*., 1966.
-11512.
Journee, J. M. J., "Motions and Resistance of a Ship
in Regular Following Waves," Laboratory for Ship
Hydrodynamics, Delft, Report 440, 1976.
13.
Korvin-Kroukovsky, B. V. and Jacobs, W. R., "Pitching
and Heaving Motions of a Ship in Regular Waves," Transactions, S.N.A.M.E., Vol. 65, 1957.
14.
Kreitner, J., "Heave, Pitch, and Resistance of Ships
in a Seaway," Transactions of the Royal Institute of
Naval Architects, 'London, Vol. 87, 1939.
15.
Lee, C. M. and Newman, J. N., "The Vertical Mean Force
and Moment of Submerged Bodies Under Waves," Journal of
Ship Research, Vol. 15, 1971.
16.
Loukakis, T. and Sclavounos, P., "Some Extensions of
the Classical Approach to Strip Theory of Ship Motions
Including the Mean Added Forces and Moments in Oblique
Regular Waves," Submitted February 1977 to Journal of
Ship Research.
17.
Marou, H., "The Excess Resistance of a Ship in Rough
Seas," International Shipbuilding Progress, Vol. 4,
1957.
18.
Newman, J. N., "The Damping and Wave Resistance of a
Pitching and Heaving Ship," Journal of Ship Research,
Vol. 3, June 1959.
19.
Newman, J. N., "The Exciting Forces on a Moving Body
in Waves," David Taylor Model Basin, Washington, D.C.,
February 1965.
20.
Newman, J. N., "The Drift Force and Moment on a Ship
in Waves," Journal of Ship Research, Vol. 11, 1967.
21.
Newman, J. N., "The Second Order Slowly Varying Force
and Moment on a Submerged Slender Body Moving,- Beneath
an Irregular Wave System," 1973, unpublished.
22.
Newman, J. N., "Second Order Slowly Varying Forces on
Vessels in Irregular Waves," International Symposium
on the Dynamics of Marine Vehicles and Structures in
Waves, London, April 1974.
-11623.
Newman, J. N., Marine Hydrodynamics, Lecture Notes,
Massachusetts Institute of Technology, Third Edition,
1974.
24.
Price, W. G. and Bishop, R. E. D., Probabilistic Theory
of Ship Dynamics, Wiley and Sons, New York, 1974.
25.
Russo, V. L. and Sullivan, E. K., "Design of the Marinertype Ship," Transactions, S.N.A.M.E., Vol. 61, 1953.
26.
St. Denis, M. and Pierson. W. J., "On the Motions of
Ships in Confused Seas," Transactions, S.N.A.M.E.,
Vol. 61, 1953.
27.
Salvesen, N., Tuck. E. 0., Faltinsen, 0., "Ship Motions
and Sea Loads," Transactions, S.N.A.M.E., Vol. 78, 1970.
28.
Salvesen, N., "Second Order Steady State Forces and
Moments on Surface Ships in Oblique Regular Waves,"
International Symposium on the Dynamics of Marine
Vehicles and Structures in Waves, London, April 1974.
29.
Strom-Tejsen, J., Yeh, H. Y. H., Moran, D. D., "Added
Resistance in Waves,"' Transactions, S.N.A.M.E., Vol. 81
1973.
30.
Vassilopoulos, L. A., "The Application of Statistical
Theory of Non-linear Systems to Ship Performance in
Random Seas," International' Shipbuilding Progress,
Vol. 14, February 1967.
31.
Wang, S., "Experiments on Added Resistance of a Ship
Among Waves," Massachusetts Institute of Technology,
Department of Naval Architecture and Marine Engineering,
Report 65-1, March' 1965.
-117APPENDIX A
CHANGES TO M.I.T. 5-D SEAKEEPING
PROGRAM USER'S MANUAL
The program is now capable of computing mean second
order force (added resistance and drift force) for any ship
heading, in addition to calculating ship motions, dynamic
loadings, and events.
The second order force is calculated
using a theory published by Salvesen (1974).
It includes
forces arising from wave-ship motion interaction and wave
reflection.
The second order force routine given here
operates properly only in English unit systems with length
dimensions in feet.
For example, it has been tested in a
system describing the ship in tons and feet; for this case,
it gives output in pounds force.
All the subroutines in the original MIT 5-D seakeeping
program remain in existence, although some have been modified as noted in Appendix C.
RESIST has been changed from
a function subprogram to a subroutine.
The input format remains unchanged.
However, there is
a new option associated with the integer, NADR, of card
set number four.
If NADR =2, only the final totals of the
added resistance and drift force are printed out in dimensional and non-dimensional form.
IF NADR =I, all eleven
components involved in the Salvesen computation (two for
-118each mode of motion and one for wave reflection) are printed
out.
Examples of each of these output forms are presented
in Appendix B.
If NADR =0, no second order force computa-
tions are performed.
The short-crested seas auxiliary program has been
modified to include a mean second order force calculation.
In order to implement this change, one integer called
NADR has been added to card set number one.
It is written
in column 40, following the standard 15 format used by the
original program.
If second order force data is to be read,
this integer should be equal to one.
The mean drift force
and added resistance will be prepared for input by the 5-D
(if requested) in the same way as the mean squares of the
other responses, so card set #3 can still be included just
as it is punched by the main program.
-119APPENDIX B
SAMPLE OUTPUT
B-1: Example of the short form of output for second order
force in regular waves obtained by setting NADR =Z.
This
follows the print out of themotion amplitudes and phase
angles.
The totals shown represent the sum of all eleven
terms of the second order force formula developed in the
main text (See Eq. (86)).
Added resistance is the compo-
nent along the longitudinal ship axis (positive aft).
Drift force is the component along the transverse ship axis
(positive in the direction of wave propagation).
The
dimensional quantities are in pounds force provided XRHO
is given in slugs/ft 3 and provided that the ship is described in English units with feet as the length dimension.
As noted in Appendix A, if this restriction is not met then
the second order force output will be meaningless.
The
nondimensional quantities have been divided by the factor:
pgct2 B2 /L where p is the mass density of the water, g is the
gravitational acceleration, a is the wave amplitude, B is
the ship beam, and L is the ship length.
-120I
I
I
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II
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Z
-121-
B-2:
Example of the long form of output for second order
force in regular waves obtained by setting NADR =1.
This
follows the print out of the motion amplitudes and phase
angles.
Dimensional and non-dimensional quantities are as
defined in B-i.
Here all eleven components that are summed
to give the final Salvesen second order force are shown.
There are two components for each mode of motion; FR-KRL
which represents the Froude-Kriloff or Havelock portion of
the motion interaction,
3jI
(Eq.
(75)
of
the
text),
WVDIFF
which represents the diffraction potential contribution of
the motion interaction,
text).
jD
(Eqs. (78) and (79) of the
The eleventh component is DIFFR.POT.CONT.,, which
represents the wave reflection, 0 ýD (Eqs. (80) and (85) of
the text).
Numerous subtotals are also given, including
the contribution of each mode of motion, the total of all
the IjI s,
the
the total of all the
.jD'sand the
3jI's.
jD's,
and the total of
-122-
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-123B-3:
Example of the long-crested irregular seas output
giving the mean second order force in a long-crested seaway (Eq. (91)).
This is given if both spectral calculations
and second order force calculations are requested.
The
(LBS) notation shown here is only correct if the variable
XRHO in the main program (See Input, Appendix D) is given
in
slugs per cubic foot and the ship
is
described in
English units using feet as the length dimension.
If these
requirements are not met, the computation will be invalid.
The two components, ADD.RES. (x) and DRIFT(Y-AX) are
always in the output.
The spectral amplitudes and asso-
ciated statistical quantities may be obtained (as shown
here) if NSPC f 0 (See Appendix D).
The integer (zero)
that appears at the end of the line, "Response spectrum
for..." appears because the SPIN subroutine (Appendix C)
is used for all the statistical calculations.
The integer
has meaning only in'the bending moment calculations, where
it transmits the station number.
The spectral amplitudes represent the value of the
integrand in Eq. (91) in the text at each spectral frequency (input by the user or given in default by the
program; See Appendix D).
The only statistical quantity
shown that has meaning for the second order force analysis
is the zeroth moment, which is the value of the integral
-124in (91) (half the second order force).
The second moment,
fourth moment, and broadness factor are printed out because use is made of subroutine SPIN already in existence
in the 5--D program.
Of course, these additional statisticdl
quantities are useful, when they are associated with a
motion spectrum in another part of the output.
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-126B-4:
Example of the output of auxiliary program Short-
crest giving the mean second order force in a shortcrested irregular seaway (Eq. (92)).
An entire set of
output including case identification, motions, and second
order force is obtained by specifying NADR =1 in the
short-crest input.
There is no full component print out
option like the one described for the main 5-D.
NADR =0
Either
(and no second order force output is generated) or
NADR =1 and the output shown is generated.
If NADR = 1 in
Shortcrest, then the user must be sure to run data on long
crested seas from the main 5-D that includes second oraer
force computations (i.e. NADR =1 or 2 in the corresponding
main 5-D data generation run).
The (LBS) notation is correct if the variable XRHO
in the main 5-D program (See Input, Appendix D) was given
in slugs per cubic foot and the ship was described in
English units using feet as the length dimension.
If
these requirements are not met, the computation will be
invalid.
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-128APPENDIX C
SUBROUTINES MODIFIED
The following briefly describes each subroutine
modified in order to implement the Salvesen second order
force computation in the MIT 5-D seakeeping program.
A
flow chart of this new version of the program is shown on
the next page so that an understanding of the position and
function of each routine is available.
Further information
on the routines not modified (and consequently not listed
here) can be obtained from the 5-D Seakeeping Program
User's Manual or Reference [3] of the main text.
-129-
NEW 5-D FLOW CHART
Il
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5-D
PROGRAM WITH SAL VEEA/
SeCOn.D
fRDfrD FORCF (1 iOn
-131C-1:
MAIN PG -
This subroutine is the actual main program
for the 5-D; it calls all the required
routines, loops on frequencies, heading
angles, and ship speeds.
Several lines
were added as part of the scheme for
computing the sectional quantity hj (x)
(Eq.
(78d)).
These new lines include
numbers:
(14)
A new common storage
(19)
Some additional output data
(25-28)
A zeroing routine
(100-102)
Three new Write statements
A listing of the modified routine follows
on pages 132-135.
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-136C-2:
INTRPL -
This routine interpolates the hydrodynamic
coefficients for the desired frequency and
calculates Froude-Kriloff and sectional
diffraction forces.
Now it also computes
the sectional quantity hj(x),
(Eq.-(78d))
needed for the second order force calculation.
The new lines include:
14, 15, 16, 26, 28, 30, 37, 42, 43, 44,
69, 72, 82, 84, 86, 88, 90, 92, 94, 97,
109, 110, 114, 118, 122, 125, 129.
A listing of the modified routine follows
on pages 137-140.
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ADDRES -
This is a new routine which computes the
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regular wave due to all five motion components and wave reflection.
This is
performed according to the 1974 theory of
Salvesen.
ADDRES is primarily intended
to handle output computing various subtotals and storing the response operators
for the statistical routines.
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pages 142-144.
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RESIST -
This is a new routine which is called by
ADDRES.
It performs the real calculation
of the second order force outlined in the
theory of Chapter III, and then returns
the resulting values to ADDRES for
manipulation.
A listing of this new routine follows on
pages 146-147.
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-148C-5:
STATIS AND SPIN - These routines perform the statistical calculations necessary for
long-crested irregular seaway response predictions.
They have been
modified to include the calculations
of a mean second order force.
The
new lines in STATIS include:
22, 115-126, 135, 150.
The new lines in SPIN include:
26-31.*
*NOTE: These new lines in SPIN exist to take care of the
problem of a negative value for the variable, SUM.
This
does not occur for motions which have R.A.O.'s which are
always positive.
However, it might occur for second order
force (e.g. following seas where the waves will help to
push the ship).
Since it was desired to use SPIN for the
second order force calculation (just as for all the other
response calculations), it was necessary to provide a way
to avoid taking the square root of a negative number and
causing an error (See line 30).
To summarize, these lines
do not affect the positive motion R.A.O.'s and they exist
only to avoid computer error messages when SPIN is called
-149for the second order force.
The quantities S, S3, S10,
S1000 are not meaningful for the second order force, and
are not written out.
A listing of these two modified
routines follows on pages 150-156.
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-157C-6:
AUXILIARY PROGRAM SHORTCREST
This program uses input prepared by the main 5-D
program (subroutine STATIS) to calculate the mean
responses in short-crested random seas according
to Eq. (92) of the text.
The conventional cosine-
squared spreading function is used.
This program
has been modified to include a calculation of mean
second order force as well as motion responses.
The
new lines include:
5, 6, 17, 58, 59, 71, 93, 148-161, 180, 181, 182.
A listing of the modified program follows on pages
158-163.
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-163-
-164C-7:
MATRIX -
In Chapter IV-A of the main text, it was
noted that, to obtain some of the computer
results shown in Figures 4, 5, and 6, the
5-D program was restrained to roll and
heave (a 2-D system).
This was done by
making a few changes in subroutine MATRIX.
The concept behind the changes can best
be illustrated by letting Djk represent
the term in brackets in Eq. (24); and
letting the terms indicated become zero:
0
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D 4 4 4 =F 4 ,
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equations for a decoupled two-degree of
freedom system.
Of course, this can be
done for any mode or modes of motion
desired.
-165The necessary changes to subroutine MATRIX
are given next.
The new lines include:
26, 27, 29, 41, 42, 44, 45.
A listing of the routine modified for a
2-D system follows on pages 166-168.
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-168-
-169APPENDIX D
LISTING OF MARINER EXAMPLE DATA
This section contains listings of data used in the MIT
5-D program to create the various graphs presented in the
text.
Two data decks are given.
The first is based on
21 stations, and it represents the data used to make the
comparisons in all the graphs except Figures 4, 5, and 6.
These three graphs (for the beam seas experiment) were
created using the second data deck based on 11 stations.
The "towing tank" data cards have a different density,
metacentric height, roll and pitch radius of gyration,
and a different number of stations.
These changes were
made to more accurately represent the model and tank characteristics as measured.
The station number was lowered
to minimize expense.
An explanation of the meaning of each number is
provided by reproducing the portion of the 5-D output that
gives the input data (See D-3).
The input shown includes
a sample selection of regular wave frequencies, ship speeds,
and ship heading angles.
Spectral information may be added
if desired as shown in D-3.
Further details may be obtained
from the 5-D User's Manual already mentioned.
This manual
should be consulted before any runs are attempted.
-170D-l:
Data used for all graphs and calculations
except Figures 4, 5, and 6 (the beam seas
experiments) .
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-172-
-173D-2:
Data used for preparing comparisons with beam
seas experiments (Figures 4, 5, and 6).
4
t,600
C0
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-174-
'D
,l aC
-175D-3:
SAMPLE OUTPUT OF 5-D USED FOR CHECKING INPUT DATA
The variables can be described as follows:
Card 1
(not
shown)
Number of cases (ships)
Card 2
NAM4E_-
Description of case run
Card 3
NSTA
Number of ship stations
NROMS
Number of regular wave frequencies
NENC
Number of headings
NVL
Number of speeds
NMOT
Number of points where motions are
calculated
NSP
Even spacing of ship stations?
0 =yes; I =no
NP
Number of points used on each ship
station
MP
Number of multipoles used in hydrodynamic potential
NTURB
Ship with bilge keels? 1 =yes; 0 =no
HASBK
Vertical motions only or full 5-D
NB
Station where bending moments computed
NBEND
Are bending moment calculations desired?
NWT
Number of weight ordinates
NPCH
Controls printout of various matrices
NFQ
Conveys form of regular wave frequency
input data
NFR
Conveys form of ship speed input data
-176Card 4
Set
Card 5
Set
NSEA
Regular wave calculations only?
NWX
Number of sea states
NSOMS
Number of spectral frequencies?
NS
Default set of spectral frequencies?
NSPC
Controls printout of various statistical quantities
IO
Specifies output device for input to
Shortcrest,
NADR
Controls added resistance calculations
NEVT
Controls calculation of event probabilities
CB
Block coefficient
XLBP
Length between perpendiculars
BEAM
Midship beam
DRAFT
Midship draft
GRAV
Gravitational acceleration
XCG
Longitudinal center of gravity measured
from )
Card 6
Set
VCG
Vertical center of gravity measured
from the waterline
GM
Metacentric height
RYY
Radius of gyration (Y-AX)
RXX
Radius of gyration (X-AX)
RZZ
Radius of gyration (Z-AX)
XZI
Product of inertia (X-ZAX)
RHO
Mass density in ship units
XRHO
Mass density in slugs/ft 3
-177-
Card 7
Set
Card 8
Set
Card 9
Set
NU
Kinematic viscosity
WSURFA
Wetted surface area
UNIT
Input units: English= 0; Metric =1
ORIGIN
Desired origin for motions calculations
ZETAA
Wave amplitude
ALFA
Maximum wave slope
XI
Distance to ship station from
YM
Sectional waterline beam
ZM
Sectional draft
SIGMA
Sectional area coefficient
ZCB
Vertical sectional center of buoyancy
GIRTH
Girth of section
RIFLR
Rise of floor of section
ALPH
Angle between ship side and vertical
at section
IWBK
Determines type of viscous roll damping
BKRAD
Geometric property of bilge keel
BILRAD
Geometric property of bilge
BKGIR
Geometric property of bilge
BKWID
Geometric property of bilge keel
PHI
Geometric property of bilge
PSI
Geometric property of bilge
LIWO
Length of bilge keel
(
-178NOTES:
1)
All of card set 9 are employed in the viscous
roll damping (quasi-linear) calculation of B44.
2)
If structural bending moment or motions
calculations are being performed, Card sets
Number 10 and 11 must be included here as
described in the User's Manual.
Card 12
Set
UOB
Ship speeds, units defined by NFR
Card 13
Set
BETA
Heading angles in degrees, ascending
order
Card 14
Set
OMEGA
Wave frequencies, units defined by NFQ
Card 15
Set
H13
The 1/3 highest wave heights for which
sea state calculations are desired
Card 16
Set
OMP
Peak Spectral frequencies for the corresponding wave heights above. If
blank, fully developed seas will be
used.
Card 17
Set
SPOMS
Spectral frequencies of each sea state
specified (Us). These values can be
given as input (NS 0) or the program
will supply a default set(NS =0) (as
was done here).
Card 18
Set
SPCTM
Spectral amplitudes for the chosen sea
spectrum. This card set may be omitted,
and the program will calculate Bretschneider amplitudes that correspond to
the frequencies in Card Set 17. Otherwise, NS may be set equal to two and
spectral amplitudes may be read in for
any type of spectrum the user may require.
In this case the default Bretschneider
spectrum was chosen.
-179NOTES:
1) Card Set 19 should follow here as described in
the User's Manual if the probabilities of the
various events detailed there ae desired.
-180MARINER SiECT'AtL A•NLYSIS
NSIA NROMS NF.C
21
21
7
NSEA
NWX
NSCM
1
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3.6203
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264.0003
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3 211.2000
184.8000
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