12 DEC. 972
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NA VAL. SHIP RESEARCH AND DEVELOPMENT CENTER
WASHINGTON. D.C. 20034
D 0 Cii H E ii I I C
THE FRANK CLOSE-FIT SHIP-MOTION
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COMPUTER PROGRAM W. Frank and N. Salvesen DEPARTMENT OF HYDROMECHANICS RSEARCH:AND DEVELOPMENT REPORT This document has been approved for public release and sale; itsdistri-bution is Unlimited.
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Report 3289't 1
t7 by BibIioth.eevn
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risch Hoqescioo, DöCUMN1ATE -DATUM:The Naval Ship Research and Development Center is a U.S. Navy center for laboratory effort directed at achieving improved sea and air vehicles. It was formed in March 1967 by merging the
David Taylor Model Basin at Carderock, Maryiand and the Marine. Engineering Laboratory (now Naval Ship R & D Laboratory) at Annapolis, Maryland. The Mine Defense Laboratory (now Naval
Ship R & D Laboratory) Panama City, Florida became part of the Center in November 1967.
Naval Ship Research and Development Center
Washington, D.C. 20007 * REPORT ORIGINATOR C EPARTNEN I OF ELECTRICAL ENGINEERING A000 DEPARTMENT OF MACHINERY TECHNOL.OGY A700 DEPARTMENT OF MATERIALS 1.ECHNOLOGY,}._ A800 OARTMENT OF APPLIED SCIENCE A900 SYSTEMS DEVELORMENT OFFICE 01101 NSROL ANNAPOLIS COIIdANDING OFFICER TECHNICAL DIRECTOR
F
F
}
MAJOR NSRDC ORGANI ZATIONAL COMPONENTS
NSRDC CARDEROS( cO,&ANDER TECHNICAL DIRECTOR DEPARTMENT DF IIYDRONECHARICS 500 DEPARTMENT OF AERODYNAMICS 651 OEPARTNENT OF STRUCTURAL MECHANICS 700 DEPARTMENT OF APPUED MATHEMATICS 800 DEPARTMENT OF ACOUSTICS AND VIBRATION
ME N SR DL PANAMA CITY COMMANDING OFFICER TECHNICAL DIRECTOR
H
H
H
DEPARTMENT OF OCEAN TECHNOLOGY P710 DEPARTMENT OF MINE COUNTERMEASURES P720 DEPARTMENT OF AIRMSRNE MINE COUNTERMEASURES P730 DEPARTMENT OF INSHORE WARFARE AND TORPEDODEFENSE P740 DEVELOPMENT PROJECT OFFICES OH. 50. 80. 90 SHIP CONCEPT RESEARCH OFFICE 01170
DEPARTMENT OFTHENAVY
NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER
WASHINGTON, D.C;20034
THE FRANK CLOSE-FIT SHIP-MOTION
COMPUTER PROGRAM
by
-W. Frank and N. Salvesen
This document has been approved. for public release and sale, its distri-bution is unlimited.
FOREWORD - IN MEMORIAM
The close-fit ship-motion program described herein is primarily the
product of Mr. Werner Frank. The most significant part of the program is
the method by which he obtained the. darnpig.and added mass of an oscillating
arbitrary two-dimensional section on or below the free surface This is
described in detail in NSRDC.Report.2357and results have been published
in the Journal of Ship Research (Vol. 11, No. 3, Sep .1967). His work
represents both a carefUl analysis for reducing the integral equations to numerical methods of solution'and an efficient digital program by which to
carry it out.
Mr. Frank died on2O May 1968, the day he was to be honored for his continued superior perfOxmance.. Dr. NilsSalvesen,the coinvestiga.tor associated with Mr. Frank at that time, prepared this document, and it was Dr. Salvesen's suggestion that the program be named The Frank Close-Fit Ship-Motion Coiuter Program.
We, Mr. Frank!s associates, concur with this .suggestion for it is
our desire that the program serve as a contthuing memorial for our friend
and colleague, Mr. Wrner Frank. .
.V.J. MONACELLA
TABLE OF CONTENTS ABSTRACT ADMINISTRATIVE INFORMATION I. INTRODUCTION DISCUSSION SAMPLE COMPARISON
II. SHIP MOTION THEORY
A. EQUATIONS OF MOTION
B.. ADDED-MASS AND DAMPING COEFFICIENTS
Close-Fit Method Lewis-Form Method C. EXCITING FORCE AND MOMENT III. IRREGULAR SEA COMPUTATIONS
IRREGULAR SEA REPRESENTATION. SHIP RESPONSE
PROBABILITY
IV. DESCRIPTION OF INPUT AND OUTPUT SCHEME A. GENERAL DESCRIPTION
Input Output
B. DESCRIPTION OF DATA CARD.FORMATS C. SAMPLE INPUT AND OUTPUT
V CORRELATION BETWEEN ThEORY AND EXPERIMENT ACKNOWLEDGMENTS APPENDIX REFERENCES LIST OF FIGURES Page Figure Figure 1 2
-. Body Plans of Davidson A and Friesland
- Comparison of Theoretical and Experimental Pitch and Heave Amplitudes and Phases for Davidson A and
4
Friesland at Fr = 0.45 6
Figure 3 - Bow Motion of Friesland and. Davidson A in a
Sinusoidal Wave of L/A 0.55 and Fr = 0.45 8
Figure 4 - Bow MOtion of Friesland and Davidson A in a
Sinusoidal Wave Of L/X 0.85 and r = 0.45 9
Figure 5 - Added-Mass Cross-Coupling Coefficient, A53 for
Friesland 17 Page 1 1 1 2 4 12 13 16 18 19 21 24 25 31 36 38 38 38 42 43 47 63 84 85 132
Figure 6 -Figure 7 -Figure 9 -Figure 10 Figure 11 -Figure 12 -Figure 13
Range of Sectional Area Coefficients for Lewis Form
The Pierson-Moskowitz Sea Spectrum for 15-, 20-, 25,, and 30-Feet Significant Wave Heights and the Newmann Spectrum for 30 Feet
Charactron Output, Computer Representation of Forward Stations of Davidson A
Cháractron Output, Computer Representation of Aft Stations of Davidson A
Charactron Output, Pitch and Heave mplitudès for Davidson A at Fr = 0.35
Charactron Output,Pitch aiid Heave Phases for Davidson A at Fr = 0.35
- ChäractrOn Output, Pitch Amplitude Divided by Wave Slope for Davidson A at Fr = 0.35
Page
22
27
Figure. Eample Showing the Flexibili.ty of the Close-Fit
Program -- 40
Figure 14 - Body Plan and Computer Representation of
Friesland...66
Figure 15
-
Comparison of Theoretidal and Experimental Heave andPitch Amplitudes and Phases for Frieslànd ...67
Figure 16
-
Comparison of Theoretical and Experimental Heave andPitch Amplitudes and Phases for Priesland
...
68 Figure 17 - Comparison of Theoretical, and Experimental Heave andPitch Ainlitudes and Phases for Friesland at
Fr=0.55
69Figure 18 - Body Plan and Computer Representation of
'Davidson A
Figure 19 - Comparison of Theoretical and Experimental Heave and Pitch Amplitudes and Phases for Davidson A at
Fr = 0.25
Figure 20 - Comparison of Theoretia1 and Experithental Heave and Pitch Amplitudes and Phases fOr Davidson A at
-Fr0.35
Figure 21 - Comparison of Theoretical and Experimental Heave and Pitch Amplitudes and Phases for Davidson A at
Fr = 0.45
Figure 22 - Computer Representation of Destroyer with Sonar Array
Figure 23 - Comparison of Theoretical and Experimental Pitch and
Heave Amplitudes for Destroyer with Sonar Array 75 58 59 60 61 62 70 71 72 73 74
Figure 24 Figure Figure Figure Figure Figure
- Comparison of Theoretical and Experimental Pitch and Heave Phases for Destroyer with Sonar Array
25 Body Plan and Computer Representation Of Mariner Hull
26 - Comparison of Theoretical and Experimental Pitch and Heave Amplitudes for Mariner Hull
27 - Comparison of Theoretical, and Experimental Pitch and
Heave Phases for Mariner Hull
28 - Body Plan and Comput:er Representation of Series 60, Block 0.70
29 - Comparison of Theoretical and Experimental Heave and Pitch Amplitudes for Series 60, Block 0.70 at
Fr=0.l5andatFr=O.20
30 - Comparison of Theoretical and Experimental Heave and Pitch Amplitudes for Series 60, BlOCk 0.70 at
Fr = 0.25 and at Fr 0.30
31 - Comparison
of
Theoretical and Experimental Heave andPitch Phases for Series 60, Block 0.70 83
LIST OF TABLES
P age
Table 1 - The Pierson-Mbskowitz Sea Spectrum 30
Table 2 - Listing of Data Cards for Sample Jüput 48
Table 3 - Sample Output so
Table. 4 - Listing of Computer Program
89 Figure Figure Page 76 77 78 79 80 81 82
NOTATION
-A, B Coefficients in Pièrson-Moskowitz sea spçctrum
A.. Mass coefficients in equations of motion (i,j = 3 and 5)
a Wave amplitude or response "amplitude"
aN Most probable largest response "amplitude" aav Average response "amplitude"
-a113 Significant response "amplitude"
a110
Aveage of one-tenth highest response "amplitude"B1 Damping coefficients- in e4ations of motion Sectional beam
Restoring coefficients in equations -of motion
d(.) Seëtional draft
E Energy of spectrum
ER Energy of -response pctrum
-Energy of sea spectrum F3 Heave excitation force
F5 !itch excitation rnrnent
Sectional exciting force
Gravitational acceleration
-h Wave height
-hN Most probable largest wave-height
1av Average wave height
-h13
Significant wave height, average- of one-third highest- wavesh110
Average ofone-tenth highest wavesI
Imaginary unitk Wave number, nondimensional
-L Length of ship between perpendiculars
1 Longitudinal distance from center
of gravity
m()
Sectional exciting moment-m th moment of spectrum
N Large number of values
-Pr{a>a1} Probability that a is larger than
a1
pv R (w) S (w) SR(w) Ss(w) T
Pressure in phase with velocity
Response operator, response to sinusoidal excitation of unit amplitude
Energy spectrum
Response energy spectrum Sea energy spectrum
Sectional area coefficient
Wave period
T Average wave period
av
T Period of maximum energy
t Time
U Ship speed
V Displaced volume
v Froude number
x3 Heave, vertical displacement of center of gravity x5 Pitch angle
z Heave amplitude at center of gravity
a()
Sectional added mass coefficient Sectional damping coefficient Heave phasePitch phase
Sectional vertical coordinate
n Sectional horizontal coordinate 0 Pitch amplitude
A Wave length
Aav Average apparent wave length
p Wave frequency, nondirnensiona].
Frecjuency of encounter, nondimensiona].
Longitudinal coordinate
Longitudinal position of center of gravity
w Wave frequency
ABSTRACT
This report is a manual for "The Frank Close-Fit Ship-Mption Computer Program." The input and output schemes and
the theory are explained, in detail. The program computes
pitch and heave motions for ships in regular aid irregular
head waves. The regular wave responses are computed
accord-ing to an improved version of the Korvin-Kroukovsky strip
theOry. The irregular seaway is represented by the,
Pierson-Moskowitz Sea Spectrum and statistical response parameters are computed for pitch and.heave as well as for absolute and relative vertical displacement, velocity and accel-eratiOn at any point along the length of the ship. A comparison between theory an4 experiment of the pitch and heave amplitudes and phases for five hull forms is
pre-sented and the agreement is satisfactory for both common
crUiser-stern ships and high-speed destroyers and also for
hulls with large bulbs. an4 sonar arrays.
ADMINISTRATIVE INFORMATION
This study was authorized by the Naval Ship Systems Corñrnand under
the General Hydromechanics Research Program, and it was funded under Sub-project S-009 01 01, Task 0100.
L INTRODUCTION
The study of the behavior of ships in a seaway is essential, for the
design of new high-spee4 naval and dry-cargo ships as well as for huge
modern tankers. It is desirable in the design of these ships both to
reduce their motions and to minimize the wave-induced bei4ing moments. The reduction of ship motions is important with respect to maintaining high
speed in seas and to operating electronjc equipment and weapons which are
sensitive to acce'lezation'; while a reduction of the wave-induced bending
nioments obviously will result in less steel weight and hence a more
in-expensive hull. It is not only for the purpose of increasing operating
speed and redUcing cQt that it is essential to have better knowledge of
the hydrodynanic loads and the motions of the 'ship, but it is even more
important from the standpoint of safety of the ship and the seamen. This is all well known, still the motion characteristics and the hydrodynamic loads are far from adequately considered in the design procedure to today's
The Naval Ship Research and Development Center (NSRDC) has devoted
a major effort toward the improvement, of methods for adequately predicting
ship motions. This effort has resulted in several' computer programs for
determining motion characteristics. Unfortunately, most of these programs
are not in a form that can be made generally available to the design
engineers. The present program has been constructed to meet the naval
architect's immediate need for a prediction scheme easily 'available to him
and which is as good as the state of the art allows. Even though the
theory is limited to pitch and heave motions in head seas, the program is one of the most advanced, accurate and flexible ship motion programs.
This report is intended to be a manual for "The Frank Close-Fit Ship-Motion Computer Program" and is divided into five sections. The first
section, the Introduction, gives in addition to general -introductory
statements a sample of computed results. These results are presented in
order to familiarize the reader with some general concepts and to give
some physical insight into the motion problem before going into the detailed
discussion of the theory and the program. The second section of the report
deals with the theory used for computing the motions in regular waves
while the statistical prediction method used for the irregular seas is
presented in Section III. A description of the input and output schemes
is. given in the fourth section together with a sample output. In 'the last
section a comparison between computed and experimental results is presented,
and a complete listing of the computer prograni can be found in the
Appendix. One should note that in order to apply the computer program it
is not strictly, necessary to go through Sections II and III, since Section
IV is complete in itself and gives the needed information for preparing
the input cards and for reading the output. It is not recommended,' however,
that the program be applied without sOme understanding of the theory and
its limitations and accuracy.
A. DISCUSSION
Today there exist several versions of the original linear strip theory by Korvin-Kroukovsky and Jacobs1 for computing the pitch and heave motions
of ships in regular sinusoidal head waves. Even though these theories are essentially all of the same form, they have different forward-speed
con-tributions which results in rather large differences in the answers for
the high-speed cases. These speed effects have been derived in a somewhat
empirical way and the correctness Of the different speed terms is
uncer-tam. Work is in progress, Ogilvie and Tuck,2
on a new strip theory with
speed contributions derived, in a sound and rational manner. The result of
this work, however, is not presently in a fOrm suitable for routine
cothpu-tations, The theory applied in this computer program is also a strip theory
similar to the Korvin-Kroukovsky theory, but with forward-speed correction
terms which differ somewhat. from the other theories. Theoretical and
ex-perimental investigations seem to indicate that these latter correction
terms are more accurate than the others.
In order to utilize a strip theory numerically, One must firt
com-pute the two-dimensional aded thass and damping coefficients for each
station along, the ship. This is a very essential ,part of the computations
and the most time consuming. There are two methods i use for computing
these quantities, the Lewis-form method and the close-fit method. The Lewis-form method is fast but it is only accurate for regular common ship
forms, while the close-fit method is time-consuming but accurate for any
section shape, The present computer program applies both methods and th
user has the Option to select the method best suited for each station. In this way, the program can be used for ships with any unusual bow or stern
arrangements with 'good accuracy and minimum time.
A comparison of the motions computed by this program and experimental
results fOr some twenty ship foins has showi -for most éases.a satisfactory
coPfidence level. The improved accuracy of the theory was such that it
became apparent that a more sophisticated testing procedure was required for high-speed ships in order to properly evaluate the theory.
This was
accomplished by using free-running models
and sonic transducers for the
motion measurements-. Through this extensi-e comparison
study we have, been
able to establish that the theory' works quite well not only for
ruiser-steti ships but also for high-speed destroyer hulls atFroude
numbers up to 0.45* as well as for hulls with large bulbs and sonar arrays.
B. SAMPLE COMPARISON
The kind of useful information which can be obtained from this program can best be demonstrated by a samplecomparison of two destroyer
forms. Body plans of the two hulls, the Davidson Type A and the Friesland
Class,.ãre shown in Figure 1. It can be seen that the Davidson Type A has
a very large bulbous bow while the Friesland Class has a more conservative
bow foiin. Both hulls have regulartransom sterns. The Dutch navy has
several ships of the Friesland Class andexperience has shown that they
are good seaworthy hulls. The Davidson Type A destroyer form, on the
other hand, is an experimental hull and no ships have been built of this
particular form. High-speed ships of the Davidson A form have great
practical interest, however. Several of today's destroyers are equipped
with
large sonar array domes, and hih-speed hulls with very large bulbs and transom sterns are being considered for the future dry-cargo ships.31P!
Figure 1 - Body Plane of Davidson A and Friesiand
Froude number of 0.45 corresponds to approximately 35 knots for a
500-foot ship.
Let u now look at the head-seas motion responses of these two hull
forms in regular sinusoidal waves at Froude number (Fr) = 0.. 45 (35 knots
for a 500-fOot ship). Figure 2 shows three cOmparison plots.: the first
plot gives the pitch anplitudes. for the two hulls, the next, the heave
amplitudes and the last the pitch and heave phases. These quantities are
plotted against the ship length divided by.th waVe length, L/A. Note
that the pitch amplitude 0 is scaled by the wave, amplitude a and multiplied
by half the ship length L/2 so that the pitch values, OL/(2a), shown on
the plot are approximately equal to the non-dimensional vertical bow
motion due to pitch.. A considerable difference in the motions for the
two hulls is seenin Figure 2, both for the amplitudes and for the phases. For the Davidson A the theory predicts in part of the wavelength range less thanhalf the pitch but more than twice as much heave as for the
Friesland.. The phase plots show that the. Davidson A. has for both pitch
and heave about 45-90 degrees more lag than the Friesland. Experimental values for both hull forms are also presented in Figure 2 in order to demonstrate the fair agreemeIt between theory and experiment. One notes
that for the pitch amplitides Of the Friesiand hull there i some
dis-crepancy between theory and experiments. For high speeds the theory
gen-erally predicts more pitch than given by the experiments for both destroyer hulls without bulbs and cruiser-stern ships Notwithstanding this dis-crepancy in pitch, the experimental points clearly show the same. trend as given br the theory; namely, that the bulb has the effect of decreasing the pitch while increasing the heave, and at the same time increasing the lag between the motions and the incoming waves.
*
The ship length L used throughout this paper is the length between
perpendiculars:. **
Pitch is conventionally scaled by the maximum wave slope; however, we have found that in comparing theory and
experiments and also in
com-paring the relative importance of pitch and heave that it is
more con-venient to present the pitch as vertical bow motion.
**.*
The phase angles express the lead with respect to
maximum wave ele-vation at midship.
3.0 2.5 :i'
f E
0.5 2.5-
THEORY 0,EXPERIMENT - -.- THEORY EXPERIMENT DAVIOON'A +45 I-, UI UI.0
z
UI -in4
UI >. -1354
UIx
0
z
4
-180 U I--225 -270 1.2 0.4 06 O8. 1.0 1/A 1.2 I FRIESLAND' O4 0.6 0.8 1.0 L/A .0.4 04. 0.8 1.0,U
.1/A.In order to get a better understanding of the combined effect of the bulb due to this difference in the pitch and heave amplitudes and phases, let us look at the total head-seas motions for a given wave
length.. At the wave length L/A = 0.55 the .pitch amplitude is approximately the same for .both hulls while the heave amplitude is more than twice as
large for the Davidson A as the Friesland. Combining the amplitudes with
the phases in Figure 3 shows the total bow motions of the Friesland and the Davidson A in a sinusoidal wave of L/X = 0.55 and at Fr = 0.45. At this particular wave length the Friesland clearly has better motion.
characteristics in head seas than the Davidson A. It is interesting to note in Figure. 3 that for the motions of the bow relative to the water
surface, the difference in the phase angles cai have as serious an effect
as the difference in the amplitudes. The. total bow motions of the two
hulls have a phase difference of about 75 degrees which results in the Davidson A having its maximum bow up at approximately the wave trough while the Friesland has its maximum bow up at about the wave crest Hence, for this particular wave length the unfavorable motions of the Davidson A are
caused both by the additional heave and by the phase. difference.
Considering now a shorter wave length, say L/A = 0.85 but with all the other conditions unchanged, it can be seen in Figure 2 that the two hulls have about the same amplitudes of heave, while the Friesland has
considerably more pitch than the Davidson A. There are also differences
in the phases. Froth the total bow motions which are shown in Figure 4 it
is apparent that at this wave length the Davidson A has the best bow
motion characteristics. Since the vertical, stern motions in head seas for
transom hull forms are relatively small both for hulls with and without. bulbs one can oiiclude from these results that in head seas at r = 0.45 the Davidson A has the more favorable motions in the short wave-length range L/A > 0.8, while the Friesland has the better motions in the long
wave-length range L/A < 0.6. This seems to indicate that for large ships
a huge bulb may result in a more seaworthy hull, since the large ships will seldom encounter sea conditions with Significant energy in the long wave-length range L/X < 0.8, which corresponds to wave .length.s larger than
FRIESLAND CLASS
FRIESLAND CLASS
This comparison in regular waves was presented mainly to give some
physical insight for the bow motion and to demonstrate how one ship may have much better motion characteristics than another ship at One
par-ticular wave length, while at another wave length the other ship is far
superior. The essential practical problem then is to determine which ship
has the best seakeeping characteristics in a realistic seaway where all
wave lengths are present. In this program the ship's statistical responses
can be computed in an irregular seaway represented by the unidirectional
Pierson-Moskowitz Sea Spectrum. The program computes the average,
sig-nificant and average of the one-tenth highest amplitude of pitch and
heave, as well, as the absolute and relative vertical displacement, velocity
and acceleration at any point along the length Of the ship. Both these
statistical values together with the regular-wave responses can be obtained
for a given ship at several shi.p speeds in about 8=10 mifiutes on an IBM 7090.
The possibility of obtaining such information with satisfactory accuracy within this short time clearly suggests considerable potential
for practical applications. However, even if the program can give the
user reasonable predictions of the pitch and heave motions in head seas, there are numerous other onsideratiOtis in the design of a seaworthy ship.
For most of the other problems related to the seaworthiness there are no adequate numerical prediction schemes and the designer must rely on results
from model testing and ful 1-scale measurements as well as empirical
results and practical experience. The seaworthiness of the final hull
design will depend to a large extent on a realization of the equal
im-portance of empirical., experimental and theoretical methods and, a
utili-zation of each of these methods in the design procedure. It must be recognized, therefore, that the present computer program is only an
additional tool which must be used intelligently and in coordination with
other existing techniques.
*
One may use any other spectrum by changing only a few statements in
It is important to hote that the wave-induced bending moments and shear forces are not included in this program since it is felt that at the present time the confidence level of such computations is insufficient
for general practical applications; however, these quantities could easily be computed with a small addition to the present progr.m. GoOd estimates of the wave-induced bending moment and shear forces is of sig-nificañt importance in the seaworthiness study, in particular since a reduction of the mOtions will usually result in an increae in the bending
moment. It is hoped, therefore, that adequate correlation studies soon
will be carried out so that thô bending-moment computations can be
in-cluded in the practical design procedure. Computation techniques also
exjst for determining the motions in oblique seas, but it is also necessary here to undertake an extens±ve correlation study in Order to establish
the accuracy of the numerical results. NSRDC is presently developing
a
computer program which utilizes an extended strip theory and which
wIll
determine not only the pitch and heave, but also the sway, yaw, roll and surge for regular waves at all heading including the wave-induced loads.*It is felt, therefore, that with continued effort in this area, we should have available within the near future theoretical prediction schemes which can give useful practical predictions in a realistic seaway fOr al.l six modes of motion and fOr the bending moments.
In today's ship design the main criteria are effective rower per-formance in calm water
aid
1aximum bending mmeht in the static"one-oVer-twenty!? wave. Ship motions and dynamic sea loads
are barely considered.
This approach has worked quite satisfactorily for more than a hundred years, but it is not adequate for the design of our future ships. The future design criterion mut be the ship's performance in a realistic
seaway. A final goal in the seakeeping work,
therefore, shoUld be to in-corporate the ship-motion computer programs into the overall computer-aided ship-design concept, and it is hoped that the present computer
program will initiate such work.
*
"Ship Motions and Sea Loads" by N. Salvesen, E.O. Tuck, and 0. Faltinsen to be presented at the SNAME annual meeting November, 1970
*
The validity of the principle of superposition is discussed in Section
**
II. SHIP- MOTION
THEORY-We are primarily interested in the motions of the ship in a
realistic- irregular seaway; however, with the now establishe4 acceptance
of the validity of the principle of superpOsition as applied to ship
motion, our complex problem is reduced to the following two solvable
problems: (1) the prediction of motions in small-amplitude regular waves,
a problem which will be discussed in this section, and (2) the statistical
prediction of random processes, which will be presented in Section III. The theory used in computing the pitch and heave amplitudes and phases in regular sinusoidal waves is essentially the linear strip theory
1.
of Korvin-Kroukovsky and Jacobs with improved, forward-speed contributions
**
similar to those originally proposed by Tuck and- With accurate close-fit
station representation according to the method by Frank.4 A comparison of the pitch and heave amplitudes and phases computed by this strip theory and results obtained by experiments in regular waves is presented in
Section V. The agreement is in general quite satisfactory.
The most important assumptions and restrictions applied in the
strip-theory- are:
I. The theory is linearized by assuming that the ship has "small't-disturbances from the equilibrium position and, that both the incomthg waves and those created by the ship are- "small."
2. The water is assumed -to be inviscid (all viscous effects
-are disregarded). -
-3. The ship length is assume4 to be much larger than the beam and draft, so that each section can be treated as a
two-dimensional "strip" with no interaction, between sections
Our results seem to indicate that these assumptions are all quite
accept-able for conventional hull forms. Experiments give evidence to believe
Furthermore, for vertical motions the damping caused by the creation of
*
free-surface waves is much larger than the viscous damping so that the
inviscid-fluid assumption is quite accurate. The third assumption would
surely be violated by applying the theory tQ wide barges, for example, with beam = 1/3 length, but it is generally satisfactory for regular ship
forms.
A. EQUATIONS OF MOTION
The coupled differential equations for pitch and heave motion of a
ship in response to sinusoidal excitation, are
(A33 + 1) X3 + B33x3 + C33x + B35X5 + C35x5 = F3
(2.1)
Ax3 + B53x3 + C53x3 + (A55 + R)x5 + B5x5 + C55x5 = F5
where x3 is the heave, the vertical displacement of the center of gravity, positive upward, and x5 is the pitch angle, positive for bow down.
Here
the equations have been normalized with respect to the
mass of the ship,
so that the mass of the ship is equal to onO an4 the length scale is normalized with respect to the length between perpendiculars L,
so that R
is the radius of gyration for pitch 1ivided by L.
The following coefficients in Equation (2.1) are dependent on the
frequency of the excitation
1 2
1e
J c()
d (2.2) cz() d -1.'_B = =
PV
$
B53 = C35 = -4(c)
-
d v A33J(
)2ç) d
+ A33 V 1e B5 =lv.
J(
)2.) d
while the restoring coefficients in E4uation (2.1) are only a. function of
the geometry of the water plane of the ship
C33 =
J
b() d
$
VL -
b() d
The excitation fOrce and moment are givn respectiyy by
F3
=Ref._J
Jf()cik(_c)
deet}
(2.4)
Re (CVLJ m() e'c d
eet}
The integrations in Equations (2.2)-(2.4) are all over the effective
length of the ship.
Here the Froude nthnber
U
V=:
3 3a() d
+ B3 (2.2) Cont'd - ç) b(,) d (2.3) (2.5)where U is the ship speed, and the nondimensional frequency of encounter
(2.6)
where
e is the dimensional frequency of encOunter. The other symbols
are
() is the sectional added-mass coefficient.,
() is the sectional.damping coefficient, b() is the sectional beam,
f() is the complex sectional exciting force, m() is the complex séctional exciting moment, V is the displaced lume,
is the longitudinal position of the center of gravity,
k = w2/g wave number,
g is the gi-avitational acceleration, and
Re means that the real part is to be. taken.
The frequency of encounter We is related to the wave frequency w by
w
=(1.!L
e \ g
Before explaining in more dOtail the computation Of the sectional added mass and damping coefficients and the sectional exciting force and moment,
let us first state the form of the solution and discuss the forward-speed
contributions to the coefficients (2.2).
For a given frequency of encounter and a given ship speed the
ste.ady-state solution to the coupled differential Equations (2.1) may be
written.
x3 =
z cos
et
6)where z and 0 are the heave and pitch amplitudes and 6 and are the heave
and pitch phases. Here the phase angles express the lead with i'espect to
maximum wave elevation at midship. The motiOn results are expressed in
the computer Output in terms of the. heave
andpitch amplitudes
andphases
as functions of frequency of encounter and ship speed.
(2.7)
'
(2.8)
The significant difference between the strip theory applied here and other strip theories is the form of the forward-speed contributions
to the càefficients (2.2). The speed ternis used here are essentially the
same as those originally proposed by Tuck. The added masscrbsscoupling
coefficient A53 contains an additional forward-speed' term not present in
other theories; and the pitch added-mass coefficient Açç lacks a linear speed term included in some other theories (e.g., Gerritsma and Beukelman).5
Recent computations have indicated that this difference in the
forward-speed contribution to the coefficient A55 has a, rather small effect on the
motion. However, the forward-speed effect on A53 is very important and
has a large effect on the motion at high speeds in near-resonant situations. The additional speed effect on A53 is believed to be correct for two
reasons. (1) Timman and Newman6 have proved that A35 and A53 must have
the same forward speed terms but opposite sign. Our coefficients satisfy
this symmetry requirement. (2) Experiments by W.E. Smith7 at Deift,
presented in Figure 5, show that A53 has a fairly strong speed dependence.
The points in' the figure represefit 'his experimental results and the two
curves show calculated values. The broken line is the computed coefficient
without the speed effect
A53
=
whereas the solid line includes the speed term
1 r v
A53
= 2
c() d
- -- BV
Good agreement between the experimental data and the solid line is seen in
Figure 5.
B. ADDED-MASS AND DAMPING COEFFICIENTS
The computation of the sectional ad4ed-mass and damping coefficients,
ct() and 8(c) 'is the most complicated and the most time consuming part in
solving the equation ,of.rnotion. These coefficients must be quite accurately
computed in order to get sa factory motion' results. This computer pro-gram applied two methods in obtaining these quantIties: (1) the close-fit
0 -2 -4 0 Fr = 0.15 Fr = 0.45 0
- _o
r-
.u.y.6
'°
-0 WITH WITHOUT SPEED EFFECT -SPEED EFFECT W.E. SMITH 7) I I EXPERIMENTS (REFERENCE 0 I I I 1 2 3 4 5 670
1 2 3 4 5 6 7 FREQUENCY OF ENCOUNTER, eFigure 5 - Added-Mass Cross-Coupling Coefficient, A53 for
method byW. Frank4 an (2) the Lewis-form method by.O. Grim.8 Since the Lewis-form method is fast but only accurate for regular common ship forms, while the close-fit is time consuming but accurate for any section shape, this program is so constructed that the user has the option to select the
thethod best suited for. each section.
The two-dimenional problem conits of determining the added mass (the hydrodynainic force in phase with the acceleration) andthe damping
(the hydrodynaniic force in phase with the velocity which is proportional
to the dissipated wave energy) for a cylinder oscillating in harmonic motion in the free surface of deep water. In solving this linear
po-tentia], problem, t.he essential difference between the close-fit method and the Lewis-form method i-s in the way the cylinder-wall condition is
satis-fied.
1. Close-Fit Method
The geometrical shape of: the section is mathematically represented by a given number of offset points (about 8-12) with straight line
seg-ments between the points (see Figure 9). The trelocity potential. is then
obtained or a distribution of sOurce singularities over the submerged
surface of the cylinder with constant strength over each of the.straight
segments. These source singularities are so constructed that each of them
satisfy the Laplace equation, the free-surface condition., and the infinity
conditions. The cylinder, condition is finally satisfied by solving for
the appropriate strength for each of the source segments. This gives the velocity potential for.the boundary-value problem and the hydrodynamic pressures are then obtained from the potential by means of the linearized
Bernoulli equation. Integration of these pressures over the immersed
portion of the cylinder yields the hydrodynamic force.
This method is very accurate for practically any. section shape but
it has two disadvantages. First, it is rather time consuming and second,
.the method breaks down at higher frequencies. For a rectangular cylinder
with beam b and draft d it can be shown that the first frequency for which the method fails to give a solution is
=
[ci
coth(Trd/b)]1/2
where the frequency, p. has been nondimensionalized with respect to ship
iength L. When the close-fit method is applied the program only computed
results for frequencies less than that given by relation (2.9).
Un-fortunately, the critical frequency may occur at values less than given by (2.9) for non-rectangular sections with ,setion area coefficients (area!
beam draft) larger than one.
Equation (.2.9) shows that the critical frequency decreases for in.-creasing beam (for a ship with L/d = 20 we. have for b/d = 0.5 that
p1 = 11.20, for b/d p = 7.95 and for b/d = 20, = 5.77) and the
minimum critical fre4uency is equal to for b - . F. John9 has
proved that this is the minimum critical value not only for rectangular sections but also for any shape with sectional area coefficient, s < 1.0 and a numerical investigation byO.M. Faltinsen1° seems to indicate that
this is also the mininium.value for sections with s > LO. Hence, we may
state that the close-fit methOd is in general applicable for frequencies
(2.10)
A more detailed description of the critical frequencies can be found in
4
Frank..
2. Lewis-Fohn Method
The geometrical shape of the section is mathematically represented by the Lèwisforni which has the same beam, draft and area as the given
section. Lewis-forms are defined as the forms which can be obtained by
mapping the circle in the -p1ane into the z-plane using the mapping
z = c1 + c2c1 +
c3
(2.11)where c1, c2, and c3 are coefficients related to the beam, draft and area.
The. Lewis-form representation of a section, therefore, only has the correct
beam, draft and area, while the shape is given by the mapping (2.. 11) aiid
not, by the actual shape of the given section. FOr most oththo regular ship
forms the Lewis-form representation is very satisfactory as can e seen
later from the computer representations of the.Fries.land and the Mariner
In the Lewis-form method the velocity potential is represented by a sum of multipoles each satisfying the free-surface condition while the
density of the multipole is.determined by mapping. thesection into the
c-plane and satisfying the cylinder condition on the circle, 1.
This method is fast, but it is not applicable to sections ith very
small sectiOnal area (fine at sections as Station 19 on Series 60, block
0.. 60) or sections with very large sectional area (sections with. very large
bulbs). A detailed discussion of the limitations of the Lewis.form
representation is given in Appendix B of von Kerczek and
Ttick.11
It canbe shown that the minimum sectional-area. coefficients (S = area/beai1i
draft) for which a Lew-is-form exists is for half beam, draft ratio, b/2d < 1
ti.
5min = .}. (2 - b/2d)
To avoid undefined Lewis forms this program is so conStructed that if the Lewis-form method is applied to sections with area coefficients less than
(2.12)
and for b/2d > 1
Smin = (2 - 2d/b) (2.13)
where b is the full sectional beazñ and d is the sectional draft.. Thern
maximum sectional area coefficient is given by
1
ri/b
2dS
= --+-+
10) (2.14)If the sectional area is smaller than given by (2.12) and (2.13) we w111 get reentry forms as shown in Sketch A and jf it is larger than given by
(2,14) we will get forms as shown in Sketch. B.
given by (2.12) and (2l3) or larger than given by (2.14) thearea
co-efficient will be automatically increased to the lower limits, (2.12.) and
(2.13) or reduced to the upper limit (2.14) and the computations will
con-tinue using the new area coefficient. The coffiputer results will have a
note printed out explaining the change. However, it is not recommended
that Lewis form in general be used for sections with area coefficients
close to the upper limit given by (2 14) igure 6 has been prepared,
therefore, to show the range of sectional areas fdrhich the Lewis form
can be used with satisfactory results.
The Lewis forms which have sectional-area coefficients close to
the upper limit (2.14) differ c.oisideiably froth coiimion bulb shapes. In
order to determine the effect on the mQtipis due to this poor bulb
repre-sentãtion the thotions were computed by applythg both the Lewis-form method
and the close-fit method for ships with regular bulbs and with large sonar
array bulbs. Te rêsultsindicated that the Lewis-form representation may
be used even for sections with rather large bulbs without much effect on
the final total motions.
C. Exciting Force and Moment
The complex sectioiial. exciting force and moment, f() and m() are
computed by two techni4ues. For those sections where the Lewis-form
method is used for computing added mass aiId danping the sectional exciting
force is obtained by
f()
I.'[ctc)
+i8()]}
d()s () k (2.15)and the sectional exciting moment by
m()
=-(c..-vp
2 [c
()
where the notation is as &efined previously, but for convenience will be
repeated here:
is the wave frequency, nondimensional,
is the frequency of encounter, nondiinenslonãl
is the longitudinal position Of center of gravity,
1.5
U.' U.. -U-LU0
cC LU -J cC0
O 5
LI4
i,#max'
EQUATIONLEWfr'NOT
2.14 RECOMMENDEDA
V(/fr7
LEWIS ( . RECOMMENDED FORM-FORII/(4
0.5
1.0.:
1.5
2.0HALF BEAM, DRAFT RATIO, b/2d
Figure' 6 - ange of Sectional Area Coefficients
for Lewis:Form
() is the sectional added mass coefficient,
() is, the sectional damping coefficient,
v is the Froude number,
d() is the sectional draft,
s() is the sectional area coefficient, and
k is the wave number.
While for the sections where the close-fit method is used the exciting force is computed by
where (n, ) are the sectional coordinates with the c-axis positive upward
and the n-axis at' the undisturbed free surface. Here
a and 'are the
two-4imensional hydrodynamic pressures in phase with the acceleration and with the velocity respectively and as Obtained by the cibse-fit method.
Of the two methods for computing the sectional excitation, the
second method (Equations (2.17,) and (2.18) is the more correct since the
vertical variation in the pressures is included very accurately by
this
method. On the other hand, computations have shown that the two
rrtetho4s
give practically the same results for sections where the Lewis form is a good representation of the geometrical shape.
The forward-speed contribution to the excitation differs fro
- 1 . .
Korvin-Kroukovsky and Jacobs by the additional damping term underlined in
E4uations (2.16) and (2.18). It is interesting to note that computations
and experiments have shown that the forward-speed contribution to the ex-citation has relatively small effect on the resulting motions while the forward-speed contribution to the coefficients in the equations of motion
has a considerable effect on 'the motions at higher speeds.
b()
f()
2' '
+ I
k(n)
dn (2.17)- ;
[p(1i)
and the exciting moment by
v.i (n)
m()
=-- f() - 2i.;_YJ
[
e 'o + i Pv(n)]e1"
dii (2.18)III.- IRREGULAR SEA COMPUTATIONS
The motions of .the ship in irregular seas are analyzed in a statistical manner following the procedure of St. Denis and -Pierson. The statistical approach does not giv,e the actual time history of the
motions but rather a statIstical description of the response.
The procedure can be outlined in short as follows. It is assumed that both the excitation (the irregular sea) and the responses (the ship motions) are random processes which can be assumed tO be stationary
Gaussian. Under this assumption an adequate statistical knowledge of the
sea elevation and the ship responses are determined completely by their
respective energy spectra. The energy spectrum is a function which
specifies the fraction of the total energy which is associated with any
given frequency band. The sea energy spectra are known functions furnished
by the oceanographer, while the response spectra must be computed.
The second fundamental assumption is that the linear superposition principle is applicable to this problem. This assumption leads to a
simple relationship between the sea spectra and the ship response spectrum. More specifically, St. Denis and Pierson12 stated that "it is assumed that
the -sum of the responses of a ship-to-a number of simple sine waves is
equal to the response of the ship to the sum of the waves." Underthis
assumption the ship response energy spectrum is
SR(w) = [R(w)]2 . S(w)
where R(w) is-the amplitude of the response to a sinusoidal wave of unit
amplitude, S5(w) is the sea energy spectrum and- w is the wave frequency.
Hence, the ship response spectrum SR(w) which completely describes
statistically the ship motions in the irregular sea is given in terms of the response amplitudes which are obtained by the strip theory as explained
- in Section II and the sea spectrum which is a known function.
Finally, the statistical response values (e.g., the average, sig-nificant, and one-tenth highest amplitudes of motion) can be easily
ex-pressed in terms of the response spectra. It is important to stress,
however, that by applying the statistical description one effectively
fre-The essential assumptions applie4 here are that the motions of a
ship are Gaussian processes ajid that the superposition principle is
applicable. Full scale measuremets give all indications to believe that
the process.is very nearly Gaussian. The superposition principle was
stated by St. Denis. and Pierson'2 only as a hypothesis and there existed
at that time practically nO evidence which could show its validity. St. Denis and Pierson stated very objectively that "if t1ê principle is
not valid, then the resultSderived in this paper is not valid." Fortinately,
since that time evidence has accumulated which demonstrates the validity and according to Ogilvie13 it "may now be cOnsidered as.proven, beyond the fondest hopes of earlier investigators."
A. IRREGULAR SEA REPRESENTATION
hi his computer program it is assumed that-the seaway is üni-directional and can be statistically represented by a Pierson-Moskowitz
energy sea spectrum of the forth
,Ss C ) A -B/w
(3.1)
U)
where L is the circular frequency in radians per second and the constants
A = 0 0081 g2 and B = 33 56/h,3 Here h1,,3 is the significant wave
height in feet. The Pierson-Moskowitz spectrum in this form was
recommended by the 11th International Towing Tank Conference (1966) ii
Tokyo for ship ioioji computations "when information, on typical sea
spectra is not available." It s assumed here that the sea is long
crested with all of the wave components travelling in the same dIrection.
The short crestedness of the sea can be represented by a distribution of the direction in which the waves travel; however, it is felt that the
uni-directional
sea
representation for most head sea conTutations.The Pierson-MOskowitz spectrui in the form (3.1) expresses the
energy in fully developed seas, and it depen4s only on the one parameter,
the significant wave height h113,. With a two-parameter speëtrum one can
The significant wave height is defued as the average of the one-third
represent more accurately the sea conditions actually encountered by the
ships which are for the most not fully developed; however, a cne-parameter
spectrum is selected here since it is easier to apply.
One may use any
other spectrum by changing only a few statements in the computer program.
The Pierson-Moskowit.z energy spectrum for different values of the
sig-nificant wave height is shown in Figure 7.
For comparison the Newinann
spectrum which also represents fully developed seas is plotte4 fot a
sig-nificant wave height of 30 feet.
It is seen that the Pierson-Moskowitz
spectrum has more energy than the Newmann spêctrumin the lOwer frequency
range (the. long wave range) but less in the higher frequency range (the
short wave range).
This causes the average frequency for the
Pierson-Moskowitz spectrum to be smaller than for the Newmarin spectrum.
As mentioned. in the introductory statement, there are several
statistical values which can be determined completely by the energy
spectrum.
Before expressing these values., let us first define the energy
of the spectruTm.
S) dw
(3.2)
Let us also define the
th
moment of the spectrum about the origin
s
Ss(w) dw
(3.3)
The zeroth moment is then for any spectrum by definition equal to the
energy of the spectrum
m =E
0S
and for the Pierson-Moskowitz sect-rum. the secondmoment is
m2-2.56 %1j
while the fourth moment, m4 for the Pierson-Moskowitz spectrum is divergent
and therefore does not exist.
Now from the theory of statistics for
3 V) 180 160
80
w
0 V)40
0.8
FREQUENCY,Figure 7 - The Pierson-Moskowitz Sea Spectrum for 15-, 20-,
25-, and 30-Feet Significant Wave Heights and the Newinann Spectrum for 30 Feet
30 30
I'
25I
I
I I PIERSON NEUMANN -MOSKOWliz
0.4
1.2 1.6T . = 1.96
'Vh
av 1/3A =1-T
2ir (3.6) (3.9)42_2.=
15.45 (3.. 4)and furthermore, if it is assumed that the wave heights have a Rayleigh distribution than the average of the one-third highest waves (the sig-nificant wave height) is
h113 = 4.00 (3.5)
Since the significant wave height is the parameter of the sea spectrum (3.1) it. is convenient to express the statisticälvalues for the irregular
seas in. terms of this parameter. The average period (3.4.) is then by
(3.5)
Froth the spectru&density function (3.1) it follows that the period
of
maximum wave energy (the frequency w for wh.ch the spectral density SC )is maximum) is
2.76 h113 .. (3.7)
One frequently refers to the significant period range for the
energy spectrum and by definition this range is obtained by cutting 3
per-- cent of the area under spectrum off the high frequencies and 5 percent off
the low frequencies. The significant range for the Pierson-Moskowitz
spectrum in terms of the significant height is
1.091 jf'h1/3 < T < 3.434 h111 (3.8)
Sometimes it is preferable to use
a
more physical quantity such as the average apparent wave length rather than the average period. Forsinusoidal deep-water waves we have the well known rel4tionship between the wave length and the wave period
however, the mathematical properties of averages are not necessarily the
matheinatical properties of the analogous quantitries for pure sinusoids.
In fact, the relation (3.9) is incor.ret for the averag apparent wave length which is in accordnce with the theory Of statistics
.L
T2 = 13.14 h av 32,i av 1/3 V 0 A = 2rrg il av i/rn4 3.10)Unfortuate'1y, this relationship cann'ot be applied to the
Pierson-Moskowitz spectrum because the foürt monlentm4 for this spectrum is
diVergei't and does not exist. On the other hand, the average wave lepgth
can be obtained 'according toPierson, Neuthann afid James14 by the relation
(3.11)
A tabu ationof the significant range of periods, the periods of maximum energy, the average periods and the average wave lengtha for given' values of thesignificant wave height are given in Table I for the.
Piêrson-Moskowitz sea spectrum. For reference purposes a sea state scale
is also gi'en in thIs table. The sea state numbers used here are defined
as given in Table 1, so that sea state 6 fOr example is by definition the
sea represented by the Pierson-Moskowitz energy spectrum and with 12,. 00
h1/3 < 20.00. The average wave height and the average of the ne-tenth
highest waves are no listed in the table, but are given in terms of' the
significant wave height by the following rèlátionships
hay 0.626 h113
and '
' (3.12)
= l.211h1/3
-it was 'previously customary to use the wthd speed as the paratheter
of the sea spectrum rather than the significant wave height which is
commonly used today. If the wind Speed is the only known inforation on
TABLE 1
The Pierson-Moskowitz Sea Spectrum
s
Ste
Significant Wave Height ft Significant Range of Periods sec Percent of Maximum Energy sec Average Period sec Average Wave Length ft 0 . 0.10 . ..34 - 1.09 0.87 0.62 1.31 0 0.15 0.42 1.33 1.07 0.76 1.97 1 0.50 0.77 - 2.43 - 1.95 1.39 6.57 1 1.00 1.09 - 3.43 2.76 1.96 13.14 1 1.20 1.19 - 3.76 3.02 2.15 1576 2 1.50 1.34- 4.2 3.38 2.40 19.70 2 200 1.54 - 4.86 3.90 2.77 26.27 2 2.50 1.72 - 5.43 4.36 3.10 32.84 2 3.QO 1.89 - 5.95 4.78 3.40 39.41 3 - 3.50 2.04 -6.43 5.16 3.67 45.98 3 4.00 . 2.18 - 6.87 5.52 3.92 .52.54 3 4.50 2.31 - 7.29 586 4.16 59.11 3 5.00 2.44 - 7.68 6.1? 4.38 . 65.68 4 6.00 2.67 - 8.41 6.76. - 4.80 78.82 4 7.00 2.89 - 9.09 7.30 5.19 91.95 4 . 7.50.99 -
9.4 7.56 5.37. 98.52 5 8.00 3.08-9.71 7.81 5.55 105.09 9.00 3.27 -10.30 8.28 5.88 118.22 5 10.00 1.45 -10.86 8.73 6.20 131.36 5 12.00 3.78 -11:90 9.56 6.79 157.63 6 14.00 . 4.08 -12.85 - . 10.33 7.34 183.90 6 16.00 4.36-13.74 11.04 7.84 210.17 6 18.00 4.63 -14.57 11.71 8.32 236.45 6 20.00 4.88 -15.36 12.34 8.77 262.72 7 25.00 5.45-17.17 13.80 9.80 328.40 7 30.00 5.97 -18.81 15.12 10.74 394.08 7 35.00 6.45 -20.32 . 16.33 11.60 459..76 7 40.00 6.90 -21.72 17.46 12.40 525.43 8 - - 45.00 7.32 -23.04 18.52 13.15 591.11 8 50.00 7.71 -24.28 19.52 13.87 656.79 8 55.00 8.09 -25.47 20.47 14.54 722.47 8 60.00 .8.45 -?..60 21.38 15.19 788.15 70.00 9.12-28.73 23.09 16.41 919.91 9 80.00 9.75 -30.72 24.69 17.54 1050.87 9 90.00 10.35 -32.58 26.19 18.60 1182.23 9 100.00 10.91 -34.34 27.60 19.61 1313.59applying the sea spectrum (.3.1) one may use an approcimate relationship
between wind speed and significant height for the open ocean given by a
curve having the following ordinates:
B. SHIP RESPONSE
By the superposition principle which, states that the ship response
to a random sea is just the sum of its responses to the various frequency components we have that the ship response energy spectrum is
SR(w) = [R(w)]2
S (3.13)
Here R(w) is the response operator defined as the amplitude of the response to a sinusoidal wave divided by the wave amplitude, S5(w) is the sea
spectrum given by Equation (3.1) and w-is the wave frequency. The energy of the response spectrum is by definition
ER
=
5R dw (3.14)
0
Assuming a Rayleigh distribution of the maxima ("amplitudes") one can express the following statistical iesponse values in terms of the energy of the spectrum:
the average "amplitude"
-aav = 1.253
significant "amplitude" (average of the 1/3 highest "aniplitfldes")
a113 = 2.000 Wind Speed knots Significant Wave Height feet 20 14.5 30. 18.5 40 26.5 50 36.0 60 48.0
and the average of the one-tenth highest "amplitudes"
a1110 = 2.546
Note that these quantities ec ès "amplitudes" while wave heights (double
amplitudes) were used for the irregular sea values.
In this computer program the average, the significant, and the average of the one-tenth highest "amplitudes" are computed for pitch and heave displacements, velocities, and accelerations as well as for the absolute and relative vertical displacements, velocities and accelerations
at any point along the length of the ship. The response Operators R(w)
used in computing the response spectra for each of these responses are
obtained from the. pitch and heave amplitudes, 0(w) an4 z(w) an4 the pitch
and heave phases and.6 computed by the strip theory as explained in
Section II. The response operator for heave (vertical displacement of the
center of gravity) is
Rhd(w) = z(w) (3.15)
and the response. operators for the heave velocity apd .accelera.tion are
R(w)
=w(w)
. z(w) (3. 16)= [W(W)]2
z(w)
(3.17)where the frequency of encounter We is related to the wave frequency w by
wU g
The first of he two subscripts to the response operators refer to heave, pitch, absolute or relative motion (h, p, a, or r) while the second sub-script refers to displacement, velocity, or acceleration (d, v, or a).
*
Heave is defined as the vertical displacement of the center of gravity,
positive upward. Pitch is defined as positive for bow down and phase
Similarly we have that the response operator for pitch displacement, velocity and acceleration are
Rd(w) = 0(w) (3.18)
R(w)
= w(w)
0(w)(3.19)
Rpa(w) =
[w(wfl2
8(w) (3.20)The response operator for the absolute vertical displaèeinent at any point
along the ship at a distance £ from the center of gravity is
[Rd(w)]2
[z(w)J
+ £2[0(w)]2 - 2z(w)2,0(w) cds (tS-c)
(3.21)and the response operator for the relative vertical displacement at any point is
rd
=
2,2[)]2
Z(w) £e()
áos (s-c)
+ 1 - 2z(w) cos(.. z
-
cS) + 2i0(w) cos(2
)
3 22)
The operators for the absolute and relative velocities and accelerations at any point are obtained by multiplications by the frequency of encounter
in the same way as for the heave and pitch motions.
Having computed the response operators, the response energy spectrum SR(w) is obtained by Equation (3.13), and the energy
of
theresponse spectrum ER is obtained by numerical integration of the response
spectrum
fmax
SR(w)dw
(3.23)where w refers to the maximum limit of integration used in the numerical
max
computation. The program is constructed so that the nondimensional
< 10.0 which should be adequate fpr most cases; however, this range may be reduced automatically by the computer when the close-fit method is
ap-*
plied. Onö should check, therefore, if the value of u is sufficient
max
for an accurate estimate of the spectral energy. The sea spectrum and the
response spectra for the heave and pitch displacements which are printed
out for the entire frequency range. may be used for this check.
The adequacy of the frequency range is also checked by the computer
program. Before the integration of a response spectrum is performed
the following two conditions must be satisfied:
S
<S
w=w max
max
(3.24)
which states that the value of the response spectrum at w = w must be
rnax
smaller than the maximum value of the response spectrum, and
E2 < 0.20 E1 (3.25)
where E is the area under the response spectrum from w= 0 to w = w
1 max
and .E is the additional area in the range w > w obtained by extending
-2 . max
the response spectrum by drawing a straight line through the two last
computed values of the spectrum. If. these two conditions are not satisfied
the statistical values will not be computed and the 'computer will print
out for this particular response "INSUFFICIENT FREQUENCY RANGE." If, on
the other hand, these conditions are satisfied, the computer Will et the
energy of the spectrum
ER = E1 + (3.26)
and use this energy value in computing, the. statistical response values. It is important to 'note that the response values for the relative
motions are very sensitive to an accurate computation in the high-frequency
range. The' reason is that as the frequency increases the response operators
As explained in Section II, the close-fit method is' limited by a critical frequency at which the.two-d1imensional added mass and damping coefficient cannot' be computed.
diminish for the absolute motions,
relative displacements and diverge
accelerations. By Equation (3.22)
the relative displacements, veloci
2
L(w)]
-- 1[Rd(wfl2 - 1
Considering the form of the energy spectrum (3.1) we have for the response spectrum that as w -4
S(w)
g (3.28) 4 S (w) ra 4 gwhile they approach a constant for they for the relative velocities and
we have for the response operator for
ties
andacceleration that as w
-2 2/ wU\2 w
=wil+-e \ g 441
wU\'.w
=wil+-V g 3Aw e
(3.27)
which shows that the spectrum for the relative velocity diminishes quite slowly and that the spectrum for the relative acceleration diverges. This implies that the energy for the relative acceleration is undefined
and
it is recommended, therefore, that the statistical responsevalues for relative acceleration computed by this program should not be applied inship motion prediction work. It is recommended also that critical judgment
be used in applying the relative velocity results, for example in
pre-dicting the occurrence of slaiwiing or the slamming pressure. Jn general,
one should be very careful in applying this statistical approach to responses such as the relative motions which are highly dependent on the
sea energy in the high frequency range (the short wave range). The
- 15
following statement by the oceanographer B. Kinsman (p. 343) in his book
"Wind Waves" is relevant.
He states that the assumption of a Gaussian
distribution "has been remarkably successful when, applied to problems for
which the high frequency clutter on the
sea surface is negligible.
The
C. PROBABILITY
The probability of occurrence of the vaioUs phenomena can be ex-pressed in terms of the energy of the respective Spectra. It will be
assumed that the maxima, the "amplitudes" of a process have a Rayleigh probability density function
f(x) = (x/E) exp (-x2/2E) for 0 c x <
(3.29) f(x) = 0 for - < x <.0
where E is the energy of the spectrum. The probability that the
"ampli-tude,1' a. is larger than the value a1 is obtained then by integration of (3.29) namely
P{a > a1} =J (x/) eicp (-2/2E)
dx = ep (a/E)
(3.30)If we write this relationship as
1/2 =
>ai})
and apply that for a large number 4 of a values the probability that one
of the a values is the maximum value is equal to ]JN, we have that
aN = )" }/ln N (3.32)
where aN is the most probable largest value of N numbers of a values Let us first apply these probability relationships to the ship
responses. Since the responses are given by the computer as the
sig-nificant response "amplitude"
a1,,3 = 2.00 (3.33)
it is most lonvenient to write the probabilities in terms .of a113. This
gives the probability that the response "amplitude," a is larger than.a1
P{a > a1} = exp (-2a/a,,3) (3.34)
and the most probable largest "amplitude"
aN = al,3 in N (3.35)
The probability values are not computed by this program but they can easily
be obtained from the two relationships (3.34) and (3.35) by applying, the computed significant response "amplitude," a113.
Similarly, we can express the probabilities for the wave height in
terms of the parameter of the sea spectrum, the significant wave height
h13 = 4.00
The probability that the wave height h is larger th.n is
P{h > h1} = exp (-2h/h3
while the most probable largest wave height is
=h113 1lN/f
(3.36)
(3.37)
IV. DESCRIPTION OF INPUT AND OUTPUT SCHEME
This section first gives a general description of the input and output system and then gives a more detailed description of how to prepare
the data cards. At the end of the section examples of .both the printed
and the Charactron.outputs are presented. A description of program links
and subprograms including a complete listing is presented in the Appendix.
A. GENERAL DESCRIPTION
1. Input
The following information about the ship must be given:
the longitudinal radius of gyration,
the length between perpen4iculars, and
the geometric description of an adequate number of ship stations.
The longitudinal radius of gyration should be given as a fraction
of the length between perpendiculars. If the radius of gyration is not
known, 0.25 is recommended. This should give adequate accuracy for most
ship forms. An investigation of several ships has shown that the motion
responses were affected only slightly by changing the radius of gyration from 0.23 to0.27.
The length between perpendiculars may be given in any units. Model length, ship length or any other convenient length scale can be used; however, the same length scale and units must also be used for the
station descriptions.
The program is constructed according to a system of twenty-One stations with the forward perpendicular FP as station 0.0 and the aft
prpen4icular Al' as station 20.Q. This length scale of 20.0 between
perpendiculars must be used in locating all of the stations. Computations
have shown that it is ade4uate for most motion computations to use only
twenty-one stations equally spaced. More or less stations may be used in
defining the ship but stations 0.0, 10.0, and 20.0 must always be included. Stations fOrward of FP as well as aft of AP can also be used. The pro-gram allows unequal station spacing; hoever, it is required that the station spacing must be ordered in pairs starting from AP and from FP with
equal station spacing within each pair. When using twenty-one stations
