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4Computer Program Results for Response Operators
of Wave Bending Moment in Regular Oblique Waves
By
Jun-ichi FUXUDA
- Reprinted from the Memoirs of the Faculty of Engineering
Kyushu University, Vol. XXVI, No. 2
Contribution to
3rd International Ship Structures Congress, 1967, Oslo
FUKUOKA, JAPAN 1966
Computer Program Results for Reponse Operators of
Wave Bending Moment in Regular Oblique Waves*
by
Jun-ichi FIJKUDA
Assistant Professor of Naval Architecture
(Received October 26, 1966)
Summary
The calculations of response operators of ship motions and vertical wave bending moment at
mid-ship in regular oblique waves were carried out on two therchant mid-ship forms of 0.60 and 0.70 block
coefficients, and the results for midship bending moment are presented here.
Numerical calculations were performed by the computer program based upon the linear strip theory given by Wàtanabe. The response operators are shown in diagrams as functions of square root of
(ship length/wave length) ratio with parameters of ship speed weight distribution and heading angle
to waves.
Short-term distributions of midship bending, moment were investigated on 150 meter long ships in rough seas by using the derived response operators and the family of wave spectra recommended by 2nd I.S.S.C., and the results are shown in diagrams as functions of visually estimated average wave period and heading angle to waves.
L Introduction
Estimation of wave bending moment act-ing upon a ship that navigates among ocean waves is the most important problem in the design of longitudinal strength of ship. In recent years, theoretical and experimental re-searches concerning the longitudinal strength of ship in irregular waves have become more
active with the progress and development in
the theories of ship motions and ocean waves, and along with the collection of full scale ship data and wave data, the possibility of long-term prediction of wave bending moment has been established. Based on these mate-rials the designing standard of longitudinal strength shall be also able to be determined
practically.
For the long-term prediction of wave bend-ing moment, there are the followbend-ing two methods at present. To explain brIefly, since the object of this paper is laid otherwise, one
of them is to depend on the statistical ana-lysis of the stress frequency derived from
* This paper is rewritten in English from the paper
published in Japanese in Journal of SEIBU
ZO-SENKAI (The Society of Naval Architects of West
Japan), No. 32 (1966)
full scale measurements and the other is to estimate short-term distribution as well as long-term distribution in irregular. seas by us-ing the response operators evaluated through model experiments or theoretical calculations.
In the method based on full scale meas-rements, as followed by Jasper", Bennet2, NordenstrOm3 et al., short-term distribution of wave bending moment is evaluated from the measured data of actual ship, and then by availing long-term distribution of ocean
waves, long-term distribution of wave 'bend-ing moment is estimated. On the other hand,
there is another method to evaluate directly
long-term distribution from the extreme
val-ues of short-term, as proposed by Yui1le4
and Watanabe5>. The wave bending moment
acting upon a ship, however, changes
cpm-plicatedly by many factors such as ship form,, loading condition and heading angle to wases as well as the sea state. In order to estimate
long-term distribution of wave bending
mo-ment on the basis of measured data of actual
ship, we have to carry out measurements on a great number of ships for many years and
at an enormous cost. Though a large number of ships of maritime countries of the world is
it seems not easy to draw a conclusion on the
designing standard of longitudinal strength based only on the results of full scale meas
urements.
Two obvious defects are found in the meth-od based on the full scale measurements, which are that it requires a period of long time and that it needs enormous cost, but the
method given here secondly is attractive as it enables us to attain object within a
com-paratively short period and at less
cost.That is, it is a method to evaluate short-term distribution of wave bending momet by using the response operator obtained by
model experiments or theoretical calculations and to estimate long-term distribution through
a statistical method. Either in the method
given first, i. e., the method based on the full scale measurements, or in the second method
to be stated here, the similar method is used when long-term distribution is estimated by
summarizing short-term distributions. There
is, however, much difference in the process
and method of evaluating short-term
distribu-tiOns, especially in the points of time and
expense, the latter is by far superior to the
former.
Now, in the case depending on the latter,
it is necessary to evaluate the response oper-ator of wave bending moment in regular waves as the first step. In that case, there
are difficulties in technique and finance when we attempt to evaluate the response operator
of wave bending moment by model experi-ments, with consideration of the effects of
ship form, loading condition (including weight
distribution), ship speed and course angIe to
waves. Since Sato's experirient6 in 1939 us-ing a destroyer model, model experiments have been undertaken countlessly, arid much have beeri made clear about the characters of wave bending moment by those experiments. When the problems in oblique waves are con-sidered, however, the available data of re-sponse operator derived by model experiments
at present are only those by Vossers et al7, and even those are not provided with suffi-cient ones as the data for response operator. At present, when we find it not easy even to
evaluate the response operator by model ex-periments, the method to evaluate the response
operator by theoretical calculations attracts
our attention as the key for solving the
dif-ficulty. Though this method by theoretical calculations is not yet perfect, it is to be highly valued as it enables us to obtain the response operator of wave bending moment with consideration of various factors, when
utilizing an electronic computer that has made rapid progress lately, within a very short time and at a rather moderate cost.
Since Sato's experiment, there was nothing
noticeable in that sort of experiment for
some time after 2nd World War, untill model
experiments on wave bending moment be-came popular after the presentation of St. Denis and Pierson's study on ship motions in
confused seas in 19538). They proposed the method to evaluate a short-term distribution in irregular waves by applying the linear superposition technique to the theory of ship
motions. Eperimental studies on wave bend-ing moment have been forwarded in great number, initiated by the experiment of T2-tanker model by Lewis9 and with the
com-parative experiments of T2-tanker model' re-commended by the Wave Load Committee, I.
S.S.C., 196110) as the peak. Simulteneously, the
theory of ship motions proposed by Korvin-Kroukovsky", and Watanabe12, in which linear Strip method is applied, was developed
practically by Grims) and Tasai'4 who
intro-duced into it added mass and damping based
on the two-dimensional theory, and then it
was extensively applied to the theoretical cal-culation of wave bending moment by Jacobs'
and the author'6.
At present, the theoretical calculationof wave bending moment based on the linear strip theory is admitted to give a result that agrees practically well with the result of model experiment both qualitatively and
quantitatively, especially it serves us to grasp clearly the effects of weight distribution, ship
speed and heading angle to waves upon the
response operator of bending moment, which
are technically difficult to be made clear by
experiment. There are, however, some in adequate points in the theoretical calculation
by the linear strip theory, which leave vari-ous problems yet to be pursued. There is sometimes a certain degree of difference which is not to be over-looked between the result of theoretical calculation -and that of
1966) Computer Program Results for Respoàse Operators of Wave Bending Moment 61
môdèl experiment because of the
incomplete-ness in the expression of ship fOrm section by the theory, the existence of the three-dimensional effect that cannot be solved, by
the two-dimensional theory and various other non-linear effects lying beyond all linear
theo-ries. Accordingly, there is still some doubt in accepting the respohse operator of wave
bending moment evaluated by theoretical
cal-ulation as it is, and' it is necessary for us to confirm the response operator or correct it by comparing it with the result of model experiment as occasioIs demand. There is however no doubt, for all these defects in the present method of theoretical calculation, that it is an extremely powerful means to evaluate the response operator of wave bend-ing moment, when availbend-ing an electronic computer effectively.
In this paper, the author has investigated the effects of weight distribution, ship speed
and heading angle to waves upon the response operator of wave bending moment by apply-ing extensively the calculation method of ship motions and wave bending moment based on Watanabë's theory'2> to the case in regular
oblique waves. There are given the theoreti-cally calculated results of the response
ope-rator of wave bending moment on two kinds of ship form among Series 6017>, which are carried out availing the opportunity of com-pletion of the computer program for response
opertors in regular oblique waves.
Inciden-tally, the author has also given some
calcula-tion examples of short-term distribucalcula-tion of wave bending moment evaluated by utiliz-ing the derived response operator.
2. Method of calculation and condition of calcu-lation
Theoretical calculations of response opera-tors of ship motions and midship bending moment were made On two kinds of ship form among Series 60'?>. In the present paper, however, oniy the results of wave bending moment are presented and the
re-stilts of ship motions are omitted. because of the limited space.
2.1) Method of calculation
There is included, in the calculation theory, the effect of ship speed and the so-called Smith effect in accordahce with the linear
strip method based on Watanabé's theory12
on heaving and pitching motiOns. The de-tails about the theory are stated in the au-thor's .aper already published'6>, and the thethod of alculatiofl extensively applied to the case in regular oblique waves is given in the Appendix of this paper. For the
cal-culation of added mass and damping
coeffi-cent, Tasai's method'4> is adopted.
Numerical calculation was carried out by the computer IBM-7040 belonging to the Nagasaki Shipyard, Mitsubishi Heavy In-dustries Co., Ltd. In the computer program,
there are stored numerous data of added mass
and damping coefficient calculated by Fujii and Ogawara8). We have only to give the data expressing ship form and weight dis-tribution, ship speed, heading angle to waves and wave length for calculation input, and we can obtain output of amplitudes and phase angles of ship motions (heaving, pitching, vertical bow motion and relative vertical bow motion with respect to wave surface) and midship bending moment. Average time required for calculation of one case in which
certain ship speed, heading angle and wave length are combined is about 6-.-7 seconds. 2.2) Ship form and weight distribution
Calculations were made on the ship forms
of 4210 W (Cb=O.6O) and 4212 W (C,=O.7O)
among Series 60', whose main pai-ticulars
are given in Table 1. Load conditidn of each ship is assumed to be full, with five kinds Of
weight distribution as shown in Table2
re-spectively.
Calculations were made on each ship form
under various cases of weight distribution, heading angle, ship speed and wave length,
as given in Table 3. Furthermore, additiOnal calculations were carried out for the cases of (çl'=15, 45, 75, 105, 135, 165') by taking the heading angle at 15' interval as occasions
demanded.
3. Results of calculation
Among the results of calculation, only those related to bending moment at midship were picked up and plotted, but those concerned to ship motions were omitted. The principal notations used in the figures are as follows:
M0: amplitude of vertical wave bend-ing moment at midship
p density of sea water
g : acceleration of gravity
L : length between perpendiculars
B : breadth of ship
h0 wave amplitude
A wave length block coefficient
longitudinal gyradius of ship
distance from midship to centre of
gravity of afterbody
distance from midship to centre of gravity of forebody
angle between ship course and re-gular wave direction (çl'=O' : head waves)
Fr. : Froude number
R : standard deviation of wave
bend-ing moment at midship
H0 : visually estimated wave height (significant wave height)
visually estimated wave period
- (average wave period)
0 : angle between ship course and ave-rage direction of irregular waves 3.1) Response operators of wave bending
moment
In Figs. l.3, the response operators of mid ship wave bending moment are plotted as
functions of V'1JTL Each figure shows the
effect of ship speed (Froude number), weight distribution (radius of gyration and centre of gravity of forebody arid afterbody) and heading angle (i = 0 is defined as head waves) upon the wave bending moment. In Fig. 4, non-dimensional amplitudes of wave bending moment are plotted as functions of
heading angle, while in Fig. 5, they are
plot-ted as functions of ship speed for the cases
of head waves (çl'=O') and following, waves (p=180').
When the calculated results on ship form
4210W (Cb=O.6O) and 4212W (Cb=O.70) are
compared, it is found that the full hull form gives generally larger value of non-dimen
sional wave bending, moment.
The tendency of the effects of ship speed, weight distribution and heading angle to waves upon the wave bending moment is similar in the cases of both ship forms.
When the effect of ship speed is investi-gated in the case of head waves, bending moment slightly decreases with increase of ship speed within the range of 0.1 Froude number as a general tendency (Figs. 1 and
5). At a higher speed, bending moment be.
comes larger with increase of ship speed, though such a tendency considerablly changes
according to the condition of weight
distri-bution. In the case of following waves (Fig. 5), bending moment decreases gradually with increase of ship speed.
When the effect of weight distribution is
investigated in the case in head waves, bend-ing moment generally decreases with in crease of the radius of gyration, as shown in Figs. 2 'and 5, if the radius of gyration is
changed while keeping the centres of gravity of forebody and afterbody constant '(to
main-tain constancy of the moment about midship of the weight of forebody and afterbody). Again, when the centres of gravity at fore-body and afterfore-body are changed (to change the moment about midship of the weight of forebody and afterbody) while keeping the radius of gyration constant, 'bending mo-ment generally decreases if the centre of gravity of each half body is shifted towards
each ship end. In the case when a ship is
loaded rather one-sidedly on both ends, that
is, when a ship is loaded in a way that a
hog-ging moment may act in still water, the
radius of gyration becomes large and- the centre of gravity of forebody or afterbody approaches toward each ship end (the momentof midship about the weight of forebody or
afterbody becomes large). Under such a con-dition, therefore, it is expected to cause
smal-ler wave bending moment than in a loaded condition resulting in a sagging still water bending moment where load is rather con-centrated amidship. Accordingly, as far as
wave bending moment is concerned, such a
load condition as giving a hogging moment in still water is more advantageous. This fact was pointed out in the author's paper -publish ed befor&6, which has been lately confirmed
experimentally again by Swaan et al'°;
While the tendency of the 'effect of weight
distribution upon the wave bending moment has been considered so far, the curve of
re-sponse operator of wave bending moment has
1966) Computer Program Results for
Response Operators of Wave Bending Moment 63
generally two peaks, as shown in Fig 2, and
these two peaks tend to become sharp under a condition where load is concentrated at midship and at high speed. On the other hand, under a condition where load is laid more on both ends of ship, the curve of
re-sponse operator has one peak, showing the tendency to make its peak to be sharp and
high in a case of high speed According to the results of model experiment lately pub-lished by Moor20), similar tendency is
ob-served It is, however, regrettable that such a tendency is not explained experimentally as the effect of weight distribution, there being no detailed account on the condition of weight distribution in his paper.
In Figs. 3 and4, we can find the effect of
heading angle to waves upon the response
operator of wave bending moment in thecase
of weight distribution that seems to be ap-propriate for a cargo boat. Generally, the
value of wave bending moment becomes large
when the course angle is taken in the range Of ±30 from head waves (cL==O) or
follow-ing waves(çt'=lSO°), but it becomes
consider-ably small when the course angle is taken beyond the range of ±60 from head waves or following waves. Again, at such a low speed as 0.1 Froude number, wave bending
moment is rather large in following waves than in head waves, and at a speed as high
as 0.2 Froude number, wave bending moment
is fairly large in head waves than in
follow-ing waves.
If we use the response operators of wave
bending moment in oblique waves given in Fig. 3, we can evaluate the wave bending moment among long-crested irregular seas or short-crested irregular seas whose wave
spectrum is given.
3.2) Example of calculation of short-term
distribution
The calculation concerning the short-term distribution of wave bending moment in
irre-gular seas was carried out by assuming ship length as 150 meters of each ship form of 4210W and 4212W, and using the response
operators of wave bending moment which are shown in Fig. 3b.
As the parameters representing a sea state, the significant wave height 11, and the
aver-age wave period 7', were adopted and the formula of modified Pierson-Moskowjtz spec-trum recommended by the Committee on
En-vironmental Conditions, 2nd I.S.S.C.21) was used so that it may satisfy with H, and T,. Namely, as the wave spectrum of long-crested irregular seas, the following
expres-sions were used,
[f(cu)] 2=O.11H2Oco,_1(w/w,)_5 X exp(-0.44(w/w,)4) (1)
w,=2rrJT,
w : circular frequency of a
compo-nent wave
and the wave spectrumof short-crested irre-gular seas was assumed as follows,
[f(w X)I2(2/ir)f(co)cos2 :-1rJ2<z<n./2
=0 Xir/2 and
X_rr/2 Tx angle between a component wave di-rection and the mean wave didi-rection Variance and standard deviation of wave
bending moment were evaluated by means
of the energy spectrum calculation method based on the theory of linear superposition
by St. Denis and Piersono) by using the re-sponse operators given in Fig. 3b and the wave spectra defined by (1) or (2).
The results of calculation are plotted in
Figs. 6 and7. In either of the diagrams,
non-dimensional value which is obtained by
di-viding the standard deviationR of wave bend-ing moment by pgL2BH, is taken as ordinate. In Fig. 6, results are given as functions of the average wave period 7', with the
para-meter of heading angle 0(O=O: head waves),
and, in Fig. 7, they are given as functions of the heading angle 0 with the parameter of average wave period T,.
In the case in long-crested irregular seas as well as in short-crested irregular seas, wave bending moment is large in headseas
or following seas, and in the
calculation where ship length is 150meters, RJpgL2BJf becomes maximum in the sea state of about 8 seconds of average wave period. In gular waves, the maximum value of re-sponse operators of wave bending moment occurs in the case of /LJX=i.i, 1. e, when the wave period is about 9 seconds, in head waves or following waves. Therefore, itmay be said that the nondimenSiOnal value of R/pgL2BH becomes maximum in the irre-gular seas whose average wave period is somewhat smaller thanthe period of regular waves where the response of wave bending
moment becomes maximum.
When RJpgL2BH in long-crested irregular seas and that in short-crested irregular seas are compared,the value of R/pgL2BH in
long-crested irregular seas is somewhat larger in
-the case of head seas or following seas where its value is large, but in- the case of beam seas where its value is small, the value in short-crested irregular seas
is larger than
the other. There is, however, not sosigni-ficant difference between the values of wave
bending moment in long-crested irregular
seas and short-crested irregular seas. If the two parameters representing a sea state, i.e., the average wave period T and the
significant wave height Iii, are given, it is
possible to evaluate from Fig.6 or Fig. 7 the parameter R (standard deviation of wave bending moment) that represents short-term distribution of wave bending moment when a ship takes an arbitrary heading angle to
waves in such a sea state.
-In Figs. 6 and 7, examples of the case when ship length is 150 meters and Froude number 0.15 (11.2 knot) are given but, even for the case- when ship speed is different, we can evaluate the parameter R representing
short-term distribution of wave bending moment
by the same
method. Also with the casewhen ship form is similar but ship length is different, the similar method is applicable for the evaluation of R.
If the response operators of wave bending moment in regular oblique waves have been obtained, we can evaluate the parameter R representing the short-term distribution of
wave bending moment for the case when the ship course has been taken arbitrarily against
waves in a certain sea state by t-he
above-mentioned method. Furthermore, it is pos--sible to estimate a long-term distribution of
the parameter R representing short-term
dis-tribution of wave bending moment by utiliz-ing long-term data of sea state on world sea
areas and routes, for instance, data of long-term wave observations in the world sea
areas by the
Committee on EnvironmentalConditionS, 2nd I.S.SC.21, those inthe North
Atlantic Ocean by Roll22 and those in the
North Pacific Ocean by Yamanouchi et al23. Thus we can also predict long-term
distri-bution of wave bending moment as well as short-term distribution by availing response operators in regular oblique waves.
4. Conclusion
The response operator of wave bending
moment can be calculated theoretically by using linear strip method. The author car-ried out calculations of respone operator of
wave bending moment by applying extensive-ly this method of calculation into regular oblique waves and utilizing an electrOniC
computer.
Furthermore, by using, the obtained re-sponse operators, he evaluated the parameter
R (standard deviation of wave bending mo-ment) that represents short-termdistribution of wave bending moment in irregular seas and investigated briefly the properties of short-term parameter of wave bending
mo-ment.
Since it becomes possible to predict long-term distribution of wave bending moment by availing such results of calculation,, it may be said that the theoretical calculation of wave bending moment utilizing an elec-tronic computer is extremely useful for the
researches on wave bending moment even if there is a little defect in the reliability of the result. It i_s needless to say that we
have to give some appropriate correction to it
by comparing it with theresults of model
ex-periment and full scale measurement.
How-ever, the superiority of this method -in time
and cost does not vanish through suc,h a little
defect
cknowledgement
The author wishes to express his deepest thanks to Dr. T. Okabe and Mr. H. Shimada for their constant support
and help in the
work. He is -also thankful to Mr. M. lizuka, Mr. G. Ogata, Mr. M. Konuma for their co-operation in preparing the comupter program and is in debt to Dr. H. Fujii and Mr. Y. Ogawara. In the arrangement -of - great64 Jun-ichi FuziDA
1966) Computer Program Results for Response Operators of Wave Bending Moment 65
number of calculation data and drawing of diagrams he is much indebted to Mr. I. Hata and Mr. S. Tsutsumi.
References
N. H. Jasper: "Statistical distribution patterns of ocean waves and of wave-induced ship stress-es and motions, with engineering applications"
Trans. S.N.A.M.E. Vol. 64 (1956)
R. Bennet, A. Ivarson, N. Nordenström: "Results from full scale measurements and prediction of
wave bending moments acting on ships" The
Swedish S.R.F. Report No. 32 (1962)
N. Nordenström: "Further analysis of full scale
measurements of midship bending moments"
Re-port from the Division of Ship Design, Chaimers University of Technology, Goteborg (1965) I. M. Yuille: "Longitudinal strength of ships"
Trans. R.I.N.A. Vol. 105 (1968)
Y. Watanabe: "On the statistical method of
a-nalysis of bending stresses of ship at sea" Bulle-tin of the Society of Naval Architects of Japan,
No. 429 (1965)
M. Sato: "Model experiments on the
longitudi-nal strength of ships running among waves" Journal of the Society of Naval Architects of
Japan, VoL 90 (1956)
G. Vossers, W. A. Swaan, H. Rijkin: "Vertical
and lateral bending moment measurements on
Series 60 models" I.S.P. VoL 8, No. 83 (1961)
M. St. Denis, W. J. Pierson, Jr.: "On the motiOns
of ships in confused seas" Trans. S.N.A.M.E. Vol. 61 (1953)
E. V. Lewis: "Ship model tests to determine bending moments in waves" Trans. S.N.A.M.E.
Vol. 62 (1954)
"Report of Commitee on Wave Loads"
I.S.S.C.-Glasgow (1961)
B. V. Korvin-Kroukovsky, W. R. Jacobs:
"Pitch-ing and heav"Pitch-ing motions of a ship in regular
waves" Trans. S.N.A.M.E. Vol. 65 (1957)
Y. Watanabe: "On the theory of heaving and pitching motions" Technology Report of the
Faculty of Engineering, Kyushu University. Vol.
31, No. 1 (1958)
0. Grim: "Berechnung der durch Schwingungen eines Schiffs-kOrpers erzeugten hydrodynamis-chen Kräfte" Jahr. S.T.G. 47 (1953)
F. Tasai: "On the damping force and added
mass of ships heaving and pitching" Reports of Research Institute for Applied Mechanics,
Kyu-shu University, Vol. 7, No. 26 (1959) and VoL 8, No. 31 (1960)
W. R. Jacobs: "The analytical calculation of ship bending moments in regular waves" J.S.R. Vol.
2 (1958)
J. FukudO: "On the midship bending moment of
a ship in regular waves" Journal of the Society
of Naval Architects of Japan, No. 110 (1961) and No. 111 (1962)
F. H. Todd: "Some further experiments on single screw merchant ship forms-Series 80" Trans. S.
N.A.M.E. VoL 61 (1953)
H. Fujii, Y. Ogawara: "Calculation on the
heav-ing and pitchheav-ing of ship by the strip method" Journal of the Society of Naval Architects of
Japan, VoL 118 (1965)
W. A. Swaan, W. P. A. Joosen: "The influence of weight distribution on wave bending moment"
I.S.P. Vol. 12, No. 134 (1965)
D. I. Moor: "Longitudinal bending moments on models in head seas" Read at the meeting of R.
I.N.A. on March 24, 1966
"Report of the Committee on Environmental Con-ditions" Proceedings of LS.S.C.-Delft (1964)
H U. Roll: "Height, length, and steepness of sea
waves in the North Atlantic" (English
transla-tiOn) S.N.A.M.E. Technical and Research Bufletin No. 1-19 (1958)
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winds and waves on the Northern Pacific Ocean and South Adjacent Seas of Japan as the
envi-ronmental conditions for the ship" Papers of
Ship Research Institute, Tokyo, No. 5 (1965)
Nomenclature B : Breadth of ship
Cb : Block coefficient
Fr. : Froude number
H,, : Visually estimated average wave height
(Significant wave height)
I : Total 2nd. moment of weight of ship about centre of gravity
Afterbody 2nd. moment of weight about midship
Forebody 2nd. moment of weight about midship
L : Length between perpendiculars
M0 : Amplitude of wave bending moment at midship
M3: Still water bending moment at midship
R : Standard deviation of wave bending
mo-ment at midship
T,, : Visually estimated average wave period
g : Accerelation of gravity
h0 : Wave amplitude
'a : Distance from midship to centre of
grav-ity of afterbody
Distance from midship to centre of
Distance from midship to centre of grav-ity of ship
Longitudinal gyradius of, ship
A Wave length
p Density of sea water
Angle between ship course and regular
Table 1. Main particulars of the model ships
wave direction
x : Angle between a component wave
direc-tion and the mean wave direcdirec-tion
4 : Weight of ship Weight of afterbody
4 : Weight of forebody
Table 2a Weight distribution of model 4210W (Cb=O.6O)
Table 2b Weight distribution of model 4212W (Cb=O. 70)
Condition number (23-..48) (23-..20) (25-.20) (27.-.-20) (27-22)
Longitudinal gyradius /L 0.2300 0.2300 0.2500 0. 2700 0.2700
Afterbody C. G. from midship 13 /L 0. 1800 0.2000
-
-0.2200Forebody C. G; from midship lj /L 0. 1800
*-
0.2000-
0.2200Afterbody weight 4 /4 0. 5411 0. 5370
-
0. 5337Forebody weight 4f /4 0.4589
-
0.4630-
0.4663Afterbody moment about midship 13 4 /4L 0.09740 0. 10741 - -0. 11741
Forebody moment about midship 4j /4L 0. 08259
*-
0. 09260 0. 10260Total moment about midship XG/L -0.01481
e-
-0.01481 -
-0.01481Afterbody 2nd. tnt. about midship 1 /4L2 0.02874 0. 02874 0.03368 0. 03902 0.03902
Forebody 2nd. tnt. about midship Ip /4L2 0. 02438 0.02438 0.02904 0.03410 0. 03410
Total 2nd. moment about C. G. 1/41) 0. 05290 0.05290 0. 06250 0.07290 0.07290
Still water hogging moment M3/4L 0.0037
.c-
0.0137--*
0.0237Condition number (23-..19) (23-.21) (25-21) (27-.21) (27-23)
Longitudinal gyradius ic/L 0. 2300 0.2300 0.2500 0.2700 0.2700
Afterbody C. G. from midship lo /L -0. 1900
.-
-0.2100-
-0.2300Forebody C. G. from midship I /L 0. 1900
-
0.2100-
0.2300Afterbody weight 4 /4 0.4871
*-
0.4883 .- 0.4893Forebody weight 4f/4 0.5129
-
0.5117-
0.5107Afterbody moment about midship 10 40 /4L -0.09254
-
-0.10254 -
0. 11254Forebody moment about midship l 4 /AL 0.09746
-
0. 10746-*
-0. 11746Total moment about midship Xo/L 0.00492
-
0. 00492-
0.00492Afterbody 2nd. tnt. about midship 1 /41) 0.02584 0.02584 0.03053 0.03561 0.03561
Forebody 2nd. tnt. about midship Ij /4L2 0. 02708 0.02708 0. 03199 0.03732 0. 03732
Total 2nd. moment about C. G. 1/41) 0. 05290 0. 05290 0. 06250 0.07290 0. 07290
Still water hogging moment
M/4L
0. 0008<-
0. 0108->
0. 0208Model number 4210W 4212W
Length/Beam ratio L/B 7.50 7.00
Length/Drought ratio Lid 18.75 17.50
Block coefficient Gb 0.600 0.700
Water plane area coefficient Cw 0.706 0.785
Midship area coefficient Cm 0. 977 0.986
LCB from midship (forward) X0/L -0.015 +0.005
Table 3.
Calculation program
Weight distribution
(Heading angle to wave)
Fr.
(Froude number)
VL/A
(Square route of ship length/wave length)
(23-48) 0'(head wave) (23-20) 0'(head wave) 0.40 (0.10) 0.90 (0.05) 1. 50 (0.10) 2.00 C) CO 0, 30', 60'(head wave) (25.20) 90'(beam wave) 0 (0.05) 0. 25 0.40 (0 10) 3.00 120', 150', 180'(following wave) (27'-.2O) 0'(head wave) 0.40 (0.10) 0.90 (0.05)1.50 (0.10) 2; 00 (27-22) 0'(head wave) (23-49) 0'(head wave) (23..21) 0'(head wave) 0.40 (0.10) 0.90 (0.05) 1.50 (0.10) 2.00 0', 30', 60'(head wave) (25'-.21) 90'(beam wave) 0 (0.05) 0.20 0.40 (0.10) 3.00 120', 150', 180'(following wave) (27.21) 0'(head waVe) 0.40 (0.10) 0.90 (0.05) 1.50 (0.10) 2.00 (27-23) 0'(head wave)
0.0! 0 0.03 0.02 0.0! 0 0.5 10
15 -(7X 2.0
Fig. 1. Bending moment amplitudes fçr different ship speeds
(Cb=O.GO) (25-2O):P =0.25 1.h'a/L.Oj/L0.20 : : = 0.! : =0.2
----(C6=0.70) (25-21): ?rIL=0.25---- :
=0.! =0.2 t10°(.Aca1Lwac,e)/7
68 Jun-ichi FUKUDA (Vol. XXVI,
0 05
!0
15iZ7x20
0.03
I
1966) Computer Program Results for Response Operators of Wave Bending Moment 69 0.03 0 0.02 0.0! 0 0.0 0 0.0 0.0! 0 05 I 0
-17x
2.0 0 05 10 15-'7:7X
2.0Fig. 2a. Bending moment amplitudes fOr differel3t weight distri-butions (Fr.=O. 1) (C6=0.60) (23-I8) (23'-20)
Fr.0.1
.(2520)
---- (2720)
(27-22)7..
,/
\'
/7
\
7/
(Cb= 0.70)---
(23-I9) (23-.21) F,-=0.i (25-2!)(272I)
(27-23).1'
//
/'/
\
0.03 0.02 0.0! 0 0.03 0.01 0.02 0 0 0.5 lO
15---'ETX20
Fig. 2b. Bending moment amplitudes for different, weight
distri-butions (Fr.=0. 2) (Cb=O.7O) V'0°(AAa4tmw) F,.=0.2 (23-49) (23-2!) (25-2!) (27-2!) (27-23
----,-1/
/j
(7'\
(Cb=O.6O) Fr. 0.2 ---(23-20) (23-18) (25-20) (27-20) .(27-22) -'/,/".J
1/'1;q
/,I\
\\
\.\
/,'
\.
70 Jun-ichi FUKUDA (VoL XXVI,
.1966) Computer Program Results for Response Operators of Wave Bending Moment 71
0
0 0.5 1.0
15 iLj 2.0
0 0.5 lO
15(L7X20
Fig. 3a. Bending moment amplitudes for different wave directions
(Fi-.=O. 1) (Cb=O.G0) (2520).J1r'L 0.25 tI0JL.e,1L0.20 0°
=150°:---,>/
,/
\
\
(Cb=O.7O) Fr.°°0.1 IXIL=0.25 ,,'=3g---::1go
::g"
0.03 0.02 0.0! 0 0.03 0.02 0.0!0.02 0.0I 0 0.02 0.0 1 0 0 05 10
15 -Z7X
2oFig. 3b. Bending moment amplitudes for different wave directions
- (Fr.=O.15) --.-(Cb=O.6O) ?c/L = 0.25
(25_20)l!a/L1fIL0.20
Fr.O.15 V'= 0 o=180°: - -0 (Cb=O.70) fac/L0.25(2S=2I).l,L
a 021Fr0.15
= 00: o18d3: ,,=1200:72 Jun-ichi FJEUDA (VoL XXVI
1966) Computer Program Results for Response Operators of Wave Bending Moment 73 0.03 1 0.0 0.01 0 0 0 0.5 10
I5(C7)20
Fig. 3c. Bending moment amplitudes for different wave directions
(Fr.=O.2) (Cb°0.60) (25=20): c/L 0.25
it/L=if/L=0.20
k= 00 ::2
= 300.... Fp°°0.2 "=150°: =1800: Oo/\
V; j'4
(C6=O.70)*=
Q°:30°:
---(25-2!)J?r/L-0.25 lIa/L=Jf/L=0.2!=,°
Fi-.=0.2 4=180°:_1i7
0 0.5 1015 -v'L7
20 00 0.02 0.0!0.03 0.02 0.0! 0 0.03 0.02 0.0!
Fig. 4a. Bending moment amplitudes as functions of wave
direction (Fr.=O. 1) (Cb=0.60) (25-20) Fr.-0.I ,i'oft.
',:
--N\
I,. ///i (Cb=O.7O) (25-2!)Fr.0.!
,.
T
74 Jun-ichi FUXUDA (VoL XXVI,
0 30 60 90 120 k 18
1966) Computer Program Results for Response Operators of Wave Bending Moment 75 .'.03 0.02 0.0! 0 0.03 0.02 0.01 0 0 30 60 90 120 180
Fig. 4b. Bending moment amplitudes as functions of wave
direction (Fr.=O. 2) (Ct O.60) (25-20) Fr. 0.2
-(Cb = 0.70) (25-21) Fi-.= 0.2Wi!!
!-0 30 60 90 120 leoCb=0.60 IL/A =1.0 = I80°(Fc/J
0.2
0.10.02
0 00
----u--- : (25-18)
----+---- (23-20)
-°---- : (25-20)
(27-20)
---A---- : (25-22)
---c--- (25 19)
----+----: (23-21)
--o---. : (25-21)
(27-21)
---A---- : (25-28)
---.
/
Q°(Head Wave)Fig. 5. Bending moment amplitudes as functions of ship speed
A
/
/
/
-...,..--.4._-,
,-0.!0.2Fr
76 Jun-ichi Fuzua& (Vol. XXVI,
Cb= 0.70 0.02 0.0I =1800 (Fcelo,,jM3Wa4e) 0 0.I 0 fr=Q° (Head Wwe) 0.1 0.2 Fi - 0.01 Wave)
0.003 O.O02 o.0oI
to
0.003 0.002 0.00l 0 t,dz4 4ea4) (bn 4t4.tctgo:t& t.2?Ja4 4tt4) 6 8 10 12 T(sec) 0003 0.002 0.00 I 0.003 0.002 b.°0I 0(4t i
csd (t2a4 4a4)
-000 -- '-OO(t-,t 4.t ctist44 bftda4 4el4)
4 6 8 SO 52
-
Tr(sec)Fig. 6. Standard deviations of bending moments in irregular seas as functions of visual wave period
L = I 5O, Cb=0.70
L I5O, Cb
0.6030 60 90 120 I80 L=I50.Cb=0.7O c,,.:(252ILFr.0.f5 (.. C.'.tc& LW4 4) 60 90 120 0.002 0.00I 0 ___44,'__ - --L=150,..Co=0.60 C.,,t.:(25-20), Fj=O.I5 (.. 4&4) 30 60 90
I20 -.
S80i966) Computer Program Results for Response Operators of Wave Bending Moment 79
Appendix
The calculation method for response
operators in regular oblique waves
The calculation method for response operators of ship motions and vertical wave bending moments at midship was conducted based on the linear strip theory given by Watanabe'2 which is similar to Korvin-Kroukovsky's theory" in principle. The application of the theory
was attempted here in regular oblique waves, though the original theory had been established
for cases in regular heading or following waves. In this calculation method, only the heav-ing and pitchheav-ing motions are considered ignorheav-ing the other motions: surge, roll, sway, yaw and drift.
Consider the case when a ship goes forward with a constant speed in regular oblique waves.
As shown in Fig. 8.9, the co-ordinate system O-XYZ fixed to the space is employed such that the XY-plane coincides with still water surface and the Z-axis indicates the upward di-rection perpendicular to the XY-plane. The co-ordinate system o-xyz fixed to the ship is determined such that the origin locates at midship on the centre line of water plane, the x-axis points out ahead the longitudinal direction and the z-x-axis upward. The ship goes straight on with a heading angle 0 to the wave coming from the positive X directionto the negative
X direction. Then, the surface elevation of regular oblique wave encountered with the ship can be expressed as
z
0
(=Ao cos((ziu,.t)
Fig. 9. Coordinates in ship centre vertical plane
x
Fig. 8. Coordinates in sea surface
OX section z PP' section
(h44 a/A)
z X,80
M0 = T./Mq,C2 + M,52, IM = '(M,,5/ M,)
ac, be,..., g,,; A,,, B1,,..-, G and Fe,, Fr3; M,,, are determined as follows by using the nu-merical calculation results described in Table 4-5.
ac =a0±a1, a0= [00](), a1= [l0] b =[20](s)
Cr =[30](s)
4', = d1 - xcaj, d1= [11]() e.1, =e1x0b--Va1, e1= [21]()
g1, .=g1xcVb,
g1=[31]()A,, A0±A1, A0 [02] XG2[OO] Ai= [12] 2x0dj+x2a1
B.', =Bl-2xGej+xG2bC, B1=[22]() C4, =Ci-2xgi+xc2cVE, C1 [32](3) Dr d4, Er =elxGbC± Va1
G =gixc
F.
h0(f1±f2+f3), F5 =h0(f1'±f2'±f31) fi =[30C1 11' = [30S}() 12 W[2OSJ 12' W[20C]) 13 =cow,[lOC]fS' =ww,[l0S]
where x is the x-coordinate ofM',=ho(rn1±m2±m3) M,',3 = h0 (m1'±m2'±m3') m1 = [31C]C5xcf1 m1' = [31S] ()xGfj'
m2 =W[21S)Xcf2
m2' =o)[2lC](xGf2' m3 = cow[1lCJ )xGf3 coV[1OS](s) m3' = _WWe[llS](3)_XGfS'+WV[lOC]Wthe, centre of .gavty qf s1ip,
Jun-ichi FUKUDA (Vol. XXVI,
(7)
h =h0cos (k*x+wet) (3)
in the vertical plane including the ship centre line.
where
= wave amplitude
k* =kcoscl', k=2n/A, A=wave length
We C0 +kVcoscl' = encountered frequency
w =/kg =wave frequency, g=acceleration of gravity V = ship velocity
The differential equations of heaving and pitching motions in regular oblique waves are obtained by the aid of the linear strip theory as follows:
a± b+c±d,qc +e4±g,q=F
whereC =heaving displacement, qS = pitching angle
F (Wet ± aFt)
M4, = M,coscot - MsinWtM4,0cos (Wt± iM) F0 = VF,2 +F52,
a =
(F5/F,)966) Computer Program Results for Response Operators of Wave Bending Moment 81,
In the numerical integrations, in Table 4, Fo, P, P2 and P3 are defined as follows: Po=w/g, w=weight of ship and load per unit length
P1=pC0K4('r/2)y2=sectional added mass
p2=(pg2/w3)A2=sectional damping coefficient (8)
P3=2pgy, y=halfof waterline breadth p ==density of sea water
The values of addes mass and damping for sections are derived by means cif Tasai's method'. Further, the following terms are introduced in the numerical integrations to determine the forces and moments induced by regular waves:
(9)
eTm=exp - (27r/A).(dc), (10)
where
d= drought at x
c=sectional area coefficent at x
As shown in Fig. 8, the surface elevation of regular oblique waves encountered with the ship can be expressed as
h(x, y, t)= hos(kxcOsç1' - kysifib +ot)
at a location p(x, y) in a transverse section Of the ship. The average of wave elevations in a transverse section of the ship is taken as
h(x, :) =
Jh(x
y,t)dy= Cehocos(ktx+wet)
where
C==sin(kysinç1')/kysinØ
Hence, Ce is the coefficent representing the average of wave elevation in a transverse section of the ship. In head waves (cb=0) or following waves (ç'='l8O) and in long waves where Icy,,, is very small, C,, come to a unity and he(x, t) given by the formula (11) coincides with the wave elevation given by the formula (3).
The term of eim represents the effect of decreasing orbital motion for the wave force which is so called "Smith effect".
The solutions of the differential equations (4) are obtained by solving the following al-gebraic equations:
where
bi*=_co,,Dc±Gc
a2*= O,),,bc;
a3*= w,,2d-i-g, b*=we2A+C,,
Thus, the solutions of heaving and pitching motions are derived in the form of al*Cc + a2*Cs+as*cb+a4*c1,=Fc
'1
M4,
I
82 Jun-ichi FuKuii& (Vol. XXIV, C 1 (14) &coscot - 3sinw0tosin(cot+ j9) i where
C0/Cc°±Cs2, d=tan1(C/c)
1 (15) qo VqS°+ ç32, j9= tan1(q3/) Then, the vertical motion at x is derived in the form ofZZccoswetZssinwetZocos(Wet+1z) (16)
where
Z==C+(xxc)cb, ZSCS+(xxG)c6
Z0 =:1/Z° +Z32, c2==tan-1(Z3/Z)
And, the relative vertical motion with respect to the wave surface at x is form of 2', = ZrccOS wt - Z7351fl0)etZroCO5 (w0t +c2) where Mc: (21) M, =
In the formulae (21), d1*, d, d3, d4, r. and r3 are given by follows:
d1*= we°pi±ps d3*= we°qi'±qoqo d2*= 0e(P2+P2'), d4*= weq2' (22) and r= h0(r1±r2+rs) r3=h0(r1'+r2'±rs') where Pi = [O1](d)+[11}(d) P2=[211(d), poI=V[10](d) q1_[02]Cd)_xc[O1](a)+[l2]&)_xG[1l] q2'= [22](')x0[21](") q3'= [32](i)_x0[31](d)_Vp2' q3=Vp2 -Z,, =Z, - cosk*x, Z,,3 = Z, - sink*x
7_,/7272
7U v c - r C, zr=t2fl'(Z./7
----'.-rCJ rU,The vertical wave bending moment at midship can be obtained by the calculation using the solutions of equations (12) and the numerical integration results in Table 4-5, in the form of
M= Mcoswt - MssinwgtMocos(wet± ô) Mo=VM2± M2, = tan(M3/M,)
(20)
where
196) - - Computer Program Results for Response Operators of Wave Bending Moment 83
Afterbody [00] a = JPodx
Forebody IOO]f = JPodx
Afterbody [10] a = JFidx Forebody [l0]-= JFidx Afterbody [201 a = JP2dx Forebody [2011 = JP2dX Afterbody [30] a = JPdX Forebody [30]= fPadx Afterbody Forebody Afteobody Forebody Afterbody Forebody = [31CJ Cd), r2 = w [21S] Cd) - [31S](''), r2r= _w[21C](d) r3 ww[11C]&)±cOV[10SI( ra'=ww6[11SJ (d)_(ov[1oC] Cd)
Table 4. Numerical integrations
[1CC] a =f C6 e-_kampicosk*xdx [105] a = 5C6 e_kdmpisink*xdx [1CC]1 = 5C. ekdmPjcosk*xdx [10S] = Jc6 e_hdmpisink*xdx [2CC] a =f C6 e_kdmp2cosk*xdx [205] a = JC6 e_kdmp2sink*xdx [20C], = Jce e_kdmpscosk*xdx [2051í = 5C6 e_kdmF2sink*xdx [3CC) = Jc6 e_kdmpscosk*xdx [30S] a = 5C6 e_kdmpasink*xdx [3CC], =J C6 e_kdmp8cosk*xdx [30S] = Jc. e_kdmPssink*xdx [01]a = [01] = [lila = [1l] = [2i] = [21] = [31] = [31]= JPocdx JPoxdx fP1x4x JFixd JP2xdx JPzxdx JPsxdx JF3xdx [021a = [02l = [l2la = [12] = [22] a = [32] a
Numerical integrations should be carried out from A. P. to midship for the afterbody and from midship to F. P. for the foredoby. F0, Fi, P2 and P3 are defined as given in (8).
fPox2ix fPox2dx JFix2dx jPix2dx fr2x2o JP2x2dx JF8x2dx [hG]0 = 5C. e-kdmPicoskx.xdx [115] = JC6 e_kdmptsink*x.xdx [hG]1 = 5G. [11S]t = Jc6 e_dmFisink*x.xdx [2W] a = JC'6 ekdmP2cosk*x.xdx [215] a = JC8 ehdmP2sink*x.xdx L21C], = 5 C6 ehdmF2cosk*xxdx [2lS] = JC. ekdmP2sink*X.xdx [31C]0 = JC6 e_kdmPacosk*x.xdx [315] a = JC6 e_kdmPasink*x.xdx [31C), = 5 C6 e*dmPocosk*x.xdx [3iSJ = JC6 ekPasink*x.xdx (25) [32]= JPax2dx
Table 5. Results of numerical integrations [00] = [oO]j + [00] a [00] [O0]j [00] a [01] ) [Ol]j + [01] a [01] (") [Ol]j [01] a [02] )= [02]i [02] (d) [02] + [02] a [02] a
[1O](3)[1O]i+[10] [l11(3)[ll]i+[l1)a [12](3[l21i+[l2]a
[10] (d) [10] [10] a [11](d)=[11]i [11] a [12] (d) [12]í [12] a
[20] ) [20]r + [20] a [21] (s)= [2l]j + [21] a 1221(3)r [22] + [22] a
[20] &) [2O] [20] a [21] &) [2l] [21] a [22] (d) [22] [22] a [30] [30]f + [30) a [31] [31]i + [31] a [32] ) [31] + [32] a [301 (") [3O]1 [30] a [31] (d) [31] [31] a [32] (d) [32] f [32] a
[1OC] C3 [1OC)j + [10C] a [11C] () [11C]1 + [lid a
[1CC] ) [bC]1 - [1OC] a [lid (d) [lid1 - [hG) a
[lOS] (s) [lOS]1 + [105] a [115] (s)= [hiS]1 + [115] a
[lOS] &) [lOS]1 [1051 a [uS] (d)= IllS]1 - [uS] a
[20C] ) [20C]1 + [20C] a [21C] (s)= [21C]1 + [21C] a [2CC] (a) [2CC]1 - [2CC) a [21C] &) [21C], - [21C] a 120S] [205], +[205]a [21S] (5 [21S]1 +[215]a [205] (d) [205], [205] a [215] (d) [21S]1 - [21S] a [30C] () [30C], + [3CC] a [31C] (' [31C], + [31C) a [3CC] (d) [30C]j - [3CC] a [31C1 (d) [31C]1 - [31C] a [305] ()= [30S) + [305] a [315] ()= [31S] + [31S] a [30S] (d) [30S] [305] a [31S] ('1) [3lS] [31S] a
