by M. Takagi Y. Nekado M. Matsui M. Mokunaka
y,
HITACHI ZOSEN TECHNICAL REPORT
NO.1
Activated Anti
-
Pitching Fins
HITACHI ZOSEN
TECHNICAL RESEARCH LABORATORY: OSAKA, JAPAN.
HITACHI SHIPBUILDING & ENGINEERING CO., LID.
A Study on
Introduction
The rolling motion of ships most influences their performance Ofl tile point of view concerning com-fort and safety. In the light of dus fact, a number of researches have been conducted for many years and many effective methods for the reduction of rolling motion have so far been exploited and
developed.
On the other hand, the pitching and heaving motions are not such as would directly endanger the safety of ships and, for this reason, researches in this particular field have relatively been neglected. However, with recent advances in theories on sea waves and wave-making characteristic of ships, the study on pitching and heaving behavior has been pursued actively. As the results, their close relation with the propulsive performance of ships has been clarified, and various attempts have been undertaken on measures to reduce pitching.
The pitching motion, in particular, is known to have much to do, besides the propulsive per-formance, with the comfort of passengers and the slamming at the bow and bottom. Thus, the
re-duction of pitching motion is considered to be
indispensable for improving tile sea-worthiness of
ships.
For the reduction of pitching motion of ships, two methods are now attempted, the anti-pitching tank1'2 and anti-pitching fjB) They, though
pro-mise to prove efficient to a certain extent, are
uncontrollable and are limited of their effects which are somewhat smaller in comparison with the size of the equipment, reducing much their practicability. In view of this, tile authors hit upon the idea of having automatically controllable an ti-pitching fins, by which satisfactory results even in a practicable size of them can be expected.
The present study concerns a theoretical and ex-perimental clarification of the efficacy of reducing pitching when the anti-pitching fin is so remodelled that it may be operated by automatic controls.
Activated Anti-Pitching
Fins
By Matao Takagi, Dr. Eng.; Yukiharu Nekado &
Masahiro Matsui: Technical Laboratory
Masaru Mokunaka, Shipbuiiding Division Hitachi Shipbuilding & Engineering Co., Ltd.
Fig. 1 Coordinate System
Equations of Motion
Consider a ship among the regular waves which progress in the opposite direction of the advance of the ship.
As shown in Fig. 1, the cartecian coordinate system is taken so that the origin O coincides with the center of gravity of the ship at rest, the axis of
x in the direction of the advance of the ship and the axis of z vertically upwards.
If the wave elevation in a place is given by
h(t, x)=h0 cos (wt+x+) (1)
the vertical velocity of the fluid by the orbital motion is as follows:
V,
-o0h0e' sin (wt+-x+e)
(2)where h0 a half of wave height,
w: circular frequency of encounter, A: wave length,
E: initial phase,
d: the distance from water line to fins, and w0: circular frequency of wave.
When the fins are fitted at a distance of i from the origin, the attack angle of the stream to the fins is given by
where z : heaving of ship,
pitch angle of ship (positive to clock-wise direction),
ô' : deflection angle of fins (positive to
clockwise direction and zero when they are parallel to the water line of ship at rest).
Then, the lift and the clockwise moment become
M1= plVs2A'1
ôa
&CL where A is the projected area of fins and is
the slope of the lift-coefficient of fins.
W.L.
0
(3)
(5)
Fig. 2 Relative Velocity of Water Particles
Now, the equations of heaving and pitching mo-tions for naked ships can be obtained by the strip method and are given as:
M,+ N, + B,,z + M + + = F2h(t) ( 6)
Mp+ N± +B,z + M4 + Nçi' +B4' = F,h (t+ r) (7)
where M, N and B correspond to the terms of
inertia, damping and restoring forces, respectively, and F,, and F represent the external force and moment, respectively. Furthermore, h(t) is time
factor of the waves.
In case the ship is equipped with the fins, the equations of motion can be obtained by adding
the lift and the moment to Eqs. (6) and (7).
Namely,
M,, -- N +B,,z+ Nfr +B4,
\
=Fh@+r) + PlVS2AaCL (8+ +
y,
)
where the added mass of fins is neglected.
Putting x=i in Eq. (1) and considering the rela-tion h(t)=h0 cos (wt+E), we obtain
V=-e
cosl./(t)a,e' sin -l.h(t)
h(t+ r) = CO5 O)T Ii(t) + - Sin iz(t)
Substituting these equations in Eqs. (8) and (9), we obtain the following equations of heaving and
pitching motions for the ship equipped with fins:
M,, + (Ne. + pV,A& )+ B,,z
+M+(N _-p1V4-!)P
+(Bz+pvs2A}P
sin I &CL O (Z lii (t) +- -p V,A---e
COS a 0)+(
i --plV,2A 2=(F
cos or---p1V,A-o)oe ' sin+(
i . 1 CL o - 27rF sin tor+plV,A
2e
Oa cos__l)h(t) =F,,h(t)_pVs2A& (s+_1_ V)
(8) +--plV,A6 8(t) (13) vs v(Vs: advance speed of ship) MZi + N, + B,z+ Mq +Nç + B4
Lf= ( 4)
---pV,A-- 8(t)
(12)Transfer Functions & Frequency Responses
Hereafter, Laplace transforms of the equations of the ship motions will be used for conveniencc of treat of a feedback control system.
Since tue free heaving and pitching motions soon damp out, the forced motions only are considered. Laplace transforms of Eqs. (12) and (13) under the condition that all the initial values are equal to
zero give (M's2 + N'2s+B')z(s) -- M's2 + N's+B'2)fr(s) =(Fzo+F1s)h(s)+ A8(s) (14) (M's2 + N's+ B')z(s) + (M.1s2 +Ns+B'>fr(s) =(Fo+Fjs)h(s)+4,8(s) (15) where B M',=M4 N'=N. + pV8A , N'z
B=B+ipV02A,
2N', =N _pl VSAL N' = N + pl2VA',
B'=B_IplVs2A 0CL 2 B '91=B2, 1 0CL 2rF0=F--pV3A
w0e ASfl 1
2 ôa À F1=-1-pT'3Aü-fd
27r 2 6a w cosTl, 2r Fo=F COS wrplVsAO Lwoe A140=_pV02AL,
z(s)=11z(t)= e_Atz(t)dt,
(s)= (t)
=
et(t)dt.
For the sake of simplicity, put
where
a(s) = /5ì(s)F0(s)
-b(s)=J(s)4z(s)4,
c(s) = (s)F2(s) + (s)F(s),
d(s)= (s)A0+(s)4.
When the fins are operated by tue signals which are composed of i, and , the fin angle b', is
represented as
fr(s)=M,s2+N's+B',
Fe(s) = F00 + F01s
F(s)=F0+F1s,
then, we can express Eqs. (14) and (15) as a fol-lowing matrix equation:
Ia(S)
7z(s) z(s) Fi(s) ii0 h(s)
= ...(16)
(s) i7s) i/i(s) F, (s) A 6(s)
of which the solution becomes
Íz(s)
i a(s) b(s) h(s)
sb(s) e(s) c(s) d(s) 8(s)
And Laplace transform of Eq. (18) gives
j 8(t)8(s) = (k3s2 + k2s+ kj>/ì(s).
From the above expression, we obtain a transfer function from the pitch amplitude to the fin angle
as follows:
F1=F-1- . 1
8(s)
w Sfl
wT+--plV0A-co0 --o 27T k(s)r----=k:352+k2S+ki.
e A
w cos À i,But, strictly speaking, we must consider the first or the second lag element of this feedback system. Therefore, the transfer function has to be written as
6(s) K3s2+K2s+K0 =G(s)
1i(s) - T3s2+T1s+l
which is called a feedback element and where
K1=kk1, K2=kk2 and K3=kk3. Eq. (19) suggests that the angle of fins can be varied in accordance with the values of (K1, K2, K3), reducing the heaving and
(17)
(18)
pitching motions. Thus, vector (K1, K2, K3) lias a very important meaning on the following
considera-tions. In this report, it is represented by the letter K and is called control vector.
Now, the question is that which value of K has to be selected for the most effective reduction of
niotions. At lirst, we are struck with an idea of
making amplitude of the fin angle as large as
possible. But it is undesirable to enlarge it so
much that the fins may increase the ship resistance or may stall. So we need to seek the most desirable value of an amplitude of the fin angle by the
method of experiments.
Now, the value of K is assumed to be varied keeping the amplitude of the fin angle constant through all the frequency region. Furthermore, since the ship is among the regular waves which have constant wave height, we may obtain the following condition:
8,
= r0 .(const.)
h0
From Eq. (17), we obtain
z(s)=h(s)+
8(s)1j(s)=
hs)+8(s)
In the above equations, if we 1)ut
=A(s), =B(s),
'=C(s)
A(s), B(s), C(s), and D(s) may be called transfer functions, which compose a feedback control
sys-tem with C(s) as shown in Fig. 3.
C(S1 At St
F-BC S) +
Dt S)
Fig. 3 Block Diagram
Next, substituting the following equation;
8(s)= G(s) fi(s)
--fr(s)
k(s) i(s) (23) where k(s)=K3s+K,s+Ki, t(s)=T2s2+Tis+1 and Ç-=D(s)in Eqs. (21) and (22), we obtain the transfer func-tions from wave height to ship motion as follows:
z(s) a(s)e(s)t(s) + ( a(s)d(s) - b(s)c(s) )k(s) h(s) e(s)( e(s)t(s) + d(s)k(s)) çl(s) c(s)t(s) h(s) e(s)t(s)+d(s)k(s) 6(s) c(s)k(s) h(s) e(s)t(s) + d(s)k(s)
As well known the substituting of jw for s in a transfer function gives a complex frequency re-sponse of motions of a ship, Z(jw) and !It(jw); that is
Z(jw) z(jw) - h(jw) ' From Eq. (18), 8Qw) =r0 exp (1g) h(jw)
By substituting the above in Eq. (26), we obtain
kÇ )
cQw) + rodaw) exp (f9)r0efw)tQw)exp(j8)From the above equation, we can obtain the opti-mum K under the condition (20) for values of a e, and it is determined from the condition that the heave and pitch amplitudes have minimum values. From Eqs. (21), (22) and (23), we get
z(jw) a&w) + r0b(fw) exp QO) (29) h(jw) ejw) e(Jw) h(jw) - eQw) e(Jw) c(joo) d(jw) +r0 exp(j9) (30) and if we put 6= 8,arg { a(jw)/b(jw)) O=Oiarg { c(jw)/d(jw)),
we can get the minimum value of heave, Z(w,K), or
pitch, (w, K) as follows:
!V(jw)
h(jw) (27)
(28)
where L and L are the values of K when the
values of e become e and O . Furthermore,
--and -- are the frequency responses of the ship equipped with the fixed fins, corresponding to the case of K=O. Hence, Eqs. (31) and (32) can be
written as
Z(w, K,)zr a(jw)
1r0
bQw) (31)e(jw) a(jw)
Jt(w, K) c(jw)
I r0
d(jw) (32)KiK3w2=R,(k(joi))
K2w = Im( k(jw))
9=arg-o'
b(jw) - d(jw) in Eq. (37), the frequency motion become minimum
H(w, K)=H(w, O) 1 r0
which suggest that the activated fins may reduce the heaving and pitching motions by X100(%)
and r0 X100(%), respectively, in comparison with those in the case of the fixed fins. Generally, how-ever, since K, is not equal to K, the said motion reduction cannot be obtained simultaneously.
Therefore, in actual case, we must consider the problem in which the resultant motion in a place is minimum. Let us consider to noinimize the re-sultant motion at a distance of from 0. Since the resultant displacement is
(35)
the frequency response at the place becomes
H(jw) = Z(jo) - ¿T(jw) a(jw) b(Jw) exp (jO) e(jw) e(1o) c(jw) d(jw)
-
+r0 exp(j(e(jw) e(Jw) O)). (36)
If we substitute
side, having symmetrical cross section.
Now, supposing on conditions that 6=30°, h0 20mm and V,=l.34m/s, the results of calculation carried out are illustrated as shown in Fig. 4 and Fig. 5.
(37)
responses of the resultant and are given as follows:
a(jw) - d(jw) b(jw)
-(38)
where R, and I,,, are to be taken as real part and imaginary part, respectively.
Examples of Calculations
On the basis of the results discussed in the fore-going chapter, theoretical calculations were conduct-ed referring to a model ship usconduct-ed in an experiment. For the particulars of the model ship, refer to Table 1. The values of the coefficients in the equations of motion are calculated by strip method. The fins used in the experiment are plane foils of square plan of which a span is 120 mm in each ship
0.5
Fig. 4 Comparison of Calculated
Val-ue and Experi-mental Value for
Frequency 15 ç 1.0 -c 0.5 1.0 0. 5
Fig. 5 Comparison of Calculated Value and Experimental Value for Frequency
Fig. 4 shows the frequency responses of heave amplitude, where the heave amplitude is divided by the wave height, and w0 represents the natural angular frequency. Also, Fig. 5 shows the frequency responses of pitch amplitude, where the pitch am-plitude is indicated in radians and is divided by the wave height.
In these figures, there exists a point where the response becomes zero. In order to realize this,
the infinite gain of amplification on the control system is necessary, but it is impossible in practice. In the range of higher frequencies than the zero response frequency, the forces excited by the fins
under =30° may be larger than those by the wave,
Z(w, K) = Z(, O). qr(w, K)=JT(w, O).
lr0
I r0
b a d C (33) (34)The elements of K (K1, K2, atid K3) can be obtained 0.5 1.0 1.5
because of too large deflection angle of the fins. Therefore, it is desirable that the fin's angle is
limited to where ô'0 is less than 300. That is, when the wave exciting forces are large enough, it is
desirable that the amplitude of the fin's angle is taken as large as possible. On the contrary, when the exciting forces by the waves are small, it is
necessary that ô' is decreased according to the
ex-citing forces. The method described in the above corresponds to that the value of K cari be varied
freely. In actual practice, however, the control sys-tem includes more or less the non-linear elements. Therefore, it is necessary to take such elements into consideration when the above theory is applied. The control system which is used for the experi-ment described in the next chapter includes also the non-linear elements. Therefore, it is supposed
that the experimental points as shown in Fig. 4 and Fig. 5 do not coincide strictly to the tlieoreti-cal values tlieoreti-calculated from the optimum value of K.
Model Experiment
Model Ship and Fin for Experiment
For the model experiment, the Ship Model Basin of University of Osaka Prefecture was used, and the model ship was tested in regular head waves, changing the period of encounter. Namely, the wave length only was changed, with the wave height and ship velocity kept constant; 3/4 full speed.
6
Projected area
Span
Maximum foil thickness
Aspect ratio
Table 2 Fin Particulars
0.035 Aw 240 mm
15mm
2.00 *
* Remarks: Value is calculated from the sum of
both chords.
A common passenger boat was selected for the model experiment, as shown in Table 1. The natural period of pitching was set at 0.9 sec. The fins for the experiment were made of a plane foil
Photo i Larger Fins Mounted in Bow Part
having a streamline section. The particulars are as shown in Table 2. These particulars were deter-mined by referring to the experimental results ob-tained by fixed fins previously tested4. Aiming at obtaining a greater effect as far as possible, the fins are made somewhat larger. As indicated in Photo
1, the fins are mounted in the bow part.
For measuring, a potentiometer was used for
heaving, while for pitching angle, a vertical gyro-scope was used.
Control Device
The control system is represented in Fig. G by a block diagram. The control device, surrounded by a chain line in the figure, is considered in terms of three sections divided according to their action. Primary means: The pitching angle p is picked up by a vertical-gyroscope, and the angular velocity by a rate-gyroscope. In this experiment, the pro-cedure to take out the signals of by using an angular accelerometer was omitted.
Controlling means: This means is composed of the following items. First, the pre-amplifier carries out amplification of the gyroscope output, and phase sensitive demodulation. The multiplier (potentio-meter) controls those output signals. And the servo-amplifier gives D.C. power.
Operating unit: This operates an actuator, which is made integral with the oil-hydraulic servo-valve (piston stroke: 25.1 mm, piston output: 10 kg, oil
Table i Principal Dimensions of Model Ship and
Conditions of Experiment
Length P.P. 2.400m TGM 0.032m
Beam, moulded B 0.412m Water line areaA 0.798m2
Draught, mouldedD 0.200m B lock coefficient CB 0. 581
Draught full load
line 0.124m df
0. 117m
Displacement 74. 8kg Draught 0. 139 m
Disturbance (waves)
-o----Photo 2 Operating Unit
Power Source for
Generating Gyro.
pressure: 20kg/cm2), and the fin. Photo 2 indicates
how this unit is mounted. The values of the fin's angle can vary from ±32.5° to 32.5° in accordance with the piston stroke. In the experiment, the move-ment of stroke was detected by potentiometer to record fin's angles.
From the point of view on the similarity law be-tween actual ship and model, the output 10 kg of the oil-hydraulic servo-valve is considered too large. But, due allowance is made for mechanical fric-tion, etc.
Besicles, the variation of the lift was recorded
Ship Motion
Amp. Source DC 24v
Fig. 6 Block Diagram of Control System
o 20 - 40 - 60 - 80 be .- 100 180 200
2
4- 6
- 8
- 10 -o 12 16 AC. 100V 60 C/5 400 C Frequency %) 0.5 t 3 10 30Fig. 7 Frequency Responses of Servo-Actuator
by using a (linear) differential transformer. The synthetic characteristic of the control device was derived by the simplest method, in which the relation between gain and frequencies or between phase and frequencies was obtained by recording the output to a sinusoidal input in various fre-quencies. The frequency response is represented in a Bode diagram as shown in Fig. 7. And, in terms of the frequency range in the experiment, its
at-tenuation and phase lag are thought to be practically
negligible.
Primary Means Controlling Means
Dil-Hydraulic Pump
-20kg cm'
3L
Vertical AC P
Idrer -
DC Multiplier (Potention Meter Gyroscope re -am p Final AC Oil-hyd-Adder S e reo-amplifier raulic Serva-valve Control Element Rate AC DC MultiplierGyroscope Preamplifier (Potention Meter
AC DC Recorder C Preamplifier Actuator AC Fin's 400 % AC DC Angle 115V Power Power
Supply Supply Pifching
Pitching Angle
'2Omnf
Heaving- Fin Control Commenced
Deflection Angle of Fin
''
i7Wave Height
Lift
Period of Encounter
Fixed Fins Pitching Angle
Deflection Angle of Fin
Wave Heigh Period of Encounter 39mm 2
30
(s) I S-Activated Fins Servoamp's Gain Small Pitching Angle63mm
Deflection Angle of Fin
eight
Period of Encounter
Experimental Results
As already described, the signal of effective feed-back of activated fins depends on the angular
velocity s. Accordingly, the experiment was carried
out with emphasis placed on tue feedback of
. To
feedback the signal of the angular accelerometer was not assumed to be much effective, and it in-volved technically difficulties. Thus, it was omitted in this experiment, which was performed by the following sequence:In case the signals of rate gyroscope only were used.
In case the signals of vertical gyroscope only were used.
Activated Finn
Servo-amps Gain "Medium Pitching Angle
1 2 3
(S)
Period of Encounter
1 2 3
Fig. 8 A Record in Comparison of
"Fixed Fin" and "Activated Fin"
(s)
Fig. 9 Examples of Experimental Records
Activated Fins
Servo-amps Gain "Large'
Pitohing Angle 0.16' 65
LifluN
er odof 1 2 3-.. (s)
(3) In case the signals of both rate gyroscope
and vertical gyroscope were used.
By a combination of the case when the gain of the servo-amplifier on the control circuit was changed to "large", "medium" and "small", and the case when each signal is regulated by means of a multi-plier, many more conditions of experiment may be
assumed. But in this experiment, sorne of them
were conducted.
First, Fig. 8 and Fig. 9 show the examples of the experimental records. The former indicates the dif-ference between "fixed fin" and "activated fin",
showing the process of the reduction of heaving and pitching. The latter indicates the conditions of change of the reducing effect, depending on the
Heaving Heaving Hoaving Heaving
7, 5mm
6mm
-4--Deflection Angle of Fin Deflection Angle of Fin
Litt Litt
E 0 1.0 0.6 0.4 0.5 08 10
Fig. 10 Frequency Responses of Pitching Motion
\
\"\
' 'SS 02\. \
1.5Fig. 11 Frequency Responses of Heaving Motion 0.8 E 0.6 0.4 0.2
Fig. 12(a) Frequency Responses of Pitching Motion
N
15
0.5
Fixed Fins
Fig. 12(b) Frequency Responses of Heaving Motion
large or small gain of tise servo-amplifier. If the gain of the servo-amplifier is small, the movement of the foil is small. If the gain is excessively large, the movement of the foil, as indicated on the right hand end of the oscillogram, is saturated and takes
the pulse form like a trapezoid, which, requires separate theoretical treatment.
Fig. 10 and Fig. 11 show the frequency response of pitching and heaving. As seen from Fig. 10, the reduction of pitching of ship with the activated fins is about 90% as compared with that of a naked ship (without fin), in the neighborhood of tuning factor e=l. However, the movement of the fins in this case is in the pulse forni.
The control constant K1', K2' correspond to the
Conditions of Fin Marks GainsServo
amp ContrOl Constants Fin s ogles K =Kfr+K4--- Medium 0.75 0.67557-03' -(3) =K +Kp---- Medium 0.50 OE675 42-90' (3) =Kp-f-Kp----a--- Medium 0.250.67542-51' (3) =K,fr±K, - Medium 0 0.675 46-48'
, (3) =Kçb -l-KØ----o- Medium -di1O.675 25- 42'
Conditions of Fin Marks Gains
Servo-amp Control Constants . Fin s Angles K K', Without Fin -°---
--
-Fixed Fins
-.--- - - -
O. (1) &=Kç0 ---d--- Small 0 1.0 37- 42' (1) ô=Ktb d Medium O 1.0 47-51' (1) 5=Kp ---*---- Large 0 1,0 65' (2) =K,ip --e'---- Medium 1,0 0 30- 34'(3) ,=K;p+K;b -b- Medium 1.0 0,9 63- 65'
, (3) =K'p+Ki v- Medium 0.5 0.9 54- 63'
(3) ô1(+ -'--O-'- Medium 1.0 0.45 54- 03'
Gains Conditions of Fin Marks Servo'
amp Control ¿o Constants K Fins An les (3) =K-i-K----x--- Medium 0.75 0.67557"- 63' Medium 0.50 8,67542- 50' .21(318 K +K6b-o-- - Medium 0.25 0,67542- 51' (3) 5 K +Kt--- Medium O 0,67546-40' ,)( (3) ô =KJi +KÇ-cs--- Medium -0,500.67525- 42'
Conditions of Fin Marks Gains
Servo-amp Control -Constants F'A -es g K K Without Fin
-Fixed Fins
-.- - - -
O' 1) &=Ki.P ---'d--- Small 0 1.0 37- 42' (1) 8=Kçb ---i--- Medium 0 1.0 47- 51' Li (1) 8K,çt, ---*--- Large 0 1.0 65' (2) ò=Kø ---x--- Medium 1.0 0 30- 34'(3) 8=çO+Rçb ---&--- Medium 1.0 0.9 63- 65'
(3) 8KØ+Kçb ---r--- Medium 0.5 0.9 54'- 03'
(3) =K0 -I-Kçb ----o---- Medium 1.0 0.45 54- 63'
1.4
1.2
10
scale factor K1 and K2 as aforementioned, but do not indicate the same size. K1' and K2' show the degrees of control of i and , obtained by
multi-plier, and, the positive indicates a negative feedback, and the negative is just the opposite.
Fig. 12 (a) and (b) pertain
to an attempt at
detecting the influence to be caused by a change of the control constant K1' of the signals due to pitching angle. No definite tendency is noted re-garding the effect of reduction here. But in regard to heaving reduction, it is conceived that the effect of fins is to a considerable degree. And it isas-sumed that K1' has a considerably large effect for tile reduction of heaving.
Conclusion
As a means of reducing pitching of a ship, the performance
of an
activated fin was examined theoretically and experimentally.It is expected from the calculation that the
maximum pitch amplitude of a ship with controlled fins as used in this experiment is about 75% lower than a ship without fin under the best condition. On tile other hand, the results of the experiment showed also that the effect of pitching reduction was exceedingly great, as indicated by the values of more than 80%. Thus, it is obvious that to activate anti-pitching fin is very effective. But, we reach
lo
the optimum state by the method of try and error in the experiment and it is very difficult to realize the optimum control system which is predicted by the theory. So, the relation between control vector, K. and control system must be studied.
Finally, the authors wish to thank Prof. Hishida of University of Osaka Prefecture for his kindness in lending a tank for experiment as well as various other conveniences.
Also, it should be acknowledged that the study has been conducted with a subsidy gi-anted by
Min-istry of Transport.
Reference
E. Tasaka S others; "An Experimental Study
of the Effectiveness of Anti-pitching Tanks",
Journal of Zosen Kiokai, Japan Vol. 117, June
1965.
Y. Nekado & M. Matsui; "On the Motion of Ships with Anti-pitching Tanks", Journal of the Kansai Society of Naval Architects, Japan No.
114 1964.
A. Abkowitz; "The Effect of Anti-pitching Fins on Ship Motions", SNAME 1959.
M. Takagi S others; "Methods for the Reduc-tion of Pitching MoReduc-tion No. 2On the Effect of Anti-pitching Fins", Hitachi Zosen Technical Report 1965.
