17 OKT.197
ARCHI EI
PAPERS
OF
SHIP RESEARCH INSTITUTE
Wave-power Absorption by Asymmetric Bodies
By
Makoto KAN
February 1979 Ship Research Institute
Tokyo, Japan
Lab. y. Scheepsbouwkune
Technische Hogeschool
DeUI
WAVE.POWER ABSORPTION BY ASYMMETRIC BODIES* By
Makoto KAN** SUMMARY
Numerical examples of the efficiency of the wave-power absorption by
the oscillating two-dimensional cylinders with asymmetric sectibnal shapes
are presented. The choice of the sectional shapes is made intuitively so that the maximum efficiency of the wave power absorption in a single mode
of heaving or rolling of the cylinder may be high accorthng to the theory by Evans Mei or Newman Comparisons with experiments on the Salter cylinder were also presented
A simple formala for the damping coefficièñt of a rolling Salter cylinder
in the limit as the wave number tends to zero i also given.
1. INTRODUCTION
When a two-dimensional cylinder with a symmétric sectional shape is suspended relative to some stable reference platform by a system of
masses, linear springs and dampers and is constrained to oscillate
in a
single mode of heaving, swaying or rolling in an incident sinusoidal wave
train as shown in Fig 1, it is known that the damper can absorb 50
per-cent of the energy Of the incident wave train under the following two
conditions The first conthtion is to make the natural frequency of the
oscillation of the cylinder with the system equal to the frequency of the
incident wave by adjusting the spring constant The second condition is
to make the damping coefficient of the damper eqi.ial tO the wave damping
coefficient of the cylinder for the frequency of the wave. The second
con-dition is identical with to make the amplitude of the, oscillation of the cylinder with the damper equal to a half of that Of the cylinder without the damper. it is also known that if two modes of the oscillation such
as heaving and swaying or heaving and rolling are allowed, the damper
can absOrb 100 percent of the wave energy by satisfying the above two
conditions for each mode. These facts seem to have been clarified first
by Bessho', then by Kato et a12, and after that by Evans3, Mei4 and
Newman5 almost at the same time for the latter three. The problem of
the complete wave absorption by a fully submerged circular cylinder
os-cillating in a circular path. treated by Fuwa
is a special case of the
above mentioned.
In the case of an asymmetric section, it was shown by Evans, Mei4
* eceived on November 9, 1978. '
Ship Dynamics Division.
c nische
SCÇO1L..
DÖCUMENTATIE
2
alid Ne*mah5 that the damper can absorb more than 50 percent of the
wave energy even in a single mode of the oscillation It is therefore an
interesting problem to study what kind of the asymmetric shape is more
efficient in the wave absorption Although the Salter cylinder" seems to
be one of the most efficient shapes, the exact calculations of the
char-acteristics such as the efficiency of the wave absorption
by using the
wave theory do not seèm to have been done. Since the exaçt
calcula-tions of the hydrodynamical forces and wave making characteristics of
the asymmetric cylinder have been already published by Kobayashf
and Katory'° ">, the subject itself is not original However, all of them
seem to have been done from the viewpomt of the motioñ of the body
In this paper, therefôre, from the viewpoint of the wave-power
absorp-tion, the exact calculations of the efficiency of the wave absorption and
the motion amplitude of the cylinder are presented by using the
two-dimensional linear wave theory, for several kinds of the asymmetric shapes
including the Salter cylinder which are intuitively considered to be
effi-cient.
There is also given a simple expression for the wave damping
coeffi-cient of a rolling cylinder with an asymmetric shape like the Salter cylin-der in the limit as the wave number tends to zeo.
2. MAXIMLM EFFICIENCY
When the cylinder suspended ,elative to some stable reference plat-form by a system of masses, linear springs and dampers is oscillating in
an sinusoidal wave train incident from the positivedirection of x as shown
in Fig. i, the efficiency of the wave-power absorption E is defined as
TRANSMITTED REFLECTED INCIDENT
WAVE WAVE WAVE
Fig. i. Wave-power Absorber
E=1I$2jTI2
(1)'
where R and T is the amplitude of the reflected and transmitted wave
respectively for the incident wave having the unit amplitude.
For a certain frequency of the wave, the maximum efficiency of the
wave-power absorption E, which can be attained under the
I+(A/A)2
where A and A re the ratio of the amplitude of the waves at x= -oc
and x= + co respectively radiated from the cylinder forced to oscillate in
the absence of the incident wave to the amplitude of the forced
oscilla-tion In the case of the rolling, the amplitude of the oscillation should
be considered as multiplied by the reference length a
Since equation (2) expresses that a highly efficient cylinder is one for
which the amplitude of the wave at x= co produced by the forced
os-cillation of the cylinder must be as small as possible compared with the
amplitude at x=+ co, for the óbjects of numerical examples we choose
several asymmetric shapes as shown in Fig 2, which may be considered
L5
a-4
V V
(1) QUADRANT (2) TRÌÁNGLE () COMPLEMENTARY QUAD.
a acos3 1.5a/cos3 a (1)-(3)FOR HEAVE (4)- (6)FOR RÖLL )6) SALTER CYLINDER
Fig. 2. Shapes for Numerical Calculations
to be highly efficient by intuition. For the heaving we choosè the
fol-lowing three shapes, (1) quadrant, (2) isosceles right triangle (hereafter we call this only triangle), and (3) a remainder of a square from which a quadrant is cut off (hereafter we call this a complementary quadrant),
and for the rolling the following three, (4) a figure which consists of a
quadrant and a triangle, (5) a figure which consists of a quadrant and a
complementary quadrant, and (6) Salter cylinder Although the detailed
shape of the Salter cylinder is not clear to the author,
we decide the shape as shown in Fig. 2 by rêfering to the papers by Katory1t> and Swift Hook et al'2.For these asymmetric shapes, the wave making characteristics and
the maximum efficiency of wave absorption are shown
in Fig 3 to Fig
4
when the efficiency of various shapes is compared with each other, in this
paper the draft of the floating cylinder is taken as the reference length
a except the Salter cylinder As for the Salter cylinder, since the radius
or diarnetér of a smaller circle in back side is often taken as the reference
length12 we choose the radius as the reference length also in this
pa-per However, for the convenience of comparisons with the other shapes,
the case in which the draft of the Salter cylinder (equal tó 1.5 times of
the radius, here) is taken as the reference length is also shown in the
figures sometimes. In these figures, Ka w2ä/g= 22ra/2, where w is the
circular frequency of the oscillation of the cylinder or the incident wave, and 2 is the wavé length of the radiated or incident Wave
FrOm Fig. 4 the efficihcy of the triangle seéms tö be hither than
the other two shapes. However, if the square rot of the sectional area
is taken as the reference length, the complementary quadrant seems to
become the most efficient among the three shapes as shown in
Fig 5
Therefore, if the comparisons between various shapes are made, attention
should be paid to what is taken as the reference length. In the case of
the heaving, there exists a limiting value K1a under which the resonançe
I. O +, -0.5 --I I I 0.5 LO [5 2.0
Ka
2a/gFig. 3. Wave Making Characteñstics (Heaving)
/
F F F 1 ai
F F LS-aT j-I F F0.9 Emax. 0.8 0.7 T K= c..2/g=2'Vx (x:wove length) KL=0.80
7
(I/oO.7O7) 0.6 1 £=/sect area a = draft 0.5 K= 2/g 2/x (X:wave length) I I I I 0.5 LO .5 2.0- Kl
Fig. 5. Maximum Efficiency vs. Kl (Heaving)
can not be realized because of the restoring
force due tp the natural
buoyancy, unless the spring constant become negative. The value of K1a
is obtained grapluca1ly such that B/Q (1 +1i1)K1a is satisfied, where B is
the breadth of the cylinder, at the water line, Q is the sectional area of
the cylinder under the water line and is the added mass coefficient for
Ka=K1a In Fig. 4 and -Fig. 5, the limiting value K1a is shown by the
1.5 2.0 0.5 1.0
- Ka
05 10 15 2.0
Ko = Jo/g
Fig. 6. Wave Making Characteristics (Rolling)
mark of a small circles Thé at of these curies for Ka lower than K1a
may be said as imaginary since E can not be attained as far as the
end of the spring is fixed to the space. There is no such a lithit in the
case of the rolling3.
Since the numerical calculations are carried out by the usual integral équation method using the usual Green function, the discontinuity of the
curvés appeared at Kal.4 in Fig. 6 is due to the so called irregular
frequency. Although it was shown that the phenomenon of the irregularfrequency could be removed by the method by Ohmatsu'4 or by the simpler
method by Ogilvie and Shin'5, the present results which were calculated
for Ka=Öd, O.2, . 1.3, 1.4, 1.5, . . , 2;O, are shown as they are. I±i Fig. 6
and Fig. 7, the results of the Saltei cylinder are shown for the both
cases in which the reference length is the radius of the smaller circle,
and is the draft of the cylinder
It seems that the Salter cylinder is
0.9 E max. i0 0.8 Q.7 0.6 0.5 1.0 0.9 E max. 0.8 0.7 / K= c2/g=2l!/x (x:wave length.) - ---1 - I I I 0.5 .0 LS 2.0
Ko
Fig. 7. Maximum Efficiency vs. Ka (Rolling)
1.134) j=J sect: area 0.6 o= draft = 2/x I-0 i/aI.I72) I i i I 0.5 1.0 --I.5 - 2.0
--lu
Fig. 8. Maximum Efficiency vs. Kl (Rolling)
as the reference length.
Although the following limit can be proved
11m EmaxO.5 ( 3),
according to Fig. 4 or Fig. 5, in the case of the triangle and the com-plementary quadrant there exists the frequency range where Emax is less
than 0.5, namely the wave amplitude at x=± oc is
ess than one at x
-
co contrary to expectation by the intuition.If we denote the maximum efficiency of the cylinder put in. the in-cident wave from x= + co and x= - co, by Eax and E;ax respectively, the
following relation can be proved easily
'- 1
max P+mazThis means that the shape which is reversed right and left is the worst
one, if the original shape is the best one concerning the maximum
effi-ciency.
3. EFFICIENCY OF WAVE-POWER ABSORPTION IN
THE CASE OUT OF RESONANCE
If the spring constant and the danipifig coefficient of the damper are
decided at a certain fixed wave number Ka=K0a such that the efficiency
become maximum as mentioned in the preceding chapter and are kept to
those values, the efficiency in waves with Ka other than K0a, namely in waves öut of the resonance, is expressed by
E
(5)
- {(m' + ,i)K2ã (rn' + p)Ka}2 ± Ka(/K2 + s/K0a20)
where
2=ß/Mw,
p=cï/M, m'=m/M
(6),
ß is the wave damping force coefficient (héreafter in the case of the roll-ing "force" should be replaced by "moe±it"), a is the added mass (here-after in the case of the rolling "mass" should be replaced by "moment
of inertia"), m is the mass of the cylinder itself and M
is the mass of
the fluid displaced by the cylinder. In the case of the heaving m' is
gen-erally equal to unit while in the case of the rolling m' can be varied by
changing the weight distribution Suffix O shows the values for Ka=K0a
The definition öf M differs from oné by Evañs.
The ratio of the amplitude of the oscillation to the wave amplitude
A is expressed by
/A 2 = E/{2(Ka)2AoS/Koa/Ka}
(7),
where S=M/pa2.Therefore in the case of the rollig, if S is taken
as the dimensionless moment of inertia M/pa4, the left hand side of equation (7) is regarded asIa/AI2, where is the rolling amplitude.
If we denote the efficiency of the cylinder put in the incident wave
from x= + co by E, and from x==
-
co by E respectively, the followingrelation can be obtained
1.0
-Fig. 9. Efficiecy of Wave-power Absorption (Quadrant, Heaving)
Fig 10. Heaving Amplitude (Quadrant)
This relation shows also that the shape which is reversed right and left is the worst one if the original shape is the best one.
3.1 Heaving
Numerical examples of the efficiency for the three shapes, (1) quadrant,
(2) triangle, and (3) complementary quadrant are shown fôr a few casés
of K0a as parameter in Fig 9, Fig 11 and Fig 13 In those figures the
curves noted as "partial turnng" are the results calculated with the zero spring constant since the resonance can not be realized at lower wave
number than K0a as mentioned earlier. The damping coefficient of the
damper, however, is adjusted to be equal to the wave damping coefficiet
of the cylinders at Ka
=K0aIn the case of the partial tumng, those
figures show that there is no much thiference even if the parameter K0a is varied largely.
The frequency range of the high efficiency, namely the bandwidth, of the triangle and the complementary quadrant is wider than the quadrant
For instance, the range of Ka of E 0 8 for K0a
=1 0, is 0 88< Ka <1 14(identical to 5 5.(À/a<7 1) for the quarant, O 79<Ka<1 76 (3 6<A/a<8 O) for the triangleand O.82<Ka<3.O (?) (2.I?<2/a(7.7) for the complementary.
lo 1.0W HeEve(EvrT.tu) --+Swy IEvof.0 OEU3 / 0.5
'r1
j.-,,t>
0.5 1.0 1.5 2.0 KoFig. Il. Efficiencj of W è-pdwer Absorption
(Tiiangle, Heaving) io E s 0.5 E s s = K0*L13 K00L4 Kó048 Heave aÇ/ Evrtiol tung K,Û5l. IO ,Ev00E41 Ç,-KoOIÖ PorEoltunin '(portoItuno9I.7 ---::. KoOL0 (I<,OEI.I0)
-
0.5 LO 1.5 - 2.0Fig. 12. Heaving Axñplittidê (Tñangle)
1.0 KoOI.4 ãI n,ng -- -- -K0L4I -- -Ev0l.8 ./ ,1 '-Kò0I.0 0.5
f,'
j P1OI?umflQ K0010 / ---- -- ---. / Po1,ol ?Uflfl9 -y.-KOvI.47 0.5 .0 L5 2.0 KOFig. 13. Efficiency of Wavepower Absorption (Comp1èméntthr Qüadraht, Heaving)
In these figures there arè also shoWÍi the efficiency of the cylinder with the symmetrical sectional shape which consists of the corresponding
two asymmetrical shapes as shown in the figures Although in general
the efficiency of the asymmetrical shape in a heaving mode is higher
than one of the symmetrical shape in a single mode (here only the
heaving is shown), the situation is reversed in a lówer frequency range.
Fig 9 shows that in the case of the quadrant, the bandwidth of the
E '.5 0L4I -Ko0O.4 - ,'1o0L4J 0.5 / ---._-.
i,1
5wa;.
/
0.5 .0 - - - .5 - 2.0Fig. 14. Heaving Amplitude (Complementary Quadrant)
and swaying is wider than that of the asymmetrical shape (quadrant) in.
a single mode of heaving
For example, the range of Ka of E 0 8 for
K0a=1.0 is 0.67<Ka<1.34 (identical to 4.7(2/a-(9.4) for the half circle
in two modes However this is not the cases of Fig 11 and Fig 13 In
the cases of the triangle and the complementary quadrant the efficiency curves cf the Symmétrical shapes in two modes of heaving and swaying seem to be shifted to the lower side of Ka from the curves of the
asym-metric shapes in a single mode of heaving. For example, the range of
Ka of E08 for K0a=10 is 062<Ka<169 (3 7.c(A/a<10 1) for the metrical double triangle and 0 61 <Ka Kl 96 (3 2< 2/a <10 3) for the
sym-metlrjca.l double complementary quadrant. These comparisons, however, are not of very significance as well as the case of the maximum
efficien-cy Emax mentioned in the preceding chapter, since they are variable if
other length such as the breadth of thé cylinder or the square root of
the sectional area is taken as the reference length
It should be said these figures have a meaning only in a sence of giving some numericalexamples.
The curves of the heaving amplitude are shown in Fig 10, Fig 12
and Fig 14 For the symmetric shapes the curves of the swaying
ampli-tude are also presented.
In the case of the partial tuning, in the limit
as Ka tends to zero the curves of I I tends to i O Elsewhere, by using
thé limiting value of À as Ka-0
liin2=B2IQ (9),
KaO
where B is thé breadth of the cylinder at the water line ad
Qis the
sectional area of the cylinder under the water line, the following can be obtained easily
hrn
lIA
I=BaI{Q(m'±po)Koa} (10).12
3.2 Rolling
As for the rolling the curves for the both shapes made by the quad-rant and triangle, and by the quadquad-rant and complementary quadquad-rant are
shown in Fig 15 to Fig 18 Although the bandwidth of the former
(quad-rant +triangle) is narrower than the latter (quad(quad-rant+ complementary quadrant), the both shapes have a qualitatively simñar tendency of the
narrow band. For example, the range of Ka Of E08 fOr K0a=1.0 and
m'=l O is O 98<Ka<1 02 (6 2(2/a<6 4) for the shape made by the
quad-rant and triangle. It can be understood that these both shapes aré not
suitable for the wave-power absorbing device Most calculations in this
case were carried out for m'=1 0, that is an uniform distnbution of mass
If m' is decreased by concentrating masses close to the center of
gyra--taon, the bandwidth of the efficiency might be fairly improved An
ex-ample for m'r0.3 is added to those figures for Ka=1.0. The frequency
range of E0.8 for K0a=1.0 and m'=0.3 are spread to 094<Ka<1.07
(5.9< 2/a< 6.7).
Judging from the comparisons of the maximum efficiency in Fig 4 and Fig 7, these shapes look like more efficient than the three shapes
in the heaving mode or at least equivalent to them However, it can be
LU ,K0I.0 -Ko0I.4 ,.-I(o0I.8
'...-á / r.'.l.O fl.LO ,«.i.o
<,0.0.4 / i \ 0.5 \ i E' - -,,, 0.5 .0 LS - Ko
Fig. 15. Efficiency of Wäve-power Absorption (Quadrant + Triangle, Rolling)
K0a1.04
m.03 /
0.5
05
i.a3
,1
-2.0
Ko
Fig 17. Efficiency of Wave-power Absorption (Quadrant+ Complementary Quadrant, Rolling) /K0004 a»1O n,.3 I(oO 1.0 // Ko0 .4 coo-l.a
said from Fig. 15 and Fig. .17 that they (both shapes (4) and (5) in Fig. 2) are not suitable for the wave absorber.
3.3 Salter Cylinder
The efficiency curves of the Salter cylinder calculated for m'= 1 0
(üíiîfòrm distribution of masses) and m'=O.3 are given in Fig. 19. Though
the accurate shape of the cylinder, and the values of the moment of inertia
CAL. m=O.3.KoQ=LO \ EXP CSWIFT-FOOKeloI) CAL. m"0.3,KoQO.3 .._.=_ -0.5: Ko
Fig. 19. Efficiency of Wave-power Absorption (Salter Cylinder, Rolling)
10 .5 2.0
Ko
Fig. 18. Roffing Amplitude (Quadrant± Complementary Quadrant)
14
of the cylinder and the damping coefficient of the damper such as the
dynarnometer used in the experiments by Swift-Hook et al12 or Salter et
al' are not reported in their papers, a numerical example for m'=O 3
and K0a 1.0 agrees very well with the Swift-Hook's experiment and an example for m' = 0.3 and K0a = 0.3 agrees fairly well with the Salter's
ex-periment. However these comparisons might be meaningless because of
the above mentioned unknown factors The efficiency curves for m' = 0 3
are improved in comparison with those for m'=l.O in this shape too. In
general it might be said that the bandwidth can be spread by reducing. the moment of inertia of the cylinder in the case of rolling
Fig. 20. Rolling Amplitude (Salter Cylinder)
Fig. 20 show the rolling amplitude. According to this figure the ratio of the rolling aplitude to the maximum wave slope tends to infinity in the limit as Ka-+0. This is the remarkable difference from the tendency that the curves for the shapes treated in the previous section tends to
the finite value as Ka tends to zero. The difference can be explained by
the following limit of the damping, coefficient 2
hm 2=(x-x)2/(4P) (11)
where X,. and x1 are the x coordinates of the right and left end of the cylinder at the water line provided that the center of rolling is located on x=0, and P is the moment of inertia of area of the cylinder under
the water line around the center of rolling. If I i =lxii is satisfied as
the shape (4) or (5) in Fig, 2, 2 tends to zero as Ka tends to zero. In
the case of the
Salter:cylinder, however, 2 does not tend to zero as Kap0 since iX,. I lxii This is the reason why there exists the difference of
the tendency of the rolling amplitude in the limit as Ka-0 between the
Salter cylinder and the other two shapes.
It is needless to say that
though the ratio of the rolling amplitude to the maximum wave slope
itself tends to some finite válue.
Furthermore the followings are also needless to be said Although
the efficiency curve for some K0a as a parameter versus Ka has the value E = Emax at Ka = K0a, the curve always does not have a maximum there.
Especially in the frequency range in which E curve varies rapidly as
a function of Ka, the location of the maximum is shifted slightly from
Ka = J10a In the frequency range in which Ema curve does not vary
rap-idly, the maximum of the efficiency curve seems to occur at aimost Ka
= K0a.
4. CONCLUDING REMARKS
By using the tWo dimensional linear wave theory, numerical examples
of the efficiency of the wave-power absorption for the cylinders of several
kinds of asymmetric sectional shapes have been given Although the
de-finite conclusions about the problems such as what kind of shapes is the best as the wave absorber, or how to determine the mechanical constants like mass, moment of inertia or damping coefficient for the best quality
of the efficiency have nct been obtained yet, the followings are summarized
as concluding remarks of this paper.
Suitable shapes for the wave absorber should be chosen not only from the quality of the maximum efficiency, but also from the total quality
of the efficiency ut of resónance.
Bandwidth of the efficiency seems to be improved by reducing the mass of the body in the case of the heaving, and the moment of inertia in the case of the rolling
Simple expression for the wave damping coefficient in the limit as the dimensionless wave number Ka tends to zero has been obtained as
equation (11).
Numerical calculations were carried out by using the electronic com-puter TOSBAC 5600 in the comcom-puter center of the Ship Research Institute.
REFERENCES
Bessho, M.:. Feasibility study on wave absorbers of floating type (in Japanese),
Paper presented at 34th meeting of Panel 2 of J;T;T.C, 1973.
Kato N et al A fundamental study on the wave absorber (m Japanese) J of the Soc. of Naval Arch. of Japan, Vol. 136, 1974.
Evans D V A theory for wave-power absorption by oscillating boches J Fluid
Mech. Vol. 77, Part 1, 1976.
Méi, C. C.: Power ertraction from water wäves, J. Ship Res. VoL 20, No. 2, 1976.
Newman J N The interaction of stationary vessels with regular waves 11th Symp. on Naval Hydrodynamics, London, 1976.
Fuwa T Wave generation and absorption by means of completely submerged
horizontal circular cylinder moving in a circular orbit, Glasgow Uthv Rep. No. NAÓE-HL-09, 1977.
16
Salter, S. H.: Wave power, Nature, Vol. 249, June 21, 1974.
Kobavashi M Hydrodynamic forces and moments acting on two-dimensional asym-metrical bodies (in Japanese) Mitsui Tech Rev No 87 1974
Kobayashi M Hydrodynamic forces and moments acting on two-dimensional
asymmetrical boches Inter Conf on Stabthty of Ships and Ocean Vehicles Glasgow
1975.
Katory M Application of theoretical hydrodynamics to the design of wave power generators, The Naval Arch., May 1976.
il) Katory M On the motion analysis of large asymmetric bodies among sea waves
The Naval Arck, Sept. 1916.
Swift-Hook, D. T. et al: Characteristics of a rocking *ave power device, Natu±e,
Vol. 254, April 10, 1975.
Salter, S. H. et al: The architecture of nodding duck wave power generators, The
Naval Arch., Jan. 1976.
Ohmatsu, S.: In the irregular frequencies in the theory of óscillating bodies in a
free suuface, Päpers of Ship Research hit., No. 48, 1915.
Qgilvie, T. F and Shin, Y S.: Integral-equatioä solütiöns fot time-dependent frée-suiface problems, J. of the Sock of Naval Arch. of Japan, Vol. 143, 1978.
Kan, M.: The added mass coefficient of a cylinder oscillating in shallow water in the limit K-.0 and K.00, Papers of Ship Research Inst., No. 52, 1977.
AP N IX
Wave I)amping Çoefficient in the Limit Ka=--O
The simultaneous integral equations for spiving the potential flow field accompanied with the wave making phenomenon are written as
fol-lows
-ør-+
J3(rGrn_ Gin)dS=f TYnGrds
(A.i),--
L
(rOin+iGrn)dS-J
GcdS (A.2),where
G+iG=ln rIii r1-2
§
x')dk±i2re' cos K(xx')
(A.3),r2 = (x x')2 + (yy')2, (xx')2 ± (y +y')2 (A.4),
and the velocity iibténtial Of tie flow field fdr thé unit velocity of
Oscil-lation is expressed by the real part of S is the body
suiface under the mean water line. Suffix n expresses normal derivatives.
Since we have the limit as
lùn G1=O and um G=27r (A.5),
bErJ ds (A.8) In the case of the rolling,
ø(Yyo)l
(A.9)where m=6y/dn= ax/as, l=ax/an=ay/as and the center of rolling is
assumed to be located at (Ô, YG). Therefore we haveL
S.s x
s-5 (Y_Ya)f_ds
=
S::
x'dx_f yy6dy=
-
(A.1O)and. hence from (A.8)
(A.11) where x,. and XL is the x coordinate of the left and right end of the
sec-tion at the water line.
Damping coefficient A is defined as
A = ß/Ma pw J ds/(Mw) (A.12.
Using (A.1O) and (A.11), we have
A = (x - x)2/(4p) (A.13),
where P= M/p, namely the moment of inertia of area of the section around the center of rolling.
In the case of the heaving,
(A.14). we have the following expression in the limit Ka=-+O from (A.2)
«+!
J
íGrds2J øds
(A.6)Since ø must be independent on the space in: the case of the limit of Ka-+O, and the relation
f
(A.7)18
TherefOre we have
(A;15),
... (A16),
s
(A.17),where B is the breadth of the section at the water line and Q is the area of the section under the water line
PAPERS OF SHIP RESEARCH INSTITUTE
Nô. i Model Tests on Four-Bladed Controlláble-Pitch Propellers, by Atsuo Yazaki,
March 1984.
No. 2 Experimental Research on the Application of High Tensile Steel to Ship Struc-tures, by Hitoshi Nagasawa, Noritaka Ando and Yoshio Akita, March 1964.
No 3 Increase of Sliding Resistance of Gravity Walls by Use of Projecting Keys under
the Bases, by Matsuhei Ichihara and Reisaku moue, June 1964.
No 4 An Expression for the Neutron Blackness of a Fuel Rod after Long Irradiation
by Hisao Yamakoshi, August 1964.
No. 5 On the Winds and Waves on the Nothern North Pacific Ocean and South
Ad-jacent Seas of Japan as the Environmental Condition for the Ship, by Yasüfuimi
Yamanouchi Sanae Unoki and TarO Kanda, March 1965.
No 6 A code and Some Results of a Numerical Integration Method of the Photon Transport Equation is Sláb Geometry, b Iwäo Kataoka and Kiyoshi Tákeuchi,
March 1965.
No 7 On the Fast Fission Factor for a Lattice System by Hisao Yamakoshi June
1965.
No. 8 The Nondestruôtive Testing of Brazéd Joints, by Akira Kannö, November 1965. No. 9 Brittle Fracture Strength of Thick Steel Plates for Reactor Pressure Vessels, by
Hiroshi Kihara and Kazuo Ikeda, January 1966.
No 10 Studies and Considerations on the Effects of Heaving and Listing upon Thermo Hydraulic Performance and Critica] Heat Flux of Water Cooled Marine Reactors
by Nàotsugu Isshiki, March 1966.
No. 11 An Experimental Investigation into the Unsteady Cavitation of Marine Pthpél-1ers, by Tatsuo Ito March 1966.
No. 12 Cavitation Tests in Non-Uniform Flow on Screw Propellers of the
Atomic-Power-ed Oceanographic and Tender ShipComparison Tests on Screw Propellers De-
-signed by Theoretical and Conventional Methods by Tatsuo Ito Hajime
Tákahashi and Hirôyüki Kadoi, March 1966.
NO. 13 A Study on Tanker Life Boats, by Takeshi Eto, Fukutaro Yamazaki and Osamu. Nagata, March 1966.
No 14 A Proposal on Evaluation of Brittle Clack Initiation and Arresting Temperatures
and Their Application to Design of Welded Structures, by Hiroshi Kihara and
Kazuo Ikeda, April 1966.
No 15 Ultrasonic Absorption and Relaxation Times in Water Vapor and Heavy Water Vapor, by Yahéi Fiijii, Jùe ]966.
No. 16 Further Modèl Tests on Four-Bladéd Controllable-Pitch Propellers, by Atsüo
Yazaki and Nobuo Sugái, August 1966. Supplement No. 1
Design Çharts for the Propulsive Performances of High Speed Cargo Liners with CB=
0.575, by Koichi Yokoo, Yoshio Ichihara, Kiyoshi Tsuchida and Isamu Salto, August
1966.
No. 17 Roughness of Hull Surface and Its Effect on Skin Friction, by Koichi Yokoo, Akihjro Ogawa Hideo Sasajima Teiichi Terao and Michio Nakato September
1966.
No. 18 Experimets on a Series 60, C5=O.70 Ship Model in Oblique Regular Waves,
by Yasufumi Yamanouchi and Sadao Ando, October 1966.
No. 19 Measurement of Dead Load in Steel Structu±e by Magnetostriction Effect, by Junji Iwayanagi, Akin Yoshinaga and Tokuharu Yoshii, May 1967.
No. 20 Acoustic Response of a Rectangular Receiver to a Rectarígular Source,
No. 21 Linearized Theory of Cavity Flow Past a Hydrofoil of Arbitrary Shape, by Tatsuro Hanaoka, June 1967.
No 22 Investigation into a Nove Gas Turbine Cycle with an Eqm Pressure Air Heater
by Kösa Miwa, September 1967.
No 23 Measuring Method for the Spray Characteristics of a Fuel Atomizer at Various Conditions of the Ambient Gas, b Kiyoshi Neya, September. 1967.
No 24 A Proposal on Criteria for Prevention of Welded Structures from Brittle Frac ture, by Kazuo Ikeda and Hiroshi Kihara, December 1967.
No 25 The Deep Notch Test and Brittle Fracture Imtiation by Kazuo Ikeda Yoshio Akita and Hiroshi Kihara, December 1967.
No 26 Collected Papers Contributed to the 11th International Towing Tank Conference January 1968.
No 27 Effect of Ambient Air Pressure on the Spray Characteristics of Swirl Atomizers
by Kiyoshi Neya and Seishiro Sato February 1968
No 28 Open Water Test Series of Modified AU Type Four and Five Bladed Propeller
Models of Large Area Ratio by Atsuo Yazaki Hi.roshi Sugano Michio Takahashi and Junzo Minakata March 1968
No. 29 The MENE Neutron Transport Code, by Kiy6shi Takeuchi, November 1968. No 30 Brittle Fracture Strength of Welded Joint by Kazuo Ikeda and Hiroshi Kihara
March 1969.
No. 31 Some Aspects of the Correlations between the Wire Typé. Pènetrameter
Sensi-tivity, by Akira Kanño, July i69.
No 32 Experimental Studies on and Considerations of the Supercharged Once-through
Marine Boiler, by Naotéüg'u Isshiki and Hiroya Tarnaki, January 1970.
Supplement No. .
Statistical Diagrnis On the Wind and Waves on the North Pacific Ocean, by Yasufumi
Yamarouchi and Akihiro Ogawa, March 1970.
No. 33 Collected Papers Contributed to the' 12th International Towing Tank Conférence, March 1970.
No 34 Heat Transfer through a Horizontal Water Layer by Shinobu Tokuda February
1971.
No 35 A New Method of C O D MeasurementBrittle Fracture Initiation Character istics of Deep Notch Test by Means of Electrostatic Capacitance Method by Kazuo Ikeda Shigeru Kitamura and Hiroshi Maenaka March 1971
No. 36 Elasto-Plastic Stress Analysis of Discs (The ist Report in Steady State of
Thermal and Centrifugal Loadings), by Shigeyasu Amada, July 1971.
Ño. 37 Multigroup Neutron Transport with Añiotropic Scattering, by Tornio Yoshimura, August 1971.
No 38 Primary Neutron Damage State in Ferritic Steels and Correlation of V Notch Transition Temperature Increase with Frenkel Defect Density with Neutron Ir
radiation, by Michiyoshi Nomaguchi, March 1972.
No 39 Furthèr Studies of Cracking Behavior in Multipass Fillet Weld, by Tàkuya Köbayashi, Kazumi Nishikawa and Hiroshi Tamura, March 1972.
No 40 A Magnetic Method for the Determination of Residual Stress by Seiichi Abuku May 1972.
No. 41, . An Investigation of Effect of Surface Roughness on Forced-Convection Surface
BOilin Heat Transfer, by Masanobu Nomura and Herman Merte, Jr., December
1972.
NO. 42 PALLASPL, SP A One' Dimensional Transport Code, by Kiyoshi Takeuchi,
February 1973.' '
No. 43 Unsteady Heat Transfer from a Cylinder; by Shinobu Tokuda, March 1973. No. 44 On Propeller Vibratory Foces of the Container shipCorrelation between Ship
Takahashi, March. 1973.
No 45 Life Distribution and Design Curve inLow Cycle Fatigue by Kunihiro lida and
Hajime moue July 1973
No 46 Elasto-Plastic Stress Analysis of Rotating Discs (2nd Report Discs subjected to
Transient Thermal and Constant Centrifugal Loading) by Shigeyasu Amada and Akimasä Machidá, July 1973.
No 47 PALLAS 2DCY A Two Dimensional Transport Code by Kiyosln Takeuchi
NvembeE 1973.
No 48 On the Irregular Frequencies m the Theory of Oscillating Bodiesin a Free
Stfrfaôe, by Shigeo Ohmatsu, January 1975..
No 49 Fast Neutron Streaming through a Cylindrical Air Duct inWater by Toshimasa
Miura Amo Yamaji Kiyoshi Takeuchi and Takayoshi Fuse September 1976 No 50 A Consideration on the Extraordinary Response of the Automatic Steering Sys
teth for Ship Model in Quartering Seas by Takeshi Fuwa November 1976 No 51 On the Effect of the Forward Velocity on the Roll Damping Moment by Iwao
Watanabe, February 1977.
No 52 The Added Mass Coefficient of a Cylinder Oscillating in Shallow Water in the
Limit KO and K., by Makoto Kan, May ]97.
No 53 Wave Generation and Absorption by Means of Completely Submerged Horisontal Circular Cylinder Moving in a Circular OrbitFundamental Study on Wave
Energy Extraction, by Takeshi Füwa, Oötobér 1978
In addition to the above-mentióned reports, the Ship Researôh Institute has another senes of reports entitled Report of Ship Research Institute The Report is
