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Reports of Research Institute for
Applied Mechanics
Volume X Number 37
1962
Published by the Institute
Reports of Research Institute for Applied Mechanics are the continuation of Reports of Research Institute for Fluid Engineering
and of Reports of Research Institute for
Elasticity Engineering, Kyushu University.
The Research Institute shall not be responsi-ble for statements or opinions advanced in the following papers.
All communications should be addressed to the director or to the librarian of the Institute.
Research institute for Applied Mechanics Kyushu University, Hakozaki, Fukuoka
Japan
Printed by SHUKOSHA PRfl'TtNG CO.
Reports of Research Institute for Applied Mechanics
Vol. X, No. 37, 1962
THE EFFECT OF DISTRIBUTION OF LOAD UPON THE
VIRTUAL INERTIA COEFFICIENT IN THE
VERTICAL VIBRATION OF A SIIIP*
By Toyoji KuMAI*
Abstract. Here is investigated the effect of distribution of load on the virtual
inertia coefficient in the vertical vibration of a ship. The author emphasized that,
since the inertia coefficient is defined as the ratio of kinetic energy of the vibra-ting water surrounding the ship hull to that of the ship itself, the velocity
am-plitude of the hull vibration should be taken into account in the calculation of
the inertia coefficient under consideration. Some numerical examples on the effects
of load distribution and the number of nodes of vibration upon the inertia coef-ficient are shown and the results are confirmed by the model experiments.
Introduction
Since the virtual inertia coefficient in the flexural hull vibration of a ship
is defined as the ratio of the kinetic energy of the added mass of the surrounding
water to that of the ship itself, the velocity amplitudeof the ship vibration should
be taken into account in computing the maximum kinetic energies mentioned above. The energy of the ship hull in the nodal vibration considerably depends on the load distribution along ship length, whereas the energy of the vibration of the
added mass of the water is influenced only slightly by the load distribution due
to a little changeof the velocity amplitude. Accordingly, the inertia coefficient
of water in the nodal hull vibration is considerably affected by the distribution of ship load along her length. Present paper shows some numerical examples on the problem mentioned above with the aid of an experiment using a model of cargo
ship.
1. Expression of inertia coefficient in the vibration of a ship
The natural frequency of the vertical vibration of a ship in which shear and rotary inertia effects are ignored is computed by
¡d2 \2
f,,2(1\2
.f.)
j Et(,Ç)")dx
/0 , (a)Jmy2dx(1±r)
* The paper was read before the spring meeting of Western So&ety of Naval Architects of Japan on May 1962.
Professor of Kyushu University, the member of the Research Institute for Applied
Me-chanics, Kyushu Unversity, Fukuoka, Japan.
(b)
where, y,,; mode of vibration of the ship model in air. Taking the ratio of (3)
to (a), the inertia coefficient r is easily obtained from the following relation,
2 T. KUMAI
where,
Yw2 dx t-n
fm y2 dx (1)
The m, and ni denote the added mass of water and mass of the ship per unit
length respectively, and y, is the n-noded mode of vibration of the ship hull in water. J, presents the three-dimensional correction factor defined by Lewis [II and
Taylor 121, and r the virtual inertia coefficient of water under consideration.
From the above relation (a), the well-established formula is derived as follows,
f _fi/
Ef0L34(1--r)
where, 4 denotes displacement and fi the eigenvalue which includes the integral value. In the expression (b), t-,, is the inertia coefficient defined by equation (1). As was seen in equation (I), the t--value is defined as the ratio of the kinetic energy of the vibration of the entrained surrounding water to that of the ship itself, but it is the ratio of added mass of water to displacement only in the fol-lowing two special cases: In the first place, the distributive functions of me equals
to that of m, such as cylindrical ship form with uniform load. Next, in the case
of translational oscillation of a hull or o-node vibration such as heaving of the
ship. In these two cases, the inertia coefficient yield
rL
J mcdx
r
,l_ 4 -J
The above expression of t- is the well known Lewis's method. In general
cases, the distributive function of me in the ship vibration differs from that of m, and o-node mode vibration is beside the question. In calculating the virtual inertia coefficient of the water in ship vibration, therefore, the equation (1) should be generally used instead of the Lewis's method. Lewis confirmed his calculation by Nicholls's experiments in three cases. This experiment was carried out using uni-form bar, instead of ship model. The formula (2) is therefore available as uni-form distributions for both of nie and m.
On the other hand, in order to determine the inertia coefficient by the
ex-periment, the natural frequency and the mode of the vibration of model ship should
be measured in air and in water. The natural frequency of the ship in air is
expressed by
fn2
-
(1)2
'LI my,,2 dx
THE EFFECT OF DISTRIBUTION OF LOAD UPON THE VIRTUAL INERTIA COEFFICIENT 3
(fj2
(4) Jm y,,2dx o dx o 5 JLmyt,,2dx JLJ(d2Yz)2dxIf we assume y,. as equaling to y,,, y takes unity. As an approach, y-value may
be assumed to be unity in practice.
As was seen in equation (1), the kinetic energy of the mass of the ship will considerably change for the distribution of the ship load, whereas the distribution of the kinetic energy of the vibration of added mass of water is almost constant.
Accordingly, it is necessary to consider the effect of the distribution of the ship load on the inertia coefficient in the vibration of a ship hull.
Numerical examples and experimental verification
Since an analytical calculation of the energies in (1) is difficult to make
even in a model ship, the numerical examples should be illustrated first by the use of the results of vibration measurements given on a model ship in the present section.
A wooden model with 1/100 scale of a cargo ship was used for the numeri-cal numeri-calculations and measurtements of virtual inertia coefficients of water in the vertical vibration of the ship hull with three types of load condition.
The principal dimensions and particulars are shown in table I, and also the body plan of the model ship is given in Fig. 1. The vibration tests were carried out in a small tank with the ratio of depth of water to draught about lO.
LWL
Fig. I. Body plan of the cargo ship used for strip calculation and
experi-ment.
The vertical vibration was excited by a vibrator of a moving coil type speaker
connected to thin rod at the end of the model, and the crystal pickup was used
for measuring the natural frequency and its mode of the vibration of the model
ship in air and in water. The inertia coefficient is computed from the formulae
(4) and (5), provided that the measured results of the mode y, and y., and the
Table 1. Particulars of the cargo ship model used
in strip calculation and experiment.
L 145 (cm) B 19.6 'i D 12.5 'i d 6.13 « 4 I O.04(kg) hull wt. 6.40 't load 3.64
4 T. KUMAI frequencies in air fa and in water f are used.
On the other hand, the inertia coefficient is calculated from the formula (1)
by means of the strip method, in which the Lewiss C-value of each section of
the ship hull is obtained using the convenient chart proposed by Prohaska [3]. The modes of the nodal vibration measured in experiments were adopted in the present strip calculation.
Fig. 2(a), (b), (c) show the computed results
of distribution of added mass of water and that of mass of the ship in which the displacement in half load takes unity with three kinds of load distribution cases I,IT and III respectively in top view. The measured amplitude of the natural
vib-ration of the model ship in air and on water, and square of y, are shown in next view, and the distributions of energies of the vibration of added mass of water
y
Fig. 2. Results of calculations of ?ne,m and mey2, my2
on the model ship in three loading conditions. Table Il. inertia coefficients obtained by calculations and
experiments.
yw
Case I Case 11 Case ill
ZJ 10 1.64 1.64 1.64 1.54 1.54 1.54 0.94 (Lamb) r (t.w1s) 121 1.21 1.21 0.74 (Taylor) 2112 1.31 106 0.755 22 0.97 0.78 0.56 0.74 (Taylor) 22exp 094 0.70 0.50 fa eps 310 242 215 fw 217 190 181 (a) (b) (c)
THE EFFECT OF DISTRIBUTION OF LOAD UPON THE VIRTUAL
INERTIA COEFFICIENT 5
and of the ship in water are presented in bottom view of Fig. 2. The
two-dimen-sional inertia coefficients under consideration are computed by integration of the energy curves obtained as above. The three dimensional or longitudinal coefficients J-value in the vertical vibration of a ship obtained by Taylor [2] was used in the
present calculation of . The numerical results are shown in table IT.
As will be seen in the results, the inertia coefficient in the nodal ship vibra-tion is different from that of the translational oscillation of the ship or that
obtained by Lewiss method in the same draught of a ship. In addition, the effect
of the load distribution on the inertia coefficiem under consideration may be clearly seen in the present illustrations of the calculations and experiments as shown in
the figure and table.
3. Estimation of the inertia coefficient in the vertical vibration of a ship
It is necessary to estimate the tendency of increase or decrease of the
value for various load conditions of a ship. So that some analytical evaluations
of inertia coefficients should be carried out under the appropriate assumptions of the distributions of added mass, ships mass and the mode of the nodal vibration of the ship.
i. Effect of load and added mass distributions on the inertia coefficient in
the fundamental mode of the vertical vibration of ships
As the first example, the ship load as well as the added weight of water are assumed to be symmetrical about midship, and the function of both distributions of me and m which are close to the end of the ship are assumed as square of a quarter length of sine wave and the length of run or entrance in both curves as parameters respectively. In the mode of two-node vertical vibration, a cosine curve with corrected base line is assumed as an approach in the mode of the nodal
vibration. The above mentioned three curves are written as follows, and also
shown in figure 3(a).
me=mosin2
o.eee
m= Sin2O<CCm
1Cm 2Cm 1Cmy=
cos2tC ± a.cmC
1/2In the above expressions, C, and Cm are coordinates of the ends of parallel
parts of the curves m and m respectively. The m0 is written by
r C,2 B
6 T. KUMA! provided that,
where,
C112 Lewis's C-value at midship
Cb block coefficient
B/d breadth-draught ratio
By substitution of the above expressions into equation (i), the r-values are an-alytically evaluated, and the parameter, a, in the above quation is determined by means of the minimum energy principle as follows,
JtI2d
= [( i r'-4 Jomdx=1a2)(lee)
-
---(1
- sin2re i I+ -
--- sin 4r
} 8l-l6çg
J'2my2de=2(1m)
[(±a2)(1_m)_
{a sin 2iri ¡
+--.j1-sin4irem}
from = O,
a-
m0 sin 2rce 1- 4e,, Sm 2r2r1+mo(1-e0)
The above results are illustrated in Fig. 3-(b), letting and crm be variable and parameter respectively. As a special case, if we put the r-value will be
presented by dotted straight line, as seen in the figure. This line just shows
Lewis's expression (2). As an example, when a fine ship (assume e,,=O.5) is fully
loaded along ship length (assume =O.l5), the r-value becomes a half of that
obtained by Lewis's method. As an extreme case, taking this value
will show that of the vibration of a uniform bar. On the contrary, if the load is concentrated almost at the midship (m=O.5) on the above ship form (ee=O.5), the r becomes maximum, it corresponds to the Lewis's result. The distribution of the ship load, however, shows no such concentrated load, because the hull weight is generally distributed along ship length. As mentioned in the above examples, the r-value is seen to be smaller than that computed by Lewis's method iii an
or-dinary type of ship.
THE EFFECT OF DISTRIBUTION OF LOAD UPON THE VIRTUAL INERTIA COEFFICIENT O 0.1 0.2 0.3 D.'I 0.5 3.0 2.8 2.0 2.. 2.2 2.0 1.8 1.6 1 1.2 1.0 0.8 o.6 0.4 0.2 o uJ2
The distributions of m, m, and y are assumed in the higher mode of vibra-tion as follows,
m = m0 sin2tre,
m = por +2 (1p)
a+(1ß)sin tre],
y =cosn3rC+a cos (n-2)tr, n=2, 3,where p is a parameter which gives the distribution of load. The n is the number
of nodes of the vibration of the ship. Substituting me, m and y mentioned above
into (1), the results will be written as
1mey2d5
m0 r sin(2n-4)r a2 sin(2n-6)or
0 4
[(1+a2a)a(1a)
(2n-4)or 2 (2n-6)orI
pa2 sin(2n-4)r JmY2d
=
(1+a2)+ ±2(1 ) (2n-4)or i2(1p)
/Atr+2(1-12){2(_3)(2n_5) ±a(
±(2n_1)(2n_3))+
2421}.
7 (10)Ìbo=2
-k
-- -
Lewis t he present strip methoU s on ca1cuittion the nodcl shipJ (9) 0.1 00 0.3 04 o 0.1 0.0 0.30.40. (a)
Fig. 3. The results of calcuations of r-value with given me, m and y.
a
8 T. KUMAL
The parameter a in the expression y should be determined according to the minimum energy principle as follows,
2(liz)
f 1 1 m011 sin(2n-4)rr±2(liz) 1
3(2n-1)(2n-3) J
4 i (2n-4)'r1 ,fprsin(2n-4)2r
2(1a)
i±moJi±sn(21z_4) lsin(2n-6)2r±,±2(1/2)t
(2n-4)tv (2n-3)(2n-5)J 2 1 (2n-4)ir2 (2n-6)tr
(12) 0.! 0.2 0.3 0. M-o '4 z 6 'I ç) (a) (b)Fig. 4. Computed r,-values in the higher modes of vibration of ships
Fig. 4(a) shows assumed distributions of m, rn and y, and a determined for m0=4. Fig. 4(b) presents the results of calculations of t-,,. It will be interesting to note that, for given value of it in the load distribution, it greatly influences to the t-,, as well as the number of nodes n, as seen in the expressions (10), (11), (12) and the figure. The illustration mentioned above will give us a clear understan-ding on the tendency of the relationship of the inertia coefficient and the number of nodes of vibration and the load distribution.
iii. Empirical formula
As was seen in the expression of (2) in the previous section, if the ship
form is uniform, the existing Lewis's method of calculation and Todd's formula
for inertia coefficient in the vertical hull vibration is available. However, in a
general case, as the velocity amplitude of the vibration should be taken into
account in a ship hull vibration, the empirical expression of the inertia coefficient
of a cylinder considerably differs from that of a ship hull as will be seen in
Fig. 5.
As a result, the formulae yield as follows, B
(13)
THE EFFECT OF DISTRIBUTION OF 1OAD UPON THE VIRTUAL INERTIA COEFFICIENT
c=O.24; for a ship hull
OCYJOSICI- ndnl
ship nrde1
o.
n.:
o.j
* u .sred ohol low noter taIlS, I, ¡duS
Fig. 5 c-values obtained by experiments on elliptical cylinders
and shipshaped models (see also [61, [71 and [8]).
Conclusions
In the present paper, the author emphasized that since the virtual inertia coefficient in a ship hull vibration is defined as the ratio of kinetic energy of the
vibration of water entrained by ship hull to that of the ship itself, the velocity
amplitude of the hull vibration should be taken into account in the calculation of the virtual inertia coefficient. As the results of the calculations of the coefficients on a model ship and the analytical evaluation of energies under appropriate as-sumptions, the inertia coefficient in a ship vibration is seen to be considerably smaller than that in the translational oscillation of a ship or the fiexural
vibra-tion of a cylinder. In addition, it s to be noted that the load distribution as
well as the number of nodes of the hull vibration considerably affects the inertia coefficient in the ship hull vibration. We may safely assume that one of the rea-sons of the discrepancy between the Lewis's r-value and the experimental result on board ship, as suggested by McGoldrick and Russo [5], Dr. F.H. Todd [4] and P. Kaplan [9], is clearly explained by the present investigation. Further study into the three-dimensional correction in the above problem is required for determining the virtual inertia coefficient of ship hull vibration.
References
[i] Lewis, F. M.: The Inertia of the Water Surrounding a Vibrating Ship. SNAME, 1929.
Taylor, J. Lockwood: Some Hydrodynamical Inertia Coefficients. Phil. Mag. 1930. Prohaska, C. W.: Vibrations Verticals du Navire. A.T.M.A., 1947.
Todd, F. H.: Ship Hull Vibration. Arnold, London, 1961.
C Lo O ewrpo stip,
t.,n ILe rCE J
I tpticwj rol rjer L/ß Io
b'-fo Ol./fll1)
t Orpodo b. t le rolo) t J * Corto ship tore,Io bot (YoInot.)[7J
9
(15)
T. KUMAI
McGoldrick, R. T. and Russo, V. L.: Hull Vibration investigation on S. S. Gopher Mariner. S.N.A.M.E., 1955, p. 490.
Terada, T.: On the Vibration of a Bar Floating on a Liquid Surface. Proc. Tokyo math. Physis. Soc., 1906.
Yokota, S.: On the Vibration of Steamer. Jour. Coil. Eng. Tokyo Imp. Univ. Vol. 5,
No. 1, 1910.
Kumai, T.: On the Vertical Inertia Coefficients of the Vertical Vibration of Ships. J.S.
N.A. Japan, Vol. 105, 1959.
Kaplan, P.: A Study of the Virtual Mass Associated with the Vertical Vibration of
Ships in Water. Report No. 734, Davidson Laboratory, Stevens Institute of
Techn-ology, 1959.
Reports of Research Institute for Applied Mechanics
Vol. X, No. 37, 1962
VIBRATION TEST IN AIR ON A 200 TON TUGBOAT AND
ESTIMATE OF VIRTUAL INERTIA COEFFICIENT
By Toyoji KUMAI**, Takeichi MATSUMOTO*
and Yukio KANEKO** *
Abstract. An experimental study on the vibration measurement in air on board 200ton tugboat was carried out and by the test on the same ship in water the virtual inertia coefficient in two node vertical vibration was estimated re-ferring the tests of the model ship. Both results in actual ship and that in model
showed good agreement.
Introduction
Since the water surrounding a ship is considered to be ideal fluid in ship hull vibration, the virtual inertia coefficient in the vibration of the model ship may
be assumed to be identical with that of the corresponding actual ship. It is
there-fore needless to attempt vibration test on actual ship in air. However, since the natural frequency of the vibration of a model ship is much higher than that of actual ship, there are some questions whether or not the water surrounding the
mo-del ship follows to the vibratory motion of the momo-del or whether there is an effect of surface tension in the small model ship as suggested by Dr. F. M. Lewis in his paper of 1929. Clarifying such physical mechanisms mentioned above, the authors attempted to obtain the inertia coefficient through the vibration tests in air and water on a 200 ton tugboat and compared the result with that obtained from 1/25 scale model ship. The three-dimensional correction factor of the virtual mass in the two node vertical vibration of the tugboat are also determined by the
experiments and calculations.
1. Vibration test on board 209 ton tugboat in air and in water
The principal dimensions of the tugboat and the conditions in the vibration tests are shown in table I. The tugboat was vertically excited by gear wheels with
excentric mass which is driven by D.C. motor of 1/4 H.P. The exciting force is
180 kg in 700 rpm of two wheels 20cm in diameter each having excentric mass. The above exciter was installed at the after perpendicular on a deck. A portable mechanical accelerometer was used for measuring the natural frequency of vertical vibration of the hull in the fundamental mode, and also for measuring the accele-* Paper will be read before the autumn meeting of Western SocieLy of Naval Architects,
Japan, on Oct. 1962.
Professor of Kyushu University, Fukuoka, Japan.
Chief Designer of 1-lull Section, Mitsubishi Ltd., Shimonoseki, Japan.
Engineer of Hull Section, Mitsubishi Ltd., Shimonoseki, Japan. 11
Photo. View of 200 ton tugboat in suspended state. length P.P L 25.5 M length O.A L 28.4 breadth Md. B 7.8 depth Md. D 3.75 draught d 2.85
condition in vibration test:
d(f) 1.20 M ci(a) 3.15 ', d(m) 2.20 " LIB 3.27 B/d 3.55 « displacement 197 tons
rations at ten points on deck moving the accelerometer the mode curves were
made. The vibration test in air was carried out using the above-mentioned
ex-citer and accelerometer, when the 200 ton tugboat was being suspended in air for
having on board the deck of a cargo ship. The test in water was taken place
when the ship floated in dock.
photo. 2. Model ship suspended in air.
Photo. 3. Mudel ship in the state of
free-free bar.
12 T. KUMAI, T. MATSUMOTO and Y. KANEKO
g1 -So -20 -n - -Io 0-5 500
VIBRATION TEST IN AIR ON A 200 TON TUGBOAT 13
On the other hand, in order to confirm the effect of supporting points of a suspended ship and of the shallow water in dock on the natural frequency and the vibration mode of the ship hull, a wooden model ship of 1/25 scale was prepared and some experiments were carried out.
General views of the tugboat in the state of suspension and the model
ship in two states of measurements in air are shown in photographs 1, 2 and 3 res-pectively.
2. Experimental results
2.1. Test results on boad ship
Fig. I and table 11 show the results of measurements of two-node vertical
vbration of the 200ton tugboat in air and in water. As be seen in the figure, the
000 700 000 900 1000 ? Jmdz m 5z z: - Jm dzJ0du7 I . 69 -O 93g.
Fig. I. Experimental results of vibration Fig. 2. Distributions ofrn 'm-, y2 and kinetic
measurements on board 200 ton energies in two node vertical vibration
tugboat in air and water. of 200 ton tugboat.
natural frequency in air is higher than that in water true to our expectation. The nodal points of the measured modes in water are located nearer to midship than that in air. It is to be noted that the damping coefficient of the vibration of the hull in water is smaller than that in air.
With regard to the virtual inertia coefficient of water, the -value agrees with that obtained by the model ship, and the three-dimensional correction factor, namely, J-value which was obtained from the ratio of the result of experiment on actual ship to two-dimensional strip calculation is determined.
14 T. KU MAI, T. MATSUMOTO and Y. KANEKO Fable Il. Results of measurements on board ship and model ship.
items
natural frequency of
two node vertical
vib-ration on actual ship,
per min. measured corrected for shailow water corrected for suspension
measured on model ship per sec. logarithmic derement of damping
coefficient, 62, on actual ship
max. acceleration of vibration, a (gal) on actual ship
exciting force, F(kg), on actual ship
virtual inertia coefficient, r2, three-dimensional correction factor, measured on actual ship measured on model ship two-dimensional calculation by Prohaska 0.524 0.510 0.834 by experiments 0.628 by Kruppa 0.574 0.564 see Fig. 4 see Fig. 3 remarks hanging state free-free state ò2 , C=4.04 [5) 4zr4, [6] at A.P V2oxp/t112 j ïo= 1.69) ellipsoid [3] ellipsoid of revolution [1] .120.38 i'L/B [2]
It should be noted that since the virtual inertia coefficient is defined by the ratio of kinetic energies of virtual mass and of ship hull, the square of a velocity am-plitude or the mode of vibration should be taken into account for two-dimensional calculation of virtual inertia coefficient, as already referred to the previous paper
of one of the authors [7]. The results of energy distributions mentioned above in
the translational motion and two-node vertical vibration are respectively shown in figure 2.
The J-value made in the present study closely approaches to that obtained from and ellipsoid with the same LIB and Bld values as actual ship under water
line. The theoretical investigation on J-value of the ellipsoid was recently psented by C. Kruppa [3] and the numerical computation with respect to the
re-In water airin 700 870 712
-
887 250 0.158 305 311 0.343 25.8 21.0 180 278 0.490 by Taylorsuits of measurements of the mode of the vibration in the present experiment was carried out by the courtesy of Dr. C. Kruppa. The J-value obtained from J. L.
Taylor [1] and C. W. Prohaska's empirical results [2], both in the ellipsoid of
revolution, are also shown in table II.
In regard to the logarithmic decrements of the two node vertical vibration of the hull of the tugboat in water, the empirical expression [5] almost represents that of experiments. The force-amplitude relationship was also obtained as a
spe-cial type ship. The result of response factor [6] is shown in table Il, Some
cor-rections for obtaining the above results in actual ship will be verified by the fol-lowing model experiments.
2.2. Experimental studies on model ship
Since measurements of vibration on actual ship in air are resulted in the state of hanging by eight points at gunwale close to the nodes of hull vibration, as was shown in photograph 1, and Fig. 2, it is necessary to correct these
restric-tion of support into free-free vibrarestric-tion of the hull. In the model tests, the first
experiment was carried out in the same hanging state as actual ship as shown in photograph 2. As the second experiments, the model was hung with its hull turn-ed sideways and the test was carriturn-ed out in air as usual as seen in photograph 3. The results of the above measurements made in air and in water are shown in Fig. 3. As will be seen in the figure, a little effects of restraint of supporting points on the natural frequency and on
the nodal points of the mode of the
vibration may be clarified in these
dif-ferences and the correction for this effect will be applicable to the result of measurement on actual ship.
The effects of shallow water as well as that of wall effects on the natural O tflflui r (i) 200 o0
- c s
A / ' / fi' / \ / /U
VIBRATION TEST IN AIR ON A 200 TON TUGBOAT 15
to jr it) in noter i n water f t) 1.00 ('fi", o.'flo LI!r..e.uInoInnat.)...i fl!j.,Ifl.,J(((!flfl&IflIflj o e/5I22
- -' :"r::
O/t - -- fi/filFig. 3. Experimental results of vibration
measurements on the model ship Fig. 4. Effect of shallow water on the vertical
1.00 .00 1.00 1.02 1.01 0.97 0.96 0.95 0.90 0.99 0. 9. 0.9
''
acK,)zl. m "'.Cí-z')-0.9 I 00 1.15 1.20 1.25 L4Fig. 5. Correction factor -value for various c anda
1.05 1.10
ç0
16 T. KIJMAI, T. MATSUMOTO and Y. KANEKO
frequency should be also corrected in the present experiments. The model test on the effects of shallow water and wall upon the natural frequency was carried out
in a small water tank. The results are shown in Fig. 4. The theoretical results
on the vibration of a circular cylinder ¡n the three- and two-dimensional flow which
were calculated by M. Yoshiki and others [4] are plotted in the same figure.
There ¡s a remarkable agreement between experimental results and the theoretical calculation for a circular cylinder in the three dimensional flow as will be seen in the figure. The natural frequency measured in the present study was corrected according to the effects of shallow water and wall from the curve in Fig. 4.
The inertia coefficient of the virtual mass in the vibration is determined by
the natural frequencies in air and in water with a coefficient as follows [7],
f2
Ja
J w
where, y is calculated from the locations of the nodal points of vibration modes in air and in water, the modes and curvature, the distribution of rigidity and that of mass of ship hull (see table III). If we assume that the mode of the two node vertical vibration is to be in the second degree parabola as is usually assumed in ellipsoid, and the mass distribution is in parabolic form, both symmetrically about the midship, the u-value will be obtained with ratio of distances between two nodal points of vibration modes in air and in water, », in variable and distance, a, in
VIBRATION TEST IN AIR ON A 200 TON TUGBOAT 17 Table Ill. Locations of nodal points in two node vertical vibration
in air and in water and correction factor p.
corrected for ai,, by the model test
parameter as will be seen in Fig. 5. By the use of v-value and the natural
fre-quencies in air and in water, the inertia coefficient in the two node vertical vibra-tion is determined from the vibravibra-tion measurements on board ship or model ship
in air and in water using the above formula.
Conclusions
Brief conclusions will be drawn from the present experimental study as follows:
The results of measured virtual inertia coefficient of the two node vertical vibration of actual ship and of its model ship almost agreed. The virtual inertia coefficient in the vibration of a ship hull may be therefore determined by the vib-ration test of the corresponding model ship.
In the calculation of virtual inertia coefficient by the strip method, it is to
be noted that a square of amplitude should be multiplied by corresponding virtual mass and mass of the hull of each strip for obtaing the kinetic energies of water and ship hull, which was noted by one of the authors in previous paper [71.
The damping coefficient obtained by measurement on actual ship in water
agrees with that obtained by empirical one, though it is seen larger in air than
that in water even in the model ship. The cause for this phenomenon is supposed
that for the restriction in hanging state.
The authors wish to express their gratitude to Dr. C. Kruppa of Portsmouth England for his assistance in carrying out the numerical calculation of J-value of the ellipsoid by the use of the results of measured vibration mode and other given data of the tugboat which was tested in the present investigation.
References
Taylor, J. Lockwood: Some Hydrodynamical Inertia Coefficients. Phil. Mag. 1930. Prohaska, C. W.: Lodrette Skibssvingninger med To Knuder. p.46. Kbenhavn, 1941.
[31 Kruppa, C.: Beitrag zum Problem der hydrodynamischen Trägheitsgrößen bei
elastischen Schiffsschwingungen. Schiffstechnik Bd. 9, 1962.
[4] Yoshiki, M. et al.: A Contribution to the Virtual Mass of a Vibrating Ship. Jour.
Soc. N.A. Japan, Vol. 84, 1952.
in actual ship, measured 0.520 0.410 1.13t 0.982
hanging state in model ship
free-free state
0.500
T. KUMA!, T. MATSUMOTO and Y. KANEKO
Kumai. T.: Damping Factors in the Higher Modes of Ship Vibrations. European
Shipbuilding, No. 1, VoI. VII, 1958.
Kumai, T.: Response of the Higher Modes of the Hull Vibration of a Large Tanker.
European Shipbuilding, No. 5, Vol. VII, 1958.
Kumai, T.: The Effect of Distribution of Load upon the Virtual Inertia Coefficient in
the Vertical Vibration of a ship. Jour. Western Soc. of N.A. Japan, Vol. 24. 1962, Rep.
Res. Inst. of A. M. Vol. X, No. 37, Ps. 1-10, 1962.
Reports of Research Institute for Applied Mechanics
Vol. X, No. 37, 1962
ON THE VIRTUAL INERTIA COEFFICIENT IN THE
VERTICAL VIBRATION OF AN ELLIPTICAL
CYLINDER OF FINITE LENGTH'
By Toyoji KUMAI1''Abstract. An investigation is made into the three-dimensional correction
factor of virtual inertia coefficient in the vertical vibration of an elliptical cylinder
of quasi finite length in water. The experimental verifications are given including higher modes of the vibration of the elliptical cylinders.
Introduction
A theorectical investigation was carried out on the three-dimensional correc-tion factor of the virtual inertia coefficient in the vertical vibracorrec-tion of an infinitely long circular cylinder floated partly immersed in water by J. Lockwood Taylor [1] in 1930, and the same problem has been recently studied by O. Grim [21 and Leibowitz-Kennard [3]. In addition, Joosen-Sparenbcrg [9] presented a theoretical study on the longitudinal reduction factor for the added mass in the vibration of infinitely long rectangular cylinder. With regard to the model experimental results,
the above-mentioned correction factor in experiments [4], [5] is considerably
smaller than that obtained by the theoretical calculation.
To explain the cause for the discrepancy between the results obtained by the existing theory and the experiments, ari approximate calculation of the three-dimen-sional correction factor of the virtual inertia coefficient in the vertical vibration of the circular and elliptical cylinders of quasi finite length is given in the pre-sent paper. Furthermore, the effect of the sectional form on the three-dimensional
coefficient is shown dealing with various elliptical sections of the cylinder of
finite length.
1. Hydrodynamical inertia coefficient for vertical vibration of a ship
The inertia coefficient for the vibration of a ship is computed as the ratio of the kinetic energy of the vibration of the water surrounding the hull to that of
the vibration of the ship hull itself. Since the two-dimensional calculation
avail-ing strip method is used for computavail-ing the inertia coefficient, the three-dimensional correction factor should be taken into account. The inertia coefficient is obtained * Paper will be read in Tokyo at the autumn meeting of the Japan Society of Naval Architects on Nov. 1962.
Professor of Kyushu University, the member of the Institute for Applied Mechanics, Kyushu University, Fukuoka, Japan.
20 T. KUMAI
as follows;
JLmy2dz J
L
where,
virtual inertia coefficient
me virtual mass per unit length of the ship
m mass of the ship per unit length
y velocity amplitude of vibration of the ship
z length co-ordinate of the ship
L length of the ship
J
three-dimensional correction factor defined by LewisFor evaluating the above expression (1), the integrands my2 and iny are
calculated by means of strip method for ship hull vibration. As to the
three-di-mensional correction factor, the well established Taylor's J-value may be used for such a fine ship as formed a half of an ellipsoid of revolution under waterline.
The above J-value is not to be applicable for a ship as full as tanker. When we
consider the three-dimensional correction factor for the inertia coefficient of the vibration of a full-formed ship, the cylinder of finite length will be referred to. While experimental results from J-value of the cylinders variously formed were
shown in the author's previous paper [4], tests were recently made by Burrill, Robson and Townsin [5]. To the best of the author's knowledge, there is no
theo-retical investigation made so far on the J-value in the vertical vibration of a cylinder except that by Taylor [I], Grim [2], and Leibowitz-Kennard [3] who
used a circular cylinder of infinite length. These theoretical results, however, show a great difference from those of experiments. On the other hand, the three
dimen-sional inertia coefficients of the vibrations of the ellipsoid were adopted from
mathematical treatise by Kruppa [6]. This result will be more useful for
con-sidering the vibration of the fine ship than that of ellipsoid of revolution.
2. Three-dimensional correction factor, .1 in the vibration of a circular cylinder of
finite length
As a special case, when the distributive function of m equals to that of in as it occurs to a homogeneous cylinder in water, the expression (1) will lead is the following equation:
= c-J,
(2)where, c is constant for the sectional form of the cylinder under water line. The
c takes unity in the case of the vibration of a circular cylinder immersed partly
in water, c=B/2d in elliptical cross section.
Let us consider a circular cylinder of length L, floated on water with draught a, where a is the radius of its section. The cylinder is now in vertical vibration
and that with the natural frequency. The mode of vibration is assumed as an
where, m mir
L
(4)
As the last condition, the normal velocity of the wetted surface of the
cyl-inder should identify that of the surrounding water, namely,
(5)
m=l,3, 5, for n=O, 2, 4,
m=2, 4,6, for n=l, 3, 5,
On the other hand, the Laplace equation with respect to the potential func-tion for the three-dimensional flow in the cylindrical co-odinate is written by
O2 i Z3ço i a2 a
ar2 + r ar + r2 Z302 + 2
The velocity potential under consideration should satisfy the following boun-dary conditions:
At the boundary of the free surface,
for 0=0 and the water parted infinitely distant from the cylinder,
for rcc
for the end condition of the quasi finite length as approximate consideration,
for z-0 and L
The solutions of equation (5) satisfying the above conditions are written as follows;
tS ArnK1(kmr) sin kmz sin 0.e1, (6)
where, Am constants
Ki(kmr) K-Bessel function or Hankel
fLinction with imaginary argument
VIRTUAL INERTIA COEFFICIENT OF ELLIPTICAL CYLINDER 21
approach to be of a cosine mode, namely, the velocity distribution is shown as follows
v=Ucos'
ze4°, (3)where, U unit velocity
n number of nodes of vertical vibration of the cylinder
The velocity y is also expressed by means of Fourier expansion as follows;
4U/ m
A comparison of the distributions of the velocity amplitude y and the pot-ential function Çlr,9J2 along z-axis which considered by Taylor with that
in-vestigated by the present author is shown in figure a. As is seen in the figure, the major difference between the above two assumptions will be found in the
distribu-tion of the potential funcdistribu-tion near from both ends of cylinder. In both cases, no
boundary conditions at the plane of flat ends of a cutting out cylinder are
satis-fied.
The kinetic energy due to vibration of the water surrounding the hull is
computed from the above potential function and its derivative, as follows;
2T= - s: s:
-pirU2a2
L .
16 ni2 Ki(krna)2
-. 2
2 (m2n2)2 aKi'(kma)On the other hand, the two-dimensional calculation of the energy of
vibrat-ing water surroundvibrat-ing the same cylinder as mentioned above is easily shown by
Lflhjfl 0,11 fl
a I wo-,l0ded
ade
J - 4
infinitely loIii eylinde
V
1 /0
K1'(ka)
z
(i L
quasi taite cylinder
Fig. a. Distributions of y and of two node vibration of cylinder cutting out from two types of infinite cylinders.
(10) 22 T. KUMAI ( ' =
at
r=ZSubstituting (4) and (6) into (7), constants A,, are determined as follows;
4m U
Am ir(m2n2)K1'(k,a)
where, dash denotes the derivative of Ki(kmr) with respect to r.
The velocity potential function satisfying above conditions will be written as
VIRTUAL INERTIA COEFFICIENT OF ELLIPTICAL CYLINDER 23 where, T01 2
L
0.11 0.80 0.7 0.110 prrU2a° L 2T11= 2-. 2
Taking the ratio of (10) to (11), the three-dimensional correction factor in n-noded vertical vibration of the circular cylinder is obtained as follows,
m2n2
i +k Ko(krna) Ki(kma)
mir B ---I;
As was seen in the above formula, the three-dimensional correction factor of the circular cylinder of the quasi finite length in two-node vertical vibration,
for an example, is obtained with LIB as variable, as shown in Fig. 1. The figure
shows J-value comparing with the existing results of calculations and of the
ex-periments. The results of the present calculation made on the number of nodes
of vibration in the parameter are shown in Fig. 2(b).
,
.,
p'0,/
12' .Fig. I. J-values of 2-node vertical vibration of infinitely and finitely long cylinders and of ellipsoid of revolution.
(12) O rxprrinieuuil, le thceeecrbç,r [4) e.xp-rinir,l:eT, ollielee by [5) Burrill enel in i i
I-0 09 0.8 07 08 0-5 0.4 0.3 02 0.1 o n 0.9 08 0.7 0.8 0.5 0.4 0.3 0.2 0-I o 5 5 8 21 24 27 30 33 38 3 (Q) Io 3 2d (C' 8 7 8 9 IO II J?' 2 0, 2,4 and 6 as parameter. o-9 0.8 0.7 0.8 0.5 0.4
Fig. 3. Experimental verification various cross sections.
-o-- Tfl.orstoaI 08 07 08 05 04 03 02 0I 0.9 0$ 0.7 0.8 05 0.4 0.3 02 0.I o 3 5 -O- Exp.rlmntal 905 6 7 8 9 lO II 2 13 (b)
8789
4 Io on J-value versus n io Ii 2 13 in cylinders of 24 T. KUMATVIRTUAL iNERTIA COEFFICIENT OF ELLIPTICAL CYLINDER 25 The experimental verification of the theoretical results in the higher modes of the vibration of the circular cylinder under consideration was carried out by
the use of a wooden model of L/B= 10. The result is shown in Fig. 3. As will
be seen in the figure, while the J-value decreases slightly by the increase in the number of nodes of vibration in the circular cylinder, the tendency in the theory agrees with that in the experiment.
3. .1-value in the vertical vibration of an elliptical cylinder
For evaluating the effect of the sectional form of the cylinder of finite
length on the three-dimensional correction factor for the virtual inertia coefficient in the vertical vibration of the cylinder in water, the calculation of the three-dimensional virtual inertia coefficient of an elliptical cylinder of the finite length vibrating in water will be shown as follows.
The Laplace equation for the three-dimensional potential flow in the car-tesian co-ordinates is
32
3x2 + 3y ± =0.
When the potential function 4 is presented by 4' sin k,z, provided that the
mode of vibration is assumed to be of the trigonometric function, the following equation is obtained,
24'. 24'
X2_± ,2 km24'O.
By the use of the following elliptical co-ordinates , t,
x=hcoshcos
y=hsinh 5sin2h: focal distance,
and also putting 4'=4142, the following two equations are obtained, d
(a-2qcosh2)4i=O,
d2çb2
+(a-i-2q cos 2)42=O,
where, a: separation constant
q = ,c =k./z/2
The solutions of the above two types of Matheu equations are easily obtained [7] and the 4 becomes,
=. CmGeki(e,
q)sej(,qm) +
DmFeki(e,qm)ceiì,qm) sin kmze0. (16) The integration constants Cm and Dm in the above solutions are determined by26 T. KUMAI
D,=O, for
C,=O, for r 3
the Matheu functions se1 (, - q,,) and
ce1(, - q)
for small value of q areapploximately written by
se1(, q,)' sin Ti,
cej(,
q)
COS .From the foregoing solutions, the three-dimensional correction factor for elliptical cylinder of the section, e=e0, is computed by using the same method as that in the previous circular cylinder and the results are obtained as follows:
16 ; /
m '\2Gekl(e0, qm)ir m2_n21 Gek1'(e0, qn)
where, the dash denotes the derivative of respective function with respect to .
The modified Matheu functions in the above formulae are approximately represented by the products of modified Bessel functions as shown by following expressions. Gek1(e0, q,n) - Gek1'(0, qm) 1+ It(u,,)Ko(vm) Io(u,,,) Kt(Vm) (Ko(Z7m) 11(u,,,)) I1(u,,)K0(v,,,) ± (U,,±Vm) lK1(zÇ
Ï,)1
Io(Um)Kt(Vm) ¡i(Um)Kø(Vm) Fek1(e0,q,)
-Fek1'(e0, -q,,,) -
!i(Um) Ko(v,,,) (Ko(Vm) fi(Um) tlo(Um) K1(v,,) "" K(v,,,) !o(u,,,) i
where, Unz=FCmC°, vp,=icme', icm=krnh/2.
The numerical values of the modified Bessel functions in the above equations arc obtained by Shibagaki's table [8].
When the sectional form of the elliptical cylinder approaches to circular
one, becomes large and then the values presented by (18) approach to that of
(12). On the contrary, when e approaches zero, the sectional form presents the
flat plate and the values are calculable by the use of (18). The computed results
of the J-values on the cylinders of typical elliptical sections including higher modes
of the nodal vibrations are shown in Figs. 2(a), (b),
(e) and (d).
As will beseen in these figures, it is to be noted that the coefficient in the high mode of the
vibration shows a higher value in the flat shaped section than that in the circular section, whereas the effect of the sectional form on the J-value in the lower mode does not change so considerably as is shown in Fig. 4. As an example, Fig. 3
shows the J-values versus number of nodes of the vibration of the cylinders of
(18)
VIRTUAL INERTIA COEFFICIENT ÖF ELLIPTICAL CYLINJ1ER 27 three kinds of the sectional form which were calculated and tested. The J-values versus B/2d of the elliptical section with L/B=l0 are shown in Fig. 4 with the
number of nodes of the vibration in the parameter. There seems to be a little
discrepancy between the theory and the experiments as is seen in Fig. 3. However, the tendency of the effect of the sectional form upon the J-value of the cylinder in the high mode of the vibration in water is shown fairy well.
Jf' o 0-9 o-a 07 06 0.5 04 03 o o-no, 2 3 4 L a- O 5
Fig. 4. J-value versus B/2d with number of nodes as parameter in L/B=lO.
4. Longitudinal distribution of the hydrodynamical pressure in the vertical vibration of the cylinder
The hydrodynamical pressure along keel in the vertical vibration of the cylinder under consideration as mentioned above is presented by
p ( 0
\g 3t O=-r/
The distribution of the above pressure is easily obtained from the previous
calculations. As an example, Fig. 5 shows the distributions of the ratios of
three-and two-dimensional pressures at the bottom centerline of the circular cylinders of three kinds of L/B in the vertical translational oscillation and the two node vibration in water with a half immersion. As will be seen in the figure, there is considerable difference between the two- and three-dimensional pressure distribu-tions especially near the end of the cylinder. It will be supposed that the above difference in the ellipsoid is not so conspicuous compared with that of the
cylin-der as shown in these illustrations. The theoretical result will be comfirmed by
the model experiment in near future.
1.0 0.8 0.8 0.4 02 o oe 0.4 0.2 o -0.2 -0.4 - 0-C -0.8 -1.0 0.I 0- nOde Z
Ir
02 03 0.4 0.5:
2-node " 982 82e 0.1 0-2 0-3 0.4 0.5 zFig. 5. Distributions of hydrodynamical pressure at the bottom cense line of the circular cylinder
Conclusions
As a contribution to the calculation of the virtual inertia coefficient of the vertical hull vibration of a ship, the three-dimensional correction factors in the
vibration of a circular and elliptical cylinders of the quasi finite length were
calculated under some appropriate assumptions and the results of the calculations were compared with that of some model experiments.
So far as the results of the present investigation are concerned, the effect of the end condition of the vibrating cylinder of finite length upon the three-dimensional correction factor in the vertical vibration seems considerably large compared with that of infinitely long cylinder, whereas there is little effect of the sectional form on this factor in lower mode of the vibration of the cylinder in
water. In regard to the higher modes of the vibration, however, the J-value
ap-pears considerably high. The tendency of this effect was confirmed by the model experiments in the present study. We may assume that one of the causes of the difference between the inertia coefficients of a slender ship and that of a full-formed ship with flat bottom in the higher mode of the vertical vibration of the model ship was clarified by the present investigation. As was suggested by Joosen-Sparenberg [9], we should bear in mind that the three dimensional influence is more important for hull forms of which bottom has a larger distance to the surface of the water. Further study is required into the present problem in ship-shaped
beam.
VIRTUAL INERTIA COEFFICIENT OF ELLIPTICAL CYLINDER 29
The author wishes to express his gratitude to Dr. Y. Watanabe for his
va-ruable advice in the present study.
References
Taylor, J. Lockwood: Some Hydrodynamical Inertia Coefficients. Phil. Mag. Vol. 9,
1930.
Grim, Otto: Elastische Querschwingungen des Schiffskörpers. Schiffstechnik, 1960.
[31 Leibowitz, R. C. and Kennard, E. H.: Theory of Freely Vibrating Non-uniform Beams,
including Method of Solution and Application to Ships. T.M.B. Report 1817, 1961. Kumai, T.: On the Virtual Inertia Coefficients for the Vertical Vibration of Ships. J. S.N.A., Japan, Vol. 105, 1959, European Shipbuilding Vol. 8, 1959, Rep. Res. Inst.
Applied Mech. Vol. VII, No. 28, 1959.
Burrill, L. C., Robson, W. and Townsin, R. L.: Ship Vibration: Entrained Water
Experiments. R.I.N.A. Paper No. 4. 1962.
Kruppa, C.: Beitrag zum Prolem der Hydrodynamischen Trägheitsgröl3en bei elastischen Schiffsschwingungen. Schiffstechnik, 1962.
McLachlan, N. W.: Theory and Application of Mathieu Functions. Oxford, 1951. Shibagaki, W.: 0.01% Tables of Modified Bessel Functions. Baifukan, Tokyo, Japan, 1955.
Joosen, W.P.A. and Sparenberg, J. A.: On the Longitudinal Reduction Factor for the Added Mass of Vibrating Ships with Rectangular Cross-section. Netherlands' Research Centre T.N.O., Report No. 40S, 196!.
Brief Summaries of Papers Published in
Bulletin of Research Institute
for Applied Mechanics (Japanese) No. 18, 1961
On the Shallow Waler Tank Newly Con-structed in Tsuyazaki Laboratory
By Kinji SHINOHARA, Shigeru IKwA
and uro ENDO
This paper presents the outline of the shallow water tank newly constructed in Tsuyazaki detached laboratory of Research Institute for Applied Mechanics, Kyushu
University. That is, it gives the principal dimensions of the tank and main equip-nients. Also it describes our plunger type wave making apparatus and waves gener-ated by it.
Tables of
and ,,J7
(Supplementary Note, Part 2) By RESEARCH COMMITTEE
FOR HYDROLOGY
By making use of the numerical tables of and (u=0.500-.. I .500) published
in BuI/cti,x no. 5 (1954) of Research Inslilute for Applied Mee/zanjes with a view to sim-plifying the computations of turbulent boun-dary layers by the method due to MIL-tIRAN, the following calculations were car-ried out:
I) Assuming five kinds of velocity dis-tributions in two dimensions along a smooth
surface, and with the aid of the above-mentioned tables, the thickness of the tur-bulent boundary layers were worked out by the method of numerical integration in order to check the tables in part. lt was verified that the result thus yielded agreed almost completely with the values obtained
directly from the NIILLIKAN'S formula,
pro-vided Ax (the step of the non-dimensional coordinate along the plate) was taken sufficiently small.
2) In the case when the velocity decreases
linearly along the plate, the calculations were repeated for miscellaneous choices of Ax, and it was proved that the accuracy of the numerical integration could be pre-served practically satisfactorily until Ax
reached the value 0.24, where x was mea-sured along the plate and x=0 and I at the front, and the rear, edge, respectively.
Fur-ther, out of this conclusion, it was sug-gested that we should be able to choose an appropriate value of Ax for a general type of velocity distribution.
3) Employing the parameter V introduced
by HOWARTH, the positions (xx) of the separation points were determined for each of the five velocity distributions from the condition V(x)=-0.06. lt was found out that the value of the ratio lij./UImr showed no remarkable variation among these five
separation points, where UI.s is the
ve-locity just outside the boundary layer at the point of separation, and UI,ma,r is the
maximum value of the velocity at the outer
edge of the boundary layer. It was remarked
that, instead of being a universal constant, this ratio could be kept nearly constant only for a series within which the form of a body was changed systematically.
If, therefore, in a series of systematic variation of flow, one of the points of se-paration is known theoretically or
experi-mentally, there is a possibility of estimating
very roughly the remaining points of sepa-ration without the labor of calculating the turbulent boundary layers for these flows.
This note was prepared by J. OKABE,
member of the committee, assisted by S. iNouc, S. HosulNo, and H. Amti.
32
NOTE
Note on the Shipping Water
By Fukuzo TASA!
The relative rise of water surface due to pitching and heaving motion of a ship con-sists of three parts : first the submergence of the bow due to the quantity of pitch and heave of a ship, secondly, the statical
swell-up of bow waves due to forward speed
of a ship and thirdly, the dynamical swell-up of water surface due to relative vertical velocity of the bow and waves.
In this note, the quantity of dynamical swell-up was evaluated making use of the
two-dimensional va1ues.
Then the results by calculation were
com-pared with the experimental results by R. TASAKL**
F. TASAI: "Wave Height at the side of
Two-dimensional Body Oscillating on the
Surface of a Fluid." Reports of Research
Institute for Applied Mechanics, Vol. lx, No. 35, 1961.
R. TASAKt: "On shipping water."
Monthly Reports of Transportation Tech
-nical Research institute Vol. 11. No. 8. 1961.
