Influence of the surface tension of the surrounding water upon the free rolling of model ships

tali

y.. ScheePsb0U

t,chiSChe

Bcçjesch00

Del(t

FUKUOKA, JAPAN.

1950

With the Compliments of the A uthor.

Influence of the Surface Tension of the Surrounding

Water upon the Free Rolling of Model Ships

By

Keizo UENO, Kogakuhakushi,

Reprinted from the Memoirs of the Faculty of Engineering, Kyushu University, Vol. XII, No. i

Surrounding Water upon the Free Rolling

of Model Ships

By

Keizo UENO Kogakuhakushi, (Professor of Naval Architeclure.) (Received September 21st 1949)

Contents

Pages

L Introduction 21

Stability couple due to the surface tension of the surrounding

water 22

Solution of the equation of motion 32

Free rolling period and loss of amplitude 36

Comparison between the theoretical values and the

experi-mental results 39

Conclusions , 56

Afterword.

Bibliography and a note. Figures.

I. Introduction

A free rolling period of a ship is almost irrespectivé of the amplitude while the amplitude is small, that is to say the so-called

"isochronism ", but it varies with the amplitude because of the effect of the ship's form on her stability couple when the amplitude is large, that is to say the so-called "pseudoharmonic oscillation ", and it is well-known that the period increases in accordance with the increase of the amplitude if a ship has a rounded bottom and it decreases with

the increase of the amplitude if she is a wall-sided ship. But we always experience by the free rolling experiments of model ships that the above-mentioned property of free rolling is well shown when the

amplitude is greater than about five degrees, but the period steeply decreases in accordance with the decrease of the amplitude when the amplitude is so small as less than about five degrees. On the cause of the last phenomenon nothing had been hitherto known. The author verified theoretically that the above phenomenon is caused by the stability couple due to the surface tension of the surrounding water

acted on the both sides of the model ship which is freely rolling. Further by the theoretical analysis the author found that the function of the surface tension affects on the curve of extinction too and the

loss of amplitude per swing increases approximately by a certain amount

on account of the surface tension effect. The free rolling experiments had been carried out on the various types of model ships under the variation of her surface properties, metacentric heights, and surface tension of the surrounding water in order to ascertain the abovetheory

concerning to the influence of the surface tension of the surrounding water upon the fre rolling period and curve of extinction of a model ship and the author could verify that the above theory was perfectly

in agreement with the experimental results and correct. In this paper the influence of the surface tension of the surrounding water uponthe free rolling of model ships is theoretically analysed and compared with the free rolling experimental results.

II.

Stability couple due to the surface tension

of the surrounding water

If a plate of glass is plunged in water with its sides vertical, it

will be found, as shown at Fig. 1, that where the liquid touches the glass it is drawn up above the level of the general surface and the

form of the liquid surface near the glass is concave upwards. If,

however, the glass is placed in mercury, the surface of the liquid near the glass is depressed below the general surface and its form is convex upwards, as shown at Fig. 2. The angle u between the tangent to the liquid surface at the point, where it meets the solid, and surface

of the solid, is called the angle of contact between the liquid and the

solid. The angle of contact between a solid and a liquid depends on

Fig. i Fig. 2

their propertiès and also on the third material, which exists above the free surface of the liquid. In the case where the angle of contact a is less than 900 (Fig. 1), the surface tension of the liquid surface supports the part of the liquid which is above the general level. In the same

way the surface tension, when the angle of contact' a is greater than 900 (Fig. 2), withstands the hydrostatic pressure due to the liquid

displaced near the surface of the solid. It is well known1 that the

relations between the surface tension, thee angle of contact, the form of the liquid' surface near the solid, and the vertical distance of the point where the liquid meets the solid above the level of the general surface can be hydrostaticall calculated.

We shall now consider a varnished wooden model ship who is freely rolling in still water. Since the angle of contact between the varnished wooden surface and water is about 900, but not 9Ø0 correctly speaking, for the sake of convenience we shall proceed with our studies under the assumption that it is 900. If a varnished wooden plate is

quietly dipped into water with its sides vertical, the water surface meets the plate horizontally because of the above assumption that the angle of contact between the plate and the water is 9O, and so, if

the surface tension of the water is H, the force exerted by the tension

of the surface on the side of the plate per unit length is H, and its dirction is horizontally outwards of the plate, as shown at Fig. 3.

If the plate which is under the condition of Fig. 3 is pulled up from the water a little distance vertically, the form of the water surface near the plate will become concave since the part of the water near the plate is drawn up above the level of the general surface by the adhesion between the plate and the water, and if the angle of contact between the plate and the water in this case is ¡9f, the surface tension will pull the plate in the direction é9i off the vertical downwards, as

shown at Fig. 4. If the plate which is under the condition of Fig. 3

is pushed into the water a little distance vertically downwards, at

that instant the form of the water surface near the plate will become convex since the part of the water near the plate is depressed below the general surface by the adhesion between the plate and the water,

and if the angle of contact between the plate and the water in this

case is (irß2), the surface tension will pull the plate in the direction A92 off the vertical upwards, as shown at Fig. 5. In these cases the

vertical distance of the point where the water touches the plate from the level of the general surface is almost less than about 0.4 cm. and generally the above distance in the case of pushing down (Fig. 5) is always less than that in the case of pulling up (Fig. 4).

Fig. 3 Fig. 4 Fig. 5

Now considering such case that a wall-sided model ship who had been floating in an upright condition in still water rolls by avery

small angle (i about her centre of gravity G, as shown at Fig. 6, we shall calculate the uprighting couple ÖM due to the surface tension H of the surrounding water everted on the ring-shaped part put between

two transverse sections apart a very small length f3L of the model

ship. In the ascending side of the above-mentioned ring-shaped part, which corresponds to the case of pulling up the plate from the water (Fig. 4), the form of the water surface near the ship's side is concave

and the surface tension H acts on the ship's side in the direction /9

off the middle line of the ship downwards. On the other hand, in the descending side of the ring-shaped part, which corresponds to the case of pushing down the plate into the water (Fig. 5), the form of the

water surface near the ship's side is convex and the surface tension

Fig. 6 Fig. 7

H acts on the ship's side in the direction ß2 off the middle line of the

ship upwards. Neglect the vertical distance between the point where the water surface touches the ship's side and the general level of the water surface since it is so small as less than 0.4 cm., as stated above

and put that the breadth of the water line at the part of the length

L under consideration is b and also that the tangent line to the water line at that part makes an angle c with the longitudinal direction of the ship, as shown at Fig. 7. Then the length of the water line of

the part under consideration is ÔL. sec o änd the transverse component of the surface tension H acted on the unit length of that part of the

water line is H cos çc, and therefore we have

ôM =

[H(cos

91+cos ¿2)-H cos

((-tan O_a.)sin

+(- tane±a)sin 192}] 3L sec °

=

H.[-(cos

1+cos

P2)sec çc-tan

.(sin 1+ sin I2)

+a(sin P1- sin p2)] 3L,

where a is the vertical distance of the centre of gravity G below the water plane. The value of a is small since the centre of gravity G is generally in the vicinity of the water plane, and it is assumed that the rolling angle / _{is very small.} Therefore, neglecting the second

and third term in the right hand side of the above equation and putting sec o = i since is small, we have

3M Hb (cos ß1+cos 92).ôL

(i)

Integrating M over the total length L of the model ship, we havé

the total uprighting couple M0 of the model ship due to the surface

tension as follows:

M₀

H A

(cos /!± COS_ß2), (2)

2

where A is the water plane area of the model ship. Now, if we give an up and down motion, namely, a dipping oscillation with a period of the same order as that of the free rolling of a model ship, that is, about two seconds, in vater to a varnished wooden plate with

its sides vertical, the angle and P2 can be measured by the obsérva-tion, but at the same time the following remarkable phenomena will

be found, too. In the case of ascending of the plate the part of the

length of the plate in which the water surface near the side of the plate is drawn up and its form is concave, spreads over the total

length of the plate,

but in the case

of descending the part in

which the water surface is depressed under the general level and its form is convex, does not spread over the total length and

occupies only some part of the total one, and the water -surface over the remaining part of the total length except the above convex part is,drawn up and its form is concave like the case of ascending. This can be explained as follows. If the plate oscillates vertically about the mean position A with the amplitude h, as shown at Fig. 8, the part of the plate between the vertical distance 2h should be wet. Therefore, even in the case of descending, if there is the part of the plate above the water surface where the water particle remains and adheres to the plate since it had been wet by the preceding oscillation,

the water surface near the plate of that part is drawn up by the

Fig. S

adhesion between this remaining water particle and the plate, and its

form is concave like the case of ascending. The perfectly same phenomena as above occur at the both sides of the model ship who is

freely rolling. At the ascending side of the model ship the form of the water surface near the side is concave -over the total length of the model, but at the descending side the form of the water surface of a certain part of the total length is convex Sand the remaining part of the total one is concave like the ascending side. The surface tension of the convex part at the descending side gives an uprighting couple to the model ship together with that of the corresponding (cöncave) part at the ascending side, but the surface tension of the concave part at the descending side gives no uprighting couple to the model ship

/

because it counterbalances that of the corresponding (concave) part at the ascending side. Namely, the length, on which the model ship is given an uprighting couple by the effect of the surface tension, is only some percentage of the total length of the model ship. Therefore, the value of (2) obtained by the integration (1) over the total length L overestimates the total uprighting couple M0, and then, multiplying (2)

by an effective factor uì, we have

M0 uìHA (cos ß1+cos é2).

(3)

There should be an error of the above theory caused by the initial assumptiòn that the model is a wall-sided ship since the type of ship is not always wall-sided. We will include this error in the effective

factor . Therefore, the value of i will depend on the nature of

surface and type of the model ship, and it will be determined by the

experiments.

Above we have simply explained the function of the surface tension, and now we will further go into details as follows.

Let the point where the water surface touches the side of the

model ship who is freely rolling be P, the vertical distance between the point P and the general level of the water surface be Z, and the

angle which the tangent line to the water surface at the point P

makes with the horizontal level be ç, where / and Z should be positive when the water surface near the model ship's side is concave upwards.

It is well known' thatthe relation between the values of and Z

when the model ship is floating in an upright condition in still water at rest is expressed by the following formula.

sin

(4)

a=frJ(5)

gp

-where H = the surface tension.

g = the gravitational acceleration, p = thet density of water.

Correctly speaking the relation between ç and Z when the model

ship is rolling does not equal to that w.hen the model ship is at rest, but the effect of the rolling upon the relation between and Z can

be negligible as small since the rolling motion of the model ship is so slow as her rolling period is about two seconds or above. Therefore

it can be assumed that the relation between and Z when the model ship is rolling is approximately expressed by the equatiön (4) obtained

hydrostatically. The value of a in (5) indicates the value of Z in the case of = r/2. In this case H = 77 dynes/cru., g = 981 cm./sec2., and

= 1 gr./cm3., and therefore we have

a = 0.4 cm.

We have been proceeding with our studies under the initial assumption that the angle of contact between the model ship's side and the water

when the model ship is floating in still water at rest is 90°, but correctly speaking it is not 90°. Therefore, now show it by (2r/2x0), then in this case ç and Z will be expressed as follows:

= X0

Z=a1/1sjn(7r/2;)=a1/1cos;Z0.

It was confirmed by the experimental results that in the case of a

varnished snrface x0>0, and in the case of a paraffine surface x0<0. Considering the model ship who is rolling, let the maximum value of ç at her ascending side be x1, the minimum value of ç at her

des-cending side be z2, and also put

Z1=a/1cosx1,

Z2 = a. V' 1cos z2.

Then at the ascending side the variation of Z between ±Z and Z1

corresponds to the variation of. between z0 and z1 and also at the descending side the variation of Z between -± Z0 and Z2 corresponds to the variation of ç between z0 and z2, where among the signs "±" annexed before Z0 the same sign as the value of z0 should be adopted. Since the approximation

irZ

s

a a2 2a

is possible, we have from (4)

irZ 2a 2ax

aVicosx=

_ir Put T where

B = the breadth of the mdel ship,

O = the transverse inclination of the model ship.

= the amplitude of the free rolling of the model ship. and also put

M=M0/i.H.A

(8)

On the other hand it was cleared up by the experiments that the

following relation exists in the cases of the varnished and paraffine

surfaces.

xj>(x2+2 z0).

Assume that the model ship is at first inclined by an angle 0, and next she is set to be perfectly free, and then she will commence a

motion of free rolling. Take this instant as the origin of the time

t. Bearing the above relation of x

in mind, we can assume that

the relations between M and t within the time elapsed from t = O to t T(= ir/w the free rolling period of the model ship) are

approximately expressed by the following equations which are divided (6 ) 02=

2(ZZ0)

2a 2 - (x2+x0), ir

(7)

D 2(Z2+Z0) T) D 2a B B

into the four cases from (a) to (d) in accordance with the values of

or 3.

(a) In the case of

for t (x1--x2) 3w ô (x1x0), M

= -

(ôwtx1) + sin (3wtx for t (x1+x2)

M= --_[sinxi+sinx2].

In the case of (01+02) 2

__(xx)

3_L(1+2),

for t = ir + X2+X0 \ \2w 3w 1 M = __[Sifl {3( __wt)x0} Sjfl(3wtr_x2)]

(ir

_x2+x0

)_

(x1+x2) \ 2w 3W 3w M= -__[sinxi+sin(òwt_-x2)],

for t

(x1-i-x2)ir 3w w xi + SinX2].

In the case of (01+02)

OO02

for t = + X2 ± )ç

)

\2W aw M = [sin

_{{ô(+wt)+xo}_Sin}

(ôwt-z)]. 1 7V X2+Xu) 7V

for t=2+

_(t)

M __[sin x2+sin

(owt_x2)j.

(d) In the case of .. _(x2+x0)

for t=

7V

(L,

[Sfl ja(-j__wt)+x0}±sin{8(

_wt)_x,}].

III.

Solution of the equation of motion

The equation of motion for the free rolling of a model ship is expressed by the following equation.

I't9+F+Wh0+M,,=O, (9)

where I' = the mass moment of inertia of the model ship about the longitudinal axis through the centre of gravity,

F = the resistance couple of water,

W = the weight of displacement of the model ship,

h0 the statical stability arm.

The values of F and h0 are given by the following equations which had been formerly used2 by the author.

h= Ii(8+k1O3±k2t),

where h is the rnetacentric height, k1, k2, J,

I, I, Ii, I and 13 are

constants irrespective of ¿ respectively, and among the signs "±

at the instant under consideration should be adopted. The value of

M0 is given by (8). Now adopt the following notations:

¡'±10=1, Lu1,

12/Im,

13/I= n, 1/1=

2;,

120/1 ¿9,

1/I

= r, Wh/1 w2, _{HAW/ Wh}

Then (9) can be transformed into the following shape.

{ii? (± )m

n0}Ö + 2aO ±í9(2 + j3

+ wo2(1 f.k103±k205)± =

0

(11)

In order to solve approximately the above equation, the author will

apply Prof. Y. Watanabe's method of solution using Hamilton's principle,

which had been formerly2 adopted by the author. Now show the left hand side of (11) by the notation (P, and put

O = sec

e' cos

(wt--c),

tan = a/w.

Calculate the value of 'P by putting (12) into 'P, and then put the

thus obtained value of (P in the following two equations.

(Pe"cos(wtc)dt

= o,

r /»

j 'Pe_sin wt dt=O.

The following approximate values of both the circular frequency w

and the damping power a will be obtained by solving the above two

equations simultaneously, neglecting the value of a _{as small compared}

with the value of w on the way of calculation.

W =

(LI(1)(1+

. (14) K1/1

C1\

C2

a

+ 1a1+ , i K9Gi

, 2a

irti0G \

'B/

where

K1 = 1+

_{2(1_elc0)}

(1e_4w)

[i

(w2_23a2)] 1,02

4(w+4a2)J

'

(13) I

+ (1e"") [i_ (3o_87w4a2_5O3w2a1_413a6) 8(w2 +a2)(w2+ 92/4)((v2±9a2)

rrr1+

(1e4w)

k10+

(1e6")

k2004

2(1e2)

3(1e2)

=1+k1002+k2t'04, (16)

K2= i + 4a(w2+ a2) (1 + e_3

/W)

3(w2+9a2)

(1e2')

+ (2+a2)2

(1e4w)

mil02

8(w2+4a2)

(1e2'»)

+ 8a(w2 + (12)6 (1 + e_OIú) nil03

15(w2+25a2/9)(w2+25a2) (1 e-21T) 8w 1 + i'0 + - m (32 + 157r n ()3, ₍₁₇₎ 3ir 4a(w2+a2) ¿01 = 3(w2+9a2) 3(w2+a2)2 + 16(w2+4a2) C2= (1+ e3aw)

(1e2'°)

(1e4'°)

_r(32 (1

e2'°)

x+cos 2)t1_cos (x1+r2)o } k2004 3

ß00+--r00,

(18)

The values of both C1 and C2 take the following approximate values which are divided into the four cases from (a) to (d) in accordance with the value of ¿.

(a) .In the case of (x1x0)

Ir = (ô-1) (ô+ 1)1 (sin x + Sin x) (xj+x2) Cl

2irôHj

1 1 27rò ±cos (0-1)

(+1

xl+SjflX)(1 (x1+x2) }

E2t

I i o

+ + sin

ii.

I i

41

(81)

+ (3+1)}(cos xi+cosx2)

.(x-i-x2)

(b) In the case of

?(x_x)6

1(xI+x2)

c_iç1

11

23

1(1-3) (1+3) j[Cos(6I2+ (x2+x0) o 1f i

i

i

_srn

(x+x) 1 1

ii

± COS X9' COS (x2+ x) ] + [2 {

f 3) + (1+ )

¡j'

[

+ (x1±x2) }sin

ti_sin

(x2--xO) X2]

ïç

i 1 + T 1(1-8) + (i + 3)} [sin - sin (ô/2 + x) (c) In the case of I (x1± x2) (x2_{+ x)}

= 2th

[2_{(i)

+ (1+8) sin x cos

(x2+x0) o 6 (1-3)

(i0)}[cos(ô_x?)+cos(o/2xo)

+{1+sin (x2--x0) }cos 2].

4 (1-3)

(Ï+ COS + xi Sin (x1+x2)

+i

X2 _GOS (x2-x0)

]

= T

((1

) -

}°

COS (x2± x)

+

+ (l+)}]{1_5mn (x2+xfl) _}

+

_(

1'

+ } sin (31rx2)sin (ir/2+x).

(d) In the case of (x2+x0)

C1

= {(1) (1)} {cos(3i-r/2x0)+cos(ôir/2+x0)}, *

c2=o.

The approximate values, of the angles x1, X2 and x0 observed

experi-mentally by giving an up and down motion, namely, a dipping escilla-tion with a period of the same order as that of the free rolling of a model ship, that is, about two seconds, in water to two wooden thin plates, one of which has a varnished surface and other a paraffine surface, with their sides vertical, äre tabulatedas follows:

The values of X

The values of C1 arid C2, both of which were calculated by using the above values of x, are expressed by the curves on the. abscissa of ô

in Fig. 9. It is shown in Fig. 9 that the values of both C1 and C2 in the case of the varnished plate are very much the same as those in the case of the paraffine plate respectively.

IV.

Free rolling peri6d and loss of amplitude

The free rolling period T is obtained from (14) as follows:

Kinds of paint z0

varnish 450 _loo 50

T= 4-=iTo,/--_//1

G2a

T0=J F:7

where G = 1± k102_{+ k2e104,} 4w w2 8w3 K2 = 1+ ¡0 + m + n 03 (20) 1+J0+jO02+f03.

In the equation (19), T0 shows the period of the free rolling which amplitude is very small in the case of no surface tension effect of the surrounding water, G, in which the values of both coefficients k1 and k2 depend upon the form of the statical stability curve of the model ship, represents the effect of the model ship's form on the period, K2 in which the values of the coefficients. _{f, ¡2} and

f

can be evaluated by the freé rollìng experiments, indicates the effect of the damping

upon the apjrent mass moment of inertia and the second term in

the denominator of the above equation expresses the influence of the surface tension of the surrounding wattr upon the period.

Let the loss of amplitude for the single roll from one side to the opposite side be 40, and then from (15) we have

40 =

w

00 + _{4 f9}f2 + 0 8

To)

Since

(1+)/K2r1

was proved by the experiments, put

(19)

a0 +

310

98 ±

--

-O = a8 + b82 ± cOo 0 (22)

8

and then

40F= a110+b002±c803+ TC2 (23)

The fourth term in the above equation represents the influence of the surface tension of the surrounding water upon the curve of extinction. Among the terms representing the surface tension effect in (19)

and (23), depends upon the form and scale of the model ship añd

also upon the nature of the surrounding liquid since it is HAW/WJZ as shown in (10), and rj depends upon the type and nature of surface of the model ship. Therefore, in the case of the free rolling experiment on a definite model ship in water, iE is constant, and the surface

tension effect is given by C1/G and C2/G, or approximately given by C1 and C, neglecting the effect of G. It is shown in Fig. 9 that the value of G1 is very large when the value of ô is very small and ap-proaches to zero in accordance with the increase of the value of 6, while the value of C2 is approximately constant except he case of the very small value of & On the other hand, since 6 is ¿ì/ -v-as shown in (7), 3 is proportionate to O, when the scale of the model ship and

the property of the surrounding liquid are definite. Therefore it is

evident from (19) and (23) that at the free rolling experiment of the model ship the influence of the surface tension of the surrounding

water upon the period T is very large when O is very small, and

almost dies away when 8 s large, and T steeply decreases in accor-dance with the decrease of O, while by the influence of the surface tension upon the curve of extinction the value of 40 increases appro-ximately by a certain amount except the case of the. very small value

of .

Since = HAW/Wh, the surface tension effect decreases in accor-dance with the decrease of the amount of the surface tension H and increases in accordance with the decrease of the amount of the mcta-centric height h.

Since H is constant for a definite liquid, and A, and Wh are

proportionate to the second and fourth power of the linear dimension of the model ship respectively, is reciprocally proportionate to the second power of the linear dimension of the model ship. Therefore it

is apparent that the surface tension effect exists in the case of the

free rolling experiment of the model ship, while it is negligible as small in the case of the actual ship.

It is evident that the value of v, for T, on which the effect of the surface tension is large when 0 is very small, should differ from the

value of for JO, on which the effect of the surface tension is large

when t10 is large, since the value of naturally should vary according to the amount of 00. Therefore, for the sake of convenience, let for

T be and for ¿10 be 12, and then

(19')

J0=aU0+bti2+c003+ '222 (23')

The values of both and must be determined by the free rolling

experiments of the model ships. But it will be cleared up by the

analysis of the experimental results as later stated that the experimental

values of are approximately equal to those of and therefore the

value of v is approximately constant irrespective of the amount of 0.

V.

Comparíson between the theoretical values

and the experimental results

The following experiments of free rolling of model ships, which are divided into two series the first and the second, were carried out and compared with the theoretical values in order to examine that to

what extent the theoretical values of both (19') and (23') agree with the experimental results, how the values of both coefficients and 9

and how the bilge keels affect on the function of the surface tension of the surrounding water.

(i)

The first series of experiments

The first series of experiments were carried out as the preliminary experiments for the second series.

The model ship used in this experiment is a varnished wooden model, and has straight frame sections, bilge circles, but no bilge keel.

The principal dimensions and etc. are as follows: Length = 150 cm., Breadth = 25.5 cm., Draught 12.5 cm., Displacement = 34.3 kg.,

The statical stability curve of this model is expressed in Fig. lo. From this curve the following values of k1 and k9, which are the values corresponding to the value of O expressed in radian, are obtained:

k1 = 1.5742,

k2 = 4.1617.

The layer of varnish on the surface of the model ship already had

partially fallen off due to the past many time experiments for the other purposes. The free rolling experiments were carried out on the surfaces of two kinds, the one being the pàrtially fallen off varnished

surface as above and the other being the fresh varnished surface newly painted. The experimental values of the period T and the loss of amplitude per swing JO, obtained from the free rolling experiments of the above two kinds, are expressed on the abscissa of O in Fig. 11 Fig. 14, respedively. The theoretical value of T = T0/1/, obtained by putting K2 = i and = O into (19'), is expressed by the curve in

Fig. 'I1. This curie can be obtained as follows. Let the mean f the experimental values of T at (=0 in Fig. 11 be T0, and the value of

T calculated by putting this obtained value of T0 in the equation KB =66 cm., BM = 4.0 cm.,

KM 10.6 cm., KG = 8.8 cm.,

T = T0/1/ó can be drawn on the base of 0 in Fig. 11. It is shown in Fig. 11 that the theoretical value of T = T,/-/ is quite a mean

curve through the experimental values of T within the large range of 00, and therefore the surface tension effect does not exist at all in the case of the fallen off varnished surface. This means that in the case of the fallen off varnished surface, the form of the water surface near the side of the model ship who is rolling is concave upwards at either side, the one being ascending and the other being descending, since this surface of the model ship is comparatively easier to be wet than the fresh varnished one, and therefore the uprighting couple due to the surface tension is zero. It is expressed in Fig. 12 that in the

case of the fresh varnished surface newly painted, the theoretical curve of T, calculated by putting K2

= i into (19') and by using the

value of T0 obtained from Fig. 11, is quite a mean curve through thé experimental values of T. This means that in the case of the fresh

varnished surface the surface tension effect perfectly arises and the equation (19') is quite in agreement with the experimental results. It can be said from the above phenomenon as regard to the period that

the surface tension effect on JO does not exist in the case of the

fallen off varnished surface (Fig. 13) and perfectly arises in the case

of the fresh varnished surface (Fig. 14). The amount of the water

resistance to the free rolling of the model ship whose surface is the fallen off varnished one is approximately equal to that of the fresh varnished one because it had been proved by the experiment&3 on the

flat plate that the frictional resistance of the former surface is

appro-ximately equal to the latter one. Therefore the difference between the amount of JO in Fig. 13 and that in Fig. 14 should indicate

quan-titatively the amount of the surface tension effect. If we draw a mean curve through the experimental values of 40 in Fig. 13,

it can be

expressed by the equation

'4O a00+b002+c003

namely, in the case of 2 = O in (23'). The amount of 46 including

the surface tension effect, calculated for the three cases of '2 = 0, 0.5 and 1.0 by putting the amount of lU of no surface tension effect

as above into (23'), are drawn by the three curves in Fig. 14.

It is

shown in Fig. 14 that the experimental values are scattered about

between two curves of r12 = O and 1.0, and that the curve of 12 = 0.5

is approximately a mean curve through them.

In summary of the above experimental results the following con-clusions are obtained:

The theory of the surface tension effect on the free rolling ex-periment of the model ship is in perfect agreement with the experi-mental results. The surface tension effect perfectly arises in the case of the Iresh varnished surface, but does not exist in the case of the

fallen off varnished surface. This phenomenon is caused by the fact that the latter surface is easier to be wet than the former one. The value of i and r12 can be quantitatively obtained from the difference

between the experimental values of the above two surfaces. From the above experiments we obtained m 1 and 12 == 0.5 for the varnished

surface, but the further experiments of various kinds should be re-peated many, time in order to determine the correct values of

'' and

l2.

(ii) The second series of experiments

(a). Kinds of experiments

It is evident from the results of the first series experiments on

the varnished model ship as above that the surface tension effect

should arise more remarkably in the case of the paraffine model ship whose surface is very much harder to be wet than the varnished one,

and should not arise at all in the case of the model ship whose

surface paint is perfectly dropped off by polishing up of sand papers and therefore who has a naked wooden surface which is very much

between the experimental results of a surface and that of the naked wooden surface should indicate quantitatively the surface tension effect on that surface if the frictional resistances of both surfaccs as above are approximately equal. In the case of the model ship, who has

varnished surfaces over her bottom and has paraffine surfaces over

the area of both sides in the vicinity of the water line only, the

frictional resistance of this model should indicate the property of the varnished surface, and the surface tension effect should indicate the property of the paraffine surface. On the other hand, since E among the terms expressing the surface tension effect in the equations of both (19') and (23') is HAW/Wh as represented by (10), E is a function of H and h for a definite model ship. The value of H represents the amount of the surface tension arid is 77 dynes/cm. for water. If we drop an oil on the water surface, the value of H of this oil surface should be about half the amount of H of the pure water surface and therefore the surface tensio'n effect of the former case should decrease so much compared with the latter one. The surface tension effect should increase according to the decrease of the metacentric height h.

Then, in order to confirm the above properties of the surface

tension effect, the free rolling experiments were carried out on the

model ships of three kinds, A; B and C, whose particulars are

re-presented in Table I, under the variation of the metàcentric height h of two kinds as expressed in Table II, Of the nature of surface of

four kinds and of the amount of the surface tension H of two kinds as indicated in Table III, in two cases of both with and without bilge keels, respectively. And the problems of that, whether the surface tension effect varies just like the theory according to the variation of the amounts of both H and h or not, how the values of the effective

factors and 2 vary according to the types and nature of surfaces of the model ships, and how the bilge keel affects on the function of the surface tension of the surrounding water, were examined. The

same bilge keel, which has a length of 68.64 cm. and a breadth of 1.20 cm., was used for the three modelships A, B and C.

The statical stability curve of A-model is expressed in Fig. 16, the statical stability arm of B-model is given by

GZ = GM{1+2BM.tan2(1}sinO and that of C-model is given by

GZ=GMsin fi.

The values of both k1 and k2, calculated from the above curve and equations, are tabulated in Table II, which shows that the values of k1 and k2 för C-model are let to be approximately zero since these values are very small. In the calculation of the value of , as the

value of A in the water plane area must be used in the case of

A-model who has a ordinary ship-shape form, but the value of

('B/2

(i+ B \

LB+4tydy=LB

_Jo

_2L)

must be used instead of the water plane area in the cases of both B-and C-model, who have the cylindrical forms, adding the allowances of the integration for both the' fore end plane and after end one of the model ship, since the integration of (2) over the total length of

Table I Kinds of model ships

Notations of model ships A B C Types of model ships Cargo ship model Box-shaped model with bilge circles Cylindrical model with circular sections Material Length (cm.) Breadth (cm.) Draught (cm.) Weight (kg.) Wood '175.5 25.9 11.5 40.06 Wood 181.0 25.0 9.O 40.12 Wood 182.3 22.7 11.35 37.04

Table II Values of GM, k1 and k..

The above values of k1 and k.. correspond to the values of o expressed in radian.

Table III

Surfaces of model ships and kinds of experiments

the water line of the model ship must be taken over the water line along both sides and also over the water line along both the fore and after end planes, where

L = the length of the model ship, B = the breadth of the model ship,

Notations of model ships GM (cm.) KM (cm.) KG k1 (cm.) k2 A 0.35 10.66 10.31 9.943 27.68& .0.88 10.66 9.78 3.861 11.345 B 0.35 10.41 10.06 8.182 4.532 0.88 10.41 9.53 3.142 1.875 C 0.15 10.90 10.75

0

0

0.67 10.90 10.23

0

0

Number of experiment Nature of surface Surrounding liquid Curves of experimental results

I Naked wood Water

II Varnish Water

1H Paraffine Water

.

-IV Paraffine An oil on water_surface Varnish over

bottom and paraffine over sides near the

water line

-y - the transverse distance froni the centre line to the

point

under consideration on water line in the fore and after end

- planes of the model ship.

(b). Experimental Results

Among the experimental results, the period curves are expressed in Fig. 17Fig. 28 and the extinction curves are represented in Fig.

29Fig. 40. The results of experiments of five kinds (no. of

experi-ment 1, II, III, IV and V) as represented in Table III are expressed in each Figure, where the experiments of no. I (the experiments on the naked wooden surface) in the case of larger metacentric height for each model, that is, in the cases of GM = 0.88 cm. for A- and B-model and in the case of GM = 0.67 cm. for C-model, are neglected, and also the experiments of no. V are neglected for B-model.

In each Figure the theoretical values of both T and 4(1, calculated

by putting 1i = = 0, 1 and 2 into the equations of both (19') and (23'), are drawn by fine lines. The process of drawing of these

theoretical curves is as follows:

At first take the case of Fig. 17 for instance in order to explain the drawing process of the period curve. Fig. 17 indicates the experi mental results of T in the case of GM = 0.35 cm. of A-model without bilge keels, where it is evidently shown in the Figure that among the experimental values of the various surfaces that of the naked wooden surface corresponds to that of ij = O since the surface tension effect of that surface does not arise at all in the Figure just as expected.

Let the mean of the experimental values of T at O=0 in the case of

the naked wooden surface be T, and the value of T calculated by

putting this obtaining value of T0 into the equation T = T0/V can

be drawn on the base of (1 in Fig. 17.. The values of coefficients J,

J and 1 in the equation

K2 = I +ffl0+f2O,2+f3(103

between thus obtained values of T = T0/V and the experimental values of the naked wooden surface. The value of T0//K2/G, calcu lated by using thus obtained value of K2, can be drawn by the curve as the theoretical value of T in the case of = 0, that is, the case

of no surface tension effect. The theoretical value of T in the case, in which the surface tension effect exists, can be calculated by putting

the above obtained value of T, and

= i or 2

into (19'). Thus

the values of jj, j and f3 in K2 can be determined by the experiments. But, since these values had been so small as cóuld be negligible within the range of these experiments with the three models A, B and C, the theoretical values of T were calculated by putting K2 = 1. ip all cases

of this paper.

Next take the case of Fig. 29 in order to explain the drawing

process of the extinction curves theoretically calculated. Fig. 29

indi-cates the experimental results of JO in the case of GM = 0.35 cm. of A-model without bilge keels just like the case of Fig. 17. It is evident by the experimental results in Fig. 17 that the experimental values of 40 in the case of the naked wooden surface in. Fig. 29 correspond to those in the case of no surface tension effect, that is, the case of

= 0. Determine the values of coefficients, a, b and e in (23') from

the experimental results of the naked wooden surface, putting 2 = O

into (23'), and then the theoretical values of 40, calculated by putting thus obtained values of a, b and c, and = 1 or 2 into (23') can be drawn by the curves in Fig. 29.

Thus the drawing processes of the theoretical values of both T and 40 on the basis of the experimental results of the naked wooden surfaces have been explained by taking the cases of both Fig. 17 and Fig. 29 as the examples. But the experiments on the naked wooden surface are neglected in the cases of large metacentric height for each model as before stated. Namely, the period curves in Fig. 23Fig.

28 and the extinction curves in Fig 35Fig. 40 does not include the experimental values of the- naked wooden surfaces. The drawing process of the theoretical curves in this case is as follows:

First take the case of Fig. ? for instance in order to explain the drawing process of the period curve. Fig. 23 indicates the experimental

results of T on the various surfaces in the case of GM = 0.88 cm. of A-model without bilge keels, where the surface tension effects of the above experimental results differ according to the natures of surface of the model. The experimental curves of T on the various surfaces differ greatly at very small amplitudes where the surface tension effects of these surfaces arise greatly, while they are almost agreed at large amplitudes where the surface tension effects are so small as can be said to exist no more. Draw an approximate mean line through those experimental curves at the largeJamplitudes, and extend it according

to its tendency to the small amplitudes. Assume that this mean line approximately shows the curve representing the value of T0/V K2/G, and also assume K2= 1, and then the value of T0 can be determined from this mean line. The theoretical values of T can be obtained by putting thus obtained value of T0, and 1j = 1 or -2 into (19').

Next take the case of Fig. 35 for instance in order to explain the drawing process of the theoretical value of JO. Fil. 35 shows the

experimentalresults of 40 in the case' of GM = 0.88 cm. oP A-model without bilge keels just like the case of Fig. 23. It is shown in Fig.

23 that the surface tension effect in the case of the experiment no. IV is least among those in the cases of other surfaces. This tendency in Fig. 23 exists in Fig. 35, too, where the experimental value of 40 of the experiment no. IV is least, its experimental curve is lowest and nearest to the O axis since its surface tension effect, is least among those in the cases of other surfaces. Now assume the curve of 40 in the case of no surface tension effect, that is, the case of 2 = O at the

proper small distance below the curve of the experiment no. IV, and then the theoretical values of 40 for '22 = i or 2 can be calculated

from (23') on the basis of this assumed line of 22 0. But strictly

speaking in this case the theoretical values of. 40 thus obtained are the imaginary and not correct values since the curve of 40 of 22 = O

The cases of Fig. 25, 26, 37 and 38 were treated as special cases, where they show the eperimental results of T and 40 for GM = 0.88 cm. of B-model both with and without bilge kedls, respectively. The

experimental curves of T in both Fig. 25 and 26 show that the surface tension effect almost does not arise in the case of the varnished sur-face, namely, the case of the experiment no. II. Therefore in the

cases of both Fig. 37 and 38 the theoretical values of 40 were calculated on the basis of the assumption that the value of 40 of the experiment no. II shows the value in the case of no surface tension effect, namely, the case of ì2 = 0.

(c) Examination of the experimental results

It is shown in Fig.- 17Fig. 40 that the experimental curves of no. J, II and III in each Figure differ, and specially, differ greatly in

the case of the small metacentric height. This difference can be considered as originated in two causes, the one being the difference of the frictional resistances of their surfaces and the other being the difference of their surface tension effects. The skin friction of the

surface of the model influences upon the virtual moment of inertia in T and also upon the resistance in 40, but the influence upon the former is so small as can be negligible compared with that upon the latter. The results of towing experirnents3> of a flát plate on the same surfaces as the experimental no. I, II and III, respectively, show that the

frictional resistances of their surfaces so little differ that very roughly speaking the frictional resistance of no. ill is about 10% greater and that of no. I, which amount of 48 is least among three, is about 20%

greater than that of no. II. Therefore it is evident that the above

differences between the experimental curves of both T and 48 originate only in the surface tension effect, but they are perfectly irrespective of the frictional resistances of their surfaces.

The experimental values of both and 2' approximately obtained

burves in Fig. 17-.Fig. 4, on the various natures of surfaces, accord-ing to the ship's types and to the amount of the metacentric height, in two cases of both with and without bilge keels, are tabulated in

Table IV. The followings are cleared up from Table IV.

If the type of the model ship and her nature of surface are

definite, the experimental value of ij is approximately equal to that of J2, irrespective of the amount of the metacentric height and of the existence of the bilge keels. This means that the effective factor for T is approximately equal to that for 40, and therefore that the effective factor is constant irrespective of the amount of the amplitude since the surface tension effect for T is great at the small amplitude, while that for 48 is great at the large amplitude. Thus we obtained

experi-mentally 1)i 2, and therefore, unifying those notations of both and

2 by î as before, we shall proceed on the following explanations.

We have known from the above experimental results that the values of j are approximately constant irrespective of the amount of

the metacentric height. This means that the experimental values of both T and 40 vary as just like the theory according to the amount of h in e among the terms representing the surface tension effects of both (19) and (23), and therefore the theory. concerning to the variation of the surface tension effect due to the variation of the amount of the metacentric height is completely in agreement with the experimental

results.

We have also found from the above experimental results that the

values of are approximately constant irrespective of the existence of

the bilge keels, which generally affect on both the virtual moment of

inertia in T and the resistance in 40, and increase their amounts.

This means that the amounts of the surface tension effects upon both T and JO are very little influenced by the bilge keels, namely, both the virtual moment of inertia in T and the resistance in 49 are in-dependent of the function of the surface tension.

Table IV

Experimental Values of and

-1) 1/1

Model GM Number

of

Bilge keel Bilge keel without with without with slips (cm.) exp I 9 0 0 0 II 0.4 2.Ó 0.4 1.2 0.35 III 2.0 2.0 2.2 2.0 IV 0.9 0.8 0.8 0.7 V 2.0 2.0 2.7 1.6 A I

-

-

-1.1 1.0 1.5 1.0 1.5 0.88 III 1.5 2.0 1.5 2.0 IV 0.4 0.9 0.4 1.0 V 1.0 1.5 0.9 1.5 I O O O O II 0.2 0 0.2 0.1 0.35 fI 1.0 1.0 1.0 1.0 IV 0.5 0.2 0.5 0.4 V

-

-

-

-B II O O O O 0.88 111 1.0 1.0 1.0 1.0 IV 0.4 0.2 0.4 0.2 V

-

-

-

-I O 0 0 0 II 0.9 0.8 0.9 0.6 0.15 111 1.0 1.0 1.0 LO IV 0 0.5 0.6 0.5 V 1.0 1.0 1.0 1.0 C II . 0.5 0.5 0.4 0.4 0.67 III 1.0 1.1 1.0 1.1 IV 0.5 0.5 0.4 0.5 V - 1.1 1,0 1.1 1.0

In no. I experiments, which were carried out on the naked wooden surfaces, the surface tension effect does not arise at all, and perfectly

'1 = O for every experiment. This means that in this case the form

of the water surface near the side of the model ship who is rolling is concave upwards at either side, the one being ascending and the other being descending, since the naked wooden surface is very easy to be wet, and therefore the uprighting couple due to the surface tension is

zero.

3. Results of no. III experiments

In no. III experiments, which were carried out on the paraffine surfaces, take a mean value of 1 for each model, and then we have

2 for A-model and i for both B- and C-model. This difference

of v betwéen A-, B- and C-model represents the so called influence of the ship's type upon the function of the surface tension, which can be explained as follows: Namely, a tangent line to the frame line at the intersecting point of the water line with the frame line in the transverse plane of the model ship is vertical for both B- and C-model, either of whom is a cylindrical type, but not vertical for A-model, who is a ship-shaped type. Therefore the values of ;, x1 and all of which are

necessary for the calculation of the valves of both C1 and C2, should show the values as just like the initial assumption for both B- and C-model, but not for A-model. That is to say, that the values of C1

and C2, both of which were obtained by the calculation using the

above values of ;, x1 and 12, should indicate the same values as the initial assumption for both B- and C-model, but not for A-model. On the other hand, the theoretical values of both T and JO in our theory had been calculated by using the same values of both G1 and G2 as the initial assumption, namely, those as represented in Fig. 9, for the three models of A, B and C. Therefore the values of both C1 and C2 used in the calculation are suitable for the type of ship in the case of B- or C-model, but not in the case of A-model. The value of should

be influenced by the above condition concerning to the values of botti C1 and C2, and should indicate the different value for each model as before stated.

Results of no. IV experiments I

In no. IV experiments, which were carried out on the paraffine models in the water having an oil surface, the value of the surface tension H of this oil surface should be about half the amount of H of the pure water surface, and therefore theoretically speaking the surface. tension effect of this case should decrease so müch compared with the latter case. It is shown in Table IV that the experimental values of i, of no. IV experiment are about half the amount of those

of no. III experiment for each of three models A,, B and C. This means that the experimental values of both T and 49 vary as just

like the theory according to the amount of H in E among the terms representing the surface tension effects in the equations of both (19) and (23), and therefore the theory concerning to the variation of the

surface tension effect due to the variation of the amount of the surface tension is in perfect agreement with the experimental results, and the

above theoryis cOrrect.

Results of no. II experiments

In no. II experiments, which were carried out on the varnished surfaces, the experimental values of _{' täke the various intermediate} values within the range between the experimental values of both no. I

and III experiments in the cases of two models of A and C, while

they show the same values as or near those of no. I experiments and therefore the surface tension effects hardly arise in the case of the

model B. Namely, in no. IT experiments, in some cases the surface tension effect don't arise at all as just like the case of the naked

wooden surface, and, in other cases arises so much amount as just like the case of the paraffine surface, while generally the experimental

54 Keizo Ueno

w

values show the intermediate values between those of the above two extreme cases, but they indicate no constant values and are unstable. Very roughly it can be said that the mean value of is about half that of the case of the paraffine surface.

In the cases of both I and III, the values of v are approximately

constant, for instance, in the case of I always = O for all models, while in the case of III always 7) 2 for A-model, and v = i for

B-and C-model, since the surface conditions of both I and III are so

steady that the naked wooden surface of I is very easy to be wet and the paraffine surface of III is very hard to be wet. On the contrary, in the case of II, the values of are very unsteady as above stated

since the varnished surface of II is easier to be wet than the paraffine surface, but is harder to be wet than the naked wooden surface, and therefore the surface condition of II shows the interriediate one between

those of the above two extreme steady surfaces of both I and III.

Moreover the unsteadiness of i of II is due to the fact that, since the layer of varnish is easy to be fallen off from the surface in the midst of the experiment, the value of ' under the surface condition of fresh

varnish newly painted, which shows the property near the parafline surface and is hard to be wet, indicates the value of ïj

near that of

III, while the value of i under the surface condition of varnish fallen off due to the many time operations of the experiments, which shows the property near the naked wooden surface and is easy to be wet, represents the- value of near that of I, and thus the value of r always varies according to the surface condition of varnish at the time of the experiment and is not definite for each experiment.

6. Results of no. V experiments

In no. V experiments, which were carried out on the model ship who has varnished surfaces over her bottom and has paraffine surfaces over the, area of both sides in the vicinity of the water line only, the frictional resistance of this model should indicate the property of the varnished surface, and the surface tension effect should indicate the

property of the paraffine surface. On the other hand, it was cleared up experimentally3> that the amount of the frictional resistance of the varnished surface little differs from that of the paraffine surface, and therefore theoretically speaking the value of V should indicate almost

the same value as that of III, namely, the paraffine surface after all. It is shown in Table V that the experimental values of 1) of V are

approximately equal to those of III for A- and C-model, where the no. V experiments were not carried out on B-riodel. Thus it was proved that our theory concerning to the surface tension effect is in perfect agreement with the experimental results, and what we call a surface tension effect certainly exists.

7. Summary

The experiments of no. I, H, HI, IV and V were carried out on three models of A, B and C. It was experimentally confirmed that the surface friction of the model is perfectly irrespective of the dif-ferences between the above experimental values. The conclusion ob-tained fròm the experimental results are as follows:

If the type of

the model ship is definite, the amounts and their order of the experi-mental values of for the experiment no. I, II, III, IV and V are

approximately equal to those of v2, respectively, irrespective of the

- exîstence of bilge keels, and also the experimental values of y of V

are approximately equal to those of III. The experimental values of both T and 41 vary as just like the theory according to the amounts of both H and h in E among the terms representing the surface tension effects in the equations of both (19) and (23), and therefore the theory concerning to the surface tension effect is perfectly in agreement with the experimental results. It can be proved by the above facts that

what we call a surface tension effect certainly exists, and the theory concerning to the surface tension effect is correct.

VI. Conclusions

By summing up the theory and the experimental results stated as above, the following conclusions are obtained.

As the results of the theoretical analysis of the influence of the surface tension of the surrounding water ubn the period and the loss of amplitude per swing in the free rolling experiments of a model ship, we found that, by the surface tension effect, the period steeply

decreases iñ accordance with the decrease of the amplitude when the amplitude is very small, and the value of loss of amplitude per swing increases approximately by a certain amount except the case of the very small amplitude. On the other hand we could confirm that the above theory is in perfect agreement with the experimental results,

and thus we could prove that what we call a surface tension effet

certainly exists and our theory-concerning to the surface tension effect is correct.

The experimental values of the effective factor concerning to the surface tension effect for the period curve, which should be great when the amplitudes are small, were approximately equal to those for the curve of extinction, which should be great when the amplitudes are large, and therefore it can be said that the effective factor is

irrespective of the amount of the amplitude.

The surface tension effect varies according to the nature of surface of the model ship, for instance, it don't arise at all in the

naked wooden surface, which is very much easy to be wet, it perfectly

arises in the paraffine surface, which is very miich hard to be wet, and it shows the intermediate property between the above both surfaces in the varnished surface, namely, the fresh varnished surface newly painted shows the property near the paraffine, and the layer of varnish

will be gradually fallen off and approach to the property of the naked wooden surface in accordance with the repeat of the experiments. Therefore the experimental value of the effective factor is zero through

L

for the ordinary ship-shaped type and 1 for the cylindrical type in the

paraffine model. In the case of the varnished surface it shows the

intermediate property between the naked wooden and paraffine surface, it varies according to the surface condition at the time of the experi-ment and its value is unsteady, but its mean value is very roughly speaking half the amount of that of the paraffine surface according to each type of the model ship.

The surface tension ffect increases very much in accordance with the decrease of the metacentric height.

The surface tension effect decreases in accordance with the decrease of the surface tension, for instance, decrease in the case, in which, an oil is dropped on water surface.

The existence of bilge keels is almost irrespective of the

surface tension effect. Therefore both of the virtual moment of inertia and the resistance are almost irrespective of the function of the surface

tension.

Since is reciprocally proportionate to the second power of the linear dimension of the model ship, the surface tension effect is great in the free rolling experiment of the model ship, but is so little as can be negligible in that of the actual ship.

AN AFTERWORD: ACKNOWLEDGEMENT

This study was carried out in the Institute of Naval Architecture of the Faculty of Engineering of the Kyûshû University.

In concluding this paper, the author wishes to express his hearty thanks to Mr. S. Uchino, S. Munakata, K. Nakahara and other members of the Experimental Tank of the University for their cooperation in carrying out the experiments.

Bibliography and a note

C. Müller: "Theoretische Physik ", 4 Auflage, s. 204.

K. Ueno: "Theory of Free Rolling of Ships', Memoirs of the Faculty of Engineerig, Kyûshû University, Vol. IX, No. 4, 1942.

3) A note: The results of the frictional resistance measuring experiments of a wooden flat plate (1.80 im length x 0.29 m. draught) towed edgewise within the range of speed between 0.1LO rn/sec. are represented in Fig. 15. The towing experiments were carried out on the following four kinds of surfaces of the plate. The first kind is the varnished surface, the second kind is the varnished surface which layer of varnish had been partially fallen off by the past many time operation of the towing experiments, the third kind is paraffine surface, and the fourth kind is the naked wooden surface which paint had been perfectly fallen off by the polishing up of sand papers. It

is evidently shown in Fig. 15 that the frictional resistances of the various kinds of surfaces increase gradually in order of the increase of the number, namely, from the first to the fourth kind within the range of speed experi-mented, but they little differ, and very roughly speaking the frictional resistance of the second kind of surface is about 5%, that of the third kind about 10%, and that of the fourth one about 20% greater than that of the

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lo

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TC'';

ny

f-L542

iiJí_-i

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t o z i-VI t.

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78 q

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01294567

q O. IN DEGgE io II 12 13 14. I5 16 17 18 iq FALLEN SURFACE OFF PAITIALLY VARNISHED

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PAINTED FRESH VARMSHED NEWLY SURFACE

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-

uI'.

__

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II

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