On the longitudinal reduction factor for the added mass of vibrating ships with rectangular cross-section

an'

e

/. Li

REPORT No. 40S

April 1961

STUDIECENTRUM-r.N:O. VOOR SCHEEPSBOUW EN NAVIGATIE

AFDELING SCHEEPSBOUIT/ - PROF. MEKEL1VEG DELFT

'NETHERLANDS' RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION)

(SHIPBUILDING DEPARTMENT - PROP. MEKEL WEG - DELFT)

ON. THE LONGITUI)INAL REDUCTION. FACTOR

FOR THE ADDED MASS OF VIBRATING SiIíP

WITH RECTANGULAR CROSS-SECTION

(Over dé Iangsscheepse reductiefàctor voor de toegevoegde

massa van triliendé schepen met rechthoekige spantvorm)

by

Ír. W. P A. JOOSEN and

_{Dr. J. A. SPAREÑBERG}

(Netherlands Ship Model Basin - Wageningen)

0

V.

7

tc)

Luued by the Council This report is not to be published unless verbatim and unabridged

Reports

No. i S The determination of the natural frequencies of ship vibrations (Dutch) . By prof. ir H. E. Jaeger. May 19 5 0.

No. 2 Confidential report, not published. July 19 5 0.

No. 3 S Pracièa1 possibilities of constructional applications of aluminium alloys to ship construction.

By prof. ir H. E. Jaeger. March 195 1.

No. 4 S Corrugaçion of bottom shell plating in ships with all-welded or partially welded bottoms (Dutch)..

By prof. ir H. E. Jaeger and ir H. A. Verbeek. November 195 1.

No. S S Standard-recommendations for measured mile and endurance trials of sea-going ships (Dutch) . By prof. ir J. IV. Bonebakker, dr ir W. J. Muller and irE. J. DieM. February 1952.

No. 6 S Sorne tests on stayed and unstayed masts ó.nd a comparison of experimental results and calculated stresses (Dutch) .

By ir A. Verduin añd ir B. Burgbgraef. June 1952.

No. 7 M Cylinder wear in marine diesel engines (Dutch).

. By ir H. Visser. December 1952.

No. S M Analysis and . testing of lubricating oils (Dutch) .

By 'ir R. N. M. A. Malotaux and, ir J. G. Smit. July 1953.

No. 9 S Stability experiments on models. of Dutch and French standardized lifeboats.

:

By prof. ir IL. E. Jaeger, prof. ir J. W. Bonebalth.er and J. Pereboom, in collaboration with A. Audigé. October 19 52.

No. 10 S On collecting ship service performance data and. their analysis.

By prof. ir J. W. Boneba/z/ter. January 1953.

No. 11 M The use of three-phase current for auxiliary purposes (Dutch).

By ir j. C. G. van Vfijk.. May 1953.

No. 12 M' Noise and noise abatemént in marine engine rooms (Dutch). By "Technisch-Physische Dienst T.N.O. - T.H." April 1953.

No. 13 M Investigation of cylinder wear in diesel engines by means of 1abo;tory machines (Dutch). 'By ir H. Visser. December 1954.

-No. 14 M The pùrification of heavy fue! oil for diesel engines (Dutch).

By. A. Bremer. August 1953.

No. 15 S Investigation of the stresS distribution in. corrugated bulkheads with vertical troughs.

By prof. ir H. E. Jaeger, ir B. Burgbgraef and J. van der Ham. September 1954.

No. 16 M Analysis and testing of lubricating oils II (Dutch).

By ir R. N. M. A. Malotaux and dis J. B. Zabel. March 1956.

Ño. 17 M 'Tle applicatiòn of new physical methods in the examination of lubricating ils.

By ir R; N. M. A. Malotaux and dr F.,van Zeggeren. March 1957.

No. 18 M Considerations on the application of three phase current onboard ships for auxiliary purposes especially with regard to fault protection, with a survey of winch drives recentlyapplied on board of these ships and their influence on the generating capacity (Dutch).

By ir J. C. G. van Wijlz. February 1957.

-'

No. 19 M Crankcase explOsions (Dutch).

By ir J. H. Minkborst. April 1957. .

No. 20 S An analysis of the application of aluminium alloys in ships' structures.

Suggestions about the riveting between steel and aluminium alloy ships' structures.

By prof. ir H. E. Jaeger. January 1955.

No. 21 S On stress calculations in helicoidal shells and propellerblades.

-By dr ir J. W. Cohen. July 1955.

No. 22 S Some 'notes on theCcalculat.ion of pitchuxig and heaving in.longitudinal wàves. By ir 'J. Gerritsma. December 1955.

.

No 23 S Second series of stability experiments on models of lifeboats.

-By ir B. Burghgraef. September 1956.

No. 24 M Outsid&corrosion of and slagformation on tubes in oil-fired boilers (Dutch).

ON THE LONGITUDINAL REDUCTION FACTOR FOR

THE ADDED MASS OF VIBRATING SHIPS WITH

RECTANGULAR CROSSSECTION*)

by

W. P. A. JOOSEN and J. A. ,SPARENBERG**)

Communication of the Netherlands Research Center. T. N. O. for Shipbuilding añd Navigation

Summary

The added mass of vertically vibrating ships with rectangular cróss-sectiòns is discussed. Thé ship is

approx-imated by a cylindér with a rectangular crosssection. 'Water motions in the direction of the axis of the cylinder

are taken into account The effect of gravity is neglected.

1. Introduction

This paper deals with the influence of the

cross-section of a ship on the three dimensional correction

coefficient for the added mass [1], when the ship

executes a vértical vibration.

The ship is considered to be a cylinder of infinite length with rectangular cross-section. Although we

do not çonsider three dimensional effects of the

bow and the stern we take into account three

dimensional effects arising from sinusoidal shear vibrations of the hull. These vibrations cause the water to move also in the longitudinal direction of the ship. Hence a theory which considers only two dimensional water motions must give a value for

the added mass which is too high.

We. discuss frequencies which are large with respect to the frequency of gravity waves with

lengths comparable to a characteristic dimension of

the cross-section. Then the effects of gravity can be neglected (high frequency approximation) arid

we may assume the velocity potential to be constant on the surf ace.

Two kinds of cräss-sections have already been

discussed in [2], viz, a flat strip and a half

im-mersed circular cylinder. Here we are concerned

with reatangular cross-sections.

First we consider the asymptotical behaviour of

the added mass for small ratios of the length of the

shear wave in the cylindrical hull and a character-istic dimension of the cross-section. Next by

nu-merical means we consider the case of finite values of this dimensionless parameter.

It is the intention to investigate other hull forms as well and to publish at the end of the research a

summarizing paper on all the results of which reference [2] and the -present article are

fore-runners.

) Risearch carried out at the Institute of Applied Mathematics of the Technical University of Deift for the Netherlands Research

Centre T.N.O. for Shibuflding and Navigation.

) Both authors are now at the Netherlands Ship Model Basin

2. Formulation of the problem

We consider the situation of Fig. 2.1, The water

occupies the half space y.> - u b. The width and

the height of the rectangle are 2 b and u b respeC-. tively. The z-axis stretches along the cylinder. We' assume that the motion of the water is irrotational

and simple periodic and can be described by a

potential O(x, y, z, t) which satisfies:.

=0.

(2.1) The velocities in the , and direction are:

ae - ao

u = -=-, y = -,u'=

. .. (2.2) ax . a az

(o,-b)

(2b-b)

/7'

(2b,o)

,..

y

Fig. 2.1. The ¡-mmersed rectan guiar cross-section

On the water surface we assume the pressure to. be zero. With the high frequency approximation.

this reads:

(2.3)

The bottom of the rectangle executes a vibration with the amplitude:

e cos a. (2.4)

-.

-=ieoicosaze"t;0<x.<2b,y=0 (2.5)

-==0;x=0,x=2b,ub<y<0.. (2.6)

ax

3.. Shear vibrations with a small period

First we consider the asyrnptotical behaviour of the added mass for shear vibrations for which the

interference of the edges is negligible. This happens

when the period of the vibration in the z-direction

becomes small with respect to the width 2 b f the rectangle.

We introduce the following dimensionless

quan-tities:

x=xa,y=ya,z=za,a=ab,

O(x, y, z, t)

=

a

O(x, y) cos z ect,(3.1)

where the assumption made for O (x, y, z, t) is a.

direct consequence of the periodicity in the

z-direc-tion (2.4). With these coordinates the boundary

value problem becomes:

(

+

) i(x, y) = 0.,.

(3.2)

i9'=0,x<0,x>2a,y=-1ua, (3.3)

=1,0<x<2a,y=0,

. (3.4)

.=0,x=0,x.=2a,O>y>,ua.(3.5).

In this case we have ab = a)) 1, hence the boundary conditions become approximately:

----= 1,0< x, y = 0;

0, X = 0,0< y.

(3.6) This type of problem viz.: seeking a solution of the

Helmholtz equation in a wedge is generally

dis-cussed in [3]. However, because our case is much

more simple we shall follow another method.

We consider the function:

VY=x.ay

(3.7)

Introducing polar coordinates (Fig.

3.1) we

obtain for the following boundary value problem:

- + -- + -- -

= 0,

'r

ar r2

I 3J) i 3/)

3yi

O, 97. = 0, ç

-Fig. 3.1. Tbc boundary vaizu' problem for a -- c

In order to avoid a source-like singularity and to

obtain a bounded function at infinity we have to

take:

.'(r, p) = Q' K,1, (r) sin 2/3 9) +

+ Q'2 K41, (r) sin 4/3 q', . .. (3.10) where the functions K21, (r) and I, (r) are' modi-fied Bessel fùnctions.

From the definition of

(37) we find by

in-tegration:

f ip(r,)d+h(y),

. (3.11) 3'

-m

+00

= - .5 p(r, q,) 'd

+ g(x), (3.12)

y

where b(y) and g(x) are arbitrary functions.

Be-cause and - vanish for r -. 0°, 0 <92

we find:

h(y) - 0, g(x) = O

(3.13')

Next we have to determine the constants Q and

Q2 in (3.10).

From the boundary conditions 'at = O 'and

97=

3 we obtain easily:

tim 1100{Qi K,,, (r) sin 2/ 92 +

y.+O -00

+. Q K4h (r) sin

4/ ç,) d

= 1, (3.14)

{Qi K,,, (r) sin '2,/s

_{9' +}

+ Q. K41, (r) sin 4/3 92) d

'- 0. (3.15)

However, the function 1(4/, (r) under the sign of

(3.8)

a2

+±

ay-ex+00 a

C_00

3

f e

₂

_aay

d+ - f e

₂

_aay

(3.25)

where A (y) and B(y) arç arbitrary functions of y.

Because - tends to zero at infmity we find from

ax

(3.22) and (3.25):

A(y) = 0, B(y) = -

±00

_f

_aay

a2

(3.26)

Integrating the integral for B(y) partially we

obtain:

a+00

B(y) =

- f ex ip

d

...

(3.27)

y-00

This integral can be calculated. Because is a

solu-tion of the Helmholtz equasolu-tion we have:

(+2_i)d=o.

(3.28)

Integrating partially we obtain: +00

y

f eid=

(3.29)

hence:

+00

f epd= ± a9 y

(3.30)

Letting y tend to ± oc we can easily estimate this

integral from which follows a2 = 0. Hence from

(3.27):

B(y) = 0.

(3.31)

The last two integrals in (3.25) can be integrated partially. Then we obtain:

f d.=-- f ed±

ax 2 x ay

ex°

a

9x+00 _ap

-- f e

d (3.32)

as the desired transformation of the first integral

in (3.21).

The second integral in (3.21) can be written

more. easily:

z _a

+00a

. ±00

f yd= f d f

_{00ay 00ay} d

ay

+00a

= - e

_{- f -- d}

(3.33)

z y

where we made use of the fact that at large

dis-tances from the corner its influence has died out.

3

integration gives rise to a singularity of the order Solving this equation we find the result:

r'" which is not integrable. We cnsider:

f(x, y) = A(y) ex +. B(y) e

+

+00 +00

f r" sin

4/3.

_{d 4 =}

f z

_{dz =}

00

Im (-3

1/s) 00

₌

(3.16)

where we have temporarily used the notation

z = x + iy. From this it follows that we này

subtract the singularity of K413 (r) without fur-ther correction and we obtain from (3.14) and

(3.1 5) the following relations between Q' and Q2;

2n

47,

Qi q

sin

_{+ Q q sin}

= 1, (3.17)

2n

Qi qi sin

± Q q sin i--

0., (3.1.8) where:

+00 Ç +00

qi = f K213 (x) dx, q

f {K13 (x) ±

210 r (4/3) _{x-4'} dx . . . . (3.19)}

From (3.17) and (3.18) we obtain:,

Q'

(v'3 qi)1, Q = - (v'3 qs).(3.20)

Now we determine the potential i. From the

Helmholtz equation we find:

+00a a

f -

_ax

dì + f

_{_00ay}

-

j

(3.21)

Herewith we have found the solution of our

problem, because each point outside the body can be reached by the paths of integration. However,. we are especially interested in the pressure under the ship, hence in values of

for y = 0, x > 0.

From this point of view we will replace the

inte-gration with respect to

in (3.21) by an

inte-gration with respect t in order to facilitate

numerical computations. We consider:

+00a

f

d37f(x,y)

(3.22)

where is a solution of the Helmholtz equation

ax

which tends exponentially to zero at infinity.

From:

+001a2

f +--

a

tax-

ay -

1j

(x,

) d = O

a

(3.23)

we obtain by partial integration with respect to :

( a

1)

t(,y=

a2

(3.24)

=

ax

Using these results we find for (x, y) the form:

+ f---{cosh(xC)-1}dC,y>,O

z Y

-

K11, (r) sin 4/g q' sin q' } For the special value y = 0, (3.34) becomes:

O(x, 0). = - i ± f

K213,(C). +

+ Q

_{K, () } {}

e

±

} d C,

...(3.36)

from which it follows that 8(x, 0) tends expo-nentially to the value - 1.

4. The case of finite periods and derivation of the

integral equation

We return to Fig. 2.1, but introduce for

con-venience another system of reference (Fig! 4.1).

Fig. 4.1. The immersed rectangular cross-section

As the unit of length we choose b and we obtain the following dimensioiess quantities:

x=,y= ,z=

[a=abO(x,y,z,t)=

= i b r w

(x, y) cos a z eb0.t.., (4.1) where x, y, z and are different from those defined

in 3. Herewith the boundary value problem becomes:

+

-

ac)

(x, y) = 0,

. (4.2)

= 0, Hi > 1, y = O ....

(4.3)

A Green function can be constructed which

vanishes at infinity, satisfies (4.2), except; for

X C, y = where it has a logarithmic singularity

and which conforms to the foliöwing boundary

conditions:

aG

---±0xO;G=Oy=O... (46)

This function has the óbvious form:

(3.35) G(x, y, C,

)=[Ko(a

(x

Cj2 + (y -

) +

- Ko(a

'( - C)2

+

(y +

) +

+ Ko(a /(x +E)2 + (y

)2) +

Ko(a

(x ± C)2 +

' +

)2)] (4.7)

In the neighbourhood of x = C, y = j

it behaves as: in

/(x - C)2 + (y

)2 _(4.8)

We assume that the point x, y is placed on the con-tour of the. rectangle. Applying Greens theorem for

the path of integration indicated in Fig. 4.2 we

obtain: B _r i

aG(x,y,C,)

an

- G(:, y, C,

a(C,)

} dC +

+ f

+

a8(C, ) i

G(x,y,C,ì).

_an

r i

aG(x, y, C, )

= -j

19(C,)

an 3'

')

1

- G(, y, C, ')

_a, j dx (4.9) where y is the small semi-circle, which encircles the

point of.siñgularity, n is the outer normai and x

the arc length along y. In the case that Q(x, y) lies on AB we.have to take the principal value f Canchy for the first integral in the lefthand part

of (4.9). Using the boundary values of (4.3)-.(4.6)

+

ay

- 1,

< i,

= p,

(4.4)

ro,1x1±1,/2>y>o

...

(4.5) (3.34) where:

ap_

12 2/3 (r) cos 5/3 q'

-- K_11, (r) sin

2/3 _{q' Sm}

₊

± Qs {

K, (r) cos 7/3 q'

±

(4.13)

Along the contour ABC we introduce the length t

or r, measured from the point B, as a parameter.

i . . . . Then we have for:

and tne iogarithmic singular behaviour of K0 we

can write (4.9) in the form: AB BC

¡3

x=t±1 C=r+i, x1

C=I

,y=t=tr

A

+ G(x,

u, _{C, ,u)) dC +} C

f(1,

aG(x,e, 1,

di = -

i(x,

B (4 10)

when Q(x, y) lies on BC we obtain:

Fig. 4.2. Application of Greens theorem

I{ (C,.)

aG(1,y,,t)

±

± G(1, y, C, t)} dC +. (1, ₎

G(1,y, 1,

)

-

(1 _y) (4.11) Differentiation of (4.7) yields: G(x, y, .C, i) a G(x, y, C,

i)

a,1 a

_{¡K1(a v'(x - C)2 + (y.}

)2)

V(xC)2+(y,1)2

+

+

Kj(a V(x + C) + (y

)2)

+

V(x ± C)2 + (y

,1)2 J IK1(a

/(x

_{- C)2± (y}

_±

,1)2)

+(y±,1)1

..

±

t

v'(xC)2+(y+,1)2

+

Ï(j(a

(x ± Cj2 + (y ±

)

v'(x + C)2 + (y ±

(4.14)

From this it appears that i on ABC depends on one variable. The two equations (4h10) and (4.11) cân be written as:

(t) =SK(t,t)

(r) dr ± ¡(t) (4.15)

where K(t, r) is defined by the scheme in Fig. 4.3

and:

f(t)= _!1[KoIt_rI) +

-

Ko(a /(tx)2+4 u2) ±Ko(a t+r+2

I)

+

Ko(av'(t+r+2)2+4du2)]dr,-1t<O

(4.16)

10

¡(t) =

-

[Ko(a /t2

+ r2) +

-JCo(a V(t2)2+r2) +

=

_{(x - C)}

_f

Kj(a V(x

¿)2 + (y

,1)2) + Ko(a

(r + 2)2 + 12) ±

- C)2 + (y

,1)2

+

Ko(a

(t -

2

)2 + (r + )2)} dr, 0<

t

...(4.17)

I f _i. .2

f

±

The integral equation (4.15) was solved

nurner-- )

F i

icaily on the electronic computer Z.E.B.R.A. of 'the

I K1 (a /(x ± C)2 + ,)2) Institute, of Applied Mathematics .at the Technical

- (x + C)

/ 2

+

Umversity at Deift The range of mtegration was

y

(x + C) + (Y - ?7)

devided into a number of intervals and the inte-Ki(a v"(x + C)2 + (y 4-

)) Ì

gration formula of Simpson was applied. Before

-

this could be done it was necessary to. subtract the

.J.. \2 J_. I J_. 2 _{J .}

.

I I i singular behaviour from the functions. For mstance

(4.12)

from the kernel K(t, r)

a -furìction. of Dirac.

t

o

+

..T {Q1 K21, (2)

+ Q2--- K41, (5)} {cosh (ax

i) dfl d

(5.3)

Changing, the order of integration and

remember-ing that (x, 0) tends exponentially to the value

1, we obtain:

+

K41, ()} {sinh4 } d (5.4)

Carrying out the integrations in this formula by

numerical means we find:

2 0.418

r .m _b

....

(5.5)

The added mass for the case f finite periods

- follows frOm: in r w2 cos a z

=

H b b r w2 cos a z.ec0t 2 f 9(x) dx (5.6) o B

_..[

t TEL +.

kj(a/t2+

+ 2 TE + J

íKu(aVt2+t2 )

+ I.

Vii,.t2

fK, CV(t_t)2..h.)

(t+2)2 )

}

\/(t_t)2+4

/t2+t+2)2

+(t._2){I(V(t2)2+t2

KCaV(t,+t_2)2+4

)

+K,CaVCt_2.O2+(t+2)2)

VCt_2i.L2+t2 VCt+t_21L)2+L l/(t_21.L)2+(t+2)2 A. 1.

2aj.ifKl(aV(tt2+4jj.2)+

B

it f

TE _L t

-.

K,(V't2

+ + ) C Ki ( Vt2+12 ). + Vt2+t2 lt

_{V(t_t)2+.L2}

-

+(t-2)2 )

.

_(t.2)

K(

K,(aVÇt+t+2)2+hj2 }

Ki((t+2)2

+

t2)

Vt+t+2)2+4J.2

. t.

Vf(t+2)2t2

2)2c2)2

.. o

t

Fig. 4.3. Scheme of the kerncl of the integral equation

This 5-function ai-ises fort and in the.

neighbour-hood of the corner

t

t = 0. After this the

solu-tion of the integral equasolu-tion was reduced to the

solution of a system of linear equations.

.5. The added mass, discussion of the results

The varying part of the force per unit length in

the z-direction, exerted by the water on the bottom of the rectangle, becomes:

K cos ae°= - f P(,

, t) d

=e5 (x,y,z,t) dx

...

(5.1)

where the integral stretches along the bottom. The

added mass in is defined by:

K cas a ze"°

m r w2 cos a z et. (5.2)

First we consider the limiting case of § 3. With the

aid of (3.1), (3.9), (5.1) añd (5.2) we obtain:

in r w2 cos a z e'°'

rw2 b

-

o

cos a z e'°' 2 f

ja

where 9() is the solution of (4.15) expressed in

the original coordinte x

In this way ve arrive at the data given in

Table I and drawn in Fig. 5.1.

TABLE i a

0.4 0.8 1.2

r

On the vertical axis is plotted the dimensionless

added mass

and on the horizontal axis the

dimensionless quotient of the width of the

rec-tangle .2 b an4 the length of the shear wave

a

The values for a = O fóllow directly from

po-tential theory [1]. For completeness we have drawn

also the results of [2], ® and

which

repre-sent the added mass of a cylinder and of a flat strip with semi-circular cross section, both with width 2 b!

As is expected and is confirmed bythe results

the three dimensional influence is more important

for hull forms of which the bottom has a larger

distance to the surface öf the water.

0.4 1.76 -1.41 1.14 0.945 0.8 1.87 1.45 1.16 0.952 1.2 1.92 1.46 1.16 0.952 1.6 1.96 1.47 1.16 09.54 2

't-

a- 't 't

Fig. 1.1. Dimensioñless added mass for several rectangular

cross-sectioñs. (I), u - 0.4; (2), u = 0.8; (3), u = 1.2;

(4). e 1.6; (5) asympto/ical behaviour.

Pefe'rences

Saunders, H. E.: "Hydrodynamics in ship design". The Society

- of Naval Architects and Marine Engineers, New York,

1957, Vol. H.

Sfsarenberg, J. A.: "Applications of the Hubert problem to

problems of mathematical physics". Thesis, DeIf t, 1958.

Laurier, H. A.: "Solutiòns of the equation- of Hélmholtz, in an

angle". Kon. Ned. Akad. Wet. Proc. Series A-, 62, No. 5,