Added mass and damping coefficients of heaving twin cylinders in a free surface

(1)

DEPARTMENT OF THE NAVY

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER BETHESDA, MD. 20034

ADDED MASS AND DAMPING COEFFICIENTS OF HEAVING TWIN CYLINDERS IN A FREE SURFACE

by

C. M. Lee, H. Jones, and J. W. Bedel

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

(2)

ABSTRACT ADMINISTRATIVE INFORMATION 1 INTRODUCTION 1 THEORY 2 FORMULATION 2 SOLUTION 4

ADDED MASS AND DAMPING 7

EXPERIMENT 9

EXPERIMENTAL SETUP 9

EVALUATION OF DATA 10

RESULTS AND DISCUSSION 12

CONCLUDING REMARKS 14

ACKNOWLEDGMENTS 14

APPENDIX A EVALUATION OF MATRIX ELEMENTS 31

APPENDIX B EVALUATION OF POTENTIAL INTEGRALS 37

APPENDIX C EVALUATION OF THE PRINCIPAL VALUE INTEGRALS 43

REFERENCES 48

TABLE OF CONTENTS

Page

(3)

LIST OF FIGURES

Page

Figure 1 - Description of Coordinate System 15

Figure 2 - Description of Boundary-Value Problem for (x,y) 15

Figure 3 - Segmentation of Cylinder Contour 15

Figure 4 - Complete Model Setup for Testing 16

Figure 5 - Block Diagram of Electric Setup on Carriage 2 for

First Series of Tests . 17

Figure 6- Block Diagram of Electric Setup on Carriage 2 for

Second Series of Tests 18

Figure 7 - Added Mass Coefficient versus Frequency Number for

Twin Semicircular Cylinders for b/a = 1.5 19

Twin Semicircular Cylinders for b/a = 2 . 19

Twin Semicircular Cylinders for b/a = 3 20

Figure 11 - Damping Coefficient Versus Frequency Number for

Twin Semicircular Cylinders for b/a = 1.5 21

Figure 12 - Damping Coefficient versus Frequency Number for

Twin Rectangles for b/a = 2 23

Figure 17- Added Mass Coefficient versus Frequency Number for

Figure 19- Damping Coefficient versus Frequency Number for

(4)

LIST OF FIGURES (CONT.)

Page

Twin Isosceles Triangles for b/a = 3 26

Twin Right Triangles for b/a = 3 . .

. ... .

. . .

. ... .

. 28 Figure 26 - Added Mass Coefficient versus Frequency Number for

Twin Right Triangles for b/a = 4 28

Twin Right Triangles for b/a 3 29

Twin Right Triangles for b/a 4 29

Figure 29 - Change of Integral Path when Re(z--1") > 0. .44

Figure 30 - Change of Integral Path when Re(z

< 0

44

Table 1 - Figure Index of Added Mass and Damping Coefficients for

Each Model Shape; Model Dimensions are Given in Inches 10

(5)

NOTATION

a Half-beam of cylinder

b Separation distance (see Figure 1)

tc.

j h line segment of cylinder contour

F Vertical hydrodynamic force

G Complex wave source near a vertical wall g Gravitational acceleration

h0 Amplitude of oscillation

k Spring constant

K /g

M Displaced mass of twin cylinders

Unit normal vector on the surface of cylinder pointing into the fluid Qi Source strength at jth segment

ni) Lefthand end point of jth line segment

z x + iy

a Tangent angle of the jth line segment 8 a2 a/g = Ka

+

Damping

Damping coefficient formed by dividing the damping A by the product of the displaced fluid mass and the radian frequency

A _{Added mass}

ji Added mass coefficient formed by dividing the added mass p by the fluid mass displaced by twin cylinders

p Density of fluid

a Radian frequency of oscillation

Velocity potential function of harmonic time dependence

0 (=Oc + j 0s) steady velocity potential of complex function with respect to j = 0c The real part of 0; subscript c indicates that it is associated with cos at

(6)

ABSTRACT

A potential flow problem, dealing with twin horizontal cylinders of arbitrary cross sectional forms vertically oscillating in a free surface is investi-gated. An associated experiment is carried out for four different sets of twin cylinders. The results from the theory and the experiment are compared and are found in good agreement.

ADMINISTRATIVE INFORMATION

The theoretical part of this work was authorized by the Naval Material Command under the in-house research project and was funded under Project R01101, Task ZR011 0101. The experimental part was funded under the Naval Ship Systems Command Research, Development, Test and Evaluation program, General Hydromechanics Research, Subproject S-R009 01 01, Task 0100.

INTRODUCTION

Investigations conducted in the past, concerning the motion of ships in waves, have demonstrated the practicability of the strip theory for obtaining the hydrodynamic forces and moments acting on ships in waves. The present work provides the means for computing the hydrodynamic coefficients associated with the motion of catamarans in regular head waves. Since the strip method is to be applied in obtaining the necessary hydrodynamic coefficients, the basic problem it reduced to a two-dimensional flow problem.

Many investigators have already studied similar problems. Potash' obtained a solution without providing numerical results for semisubmerged twin circular cylinders, rigidly connect-ed from above, heaving in a free surface. Ohkusus investigatconnect-ed the same problem as Potash and obtained the values of added mass and damping for various frequencies and separation dis-tances between the two cylinders. Later, Ohkusus studied two or more rigidly connected cylinders heaving, swaying, or rolling and applied the results obtained from this investigation to the motions of multihull ships in waves. He used an approximate scheme utilizing the results of his previous works on the twin semicircular cylinders to obtain the added masses and

dampings of the noncircular cross sections of the ship. Wang and Wahab4 also investigated theoretically and experimentally the problem of twin semicircular cylinders heaving in a free surface. They showed excellent agreement between their theoretical and experimental results;

References are listed on page 48.

(7)

de Jong5 derived solutions without providing numerical results for heaving, swaying, or rolling twin cylinders of symmetric cross sections, using conformal mapping.

The previously discussed investigators used the method of multipole expansion to determine the unknown velocity potential. This method was first introduced by Ursell6 in the solution of the problem of a semicircular cylinder heaving in a free surface. Ursell's method I assumed a series of singularities of increasing order placed at the intersections of the midplane

of each cylinder with the free-surface plane, with each of the singularities independently satisfying the free-surface boundary condition. The unknown coefficients of the series were obtained by satisfying the kinematic boundary condition on the cylinder surface. In

prin-ciple, this method may be applied to any shape that can be mapped from a circle. However,1

the investigators cited previously chose only those cylinders which were symmetric about their own vertical midplanes.

The present investigation deals with twin cylinders of arbitrary cross sections which do not have to be symmetric about their own vertical midplanes. (The problem still assumes

the two cylinders to be of identical shape.) The mathematical tool adopted in solving the problem is the method of source distribution on the cross sectional contours of both cylin-ders. The same method was applied for an oscillating single cylinder by Frank'.

Since the heaving twin cylinders constitute a symmetric flow about a vertical mid-plane, the problem can be reduced to the case of a single cylinder heaving near a vertical wall. In fact, the problem is treated in this fashion.

Tests for four different shapes of twin cylinders were performed by vertically oscillating the twin cylinders in a calm free surface. The four cross sectional forms chosen

were shaped as a semicircle, rectangle, a right triangle, and an isosceles triangle. Several

separation distances between the two cylinders were chosen, and the oscillation frequencies were selected to cover the practical range of catamaran motions in waves. The results from the theory and the experiment were compared and were found in good agreement.

THEORY

FORMULATION

Two semisubmerged identical horizontal cylinders of infinite length, connected above

the waterline, are vertically oscillated in a calm water surface with an amplitude which is small compared to the beam of the cylinders. The fluid in which the cylinders are immersed is assumed incompressible; its motion, irrotational; and its depth; infinite. It is also assumed

that the oscillation has been going on long enough for the initial transient effect of the fluid to be completely phased out.

The x-axis is taken to coincide with the undisturbed free surface and the y-axis is directed vertically upward. The origin is taken at the midpoint between the two cylinders.

(8)

The distance from the origin to each half point of the cylinder beam is taken to be b; the cylinder beam is taken to be 2a; see Figure 1. Since the problem described previously

dictates an obvious hydrodynamic symmetry about the y-axis, the problem can be reduced to the right half of the plane only.

By introducing a velocity potential function 4) (x,y,t) and by properly prescribing the necessary conditions on the fluid boundaries, a bdundary-value problem in terms of 4) can be formulated. With the assumption of small oscillation, only the linear frequency response of the fluid to the disturbance will be considered. Thus the velocity potential can be written as

4)(x, y, t) = Rei 10(x,y)e-i7t 1 = 0, cos at + 0s sin at (1)

where

=c + jos

(2)

The motion of any point on the surface of the cylinder is expressed by

y(t) = h0 sin at (3)

Continuity of mass implies that

V2 95(x, Y) = 0 (4)

in the fluid region.

The assumption of a slight disturbance on the free surface leads to a linearized form of the free-surface condition, which is given in the form of*

Oy(x, 0) - KO = 0 (5)

where K = cr2 /g. The derivation of the expression is given in Wehausen and Laitone8.

The linearized kinematic condition on the cylinder contour is given by

0. =V0 - n = Vn (6)

at the mean position of the cylinder. Here n is an outward unit normal vector on the cylinder contour, and V. is the normal component of the velocity of the cylinder contour. It can be readily shown that

when the space variables x and y and the time variable t are used as subscript, they indicate partial derivatives.

(9)

Vn = h a cos (n y)_o

where cos (n,y) means the direttional cosine between the normal vector and the y-direction. Due to the symmetry of the fluid disturbance about the y-axis, there cannot be flow crossing the y-axis. That is,

0.(0,y) = 0

This condition implies that the plane x=0 can be regarded as a rigid wall.

The expected decay of the fluid disturbance as y can be described by

N70(x,w)=0 (9)

The far-field behavior of 0 as x oe should represent outgoing waves, i.e.,Sommerfeld's radiation condition,

Ihn (0.jK0) = 0

10)

x

This completes the statement of our problem, and the boundary conditions are shown in Figure 2. The solution of this boundary-value problem will provide the sought hydrodynamic quantities such as pressure distribution, hydrodynamic force, added mass, and damping co.-efficients.

SOLUTION

The solution of the velocity potential 0(x,y) is assumed to be represented by a distribu-tion of source singularities over the immersed contour of the cylinder; see Reference 9.

0(p) =f Q(s)GR (p; s) ds (11)

co

where p = (x, y) is the field point,

Q =Qc1- jQs is source density,

GR = GRc + j G is the souree, and

Rs

Co is the immersed contour of the cylinder in y<0.

A source of unit strength below a free surface can be expressed in the form of

(10)

where r1 Rx-t)2+(y-77)2 (t,n) is the point of the location of the source, and V2 H=0

everywhere in y<0. It is further required that GR satisfy the free-surface condition4 GRy (X,0) KGR 0

the radiation condition

lim (GRx jKGR ) = 0 x+

and the deepwater condition

VGR (x,-*0)= 0

The solution for GR is given, for example, in Reference 8 in terms of a complex

velocity potential G (z;/-) which is defined by

5

F(z;) = G(z;) + G(z;- r)

.(14)

it can be shown that

d .

Re- i

_dz F(0+1 y-g-)_' =0

Thus, by adding the term G(z;-1-7) to the right-hand side of Equation (13), the function G is

redefmed as

G(z; GR (x,y;E,n)+ i Gi(x, y;t,

= i log (z - log (z

-+ 2

foo

K-k

dk-j2ire" iK()

(13)

-Here

f

indicates a principal value integral, and ? =

It is desired to have the function GR satisfy the symmetric condition

GR x (0, Y; n) = 0

By use of the well-known reflection principle, GR can be made to satisfy the previously shown symmetric condition. That is, if a new function is formed by

(11)

it will be assumed further that the variation of source strength on each segment is so small that it can be treated as constant on each segment. The latterassumption yields the

ex-pression

(p) = E Qi

f

GR(p; s)ds

j=1 ci

When the kinematic boundary condition given by Equation (6) is applied on Equation (16) it follows that

40n(po) = EN Q.;

(nV)/

GR (1); ciS I

j= I c. P =P0

=h a cos a

₀

where a is the tangent angle of the contour of the cylinder at the point p.. By taking N

number of points on the contour c0 and by assuming that these points are located at the midpoints of the line segments 9 (Figure 3), it can be shown that

EQ

(n-V)f

GR(p; s) dsI = h. a cos ai

,p-piec.

G (z; GR (x, y; E, n) + i GI= +j GR5 (GIG ) CO

=

bog

1 (4- (z+F) + 2f e-ilc(z-r) +e-lic(z +0_ dk

27r

(z-)(z+)

K-k

0

-j2ir(e-iK(z--37) + e-41( (z+

0)

(15)

The function GR given in Equation (15) satisfies the same boundary conditions as

imposed on 0, except for the kinematic boundary condition on the cylinder contour. Thi

remaining boundary condition is used to obtain the strengthsof the sources Q(s) given in Equation (11).

It will be assumed that the contour of the cylinder c0 can be approximated by N

number of straight-line segments, each ofwhich is denoted by 9, j = 1, 2, N. Thus,

Equation (11) can be written as

N,

'(p)= E

Q(s) GR (P; S) CIS

(12)

By definition the function Q and G are made of real and imaginary parts with respect to

the complex number j = IFT. Thus, the separation of the real and the imaginary parts of

Equation (20) yields N N

E

_Q

E Q ji

= h° a cos ai J=1 J j=1 3J,

E

y+E Q Ijj =

J=1 '4 j=i j for

i=

1,2, N, where = _{(_1V)./GRc} (p;s) ds I c. p = pieco j = (11-.V)

ARs

(I); S) S1p = pieco c.

and the derivation of these matrix coefficients is given in Appendix A. Equation (19)

repre-sents 2N simultaneous equations from which the unknown coefficients Qoi, Qsj, j = 1, 2, N,

can be obtained.

ADDED MASS AND DAMPING

The hydrodynamic pressure at the point (xo 0) on the cylinder is obtained from the

linearized Bernoulli equation by

P (xo, ,t) = pckt (xo, yo ,

= pRei

a0(xo,yo)e-i°t}

= pa (Os cos at sin at) (22)

The vertical hydrodynamic force acting on the twin cylinders can be obtained by

F =

2

f

P cos a ds

cc,

= 2pa (cos at

fOs

cos a ds

sin at

fOo

cos a ds) (23)

0 0

(13)

and

, y.;

s) ds

(x. , y. ; s) ds)

The separation of Equation (16) into the real and the imaginary parts yields

oc(1(09 yd= E (Q0' GR J=1 cj

Q0

fGRs

(x.,

ci N Yo) = E GRs .1=1

9

C251

fGRc

9

fGRs

ds are evaluated in Appendix B. c.

where Gitc and GRs can be obtained from Equation (15). The integrals

fGR

ds and

c.

If we let the total hydrodynamic force be expressed in the form

F =

(0AY,

_ we have by substitution of y(t)= h. sin at

F = h0a2p sin at hoaX cos at By equating Equations (23) and (27), we find that

Added Mass = =

_h.a

fib,

cos ads

E (Qci Aik

si Bik)COS ak

j,k=1

Damping =A

J.;

_:s cos ads

Co

_-2p

_{2, (Qcj Bjk+Q Ajk}

_sj _{)cos a}

0 j,k=1

;s)ds yo ds) (24) (25) (26) (27) (28) (29)

(14)

where jk

1=1Sk+iSk1=Rtk+iy2+04

G_Rc _{(xk' yk'}s) ds fcrts (Xk, yk;s) ds c. EXPERIMENT EXPERIMENTAL SETUP

Cylindrical-type models, each consisting of two wooden hulls 7.5 ft in length, were tested to determine their heave added masses and damping coefficients. The twin-hull configurations with their dimenSions are shorn in Table 1 along with the hull separations (b-to-a ratios) which were tested for each set. The dimensions b and a are indicated on the

twin-cylinder model in Figure 1. The tests were conducted in two series, the first was

concerned with the semicircular cylinders, and the second comprised the remaining cylinders, includirig some repeated tests on the Semicircular cylinders for checking purpOses.

In order to approach the desired tvio-dimensional case; a piece of one-half inch plywood (3 x7.5 ft) was attached vertically to each of the twin-hull configurations. This also served as rigid coupling between the hulls. Except for the tests of the semicircular cylinders for b/a = 2, 3, and 4, to minimize oscillation of these end boards and to improve rigidity, the boards were reinforced with aluminum angles on the outside as shown in Figure 4. The angles were mounted with head bolts countersunk through the boards into tapped holes in

the angles to minimize forces Which might result from adding the angles. Also shown in

Figure 4 with the complete model setup is the X-frame, used for attachment to the oscillator. For measuring the force required to oscillate the model, four ±100-lb block gages were used to obtain most of the data for the semicircular cylinders, While ±25 lb block gages, one at each end of the cylinders, were used for other cylinders.

The heaving frequency of the model was dependent upon the voltage input to the oscillator motor. This allowed essentially any frequency to be run within the desired range from 0.5 to 3.0 cps.

Two heave amplitudes were used during the tests. Generally the smaller amplitude of 0.25 in. was used at the higher frequencies, while the 0.50-in, amplitude was used at the lower end of the frequency range. In the midfreqUency range, tests were made at both amplitudes to check linearity of the forces with the motion. As a further linearity check for the triangular

(15)

Table 1 - Figure Index of Added Mass and Damping Coefficients for Each Model Shape; Model Dimensions are Given in Inches

models, tests were made: over the midfreqUericy range at an amplitude of 0.75 in. The range I

of frequencies tested at each amplitude was extended to provide additional checks in some

cases.

The tests were conducted at zero speed on Carriage 2 with the carriage at the mid-station of the center deep water basin. The plywood ends were parallel to the length ofthe

basin. This was to prevent the waves generated by the oscillating model from being reflected back onto the model. Sufficient time was allowed between tests for the water to calm

completely as an additional precaution against undue forces on the model.

EVALUATION OF DATA

To determine the hydrodynamic forces acting on each configuration heaving onthe

water suiface, a harmonic heaving,motion was imposed on the mddel floating on the surface.

The equation of motion in this case is

(M + R Xi kx F(t) MODEL '13/a 10-12 1s1

--el

COEF. _1.5 l'.2 12

T

,

Figure 7 Figure 8 Figure 9 Figure 1

I\...__I

X Figure 11 Figure 12 Figure 13 Figure 14

T/

$T Figure 15 Figure 16 Figure 17

6

19 20

i

...--=-1

I

K Figure 18 Figure Figure

Tii

Figure 21 Figure 22

12

1

K Figure23 Figure ;24 _ 71-, _.. 3 Figure 25 Figure 26' Figure 27 Figure:- 28 12

II

Irc,=---. 12 --01 I.'s-12 ' .""'l .

(16)

where F(t) is the force needed to impose the prescribed motion

x = h0 sin at is the prescribed motion M is displaced mass of the model

p is added mass

X is the damping factor

k is the spring constant

When data were being analyzed, it was assumed that the effect of the end boards on the added mass and damping of the cylinders could be neglected.

To obtain the added mass and damping, the forcing function was reduced to its fundamental components, which were in phase and 90 deg. out of phase with the displace-ment motion. This put the forcing function in the form

F(t) = A sin at + B cos at

Taking the first and second derivatives of x and equating the coefficients of like terms,

the added mass and damping become

h k -A₀

h₀

/ B

A -

_{h a}

0

The spring constant k was calculated by taking the product of the waterplane area of the Model and the specific weight of water. This Was assumed constant over the range of

amplitudes tested.

The previously described force coefficients A and B.were obtained by analyzing the

data in analog form during testing, using the electronic setup shown in Figure 5 for the first series of tests and Figure 6 for the second series of tests. This was done in the first case by summing two of the force signals and multiplying by 2 and in the second case by summing the four force signals and multiplying the sum by sin at and by cos at, then integrating over the run time T. The result was multiplied by 2/T to determine the A and B coefficients as

follows. 2 A =

f

F(t) sin at dt B

21TF

(t) cos at dt 0

I'

(17)

The total run time was taken during 30 heave cycles. The sine and cosine signals used for this analog Fourier analysis were obtained from a potentiometer which was

mech-anically coupled to the oscillator.

For the first series of tests with the circular cylinders the data were also reduced digitally. The reduction was accomplished during testing by first recording the force and motion signals in analog form on magnetic tape for the post-test analysis of the data. The data were then filtered, digitized, and fed to a computer program to determine the Fourier transform coefficients of the fundamental signals. The components in phase and 90 deg. out of phase with the displacement motion were derived from these coefficients.

RESULTS AND DISCUSSION

The added masses and the dampings obtained from the theory and the experiment are shown together for the purpose of comparison in Figures 7 through 28. The nondimen-1

sional parameters used in the graphs are

T.i = added mass coefficient = X = damping coefficient _Ma

6 = frequency number

0.2a

The difference between the two experimental data, one obtained by the analog method and the other by the digital method, was insignificant. The experimental results shown are mostly from the analog method. The experimental data for the semicircular cylinder are identical to those presented in Reference 4.

As mentioned earlier the theoretical approach to the solution of the problem employed in this work differs from the one employed in Reference 4 in which only semicircular

cylinders were investigated. Thus, both theoretical results are also shown for the case of the semicircular cylinders. Except in the low frequency range and at some frequencies at which hydrodynamic discontinuity occurs, both results are in good agreement.

The results of the linearity check with the different amplitudes of oscillation show that the linear relation between the forcing motion and the resulting hydrodynamic force is valid for the semicircular and rectangular cylinders. But for the triangular cylinders, particularly for the isosceles triangular cylinders, the results from the different amplitudes show some

disagreement in tlrAOW frequency range. The previously described fact suggests that

nonlinear hydrodynamic effect could be caused by the sloping sides of the cylinders.

Cylinders having sloped sides may create more free-surface disturbances than the wall-sided

cylinders at lower frequencies so that the assumption of slight fluid disturbance to support the linearity relation may no longer be true for these cylinders at lower frequencies.

(18)

The nonsolid symbols shown in the results for the circular cylinders are the experi-mental data obtained with the end boards without the reinforcing angles.

Except for some discrepancies in the damping coefficients in the very low frequency range, both theoretical and experimental results are in good agreement.

At certain frequencies, the source distribution method used in solving an oscillating body problem causes a mathematical discontinuity. These frequencies, called irregular or critical, were first pointed out by John." Frank' gave an approximate method for calculating these critical frequencies in terms of the beam-to-draft ratios of the cylinder. The approxi-mate method equally applies for twin cylinders. The irregular behavior of the theoretical results at the critical frequencies is not shown in the figures. Smooth connections of the

curves are made at the critical frequencies. A more satisfactory method of eliminating the

critical frequencies is being investigated further. The mathematical proof for 'possible elimination of the critical frequencies has been established, and the computational implementation of the elimination technique still remains to be achieved.

tam frequencies in the solution of the twin-cylinder oscillation problem. These frequencies

closely correspond to the gravity wavelength for deep water which satisfies the following relation

n x (wavelength) = 2 (b a) for n= 1 (32)

In terms of a frequency number, the relation becomes

13

nir

(b/a-1) (32a)

The situation described previously is analogous to the breakdown of a periodic solution at certain frequencies for the problem of a wavemaker in a finite rectangular tank. The break-down of the solution occurs when the relation given by Equation (32) is satisfied in which (ba) corresponds to the length of the tank. In this work the y-axis can be regarded as a rigid wall, and the wavemaker is situated at a distance (ba) from the wall. Both theoretical and experimental results prove that Equation (32a) provides fairly accurate values of the "resonance" frequencies. The breakdown of the solution at these resonance frequencies may be prevented by seeking a time-dependent nonperiodic solution. However, we shall not attempt to do this here.

Large negative added masses are obtained in the low-frequency range for all four

cylinders, except the experimental results for the right triangular cylinders. For heaving

two-dimensional single bodies in a free surface, no negative heave added mass has been reported. Thus, the existence of negative added mass for twin cylinders strongly suggests the effect Of hydrodynamic interaction between the two cylinders.

(19)

CONCLUDING REMARKS

Good agreement of the theoretical results with the experimental results confirms the validity of theory developed in this work.

The abrupt discontinuity Of the results at certain frequencies found in the theory is also indicated by the experiment The frequencies at Which the discontinuity occurs, termed the resonance frequencies, are given by a. =

tifre(ba) for n=1, 2

,--I The linearity between the forcing motion and the hydrodynamic force for the entire frequency range is confirmed for the tWin cylinders having vertical sides, i.e., the

semicircular and rectangular cylinders. With the exception of low frequencies, this linearity

is also indicated for the twin cylinders having sloped sides, i.e., the triangular cylinders.

The variation of the values of added mass and damping is greater at lower fre-quencies (8<l) than at higher frefre-quencies. For the range of separation distance considered in this work, the numerical results indicate that as the frequency approaches infinity, the

mutual hydrodynamic interaction between the two cylinders disappears.

The decrease of the separation distance between two cylinders results in (1) an

increase of the lowest value of the resonance frequency, and (2) an increase of the absolute value of the negative added mass coefficient.

The validation of the theory found in this work for the case of heaving oscilla-tion suggests extension of the theory to the cases of swaying and rolling oscillations.

ACKNOWLEDGMENTS

The authors would like to acknowledge the incorporation in this report of valuable

(20)

Figure 1 - Description of Coordinate System

= hoc, cos(n, y)

1 im( - j KO) = 0

x

VO (x, = 0

Figure 2 - Description of Boundary-Value Problem for 66(x, y)

(&j,ni)

Figure 3 - Segmentation of Cylinder Contour

(21)

(22)

F.1COS)

5 CPS FILTER

Figure 5 - Block Diagram of Electric Setup

on Carriage 2

for First Series of Tests

TAPE RECORD CYCLE COUNT IL TE 110VM SCAN PRINTER 'CLOCK RESET I '(S'ART- STOP) MANUAL ?ER 100 TRANS. SIGNAL COND. DANA TAPE CAL. SAN RECORD SINE 8 C D COS POT POWER SUPPLY SINE COSINE FORCE BLOCK 3 5' CPS NO.1 FILTER

FORCE BLOCK NO2

4

BRUSH

5 CPS FILTER

OSINE)

FORCE BLOCK NO.4

MODE CM.1 0.5V STEP S 9 5 CPS, FILTER A FILTER IDVM SONIC SONIC WAVE ELECT HT, UNIT FOR CE BLOCK BRUSH NO:3 COUNT. COMP. ILTE lovid

(23)

SINE COS POT MODE FORCE BLOCK NO.4

DANA 'PRE-SET COUNT AMP TAPE CAL. SAW 'RECORD RECORD.

Figure 6 - Block Diagram of Electric Setup on Carriage 2

for Second Series of Tests

,MANUAL RESET 0 SCAN PRINTER FILTER

801-uoNITR. y/Flei- DI R. COUNT

I

(24)

4 2 2 -4 -6 2 1 1 2 -3 19 -b/a = 1.5 , II

U U

0.25 in 0.50 in Present Theory ---Theory14] . . b =2

U

0.25 in OH 0.50 in Present Theory ---Theory[4] 0.5 10 1.5 2.0 2.5 8

Figure 7 - Added Mass Coefficient versus Frequency Number

for Twin Semicircular Cylinders for b/a=1. 5

2 3

8

Figure 8 - Added Mass Coefficient versus Frequency Number

for Twin Semicircular Cylinders for b/a=-2

(25)

1

-2

a

Figure 9 - Added Mass Coefficient versus Frequency Number

for Twin Semicircular Cylinders for b/a=3

U U

0.25 in b 0.50 ih Present Theory ---Theory[4] b/a a 3 2 -1 0

U _U

0 0.25 in Present Theory Theory[4] b = 4

Figure 10 - Added Mass Coefficient versus Frequency

Number

for Twin Semidircular Cylinders for bia=4

6

4 5

2

(26)

10 6 4 2 3 2 1 0.5 1 10 15 2.0 2.5 2 8

Figure 11 - Damping Coefficient versus Frequency Number for

Twin Semicircular Cylinders for b/a=1. 5

4

Figure 12 - Damping Coefficient versus Frequency Number for

Twin Semicircular Cylinders for b/a=2

21 b/a = 1.5

U -U

IP 0.25 in 0.50 in Present Theory --- Theory[4] me

-111----111----16---,

U U

b a =2 ICI 00 0.25 in OE 0.50 in Present Theory Theory[ 4l U.' U. ., _i (

(27)

4 3 1 2. 1. o. 1 2

Figure 13 - Damping Coefficient versus

Frequency Number for

Twin Semicircular Cylinders for

b/a=3

8

Figure 14 - Damping Coefficient versus

Frequency Number for

Twin Semicircular Cylinders for

b/a=4

U U

b/a = 3 o 0.25 in CI 0.50 in Present ----Theory --- Theory[4] . :...T. 1., .. 511.0...rt0..0"

U U

b a = 4 10 -0.25 in Present -- Theory Theory[4] I I 1 I t (it) 43/ ; I . o

(28)

1 o T6 -1

LJ LJ

0.25 in 0.50 in Theory b a = 2 23 b/a = 3 a .

..

fra"---r---LJ fra"---r---LJ

0.25 in MI 0.50 in

_Theory

0.5 1.0 15 2.0 2.5 8

Figure 15 - Added Mass Coefficient versus Frequency Number

for Twin Rectangles for b/a=2

1.3 1.5 20 2.5

Figure 16 - Added Mass Coefficient versus Frequency Number

for Twin Rectangles for b/a=3

(29)

2 0..25 in 0.50 in

--

Theory b/a = 4 a Z b/a = 2 m

LI

0.25 in 111 IP

, a

1

---theory

a 0.50 in

Theory a ^ ', 0.5 1.0 1.5 20 2.5

Figure 17 - Added Mass Coefficient versus Frequency Number

for Twin Rectangles for b/a=4

0.5 10 15 2.0 2.5

a

Figure 18 - Damping Coefficient versus Frequency Number for

Twin Rectangles for b/a=2

(30)

4 3 2 1 0 4 3 2 1 25 II b a =3 . MI

LI LJ

0.25 in II 0.50 1,, --- Theory 11 MI -II 'I fo. b/a = 4 /

LJ LJ

0 0.25 ii 111 0.50 in ___ Theory 1.0 1.5 20 2.5

Figure 19 - Damping Coefficient versus Frequency Number for

Twin Rectangles for b/a=3

0.5 0 1.5 2.0 2.5

Figure 20 - Damping Coefficient versus Frequency Number for

Twin Rectangles for b/a=4

(31)

2 V b/a = 3 v 9

.

...

a . le V w V 111

V V

0.25 in M 0.50 in 0.75 in

Theory

V e_ -b/a = 4

vv

_0.25 _in II 0.50 in V 0.75 in L I S ---Theory . o. 0.50 0.75

Figure 21 - Added Mass Coefficient versus

Frequency Number

for TiVin Isosceles Triangles for b/a=3

25 0.50 ₆ 0.7

Figure 22 - Added Mass Coefficient versus

Frequency Number

(32)

4 3 2 1 4 3 2 1 27 b/a =3

.vv

_0.251n II 0.50 in 0.75 in Theory - gi -1 OM II ' .

__

1 1 b/a =4

V

0.25 in 0.50 in V 0.75 in

/

i

.5

- _---- Theory 0.25 050 a 0.75 1.00 1.25

Figure 23 - Damping Coefficient versus Frequency Number for

Twin Isosceles Triangles for b/a=3

0.25 0.50 075 1.00 1.25

Figure 24 - Damping Coefficient versus Frequency Number for

Twin Isosceles Triangles for bia=4

(33)

1 1

\\\///

_{0.25 in} 0.50 in 0.75 in =-- TbPPrY b/a = 3

L//'

0.25 in 0.50 in 0..75 in 7___Theory 13 = 4 025 0 50 0.75 1.00

Figure 26 - Added Mass Coefficient versus Frequency Number

for Twin Right Triangles for b/a=4

1.25

0,25 0.50

is 0.75 1.00 1 25

Figure 25 - Added Mass Coefficient versus Frequency Number

for Twin Right Triangles for b/a=3

(34)

4 3 4 3 2 1 29 0.25 0.50 8 0 75 1.00 1.25

Figure 28 - Damping Coefficient versus Frequency Number for

Twin Right Triangles for b/a=4

M. , b/a = 3

NV

GO 0.25 in MI 0.50 in V 0.75 in ___ Theory S

S.

b/a =4 le 11

V

0.25 in M 0.50 in 0.75 in --- Theory m !II V e

.

_ 025 0.50 8 0.75 1.00 125

Figure 27 - Damping Coefficient versus Frequency

Number for

(35)

From Equation (15) we find that (z -GRc (z; 0(z + ._fcc e-ik (z - e-ik (z + = Rei +2 K - k GRs (z; = +e- il((z

n}

If we let a. denote the tangent angle of the ith segment c1 of the cylinder contour;

APPENDIX A

EVALUATION OF MATRIX ELEMENTS

The evaluation of the influence coefficients given by Equations (21) and (22) are

=(n7)

fGRc

(p;s)ds1 I/2 i '1 ) c. P = Pi (33) _ a ( A 21 R7'd = (n-V)

fGRs(p;

s) dsl c. P = (34) (Yi+ 3;i) a. = tan-1 (xi+, xi) we have

= (sin a., cos ad

ci

Thus, for any analytic function f(z)

dk} (35) (36) Reif(n-v) f (z) )=Re. z =z. a a

fin

a

_ax cos a _ay f (z) z = z1 a= ai

= Rei(

A.,.f(z)l ) (37) z

We can also show that for t'e co

(36)

where a is the tangent angle at the point an

= eiads

We substitute Equation (35) into (33) to derive

= R )

flog

(z- ds1 z zi Cl

(n-v)..fog(z-bds1

c. iz = z. +2(n-v)

fdsf

- - dk z=z. ik(z+ c. o

where zi = xi +tyi is assumed to be located at the midpoint a the ith segment. Utilizing Equations (35) through (37), We can show that

1,1 Rei (n-V)

flog

(z -) ds

I

Ci

=

Re.{ iei

If

1 log (z- r)

iajdr

iei -a4 {(Z -)

(z - log (z - tr..j+1

Si

= Rei _iei - a.)1 z.- J+1).

spg

3

(XiA)2 =Vz sin (a1-a.) log

ix1

+(y i.:11.4. )2 + cos (cti-1,2 Rei

{(nT) flog (z-) dsz

c. z.

iei (al+ aj) f

log (z- r)dr

Z . -1 _

tan

-1 17.1+1 x1.11 xi- to, 32 (39) (40) ,,<

(37)

L4

(x1-)2 +(y.)2

= sin (a.+ a.) log

(xi+1)2+(Yi nj+ I)2

Y.+ 77.J Yi+71i+1 + cos (a.+ a.)_j {tan"- tan"'

j+ I

L3 = Rei{(n-V) Jog (z +f) ds

z = zi} (x1+ ti)2+ (3/i nj)2

= 1/2 sin (ai+ a)log (xi÷ ti+ 1)2 cyi_ ni )2

cos (a.+ a.) {tan"'

x:+ t._j

= sin (ai+

a)f

_K-kdk

tan"'

'

j+1}

,(1-.1V)

fl°g

_(z _ds _z

zi

(xi+ ti)2+(yil-ni) = _ 1/2 sin (ac a.) log

_t

₎₂ _{(yi+ ni+ )2}

I I y.-F Yi+71.-i-}. = Rei ai+

(_

1 CH.

-(Z- dkI K-k z =

z./

= (ai+ api_f cc'

(zK

(zi -17P-1) _dk Yi+ ni)cos k (42) (43) (44)

- cos (a.-a.)_J

itan4

tan4

(z-D e-ik .1+ -L5 Rei

Tv)

fdsf

c. o K- k dk z = z.

(38)

-cos (ai+ dk fel( (Yi+ n j) sin k (xi- t;)- ek (Yi+ Vi+ )tin _{MN- Ei4.1)} K- k co L6 Ei

Reil(msf

v)fd

lic (z+D I. c_j ₀ K-k

f

dk

=

(ar ad/

e `,Pcos k(x1+ ek (y1+ 71j+ )cos k(xi+ti+3)J

+ cos (a.- a.)./.

dk i

ao ,

1 j

0

K k

.. lek(Y1+ ndsin k (x.+ t.)--ek(Yi+I .1 )Sin k (x + E.

)

j

(46)

1

i 1+1

SllbStitUting Equation (36) into Equation (34), we obtain the following intevals:

i ,

L7 == Rei_{(flp)

/04441

ds

I

)

c.

= Re.

{-iei 62 + ad'- St,' I.

r.i+1

r

1

k

e(z"

'

) dr,

3--;

= in &+a

feigYi+Vcosk(xi- V- eK (Yi+77p-i)cosK

(xi-- cos (ai + adlel( (yi+ 71;)sin K (xi- ti)-e1C(Y1+ Ili+ I )sinK(xi- C.,

)1

(47)

L3 2E---

Reif

(17)

jeliK(z+r)ds I

c '.1"-- 2

}

i

--sin (ai-ai)feK(Yi+ dcosfaxi+ ti)--e1C(Yi+ )cosK(xi+

+ cos (ai- _{feK(Yi+ VsinK(xi+ ti)-eK(Yi+ 14+1 hinK(xi+}

))

₍₄₈₎ i,

Combining the previous results, we can finally show that

lij 1 (1../ L2 L3 - L4 + 2Ls + 21.6) (49)

(39)

J.. = L L

_1J ₈

The evaluation of the principal value integrals in Equations (45) and (46), which can be converted to the exponential integral, are shown in Appendix C.

(40)

= Rei[

APPENDIX B

EVALUATION OF POTENTIAL INTEGRALS

To obtain the hydrodynamic coefficients, we must know the values of the velocity potentials on the cylinder surface. The expressions for the velocity potentials are given by

Equations (24) and (25) which contain the integrals,

fGR

(p.; s) ds

c. /9' ,

I2

fGR

(p0; s) ds

C.

where po is a point on the cylinder contour and 9 is a line segment of the cylinder contour.

We shall use the complex expression given by Equation (15) to evaluate the integrals

Ii and 12. Thus, for the point ; = xi+ iyi which is located at the midpoint of the line

seg-ment ci, we have

1 Kr le z_ ,e3

I = -;.Tr Rei

f

ds{log (zi- log (zi- F) + log (zi+ ) C. 1 2 i log (zi+ t") +2 k.-41( f c° e- ik (zi- 3') ' c ik (zi+ t')

1 ]

o K- k

dk+2f

o K- k dk j iic-/? _fCi)

12 = Re.

{ fe41((zi-Dds +fe4K(zi÷ t.)ds} (52)

We can easily show by using the relations given by Equations (38) and (39) in Appendix A that

K1 E Rei

f

log (zi- t") ds

c.

g.) (zi- /1 log (z-DI:j+1}

37

(41)

where If we define I (xi,

; r I

ili+1; K4 E= cos a fit -1 e 1- ) log I z (

- ) log I zi- (yi-ni) arg

(zi--E(Yi-ni+ arg )1

+ sin ai nj+ + 'ii) log I"; I

(yi- ni+1) log I i=1"j+1 I + (xi- ti) arg

(x - t. ) arg (z.- ₎₁

C.

z, I f(x- + (y-

02}

arg (z --3") = tarfl

= Re

Jiog

(z.- ds

the remaining integrals involving logarithmic functions in Equation (51) can be given by

1(2 a: Re.1

f

log (zi-T) ds = I (xi,yi ; el+ 1 n 1+ 1 ; a.),

c

i

1c3 aRe1

f

.+ ) cis = --I (x.,Tlog .;

(z1 1 i+ 1 i ' j+ 1,-ai) Del ,

log (z.+)ds = I (x y.;

- j+ 1 -n i+ 1₉ e. If we again define K5 a

Reif

ds

f

_K1

k c. o

we can Show that

(51)

(54)

(55)

(42)

Ks = Rei

[

By use of partial fractions

1 1 1

+1)

(K- k)k K k k we get

fo

JL fj+1 _ik (zi- r) dr Kdk-k e eazi e ilk -eikTi _dk K- k ie K e '+1 dk . ) -ik

-

(zi- Le-ik (zi-T.;) 1

Jo

JK-k - dk

The first integral in Equation (57) can be regarded as a function of z and can be written as

F(z) 0-ik (z 3 e-ik (z dk Then, we have

F'(z) = if

-ik (z (z - _dk 1 1

_

z z-3'j+1 from which z F(z) = 1og 39 (57)

(43)

where we let the arbitrary integral constant be zero to satisfy the deepwater condition as

given by Equation (9).

Substituting the previous results into Equation (57), We have

K5 Rei zi-fi + Joc 12° e-ilc(zi-fi+i)_e-ilc(zi-T.i) log dk K K- k

_

[

(2- _{(xi- ti+ )2 +(yi+ ni+i} 2

. _- _j _j (x.- + (Y.+ n.)2 dk 0.

I

ek (yi+nJ.+1)cos k (x- E ) K-k

-

ek (111+ ni)cos k (Xi- y}}

4

{ - , Yi+ ni , Yir77; 1

+ cos a.

tan-

tan- - '4

-.1 x.- E.

x1-+1

i

"-L-1 (Yi+ n.sin k (xi- Ei)

-

ek(Yi+ nj+1) sin k

k

If we let

Ks = L 1+1,71 1+1 ; ad,

we can show that

f

(zi 4:

K6 E Rei

dsf

_{K- k}

c- o

The integrals in Equation (52) can be readily evaluated as

K7 E Rei{ fe4K(71-1-)dS}=

Re.ff ri+1

T..

-C.

5i

1 [eK(yi-l-_ni)sin

{K(xi-- 13

(

d.k = ;- ti+ 1 ,n I+ 1

))

elC(Yi+ ni+ )siniK (xi* Ei+

_{) -}

_ai)

(60) H

(44)

K8

f

e-ilc( Ods1= 1..+1 1K(zi+

+ aI

dr}

9

= 1 eK (yi+ nj+i)sinIK (x.+ ) + ai --eK(Y+ ni) siniK(xi+ +

ail

i 1 (61)

Combining the results obtained from Equations (53) through (61), we can show that

1 1

I = _(K1 _{K2 + K3} K4 ) + _{(K5 + K6 )}

1 z7 7 (62)

41

12 = - K7 - K8 (63)

The principal value integrals in Equations (58) and (59) can be expressed in the form of the exponential integral, which can be easily converted to an infinite series. The details

(45)

APPENDIX C

EVALUATION OF THE PRINCIPAL VALUE INTEGRALS

In the derivation of Green's function G(z; , which can be interpreted as a wave

source near a vertical wall, the principal-value integrals were encountered as shown in Equation

(15). These integrals can be reduced to the exponential integral which can be expanded to

an infinite series.

We show below how to make this conversion. We have

fcce-ik(z-t)

_{K- k} dk =

jee

e4)

_{K- k} dk + CiK(z-T) (64)

0iwhere indicates that the path of the integration is indented above the pole at k=K, and

-at-,

the last term above is the residue value at the pole. Let us first concentrate on the integral on the left-hand side of Equation (64). If we make the transformation

w = i (k- K)(z- (65)

where k = kR + and 'w = wR +

we can show that

w = -ki(x-E)-(kR- K)(y + n)

+ {(kR- K)(x - E)-ki(y+ n) (66)

Noting that the path of the integral limits the values of k confined to kR , k1>0 and that y + n<0, we apply the transformation of Equation (65) into the integral

(z4") dk

K- k

As shown in Figures 29 and 30, the path of the integral in the k-plane changes to the two different paths, depending on whether x - E >0 or x - E<0.

(46)

Figure 29 - Change of Integral Path when Re (z-)

> 0

Figure 30 - Change of Integral Path when Re (z-)

< 0

44

0 _{k= K}

kR

(47)

When x-E >0 (Figure 29) e-ik(Z-T) fc°1+1cc K - k dk o -iK (z-f) w 00 _ic. e-ik(z4) _-w dk = -e4K(z41 e dw K-k _{_ilc} = ea( (z-

) f+f

(

dw = e-11( (z - dw

r

.

-iK(z-F) -w w ₀₀ w -iK (Z-f) _w (68)

Note that the singularity at w = 0 is outside of the closed path of the integration in this case; therefore, there is no residue value involved. The substitution of Equations (67) and (68) into (64) yields

.e° e-ik(z-r)

_dkf

e-ik(z-17)

dki

(z-02

41""

dw ± 71- e-iK (z- for x-E (69)

-iK w

= e-iK(z-t)

(f+

We make use of the exponential integral which is defined by"

(z)

=f

dt (for I arg(z)

.)

z t

= -7- log

z-n=1

E eire

nn! (iTt)e-w

dw+27r0

_w -(67) -w + 2

7r)

(z-sincef

vanishes when we let R + ....

r =Rei0

(48)

where y= 0.5772 is the Euler constant, in Equation (69) 'to derive

°°,41( (Z-15

= -e-iIC(z-D Ei (-iK(z-1-1; ±iireiK (z-17)

K k

= eK (Y+10 {cos 1,((x- t) -i sin K(x-

t)[

{7+log r

+

r_P_cos nO I+ iio E

}

n n! f nn!

n=1 ' n--=1

where r = K {(x,E)2 (Y+77)2} 1/2 and 0 = tan-1

-(Y +1) If we let r n CQS n 0 A(r, 0) =7+ log r + and and - sin nO B(r,O) = 0 L., n n! n=1

we can separate Equation (70) into its real and imaginary parts

r eiC (Y+ 71)ccks kl x7 0 _{= eK (Y} _{mr,O)cos _K V .7- _k -o + (r, 0) K (x-t)} A. 0=1 c° dk (Y+

sin

k(x-)

_dk _{eK (Y÷ 7?) IA (r, 0) sin K}

(x-fo

K- k

B (r, 0) cos K (x- (71)

(49)

Exactly identical derivations as shown in Equations (73) and (74) can be applied for the integral

eilc dk

K- k

except that r and 0 for this case are defined by

_ire

1.v)

(70)

r = K {(x+ + (y + 77)2} 1/2

(50)

REFERENCES

Potash, R., "Forced Oscillation of Two Rigidly Connected Cylinders on a Free

Surface," M.S. Thesis, University of California, Berkeley (1967).

Ohlcusu, M., "On the Heaving Motion of Two Circular Cylinders on the Surface of

a Fluid," Reports of Research Institute of Applied Mechanics, Kyushu University (Japan) Vol. 17, No. 58 (1969).

Ohkusu, M., "On the Motion of Multihull Ships in Waves (I)," Reports of Research

Institute of Applied Mechanics, Kyushu University (Japan) Vol. 18, No. 60 (1970).

Wang, S. and Wahab, R., "Heaving Oscillations of Twin Cylinders in a Free

Sur-face," Journal of Ship Research, Vol. 15, No. 1(1971).

de Jong, B, "The Hydrodynamic Coefficients of Two Paralleled Identical Cylinders Oscillating in the Free Surface," International Shipbuilding Progress Vol. 17, No. 196 (Dec

1970).

Ursell, F., "On the Heaving Motion of a Circular Cylinder on the Surface of a

Fluid," Quarterly Journal of Mechanics and Applied Mathematics, Vol. 2, (1949).

Frank, W., "Oscillation of Cylinders in or below the Free Surface of Deep Fluids," NSRDC Report 2375, (Oct 1967).

Wehausen, J. V. and Laitone, E.V., "Surface Waves," Encyclopedia of Physics, Vol. 9, Springer-Verlag, Berlin, (1960).

Kellog, 0. D, "Foundations of Potential Theory," Dover Publications, Inc., New

York (1929).

John, F., "On the motion of Floating Bodies, Part II," Communication of Pure

and Applied Mathematics, Vol. 3 (1950).

Abramowitz, M. and Stegun, I. A., "Handbook of Mathematical Functions,"

National Bureau of Standards, AMS 55, (1965).

(51)

Copies Copies

1 NAVMAT (Code 0331) 1 DIR, APL, JHU, Silver Spring

9 NAVSEC 1 DIR, Fluid Mech Lab, Columbia Univ

1 SEC 6110

1 SEC 6114D 1 DIR, Fluid Mech Lab, Univ of Calif,

3 SEC 6114H Berkeley

2 SEC 6136

2 SEC 6111B 1 Prof E.V. Laitone

Aero Sci Div

NAVSHIPS Univ of Calif

3 SHIPS 2052 Berkeley, Calif 94720

1 SHIPS 034

1 SHIPS 525 2 DIR, DL, SIT

1 Dr. C.H. Kim

3 NAVORDSYSCOM

1 Aero & Hydro Br (Code RAAD-3) 1 Ship Instal & Des (Code SP-26) 1 Loads & Dyn Sec (Code RADD-22)

4 CHONR

1 Nay Analysis (Code 405) 1 Math Br (Code 432)

2 Fluid Dyn (Code 438)

1 ONR, New York 1 ONR, Pasadena 1 ONR, Chicago

ONR, Boston ONR, London

1 CDR, USNOL, White Oak

2 DIR, USNRL (Code 5520)

1 Mr. Faires

1 CDR, USNWC, China Lake 1 CDR, NUR&DC, Pasadena

1 CDR, USNAVMISCEN, Point Mugu 1 DIR, Natl BuStand

Attn: Dr. Schubauer

12 DDC

INITIAL DISTRIBUTION

3 McDonald-Douglas Aircraft,

Aircraft Div, Long Beach, Calif 90801

1 Mr. A.M.O. Smith 1 Mr. J.L. Hess 1 Mr. J.P. Giesing

1 DIR, Scripps Inst of Oceano, Univ of

Calif

1 DIR, Penn St Univ, Univ Pk 1 DIR, WHOI

1 Prof L. Ward

Webb Inst of Nav Arch

2 DIR, Iowa Inst of Hydrau Research 1 Prof L. Landweber

2 DIR, St. Anthony Falls Hydrau Lab I Prof C.S. Song

5 Head, NAME, MIT, Cambridge

1 Prof Abkowitz

1 Prof Kerwin 1 Prof Newman 1 Prof Lewis 1 Prof Mandell

Inst of Mathematical Sci, NYII, New

York

(52)

Copies

3 Dept of Nay Architecture, Univ of Calif, Berkeley, Calif 94720

1 Prof J.V. Wehausen 1 Prof W.C. Webster 1 Prof J.R. Paulling

Prof Finn Michelsen, Dept of Nay Arch, Univ of Mich, Ann Arbor

Prof T.F. Ogilvie

Dept of Nay Arch, Univ of Mich, Ann Arbor

1 Prof Richard MacCamy, Carnegie Tech

Pittsburgh 13

2 Hydro Lab, CIT, Pasadena

1 Prof TN. Wu

1 Dr. Hartley Pond, 14 Elliot Dr

New London, Conn

Prof M. Van Dyke, Dept of Aero & Astro Stanford Univ, Palo Alto, Calif

Prof B.V. Korvin-Kourkovsky, P.O. Box247,

East Randolph Vt. 05041

Dept of Aero & Astro, MIT

1 Prof Widnall

1 DIR, Inst for Fluid Dyn & Appl Math

Univ of Md

DIR, Hydrau Lab, Univ of Colorado

1 Pres, Oceanics, Inc

Technical Industrial Park

Plainview, N.Y.

1 HydrOnautics, Inc., Pindell School Rd

Laurel,'Md

Applied Mechanics Review

Editorial Office

Southwest Research Inst.

P.O. Box28510

San Antonio, Texas 78228

(53)

CENTER DISTRIBUTION Copies Copies 1 15 2 1735 1 1506 1 Genalis 1 1521 1 174 3 1524 1 lin 1 1802.3 1 Neal 2 1843 1 Day 1 Schot 1 1528, Tejsen 1 Oh 4 154 1 1541 4 19 1 1542 1 1903 1 1544 1 194 9 1552 1 1966, Liu 1 McCarthy 1 Chang 1 Huang 1 Kerney 1 Langen 1 Peterson 1 Salvesen 1 Wang 1 Von Kerczek 8 156 1 1561-38 1564 1 Monacella 1 Wolff 1 Sheridan 15 Bedel 20 Lee 17 1568 1 Cox 1 Zamick 15 Jones 3 1572 1 Feldman 1 Gersten 1 Fein 3 1576 1 Smith 1 Russ 1 Livingston

(54)

UNCLASSI Fl ED

-,..,

2-DOCUMENT CONTROL DATA - R & D

.(Sreurity. classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) I. ORIGINATING ACTIVITY (Corporate author)

Naval Ship Research and Development Center _k

Bethesda, JVIgyland 70034

241. REPORT SECURITY CLASSIFICATION

UNCLASSIFIED

2b. GROuP

3. REPORT TITLE

.

ADDED MASS AND DAMPING COEFFICIENTS OF HEAVING TWIN CYLINDERS INA FREE SURFACE

DES.DRIPTI-VE NOTES (T)-pe of report and inclusive dates)

P

5. AUTHOR(S) (First name, middle initial, last name)

C. M. Lee H. Jones

J. W. Bedel

13. REPORT DATE

August. 1971

7a. TOTAL NO. OF PAGES

56

7b. NO. OF REFS

11 Oa. CONTRACT OR GRANT NO.

b. PROJECT NO. _{Subproject S-R009 01 01}

Task 0100

C.

d.

9e. ORIGINATORS REPORT NUMBER(S)

3695

ob.OTHER REPORT NO(S) (Any other numbers that may be assigned

this report)

10. DISTRIBUTION STATEMENT

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

.11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY AC.TIVITY

Naval Ship Systems Cornmand

13. ABSTRACT.

-A potential flow.problem, dealing with twin horizontal cylinders of arbitrary cross sectional forms vertically oscillating in a free surface is investi-gated. An associated experiment is carried out for four different sets of twin cylinders. The results from the theory and the experiment are compared and are found in griod agreement.

, .

S/N 0101- 807-6801 Security Classification

DD

(PAGE 1)

(55)

UNCLASSIFIED

Security Classification

WORDS LINK A - LINK 8 LINK C

00 LE WT ROLE WT ROLE

Catamaran Motion

Twin Cylinder Added Mast and Damping Free-Surface Effect