A procedure for determining the virtual mass j-factors for the flexural modes of a vibrating beam

HYDROMECHANICS 0 AERODYNAMICS 0 STRUCTURAL MECHANICS 0 APPLIED MATHEMA1ICS PRNC-ThB-648 (Rev. 3-58) Lab. v.

Schpsbou;kiø

Technische Hogschoo Deift A PROCEDURE

FOR DETERMINING THE VIRTUAL MASS j-FACTORS FOR THE FLEXURAL MODES OF A VIBRATING BEAM

by

G. Wayne Dutton

and

Ralph C. Leibowitz

Structural Mechanics Laboratory RESEARCH AND DEVELOPMENT REPORT

A PROCEDURE

FOR DETERMINING THE VIRTUAL MASS j-FACTORS FOR THE FLEXURAL MODES OF A VIBRATING BEAM

by G. Wayne Dutton and Ralph C. Leibowitz August 1962 Report 1623 S-R011 01 01

TABLE OF CONTENTS Page ABSTRACT 1 INTRODUCTION 1 METHOD OF ANALYSIS 2 RECOMMENDATIONS

APPENDIX A - CORRECTIONS TO APPENDIX B OF

REFERENCE 1

APPENDIX B - MATHEMATICAL DETAILS OF THE

DERIVATION FOR j OF THE INTERNODAL

SECTIONS OF THE SHIP .

APPENDIX C - MATHEMATICAL DETAILS FOR THE

DERIVATIONOF j FOR THE SHIP'S END

SECTIONS

-- - 1 2

APPENDIX D - DERIVATION OF k' USED IN T

k' I

APPEARING ON PAGE 241 OF APPENDIX

Bi OF REFERENCE 1

APPENDIX E - SOLUTION OF LAPLACE'S EQUATION IN

CYLINDRICAL COORDINATES FOR , THE VELOCITY POTENTIAL OF FLUID

SURROUNDING A VIBRATING CYLINDER,.

APPENDIX F - BOUNDARY CONDITION AT THE

CYJNDERtS SURFACE . . . . .

APPENDIX G - DERIVATION OF T =

Si

THE KINETIC ENERGY OF THE FLUID

SURROUNDING A VIBRATING INFINITE CYLINDER . . . ,

APPENDIX H EXPERIMENTAL AND THEORETICAL

FREQUENCY DATA FOR SS E.J. KULAS

FOR TH VERTICAL MODES OF

VIBRA-TION IN DEEP WATER

11 8 9 15 21 23 27 29 32

REFERENCES

BLBLIOGRAPHY . . .

LIST OF FIGURES

FIgure 1 - Plots of the Virtual Mass Factor j versus

the Ratio of Length to Half-Beam . . , . . . 4

Figure 2 - Representation of the Hull Model for Vertical

Vibration(3-Node Mode) . .

f ..

. . . ₅

Figure 3 -

Plots of K () and K1 () versus

p. . 11

Figure 4 - Ellipsoidal Model of HuLl E*d Segment . a . 15

Figure 5 - Beam Displacement and Velocity and Fluid

Potential and Velocity ₂₇

LIST OF TABLES

Table 1 _{Experimental and Theoretical Vertical Vibration}

Frequency Data Taken in Deep Water for

SS E.J. KULAS ₃₃

Page

34

This report presents a procedure for calculating the virtual mass of a ship using different j-factors for each mode of vibration. Comparison

between theoretical and experimental frequencies for SS E.J. KtJLAS indi-cates that better agreement is obtained up through the 5th mode for the light condition and up through the 4th mode for the loaded condition by using different j-factors for each mode rather than by using the constant

J-factor, determined by Lewis, for all modes. However, the ratios of

loaded to light frequencies which determine the accuracy of the calculated

virtual mass, using variable j, correspond more closely to the experi-mental ratios for all eight modes tested than do Lewis's ratios.

INTRODUCTION

Up to the present time, a constant J-factor (e.g., Lewis's) has been

used in the calculation of the virtual masses for each section of a ship in flexural vibration. Hence, the several normal mode frequencies and mode

shapes have been calculated using this J-factor rather than one that varies with the mode. Since the virtual mass of a ship decreases with increasing

mode number, however, a more accurate prediction of the modes and frequencies will occur by performing calculations that use j-values

appro-priate to the mode. The theory for establishing the relationship of the

j-factor, virtual mass, and mode shape was presented by Leibowitz and Kennard in Reference 1 . * This report elaborates upon the analysis given in Appendix B1 of Reference 1 and, using this theory, presents a practical procedure employing a digital computer for calculating the virtual mass of a ship for any mode of vibration. The limitations of this method for

the higher modes and its greater utility for the lower modes are also discussed.

* References are listed on page 34.

** t

3=

METHOD OF ANALYSIS

In Appendix B1 of Reference 1 ,* a theoretical derivation of the virtual

mass correction factor j for a±y mode of vibration is obtained for the in-ternodal sectiOns of the ship's hull treated as an infinite circular cylinder

and for the end sections treated as prolate ellipsoids Of revolution. For the shipts internodal sections in any mode,

/b

/b\ b

/b

Kii)

+IT _K0tir

where K0 and K1 are the modified Bessel, Functions of the second kind,

of zero and second order, respectively.** These particular Bessel

functions were chosen because they vanish as their arguments approach

infinity. L is the particular internodal length for a given mode and b can be taken as the average half-beam of the segment at the waterline. A

plot of j ve±sus is given in Curve 1 of Figure 1. The virtual mass of

b

a particular internodal segment is then jm, where m is the virtual mass

per unit length it pb2 for that segment corresponding to purely transverSe

water motion.t As stated inAppendix Bl of Reference 1, j for each mode

varies with. L; and L, besides varying from one internodal length to the next, is not even known exactly until the calculation of the normal modes

is completed. A practical procedure might be to make a first calculation using CureS 2, 3, or 4 of Figure 1 to find a constant Jva1Ue for the

2-node mode. Calculation of modes is then made in the usual manner

using thisvalue of J. Then.Curve 1 can be used to find j-values corres-ponding to all internodal .'s for each mode thus calculated (Figure 2).

Corrections of certain errors in Appendix Bi of Reference 1 are given

inAppendixAofthis report.

Plots of K0 (j.L) and K1(), along with the derivation of Equation [1],

are given in Appendix B.

If the beam is fully submerged, the virtual mass per unit length for

transverse fluid motion is it pb2, but if the beam is floating half

From these j's, new 'S can be recalculated, etc., until j and _converge

to essentially constant values. A corn ptr can be used to

carry out

auto-matically this proceSs of successive, approximations. *

For the ship's end sections,

(2

j=k"

e

-where Le is the eflip.soid's' semi-major axis 'Which corresponds to the di'stance from the ship's end (either bow or stern) to the first nodal point, and b is the semi-minor axis which can be taken as the half-beam of the

hull at waterline _{easu'±ed at the terminal 'nodal points}

e is assumed

to .exceed b). k-' 'is a constant that depend's solely upon the eccentricity of the ell'ipsc'id.t A computational method similar to that suggested

previously for the internodal sections may be used to calculate the j-values fOr the end sections.

The circular cylinder model for the midsections is particularly well adapted to the long slim hulls of' submarines. However, it is likely that a more accurate mathematical model for surface ships' interr'odal

sections is obtained by iepre;senting these sections, of the hull,

_{as eilptic}

cylinders of appropriate dimensions. A starting point for the

corres-ponding virtual mass calculations would be to use the velocity potential for a deformable infinite elliptic cylinder.

Appendixes E, F, and G give various mathematical details in the

derivation of Equations [].] and [2],. Appl'ication.of the 'method of

computation of virtual mass for the higher modes derived in Appendix B

of Reference 1 is found in Reference 2, A comparison of the theoretical and experimental results given in Reference 2 is presented in Appendix H

of this report.

The computations of the frequencies and mode shapes for variable j

as described in this report are presently being coded on a digital

computer at the Applied Mathematics Laboratory of the David Taylor

Model Basin.

See Appendix C' 'for a' detailed derivation of Equation [z]

See Appendix D for a deviation of the k' _{expression given in Appendix}

RECOMMENDATIONS

It is recommended that:

The procedure given in this report for determining the virtual

mass be coded for digital computation.*

Comparison be made between theoretical and experimental frequencies and mode shapes for several ships using (a) different

j-factors for each mode and (b) a constant J-factor for all modes.

The results of such a comparison will indicate whether a variable

j or constant J-factor yields optimum prediction of the vibration characteristics for a particular mode.

A theoretical study be made of the virtual mass of a deformable elliptic cylinder of finite dimensions.

0.9

0.B

0.7

0.6

* This coding is presently underway at the Applied Mathematics

Laboratory of the David Taylor Model Basin. The results for the

vibration characteristics of several ships will be the subject of a future TMB report. LEIB0YITZ LEWIS' TAYLOR4' LEYIIS' 2-NODE 2-NODE 3-CODE - KEPINARD (ELLIPSOID (ELLIPSOID (ELLIPSOID (CIRCULAR OF CF REVOLUTIOII REVOLUTION) OF REVOLUTION CYLINDER) 2

r1

3 4 6 7 8 9 10 02 13 L/

Figure 1 - Plots of the Virtual Mass Factor j versus

the Ratio Of Length to Half-Beam

Note that for Curves 2, 3, and 4, the ratio is the total beam length

divided by the average half-bea.m at the

waterline. For Curve 1,

is

the particularinternodal distance divided by that section's half-beam

measured midway between the nodes at the waterline (see Figure 2).

Thus no direct comparison shOuld be made between Curve 1 and the other Curves.

One may also use a plot of Lewis's J-factor for the 3-node mode

versus (Curve 4) as an alternative to obtain specifically the

inter-nodal j for the 3-node mode by the same procedure a.s outlined in the text. STERN NODE ELLIPSOID FOR END SECTION

where p

3ab Hab HULL MODE SHAPE ELLIPSOID FOR END SECTION m. n. C. C. C1, 01 4x 40. 401, L,b Hb 5 mL. m., 01,. Ci C C. 4x. 4j40 k' _0' LbC Hbc bc BOW NODE CIRCULAR CYLINDER 0, 0', P. P'

FOR REGION BETWEEN TERMINAL NODAL POINTS (i.e., a to c)

Figure 2 Representing of the Hull Model for Vertical

Vibration (3-Node Mode) NOTE TO FIGURE a

The ship is divided into 20 sections. For a particular mode of hull

vibration (3-node mode is shown in Figure 2), a virtual mass value is computed for each hull section. x1 within the internodal segment "a" to

that does not contain anode by jabmi where

1 2

m..rpC.H Lx

_{1 i ab}

is the mass density of the w3ter, .

is the virtual mass coefficient associated with the particular

hull section

is the longitudinal virtual mass fact3r associated with the internodal segment "a" to "b", and

is the half-beam associated with the internodal segment

Referring to Figure 2, in general, Jabmi. _{Jabmj ,Jabmlc etc.,}

because C1 C C.1, etc., for each i.nternodal segment. Also,

j m.t, ab i bc i

etc., because C.

1 C.'1 and 1 1 and 3i

abi'

3bc

and so forth, for all virtual mass values in all internodal segments. The

virtual masses for the ship sections containing nodal points must be

treated separately since these nodal sections lie partly in one internodal segment and partly in another. The virtual masses correspottding to

these nodal sect:ions rna.y be 4etermined by various methods of varying

difficulty or mathematical convenience and accuracy Four possible

methods are outline4 as follows (a computer can be used for this work): Exclude computation of the foregoing type for the sections that include the nodal points. The virtual mass at these points ma.y be

ob-tained by interpolating the other virtual mass values.

If more than onehalf of the nodal section lies in one of the

inter-nodal segments, calculate its virtual mass by assuming that the inter-nodal

section lies entirely in that internodal segment

Compute the virtual mass for the nodal section first by assuming that it lies entirely in one of the adjacent internodal segments, and then

by assuming that it lies entirely in the other internodal segment Then

determine a final value for the virtua.l mass of the nodal section by adding

theproper proportions of each virtual mass so computed, this proportion

is the distance from the node tO the end of, the section divided by the total section length.

Divide the nodal sections into two parts through the nodal points.

Calculate the virtual mass of each part of the section by using the C's, H's, L's, and j's corresponding to that particular internodal segment in which it lies. Add the two virtual masses thus obtained to get a total

Method 1 is probably the easiest and most practical since the

virtual mass curve is smooth and easily interpolated. Methods 2 and 3 are rather easy and may be adequate, but they should be tested further. Method 4 is not very practical since the physical parameters vary from

one internodal segment to the next.

It should be noted that if a different mode is excited, the resulting

virtual mass values will be entirely different from those obtained for the

particular 3-node mode shown in this example; however, the procedure will be the same.

The section in each ellipsoidal segment adjacent to the node has

already contributed to the virtual mass if Methods 2, 3, or 4 are used.

However, the method for obtaining the masses for these end sections, described in Appendix C, need not be modified because these contribu-tions will not significantly affect the final results for the frequency and

mode shapes. For the ellipsoidal segment (see Appendix C):

2 rp b (1

where jm.1 is the virtual mass of the x.₁ section, and x. is the₁

distance from the center of the ellipsoid to the midpoint of x1.

7 x. 1

-- LXX.

L2 1 e

APPENDIX A

CORRECTIONS TO APPENDIX B OF REFERENCE 1

Page 237

Insert a minus sign on the right side of Equation [B3]

Page 238

After first equation for T1, write "p being the mass density of water.

Insert a minus sign on the right side of the second equation for T1.

The left side of the third equation should read:

(V)

_{= o}

The right side of the fOurth equation should read:

r

I 1 irpb2 I (V

J02.

L r

Page 239

Change sentence after Equation [B5J to read:

Since eVery K( ) -+oo as 0, ."

Page 242

2

The equation for m should read: m = rp --

(L2

- x2).

L e

APPENDIX B

MATHEMATICAL DETAILS OF THE DERIVATION FOR j OF THE

INTERNODAL SECTIONS OF THE SHIP

As written on page 238 of Appendix Bi of Reference 1, the kinetic energy of the water per segment of length L (L = internodal distance

T1 = - _{p J'5 (4}

_{.) dS}

[Bi] *

where p is the mass density of the water, and the surface integral is to be

taken over the surface of the cylinder of radius b.. (Note that in Appendix

Bi of Reference 1 p is mass density, whereas in Chapter 3 of the same

report p is weight density.)

For the cylinder, dS bddx with 4 varying from 0 to Zir and x varying from 0 to L. Thus the surface integral becomes

T1---p

{L

f(±)bdi

0

rb

Substituting Equation [E13] of Appendix E and its derivative into Equation [Ba], we obtain

T1=

_pb1

sint

cos2kx cos _{[Ki(kr)K1 (kr)]kd}

0 ."O

L Zir

T1 _

pbAsin2tK1(kb)K (kb)k

f cos2kxdx

f

cos2d

0 U0

* This equation is derived in Appendix G. For further physical

interpre-tationofTi.,andTl , see Appendix Bi of Reference 1.

r

b

These integra.ls are easily evaluated: rL rL rL

.1

2

Ii

Idx

cos kxdx i - (1 + cos2 kx)dx = ---- + Jo Jo Jo ₀ L 1 iT

r

sin 2 kL (since k =

SimilaIy,

Zir I

2d

Thus --.Trp bkLA2sin2tK1(kb)K (kb) [B3]

By using the following recusiorL relation for K(i) and its derivative,*

the product K1(kb) K1 (kb) can be shown: to alWays rernain egat'ive so

that the kinetic energy T1 of Equation [B3] is always positive, as it

shOuld be. 1.LK (p.) =}c](p.) - p.K0(p.Y [B4J** _K1' (p.) - --K1(p.) - p.K0(p.)

(K1() +

p.

* _{p. is} the argument of the Bessel function and is equal to kb in this case

** See Chapter III, Sctiôñ 2, Equation

[17] of Referene 6.

p.

Then

K1(.j K1t (EL) - K1(jj.)

=

[K12

+K0()K1()

_[B5]

The expression -K12(L,) _{is always positive for j>O, and according}

to the curves of Figure 3 _{K0 () K1 () is always positive for > 0.}

K1) + .LK0(.L) 2.0 1.5

31.0

0.5 0 LU

Figure 3 - Plots of K0(p.) and K1

_{() versus p}

Thus the term in brackets on the right side of Equation [B5] is always

positive for p.>O and, hence, K1 (p.) K1t(p.) is always _{negative for p. >0.}

Thus the right member of Equation [B3] is always positive.

* From Table IV, Page 737 of Reference 7.

Now assume that the 'ater flows irrotationally in transverse planes

only and alculate the corresponding kiretic energy T. This n.y be

accomplished by considering an infintè cylinder moving transyersely

at a velocity -

-(v)

kA [K(kb)] sint cos kx

[B61

The virtual mass per unit length for Such thotion is

dm 2

i'rpb

Thus, the differential lcineic energy per segment is

dT 1

v)

2

dm

4Pb[V)

4 ,TpbkA

[K (kb)]

sihwt

cos2kcdx

* 'See Chapter IV, Section 71, Equation [11J of Reference 8.

dx

Then T1' becomes

L T1'

b2k2A2 [K1tkb] sth2t

J

?

rrpb2k2LA2sinot [K)]

[p8]

The j-value which corrects for the longitudinal motion of the, water alpng the cylin4er is thus given by

T1

-.ii,p

bkLA2sin2 wt K1(kb) K](kb) - 1

2 2

2 2 r 12 1

-Trpb k LA sin:

_tLK(kb)J

K1(kb) 3=

-kbK(kb)

By using quation [B4], we can eliminate K1' (id):

3-Kj(kb) + kb K(kb.)

and since k= -,

K()-+oo as

-+O,frómFigurè 3. 13 [Blo]

[Bil]

Now

lim urn K urn 1

= _{-*O K1(p.) +p.(p.)}

_{- p.-O}

_K0(p.)

K(p.)

lim 1

* The fact that 1 ,

-

.1 can also be ascertained more

p.-+0K1tp./+p.K0p. .. .

rigorously by taking limits of the. actual series expressions for the

Bessel functions involved.

1

1

K0(p.)

Also, from Figure 3, the ratio in the denominator of the above

limit goes to zero as

(i.e.

since for p. >0

K0(p.) then K1(p.) K0(p.) Thus, K1(p.) as p. Hence Urn 1

L-oo

1+0

-as it should.

Tables of. K0(p.) andK1(p.) are given in References 6 and 7.

Some values of j calculated from Equation [Bli] are given in Appendix

APPENDIX C

MATHEMATICAL DETAtL FOR THE DERIVATION

OFj FOR THE SHI'S END SECTION

Since both Lewis's J and j f Reference 1 for internodal Segments

become inaccurate for the end sections of a Ship, it is better to derive a special j-formula for these sections To do this, the end sections may be considered as prolate ellipsoids of revolution vibrating in

rotation about their minor axes (see Figure 4).

= (z-y 4x 4X3 Xk Tfl. WATER SURFACE

Figure 4 _{Ellipsoidal Model of Hull End Segment}

The kinetic energy T of the water surrounding such _{a vibrating ellipsoid}

is

T =--k'

i.c22

[clj*

where is the angular velocity of évo1utiOn,

I _{is the moment of inertia about the minor axis of the ellipsoid if}

its density is equal to that of water, and

BOW NODE C

[3eze4(1-e2)

2) [7e

_{6e+(2_e2) (1_e2)}

[ca]

* See Section 1l5,Equations _[7

_{[8], [13], and [14] of Referene}

_8.

is an inertia coefficient. * In Equation [CZ], e is the eccentricity of a

section through the major axis Le of the ellipsoid (i.e., the eccentricity

of the ellipsoid shown in Figure 4). The eccentricity is defined by

Now wile re Thus I =

f PbLe

(b2 -i-L2 ) [C4] e rn

(C = distance from center to focal

point)

4

at

=p-nb L

**

* _{See Appendix D for a derivation of Equation [Ca].}

** See table of moments of inertia on page 260 of Reference 9.

See Reference 10,, page 216, No. 8 for the volume of an ellipsoid.

As before, recalculate the kinetic energy T' assuming that the water moves only in transverse planes. The virtual mass per unit length is

dm 2

--Trpy , where y is the ellipsoidal half-width at x (see Figure 4).

The equation for the ellipse in Figure 4 is

thus 2 2 V X S.- + b2 L2 2

Yb(1fr)

17 - 1 * ¼

The transverse velocity of the ellipsoid at x is x:2 . Thus the differential

kinetic energy is

12

dT' =--v dm

122

'2

-x 2py dx

irpb2

x2dx

Hence 1

22/x3

2

\3

eL

Ec5]

* See Reference 11, Chapter 6, Section 3, Equation [6] forthe equation of an ellipse.

= p

bYJ(1

4)

2

Finally

T'

PbL2

Thus, _j for the terilinal sectiOn of length Le

J9T

kt

fpbZL(bZL2)ç

/

2 k' (i + L e

.dm

The virtual mass per unit 1engtI for the end sections is then , where

dm 2 b

12

2

-lTpy Tp L

-x

L e

e

It maybe noted that as e -1 (i.e. ,- -0), k'

1, and that as e -±0

e

b --

lt

(i.e., -

1), k' ±0. Thus j 1, as it was also shown for the

e _e

circular cylinder model of the iLidsections.

[C9]

[c7]

Referring to Figure 4, we may write Equation EC9] as in rp y cix

rn +m.frn

_I j k Trp Hence, in general, .As x

nq1rP

₅ y2dx 2

(2

= jrpY 19

[b2 (i

J2

+ y dx + y dx + x. 3 -x q

* For simplicity of. calculatioxi, the point x 0 was chosen at the center

of the ellipsoid; however, for convenience of digital calculation for an actUal ship, a single origin may be taken at the stern (after perpendicular)

1, j, k,

e

[cill

[C 12]

Then the virtual mass of the qth section of the ellipsoid is

where Xq is the distance from the center of the ellipsoid to the center

of section Xq

2

APPENDIX D

DERIVATION OF k?.:USED IN T

4k!

I APPEARING ON PAGE 241

OF APPENDIX B1 OF REFERENCE 1

The inertia coefficient k' for a prolate ellipsoid rotating about an equatorial diameter is derived y Lamb * and found to be

where

and

-(2- e)

[Ze

-(a-

e2)

(Z)

ciO e 2

1-e

3 Ze (! 1+e

1-e

1 +e

1-e

If Equations EDZ] and ED3] are sdbstltuted into Equation ED 1], we get

r

1 1-e2 p l+e 2(1-e2) (1

3 - 3 1-e Le 2e. e 4. (-ze4' 3-3e2 e - 3 Ze

(2-

e2) (e2

- (2-e2) 13 _{[3e - 2e3}

21

* See Reference 8, Chapter 5, Sections 114-115..

}(1_eZ)

)

[Dl] [D3] 1-e 2(1_e2) 1 (2_eZ) - e) [DzJ

Finally,

4r.

3 3 2

e

3Ze

(l-e) n

Equation D4] is the relation for k! appearing at the top of page. 241 in

Appendix.Bl of Reference 1.

APPENDIX E

SOLUTION OF LAPLACEtS EQUATION IN CYLINDRICAL COORDINATES

FOR c, THE VELOCITY POTENTIAL OF FLUID SURROUNDING

A VIBRATING CYLINDER

The LaPlace equation is:

!.L(

2 r ôr

r/

2 2

r

[El] * 1 d2X 1 1 d _{( dB\} 1 1 _d2'V X dx2

- R

r dr

r

_dr)

r2 [E4] Z3

See Figure 73 in Appendix Bi of Reference 1 for the geometric interpreta-tion of the symbols in this equainterpreta-tion.

Assume a solution of the form **

c(r,x,)

R(r) X(x) V() [E2]

SubstitutionOfthiscj into Equation [El] gives

_+x!Lf

(r)+JXáO[E3]

Divide Equation cE3] by R X ! and transpose:

* _{See Reference 12, Chapter 17, Section 2, Equations}

_f

_{2.2] where v=c1,}

z=x,andQi11,.

** Actually varies harmonically with time also, but for mathematical convenience the time variation is introduced later.

The right side of Equatipn [E4] is nOt a function of x. Hence both

members are equal to a constant.

Iet

where k

constant. Or

1 dX

x2

_4x

+k2XO

quation EE6I has as a solution *

X=C1coskx

From Equations 5J and [E6] we get

r2

+---kr

_r

_{ctR 2 2 1}

By the sathe. reasoning as above, each member of Equation [E8] is

a constant. Let

. . . .

ia__

z

!az

where n

constant. Or

d2

= 0 [Elo]

Eauation rEloJ has. as a solution *

.= C2cos nlJ)

* The general solutions of [E6J and [Elo] are X =

eC+

an.d' re1 +

e1', respectively.

However, the boundary coiidition that X-+O as x -+ etc leads bo the physically realizable solution X= C1

cos kx and != C2 cosn, respectively.

.

rE7]

E8]

Then Equatiqx 1E4] can be. written as (using Equations rE5J and 2 i. 1 I

/

d.R\ 1 2

-k +

_Rrdr

- - r -

_dr/

j - -

₂ r 2' 2 p R + r 2 dZR r

-i-

+

r

dr

Finally, d

dr

d2R (kr) + ..L dR (kr) d(kr)2 icr d(kr) (r

dkc1 dK1

_/

-

Il

j dp. 25 cR\

_{-Rn =0}

2

dr/

R(kr)K1(1r)

K1(p.) then Equ3tion [E12J becomes

2_. K1 = 0

(zkz 2)

_{R 0} (1 +

R(kr)

= 0 [E12]

Equation rE12j has as is solution R(icr, _{a Bessel function of the. second} kind and first order To make the solution vanish as kr* , the modified

second kind, first order Bessel function, commonly denoted_{by K1 (n),}

is chosen as the solution That is, let

[D13]

Where n has been set equal. to unity to satisfy t1e boundary _{condition at the}

cylinder's surface that the normal component of the velocity of the cylinder's surface be equal to the normal component of the water velocity (see

Appendix F) _{Equation [E13] agrees with the equation at the bottom of page}

237 in Appendix Bi of Reference 1. _{The required sOlutiOn,, 'is then}

c

CK(kr) cos kxcos

The temporal variation of cf is now conveniently included, giving

= A sint K(kr) coskx cos

[E14] where A is an arbitrary constant and the time factor sint was chosen

to correspond to a free vibration. Equation [E14j agrees with Equation IBZ] of Appendix Bi of Reference 1.

APPENDIX F

BOUNDARY CONDITION AT THE CYLINDER'S SURFACE

This is a clarification and elaboration _{the discussion in Appendix B}

of Reference 1 (page 237) on the boundary condition that the normal component of the vibrating cylinder's surface velocity must be equal to the normal component ofthe water velocity at the cylinder's surface.

Curve (1) in Figure 5 represents the deflection y of an. internodal

segment of a free-vibrating beam _{Curve (2) represents the velocity}

potential of the surrounding fluid. Curve. (3) is a plot of the cylinder's

lateral velociW which must also be. equal to the normal component

of the water velocity (-)

_{in the lateral direction at the cylinder's}

surface. NODE (3) MIDNODE (X=O) - Idy\

BEAM VELOCITY 1 I. = FLUID VELOCITY I

-dt/ _{\th r=b} i=O 27 FLUID POTENTIAL () BEAM DISPLACEMENT(y) NODE

POINT OF MAXIMUM FLUID ADBEAM

LATERAL COMPONENTS OF VELOCITY

Figure 5 - Beam Displacement and Velocity and rluid Potential

and Velocity

The beam velocity and the fluid velocity

b

are maximum

at midnode (x 0) a when y 0. 0

This graphcal representation illustrates the boundary condition that

tbe normal component of the velocity of the surface of the cylinder ±nust be equal to the normal component of the water velocity

r 0

for the solution

A sin cos kx K (kr) cos

as derived in Appendix E (see Equation [BZ] in Appendix B of Reference

APPENDDC G

DERIVATION OF T= -

_{p JJ-}

d S , THE KINETIC ENERGY

OF THE FLUID SURROUNDING A VIBRATING INFINITE CYLINDER

According to Gauss,

[G1]*

where A is any vector whose Cartesian components are U, V, W, andn is a unit vector directed inward through dS (dTis an incremental volume). That is,

Then

-- - - -

-nA

n.i U+n.j V+n.k W

.U+mV+n1W

where , m, n1 are the direction cosines of n. Also,

x _y z

Thus Equation [Gi] becomes

+rnV + n1W)dS -

J5J(-+

- +

f) dx dy dx

[G2] Now let TJ= V -,

and W

4i - [G3] **

* _{See Reference 13, Chapter 3, Section 3-6, Equation [3-115].}

** See Reference 8, Chapter 3, Section 43.

so that or .tt+ rnV + n1 11 j J ci 8n Jf5

[z

T5i5 L

az JJ J

\

Z

ax ox Oy Oy +ci) dxdydz 2 Oz Oz

=-ssi

[()z()

-

J5Jci9dxaYdz

[G5]

Now, for irrotational flow, V c 0

ii

is t1e velocity potential of a fluid

(i.e., if'

).*

Thus Equation EG5J becomes

()

+

a]

dxdydz =

-See Reference 8, Chapter 3, Section 44, Equation [1].

(

)

(ci)]

dxdydz

jf.

[G6J

+ mV + n1W = ci) - [G4]

31

Now, the Kinetic energy Of a fluid having a velocity potential is

T = -

fffv.v

ciT (p mass density of fluid)

Vdxdydz

.ifff

[(..)2+(e) 2(8)2]

which can be compared with the left side of Equation [G6]. Thus a fina. relation for the Kinetic energy of the irrotationally moving fluid is

T=PJfctdS

[G7]

Equation [G7] agrees With the expression used in Appendix Bl of Reference 1.

APPENDIX H

EXPERIMENTAL AND THEORETICAL FREQUENCY DATA FOR

SS E. J. KULAS FOR THE VERTICAL MODES OF VIBRATION

INDEEP WATER

The theoretical data recorded in Table 1 were obta.i.ned .by analog

methods using Lewis's constant J-factor and Kenna.rd's variable j-factors,

as discussed in this report, to determine the virtual mass and the

fre-quencies of vibration.

The error associated with Kennard's method for frequencies up through the 5th mode for the light condition and up through the 4th mode for the loaded condition is substantially less than that associated with Lewis's

method for the corresponding modes. For higher modes, Kennard's

method is less accurate since the assumption of simple beam vibration

for the ship's hull becomes less valid for these modes.

In practice, prediction of the higher modes is less important than

prediction of the lower modes because the resonance response of the hull and the response of the coupled hull-sprung inertia (e.g., appendages,

machinery, equipment., etc.) systems are usually greatest for the lower

modes.

McGoldrick2 states that if the theoretical virtual mass values were

correct, the theoretical frequency ratios should be equal to the experimental

ratios. The fact that Kennard's ratios correspond more nearly to the ex-perimental ratios than Lewis's ratios indicates that Kennard's j-factor will

give better results.. It also seems to substantiate the decrease in virtual

* _Thevalues in Table 1 were taken from Tables 1, 4, 5, and 9 of Reference 3.

Error theoretical frequency - experimental frequency _x

- experimental frequency

TABLE

Experimental and Theoretical Vertical Vibration Frequency Data Taken in Deep Water for SS E.J. KULAS

Mode

Natural Frequency, CPS Error ** Ratioof Loaded to Light Frequencies, percent Experrnental Theoretical _{2-node Virtual Mass}Using Lewis s

-Using Kennard s

Virtual Mass Experimental

Theoretical Using Lewisis

2-node Virtual Mass Using Kennard'sVirtual Mass Using Lewis 's_{Virtual Mass}

--Using Kennard's Virtual Mass

Loaded Light' Loaded Light Loaded Light Loaded Light Loaded Light

Condition Condition Condition Condition Condition Condition Condition Condition Condition Condition

2nd 9Z 98 74 80 83 89 -20 -18 -tO - 9 94 92 93 3rd 150 168 126 152 143 156 -16 -10 - 5 - 7 89 83 91 4th 200 230 182 199 206 230 - 9 -14 + 3 0 87 91 90 5th 246 285 237 258 279 306 - - 4 -10 13 + 7 86 - 92 91 6th 285 330 294 317 345 384 + 3 - 4 +21 +i6 86 93 90 7th 304 375 348 372 417 474 +14 - 1 : +26 81 93 88 8th 360 435 396 426 483 546 +10 - 2 +34 +26 83 93 85

REFERENCES

Leibowitz, R.C. and Kennard, E.H., "TheOry of Freely Vibrating

Nonuniform Beams, Including Methods of Solution and Applications to Ships," David Taylor Model Basin Report 1317 (May 1961).

McGoldrick, R.T., "Determination of Hull C-ritical Frequencies

on the Ore Carrier SS .J. KULAS by mea.ns of a Vibration Generator,"

David Taylor Model Basin Report 762 (Jun 1951).

Lewis, F.M., "The Inertia of the Water Surrounding a Vibrating Ship," Transactions Society of Naval Architects and Marine Engineers,

Vol. 37 (1929).

Taylor, J.L. "On Some Hydrodynamical Inertia Coefficients," Philosophical Ma.gazine and Journal of Science, Vol. 9 (1930),

Kaplan, P., "A Study of the Virtual Mass Associated with the

Vertical Vibration of Ships in Water," Stevens Institute of Technology

Report 734 (Dec 1959).

Gray, A., et al., "Treatise on Bessel FunctionS and Their

Applications to Physics," Second Editjon, Macmillan Co., Ltd., London

(1952).

7. Watson, G.N., "Theory of Bessel Functions," Second Edition,

Cambridge University Press, Cambridge (1948).

Lamb, H., "Hydrodynamics," Sixth Edition, Dover Publications,

New York (1945).

Eshback, O.W.E., "Handbook of Engineering Fundamentals,"

Chapman and Hall, Ltd., London (1936).

"HUtte, Des Ingenieurs Taschenbuch," Wilhelm Ernst and Sobn, Berlin (1942).

Thomas, G.B., "Calculus and Analyic Geometry," Addison-Wesley,

Reading, Mass. (1956).

Pipes, L.A., "Applied Mathematics for Engineers and Physicists,"

Second Edition, McGraw-Hill, New York (1958).

BIBLIOGRAPHY

Todd, F.H., "Ship Hull Vibration," Edward Arnold Ltd., London

(1961) _{An extensive discussion, a historical survey, and formulas for}

virtual mass for various conditions of. vibration are given in this book. McGoldrick, R.T., "Ship Vibration," David Taylor Model _Basin Report 1451 (Dec 1960). Discussions and methods of calculation of

virtual mass are given in this report.

McGoldrick,R.T., "Comments on Some of the Fundamental Physical Concepts in Naval Architecture," David Taylor Model Basin Report 1609

(Apr 1962) The concept of virtual mass is discussed in this report

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