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, .:..

DEPARTMENT OF THE NAVY

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTERWASHINGTON, DC. 20034

THE INFLUENCE OF ADDED MASS ON THE NATURAL VIBRATIONS AND

IMPULSE RESPONSE 'OF LONG, THIN- CYLINDRICAL SHELLS

by

E. W' Palmer

This document has been approved forpublic release.and sale; its distributionis unlimited.

October 1970 -Report 3395

-

TABLE OF CONTENTS

P [Igo

ABSTRACT ........................- .......... ............................... .................... I

ADM1INISTT IVE I NFORM ATION ....................... ......................................

1. -INTROMUCTiON AND-IAlTER'ATu,?ESU RVEI .............. .......................... 2

11. STATEMENT OF 'I'l WP ROM., EM ................ .....................................

ill. NATURAIYvuzIBrTioNs.......................................................................

A. G(iOV'ERN-ING EFQUAT11INS...... ..... ............ ........................................ 7S3 OLUTION 0 F "'llE-,GO VER NIN G EQUATIONS ...................................... S

C,- CONSIDERATION OF THE ADDED MASS ............... ........................ ...... 9I.NATURAL FREQUENCIES ......... ................ .;.................................. 12

E. -NORMALAMODE SHAPES.........I......................................................... 10F. NUMERICAI, SOLUTIONS ................... ............................................ 21

1. 'In fluenoe of Mass on Frequenekis .. ....................................... !............ 212. 1I it oun ev of'NI ass on-Mode, Sit apes ............. ......... I........................ 28-

G. -COMl'AliSON'WIT wExP ERIMIENT ............. .......................... .......... 28-

II. 6DISCUSSION ........................... .................................................... 34I. influence of Mass on Frequeh6ics ........................ .....................I........... 342. Influence of- Niasson ModoeShapes . .... ........... ................ ........ 34'

I V. IMPULSE RESP )NSE............................................................................:36)

A. GENERAL SOLUTION .e.................................................................. .30w. soL~uTION FOR SPECIFIED INITIAL, CONDMTONS .. ............................ 4-3

=C. COMPARISON WITH EXPERIMENTAL RES&ULTS, .................................. 46

1. C'yind,-rwilh Added Mass ............................. .... ............................. 40

-2. Cylinder wit-hout. Added-Mass ...... .. ................................................ 50=3. Ring without. Added Mass.............. ....... !.............. ....... .............. 5

D . DISCUSSION .................................... ....................................

V.DSUSO .................................................... ................................ 8

VthU M A Y ............ ........S........MM A....................................... 22

ACKNOWLEDGMENTS ................ ................................................... ............ 63

R E F E R E N C E S ...... .... ..... ... ........................................................ ....... o. 6 1

LIST OF FIGURES

Page

Figure 1 -Coordinate System 6 F

Figure 2- Dynamical Equilibrium of the Added Mass . . . . . . . 0

Figure 3 - Variation of Frequency with Mass, Zeroth- andFirst Modes, Extensional Class, Z = 0.0001 ....... 24

Figure 4, - Variation of Frequency with Mass, SecondMode, Extensional Class, Z = 0.0001 . . . . . 25

Figure- 5- Variation of -Frequency with -Mass, SecondMode, Flexural Class, Z =,0.0001 . . . . . .. 26

Figure 6- Variation of Frequency with Mass,-,ThirdMode-, -Flexural- -Class, Z =- 0,0001 -... . . ?.. ..... . 7

-Figure 7 - Normalized Mode 'Shapes, Extensional Class,Z 0.0001 * . .. . . . . . . . . . . . ... . . . 29

-Figure 8, - Normalized -Mode Shapes, Flexural Class,

Z = 0.0001 . ... . .. . . . . .. . ... . . . ... . 30

j igure 9 - Experimental Mode Shapes, Second Mode,Flexural Class, Symmetrical Branch,

Z 2.6 x 106 32

Figure 10- Experimental Mode Shapes, Third Mode,Flexural Class, Symmetrical Braich,Z 2.6 x 10-6 33

Figure- 11- Theoretical>-and-:Expdrimental VelocityHistories, 0 Z ,Z 0.00194 ........ •• 9

Figure. 12- Theoretical and Experimental Stress'Histories, -R 0, Z =-OO0011 * .... . .-..9.011 5,4

'LIST, OF TABLES -

Page

Table 1 'Natural Frequencies, Extensional Class,0 Z 0,0001 .. . ... 9...... . . ..... 22

Table 2 - Natural Frequencies, Flexural -Class-,Z 6.0,001 . * *--* * * . . . . . .- . 23

Table 3- Velocity Mode Amplitudes, 0 = ,-Z 0.0194 ... . . . . . . ... . .9 .9 9 • . . 1 8

K _I

LIST OF SYMBOLS

A Cross section area of unit width ring.

A Mode constants for antisymmetrical branch'.k

A l Normalized 1th modb constant, antisymmetrical branch.

a Radius of neutral surface, or subscript meaning antisymmetricalbranch.

a. tlekural, symmetric, i th mode constant from initial conditions.

.tha. Flexural, symmetric, i mode constant in term -of* a..a. Flexural, antisymmetric, V h mode constant from initial conditions.

Flexurl, . th

a Flexural, symmetriC,n mode constant from initial donditionsR: O

* tha Flexural, symmetricn h mode constant from specific initial

n conditions,R = .

Bk Mode constants for symmetrical branch.

Bl ' Normalized ith mbde constant, symmetrical branch.

b., Flexural, symmetric, I th mode cbnstant from initial conditions..th

b. Flexural, antoisymmetric, I mode constant from initial conditions.

ma2 2C Frequency parameter, -

C Extensional frequency parameter.e

Cf Flexural frequency parameter.CM Frequency parameter for case of added mass.

C Frequency parameter for case of no added mass.m

C. Material.sound velocity, .

thc. Extepsional, symmetric, i mode constant from initial conditions.

.thC. Extensional, anisymmetric, i mode constant from initial

Sconditions.

c! Extensional, symmetric, ith mode constant in terms of c..

___ -Ai v 1

5I

thc, Ektensional, symmetric, n mode constant from initial conditions,

* tth-c Extensional, symmetric, n mode constant from specific initialn conditions, R = 0.

D: Determinate of coefficients of constants Ak and B

\ D Determinate of coefficients of koantsAA ofcntat *

DB Determinate of coefficients of constants Bk .EliD Stiffness parameter,T--.

d. Extensional symmetric, ith mode constant from initial conditions.

d. Extensional, antisymmetrici ith mode constant from initial

1. conditions.

E Young's modulus.

Ev -l

Eki Constanv. in ci.

e Base of natural logarixhms, or subscript meaning extensional class.

F Constant in a..ki

f Subscript meaning flexural class.

f Rigid body zeroth mode constant.0

fl Rigid body first mode constant.

G Moment stress resultant.

go Rigid body zeroth mode constant.

g1 Rigid body first mode constant.

h Shell thickness.

h Rigid body zeroth mode constant.0

h Rigid body first mode constant.

I Moment of inertia of unit width ring cross section about neutralaxis.

I. Maximum impulse amplitude.0

v

i - or integer subscript denoting mode number or solution state.

j Integer subscript denoting solution state.

Eh 3

K Stiffness parameter, 12(1 - 2

k Integer subscript.

M Mass per unit length of the added line mass.

m Mass per unit length of ring of unit width, or integer subscript.

N Shear stress resultant.

n Integer subscript denoting mode number, R 0.

nk Mode parameters from equation (12).

MR Mass ratio, 2 ma

r Radial coordinate.

s Subscript meaning symmetrical branch.

T Thrust stress resultant.

t Time variable.

u1 Axial displacement.

u Axial acceleration.

V Tangential displacement, time independent.

thV. Tanential displacement, -time independent, i mode.V.

V.ia All

S ViV. .

4 Tangential displacement,.

V Tangential acceleration,

v. Tangintial displacement, i mode.

v Initial tangential displacement.

?AVi

A

v. Tangential velocity,, i t h mode.1

v Initial tangential velocity.

W Radial displacement., time independent.th

W. Radial displacement, time independent,, i mode.W_ W.

w', l .

W'. Wi .is -- 1

B li

-w Radial displacement.

w Radial acceleration.

w. Radial displacement, th 'mode.1

w Initial radial displacement.

v1. Radial velocity, i th mode.

Iw Initial radial velocity.

w Maximum initial radial velocity.

X. Constants in A

-Y. Constants in B

IZ Stiffness parameter, --

Aa

z Axial coordinate.

'k Circular argument, nk r , for coefficients of A.k and Bk.

r Mode parameter for stress, R = 0.

r' Mode parameter for stress, R = 0.n

0 Angular coordinate.

eM Angular location 9f added mass.

Ii "i

iw Circular frequency of vibration.

i Circular frequency-of vibration, mthmde or solution state.

Wml circula, -frequency with added mass.

tw- Circular frequency:, R = O.

V Poisson's ratio.

p Material mass density.

a Stress.

a G Stress due to moment.

o Stress due to thrust.oCt

I Non-dimensiofidi- time variable,-!-a

-V i

-- -- --------

_F

ABSTRACT

The plane strain solution is-obtained for the natural vibrations and im-

pulse response or-a thincircular cylinder containing-an added line mass. The

solutioh for a uniform cylinder is derived by taking the added mass to be zero.

Numerical calculations of the freqfencies and mode shapes for several of thelower modes are presented in graphical form for various values of the added

mass. The general impulse response-solution-for irbitrary initial conditions

-is obtained by normal mode theory. For both.the natural vibrations and im-

pulse response, the theory is found to be in reasonable'agreement withavailable experimental results.

In a particular mode, four distinct solution states'are found to exist:a symmetrical and anti-symhetrical branch for each class of vibration,.flex-

ural and extehsional. Noteworthy features revealdd by this investigation are

the difference in frequency and mode shape of each solution state andthepresence of coupling-between the flexural and extensional classes, particular-ly noticeable in the extensiofnal class mode shapes. In comparing]ipulse re-

sponse solutions for velocity with and without the added mass, the major in,

fluence of the added mass is found to be an increased participation of the

flexural class modes, including the rigid body translation, and decreased,

participation of the extensioral class oscillatory inodes.

ADMINISTRATIVE INFORMATION

The research reported herein formed a thesis submitted to and approved bythe Graduate

Faculty of the Virginia Polytechfiic Institute in partial fulfillment of the requirements for the

degree of Doctor of Philosophy in Engineering Mechanics.

This report is relatedto the program entitled "Hull Response-Inputs to Equipment,"

NAVSHIPS Task area SF 35.422. 110, Task 15041.

Ii

1. INTRODUCTION AND LITERATURESURVEY

The vibrations of mass loaded cylindi-ical shells are of interest

in many engineering situations; e.g., in the dynamic behavior of hydro

and aerospace vehicles with equipment attached to their cylindrical

walls. With the added mass considered as aq imperfection, such a study

should also be imoortant in fundamental studies of vibrations of

imperfect bodies of revolution inasmuch as the effect of such imperfec-

tions can be quite, distinct as has been shown by several experimental

investigations [1,2,3]?. A study of the impulse response of such

structures is of particular interest in military defense applications,

both in response of the. shell itself and of attached equipment repre-

sented by the added mass.

The first formal analytical work in this area-apparently was by

Den Hartog [4] in 1928. In studying the natural vibrations of ring-like

frames of electrical machines containing lumped masses, he derived

approximate fundamental flexural frequencies for these frames by the

Rayleigh .method. In 1962, Palmer [5] investigated the variations in the

natural flexural vibrations without extension due to a lumped mass on a

ring by solving the classical eigenvalue problem. A significant result

of that study was the difference in frequency and mode shape found

between the symmetrical and anti-symmetrical branches of each mode,

Michalopoulos and Muster [6] in 1966 obtained the natural vibration

solution of a ring-stiffened, mass-ioaded cylindrical shell using an

energy formulation with the displacements represented approximately by

Numbers in brackets, C ], refer to references listed on page ,64.

2

I

finite Fourier series. The natural vibrations of a stiffened pressur-

ized cylindrical shell with an attached mass were considered by Ojaivo

and- Newman [7] in 1967 by an approximate energy technique. They

determined mode shapes for the shell without the attached mass and used

these as assumed mode shapes to determine the natural vibrations for

the shell-mass system. Lee, et al, (8) in 1968 analyzed the natural

vibrations of mass-loaded-ring structures by two approximate methods,

finite-element and transfer matrix. Using essentially the same approach

as ichalopoulos and uster, Lee, et al, also studied the natural

vibrations of mass-loaded shell structures. -Although the natural

vibrations of mass-loaded beam and plate structures are well established,

e.g., Thornton [91, a literature search has failed to reveal a complete

rigorous solution for the- natui-al vibrations of mass-loaded -ring and

cylindrical shell structures.

Previous solution attempts to the subject problem can, with one

exception [5], be characterized by the approach used by Ojalvo and

Newman [7], in which the mode shapes for the uniform shell without

added mass were used as trial functions in an energy formulation for

the solution of the complete shell-mass system. A rigorous solution as

presented herein is desirable since it is well known that even crude

trial functions used in an energy- formulation can give quite good results

for the natural frequencies, at least for the lower modes, but mode

shapes, and response parameters computed from them for a specified

problem, may be subject to question.

The problem of impulse response of a mass-loaded cylindrical shell

has been treated by Michalopoulos and Muster t[]. They used- modal

3

analysis to formulate the response of the structure to a band of-uniform

internal pressure applied as a pulse in time. However, numerical results

were not presented. With the above excep'cion, the impulse response of

mass-loadedsheIl structures apparently ,has nbt been treated in the

literature. The simpler, degenerate problem of the responsi of uniform

thin rings and cylinders to non-moving impulse loads, has, however,

received somo attention. Palmer [10] used-modal anailysis to obtain the

response of a ring to a side pressure pulse represented as an initial

velocity distribution. in that study, the effects of flexure and exten-

sion were completely uncoupled. Humphreys and Winter [111 employed the

Laplace transform approach to solve slightly simplified thin shell

equations for an infinitely long cylinder under a transverse pressure

pulse. Sheng 112] solved Donnell-type shell equations for the response

of a cylindrical shell to a semisinusoidal surface pulse. Johnson and

Greif (131 used two differefit numerical methods of timewise integration

to obtain the response of a cylindricAl shell to a blast loading. Their

solution was completely numerical and involved expressing Sanders'

shell: equations in finite difference form.

kI)*

* I-

i4J .4

II. STATEMENT OF THE PROBLEMIThe plane strain solution for the natural vibrations and impulse

response of infinitely long, thin\ circular cylindrical shells containing

an added-concentratedline mass" will be obtained. The shell material

will be- isotropic, homogenous and the analysis will employ linearized,,

small deformation, thin shell theory, neglecting the effects of shear

deformation and rotatory inertia, as formulated by Flugge [14].

The midsurface displacements u, v, and w and the coordinate system

for- the- shell are defined in Figure 1. The line mass, M, is concen-

trated at the midstirface and is parallel to the z-axis. Rotatory

inertia of the concentrated mass is neglected. Motion will occur only

in planes perpendicular to the z-axis.

The solution for the natural frequencies, and mode shapes will be

obtained in the classical manner by describing continuity and equilibrium

conditions at the mass. The influence of the mass will be ascertained

by-comparison with the vibrations of a cylinder without added mass.

The expt solution to the governing equations will'be obtained for both

extensional and flexural classes of vibration with no approximations

made on eithir class. In a previous solution by Pilmer 153 only the-

flexural class was considered in an approximate manner, i.e.,, flexurev

with the midsurface extension everywhere zero, or w = . The

analytical frequencies will be compared with those obtained experimentally

by Lee [8].

The solution for the natural vibrations will be used to- determine

the response of -the shell to a transverse impulse. 'The impulse 'will be

-specified in terms of initial conditions on the displacements and-

kJ 5

velocities,. and then the mode constants will be determined through the

orthogonality relation. The influence of the mass- will'be determined

'by comparing the response with and without the mass. As noted in the

literature review, Palmer [10- obtained the response without the added

mass to an- initial velocity distribution by modal analysis. Although

'bothextensional andflexura.l classes were considered, that solution

was an approximation since extension was assumed to occur without flexure

and flexure without extension. Such simplifications permitted a solution

to be obtained in a relatively straightforwardmanner. The impulse

response solution will be compared with experimental results of Navy

tests, reported in [i5j and [20], and Sandia Corporation tests [16.

Figure 1 - Coordinate Sytem

[

?6

III. NATURAL VIBRATIONS

The general 'solution will be obtained for the natural frequencies

and noxiial mode shapes. Numerical calculations of the ,frequencies and

niode .shapes for several of the lower modes will be presented in graphi-

cal form' and compared with experimental results [8).

A. Governing Equations

The complete-thin cylindrical shell theozy equations of motion

presented by Flugge [14], using the coordinate system of Figure 1,

adding inertia terms and omittiig load terms for natural vibrations, are

1zz+ - U 00 + . r1-wV+%

I-5 .1. o " . Do Yzzz -, o,- /- ,

I+P + voo +- .v K _ 3-2 w D(-110+z _ . Wz 0 ( 2)

K I-v 3-P wv1'0 wa2 = 0D S+ I - 00z - zzz--.V0 z- .zzz -WOOzz- ,o0000 ,2w 00 i -

For the case of plane strain, u=0 and ( ),= , these three

equations reduce to two in the planar displacemenit components v and w

Thus

2-3aV W = (4)Ds 00 0

Ma2l + Vo w - K [,Woo00 + voo + w] 0 (5)Ds Dsa2

where

()o=-

and-

= t

r 1

Since it is desired to treat rings as well as stiffened cylinders,

-it is convenient to change the fon- of the stiffness parameters Ds

and K for a ring of unit width as, follows:

Ds j) = lI h = -

1 -V2

a n d K FL 3.-

D a2 Da 2 L12(I2)- Aa2

Wit h these substitutions, equations (4) and (5) can-be written in

the form

- 11V +I WOO I A0) 0 (6)

I'A F!1

-InI + ( - wV0 ) - ,W 0 0 0 + 2w 0 0,v + w) 0 (7)a

It is noted that equations (6) and(.7), also were derived by Baron

and Bleich [17].

B. Solution of the Governing Equations

Suitable solutions to equations (6) and -('7) for harmonic vibrations

are, for a partitular mode,

v = V(O) eiW t (8)

and W = W(0 C (9)

'where V(O) = , k (Ak c(1 nkO + Ik sin nk0) (10)

and Za h1 k si' nkO- BkCOS1nk (Ii)

Substituting (8) and (9) into (6-) and (7), it is found that

equation (6) is identically satisfied, whereas- equation (7) will be

-,,

satisfied if the following relation is in turn satisfied:

C, [(n2-1)2 Z + 112 + I]C + n2'(n2 - )2 Z0 (12)

where ) 1)En w (13)

and Z= (14)Aa2

The nl, n2, and n, of equations (10) and (11), are roots of equation2

(12), which is cubic in n As expected, equation (12), the frequency

equation, is quadratic in the frequency parameter C. It will be shown

shortly that for a given value of n the lesser value of C represents

primarily flexural vibration and the greater value represents primarily

extensional, or membrane, vibration.

C. Consideration of the Added Mass

As the solutions (10) and (11) contain six constants of integration,

Ak and Bk , at least one always remaining arbitrary since (6) and (7)

are homogeneoits, and since the frequency parameter C must also be

determined, then six relations are heeded at the added mass.

From continuity conditions, -the following equations can be written

io 0NI

lO v(OM)-V (-OM) 0 (15)Li N

.[,]~ 0 0 (16)-o0

[Vol ON I =0 (17)

d rom 'equilibrium, see Figure 2, summing forces in the radial

direction gives

K0 Mj (- =0

-01

9t

N~o ) MI DSURFACE

M OM)

T(M)

MT(OM) G(oM)

Figure 2 -Dynamical Equilibrium of tileAdded 'Mass

10

and in the tangential direction-gives

FOMNf-(om)+IT] = 0 (19)

and summing the moments gives

OG =0 (20)J-OM_

where N, T, and G are the respective shear, thrust and moment stress

resultants. rrom Flugge [14), with present notation, these are

N El (0oo0 + w0 )- (21)

TA; (v0 - w) - (w +--Woo) C(22)

aa

G= (w + woo)= (23)

Evaluating (15) through (20) with expressions (8), (9), (21), (22),

and (23) gives six linear homogeneous algebraic equations arranged as

follows:

from (16)

Ak (C- n2) sil ak0 (24)

from (20)

Ak 112 (C- 11)si ak0 -(25)

from (19),

3

Ak "k (RC cos ak+ "k sin ak) = 0 -(26)

*11

k2

from (15)

Iok "k Sn (1 k 0 (27)

from (17)

B nt si ak= 0 (28)

and from (18)

I3k [TR- (C- nF) cos ak (29)

where I , the mass ratio, (30)

and "k( k0 k = 1, 2.or 3 . (31)

Without any loss of generality, ON, will be taken as n for the

remainder of this dissertation.

D. Natural Frequencies

Since (24) through (29) are homogeneous, the determinant of the

coefficients of the Ak and Bk terms must vanish in order for non-

trivial values of these constants to exist.

The sixth order determinant formed from (24) through (29) is

DA O =0 (32)

0 DB

12-

where

(C - ) si a , (C - n 1),s l a ( (C n ) Sill a3

A 1 1 2 2 3nj= 1 (C - inll ,a.n5_(C -,lJ '%illl.) ,2| (C 11j) sil a:. 3.

n (l0RC Cs I + 11 Sill ) I 2 (nRC cos " 12 %'1) ") 113 (nRC Cos a3 + 113 11 a 3 )

and

III sll a IIn2 Sill a- 3 sin a3

3- 3-11l) Sill (Lj 12sill-a1-, 1)3S 1 3 (34)

C 2)T 3O sill ~~ (13~)CS 1

'n67 (C .- 112 o , r ,IC (C

-Al I sll a 1 sil a- - 13 sill a3

Equation (32)- is expressed as -a Droduct of two third-order determi-

nants, one containing only coefficients of Ak and the ,other containing

only coefficients' of ,Bk . Thus

D=DADB =0 (35)

:Equation,(35) may be-satisfied in three ways

DA. =0 (36)

D,= 0, (37)

or!)A = DBO -0 (38)

In the first of these, (.36)., nontrivial values are possible for the

Ak constants since DA=0 . But if |)A=0 and D13 0 , then

all the Bk constants will be zero,, since by Cramer's rule nontrivial

-1r

4[

values cannot exist. The reverse is true for the second of these, (37).

heferring to the expression for the radial displacement function, (11),

we find that the Ak constants are associated with- sine functions which

are antisymmetrical, whereas the Ilk constants are associated with

symmetrical cosine functions. Condition (36) then would result in a

solution containing only anti'symmetrical terms and the solution from (37)

would contain only symmetrical terms.

In either case, (36) or (37), both the governing equations and;

the continuity and equilibrium conditions at the mass will be satisfied.

The frequency equations may be found by evaluating D and DI,

rrom -(33)

) 11, (C ,n[) (C - u1) Si COS a2 si, a)+1, 3 (C ) - ( - 25) Sin l 1 Sil (L 2 (RC COS a 3 +13 Si a)

2 2 2.n1121 (C -11) (C li2)Sll 2Sill CL 3 @-, rRC os + 11 Sill j )

+ 1211 (C - Ih2) (C1. 2),ISill (t,) Sill a3 (irRC COS a, + ii ISill a,)1 li (C - Ili) (C - 11) Sill a,) Sill a3 (iTRC cos a, + iiI sin a,)W-,113 (C - it ) (C - '13) Sill (LI Sill L3 (r;RC Cos a.) + ii- Sill a,,)

113111 (C - Ii) (C - 112) Sill a Sill a2 (rRC COS 03 + n3 sil a3 )

Se tting 1) 0 the frequency equation for antisymmetrical vibra-

tions is obtained. Solving this expres,.;ion for the mass ratio and

letting R = RA for DA=O gives

- F _l)( 22 ' . ") (5-AA Ai +,s( ,)(

+n 2) ( C-!1)(n n) +11 (C 11i (C- ) \3 "1L -L- 1 -3,2 3 2()(3,l-

2(C (C 2) 1 2 2 3S3 2

~~1)CO ((LI

+1.t2 (C - 12)( - 112) (12 - '11 . (-2 11 1 COS (131 (40)n1)( 3, lI 13) +ll- 113 (C ) (C 115 (i IT) Siln a 3

i-t.~3 .. 1.--. - - - -

By the same procedure, we find from (37)

'DB nin2n3 [2112 (11 - n2) + 11,1 (n2 - 112 + I ]

cos a3 2 2cosa

2 - 1) 3 s i a 3 s( n a,

112) cos a2 ]

+113 3i) a2 (41)

from which- the mass -ratio for symmetrical vibrations, RB , is2 2 ' 2 2 2 2 2.)

1RB = nln2n3 [nn3 (,, - ns) +-11'fn2 (111,- 11) +-nln(nl -1 )

,Cos a I cos a2

c ( - 9n c 1") (C- 11) + "I n3 (" -n 0-si- a2

, cos a3 ' (42). n (2_ 1) (C _n) J_

in general, both DA and, DB will not vanish for the same nk , C

and R values since expressions (40) and (42) are-not the same. Nc'W

referring to equations (12)., (40), and (42)-, it is apparent that for

natural vibrations with an added mass-four solutions are possible; one

consisting of symmetrical vibrations and another of antisymmetrical

vibrations for each of the two classes, flexural and extensional. An

illustration of this is provided in the next section where fiumericl

solutions are obtained.

The frequency equation for the case of -a uniform cylinder or ring

may be obtained by letting the mass ratio become zero in (40) and (42).

From (40), multiplying through by sin -a sin- a2 sin a3

( 112) (C - n) ( i - 11) + n1 (C )(C- 2 n+ .1 ,i - 11)( -2 (11 -11) ' (43)

+ nj,,(g - L)(C -- 2n) (n ]-n) sin a I sin a2 sin a3 = 0

and -from -(42), again. multiplying by -sin a, sin a2 sin a3 V2n 2(112-,122 2-

13 2 [ - ,33 ) + ,p( -P2

+ i .2-2 (112'-. 2 sin a, sin a2 sin a3= 0- (44)

T3 1 +5 3

.... +'5

In this case, with 0M=i , if one of the il, is an integer,

then (413) or (44) will be satisfied. Actually, for this situation, both

(03) and (44) are identicaily zero, which is condition (38)-. Then

equation (12) suffices as the frequency equation, with one of the "k an

integer.

i L. No rmal Mode Shapes

Tie antisymmetrical branch mode constants Ak may be found by

solving -(24) and (25) simultaneously; the solution is

A, (112 - 1) (C - (45)3 - (45)A,~~ ~ "1211 ) )

I

A 2 ( 3 - ) (C - 115 sin (1,

and

A3 _ (n2 - n) (C - 2)sin a(46)

A, (12 - 112) (C - 112) sill a3 (6

and the symmetrical constants 13k from (27) and (28) are

12i~ n - 112) sill aL (7." =10 (47)

1 i l 112(n -12) sin a,)

and

B3 2 (? - n11) sill (48)

13B 3 - 11 ) sil a3

where A and B are arbitrary. From (10) and (11) the tangential and

radial displacement functions for the antisymmetrical bianch are

V=A I lCOSl0+ n--l'COSn20+ ll3 o 3 (49), ,Aoi1 X02o ~ A3 113 cos 13ol

and

A( A 91A 3 0l

\ A C- ) sin l + ii (C A) sill 112 0 + (C - ni)sii (50)L

16j

and for the symmetrical branch are

>'V B B I I sill n i0 + l 1.,bi 1120 + -3 si n 0(5 1i)= ~ sin B-~ B3 113 sill 1130](1

[I 01 ~i72 "'2 1 01

and

i2 [02 13' 2 Cl 3 .I -C) coS 11 + =(")- C) Los"20+ T1 (I,+ C) os1 3 (52)

With n, an integer, then (145) through (48) are zero and the mode

shapes without the added mass are for antisymmetrical vibrations

V =A1n1 cos 0 (53)

and

W= A, (C - h1)Sill n10 (54)

and for symmetrical vibrations

V= Bi ll Si lO (55)

and

W B1 (n2 - C) cos nO (56)

In modal analysis problems where the orthogonality condition must

be used to obtain a solution, it is convenient to normalize the mode

shapes. The orthogonality condition between the modes, from Love -[8,1,

can be written as IzI =((J2 - ) IT (ViV. + WiX)a'dO + NI (,ViV + WiWj)= = 0 (57)

where i and j are indices representing two solution states, or:normal

modes, with corresponding frequencies wi and j . f j

then the brackdted-quantity of (57) must vanish, and if wi=1j , (57)

17

----------------------- - - -

is identically satisfied. In the 1-atter situation, the value of the

bracketed quenitity is then arbitrary. However, it is customary to

normalize this term by making its value some convenient quantity. For

the case of i j, the bracketed quantity of (57) can be taken as

m [+V adO,+M N[+W?7 =21nna (58)

Dividing (58) by 2nl , we have

2+WJ1 [~~ I+ R 2V + W2 -6(9)

With equations -(-49-)- and (50) ,written as

V =A Vi aiid i,= A i,iW!i (60)

for the ith mode, and substituting (60) into (59) and solving for Ali ,

we have

2 . (61)

f'Via+W i 2d0+R[ia + w ia ]

Similarly, for 'lli we obtain

12 -- (62)

.- ITV is iJ dO + R V is + wiT

where the indices a and s represent the antisymmetrical and symmetrical

branches respectively.

Evaluating the integrals of (61) and (62) is a straightforwjard

but somewhat lengthy process. The resulting expressions are as follows:

A2i_ (63) "

-- 1 Xim + R !2+ Wi

-IV IIIr 4n

Ll

3i

Where tile Xi arce

= Ca-n2 si 2n I jal (64

x1 A2\~C si ~l2ia171 (65),

Xi2~ ~ 2'_ 2ia - J21,i

2 - sin 23a7(6

Xi 3 : = iA, a - 3ia){ 2 1 (66

14=- h 2a -n- I1 i fl 1j~ 20 17 sillkl i3 a + n2ia) Ii(67)

(C3 11n~I ) C 11 L 7- - I t I s ia1 + ,ja

Xi 2 A ia - i ni Cia - a ", I lia - nia nl 1~a 1 -6)

IV3 2 2 rsin (n ia - n3 a) 1 si n ia + n3ia) -(68)

"' - n ) L '1 ia - 3ia) "i + 3ia) -' 69

2 'in (nna r in n1a 69

(0 2)(21 - - -

' sin 21W (71)

J A3Y 2 sin+ 2"13ia-9 (2

Xi A 1 1/iaa L nha '2 a+ "2ja j,

/A )sin (i t ~ n;j 1 7)a ?T sin (111 a + 3ia~ (73)

\Ao 2 1 11 ia L ha"ia + "1 ia + It') a j

, 3 Fe" ("2l1ia i) + sll n (111 ia + 13 3) 17 (74)

Xil 1 2 A nli) "ia fliV a- "3ia -If + ia + 3ia j

sin. -12i

J ~)"

FS'l ("ia--n~id 7r(75xi 2 LA24

111 11 iI (76)-~nd 1V =2 + W.,~

21T- yul+11I1

(12ir 2 ~ + (77)

Y i I I ]C s [

Yil (niCs)Lis+ _2 is (78

- /B 3Yj 2 , C r sin 2n131i 9- "(nC 1) - + 2n3 79)

/B,\ FSi (i 2s) nT Sill (11 is + '12is)\j4 2Ij( 1 5 N i- c,5)W(-i - cis) I - +. nI 1'i (80)

~ ISL ~is 2is"us+ "is

11 2 2 - sill ("I~ is - 1 3 is)T Sill 04I is + 113 is) (8 -

li s - c ' is is- is)L "I is - "3is 111 is + 113is

Yi6~~~~~ 2 3 3 12 C.) (112 is CIS sil Ois - "3is) 7r + Sil("'-)s+ ± n3i) Ir(1~ . lis "3is I (82)

\~i ''J L 1 is ~ 2s + "3isJ

' I 2n , is' -"17 111iI - 1 is ](83)

s 12

(L2 [ i j2 (84)

fi 131 1 i s i

= I7 13i 2 Il (85)

/( 1,\ Sil II "is,- n2,) 5) IT sill (II is + 1121s)

iso 2 13i1 is" is nl Iis~ 3s

= \I'1/ 'I i -i i ",-)s is s + 0 is j(7

Yi 1,\ Ii 2"-j 3 i Si 111 is - "3i ) ' sn( II ("is + -ji (87)2 '2is "A3is -"is - "3 I ~is+ 3is

Ir II111,~ iNSil O"I - 13i) jl 02is+ 11is)(88

yi 12 2 "2i"3is n'i

F. Numerical Solutions

To illustrate the influence of the added mass, frequencies and mode

shapes were calculated for several of the lower modes by the following

prozedure:

1. Influence of Mass on Frequencies

a. After selecting a value of the stiffness parameter Z,

values of nI were chosen and (12) was solved for the flexural frequency

parameters Cf and the extensional frequency parameters C . Then forf e

each frequency parameter, Cf and Ce, (12) was solved again for the modal

parameters n2 f and n f for the flexural class, and n2 and n3 e for the

extensional class. Integer values of n1 will give solutions for the

problem without the added mass.

b. The nk and C values were then inserted into equations (40)kI

and (42) to determine values of, the mass ratio, R. Note that for each

n chosen, four values of R will be obtained, Rfal Rfs, Rea and Res

where f corresponds to flexure, e to extension, a to antisymmetric and

s to symmetric.

c. Frequency ratios of the structure with added mass to that

without added mass were obtained from equation (13):

WM 1 C) / 2

. .. . (89)

For several individual lower modes, the above parameters were

calculated and are given in Tables 1 and 2. The relationship between

frequency and mass is also shown graphically in Figures 3 through 6.

21

TABLE 1- NATURAL FREQUENCIES, EXTENSIONAL CLASS, Z 0.0001

n "--"i R R12 "3 ea es

ZEROTHI MODE

0.0 1.000100 9.924777i 10.074785 1.0 0.0 0.00.093543i 0.991352 9.902667i 10.053005 0.996 -- 0.100.096365i 0.990817 9.901308i 10.051667 0.995 -- 1.00.096652i 0.990761 9.901168i 10.051528 0.995 - 1 10.0

FIRST MODE

1.0 2.0 11.828873i 11.955009 1.0 0.0 0.00.920444 1.847218 11.593651i 11.722319 0.962 0.086 0.100.909462 1.827-122 11.561633i 11.690650 0.956 0.10 0.1780.895314 1.801590 11.520569i 11.650043 0,.950 0.118 1.00.892210 1.796041 11.511586i 11.461161 0.948 0.123 10.00.685508 1.469940 10.942573i 11.078805 0.858 1.0 --

0.668459 1.446858 10.898814i 11.035586 0,852 10.0 --

SECOND MODE

2.0 5.000180 14.903623i 15.003939 1.0 0.0 0.01.923135 4.698604 14.672060i 14.773947 0.970 0,04 0.101. 9050641 4.629420 14.617369i 14.719634 Q.962 0.051 1.0

1.902883 4.621112 14.6107601 14.713071 0.961 0.052 10.0

1.832289 4.357412 14.3961911 14.500016 0.934 0.10 --

1.673354- 3.800200 13,908495i 14.015931 0.872 1.0 --

1.671937 3.795457 13.904119i 14.011589- 0.871 10.0 --

22

-Y

TABLE 2 - NATURAL FREQUENCIES, ILEXURAL CLASS, Z 0.0001

n C n 2 n3 R fa RfsWill!

SECOND MODE

(n abi, n a-bi)2 3

a b

2.0 0.000720 0.4132 I 1.081867 1.0 0.0 0.0

1.988584 0.000697 0.418348 1.073266 0.984 0.10 0,021

1.956736 0.000634 0.431774 1.049056 0.939 1.0 0.089

1.952292 0.000626 0.433560 1.045649 0.933 1.43 0.10

1.939996 0.000603 0.438395 1.036185 0.915 10.0 0,132

1.798911 0.000382 0.483953 0.923071 0.728 -- 1.0

1.705399 0.000271 0.505549 0.842400 0.614 -- 10.0

THIRD MODE

3.0 0.005760 1.036841i 2.432945i 1.0 0.0 0.0

2.989798 0,005668 1.037634i 2.420029i 0.9921 0.10 0.009,1

2.963754 0.005439 1.039746i 2.386907i 0.972 1.0 0.035

2.951385, 0.005333 1.040797i 2.371093i 0.962 10.0 0.048

2.911696 0.005002 1.044397i 2.319959i 0.932 -- 0.10

2.737457 0.003720 1.065871i 2.086593i 0.804 -- 1.0

2.680194 0.003356 i.075971i 2.005585i 0.763 -- 10.0

23

L)O

ena -icco

-i-

L)I14 W 0

C) L4

Cl j LC) C3 -C3

oM 0 0 . -

0C0 LO p 0D

ClC)

0

4 -)

00

0

.0,

C))

2.1.

LO

4-

(1) C)

LLI V)C)

(n0

'-4 I-4

0C) 0

LO -

o

C))

-ja

< 00

I-

F- - -4LiX LiiL.

In xI

w C)

co ci -

to 0

4--- 4-J 11 -0

4

44

(10

0

II

'-4

-D 4.)C

262

-LJ

LI-

I-IZ 0

'-4-

w --)

VU)

U " m

*44

L/)C 0~>.

CI

-4

4

- n >

0%

00

C) M- 00 k p

*a a

27

A:s noted on the figures, the stiffness parameter Z was taken to be

This -articular value of Z represents a ring of unit width, 12

inches in diameter, with a thickness of 0.21 inches. Calculations using

,thofr vilues of ranging from 10- 2 to 10- 6 showed less than 10% diff6r-

*-nco. with those represented by the figures, with the largest differences

occurrin- in th, extensional class modes and virtually no difference in

-he klexural class particularly for smaller Z values.

..n~'... of :lass on Mode Shapes

Equations (4 9) through (52) were used to determine the normal-

imed node shapes for desired values of the mass ratio, R. The parame-

ters, n k and C, for particular values of R were -found by iteration on

the process described previously in 1.-a. and l.b.

The normalized displacement function W is shown in Figures 7

and 8 for 0 between 0 and 180 degrees.

G. Comparison with Exnrifiment

Lee, et al, [8] conducted an experim mtaX study. to determine the

effects of added mass on the vibrations of.ring structuzres. By reson-

rnce testing, they found flexural 'symmetric natuiral frequencies- and

node zhapes for several values of the mass ratio, R. Their frequency

results are shown on Figures 5 and 6 for comparison with present theory.

The exoerimental results show the same general trend as theory for

increasine F, but the experimental frequency ratio is, on the average,

-3bout 10; hidher than theory. For a variety of reasons, precise agree-

ment between theory and experiment generally cannot be expected,

os-ecialy fror vibratory and transient phenomena. In this case the

.-ncival cause of disagreement can be att-ributed to the fact tho.t in

O=

ECOND MODE, SYMMETRICAL BRANCH

-R=1

Z = 0.0000

29 j0

0____ rI7

R=1

R=1

+1

0=1

w

0

+1

THIRD MODE, ANTISYMMETRICAL BRANCH-1=

0 0 60 90 120 150 1800

Figure 8 -Normalized M1ode Shapes, Flexural Class,7, 0.0001

I

the conduct of the experiments the ring was driven eccentrically with-

respect to the attached mass, thus permitting some participation of the

antisymmetric branches. As shown by present theory, this would account

for somewhat higher frequency ratios than those that would result from

pure symmetric excitation. Experimental mode shapes are shown in

Figures 9 and 10 and show the same general trend of reduced amplitude

at the mass for increasing R as the theory.

.i

31

.........

.10'0

P 0 R m0. 82

R 1.58 R 3.52

Figure 9 - Experimental ['81 Mode Shapes, Second Mode,Flexural Class, Symmtrical Branch, Z 2.6 x 10-6

A

R 0R 0. 82

R =1. 58 R =3.52

Figure 10 - Experimental [8] Mode Shapes, Third Mode,Flexural Class, Symmetrical Branch, Z =2.6 x 10-6

33

II. Disecsjlon

The infIuence o± the added na-_, on the natural vibrations will be

discussed in terms of the solution without added mass.

1. Influence of 'lass on Frequencies

An obvious feature here is the difference in frequency between

the symmetrical and antisymmetrical brancnes of a normal mode, as shown

in rigures 3 through 6. This holds for both classes of vibration,

extensional and flexural. For no added mass, R = 0, both branches of a

particular class have the same frequency, but as the added mass increas-

as, the difference in frequency also increases. Note that the zeroth,

or breathing, mode has only the symmetrical branch. This is an expected

result since for the case of zero added mass, only the symmetrical

branch exists, as shown by equations (51) through (56).

For the extensional class modes, as the mass ratio increases,

the reduction in frequency is greater in the antisymmetrical branch

than in the symmetric-il branch. The reverse is true for the flexural

class.

With the exception of the zeroth mode, it is noted that for

either class, the reduction in frequency is less for both branches for

the higher modes than for the lower. The zeroth mode shows very little

reduction in frequency regardless of the amount of added mass.

2. Influence of :tass on Mode Zhapes

Fiegures 7 and 8 indicate that the displacement of the added

mass decreasc; as the mass ratio increases. A l-arge added mass tends

to become a node. In the flexural class, the mass has appreciably less

influence on the antisymmetrical shapes than on the symmetrical, as

shown by Figure 8. The effect of added mass on the extensional class

mode shapes is striking. Figure 7 shows the presence of coupling with

higher flexural class modes. Even for a very small value of the mass

ratio (R = 0.01) this coupling is quite severe, particularly for the

symmetrical branches. Observe also that even with this coupling the

true character of the extensional shapes is maintained since the flexure

is superimposed on the primary extension. This behavior is especially

noticeable in the zeroth mode shape shown in Figure 7.

For the uniform cylinder without added mass, it is interesting to

note from equations (54) and (56) that the only distinction in shape

between the symmetrical and anti-symmetrical branches of a particular

mode is that the axis of symmetry of the anti-symmetrical branch shape

is rotated radians from the axis of symmetrj of the symmetrical211

branch shape-. As shown in Tables 1 and 2, for R = 0, both branches

of the same class of vibration have the same natural frequency. Such

degeneracy does not exist when the added mass is present. Not only are

the anti-symmetrical branch shapes completely without symmetry and

different from those of the symmetrical branch, but the frequencies

associated with each branch are different as well,

The zeroth and first flexural class modes are non-oscillatory,

rigid body modes. They will be treated in the next Chapter.

35

IJ

IV. IM4PUlor P£lf ONS

The Zeneral impulse response solution will be obtained by modal

analysi. for arbitrary Cpatial variation of the initial conditions.

This reneral solition will then be specialized to a particular form of

the initial conditions for comparison with experimental results. The

impulse response of a uniform cylinder (R = 0) will be obtained as a

degenerate case of the general impulse response solution.

A. General Solution

As was shown in Chapter I1, for each mode of vibration (zeroth,

first, etc.), four solutions are possible, symmetric and antisymmetric

branches for each of the two classes, flexural and extensional. Since

each of the four is a solution and since the problem is linear, then a

linear combination of the four solutions is also a solution. There-

core, the displacement -.omponents for the i th mode can be written as

i = Veli + V es i + vlhi +* Vfs i (90)

andw t = W + W + W 't - Wfs iCal Csi (91)

It is convenient to express the spatial and temporal variables of

each of the components of equations (90) and (91) as follows

Vea lV (c1 si=V E I t T di cos Weait) (92)

Ves t Ves t (c. sill Wesit + di 0;co esit) (93)

% V1,1 01 Sit L..,fait + b t co ) (94)

V't -i sin +Ifsit + bi cos wl~sit) (95)

U

Veali =Wea ( i sill wcait + di cos weait) (96)

Wes i Wes i (ci Sit) Wesit + di Cos W fesit) (97)

Wfai =Wfai ('ai sinl Owfit + b cos wfait) (98)

Wfsi (ai Sill wl-si t + bi cos fsit) (99)

where Veai, Wea , etc., are the mode shapes obtained previously, andealeai.

ai, ai, bi, etc., are constants to be obtained from the initial

conditions. Summing all the modes gives the compi;te solution which is

v (100)i

and ow =twi (101)

i=0

For given initial conditions on the displacement and velocity, the

constants ai, a, bi, etc., can be obtained through the orthogonality

relation. Let the initial conditions be stated as

v(OO) =V

w(O,0) = W(102)

,(o,o) = T

v(Oo) = w

Then, from equations (1100) and (101), with (102) we have-

L7

C0

ZVsO0oj (103)I-"

00

wi(O.0) w (104)i-0

00,01O =v (105)

is0

00

and Zvi(O.0) %v (,06)i'-0

The object now is to manipulate equations (103) through (106) to

forms such that the orthogonality condition, equat-ion -(57-), can be

employed. To obtain ai and bi, for example, multiply (103) and (105)

by VfmadO , and (104) and (106) by Wnsi1adO and integrate,

which gives

vi(O.O) V-sitiadO = vVfsi'I(IO (107)

vi(O.0) VfsmadO J VIimlO (108)

wi(O ,0) WfsjmadO W \VfjnmdO (109)

%('i(O.0) WfsjInadO = %.InadO (110)

For Or , the location of the added mass, equations (103)

through (106) become

Vi(r,0) V(7r)(I)i0O

WiV7T,U (7) (112)

i0O

V(70 (113)

i=O

VPTA WO (11)

Multiplying equations (111) and (113) by %IV rj and -(112) and

(114) by Mw(') gives

Z Wi(10) V\~ = MV(r) Vr} 1151=0

=w(TO NIV(ir) W T (1 17)

(IT) - OiT)

E 41 %(r.O) Wf-j = MW(r) wfsj(18

Adding equations (107), (109), (115) and (117) results in

~ {~f [vj(O ,0) Vfj+ wv1(O,O) W midO + MI[~0 1()Wj

-I J V [.vj + w madO +i Mi [;VV + WW j] (19

Using equations ('90) through (09), and the orthogonality condition,

equation (b 7 ), it tollows that all the terms on the left hand side of

equation (119) are zero except those in which i j and are of the

flexural class, symmetrical branch. This latter condition is an import-

ant point since even for the same mode number of vibration (i = j),

there are four solution states, each in general having a different

natural frequency and normal mode shape. Thus equation (119) yields

the constants b.,

f1 L\rI'Si + \-\js mad f- Mf WVI~sj + WVVIsj]i .. (120)

" I s+\\Ifsi adO , M V2i + w i2

Using normalized functions as given by equation (58), equation (120)

may be written as

b I IVfsi+\TNVsi]dO + + (121)

Adding equations (108), (110), (116) and (118), and following the

same procedure, the constants ai ave found to be

2rr [V~Vs, +tw i ] (10 + R [tVfsi + Vfsi]W ... I (122)

Through a procedure similar to the above, the other constants in

equations (92) through (98) are found to be

tv +W i d + R V +W2r, I [CS1 PSI Lsi + Cs]

"I -- (123)

40

d. = V e + WWs (10 + R VVesi + wMcsi (124)

~ ir~j~

_N l['V dai + RW:i] df +R ['V fail

i 7?(125)

b i~I IVfaiWfaiI dO + R -VVfai + -VfaiI (126)

27? Vveai + WeeaiJ d + R Lveai + Weaij7i= (127)

f7r -Vi + w-ea dO + R rVVca + -Weai l (128)

Equations (123), (124), (127), and (128) associated with the

extensional class are valid for any mode number, i, equal to or greater

than zero. Equations (121), (122), (125) and (126) associated with the

flexural class are valid only for mode numbers equal to or greater

than two. As shown previously in Chapter III, harmonic solutions are

not possible for the zeroth and first modes of the flexural class.

Therefore these must be treated separately. Previous analytical studies

[10,19] of the flexural response of uniform rings have shown that the

zeroth and first flexural clas,; modes are associated with rigid body

motions. The- solution for the zeroth mode can be taken as follows

vf0 f0 + liot and wf0 go + k0 t (129)

41

Substituting (129) into the ,oveming equations (6) and (7), it is

found that o k 0 and f0 and h0 are arbitrary, which represents a

rigid body rotation. For the first mode, the solution is taken as

%I, (it hlt) an'0 1 1 ' (gl +klt0CoO (130)

tipon -;ubstituting (13J) into equations (6) and (7), it is found

that the Iovernini equationt*, Are satisfied by equations (130) if

t .' and hN = . Note that this solution represents a rigid body

translation. The non-harmonic or zero frequency solutions for the

zeroth and first flexural class modes given by equations (129) and (130)

also satisfy the equilibrium and continuity conditions, equations (15)

through (20), thus are valid solutions to the given problem. The

orthogonality relation, equation (57), also is apDlicable for these

solutions.

Froceedini; in the 3arie manner as before, the constants fo' ho, f0 0 1

and h are found to be

"dO + RI )rl

Ho 1 4 R (131)

± JVdO + Zv(-ir)

ho I(132)+ R

[Vsill 0 -f 7 Cos 0] (10 - m~

I . R (133)

II 0 + Cos 0] (10 - R (r)

(134)1 *1- R

42

Thus, with all the constants determined from equations (121)

through (128) and -(131) and (134), the complete general solution is

v fo+ hot + (F1 h1 sin 0 +,v 1 +Evei (135)i-2 i.O

and

w (r'1 11 t) cos 0 +Ew,+Ewci (136)i.2 i-O

where vn = Vfai + Vfsi

Vei = Vea i + Ves i

wi= -a I+ f'si

and Wei =Wcai+ Wesi

The impulse response solution for a uniform cylinder without added

mass has the same form as above and may be derived from the above gen-

eral solution by setting R = 0 and using the mode shapes given by

equations (53) through (56).

B. Solution for-Specified Initial Conditions

For later comparison with experimental results, the impulse

response solution for- a particular set of initial- conditions will now

be obtained. Let the initial conditions be of the following form:

V=v=w=O , forallO

w 0 WOCos 0 for 7 > 0 >--(137)

-and for- -!>O>-IT2

w= 0 for--! <

43-

A

who ve is .i s~pecified amplitude =ostant.

AJth die abive initial conditions, the mode constants are detern-

inpi vvir ejuwti r.' (121) through (128) anJ (1-31) through (134) and are

jIq I )

i ti t 1 1 8

III' H, C F A.=1

-~~~ ~~ b- - 1 d1 -U 1

1itvj Is It 2~> B 2tj3

1L B A'I, 14

(nBd 3_n: . 1 7 (140)psij, )7" *

where

sil ( l(- kfsi)n sil (I + Nks r sm(I - nkfsi) g sill (I + Ilkfsi)P + 2( 2( "Fki 2 ( - "krsi) + +i) -2(nks) - 2(i+ n)

sil ( - Nkest) sin (I + 1kei) 7r sill-( - "kesi) sin (I + nkesi) n

i 2 ( - Yk{esi i + "kesi) 2 (I - kesi)

k= 1.2,3

Wfs i and W . are obtained from equation (52). The constants BI

and Biesi are found from equation (62), and Blfs Blesi'~f I ei1

I esiB3fsi B3esiand -- are given by equations (47) and (48). TheBlI si Blesi

natural frequency, coi , is found from equation ('13), and is

a IIn

The complete solution is

00 00

V= li t sin 0 + E aiVfsi sin t + E CiVesi Sin W ~sit (141)i=2 i=O

oo 6o

SlIlt cos0 + E aWiVfsin )fsit + C sin ei t (142)i=O

In the above treatment, it is emphasized that the mode shapes V. and W.

are normalized in accordance with equation (59).

For the case of a uniform cylinder, R = 0, the form of solution

given by equations (141) and (142) remains the same; however, the

expressions for the constants hl, ai and ci are simplified. Equation

3

415

(1.3) becomes

.1 (-143)

equation (139) becomes

0 1I hnl .2(14

and equation (1410) becomes

_ i- t( - Ces n) Iii i t 0 (145)

Note that the mode number now is the same as the integer n.

Other response parameters, such as velocity and stress, can be

obtained by appropriate temporal and spatial differentiation of the

displacements Piven by equations (141) and (142-).

C. Comparison with Experimental Results

Several specific numerical solutions will now be obtained for

comparison with available experimental results.

1. Cylinder with Added Mass

The cylinder is impulsively loaded by a shockwave from a near-

by underwater explosion. The pertinent parameters for this problem are:

E = 29 x 106 (psi)

, 0.3

a 8.65 (ft)

2lb -secin : 0.0502 ( -n22

in

Z 0.00101

R 0.18

16

It is desired to compare with Navy experimental radial velocity

measurements recorded at the location of the added mass, 0 z 77

Differentiating equation (21.2) with respect to time, we obtain the

following expression for the non-dimensional radial velocity at 0 I

O0 0

w- I; Co ),I + (14

where\\ I'i Ils

w0

The amplitude constants a! and c' are listed in Table 3 through1 .

eight modes (i = 8). A measure of convergence is indicated by thevalues shown in Table 3 since both a' and c' become very small for the

higher modes. In fact, these amplitude constants are insignificant for

i > 4. Figure 11 shows the solution, equation (146), summed through

eight modes. Shown also in Figure 11 is the experimental velocity

history from reference [20]. The high frequency oscillations on the

experimental histories are due primarily to the response of the velocity

meter -elements. The fundamental natural frequencies of the -elements

were measured and found to correspond with the superimposed high-

frequency oscillations of the experimental records. This is discussed

further in reference [21], together with other details of the Navy

experimental program.

47

- TABLE' 3 -VELOCITY -MODE- -AMPLITUDES, 0i ,Z z0.00194

R 0 -R a 0*18

i ~n a' 4,1

0 0 0-031:83' -0 0 -0-42112-

1 0'.25 0.-25 0,3644 0.*0917

2 0.1696 0,0424 0.2534; 0.0067

3 '- 0 -0.0829 -0*0004

&4- 0.0398 -0.002t 0$.0652 .;0*0013,

5 0 0 0 .0008,

6 0,.0176 00-0006 -0.0289 -6.0005

7 ;0 .0 -0,0064 10

8 -0.0099 -0.0002 -0.0084

20INWARD

A

1.5 +'I

I I j~ .EXPERIMENT [20]

-~~~ I tR0.18: 1 .0 1 1

I D till ~ "'if A THEORY

I~ ~ I '

0 12 3 -4 5

INWARD1.5-

AEXPERIMENT,[,15]

IIIA

0. 21 A

TIME(MSEC)

Figure 11 - Theoretical and Experimental Velocity Ifistori~s,0 =iZ =.0.00194

49

2. Cylinder.without Added.Mass

The parameters of the previous problem apply here, except that

R 0. The solution in his case is

ULO 00(n~ + E a;, cos wfsn, icgast (147)"-- -4 , ,ta .c ' c e n

__ _ I =2 11=0

where,at 2 (-1I)T (12 - fsn 2

2() C~~ - s12-12an (1 - i12) 1(I',- Crsn)" + in21 i=24,6

0 , 3 = 5,7,---

I

2 (-les) 2,='

4, 6. - -,-en = 112) [ C'2 n)2 +, 6

S(17 l(,I 2 C n IS

e' - O ,- 11= 3, 5, 7, -.-

The above anIplitude constants -are listed in Table, 3 for eight modes-

(n = 8). Both of the above expressions for a' and 'c' rapidly become

very small for large n as is shown by the values in Table '3. The

velocity history computed from-equation (147) summed through eight modes

is shown in, Figure 11, together with the experimental velocity history

reported in [15] for the case of R = 0.

Both of the theoretical velocity histories showh in Figure ll ,are

in reasonable agreement with the p rimary characteristic& of the ixperi-

mental histories. The influence of the added mass can be seen by

comparing the upper and' lower histories of Figure 11. For the case of

50-

.I

added mass, tht- velocity generally is at a-higher level and does not

oscillate to the-degree observed for the case of the uniform cylinder

without- added mass. This can be attributed to increased amplitude in

the rigid -body transl-ation mode when the added mass is present and

reduced participation of the ,extensional class oscillatory modes, as

shown by the mode amplitude cdnstants given in Table 3.

The theoretical solutions shown in Figure 11 were obtained-by

summation through the eighth- mode Ci, n -8). Solutions were also

-obtained ,by suifimation through the sixth ,mode and the difference between

the sixth and eighth mode solutions were found to be on t' order of

one- percenti

3. Ring without Added Mass

The .theory will now be compared-with experimentai data [16]

obtained from strain gages on ,he iiside surface of a 6061 aluminum,

7 incfi outside diameter, 178 inch thick, one inch wide ring impulsively

loaded by a magnetidally-driven flyer plate., The pertinent parameters

are:

10 x i0 (psi)

a 3.- 4375 (inches-),2

m= 3.1703x - ( secin

Z 0.=0011

R 0

Substituting -equations (141) and (142), with appropriate spatial

-differentiations, into--equations (22) and (23), the stress resultants

are obtained,

51

7?

. I- A,( I + Z cos 0 sin WeslIT - Sill W

( cos I (48)+ '121, + IZ(l2 -,I)-I (148)iE,. (I- ) •

G = O0 -AZ C + 7 2 .eds T l0s110 (149)ITC /' O.) , f ,- ,6 7"

where(n- Cfs) Sill 'fsnt (112 - Cesn) sin WIn ~ ~ ~ ~ ~ s - es""9n4.

Wf1 s (n2" - Cfsn) 2-+ 2 ] esn[(n,2 - Cesr)2 + n2l

and~ , 2 1 .2 'C 2

1and Cf0 l12 - Cs-s Wfsnt .- ( .c( 2 Csill) 2 esnt -r,2 + , C. , ,W.fs nl [(i- _ Cfsnl)2- , 2 1 Jcsnl [(| Csn1)" -I

To, obtain :non-dimensional stresses on the inside surface of the

ring, the following expressions are employed:

UTIh T

IC ptOAc

0Gh_ 6Gb oh 0hAc

where aT is the stress due to thrust

o is the stress due to bending

It is; the ring thickness

c is the material sound velocity,

is the maximum impulse amplitude, 10,=hp1i 0

52

- -- -

Thus, the total stress in non-dimensional form at any position

on the inside surface of the ring is given by

Soh a~h oc, hIO= I+ (150)

. loc1loc 10 c

It is also desired to use non-dimensional time as follows:

Cta

Equation (150), summed through eight modes, is shown in -graphical.

fo~m on Figure 12 f6i 'two spatial positions, 0 = 0 and 0 = -2

Shown -also in' Figure 12 -are- the -experimental results from [16]. The;e

is good agreement between theory and experiment, pay.;ticularly at the

0 =0 position. As an added observation, the initial portion of the

stress history at the 0- 0 position indicates the effect of

"destructiVe interferenc&" of the modes which can be construed as an

index of solution convergence. Note that there is no significant stress

,until after T - , which, based on- elementary wave propagation

theory, is the time required for a disturbance to travel 1/4 of the

circumference of the ring.

I

53

-

MIrS ION0.6

0.4 -THEORY

0.2 - \ V.-EXPERIMENT

0

-0.4

-0.610 2 4 6t 8 10 12

o=0, INSIDE RING

0.4

0.2 -TER

ch _

-0.2

\% /~-- EXPERI MENT \ I

0. 2 4 ct 6 8 10 12

o .~ INSIDE RING

F-igurc 12 m Teoretical and Experimental [16] Stress Histories,11 O 0, Z 0.00011

54

D. DiscUs6sion

In the above analytical solutions the loading given by equations

(137) is assumed to occur insiantaneously over half the circumference.

Such conditions are never physically attainable in experiments since

the loading always occurs over a finite period of time. However, if

the duration of load application is small in comparison with the period

of the lowest natural frequency of the system, then an instantaneous or

impulse type load condition is usually considered to be a valid

approximation to the physical problem. In Case 3 above, where a flyer

plate impacts a ring in air, a assumed instantaneous loading is valid

as evidenced by the agreement between theory and experiment shown in

Figure 12. For the Navy experiments, Cases 1 and 2, an impulse type

loading seemingly is more of an approximation to the actual conditions

and further explanation is necessary. A strong, rapidly decaying shock-

wave interacting with a structure submerged in P fluid medium is an

extremely complex non-linear problem in the complete sens . 'The

principal cause of non-linearity is the occurrence of cavitation or

separation between the structure and surrounding fluid[ after the shock-

wave arrives at the structure. Cavitation occurs because 6f the

inability of the fluid to withstand any appreciable negative pressure

resulting from the radiated rarefaction wave. It is largely the

presence of cavitation, however, that permits the application of an

impulse type solution to describe the response of the cylinder for

early times. From experiments [21], it has been observed that the

first peak velocity, w0 0 occurs very early (on the order of 1/2 msec)

after shockwave arrival and can be predicted solely from the shockwave

1 55

niarneter- and ,ti ell mass propemlez wittout regar4 to shell stltfnts

characteristics. During this early time period after shockwave arrival,

the first disturbance in the shell has propagated along the circumference

in both directions about the initial contact point a distance nearly

equal to one-quarter of the circumference of the cylinder. By the time

the peak velocity w is reached at the first contact point, a spatial

distribution of velocity is present over about one-half the circum-

ference of the shell. For analytical convenience, it is assumed that

this initial velocity is spatially distributed in the manner described

by equations (137). It is noted that the time required to develop this

spatial velocity distribution is small in- compcrison- with the periods

of the lowest natural frequencies of either the extensional or flexural

classes of vibration. An impulse respopse solution for this pazticular

problem involves another approximation, which is that the influence of

the shockwave in the surrounding water is not accounted for except in

the determination of . he effect of the shockwave as it engulfs

the cylinder is considerably reduced because the faster moving dis-

turbance in the shell creates a precursor or a radiated wave in the

water ahead of the free-water shockwave. Thus the water is disturbed

ahead of the propagating shockwave which lessens the shockwave

influence on the cylinder. Two other factors which tend to reduce the

effect of the shockwave are the natural decay in amplitude of the wave

front as the distance from the source is increased, and the changing

angle of incidence of the wave front with the cylinder. On the far

side of the cylinder, or shadow zone, the wave is considerably diffused

due to diffraction over the cylinder as an obstacle, and due to the

56

effect of the radiated wave as described above. From experimental [21]

pressure measuremens taken near the fluid-cylinder interface, cavita.-

tion has been observed to begin at the time 0 is reached and

persist for a duration of several milliseconds. During this early

period of time, the cylinder behaves as though it were decoupled from

the surrounding fluid medium. It is this decoupling that permits the

early response features of the cylinder to be predicted reasonably well

(see Figure 11) by the theory described herein. In other situations

where little or no cavitation occurs the present theory would be less

applicable, if at all.

07

V. DISCUSSION

Technical aspects of the natural vibration and impulse response

solutions have been discussed as appropriate in previous chapters. The

discussion in this chapter will pertain to implications of the theoreti-

cal results with respect to state of the art, current applications and

future research.

The solution of a fundamental problem has been obtained herein,

thus the state of the art in the area of vibrations of mass-loaded

structures has been advanced. Heretofore, as noted in the literature

survey, the only solutions available, other than a simplified theoreti-

cal treatment, were those obtained by approximate methods of analysis.

With the' solution for the natural vibrations,a variety of initial

value and forced vibration problems can now be approached in a more

rigorous manner. An application of the theory to an initial value

problem was demonstrated in the previous chapter and judging by the

recent increased interest shown in the literature, there are several

other problem areas in which the present theory would be, directly

applicable.

The effect of imperfections on the vibrations of bodie§, of

revolution is a problem area of basic physical importance. Preliminary

experimental investigations [1,2,3] into the vibrations of imperfect

bodies of revolution have shown that the presence of a small asymmetry

tended to resolve a single natural frequency into two nearby values.

The larger the asymmetry or irregularity, the farther apart and more

distinct the frequencies. Resonance tests have revealed the existence

58

of so-called preferential planes, i.e., if the body was excited in one

of these planes, only one resonance peak occurred, but if excited at

another position around the body, two peaks were observed foi- the same

mode number. For a body of revolution with a mass imperfection, the

results of this .nvestigation tend to support such experimental

phenomena since the symmetrical and anti-symmetrical branches of the

same mode were found to have different natural frequencies. A

systematic combined theoretical and experimental study of such phenomena

would be a contribution as apparently no such study has been reported in

the literature. The governii.g equations and their solution given

herein could be utilized in a theoretical study of the effects of

combined mass and elastic imperfections for the two dimensional

problem.

With regard to the problem of an underwater shock wave interacting

with a cylinder, some further effort is needed to establish the range

of applicability of an impulse type solution as presented herein. If

large non-linear cavitation effects are not present, then linear

acoustic theory, recently extended by, Huang £22), would be more

-appropriate to this problem. To determine their range of validity,

parametric studies should be made withboth theories, impulse response

and acoustic interaction, for comparison With various experimental

results.

Concerning current applications of the theory, perhaps the most

important aspect from an engineering standpoint is that the solution

can serve as a base for comparison with approximate analytical methods,

such as the finite element. Such methods offer a means for obtaining

59

practical solutions of more complex problems. For example, an analyti-

cal study of the vibrations of a cylinder containing several lumped

masses and attached mechanical systems would be intractable except

by an approxmnatg approach. In this connection, it is noted that in an

approximate energy approach to such problems, improved trial functions

would be available since the general character of the solution would

be known from the- present theory.

As the first extension of the theory for future research, it

would be desirable to include rotatory inertia of the added mass. In

principle, this additional feature would offer little analytical

difficulty, and the- solution would be more realistic, particularly- for

anti-symmetrical branch motions. Another future research effort that

should be considered is extension of the theory!to include rotatory,

inertia and shear deformation effects in the shell. Extensioni of the

theory in this manner, whether done approximately or by using higher

order shell equations,- would involve considerable analytical effort;

however, the inclusion of such effects would offer a more accurate

account of the coupling of the extensional and flexural motions as

shown in Figure 7.

To obtain a better physical representation of a large added mass,

the theory should be extended to include the added mass distributed

over a finite length of the circumference, or a finite length rigid

inclusion, rather than concentrated at a point on the circumference.

If considering the problem of a mechanical system attached to the

cylinder, then a more accurate treatment would be to model both mass

and elastic properties of the system, instead of just the mass alone.

60

For each of the problems mentioned above, th;M governing-. equations and

their solution, equations (10) and (11J), would be applicable, although

the eigenvalue -problem would change due to differences in boundazr

conditions.

The theory could perhaps be extended to other more complex

problems, such as a finite cylindrical shell with an attached mechanical

system, to mention one, but an extensive preliminary effort would have

to be made in each case to determine if a rigorous analytical treatment

would be tractable.

VI. SUMMARY

The complete plane strain solution was obtained for the natural

vibrations of a thin circular cylindrical shell containing an added

concentrated line mass. For a particular mode, four distinct solution

states were found to exist-, a symmetrical and anti-symmetrical branch

for each class, flexural and extensional. Significant features

revealed by this investigation were the difference in frequency and

mode shape of each solution state and the presence of coupling between

the flexural and extensional classes, particularly noticeable in the

extensional class mode shapes. The solution for the case of a-uniform

cylinder was obtained from the general solution by taking the added

mass to be zero. For the uniform cylinder, the frequencies of each

branch of the same class were, as expected, the same. The influence of

the added mass was determined by comparing numerical solutions with

and without the mass. It was found that with the added mass the

natural frequency was reduced for each solution state with greater

reduction occurring in the flexural class than in the extensional classi

for a particular mode. The results of an experimental natural vibra-

tion study [8] were found to be in general agreement with the theory.

Using the natural vibration solution, the general impulse

response solution for arbitrary initial conditions was obtained by

normal mode theory. From the general solution, numerical solutions

for specified initial'conditions were obtained and c6mpared-with the

results of several experimental studies [15,-16,20]. In all cases, the

theory was found to agree reasonably well with experimenti The major

influence of the added mass on the impluse response was found to be an

62

increased participation of the rigid body translation and flexural

class modest and decreased participation of the extensional class

oscillatory modes.

In the discussion of the previous chapter, several direct practi-

cal applications of the theory were mentioned. The theory would be

applicable in basic studies of the vibrations of bodies of revolution

containing certain imperfections, in predicting the early response of

submerged cylinders to shock waves in the presence of cavitation, and

as a base for comparison with approximate analytical methods. Dis-

cussed also were areas of future research for extension of the theory

so that it would be physically more meaningful in certain future

applications. In this regard, it was pointed out that an effort

should be made to include rotatory inertia of the added mass and to

consider :rotatory inertia and shear deformation effects in the shell.

It was also pointed out that for some situations, more realistic

models of the attached mass other than as a concentrated line mass

should be considered as topics for future research.

ACKNOWLEDGMENTS

The author acknowledges the staff of the Na~al Ship Research and Development Center,

in particular Dr. 11. M. Schatier-and Mr. R. D. Short, for initial guidance in this reseaich effort.

and Dr. I. Chicurel and-othor facult members of the Engineering Mechanics Departnie,

Virginia Polytechnic Institute, -for their suggestions and guidance in the preparation of this

thesis.

63

REFERENCES

I. Tobias, S. A., "A Theory oflinperfoction for.the Vibrations of Elastic Bodies ofRevolution," Engineering, Vol. 172, No. 4170, pp-409-411 (1951).

2. Fung, Y. C. et al., "On the Vibrations of Thin Cylindrical Shells under Internal

Pressure," Journal of the Aeronautical[Sc-ence-; Vol. 24, No. 9, pp.650-670 (1957).

31. Cohen, D. S., "Vibration and Sound Radiatio-: Measurements of Stiffened Cylindrical

Shells," David Taylor Model Basin Report C-13)5 (Apr 1962) CONFIDENTIAL.

4.. Den tlartog, J.. P., "Vibrations of Frames of Electrical Machines," Transactions of

the ASME, Vol. 50 (1928).5. Palmer, E. W., "The Influence of a Mass on the Free Flexural Vibrations of a

Circular Ring," Master's Thesis, Virginia Polytechnic Institute,,Blacksburg, Va. (Dec 1962).

6. Micihalopoulos, C. D. and Muster, D., "The Effect of Plane-Symmetric Mass Loading

on the Response of Ring-Stiffened'Cylindrical Shells," Univ. of Houston, Dept. of MechanicalEngineering, Tkch. Report 4 (Aug 1966).

7. Ojalvo, I. U. and Newman, M., "Natural Vibrations of a Stiffened PressurizedCylinder with an Attached Mass," AIAA Journal, Vol. 5, No. 6, pp 1139-1146 (Jun 1967).

S. Lee, S. Y. et al., ""Mass-Loading Effects on Vibrations of Ring and Shell Structures,"Space Division, North Nmerican R~ckwell Corp., SD 68-29 (Feb 1968).

9. Thornton, E. A., "Applications of a Generalized Integral Transform toVibrations of

Continuous Media," Ph.D. Dissertation, Virginia Rolytechnicinstitute, Blacksburg, Va.

(May 1968).

10. Palmer, E. W., "The Shock Environment of Submarine Pressure-Hull Penetrations

underExplosion Attack," Shock and Vibration Bulletin No. 33, Part I pp 57-63 (Feb 1964).

i1; lumphreys, J. S. and Winter, R., "Dynamic Response of a Cylinder to a Side Pres-

sure Pulse," AIAA Journal, Vol. 3, No. 1,- pp 27-32 (Jan 19.65).

12. Sheng, J., "The Response of a Thin Cylindrical Shell to Transient Surface Loading,"

AIAA Jo!rnal, Vol. 3, No. 4, pp 701-709 (Apr 196,).

13. Johnson, D. E. and Greif, R., "Diynamic Response of a Cylindrical-Shell: Two

Numerical Methods," AIAA Journal, Vol. 4, No. 3, pp 486-'494 (Mar 1966).

14., Flugge, W., "Stresses in Shells," Springer-Verlag, Berlin,-p. 219 (1960).

15. Bloodgood, V. G., "Theoretical Investigation of the Initial Response of a Thin Ringto a Radial Shock Pulse," Master's Thesis, Virginia Polytechnic.lnstitute, Blacksburg, Va.

(Jun 1967).

16. Private correspondence from W. E. Alzheimer, Div. 1541, Sandia Corporatior,Albuquerque, New Mexico (Aug 1969).

64

17. Baron, M. L.,and Bleich, H. H., "Further Studies of the Response of a Cylindrical

Shell to a Transverse Shock Wave," Columbia Univ. Tech. Report 10 (Dec 1953).

18. Love, A. E. H., "A Treatise on the Mathematical Theory of Elasticity," Fourth

'Edition, Dover Publications, New York (1944).

19. Timoshenko, S., "Vibration Problems in Engineering," Third Edition, D. Van

Nostrand Company, New York (1955).

20. Test 5961, .Velocity'Meter 2, Underwater Explosions'Research Division (Code 773),

Naval Ship Research and Development Center, Portsmouth, Virginia 23709.

21.. Palmer, E. W., "A Method for the Shock Design of Submarine Pies.ure ttull

Attachments (U)," NSRDC Report C-2537 (Jan 1968) CONFIDENTIAL.

22. -,Huang, H., "Exact Linear Analysis of the Transient Response of an Elastic

Cylindrical Shell of Infinite Length to Plane Underwater Shock Waves," NAVS|IIPRANDLAB

Annapolis Report MACHLAB 163 (Jul 1969).

65

UNCLASSIFIEDSA.Lont Classification

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Washington,-D. C 2-00343 REPORT TITLE

THE INFLUENCE OF ADDED MASS ON THE NATURAL VIBRATIONS AND IMPULSE RESPONSEOF LONG, THIN CYLINDRICAL SHELLS

4DESCRIP TIVE NOTES (Ty~pi of report and tncluive dates)

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Ii, ABSTRACT

The plane strain solution-is-obtained for the natural vibrations and impulse response of a thin

circular c) linder containing an added line mass. The solution for,,a uniform cylinder is derived by taking,the added inab tu be zero. Numerical calculations of the frequencies and mode shapes for several of the'

lover modeb are prebented in graphical form for various values of the added mass. The general impulse

response solution for arbitrary initial conditions is obtained by no mal mode theory. For both the natural

'.ibrationb and iipulse response, the theory is found to be in reasonable agreement %v'ith available

exp~erimental results.

in a particular mode, four distinct solution states are found to exist: a symmetrical and anti-

b,.mmetriL.al branch for each class of '.ibration, flexuial and extensional. Noteworthy features revealed

b.N this in'.estigation are the difference-in frequency and mode shape of each solution state and the

presence of coupling bet%%'een the flexural and extensional classes, particularly noticeable in the

exten.,ional class mude ,hapes. In comparing impulse response solutions for velocity with and without

the added rna~b, the nmajor influence of the added mass i~s found to be an increased participation of-the

flexural t..labb modeb, inc~luding the-rigid bud) translation, and decreased participation of the extensional

class oscillatory modes.

DDI/N J 7 UNCLASSIFIEDSN0101.807-6801 Security Clasiffication

UNCLASSIFIED

4 IYWAO LIPAK A LINV II LinK c

ROLE WvI ROLt.E WI ROI-E w T

Natural vibrations

Imp)ulse response

Cylinder with attached mass

Thin shells

Comparison with experiment

FOR

DD. iHqve1473 (BACK) UNCLASSIFIED(PAGE. 2) euiyC hUn