Axisymmetric Plane-Strain Vibrations of a Thick-Layered Orthotropic Cylindrical Shell

Received 2 May 1966 12.7

Axisymmetric Plane-Strain Vibrations of a Thick-Layered Orthotropic Cylindrical Shell

NAZEER AHMED

Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York 14850

The axisymmetric plane-strain vibrations of a thick-layered orthotropic cylindrical shell are analyzed, using a generalized Fourier-series technique that allows the determination of the frequencies and vibration modes of the composite shell in terms of those of the individual layers. These vibration modes are used to study the forced vibrations of such a shell subjected to an internal harmonic pressure. Numerical results are presented for a two-layered orthotropic cylindrical shell.

LIST OF SYMBOLS

c phase velocity of a compressional wave u• Din, KNN, Mx constants defined in the text M•., QN, R constants defined in the text Er, Eo, E, moduli of elasticity in radial, circum-

ferential, and axial directions, respectively

f frequency of internal harmonic applied v, w pressure

Gij sh ear moduli z J, Y Bessel functions of first and second kind N-+- 1 total number of layers Pi magnitude of internal applied pressure

)knm Po external applied pressure 0 q• orthogonality factor r radial coordinate measured from axis of

symmetry ri, ro internal and external radii, respectively,

of layered shell a, b, d, e, f, K elastic-stiffness constants co S• elastic coefficients (), t time

displacement function of composite shell for the nth eigenvalue displacement function of the mth layer for the nth eigenvalue boundary function at zero time of the ruth layer for the nth eigenvalue displacement in the circumferential and axial direction respectively axial coordinate Fourier coefficients

strain components Poisson coefficients

nth eigenvalue for the mth layer circumferential coordinate

time-dependent part of dynamic response

density normal-stress components shear-stress components frequency of vibration of composite shell denotes differentiation with respect to argument

INTRODUCTION

ANY advanced aerospace structures employ a composite-shell construction that may consist of a load-carrying substructure, an insulation, and an outer heat shield. Each layer is constructed of reinforced materials that have essentially orthotropic properties. The ratio of thickness to radius of curvature of the shell

may be such that the shell may no longer be considered thin. These shells are often pressurized and have applications in reentry and space logistics vehicles. It is necessary, therefore, to develop a method for pre- dicting the vibration modes and frequencies of such shells. These characteristics could be used to determine the vehicle response to various dynamic loads.

The Journal of the Acoustical Society of America 1509

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 140.254.219.138 On: Thu, 04 Dec 2014

01:54:27

N. AHMED

Vibration characteristics of shells of sandwich con-

struction have been studied extensively. i." Such studies have been limited, however, to thin isotropic shells. Vibration characteristics of single-layer thick shells have also been studied? Vibration problems of thick- layered shells could be solved, using Green's-function techniques and transform methods. The inversion of these transforms is often difficult and requires the use of approximate methods. In this paper, the axisymmetric plane-strain vibrations of a thick-layered orthotropic cylindrical shell subjected to internal and external pressures are studied by a generalized Fourier-series technique that allows the determination of eigenmodes of the composite shell in terms of the eigenmodes of the individual layers. The forced vibration problem for such a shell subjected to a harmonic internal pressure is solved using these eigenmodes. The results are illustrated by application to a two layer shell. It is found that this technique is considerably simpler than those of Green's functions or Laplace transforms. The analysis is applicable in general to orthotropic layered cylindrical shell structures whose longitudinal, radial, and circumferential directions coincide with the principal direction of elasticity in each layer.

I. THEORY

A. Vibrations of a Thick-Walled Layer

It is assumed that, for each individual layer of the cylindrical shell, the strains are small. The cylinder is considered long so that edge boundary conditions have little effect on its axisymmetric vibrations and the problem is one of plane strain with cylindrical symmetry. With reference to a cylindrical coordinate system (r,O,z), the displacement field is specified by u, v, and w, the components in the radial, circumferential, and axial directions, respectively.

In an orthotropic elastic body, the strains are related to the stresses by

•= S•1•+S•00+S•a•,

•oo= S,.•+ &,.•oo+ S,•a• •, (1)

ezz = Sal•rr+ Sa2•oo+ Saa•zz,

•Oz = 844ff Oz• •zr = 855ff zr• •rO = 866ff rO•

where, because of the symmetry of the stress-strain matrix, S•=S•. The material constants So may be stated in engineering notation as

Sn=I/Er

S,.•= --•o/E•,

Sa•= -- rrz/Er,

S44-- 1/Go•,

S•.=--uor/Eo, S•a =--

S22 = 1lEo, S•.a = -- u•o/E•

Sa•.= -- voz/Eo, Saa- 1/Ez,

S55 = 1/Gzr, Soo =

• Y. Y. Yu, j. Appl. Mech. 27, 653-662 (1960). •- L. M. Habip, Intern. J. Mech. Sci. 6, 483-487 (1964). a I. Mirsky, J. Acoust. Soc. Am. 39, 549-555 (1966).

(2)

For plane strain in the r-O plane, the components of strain ez•, e0•, ezr vanish, and this is used to eliminate a•, ao•, a•r from the expressions for err, coo, erO, which are now written

err= arrrrq- baoo, (3)

where

a- S•-- (S•aSa•//Saa),

b= S•.-- (S, aSa•./Saa), (4)

d= &,.- (S•aS•/&•),

The strain-displacement relationships are

•oo = u/r+ Or/tO0,

err = Ou/ Or,

• = Ow/ Oz,

ero = Ou/rOOd- Ov/Or-- v/r, (5) er•= Ou/Oz+ Ow/Or,

e•o= Ov/Oz+ Ow/rO0.

For an axisymmetric plane-strain situation for each individual layer,

Ou/ Oz= Ou/ 00= Ov/ Oz= v = O,

erO • erz-- eOz -- ezz -- O,

O'rO--O'rz--O, (6) err=Ou/Or, eoo=u/r.

Under these restrictions, the equations of motion reduce to the single equation

r0.u+(0u)] ,u 0.u ½ ..... , (7) L OF' r-•rr r • Ot •

whose solution is

where

u(r,t) = (AJ•(Xr)+ B ¾• (Xr))e g•t, (8)

e= d/ad--b • g= a/ad--b • •= b/ad--b • fi:•= g/e, c •= e/p, X = co/c, (9)

and A and B are arbitrary constants to be determined from boundary conditions.

From Eq. 3, 5, 6, and 8, the radial and the circumferential stresses may be written as

(rrr = e{ AJ•' (Xr)+ B ¾•' (Xr) } -- b{ AJ•(Xr)-I- B ¾• (Xr) }/r,

(too = g{ AJo(Xr)+ B ¾• (Xr) }/r (10) --b{ AJ•' (Xr) -+- B¾o' (Xr) }.

For the ruth layer of a layered shell subjected to internal and external radial pressures, the boundary conditions

1510 Volume 40 Number 6 1966


01:54:27

VIBRATIONS OF A THICK-LAYERED SHELL

are

O'rr--- --Pm-1 at

and (11)

O'rr• • Pm at r-' rm,

and, for the (m+ 1)th layer,

O'rr• --Pm at r-- rm,

and (12)

O'rr-- -- Pm+ l at r-- rm+l.

Substitution from Eqs. 10 results in the following values for the constants A and B for the mth layer:

L PmgmYO(Xmr,,,-1) A ,,= P..e.•Yo' (X•r•-i)--

P'•-IK'•Y•(X'•r'•),I/ D,• ' rra

rm

where

Eel; X [emYo' (X •r•) -- g•Y0 (X •r•)/r•

X [ed •' (X •r•) -- gd • (X •r•)/r•.

(13)

Similar expressions for Am+l and Bm+l are written by changing the subscript m to m+ 1.

The compatibility condition on displacements requires that

u,,=u,•+i at r=r,•. (14)

Substitution for u from Eq. 8 and for A and B from Eqs. 13 gives the following relationship between the interface pressures P•_•, Pro, and P•+•:

--P,•-l[Jt•(X,•r,•){e,•Yt•'(X,•r,•) g•Y• (X.,r•) } + Y• (X•r•) rm

KmJO(Xmrra-1) e•J o' (X mr m--1) --

q- ¾• (X,•r,•) e,•Jo' (X,•r,•) --

_jl_pmIj•(Xm+lrra) {em+lYOt(Xm+lrm+l) Kmd-lYO(Xm+lrrn+l) } rm+ t

-Jv YO(Xm+lrm) em+lJ•t (Xm+lrm) Km+lJ•(Xm+lrm) ]/ -- Din+ 1 rm

(•5)

When there are N-¾ 1 layers, there will be N such linear equations, which can be solved simultaneously in terms of the internal pressure Pi and the external pressure P0. These equations may be written in a matrix form

K!! KI•. 0 0 0

K•.• K•,. K•.a 0 0 Kay. Kaa Ka4 0

0 ß ß ß 0 K(N-•) (N--2) 0 ...... 0

0 0 '

0 0

0 0

K(N--D(N--D K(N--DN KN(N--•) KNN

rl P2 Pa

PN_•

kr•J

(16)

where K•, K•(N-•), K•(N+!) are, respectively, the coefficients of P,•, P•_•, Pm+l in Eq. 15 and

glYo(Xlrl) glJo(Xlrl) (Xlrl)l/D1 ' Ol=-Pi[{ elYt•'(xlrl)- rl }Jo(Xlrl)-{elJo'(Xlrl)- rl (17)

Q•r=--Po[{e•r+!Y•'(X•r+lr•r) gN+IY0 (XN+lrN) } J0 (XN+lrN) rN

-- { eN+lJo' (XN+lrN)-- YO(XN+lrN) DN+I. rN

(18)



01:54:27

N. AHMED

To determine the eigenvalues of each layer in the shell, the determinant of the K matrix in Eq. 16 whose typical element is KNN is set equal to zero. This determinant is solved subject to the constraint m= 1, Nq-1, so that the time behavior of two consecu- tive layers in each eigenmode at their interface is the same. The frequency of vibration of the composite shell is then given by •0=CmXm. Corresponding to each frequency •0n, a solution of the N linear simultaneous differential equations (Eqs. 16) yields the interfacial pressures P•, P2, .-., PN. Substitution for these pressures in Eq. 13 then gives the constants of integration Am, Bin. The radial displacement in each layer as given by Eq. 8 is thus completely specified for each frequency.

time t=0 is a solution to

e(O2u/Or2+r-•Ou/Or)--gu/r2=O, (22)

with the boundary conditions stated in Eqs. 11, 12, and 14. This boundary function is found to be

an(r,0)=• anm(r,O)•I(r--rm_O--I(r--rm)•, (23)

where

= (timdin+bin) (Pmrm•m+•+ P,,•_•r•_•m+l)r•/r,,_.•2•

+Pm-•r,,-l•m+lrn•)r-•/rm_•'"--rm •'". (24)

The Fourier coefficients a• are then determined from

B. Vibrations of a Composite Shell

In each layer, the eigenmodes are those represented by Eq. 8. For the nth eigenvalue, the eigenmode of the ruth layer is written

unm(r,t)=•AnmJ•(Xnmr)+B,•mY•(Xnmr)•d •'•. (19)

The eigenmode of the composite shell corresponding to this eigenvalue is then written as the sum of the eigenmodes of each layer:

u,(r,t)=E Unm(r,t)•I(r--rm_O--I(r--rm)], (20)

where I(r--rm) is a unit displacement function, which is zero for r<rm and is equal to unity for r>r,,. It is found that the eigenmodes un are not orthogonal over the interval r•<r<ro because the conditions of the Sturm-Liouville problem are violated in that the coefficients of the differential Eq. 7 for a composite shell are not continuous over the interval rg<r< ro.

The complete displacement function at any point in the composite shell may be expressed as a Fourier series in terms of the eigenmodes Unto:

u(r,t)=5-'. E anUnm(r,t)['I(r-r,,•-O-I(r--rm)-]. (21)

The determination of the Fourier coefficients an is

complicated by the nonorthogonality of the eigenmodes un over the interval rg<r<ro. However, an orthogonal set can be constructed from the nonorthogonal eigenset un by multiplying the eigenmode Unto in each region m by an orthogonality factor qm. This procedure, as proposed by Tittle •, is used to make a Fourier analysis of the forced vibrations of a thick-layered shell subjected to an interval harmonic pressure.

C. Forced Vibrations

Let the internal harmonic pressure be specified by p=P•e •. The external pressure Po is assumed acting at r=ro, as before. The boundary function u(r,O)at

4 C. W. Tittle, J. Appl. Phys. 36, 1486-1489 (1965).

an= E q anm (F,O)Ignm (F,0)FdF m • •m--1

E q• u,d(r,O)rdr, (25)

where the quasiorthogonality property

• r{q•u•(r,O)}{q•u•(r,O)}dr=O (26)

has been used.

The differential equation for forced axisymmetric vibrations is

e (O•u/Or • + r -• Ou/Or) -- gu/r • + P •e • rs( r-- r •) = pO•u/0 F. (27)

A series of the form

u½,t)= E (28)

is assumed. Substitution into Eq. 27, multiplication of both sides by rut(r,O), integration from 0 to r0, and use of the quasiorthogonality property (Eq. 26) yields

R[•t+•& = P•e •, (29) whose solution is

Pieif t •(t) = D• sinwtt+D• cosw•tq (30)

where

and

R (r) = • { •q•r•J/•(X•r)+ Y/•(X•r)

+ (1--•/r•X•)J•(X•r)+ Y• (X•r)

+Bt•rgY•(X•r•)}. (32)



01:54:27


TABLE I. Elastic coefficients.

Sii (10 -•a cmydyn) Sll S 22 S33 S1 •. Sla S •.3 S44 S55 S66

Barite 18.4 17.36 10.96 -- 9.45 -- 2.68 -- 2.73 83.33 34.84 36.50 Topaz 4.43 3.53 3.84 -- 1.38 --0.86 --0.06 9.23 7.53 7.63

The initial conditions on •(t) can be specified in terms of the initial conditions on u•(r,t) by

f0 r0 •(0) = ru•(r,O)a(r,O)dr

and (33) =0,

from which the constants of integration D1 and D•. are readily obtained and the forced vibrations are completely specified from Eq. 28.

D. Application to a Two-Layer Shell

For a two-layer shell, the determinant of the K matrix in Eq. 16 reduces to

[Ml{elJl' ()tlrl)--KiJ1 --[- { el Y1 t (k 1•'1) -- K1 Y1 (k 1•'1) } •[M2J1 (• 2•'1)- Y1 (• 2•'1)'] -- [-MiJl(Xlrl)q- Yl(Xlrl)-] • E-- M2{ e2J1 ! (•2•'1) -- K2J1 (•2•'1)/•'1}

--{e2Y1'(X2•'1)--K2Y1(X2•'1)/•'1}']--O, (34) where

--- --

e•Y[ (X•r•)--g•Y•(X•r•)

exJx' (Xxr,)-- •zJx (Xxr,)/r,' e•.Y;' (X•.ro)-- g•.Yx (X•.ro)

e•J z' (X •.ro) -- •.J • (X•.ro) /ro'

which has to be solved simultaneously with

(C/p) lXnl 2--' (e/p)2Xn2 •' (35)

for the eigenvalues M1 and X•.. The orthogonality factors are obtained from Eq. 25 as

q•?./q•.= (e•/e•) [- (X•.•-- X •?-)/(X• •'-- X n•)]. (36)

The orthogonality factor for the inner layer may be assumed to be unity. The orthogonality factor for the outer layer is then given by Eq. 36. Substitution of m= 1 in Eq. 15 gives the interface pressure P• in terms of P• and P0 so that the eigenmodes are completely defined from Eq. 8.

Numerical results for a two-layer cylindrical shell whose inner layer is made of barite and outer layer of topaz were obtained on an IBM-7094. The data shown in Table I, obtained from Trent and Stone, • were used

• H. M. Trent and D. E. Stone, in American Institute of Physics Handbook, D. E. Gray et al., Eds. (McGraw-Hill Book Co., 1963), Chap. 2, p. 58.

in these computations. The first five eigenvalues X• and M of Eq. 34 for radii rl/ri= 1.2 and ro/ri= 1.5 are given in Table II. The corresponding Fourier coefficients are shown in Table III for applied pressures Po/Pi=2.0 and Po/P• = 10.0. The eigenmodes u,,(r,O) are plotted in Figs. ! and 2. The complete displacement function u(r,O) as given by Eq. 21 is obtained by superposition of the Fourier components of the individual eigenmodes and is shown in Figs. 3 and 4.

For an internal applied pressure P•=sin ft when the external applied pressure P0-0, the initial displacement and velocity may be assumed to be zero. The time-dependent part of the dynamic response •(t) in Eq. 28 then becomes

[' sinwt--sin fz(t) = -- . (37)

The dynamic response, Eq. 28, of the two-layered shell to such an applied pressure for the special case f=wl/2 is plotted in Fig. 5 at time t=•-/2f.

It is seen that convergence is rapid in all cases considered. The sum of the first three Fourier com-

ponents is less than 5% different from the sum of the first five Fourier components.

II. CONCLUSIONS

By a simple extension of the Fourier-series technique, it has been possible to represent the dynamic response

TABLE II. Eigenvalues for a two-layer cylindrical shell of barite and topaz.

n-1 r•--2 n=3 n=5 n=5

rl/ri= l.2, ro/ra= l.5

1.066 0.51 2.319 1.11 4.034 1.93 5.685 2.72 7.315 3.50

TABLE III. Fourier coefficients a,, corresponding to eigenvalues

0•1 0•2 0•3 0•4

Po/Pi=2.0 --1.85 --1.28 --0.517 --0.351 --0.126 Po/P•- 10.0 -- 2.4 -- 1.25 --0.92 --0.368 --0.153



01:54:27

N. AHMED

IO

-2

-4

-6

.0 I.I I • I.• 1.4 1.5

FIO. 1. Eigenmodes vs radial distance, Po/P•=2.

FIo. 2. Eigenmodes vs radial distance, Po/P• = 10.

50

40

I0

1514 Volume 40 Number 6 lC)66


01:54:27


FIG. 4. Displacement function for free • vibration vs radial distance, Po/Pi= 10. o_ ...... Sum of the Fourier components • of the first three modes. •- Sum of • the Fourier components of the first five • modes.

160

140

120

I00

80

1.0 I. 12 13 1.4 15

-40

-30

-20

-I0

.0 I.I 1.2 1.3 1.4 1.5

FIG. 3. Displacement function for free vibration vs radial distance, Po/Pi=2.

ß Sum of the Fourier components of the first three modes. •: Sum of the Fourier components of the first five modes.



01:54:27

-40

-3O

-20

-I0

N. AHMED

ß 0 I I 1.2 I :3 1.4 i 5

r

Fro. 5. Dynamic response for f-•/2 at t- •r/2 f vs radial distance. - .... Sum of the first three Fourier components. •' Sum of the first five Fourier components.

of a thick-layered orthotropic cylindrical shell to an applied pressure as a Fourier series of eigenmodes of the individual layers. For a two-layered cylindrical shell made of barite and topaz, the Fourier series is found to be rapidly convergent. Numerical examples for an arbitrary number of layers may be worked out along

the same lines given the material properties of the individual layers, their radii, and the applied pressures.

ACKNOWLEDGMENTS

The author is indebted to Professor H. D. Conway, for his valuable suggestions and encouragement during the course of this work.



01:54:27

Axisymmetric Plane-Strain Vibrations of a Thick-Layered Orthotropic Cylindrical Shell

Documents

Transcript of Axisymmetric Plane-Strain Vibrations of a Thick-Layered Orthotropic Cylindrical Shell