Effect of heterogeneity on the axisymmetric vibrations of cylindrical shells

J. Sound Vib. (1971)14 (3), 325-338

EFFECT

OF HETEROGENEITY

ON THE AXISYMMETRIC

VIBRATIONS OF CYLINDRICAL

SHELLS

J. TASI Department of Mechanics, State University of New York, Stony Brook, New York 11790, U.S.A. (Received 12 May 1970, and in revisedform 27 July 1970) The influence of large heterogeneity on the axisymmetric vibration characteristics of thin, composite cylindrical shells is studied, both analytically and numerically. In the neighborhood of the axisymmetric breathing mode, frequency spectra for shells of indite and finite length are shown to be influenced qualitatively as well as quantitatively by large deviations from material and geometric symmetry in layer arrangement. A study of mode coupling in a semi-infinite shell is made for both end modes and modes with stationary frequency with real finite wave number, the latter being uniquely generated by a special class of hetero-

geneity. In each case, analytical estimates are given for frequencies, wave numbers, and modal amplitudes as functions of material and geometric properties of the shell.

1. INTRODUCTION This paper is concerned with the influence that heterogeneity has in changing the axisymmetric vibration characteristics of a thin multilayered cylindrical shell. The frequency

spectrum for an infinitely long shell, the spectrum for a shell of finite length with mixed end and the mode coupling at a traction-free end of a shell of semi-intiite length are each studied for heterogeneity arising from the use of layered composite materials. Particular emphasis is placed on the possible spectra that can occur with the existence of large deviations from material and geometric symmetry in layer arrangement of a shell. /&symmetric motion of infinitely long homogeneous cylindrical shells is discussed in references [l-5], based on theories with various degrees of approximation. In a thin homogeneous shell, the spectrum for frequencies below the radial breathing mode is characterized by a pair of complex branches and one real, propagating branch. Within the same frequency range, it is found that heterogeneity can have a qualitative as well as quantitative effect on the spectrum, in that it is possible for heterogeneity to create two pairs of complex branches separated by three real branches. Simple, explicit formulas are given which determine the conditions under which this can occur and the characteristics of the spectrum. Solutions for a finite shell with mixed end conditions are given to indicate the extent to which heterogeneity can increase or decrease the density of resonant frequencies that would exist for a homogeneous shell. For a semi-infinite shell, mode coupling at a traction-free end is studied, with particular attention paid to the resonant response that can exist at the end of a shell due to the existence of complex branches, and the resonances that arise when complex branches intersect the real plane of the spectrum at stationary points with non-zero wave numbers. For frequencies in the sonic range of shell vibrations, the latter resonances exist only under special conditions of heterogeneity. In each case, analytical estimates are given for the resonant frequency, wave numbers, and amplitudes of the modes as functions of material and geometric properties of conditions,

325

326

J. TASI

the shell. The estimates are quite accurate for the end mode. The estimates for the modal displacements at the frequencies of intersection of the complex branches into the real spectrum plane are not as accurate, however, because of their sensitivity to higher-order terms in the wave numbers at the intersection points.

2. FREQUENCY SPECTRUM FOR AN INFINITELY LONG CYLINDER The composite shell to be considered has in-plane, coupling and bending stiffness matrices A, B, D defined, in terms of the individual layer elastic moduli C(I), by

h,/Z A = 1 C’” dz,

h,/2 j. Wzdz,

B=

-ht12

D = 1” C”’z2 dz,

-htl2

-ht/2

where h, is the total thickness of a layered shell. The stiffness matrices will be restricted to the form A=[::’

$::

8,,1

B=[i::

5:

i.],etc.

No consideration is given to coupling between extensional strains and shearing strains. As a result, the axial and radial modes of motion can be studied without consideration of the tangential modes. For a homogeneous shell, D=lhZA 12L

B = 0.

5

(1)

In a layered shell with geometric and material symmetry of the layers with respect to the mid-surface of the shell, the first of equations (1) does not hold, but one has an orthotropic shell in the classical sense with B = 0. The heterogeneity effect of interest here is that of cylinders constructed without a symmetric arrangement of composite layers, thereby introducing coupling between extension and bending, represented by the B matrix. For the frequency range of interest, classical linear shell bending theory provides a sufficient analytical framework. Therefore, Fhigge’s system of equations 16, 71 will be used. Considering harmonic axial and radial displacements (u, W)of the form u = j&, ei(W+Ax/R) w = Co eUW+~xlR)

,

(2)

,

one obtains a characteristic equation relating frequency o and dimensionless wave number A given by b6X6+b4X4+b2X2+bo=0.

(3)

The coefficients in equation (3) are defined by b6=(A,,-B111R-011/R2)0,11R2-(B,I/R)’, b4 =-@

+~AIz)D,,IR~

+~U,I

bz = (A, 1 + B, JR) (An-Bn/R b,, = -Q(Az2

- BzzIR + WR2

+BI~/R)B~~/R-A~~B,,IRI,

+ &z/R2

- 1;2>- Q2BulR

(4)

- A:,,

- J%

where Q = R2 ph, co* is a frequency parameter, and subscripts 1 and 2 in the stiffness elements

refer to the axial and circumferential directions, respectively. For long wavelengths (A 4 l), and with h,/R Q 1, the solutions to equation (3) are given by Q= VII -t-&rIR--(~:,l~22)U +B2~/~22)1~~~

(5)

327

EFFECTOFHETEROGENEITY ONVIBRATIONS corresponding to an extensional wave, and Q = A 22

-

B22IR

+

KA:2/A22)

(1

+

B22IRA22)

+

24,lRl X2,

(6:)

which yields the cut-off frequency of the radial breathing mode when h = 0. In both expressions, terms of order (/z,/R)~ are neglected. Heterogeneity, i.e. the coupling matrix B, affects equations (5) and (6) by terms of order @JR), and is therefore negligible. This arises because of the predominantly membrane response of the shell when the axial wavelength is great. As the wavelength diminishes, the branch of the frequency spectrum corresponding to extensional motion changes to a predominantly flexural branch. The influence of heterogeneity is essentially felt in the transition region. The influence of heterogeneity will be numerically illustrated for the three cases of layered composite construction given in Figure 1, consisting of axially and circumferentially oriented fibers. The three cases, each with the same in-plane stiffnesses, differ only in layer arrangement. Computation of the stiffnesses in terms of the properties of the individual layers is indicated in the Appendix. As in the notation of [8], lack of geometric symmetry in the layer arrangement is defined by the parameter 6 = (h,, - h,i)/h,, where h,, and h,.*are the thicknesses of the

(a)

(b)

Figure 1. Compositelayer construction.(a) S = 2/3, (b)

(cl 6 = 0, (c) 6 =

-213.

outer and inner circumferentially oriented layers. The elastic constants used in the computations in this paper are those for boron-epoxy composite in the axially oriented layer, and glass-epoxy composite in the circumferentially oriented layers. The characteristics of the possible frequency spectra are illustrated in Figure 2, in which frequency is made dimensionless with respect to the radial breathing frequency wR. With 6 = 2/3, the frequency spectrum is essentially similar to the results for purely orthotropic construction (6 = 0). In both, the extensional branch deviates markedly from its initial linearity as bending becomes important, while the complex branch (and its complex conjugate pair, which is understood) intersects the imaginary plane and then divides itself into two imaginary branches. In the same frequency range with 6 = -213, however, two pairs of complex branches exist, separated by a frequency interval in which three real propagating modes arise. The conditions under which the characteristics of the spectrum change can be analyzed by determining the frequency of intersection of the complex branch or branches with the real or imaginary planes. As shown in reference [5], the intersection frequencies occur at stationary points in the branches defined by dw/dh = 0. From the characteristic equation (3), do dh

-=

22

-X(6& h4 + 4b4 h2 + 2b,) 20R2 @,(X4db4/dQ + X2db,ldQ + db,/dQ) *

(7)

328

J. TASI

0 corresponds to the cut-off frequency for the radial mode. Other possibilities for a solution are obtained from the roots of

h =

6b6h4+4b4h2+2bz=0 or A2= 3[*(Ij2 - 3gy

-PI,

(8) wherep = b4/b6 and 4 = b2/bs. With ? = -bo/b6, substitution of equation (8) into equation (3) yields 2p3 - 9jkj ‘F 2(jF

- 3~j)~‘~ - 27i = 0,

(9)

governing the frequency of intersection, say wI. of a complex branch into a real or imaginary plane. It is impossible to solve equation (9) explicitly for the intersection frequencies for all conditions of heterogeneity. However, simple analytical approximations can be obtained for certain limiting cases. Since the intersections occur in the frequency region where membrane theory breaks down, we shall look for perturbation solutions with R2 ph, 0: r (A,, -

with the perturbation frequency:parameter

42/A,

1) (1

+

4

(10)

E governed by

1 > 1~1> O(h,/R). The lowest-order terms retained for the parameters in equation (9) are j r p = ‘;;I 1;12;;2;-,$ 11

-44*,4422 4 =

J_

,,=(AllA22-A:2)(A,2/A~d2=O(R,,$)2 P

1 D1

1 I Dl

= O@R/h,),

-42)

4 ’

-

[A,

4;;’ 11

11

JR2

_

JR2

(B,

-

(4

JR)21

=

(11)

co(R/hd2,

t *

JW21

Equation (9) can be put in the form 43 - @‘) @ + (SJJ) 4 + (Yi’ - iJj3) = 0,

(12)

which, if the approximations of equations (11) are used, is a cubic equation governing E. If (~/3)~ f I is greater than zero there will be only one real solution for E from equation (12), hence only one intersection frequency as in Figure 2(a) and (b). However, if (~/3)~ + r is less than zero, there will be three real solutions for E, hence three intersection frequencies as in Figure 2(c). The governing heterogeneity parameter, reflecting the lack of material and geometric symmetry in layer arrangement, is given byp, and there is a noticeable influence on axisymmetric motion of a cylindrical shell when that parameter is such that 1pi 3 s r. Therefore, large heterogeneity in a thin cylinder can be defined for axisymmetric motion by the satisfaction of such a condition. The solutions of equation (12) are given below for the following three limiting cases.? Case I : Orthotropy (B = 0). p=o

and 3 2(A 12/A 9

[ =

I d4

D,

“3

I/R’

(A22-42/A**)

(W I

0(h,/R)2’3.

t Solutions can be obtained for small heterogeneity apparent from the three cases presented.

parameter,

but the results simply co&m

the trends

329

EFFECT OF HFiTJ3ROGENEITY ON VIBRATIONS

The real and imaginary wave numbers corresponding to the intersection frequency are x: z (4r)“6,

(13b:)

;\:,3 z i(4r)“6/2/2.

(13C)

Case II: Large heterogeneity parameter (p’ 9 r) with p positive.

--(AI~B,~/R-A,zB,,/R)* (All A22 -

’ =

42)

[Al,

4,/R*

-

(&,lR)*l

(14a)

= O(P),

with the corresponding wave numbers given by hi E 2(rp-* + *d)“*,

(14b)

with d=&IA,,

-2(A,,/A,,)*,

and Ai.3 z i(p/2)“‘.

(14c)

Case III: Large heterogeneity parameter (lpi3 3 r) withp negative. Three solutions, with corresponding wave numbers, are (15a) -

W2),

as in (14a), with hi z 2(rp-* + *d)‘/*,

(15b:l (15c)

Also, E2,3~

F22/z2

-(AI,W~-~~B,,IR) 11

=

AllA22-Ai2

0(&,/R)"*,

112

I

(16a)

with hi 2 (-p)“*[l F (-r/p’)“*], A;

3~

’

edP>1’4 i(-r/p)‘14.

1

(16W

UW

A comparison of the results of equations (13)-(16) with the numerical solutions represented by Figure 2 is given in Table 1. The results of equations (13)-( 16), when used in conjunction with solutions for w/wR < 1 and solutions in the neighborhood of the radial breathing mode, enable one to determine the character of the frequency spectrum. More importantly for the purpose of this paper, they either provide, or aid in obtaining, analytical estimates of certain resonant frequencies for both the finite and semi-infinite shell. The frequency spectrum for a finite length cylinder is simply obtained from the results for an infinitely long cylinder when mixed end conditions exist. That is, with N,,=M,,=w=O

at

x=O,L,

where N,,, M,,, w are the axial stress resultant, axial bending moment and radial displacement, respectively, each real branch yields a set of admissible wave numbers given by /\LIRzT= an integer. The integer represents the number of half-waves that exist along the length of the shell, with odd integers representing modes symmetric about the middle of the

330

J. TASI

shell and even integers representing antisymmetric modes. For the three cases of layer arrangement considered previously, solutions are given in Figure 3 for odd numbers of half-waves. The solutions that remain above CO/W~ equal to unity are contributions from the radial branch and are unaffected by changes in layer arrangement. However, solutions

Figure 2. Frequency spectra for infinitely long cylinders with composite layer construction as in Figure 1. The elastic constants used in the computations are those for boron-epoxy composite in the axially oriented layer, and glass-epoxy composite in the circumferentially oriented layers. Solid lines refer to real and imaginary branches, dashed lines refer to complex branches. (a) S = 2/3, (b) S = 0, (c) S = -2/3. h,/R = OQOfi TABLE 1

Comparison of analytical and numerical solutionsfor intersection frequencies and corresponding wave numbers?

Case I. Orthotropy II. Large heterogeneity with p positive and p’ P r III. Large heterogeneity with p negative and jpl3 ~-r

Number of intersections of complex roots into real or imaginary planes One One

0.9971 (0.9971) 0.9900 (0.9902)

4.29 (4.29) 0.822 (0.815)

i 3.04 (i 2.98) i 8.08 (i 8.04)

Three

0.9900 (0.9898) 0.9958 (0.9956) 0.9980 (0.9979)

0.822 (0.820) 11.20 (11.25) 11.66 (11.71)

8.08 (8.13) 1.62 (1.78) i 1.62 (i l-47)

i_Comparison is made for the three cylinders considered parenthesis.

in

Figure 2, with the numerical solutions in

arising from the longitudinal branch indicate that, within the frequency interval considered, heterogeneity can have a large influence on the density of natural frequencies in a shell, tending to increase or decrease the number of natural frequencies within that interval depending on the layer arrangement considered. Of particular interest is the frequency interval in Figure 3(c) between the two stationary frequencies below the radial breathing mode. For long shells, the stationary frequencies corresponding to the intersections of the complex branches into the real plane, become natural frequencies of the shell, independent of length.

3.31

EFFECT OF HETEROGENEITY ON VIBRATIONS

IO1

100

099

0 98 101

100

i? 3

095

0 9E

IO1

IOC

091

0 91

I

II

2

I

I

I

4

I

6

I

I

8

I

IO

L/D

Figure 3. Frequency spectra for symmetric vibrations of finite length glass-boron-epoxy cylinders with mixed boundary conditions. Integers represent number of axial half-waves in the mode. (a) 6 = 2/3, (b) S = 0, (c)S = -213. h,/R= 0.006.

With a change of boundary conditions to traction-free conditions, mode coupling at a free end modifies the spectrum. An addition to the natural frequencies for a finite shell is an end mode arising from coupling between the real and complex branches. An exception to the frequencies being modified by mode coupling occurs at the frequencies of intersection of the complex branches into the real plane. Both cases are studied in the next section.

I. TASI

332

3. MODE COUPLING IN A SEMI-INFINITE

CYLINDER

The resonant response corresponding to an end mode occurs at ultrasonic frequencies in plates and cylindrical rods [9-l 11,but, in a thin homogeneous shell, the same type of mode can occur for axisymmetric vibration within the sonic range of frequencies [12]. Heterogeneity does not materially affect the frequency of the end mode, but it can produce marked changes in both the amplitudes and wave numbers that contribute to the mode. The procedure used in formulating the problem is given in references [9-l 11.A semi-infinite cylinder (x 2 0) is subjected to a harmonic wave impressed at infinity in the form given by equation (2), where h = h, is a real wave number. Stress resultants generated on any section normal to the incoming wave are IV,, = -@/RX,) A0 ef(wt+Alx’R), M,, = -Is, &, ei(wt+AlxlR), Q = -iA,

R-1

fi, A0 e[(wt+htWR),

where

and Q is the transverse shear stress resultant. With traction-free specified at the end of the cylinder, namely NXX=MX,= Q=O

boundary

conditions

at x = 0,

each of the three reflected wave forms, admitted by the bicubic characteristic equation, are excited by the incident wave. The amplitudes of the reflected axial waves AR,k = 1,2,3, are governed by

with the radial amplitudes given by C, = aI A&, and C, = -ak A&. Restricting ourselves to the frequency region(s) with one real wave number and two complex wave numbers, A2and h, are of the form A2= a - ib A, = -a - ib a’b ’ O9

to satisfy the condition of bounded displacements. From equations (17), the solution for Al/A0 is of the form Al/A0 = (98 - id)&% An end resonance occurs when AJA,

+ if).

= -1, or

@;I$ = Re{(&/U (G El& P, - 1) (A2BGm(X2 B2))>= 0.

(18)

The displacements for the end mode, leaving out the eiwt term, are given by u/A, = -2{sin (h, x/R) + e- b”‘R[Im(A2/A,,) cos (ax/R) - Re(A,/A,,) sin (ax/R)]), w/A, = 2(C,,/A,) {cos (X, x/R) + e-bX’R[Re(C2/C,) cos (ax/R) + Im(C,/C,) sin (ax/R)]}.

(“)

For the three cases considered previously, the frequency of end resonance obtained from equation (18) will be given in terms of material and geometric properties of the shell with corresponding expressions for the modal contributions to the displacements in equation (19).

EFFECT OF HETEROGENEITY ON VIBRATIONS

333

Case I: Orthotropy (B= 0). It has been shown [12] that the frequency, wE, of the end mode for an orthotropic shell is effectively the generalized plane stress frequency of a thin ring, WEz (422 - A&/A, ,)“‘/(R’ph,)“2.

(20a)

The corresponding wave numbers and modal amplitudes are Xi 2 r116, A,, 3 g

Wb)t

4(&l - iti)

r 1/6,

(204i

AI/A0 r -1 - id?, Co/A, z (A,JA,J c&J

(204 r1j6 = 0(R/ht)‘13,

(2Oe)

(20f)

z 2.

The solution for the frequency of an end mode for an orthotropic shell corresponds to 15~ equal to zero, neglecting terms of order @JR)*. Also, at that frequency, the group velocity of the real branch of the frequency spectrum, given by (C,)w, z W,JA,,)

(Q,lR*~hr)~‘*,

is essentially a minimum for that branch, and varies with the geometry of the shell as h,/R. None of these conditions hold for B unequal to zero but one can use equation (18) in conjunction with the results of equations (14)-( 16) to determine wE Case II: Large heterogeneity parameter (p3 % r) withp positive. The end mode has to occur at a frequency below the intersection of the complex branch into the imaginary plane. Therefore, R*ph,dz

(A,,-&A,,)(1

(21a)

+~+4,

with E given by equation (14a) and Id 1< 1. It can be shown that d = O(h,/R). Therefore, the wave numbers corresponding to equation (21a) are, retaining lowest-order terms, Al

z

A,,3 2

equation (14b),

&

~[_34rp-1(A11/A12)2

+ rp-* + +d]“* - i(+p)“*. I

(21b)

Substitution of equations (21a) and (21 b) into (1S), yields (21c) Therefore, a, the real part of h,, is a z [2rp-* + W-

(r~12All)l(p~:2)11’2[1- (~l~~/2RA12P~2,

and A2/A0 z -i(+p)“*/h, Co/A0

=

(AlllAld&‘[4rp-*

G/CO

=

3p/Ww*

(214

= O(R/h,)‘/*, -

(A

,2/A

(AdAll)* = O(l), 1 l)*)

(1

-

4

pI2RA

1

The group velocity for the real branch at wE is (cu)aE=

(1 + 8:$.,b6p4)

-*++d (A,, yAf2,AIl)pht

(21e)

,*)I= W/W.

112 1 ’

t Somewhat more accurate estimates for wave numbers are available in reference [12].

Gw

334

J. TASI

which is independent of the overall geometry of the shell, i.e. independent of h,/R, but is partially dependent on the detailed layer arrangement. Case III : Large heterogeneity parameter (I p 13 + r) with p negative. Of the two frequency regions with complex branches, it can be shown that an end resonance cannot occur in the lower region. If we look for an end resonance at a frequency within the second complex branch, then (22a)~

R2 ph, & z (A,, - &,/A, ,) (1 + 0) with 1~31 ’

IdI*W/R),

and E~,~given by equation (16a). The lowest-order expressions for the corresponding wave numbers are hi z (-p)“*[l X2,, 2

3~~P-*(~,J-41*)*1,

+

f[*(-rp-1)1’2

+ *~rp-‘(ftl,/A,,)*]“*

- i[$(-rp-‘)‘I*

- ~drp-‘(A,,/A,2)2]1’2.

(22’9

Substitution of equations (22a) and (22b) into equation (1S), yields d z -(A,,/A,,)*(-p/r)*‘*

= 4~~.

Gw

Therefore,

GW

x1 r (-pP2,

A,,3

Z

$(*I6 -

A2/Ao z -(l/G

i) (-r/p)‘/4,

t2W

+ i) (-p3/r)1’4/(1 -PB,,/RA,~)

Cd.& z (-~>“‘/6412/~

I I - PB, *IRA I I)=

= 0(R/h,)1’4,

W/h,)*‘*,

C,/C, z 1 + i/d%

W) (2%) Wh)

The group velocity for the real branch at wE is (C,),, E (WA I I> [-~~/(A22

-

&,/A

1I>

PM’*,

which varies with the overall geometry of the shell as (h,/R)“* and is directly dependent on the detailed layer arrangement. The radial displacements in the end mode are illustrated in Figure 4 for the three layer arrangements considered previously. The axial displacements are an order of magnitude less than the radial and, hence, are not shown. The difference between the displacements calculated numerically and those given analytically by equations (20)-(22) are generally very small, the two sets of maximum displacements differing, at most, by 4 %. The maximum displacement in the end mode, and the wavelength of the propagating part of the mode, are both noticeably influenced by heterogeneity and increase as 6 is increased. Finally, for mode coupling near the frequencies of intersection of the complex branches into the real plane, equations (17) yield A,/& Cl/&

--f 1,

-+ -G/A,,

A,/&

-+ AZ/&,

C31-40 -+

-C,I&

as w approaches wr. A Taylor series expansion about an intersection frequency gives w(h) = WI+ $(d* w/dh*),,@;, 3 - &, 3)2+ * - -.

335’

EFFECT OF HETEROGENEITY ON VIBRATIONS 700, \ t 600

-40 ii! (X/R=0)=1055

500

$400 s E

300

\

E :: E 200 E 2

\

100

’

B

’

,/-‘

\\

‘\

I

I

00

02

1

04

I

I

06

08

I

I

I

IO

12

14

I

I.6

18

I

20

X/R

Figure 4. Radial displacements 6 = -213.

in the end mode for semi-infinite shell. -,

6 = 2/3; ---,:a

= 0; ---,,

Let Ao = w - wI, Ah = b, 3 - %, 3, with Aw such that three real propagating waves exist, and the signs of hZ,,Vchosen such that each of the waves has a positive group velocity. Therefore, 2Aw A’ = ’ J (d2 w/dX2),, ’ with the negative sign for the higher intersection frequency and the positive sign for the lower intersection frequency. For Aw # 0, the existence of two wave numbers h, ) of slightly different magnitude creates a spatial beating displacement superimposed on the contribution from the remaining real branch. The displacements, omitting an ieiWtfactor, are

For the upper intersection frequency, given by equation (10) and c2 in equation (16a), the ingredients for equation (23) are approximately given by hi, Xi = upper expressions in equations 16(b) and 16(c), (d2 m/dh2),, z 4pW(R2 @I,WIA I I>, C42/-40),,

z

(-~~/r>“~l(l

GI&)w,

z

L4

11/A

r

_

[(-rlp)“2

(c,,~, I

- PB,

12)

(-P)“~/U

[1

+

-

A221All

(244

1IRA12)

-

=

PB, +

,IRA,2)

WlW*‘2,

Wb) =

(A12/‘11)2]~_p~,y3)1/4

(24~)

W/~J”2,

=

(-r/p)“’ BI JRA121

Similar expressions can be obtained for the lower intersection frequency.

O(R,h

t

)I/4

*

G’4d)

336

J. TASI

A comparison of radial displacements computed numerically with those given approximately by equations (24) is in Figure 5. The maximum displacement is given accurately by the analytical approximations, but the displacements then grow out of phase because of the omission of second-order terms in Xi (see Table 1). Second-order terms in Xi also influence the displacements at the lower intersection frequency, but in a converse manner; the maximum displacement is inaccurate, but the displacements have the proper phase.

I

I

I

00

02

04

I

06

08

I

I

I

I

I

IO

12

I.4

16

18

;

3

X/R

Figure 5. Radial displacements in a semi-infinite shell with traction-free end conditions, for the upper stationary frequency with real finite wave number generated by heterogeneity (8 = -2/3). -, Exact; ----, approximate.

Interestingly, although end modes are generally noted for their large displacements, a comparison of equations (22) and (24) shows that the upper (but not the lower) intersection frequency generally will have a higher radial magnitude of response by a factor of order (R/h,)1’4. Also, the displacements for the two intersection frequencies simultaneously satisfy both mixed and traction-free end conditions. In general, a change in boundary conditions for such modes simply produces a change in mode shapes, but not frequency. 4. CONCLUSIONS In the frequency

region below the radial breathing mode of a thin cylindrical shell :

(1) the frequency spectrum can be influenced in a qualitative as well as quantitative manner by heterogeneity; (2) heterogeneity has only a slight influence on the frequency of end resonance, but noticeably influences both the amplitude of displacement and the wave numbers in the mode; (3) for a special class of heterogeneity, two stationary frequencies with real finite wave numbers can arise, which do not exist in a homogeneous shell at corresponding frequencies. REFERENCES

1. M. C. JUNGERand F. J. ROSATO1954J. acoust. Sot. Am. 26, 709. The propagation of elastic waves in thin-walled cylindrical shells. 2. P. M. NAGHDI and R. M. COPPER1956J. acoust. Sot. Am. 28,56. Propagation of elastic waves

in cylindrical shells, including the effects of transverse shear and rotatory inertia.

337

EFFECTOF HETEROGENEITYON VIBRATIONS

3. G. HERRMANN and I. MIRSKY 1956 J. uppl. Mech. 23, 563. Three-dimensional and shell theory analysis of axially symmetric motions of cylinders. 4. H. D. MCNIVEN, A. H. SHAH and J. L. SACKMAN1966 J. ucoust. Sot. Am. 40, 784. Axially symmetric waves in hollow, elastic rods: Part I. 5. H. D. MCNIVEN, J. L. SACKMANand A. H. SHAH 1966 J. acoust. Sot. Am. 40, 1073. Axially symmetric waves in hollow, elastic rods : Part II. 6. W. FL~GGE 1962 Stresses in Shells. Berlin : Springer-Verlag. 7. S. CHENGand B. P. C. Ho 1963 AZAA J. 1,892.Stability of heterogeneous aelotropic cylindrical shells under combined loading. 8. J. TASI 1966 AZAA J. 4, 1058. Effect of heterogeneity on the stability of composite cylindrical shells under axial compression. 9. D. C. GAZIS and R. D. MINDLIN 1960 J. uppl. Mech. 27,541. Extensional vibrations and waves in a circular disk and a semi-infinite plate. 10. H. D. MCNIVEN 1961 J. acoust. Sot. Am. 33,23. Extensional waves in a semi-infinite elastic rod. 11. H. D. MCNIVEN and A. H. SHAH 1967 J. Sound vib. 6,8. The influence of the end mode on the resonant frequencies of finite, hollow, elastic rods. 12. J. TASI 1968 J. acoust. Sot. Am. 44, 291. Reflection of extensional waves at the end of a thin cylindrical shell.

APPENDIX With the mid-surface of a cylinder matrices are given by [8] A=

dhosen as the common

reference

surface, the stiffness

ht12 I‘ C”’ dz = ht{sC(=) + (1 - s) C(=)), -hclZ h, /2

B=

D=

j. C”‘z dz = 3/z; 6(1 - s) {C’“’ - C@}, -&I.7 ht /2 1 C (l)z2 dz = _th:{C’“’ + [s(3 - 3s -k s2) - 3?i2(1 - s)].(C”’

- C@))>,

-h,iZ

where

6 = &, - U/h,. The superscript (a) is used to indicate moduli of the axially oriented layer, and the superscript. (c) is used to indicate moduli of the two circumferentially oriented layers, each of which is assumed to be composed of the same fibrous material. For a given fiber-reinforced layer, let C’ denote the orthotropic elastic modulus matn’x referred to in-plane axes along, and perpendicular to the fiber direction. The elastic moduli referred to the principal in-plane cylinder directions, which are at an angle 4 with respect to the fiber axes, are given by Ci, = CL,(C;j,4) according to standard transformation laws [7]. For the axially oriented layer, q5= 0, and for the circumferentially oriented layer, + = 90”. In terms of Young’s moduli and Poisson’s ratios G,

=&,/(1

- ~,2~21),

G2

=~22lU

- y12y21),

c;, = c;, = v12c;2

=v21

c;,.

J. TASI

338

The numerical values for elastic constants used in the numerical illustration are given in Table Al. TABLE Al

Composite mod&

E1,, lb/in2 Ezz, lb/in2 VIZ

Glass-epoxy composite

Boron-epoxy composite

9.0 x 106 3.0 x 106 0.3

40.0 x 106 3.2 x 10’

0.2