Dynamic analysis of long composite cylindrical shells submerged in an acoustic medium

I 4

National ResearchCouncil Canada Institute for Marine Dynamics Conseil national de recherches Canada Institut de dynamique marine SYMPOSIUM ON SELECTED TOPICS OF MARINE HYDRODYNAMICS St. John's, Newfoundland August 7, 1991

DYNAMIC ANALYSIS OF LONG COMPOSITE CYLINDRICAL SHELLS

SUBMERGED IN AN ACOUSTIC LEDIUM

P. Twnprawate M. Cheniuka" C.C. Haiuig

* Department of Mechanical Engineering, Technical Univsity of Nova Sca. Ealifax, N.S.

** Martec Limited1 HiJc, N S

.I.

I

. I

National Research Council

?'

Canada

Institute for MatTe Dynamics

Conseil national de _recherches

Canada Institut de dynamaqe manne

SYMPOSIUM ON

SELECTED

_{TOPICS OF}

MARINE

_{IIYDRODYNAJ4JCS}

St. John's,

_Newfoundland

August 7, 1991

t!

_Qç-..

_c

/

DYNAMIC ANALYSIS OF LONG COMPOSITE CYLINDRICAL SHELLS

SUBMERGED IN AN ACOUSTIC ILEDIUM

P. Twinpiawate M. Chernuka" C.C. Hsiung

Department of Mechanical Engineering, Technical Univity of Nova Scia.. Halifax, N.S.

Martec Limited, M'ic, N.S.

ABSTRACT

A theoretical method for analyzing the free and forced vibratiónal behaviòr of long composite cylindrical shells in an acoustic medium is presented. The claical laminate theory is used to derive the 1aminite properties and for-mulate the equations of motion. The Íiatural frequencies

and the mode. shapes of vibrations for the cylindrical shells

in a vacuum are obtained. The dynamic fornes exerted by the surrounding infinite acoustic medium are derived in terms of a series of the vibrational mode slapes. Modal analysis techniques are then used to decouple the equa-tions òf motion. Finally, the dynamic responses of the cylindrical shélls in an acoustic medium are investigated. 1. INTRODUCTION

The importance of compoSite materials in hydrospace and aerospace applications is rapidly increasing.

The cylindrical shell configuration is widdy used in the

area of marine and aodynamic structures.

Thus, the dynamic response of the composite cylindxal shells is of great interest to na'al architects and aerodynarnicists.

The laminate properties of composite cylindrical shells usually are derived froni the classical h1nAte theory. This method can be readily applied to several types of shell construction, including beam stiffeneri, corrugated and sandwich shells [11. Isotropic cylindrical shells with closely spaced ring and/or stringer stiffeers can be ap-proximated as specially orthotropic cylindrical shells, which represents one kind of composite cylindrical shell [2,3]. Dong analyzed the free vibration cf laminated spe-cially orthotropic cylindrical shells under an arbitrary set

of homogeneous boundary conditions [4J.

For marine applications, it is important to account for the fluid medium when studying the dynamic characteris-tics of elastic structures. These characterisaics may differ considerably from those in vacuo. The dynamic analysis of isotropic long cylindrical shells in an acoustic medium for the three-dimensional case has been studied by Junger ¡5]. A similar study in which the modal analysis approach has been applied is described by Bleich and Baron [6J. However, in the latter-investigation the primary emphasis was placed on the structural response of the shells and second emphasis on the acoustical consiations. More recently, Geers and Felippa used the Doub'y Asymptotic /pprdmations (DAA) method [7] for vibration analysis o! submerged structures. Their analysis utilized the finite element method to discretize the structure..

In this paper, the dynamic analysis ofa long

compos-ite c7hndxicai shell with or without simpy supports at equal support spacing sumerged in an acoustic medium is carried out by the modal analysis as described in [6]. The

43

natural fr quicies and mode shapes of the vibrating sys-tem are readly obtained from a structural eigen-analysis. This modal approach is approate and relatively inex-pensive for thé special cylindrica geometry and boundary conditions comsidered in this paper. However, if the struc-tures are more complex or the boundary conditions are more general. numerical methods such as DAA appear to be more appropriate.

2. FORMULATION OF EQUATIONS OF MOTION FOR FItEE VIBRATION IN VACUUM

The differential equations cf motion are formulated in terms of the three middle surface displacement compo-nents u, s, u. The displaceme.s are assumed to be small compared to the shell thicknc which makes the problem linear. The Llne elements normal to the undeformed mid-dle stírface remain straight and normai to the deformed

n,.iddle surfa and they undergo no change in length. 2.1 Stress-Strain Relations

From the classical lrninte theory, the stress-strain relations in peincipal material coordinates for a lamina of an orthotroçc composite under plane stress are given by:

(

Q

011(1

'i

\

IQii = IQii n121

10

Qi,

Olici

O

Qj

(1)

where _{Q, are reduced 5tiffccL coefficients.}

In a more general case of orthotropic lamina with an angle O between the principal material axis and the

ge-ometrical body axis, the stress-strain relations are given as:

I'.\

f&1

L2 16

f c,\

,

I = IQis Qu Qie I e, (2)

r,,J

_IQ1. Qii Qss

\7sp/

where ¿,, are transform redud stiffnesses.

The valmes of i, are expressed in terms of_QJ and

angles [8]. The values of Q,, can be written in terms of the engineig constants as follows:

Qn = (I - l'3V2i) = (I

- t,'n)'

= (1

-'13

£

-2.2 Strain and Displacement of Shells

The liner strain and displacement relations can be written as

-(.Er0+ZXxO ;E,=e,+z

;7,=e,+zr,o

(3)

where the z rdinate is measured from póints on the middle surface in the direction normal to the middle surface,

x,y the adal and circumferential coordinate, respecticly,

= =

-

and e, = + are

mid surface strains,

z,, =

= -

and x,o = -2 the middle surface curvatures and

., w. w e the dis IAcements at the middle surface.

2.3 FoÑe. Resultants and the Equations of Motion

The force and moment resultants for n layered innii-nate are d,4niid as:

s

(N,,NpNgp..Wr.Jfy.Ma,)₌

E

_J

[,zzc,:r,]dz

(4)

Using Ec.(1.(3) and Eq.(4), integrating through the thickness, and sibstituting in the relevant equilibrium equations, we obtain the equation of motion in terms of

the displace2ts as:

82. 82v _10w

_8u

_82v A41 ₊

-

A4e(-f + = 82. 82w 18w 83u 82v A1 ₊

+A«(-

_{+ = vn,v (5)} 84w 84w

-

(2D13 + 4D0)8 ö

-I 8v ₊ 0v w -= m,w

where _{A,, = (hk-h$_i) are extensional stiffnesses,}

B,, = L EI

(hi,'-hL.1) are coupling stiffnesses,

D., = : 3 _{(h3-h_1) are bendingstiffnesses,} ra, is he mass per unit area of the shell.

For the tynmetric laminate

case, the values of the coupling st__-.aes B1 will be included in the governing. equations of ocEon which _{are not considered in this}

pa-per.

2.4 Free VibrszionofLong Composite _{Cylindrical Shells in}

Vacuum

The cylindr:caj shells _{are assumed to oscillate with} a natural frec2ency w and the displacement components

u,y._{w are propxtional to the harmonic function}

in 44.

Consequentl'r & explicj _{forxxz of the solutions are:} s

_-

-

_{_,}

_,

_{UPnNcoJ7cOs--e'}mix

np ,

w w

_ç-'ç.'

.mrz.-np,,

T

- ¿

_L,

4 (6) 'L mix L., ¿,W,,,,,fln_-_cos_e1w4 ¡L

Where U _{- V. W,. are coefficients of the series,}

m is & number of the deformation_{half waves} in thez direction,

n is the number of the deformation full waves in the y direction,

R, Iare the radius and length of support spadng.

respectively.

It

readily demonstrated that the simply supported

boundary condition is satisfied by Eq(6). This soluti

can al be used for long cylindrical shells with no support by substituting the length of the longitudinal half wave L in place I/rn.

B substituting Eq.(6) into E4.(5), we obtain the m-triz xm of equations of motion as

where T = A11(!!)3 +A64(j)3. =

.41)(j) +

Al2 mi

T3T(T)=T3i.

= 433()2+ A23n T::i =

= T,

=

D11(!)

_{+ (2D12 + 4Dss)(2)'(j) + +}

To obtain the nontrivial solution in Eq.(7), the

de-tirknt of the square matrix must be zerO which leacs

toe

wbe

A3+Ci7i2+C2À+C30

C1

=-(T+TT11),

C5 = -(T11TT+ 7T132T31 - TT23

_{- rL}

T11 -

TT).

The three natural frequencies can be obtained frxn Eq.(8} for each ra, n and the mode shapes of correspondig thr natural frequeúcies are obtained through the eigen function solution of Eq.(7).

In the case of mode n = O, the displacement w beco

zero and therefore only two natural frequencies are oo-t

3. DYNAMIC ANALYSIS OF LONG COMPOSITE CYLD(DRICAL SHELLS IN AN ACOUSTIC MEDIUM

3.1 Equations of Motion in Terms of Generalized Coor. na

The displacements cOrresponding to3 natural frequen-cies and 3 eigenmodes for each rn n can be written as:

woos

_mix U n'OsO.1

wooS

-mix =

E E E

Vrnnsi sinfl9

.sii lui h-I

as w _mix

w= E

EEWoon5t00si-1-mt 'OIi

= T31,

1Tu-J -T13 L T31 -T12

Ta-A.

-T

T13 T I

T-AJ

J V

\w/

J 0 (7)

where

'=

, k =1.2,3 foreach modes m and n. In order to simpify the following derivations, all sumrn:ions and the subipts m,n will be dÈopped.

The eigenmocies contain arbitrary factors t.. V, and

IV which are the function of time, and only the ratio

of these factors can be determined. By selecting the co-efficient w as generalized coordinates qe and using the abbreviations: (J v.hWC01119COS _s

V.

.mtz

=

w;

sin nO sin w

coen9sin,

nt:

the displacements u, y and w canj be expressed as the

func-tions of three generalized coordinates as u = giô.,.i + frø..3 + 33

= qi#.j +q2#.3 +3,3

w (q + q3 + q3)ö

By exploiting the orthogonality condition associated with the generalized coordinates,the equations of motion can be decoupled for each m, n and k a

Mj + Meie2qe = Qe (12) where the generalized masses Mà₌

+

A is the surface area of the shd]s,

m4 is the constant mass per im.i area of the shells.

3.2 Free Vibratioú Analysis in an Acoastic Medium When the cylindrical shell is subznesged in_{an acoustic}

medium, the generalized forces Qe

_{are only due to the}

radial pressure P of the fluid medium _{on the cylinder of} radiùs R. Eq.(12) can be writt _{in the form:}

je+iaie2qe_ fIA P.*S..4 (13)

The potential of an acoustic medium_{is defined by the} wave equation:

+C3+C3)0

(23;

Expressing Eq.(23) in the matrix form and setting th determinant of the square matrix eqtál to zero, we obtait

the frequency equation a

+ 03 + 03 7714

Wi3 Q3

-

2 _{fl. pQ2ÇL)}

In the case of a =0, we have orly2 natural frequencies corresponding to 2 mode shapes so that the left side of

Eq.(24) will have only 2 tmS.

3.3 Forced Vibration Analysis in an Austic Medium An arbitrary radial harmonic for P(t,9.z) can be ex-panded in a Fourier series

P(t. 9,:)

_{= E E P}

9 (25)

where P are arbitrary stans..

The generalized extnaI. forces _{Qe for each m, n}

be-comes:

Qe j P4

L4_Priuij .,,25f1*dA

(16) A A

Similar to the case. of free vibration, _{we obtain the} equations of motion as:

where _{c is the sound velocity in. an acoustic medium,}

4 is the velocity potential in an acoustic medium. The solution of the steady_{state responses of Eq.(14)} for each mode m, n can be written a

4 = A,,.,F(r)ccsn9sjn (15) where _{A, are arbitrary constants,}

the frequency of vibratioainan acoustic medium.

Substituting Eq.( 15) into Eq.(14),we obtain: P'(r) + !F'(r)_{+ (pZ - ()')F(r) = O} where

The solution of Eq.(18) _{which tiofles die coiidition} that f(r) should remain finite

_{as r -}

_is:

(14)

45

o.

41

r=R

_(U

(10) By substituting Eq.(11) and E. :5) into Eq.(18),

will obtain the value of_{___ and the velocity potential} bècomes:

(l F(R)

where _{F(r) can be defined for three eases as in Eq.(17)} The radial pressure exerted_{on the shells by surround} ¡ng fluid medium at r= R can be ¿e6ed as:

ai

=R

(2C

where p is the fluid densitr.

By substituting Eq.(19) into Ec20) and introducini

a

_{f f #'dA}

2

(21

rn, -

.V - n.(Ue1 _{+ W2)'}

the equations of motioü can then be rewritten _as: ¡k + fe P(tz 2+ 93)0e

F'(R) (22

For steady state response, we hare_{q = Cee"' and re} placing qe in Eq.(22), the eqations of motion willbecome

(24)

(26)

(Ifl31').

if > L;

F(r) _{= C}

H4t(iir),

j! ,c '(1

(.1. ¡

The matching condition at the srface of the shells

that the radial velocity cf die dispement,

ib, must b

equal to the velocity of the particles ai uid medium. Th can be expressed as:

The solution of steady state response for Eq.(7) can be written as q = Che'°. By subs itz±in q into Eq.(27),

we will obtain 3non-homogeneous eçations which define

the value of C

where N =

+ ,,, +

Substituting Ch into q, we will obtain the dispiace. menta u.v and w from Eq.(13) as:

a FtR) flhf2 _m4F'(R) -P,,,e''3isn9sia e Y1 e V3 F(R)

rflS(PNuI(R)

1

03V3

,,2 Q2 3

PCO9CL

01 V1 1

N_

m.F'(R)

-05V3

+ W32 -Q W3

The behavior of the radial dispLaenent due to the radial harzxxnic force will be studiéd. By letting Q O,

the value of displacement under the sazic load, w11, will

be obtainecL The absolute value of .1 can be derived as:

(29)

i

II

=

f_PQ2j)

f22 fl + W3 Q 4. NUMERICAL STUDY AND DISCVSSION

The five layered symmetric cross-ply cylindrical _sheU made from boron epoxy composite is used in this study. The shells have the following material properties:

Major Young's Modulus,31.Ox_{¿o psi. Major Poisson's}

ratio, 0.28

Minor Young's Modulus,2.7x 106 psi.

Shear MOdulus,O.75 x 106psi. Density, 192 x 106 k.Zc

The shell geometric properties indiide: length of

sup-port spacing =300 in, radius=90 in.

_{thickness of each} Iayer=.394 in. The isotropic steel sbdls with the same overall thickness are also considered _{comparison pur} poses. The density of the acoustic fluid

_{is .00112 "Ij}

and the sound velocity ¡n the fluid is 59604 in/sec. 4.1 Free_{Vibration Study}

The frequency equation (Eq.24) s solved graphically for mode m=1,n=1 _{in Fig.2 for composite shells and ¡ri} Fag.3 for steel shells. The results suggest similar behavior ¡n the composite shells and steel sheils. There is only one real root of frequency _{fl. In this de, the values of} the right side of Eq.24 _{are real until Cl} = !!1fL _{,then the}

values become complex which are not hown in the real plane graphs. The limit value of Q for which real root was obtained for mode gn=1,n=1 is 624 rad/sec. The valut of Q beyond these limit values are associated with the cp1ex roots which represent an outgoing wave which

st decay with time[6J.

4.2 !or

Vibration Study

The steady state response due to the harmouic ex-t.ernal force was also studied. The absoluté ratio ci L is calculated for the mode as ¡n free vibrations. The results show resonance when the frequency in the thud 0.. is equal to the real root of the frequency equthoa. Nondunensionalized displacements for forced vibrations are shown in Fig.4A (m=1,n=1) för composite shefls and Fig.5A (m=1,n=1) for steel shells. Beyond the limit

val--

Q = _{in Fig.4B for composite shells and Fig5B. for} steel shells, there is no resonance due to the dampig im-p&d by the complex roots. Another point to note is that there are no responses in some frequencies Q in the fluid. At such frequencies, the system vibrates like one with a vibration absorber..

5. CONCLUSIONS

The paper demonstrates that the dynamic behavior of composite shells in an acoustic medium resembles that of the steel: shells even though the material properti and

a2 _...L

the _{stiffnesses of composite material are quite different}

- Q' W3 _{from steel. However, the dynamic behavior can be quite} different if the properties of shell material or fluid are

al-tered. Two extreme cases should be noted:

a. When the stiffness of the shells is very low or the density of the shell material is very high ,the 3 naturá,! frequencies will be lower than the limit value, , and 3 real roots of Q will be obtained. The cylindrical shes will then vibrate with three resonances in the same man as

they do in a vacuum. r1L + .!L+

II)

&2 C.P3

(30)

b. When the density of the fluid is sufficiently low, the natura! frequencies of the shells will exceed thé limit

value, _{arid there will be no real root of Q at alL Iii}

this case, no resonances are expected.

as:

Cb= - f2' (28)

REFERENCES

R..F.P. van Zelst, "Hand Calculation Metd for

9ucklin of Composite Shell Structes", Paper Presented at ESA Workshop on Composite Design for Space Appb-cations. OctlS-18,1985.

W.H. Hopprnann,ll," Some Characteristi

ci the

Flexural Vibrations of Orthogonafl

Stiffened CyLccal

Shell. J. Acoustic. Soc; America. VöL3O, pp.77 '958).

Y.C. Das, "Vibrations of Ócthotropic Cy1indcal Shel]s.', Applied Scientific Resear'th, Vol. 12a.. pp317

1964).

S.B. Done, " Free Vibrazn of Laminated

Or-;hotropic Cylindrical Shells", J. Austic. Soc. America, Vol. 44, pp.1628 (1968).

5., M.C. Junger, "The PhysiCal Interpretation of the Expression for an Outgoing Wave Cylindrical

Coorda-nat&. J. Acoustic. Soc. America, Vol. 25, pp.40 1953). 6. H.H Bleich and M.L. Baron., "Free and Forced Vi-brations of an Infinitely Long Cylindrical Shell in an Infi-nite coustic Medium", J of App&d Mechanics Sol 21, pp.167 (1954).

'T.L. Geers and C.A. Feippa.. "Doubly Asynpotic Approximations for Vibration An3t7Sis of Submerged

Structures", J

Acoustic Soc America, Vol 3 o 4,

pp.1152 (1983).

8. R.M. Jones, "Mechanics of Composite Maer-.aW, Heniiphere Pub1ihing Comp., 195.

47

Fig.1 :Composite Cylind.icai Shell a.n I Crosssection

g.,

g'

ii

I 3 2 s -I -2 -3 -4 ¶4 L 4Tq.O4 3 4

Fig.3 :ROots of Freqnescy Equation foc Free Vibration_of

a Steel Shell(m=1,fl=1).

JIi)

T

2 4 _s

Fig.2 :Roots ofFrequcy Equation for Free Vibration o

Fig.4A: Ratio of Absolute Radial DisplacnentfOr.Forced Pig.5A: Ratio of Absolute R.aal Displacement for Forced Vibration of a 5-layered Composite Shefl(m=1,n=1) for Vibration of a 5-layered a Steel Shell(xn=1,n=1) for low

low frquendes. frequencies.

I

i

Fi.4B: Ratio of Absolute Radial Dispi nent for Forced Fig.5 B: Ratio of Absolute a'al Pisplacement for Forced

Vibration of a 5-layezed Composite She1(m=1n=1) for Vibration of a 5-layered a St Shell(m=1,n=1) for nigh

high frequencesf. frequencies.

su 7- s- 4- 3- s-s- --U