Static and dynamic instability of panels and cylindrical shells in subsonic potential flow

Journal of Sound and Vibration (1974) 32(2), 251-263

STATIC AND DYNAMIC INSTABILITY OF PANELS AND CYLINDRICAL SHELLS IN SUBSONIC POTENTIAL FLOW A. KORNECKI']" Department of Aerospace and Mechanical Sciences, Princeton Unirersity, Princeton, New Jersey 08540, U.S.A. (Received 8 May 1973, and in revised form 7 August 1973) It is proved that cylindrical shells and two dimensional flat panels constrained to zero displacement at their leading and trailing edges, and exposed to subsonic flow, can lose their stability by divergence (buckling) while in supersonic flow two-dimensional panels can only flutter. Moreover, it is proved that in incompressible flow flutter can only occur--if at all-above the critical divergence velocity. The proofs are based on a qualitative analysis of the expressions for generalized aerodynamic forces derived in references [I]-[3], and on the assumptions that linearization of the stability problem is admissible and that Galerkin's method is convergent.

1. INTRODUCTION Most ofthe literature on panel flutter is confined to investigations of panels in supersonic flow. In recent years, however, interest has increased in the dynamic behavior of panels in subsonic flow. A controversial problem in this context is whether the elastic panels incur divergence (static instability) or flutter (dynamic instability) at some value of flow velocity. An attempt is made in this note to clarify this point, on the basis of a qUalitative analysis of the governing equations of motion without resorting to detailed calculations. The method employed is due to Leipholz (e.g., see references [4] and [5]) involving the use of Galerkin's discretization technique and using only two admissible functions. Leipholz used his method to investigate stability of columns loaded by time-independent non-conservative forces. In a previous note the author [6] employed Leipholz's method for certain aeroelastic systems and the results were encouraging in that they agreed well with the known exact solutions. In this note the method is applied to a broad class of aeroelastic systems, and the results obtained conform with those of more exact analyses as well as with experiments.

2. FORMULATION OF THE PROBLEM Typical problems of panel flutter refer to thin-walled structural elements with one surface exposed to fluid flow parallel to it. The geometrical configurations discussed in this note are the following: a flat panel of infinite span, with a finite chord of length/, embedded in a rigid surface; a cylindrical shell bounded beyond its ends by rigid cylinders and either surrounded by a moving fluid (hereinafter called the "external case") or containing internal flow ("internal case"). t On leave from Technionr Israel Institute of Technology, Haifa, Israel. 251

252

A. KORNECKI

The governing equations of the lateral vibrationst W(.f, t) or W(~,, O,~t) of the panel or shell can be represented in the framework of the linear theory as az W E(W) + t,,h-~-+.

P ( W ) = O,

(1)

where E ( W ) is a (structural) linear homogeneous operator containing derivatives with respect to the space coordinates but not time, and P ( W ) is an (aerodynamic) homogeneous linear operator containing integrals and derivatives with respect to both the space coordinates and time. The lower sign preceding P ( W ) corresponds to the case of internal flow in a cylindrical duct. Hereinafter, whenever both signs appear, the lower one corresponds to the "internal case" unless otherwise stated. Introducing the dimensionless deflection amplitude w(x) and streamwise coordinate x = .~/I, and assuming for the panel or shell, respectively, 1

7 w(~, t) = w(x)e'%

(2)

or W ( ~ , O , t ) = e'~ ~ W.(~)cosnO, n-O

1

7 w.(~,) = w(x),

(3)

E(w) - pshlco 2 w + p U 2 Q(w, o9) = 0,

(4)

P ( W ) = p(w, o9) e z~ = p U 2 Q(w, og) e I~'.

(5)

reduces equation (1) to

with P ( W ) rewritten as

A s for the boundary conditions; the lateral deflections are assumed restrained at the edges, w= 0

at

x = 0, 1,

(6)

the remaining conditions being arbitrary, provided E(w) is a positive definite self-adjoint operator with respect to them. With aerodynamic forces absent, equations (4) and (6) at p = 0 represent free vibrations of panels (shells) in vacuum:

) The eigenvalues o92 of E(w) in equation (7) are real and positive; hence in view of equation (2) the motion is (neutrally) stable. With varying flow velocity the o92values change continuously; they can either remain real and positive, or become negative, or else become complex, depending on the nature of the aerodynamic forces, p U 2 Q. The limit of the stability domain is marked by their transition from positive to negative, or from real to complex. In the first case one has divergence, and in the second flutter. Since o92 governs the stability, while its own behavior depends on U, the relationship

No9 ~, u ) = 0 t A list of notation is given in Appendix 2.

(8)

I N S T A B I L I T Y OF PANELS A N D SHELLS

253

should be investigated in predicting the instability. The aeroelastic system is of the divergence type if at certain flow velocities it is subjected to divergence (buckling): i.e., if equation (8) yields real values of U for to = 0.t A practical method for determining the relationship (8) consists in systematic application of Galerkin's technique. It is assumed that the general solution of equation (4) can be represented as a convergent series

w(x) = ~. aJj(x),

(9)

J=*

where the aj are constants to be determined and each of the functionsL(x) satisfies all the boundary conditions, and furthermore 1

L(x)f.,(x)

d x = 6~,..

(10)

0

As fj(x) one can take, for instance, the normalized modes of free vibrations in a vacuum, satisfying equations (6) and (7) and the remaining boundary conditions. The aerodynamic operator can be represented, consistent with the deflections, as

Q(w, o9) = ~ al Q~(L, o9),

(11)

J=Z

where Qj(fj, og) denotes the aerodynamic pressure corresponding to fj(x). Substituting equations (9) and (11) in equation (4) and applying Galerkin's method yields

~aj[E,,j+pU2Q,,j-pfldo926,,j]=O,

m = 1,2 . . . . .

(12)

J

with the notation 1

E'~ = I f.(x) Ej[fj(X)] dx,

(13)

0 1

O"'f = I f,,,(x) Qj(f~, o)) dx = Qm.t(k, M),

(14)

0

k = ogl/U denoting the reduced frequency and Q,,j the generalized aerodynamic forces. Equations (12) have non-trivial solutions when their determinant vanishes: det ]Emj + pU2 Q,,o - P, hIe~ 6,,j[ = 0.

(15)

The series in equations (9) and (11) are in reality infinite and consequently equations (12) are an infinite set, but by virtue of the convergence of Galerkin's method (e.g., see reference [7]) this can be ignored and the discussion confined to j, m = 1, 2 only. Setting p = 0 in equation (15)--which is equivalent to reverting to equation (7)--and solving the corresponding characteristic equation with j, m = 1, 2 yields the frequencies of the free vibrations in vacuum:~

co~.2=

1

2--~/~/{E,, + E2z -T- Vt(EI1 + s

2 -- 4(Eu E22 - E~2)}.

(16)

I"Strictly speaking, in the general case the relationship (8) assumeffthe form F(o~,U) = 0. At small flow velocitiesthe eigenvaluesm are real or complex with positive imaginary parts. The limit of the stability domain is marked by the transition of the imaginary part of one of the m's from positive (or zero) to negative. If at the stability boundary the real part of the critical eigenvaluevanishes one has divergence,otherwise one has flutter. In all the cases considered in this note, however, the general frequency-flowvelocity relationship turns out to be of the type (8). .*The :F sign preceding the radical in equation (16) corresponds to to] and to2, respectively.

254

h. KORNECKI

The expression under the radical is positive, since it can be rewritten as ( E t t - E22) z + 4E~2. Moreover the inequality Ett E2~ - E22 > 0,

(17)

E(w) is a positive definite operator, i.e.,

holds, since by assumption

I

I wE(w) dx > 0,

(18)

0

and equation (17) is the expression of equation (18) by means of Galerkin's method (cf. reference [5]). Finally, since the quantities cox.+ are positive, equations (16) and (17) imply that Ejj > 0, j = 1, 2. (19) 3. POSSIBILITY OF DIVERGENCE Qualitative analysis of equation (15) is highly involved (if at all feasible) in the general case, because the frequency o9 appears in the generalized aerodynamic forces Q,~j (listed in Appendix 1) in a complicated form. If however one is concerned with the static instability only it suffices to consider the steady aerodynamic forces obtained from Qm~ by setting k = 0. Denoting these forces by qm~, qr,J =

Q,~jl~=o,

(20)

and substituting them in equation (15) yields det [Em~ +

pU2qr,,j -- pshlo9z t~,~j[ = 0.

(21)

Setting k = 0 in the expressions for Qm~in Appendix 1 (equations (A2)-(A7)) and separating the even and odd functions of a gives, for the fiat panels, co

1

I aCmj(e) d~, co

_

1

at

M < 1,

0

~ aS=~(a) de,

at

M > 1,

(22)

0

and evidently, in view of the properties of the C.,j and S~,j functions, q=~ = qjm,

q~ < O,

at

M < 1,

(23)

qml -qjm,

qjj = 0,

at

M > 1.

(24)

=

Similarly for the cylindrical shell in external flow equations (A9), (A10) and (A13), with k = 0, yield co

qmj . . . .

a2 Fe(~) Cm(e) de 0

where R

at

M < l,

(25)

INSTABILITYOF PANELSAND SHELLS

255

Since FeOls) is positive (see the note after equation (AI3)) and in view of the properties of Cm#(~), equation (25) implies qmj = qjm,

qjj < 0,

at

M < 1.

(26)

In the case of a shell containing internal flow equations (A13) and (A14) with k = 0 yield cO

1R

qmJ=

n i

/ ctz Fl(r/s) Cmj(ct)dot,

at

M < I.

(27)

O

Since the function Fl(r/$) is negative (cf. the note after equation (A15)), equation (27) implies qmj = qj,,

qjj > 0,

at

M < I.

(28)

Once the properties of the generalized steady aerodynamic forces are established one can proceed with the analysis. Solving the characteristic equation corresponding to equation (21), with j, m = 1, 2, givest o9~.2 = 89

+ L22 -T-"X/(LH + L22)2 - 4(LH L22 - L,zL2,)},

(29)

where l

Lm.j(U) = ~shl(fmj +_pUZ qmj).

(30)

Divergence occurs at a flow velocity U = Uo at which 092 vanishes: i.i~., at U = Uo,

A( Uo) = Lll L22 - Ll2 L211o=v, = 0.

(31)

To show that divergence is possible, it must be shown that equation (31) can be satisfied at a certain real value of the flow velocity. This can be readily shown for subsonic flow. Indeed, at U = 0, Lmj = E,,~; hence by equation (19) Lj~(0) > 0 and in view of equation (17) A(0) > 0.

(32)

Now, in subsonic flow the qjj are negative (positive in the internal case; cf., equations (23), (26) and (28)); thus the Ljj decrease hi all cases, with increasing flow velocity, and at some velocity U = Uo one of them, say Lit , vanishes, rendering A(U) negative:

A(Uo) =-L,2.L2tlu~t,o = -L~2 ~<0,

(33)

where use was made of the equality L12 = L21 resulting from equations (30), (23), (26) and (28). Comparing the inequalities (32) and (33) leads to the conclusion that there exists a flow velocity 0 < Uo <<.Uo (34) at which equation (31) is satisfied, indicating that in subsonic flow two-dimensional panels and cylindrical shells can lose their stability by divergence. The critical (divergence) flow velocity can be evaluated approximately by solving equation (3 I). It should be noted that, since divergence occurs at co2 = 0, the critical velocity Uo is obtainable by solving the static eigenvalue problem. Thus setting co = 0 in equation (4) and denoting

Q(w, to = O) = q(w),

(35)

E(w) + pU2q(w) = 0.

(36)

one has

t The q: sign preceding the radical in equation (29) corresponds to ta] and eg], respectively.

256

A. KORNECKI

U 2 may now be regarded as the eigenvalue of this static problem. On application of Galerkin's procedure, equation (36) reduces to equation (31). Similarly it can be shown that divergence of two-dimensional panels in supersonic flow is impossible. Indeed in this case equations (31) and (24) give the criterion of divergence as A(Uo) = EH E22 - E~2 + (pU2 q12) z = 0

This equation cannot be satisfied for any real values of U (by virtue of equation (17)); thus divergence of a two-dimensional panel is ruled out in supersonic flow, and the panel can lose its stability only by flutter. Summing up, one can conclude that in subsonic flow both the panel and the shell are aeroelastic systems of a divergence type (in the sense defined above after equation (8)). In supersonic flow the two-dimensional panel is a flutter type system, and as to the shell no general conclusions can be made. 4. FLU'rq'ER IN INCOMPRESSIBLE FLOW The possibility of divergence in subsonic flow does not exclude flutter. It should thus be checked whether flutter is possible in this range of flow velocity, and, if it is, whether the corresponding critical flutter velocity is below or above the critical divergence velocity. To examine the dynamic instability, unsteady aerodynamic forces must be taken into account. Since the forces Q,~j are in general complicated functions of to the analysis here is confined to incompressible flow--the only case where the frequency can be isolated explicitly from the expressions for the generalized forces. Indeed, upon setting M = 0 in equations (A1), (A2), (A8) and (A13) it is evident that in all three configurations considered, the total aerodynamic pressure can be expressed as p = p U 2 Q(w) = pUZ[q(w) + k 2 H(w) + 2ikF(w)],

(37)

where the individual functions are independent of the frequency. Thus, for instance, for the flat panel equations (A1) and (A3) with M = 0, on separation of even and odd functions of 0r, give

q(w) = -- - | txC(x) dot, n,J

H(w) = - -~ 0 ao

F(w) =

-~ I S(x) dot, 0

where 1

C(x) = I w(u) cos ~(u -

x) du,

0 1

S(x) = ~ w(u) sin ct(u -- x) du. 0

(38)

INSTABILITY O F P A N E L S A N D SIIELLS

257

Similar expressions for the shell can be readily obtained from equations (A8), (AI0), (A13) and (AI4) and need not be written out here. Substitution of equation (37) in equation (4) yields E(w) - p~lzlo92[w -T-fill(w)] + pU2[q(w) + 2ikF(w)] = 0,

(39)

where it denotes the mass ratio pl It = P s h

Setting U = 0 in the left hand side of equation (39) yields the governing equation of free vibrations of a panel (shell) in an incompressible fluid at rest: [p--~d E-o92(1-Y-pH)] w = 0 .

(40)

The term ItH(w) embodying the inertia force, orthe "apparent mass" of fluid set into motion by the oscillating surface, contributes only to the kinetic energy of the system (see, e.g., reference [8]) and cannot render the system non-conservative (see, e.g., reference [9]). Equation (40) can be looked upon as a generalization of equation (7) of free vibrations of a panel (shell) in a vacuum. To solve equation (40) by Galerkin's method, expand the deflection w(x) in a convergent series of modal functions, (41)

w(x) = ~. ajfj(x), J=l

where eachfj(x) satisfies all boundary conditions and (unlike equation (10)) the "generalized" orthogonality conditions, 1

.

f,.(x)[fj(x) T- lt H ( f j)] dx = 6mj"

(42)

0

Introducing series (41) in equation (40) and applying Galerkin's technique yields, in view of equation (42), det I/~'mj- p, hlw 2 6mjI = 0, (43) where, in the same manner as for equation (13), 1

= J A (x)

dx.

(44)

0

Solving th e characteristic equation corresponding to equation (43) with j, m = 1, 2 gives two real and positive frequencies of the free vibrations of the panel (shell) in still fluid, in the form given by equation (16) with E,,j replaced by E,,~ and, similarly to equations (17) and (19), E~, E22 - E~22> 0,

/~,j > 0.

j = 1, 2.

(45a, b)

Applying Galerkin's method tO equation (39) and using the modal functions~(x)--instead of fj(x) as in the previous section--results in det

-+ pu2(r

+ 2ik~mj) - pshloJ 2 3mJI = 0,

(46)

where the generalized steady aerodynamic forces, qmJ, are determined by equations (22a), (25) and (27) with M = 0 andre(x) replaced with~(x). The (gyroscopic) forces ff,,j are 1

~,,.~ =

~ f,.(x) 0

F(fj) dx,

(47)

258

A. KORNECKI

or, in view of equation (38) and (A7),

}

l~tXl

~,.j=-] {ff fm(x)~(u)sinot{u-x)dudx d~

(48)

~ 8 ~oo in the case of a flatpanel; similar expressions can be written out in the case of a shell.Since the properties of the generalized aerodynamic forces are independent of the modal functions, one has, by equations (23) and (26),

q,,j = qjm,

qjj < 0,

(49a)

for the panel and the external case of a shell, and, by equation (28), qmj = ~j,,,

qjj > 0,

(49b)

for the internal case of a shell. Moreover equation (48) implies that ~ltlj = --~jtrl,

'~Jj = O.

(50)

Now, upon denoting (el. equation (30))

o ) i ( "j -+ v~J~,Tmj)= r.mAu),

(50

where evidently ~.,,j = T.j,,, equation (46) becomes

det

. U

Lmj +_ 2qlco~-ymj- to 2 6mj = 0.

(52)

Solving the corresponding characteristic equation with], m = 1,2 givest

co~.~=l{f.. + ~ +

v 2 ~- ~/(E,, + L ~ + v ~ ) ~ - 4 C H L , -I.D},

(53)

where U V = 2/t T~,2.

(54)

Equation (53) implies that divergence occurs when 2(U,) = L,, L22 - L~2 = 0.

(55)

Equation (55) yields (approximately) the same critical flow velocity U~} as equation (31), both being solutions of the same static equation (36)--except for the different modal functions. Flutter occurs when the expression under the radical in equation (53) vanishes: i.e., at

U=Ur, (L~, + r.2, + V2)" - 4(L,n L22 - L~2) = O,

(56a)

(T.,, - l_.a,)2 + V2[2(Ln + T-,,2) + V 21 + 4r.~a2 = O.

(56b)

or in another form

Comparing the above two expressions shows that equation (56) can be satisfied only when both functions T.H and L, 2 are negative: Lgj(Ur) < O,

j = l, 2.

t" The :F sign preceding the radical in equation (53) corresponds to o9~and {o], respectively.

INSTABILITY OF PANELS AND SHELLS

259

Since, however, Ljj = L,jj(U) are continuous functions of the flow velocity U and in still air both are positive, Z,jj(O) > 0, j = 1, 2, by equations (51) and (45b), then at some flow velocity Uo lower than the critical flutter speed, 0 < Uo < UF,

(57)

one of the T_.jjfunctions, say T~H, must vanish: Z , , ( U o ) = O.

On the other hand the expression A(U) is also a continuous function of U. In still air ,d(0) is positive (cf. equations (51) and (45a). At the criitcal divergence speed A(Uo) Vanishes (equation (55)), and at Uo it is non-positive,

2(Uo) -Z~ ~
hence Uo/> Vo.

(58)

Ue > Uo >i Uo:

(59)

The inequalities (57) and (58) imply

i.e., flutter of panels and shells in incompressible flow can only occur (i fat all) after divergence. 5. CONCLUSIONS AND COMPARISON WITH OTHER RESULTS It was shown above that in incompressible flow divergence is critical for shells and twodimensional panels, and flutter can only occur (if at all) after divergence. In compressible subsonic flow divergence is possible, but it is not clear whether it is critical. The proof can be extended to panels of finite span. These results are in agreement with all the available "exact" analyses and experiments, which indicate that in compressible subsonic flow divergence is also critical. Indeed, numerical results reported in references [10]-[15] show that dicergence is critical in incompressible as well as in compressible subsonic flow, for flat panels of arbitrary (even very low [10, 12]) aspect ratios. This is also confirmed by experiments [14-18]. Flutter of panels can occur--according to the linear theory--after divergence [10, 13, 14] or even as soon as the divergence speed is exceeded [12], but non-linear analysis (structural nonlinearities only being considered) shows that post-divergence flutter is impossible I1 I, 15]. The results of experiments [I 5-17] are inconclusive. As to the cylindrical shells numerical results [19, 20] confirm that divergence is critical in the case of internal, incompressible flow and post-divergence flutter can occur. This is confirmed also by experiments [19, 21]. Experiments have revealed that divergence is critical for cylinders in external subsonic flow up to M = 0.9 [22, 23], and post-divergence flutter was never observed. These results however should be accepted with caution, because theory indicates that in the case of subsonic external flow an out of phase aerodynamic force appears [2] acting like "negative damping". It has been shown in this note that in supersonic flow panels can lose their stability by flutter only. This is confirmed by numerical and experimental investigations. Finally reference [24] should be mentioned, where an interesting physical interpretation was made of the fact that panels and shells in subsonic flow lose their stability mainly by divergence while in supersonic flow flutter is the most common form of instability.

260

A. K O R N E C K I

Although these conclusions are in agreement with all the available solutions obtained by complete analyses of the individual systems involved it should be underlined that the p r o o f here is incomplete. Indeed, from the strict, mathematical point of view one should also prove that the convergence of the approximate solutions is uniform so that the sequence of occurrence of divergence and flutter (in all configurations considered) is preser~,ed down to the two-admissible-function approximation (see, e.g., reference [25]).

ACKNOWLEDGMENTS I would like to express sincere appreciation to E. H. Dowell for encouragement, valuable comments and stimulating discussions. I am also grateful to H. Leiphoh for making his paper [4] available to me prior to publication, to T. Kai for translation of reference [15] and to the anonymous reviewer for valuable remarks.

REFERENCES 1. E. H. DOWELL1967 Quarterly of Applied Mathematics 24, 331-338. Generalized aerodynamic forces on a flexible plate undergoing transient motions. 2. E. H. DOWELLand S. WlDNALL1966 Journal of the American Institute of Aerodynamics and Astronautics 4, 607-610. Generalized aerodynamic forces on an oscillating cylindrical shell. 3. S. WIDNALLand E. H. DOWELL1967 Jottrnal of Sound and Vibration 6, 71-86. Aerodynamic forces on an oscillating cylindrical duct with an internal flow. 4. H. LEIPHOLZand R. N. D U B E Y 1972 SolidMechanivs Division, University of IVaterloo, Canada, Paper No. 104. Qualitative stability theory of completely supported rods subjected to follower forces. 5. H. LEU'HOLZ1972 Journal of Applied Mechanics 39, 717-722. On the sufficiency o f the energy criterion for the stability of certain non-conservative systems. 6. A. KORNECKI1973 Journal of Applied Mechanics 40, 616-617. On the character of instability of certain aeroelastic systems. 7. H. LEIFHOLZ1970 Staiblity theory. New York and Londoni Academic Press. See p. 192-230. 8. H. LAMB1945 Hydrodynamivs (sixth edition), New York: Dover Publications. See p. 163. 9. F. KITo 1970 Principles of Hydroelasticity. Tokyo: Keio University. 10. E. H. DOWELL 1966 Journal of the American Institute of Aeronautics and Astronautics 4, pp. 1370-1377. Flutter of infinitely long plates and shells. Part 1 : Plates. 11. E. H. DOWELL 1967 Journal of the American Institute of Aeronautics and Astronautics 5, pp. 1856-1863. Non-linear oscillations of a fluttering plate II. 12. C. H. ELLEN 1973 Journal of Applied Mechanics 40, 68-72. The stability of simply supported rectangular surfaces in uniform subsonic flow. 13. D. S. WEAVERand R. E. UNNY 1970 Journal of Applied Mechanics 37, 823-826. The hydroelastic stability of a flat plate. Discussion by E. Dowell ibid., volume 38, p. 565. 14. T. ISHU 1965 American Institute of Aeronautics and Astronautics Meeting, Paper 65-772. Aeroelastic instabilities of simply supported panels in subsonic flow. 15. T. ISHU 1967 Journal of the Japanese Society for Aeronautics and Space Science 15, 154--165. Flutter and divergence of two dimensional panels in subsonic and transonic flow (in Japanese). 16. J. DUGUNJI,E. H. DOWELLand B. PERKIN1963 Journal of the American Institute of Aeronautics and Astronautics 1, 1146-I 154. Subsonic flutter of panels on continuous elastic foundations. 17. T. GISLASON1971 Journal of the American Institute of Aeronautics and Astronautivs 9, 2252-2258. Experimental investigations of panel divergence at subsonic speeds. 18. D. J. JOHNS1964 AGARD Report 484. The present status of panel flutter. 19. M. P. PAIDOUSSlSand J. P. DENISE1972 Journal of Sound and Vibration 20, 9-26. Flutter of thin cylindrical shells conveying fluid. 20. D. S. WEAVERand R. E. UNNY 1973 Journal of Applied Mechanics 40, 48-53. On the dynamic stability of fluid conveying pipes. 21. H.L. DODDSand H. L. RUNYAN1965 N A S A T N D-2870. Effect of highvelocity fluid flow on the bending vibrations and static div~:rgenceof a simply supported pipe. 22. H. HORN 1972 Ph.D. Thesis, The University of Texas at Austin. An analytical and experimental

261

INSTABILITY OF PANELS AND SHELLS

23. 24. 25. 26.

investigation of the aeroelastic stability of cylindrical shells at the subsonic and supersonic Mach numbers. W. E. WHITE 1971 ArnoM Engineering Development Center, Tennessee Report AEDC-TR71-173. Investigation of the aeroelastic stability of thin cylindrical shells at subsonic Mach numbers. W. J. ANDERSON1970 Developments in theoreticalandappliedmechanics (edited by D. Frederick) 4, 319-328. Aeroelastic stability of plates and cylinders. Oxford: Pergamon Press. A. KORNECK!1973 Journal of Applied Mechanics 40, 829-830. Discussion of reference [20]. E. H. DOWELL1973 Personal communication. APPENDIX I THE GENERALIZED AERODYNAMIC FORCES

Two dhnensional flat panel [1, 26]

. . . . .

t- (k, So

w(u) e'~r

du

d~,

(A1)

where V =

~b(k, ct) = V ~

O~2 - -

for

= +iV~M

-

tx)2,

(A2)

v > 0, for

Qm~(a) = - 2-n

M2(k

v> 0

(A3) and

k - ct ~ 0;

[ ~b(k, ct) [C,,j(ct) + iS,,j(ct)] d~t,

(A4)

(A5)

-co

where 1 I

cm@c-)= f f f (x)f cu) cos =cu- x)dudx,

(A6)

O0

11

Smj(a) = f f fm(x)fi(u)sina(u-x)dudx.

(A7)

O0

Cylindrical shell [2, 3] Q(w) = pU 2 = - 2n 1

(-~)2F(ct) --00

Q,,I = -

w(u)ef~t"-X~du d~t,

1 R f {(k - a) 2 F(ct)[Cmj(~) + iSmj(a)]} d~t, 21r 1

where F(a) is defined by equations (A10)-(AlS).

(A8)

0

(A9)

262

A. KORNECKI

(a) External flow [2, 26]

/(.(7)

F(~) = F,(q) =

~K~(~)

for

H{,')(~)

= Fe(~) =

= Fe(~) =

cn~"(~)

v > 0,

(A10)

for

v< 0

and

k - ~ < 0.

(AI 1)

for

v< 0

and

k - cc > 0,

(A12)

where R 1"/= T V%,

R ~ = 7 V'=-~.

(AI3)

It should be noted that the function F,(q) is positive, because the K,(q) are positive and K'(q) are negative (for n/> 0 and q > 0). (b) Internal flow [3] /.(,i) F ( ~ ) = F~(q) = - ~

for

v > 0,

(A14)

J.(~) = Fl(~) = - - URn)

for

v < 0.

(a15)

It should be noted that Fl(q) < 0 since 1,(r/) and I'(q) are both positive.

A P P E N D I X II E(w)

structural operator

~

streamwise mode shapes satisfying equations (10) streamwise mode shapes satisfying equation (42) (p = 1,2) Hankel functions of order n thickness of plate (shell) nth order Bessel functions, modified Bessel functions of first kind, and third kind, respectively

Ej(~) structural operator corresponding t o j t h mode E~j ~ f,.(x) E~(fj) dx (x) (x) H~V~(x) h

J.(x), 1.(x) }

U

ZmJ 1 [Emj+ pU2qmj] (see equation (30)) 1 plate (shell) length M Mach number p(w) aerodynamic pressure amplitude -

Q(w) Qj Qmz q qmJ R t U W

p(w)

dimensionless pressure amplitude due t o j t h mode f~A,(x)

O~(x)dx

QIk-o radius of cylinder time free stream velocity lateral (radial) deflection of panel (shell)

INSTABILITY OF PANELS AND SttELLS

w(x) dimensionless amplitude of deflection; see equations (2) and (3) 27, X streamwise coordinate, x = ~/1 Fourier transform variable aCU) L11 Lz, - Lt2 Lzt ; see equation (31) &,s Kronecker's delta P V

P p~ CO

pl p~h a= _ M Z ( k _ a)z; see equation (A2)

air density density of plate (shell) material frequency

Superscripts

indicates that modal functions.l~(x) should be taken instead offj(x) Subscripts j,m tt

D F

streamwise mode numbers circumferential mode number on the shell divergence flutter

263