Flutter analysis of flat rectangular panels based on three-dimensional supersonic potential flow

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*- . , r AUGUST 1963 AIAA JOURNAL VOL. 1, NO. 8 I

Flutter Analysis of Flat Rectangular Panels Based on

Three-Dimensional Supersonic Potential Flow

H. J. CUNNINGHAM /

N A S A Langley Research Center

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1 ' 31

A procedure has been developed for computing flutter sties of single finite panels, particularly a t low supersonic Mach numbers where static and quasistatic aerodynamic ap- proximations are not valid. Air forces from exact linearized potential-flow theory are used.

The panel is considered a s finely divided into many boxes, and the aerodynamic influence coefficients between all phirs of boxes are computed by numerical integration. The flutter analysis is a modal type of analysis which can be used with the box method for calculation of the flutter stability of any flat or nearly flat panel, whether of isotropic or anisotropic stiffness, and even for buckled panels for which the flutter is a small-amplitude, simple harmonic, superimposed motion to which linear theory would apply. Certain results are presented for flat unstressed isotropic panels. Variables whose effects were studied are panel length-to- width ratio, Mach number, and the air cavity behind the panel. Panels with simply sup- ported and with clamped edges were studied.

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Nomenclature Mi = generalized mass for mode i = MJlwmA (applicable for uniform panel only) Mi* a . = speed of sound N = number of modes in a flutter analysis = perturbation pressure due to mode j [Eq. (17)] = aerodynamic influence coefficient giving the ve- Apj A , = dynamic pressure of airstream locity potential at a box due to unit downwash q = time-varying generalized coordinate of motion for on another box [Eqs. (3) and (911 pi( T ) , q j ( T )

I

B. = number of boxes in stream direction modes i and j i = complex amplitude of qj [Eq. (14)j B,. = number of boxes in cross-stream direction 4 7 i = generalized aerodynamic force from the pressure d = depth of cavity behind the panel Qii due to mode j and the modal deflection of mode B = structural damping coefficient of the panel = distribution over the panel of mode shape for hijhj i [Eq. (1811 modes i and j [Eq. (14)] &..* = nondimensional computational quantity con- = distribution of time-varying vertical deflection of tained in Qij [Eq. (21)] H ~ ( x , Y , T ) panel for modej [Eq. (14)] t = thickness of panel a = ( - l)l'* = unit of imaginaries = transformed panel coordinates in x and y direc- U P Z G 1 , I ~ ~ , Z G ) = integrals in A, [Eqs. (10-12)] tions, respectively, based on 4 2 aa a reference k, = U B / V = reduced frequency with reference length e length [Eq. (5)] kl = ol/V = reduced frequency with reference length 1 UI,U~,UI,UZ = lower and upper limits of integration with respect = length of panel in stream direction to u and to u [Eq. (S)] mA = maas per unit of surface area of panel W i = downwash velocity at the panel surface for modej M = V / a , = Mach number W = width of panel in the cross-stream direction X,Y = panel coordinates in stream and cross-stream directions, respectively (see Fig. l), baaed on B as a reference length in Eqs. (3) and (5) * Aerospace Engineer. Member AIAA. Xm,Yn I KeprpPeted from AIAA <Q.ULy&- Copyright, 1963, by the Anierican Institute of Aeronautics and Astronadtics, and 1796 H. J. CUNNINGHAM AIAA JOURNAL

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such a panel with added supports that divide the length of 3 the panel into equal bays. For this type of panel the general- . ‘ ized aerodynamic forces needed in the flutter determinant

lil

can be obtained by evaluation of closed-form expressions.

For finite-width panels, however, closed-form expressions have not been found for the aerodynamic force terms even for simple mode shapes of the panel.

The subject of the present paper is the determination of the aerodynamic forces by, in effect, a numerical integration process and the subsequent use of the air forces to compute flutter boundaries. For this problem, current high-speed computing machines make practicable the obtaining of solu- tions by such “brute force” methods that would not be attempted otherwise. The panel is considered to be finely divided into a large number of equal-size boxes. The aero- dynamic influence coefficient of each box on each other box is calculated and used in a modal-type flutter stability analysis. The mode shape properties can be supplied from either calculation or experimental measurement.

A number of flutter results are presented for flat unstressed panels with isotropic stiffness and by use of calculated mode shapes. Calculated modal frequencies mere used, with one exception. The principal variables studied are the panel Fig. 1 Plan view of panel divided into boxes with coordi- length-to-width ratio and the flow Mach number. Effects nate system, dimensions, and a forward-facing Mach cone of the air cavity behind the panels also are examined in a with apex at b o x center xm,yn.

qualitative sense.

Although the results presented are limited to simple panels, = panel coordinates based on reference lengths 1 a,? this modal analysis with a box method has a broad applica- and w, respectively tion and lends itself to the calculation of the flutter stability ff,ff* = see Eqs. (26-28) of any flat or nearly flat panel, whether unstressed or stressed B = ( f i f 2 - 1 ) 1 / 2 (as by thermal expansion), whether of isotropic or anisotropic 6 = w/Bz. = width of box stiffness, and even for buckled panels for which the flutter , 7 = dummy variable of integration for y

is a small-amplitude, simple harmonic, superimposed motion ’

p = ma/pl = panel-to-air mass density ratio to which linear theory would apply.

E = dummy variable of integration for z P = density of airstream U = lBz8/wE8, length-to-width ratio of box 7 = time Analysis cp = cp(m,n) = velocity potential at center of box m,n = velocity potential a t center of box m,n due to unit The panel to be analyzed and the coordinate system are +(rn,n) downwash over box p,v shown in Fig. 1. It is a single rectangular panel of length w = frequency of flutter motion 1 , width w . and the side edges are aligned with the remote w i J w ~ , w S = natural frequencies of panel modes i and j and of wind direction. It is a flexible part of an otherwise inflexible a chosen base or reference mode, respectively surface which extends at least far cnough so that upper and

D = ( w J w ) * ( 1 + i g ) , flutter eigenvalue parameter

lower surfaces of the panel induce no aerodynamic effects i j = M 2 k , / g 2 upon cach other. The supersonic stream passes over the Subscripts upper surface. No account is taken of any perturbation B = base or reference value pressures on the panel lower surface except for some re- = indices denoting mode numbers i,j marks regarding cavity effects near the end of the paper.

m,n = indices denoting streamwise and cross-stream The panel is divided into a number of equal-size rectangu- numbered location of a box lar boxes. The number of boxes in the stream and cross- T,S = indices denoting rclative locations of influencing B. and B,,, respectively. For con- I and influenced box stream direction are venience and for purposes of reference in the computational t e = value at trailing edge of panel procedure, the boxes are numbered in sequence, beginning at P,V = indices with same function a~ m,n the box nearest the origin of coordinates. In the stream

direction the index m = 0, 1, 2, . . . B, - 1, and in the cross-

Introduction

stream direction n = 0, 1, 2, . . . B,. - 1. A second set of

box index numbers is needed, and these are p = 0, 1, 2, . . .

HERE has been considerable interest and activity in the

B, - 1 and v = 0, 1 , 2 , . . . B,, - 1. Thus, the aerodynamic

T a n a l y s i s of panel flutter. Perhaps the most recent re- influence upon any box m,n due to any other box p,v can be views of the work done are those by Stocker’ and by Fung2 referred to.

which cover both theoretical and experimental investiga- tions. The present paper is concerned with an important type of panel and speed regime for which analysis has been Aerodynamic Influence Coefficients lacking. That type of panel is the single rectangular one (embedded in a rigid surface), and the speed regime is the It is assumed that the panel is divided into a sufficiently low supersonic, especially below Mach number 21/2. For these fine gridwork of boxes so that the downwash over any one Mach numbers, simple approximations for the aerodynamic box can be taken as uniformly distributed a t any instant forces, such as from Ackeret theory or piston theory, are and that the resulting pressure perturbation a t the center not satisfactory. Use of the exact theory for linearized of each box is a sufficiently accurate average of the pressure supersonic flow is a logical step.

This exact theory was distribution over that box.

applied by Luke and St. JohnJ to an effectively infinite- For convenience in the computational procedure, a refer- width panel separated into bays by supports equally spaced ence length is choscn as e, the width of a box; that is, e = along the width.

Zeijde14 obtained additional results for w/B,.. The velocity potential a t the center of box m,n due AUGUST 1963 FLUTTER OF FLAT RECTANGULAR PANELS 1797 to a uniformly distributed downwash w ( p , v ) over the box both sides of the Mach cone). For this condition p,v can be expressed for a simple-harmonic time variation as u' - (10) d m , n ) = E W ( M , V ) . ~ , ( ~ , S ) ' (1) l o l = ill e-(@n/z)u J,, where The second form of Eq. (9) occurs for portions of a box cut r = m - p s = n - v

.. (2)

by one side of the Mach cone (v # n), so that the limit v2 = . , : .

and, from Eq. (34) of Ref. 5, _ + _ u, v1 = 2s - 1 2 1. Since v2 = u, then COS-~(V~/U) = 0 and , in which the area of integration S,,," is the area of box p,v.

i n the denominator, the indicated real part of the radical term has a nonzero value for any part of SP," which lies ahead of the forward facing Mach cone with apex a t zm, y,, (see 1). All boxes and parts of boxes lying behind the Mach Fig.

cone have a zero contribution to Aq(r,s). The frequency Mach number parameter is The third form Io3 occurs for boxes that are completely ahead of the Mach cone and also for portions of boxes ahead of the = M2k,/P2 (4) point where the Mach cone passes out through the side of the box : where the reduced frequency k , = wt/V contains the chosen reference length e .

As in Ref. 5, a transformation of variable is made as follows :

5 , - E = pu/2 y* - f = v/2 utcos(PQ/2M)(u2 - v 2 ) ' / 2 - 1

( 5 ) du du (12)

L ( u 2 - u2)1/2

)

Then Table 1 shows all possible relative locations of a box and the Mach cone. The applicable integral, whether Io1, l a 2 , or To3 is shown, and the limits of integration ul, v 2 , ul, and uz are listed. The limits shown are consistent with the refer- ence length e chosen for Eq. (1) and with the transformation of Eq. ( 5 ) . The quantity u = ZBJwB. is the length-to-width ratio of a box.

where the upper and lower limits of the surface integrals, With the assumptions that have been made, the aero- indicated by vl, vz, ul, and up, are determined for any given dynamic influence coefficients have right-left symmetry box by its edges except where the forward Mach cone emanat- about s = 0; that is, ing from x , , y . passes through the box.

The integral of Eq. (G) is the one to be evaluated nu-

A,(r,s) = 4 b . 7 - 4 (13)

merically. I t is to be noted that the integrand contains a Thus, all of the A,(r,s) which are needed for any box are

singularity of order (-3) a t u = v , that is, along the Mach

obtained for either one of the rear corner boxes.

z , , y , , . In order to avoid the potential inaccuracies lines from Once all A,(r,s) are computed, the total cpl(m,n) for any involved in integrating in the vicinity of this singularity, downwash mode j a t the center of any single hox m,n is ob- the numerator of the integrand is replaced as follows: tained by a matrix multiplication: C O S ( ~ Q / ~ M ) ( U ~ - 02)'/2 E

[cos(pQ/2M)(u2 - v 2 ) ' / 2 - I ] + 1 (7)

The integration with respect to v can be partly carried out formally: cos(pQ/2M)(u2 - v ? ) ' / ? - 1

(u' - 2 1 / 2 2) dv =

+

A , is zero where m - p = r would be in which any element v ) negative.

Downwash The dovrnmash w, is employed as follows. Let the panel The integrand of the remaining integral with respect to v is deflection in mode j be expressed for simple-harmonic time zero a t u = v , and the integral can be evaluated more ao- variation as curately. Substitution of Eq. (8) into A,(r,s) gives the re- sult (15) H ~ ( x , Y , ~ ) = !11(7)h(z,~) = ~ i ( ~ ) e ' ~ ~ h ( ~ , ~ / ) where Q, is the complex ampiitude of the generalized coordi- nate of motion, and h,(z.y) is the shape of the deflection mode.

The downwash ratio is U,cos(/3Q/2M)(u2 - v2)"2 - I dv du (9)

L (u2 - v2)l/2 )

There is a need for only t,hree forms of Eq. (9). The first is for the condition that v1 = -02 = -u (which can occur only for v = n, and then only for the portion of n box p,v cut b y 1'798 H. J. CUNNINGHAM AIAA JOURNAL where f is a convenient streamwise coordinate that is zero at flutter determinant is the integral over the panel surface of the panel leading edge and 1.0 at the trailing edge. Substi- the product of the deflection mode shape hi times Api; that is, tution of Eq. (16) into Eq. (14) gives the result

~ < i = - C J J h i ("" + i d(l*)d(Wg) (19)

1 b* v

The coordinate fi ranges from zero at the left edge to 1.0 at in which hi is hj(z,,,yy). Thus, the cp,(m,n) are obtainable i n the right edge. If the term hi(bcpl/bf) is integrated by parts, a systematic manner for each modej used in a flutter analysis.

there is obtained Generalized Aerodynamic Forces The perturbation pressure Apl is obtained from the velocity potential by

J cpi ($ - i $ hi)d(lz) ] ~ ( w z ) (20)

Since hi(zt,,y) is zero for panels supported at the trailing edge, The generalized aerodynamic force term &i, needed in the In Eq. (21) the quantity [(dh;/b$) - i(wl/V)hi] readily is seen to be the complex conjugate of the amplitude of the Table 1 Types of integrals and limits of integration for downwash ratio for mode i [where the subscript i is not to computing A , (r,s) for all possible relative locations of the be confused with the unit of imaginaries i = (- l)l'z].

box p , Y and the Mach cone from box m,n I n practice the integral of Eq. (21) is evaluated by a sum- mation or matrix multiplication that can be put in the con- L i n i t S Of integration venient form " 1 "2 Ul u2 where U 1 1 (2r+l) O/B Solution of the Flutter Determinant with mA representing the panel mass per unit area.

" 2 " 1 u2 U where 21-1 2S+l 2S+l {2r+l)./P a ! = p - ( - ) 1 w z

E::: 25-1 u 25-1 2S+l

Ba Bza and k l = wl/V is the reduced frequency based on the reference length 1. For panels with a uniform mass distribution mA, the quantity a/Mi in Eq. (26) can be replaced as follows: a/Mi = (a*/Mi*)(l/~) (27) AUGUST 1963 FLUTTER OF FLAT RECTANGULAR PANELS 1799 where a* = W/ZB,B,.~ and 1/p = pl/m, is the air-to-panel mass ratio. Thus, one now has all the necessary quantities for. computing flutter boundaries. Panel dimensions and natural frequencies are employed only as ratios. The downwash ratios and other - 0

_ -

0 0 1 parameters involving the mode shapes are in forms that are independent of the panel size.

Stoble

I

As is commonly the case with exact air forces, the flutter I boundary cannot be computed directly, and an indirect - P method of computing and cross-plotting is necessary. In advance of a flutter solution for any given panel and Mach C number, choices are made for B., Bz,, and the number of

modes i n the analysis. The downwash quantities ah/&

and hi are computed or obtained from experiment at the center of each box for each mode i (or j ) and arranged sys- tematically for use in the matrix multiplications. The quanti- Unstable

p ’

ties Mi and a (or Mi*, 1/p, and a* for panels with uniform mA) are computed, except that allowance for varying the air density in a (or 1/p) is retained.

Following judicious choice of the reduced frequency kl, the block of values of A&,s), then the (pj(m,n) at each box center for each mode, and then the generalized aerodynamic CI forces Qij are computed and employed in the flutter deter- minant. For each of a number of values of the air density, the complex eigenvalues Q are computed. There is one

. eigenvalue corresponding to each chosen panel mode. By

Fig. 2 Three general types of stability boundaries for g = plotting curves of g against l / p for a sufficient range of air 0 and for g = 0.01. Arrows indicate direction of increasing densities, the existence or nonexistence of an eigenvalue with reduced frequency k I .

g = 0 for some air density is established for each mode. An associated curve of the stiffness parameter w1 1/V or w1 &‘a, in the panel flutter parameter ( E P / ~ ) l / ~ ( t / l ) , which has been is plotted against mass ratio-one curve for each eigenvalue.

evolved and used by several investigators. The fourth (The use of the speed of sound a. rather than V permits direct possible type of boundary found from a flutter determinant for various Mach numbers.)

comparison of flutter boundaries is one that falls entirely in the negative 1/p region, with On a plot of l/p against stiffness parameter, points now the positive l/p region being stable with respect to it.

can be placed which represent g = 0 or some other small value For any given panel and flow parameters, a set of stability g believed representative of the actual hysteresis structural of boundaries is computed according to the number of modes damping of the panel. Each point is on a separate stability used in the analysis. The flutter boundary separates the boundary.

region that is stable with respect to all stability boundaries By successive judicious choices of k r , a series of points can from the region that is unstable with respect to at least one be placed on the plot of l / p against stiffness parameter and stability boundary. It is the duty of the investigator to stability boundaries drawn through the points. Four differ- establish that the flutter boundary is converged, that is, that ent types of stability boundaries have been found, in general, enough modes have been used in the analysis so that addi- and three of them are illustrated in Fig. 2. For each type, tional modes do not alter the flutter boundary in any im- a boundary for g = 0 and one for g = 0.01 (as illustrative of portant respect.

some small value) are shown, and the arrows point in the direction of increasing reduced frequency k l . The type of Results and Discussion boundary of Fig. 2a predicts that, if g were zero, a fixed panel thickness would be needed as the air density tends toward Effect of Panel Length-to-Width Ratio on Flutter zero, but if g is not zero, panel thickness goes toward zero as Results have been calculated for several panel length-to- air density does.

width ratios in the range from l/w = 0 (the two-dimensional For the type of boundary of Fig. 2b, the unstable region panel) to 10, for M = 1.3, and with g = 0. The panel edges shrinks with increasing g and vanishes for some value that were either clamped or simply supported (pinned). The may be extremely small or as large as g = 0.05 or possibly results for an aluminum panel at sea level are shown on Fig.

larger, depending upon the panel and flow parameters. The 3. The abscissa is the length-to-width ratio, and the ordinate crossing point of the boundary for g = 0 in both Figs. 2a is a panel flutter parameter ( / 3 E / ~ ) * ’ ~ ( t / l ) . Analytical re- and 2b through l/p = 0 occurs a t a value of k l for which the sults calculated for M = 1.3 and with g= 0 are stown for imaginary part of Qii(kl) passes through zero. For the type panels with clamped edges by the square points connected of boundary of Fig. 2c, small increases of g have little or no by the long-dash line snd for panels with simply supported effect, and this small effect can be either stabilizing or de- (pinned) edges by the round points connected by the short- stabilizing, depending on the particular panel and flow param- dash line. Also shown for comparison on Fig. 3 is the ex- eters. An important point regarding the boundary of Fig.

perimental flutter envelope from NASA T N D-451e which was 2c, as well as the dashed boundary of Fig. 2a, concerns their obtained from tests on many panels with different sizes and resemblance to a parabola. For any boundary or portion of materials, for Mach numbers from 1.6 to 6, a t different tem- a boundary which is a parabola accurately described by peratures, and with various compressive loadings and amounts 1/p = (wl Z/a.)2 times a constant, the air density and the of buckling. The analytical results fall below the experi- t / l to prevent flutter are related by the panel thickness ratio mental envelope as would be expected, since the analysis formula t / Z ( p ) ” 3 = const. Such a relationship is contained 1800 H. J. CUNNINGHAM AIAA JOURNAL M = 1.3, g = 0 be “two-dimensional.” In Fig. 21 of Ref. 7, the experi- ASYMPTOTE INFINITE SPAN ARRAY, EDGES PINNED Q , mental thickness ratio to prevent flutter is shown as a func- , X LMODES tion of Mach number for a condition of no pressure differ- A 4MODES ence through the panel. The increase of thickness as i M decreases below 21/2 was found to be much smaller than predicted by analysis of a two-dimensional panel. Even though the panel motion showed no cross-stream variation, it sermq evident that, the aerodynamic forces exerted on the panel are not truly two-dimensional, especially near the panel side edges. The reason is that from the panel side edge to the tunnel side wall there is an area of nonmoving tunnel surface of width equal to about 6.2% of the panel width. If the tunnel side wall acted as a reflector, the image of the PANEL LENGTHlWl DTH panel beyond the side wall would be a t a distance of 12.4% of panel wXth frcm thc panel.

Fig. 3 Analytical variation of panel flutter parameter with Since the images could not be accounted for with the l/w for aluminum panels at sea level compared with the ex- perimental envelope of Ref. 6. Present results for a single present analysis, a single square panel was analyzed as a panel and previous results for an infinite span array of matter of interest. Six modes were used for the panel panels are shown for M = 1.3 and g = 0.

clamped a t leading edge, simply supported at trailing edge, and free a t side edges. The measured frequencies for the first two modes (the only ones available) and calculated fre- is of flat unstressed panels, and the experimental envelope is quencies for modes 3 to 6 were used. Flutter calculations established largely by marginally buckled panels. Also were made a t three RiIach numbers: 1.16, 1.20, and 1.30.

shown as a matter of interest are a few results from Refs. 3 The results are shown on Fig. 4 for comparison with the ex- and 4 for an infinite span array of panel bays.

perimental results from Ref. 7 . The upper curve is for g = 0, A point of interest concerns the flutter frequency and the next lower one for g = 0.01, and below that one for g = flutter mode. For all the length-width ratios at least up 0.015. Thus, in terms of thickness ratio the analytical re- through 4 on Fig. 3, the flutter frequency was near and usu- sults with a small value of g agree rather well with the ex- ally slightly above the firsbnatural-mode frequency. But periments. However, there is a significant difference in the even where the flutter frequency was almost exactly equal flutter frequency. The experimental frequency was below to the first natural frequency, the flutter mode was a coupling the first natural frequency and within about 15y0 of it.

of more than one natural mode. The degree of coupling was The analytical frequency for a single panel was slightly below small for the low values of l/w but increased with increasing the second natural frequency for A i = 1.16 and 1.20 and was 1/w. For I/w = 10, there was a great change in the flutter slightly below the third natural frequency for M = 1.30.

mode. Twelve modes were required in order to obtain a Obviously, some factor or factors are not accounted for in converged flutter boundary. The flutter frequency was the analysis to match the experiments, whether it is the between the eighth and ninth natural frequencies. The boundary layer over the panel (as suggested in Ref. 7), the largest component present in the flutter mode was the ninth reflected images of the panel through the boundary layer on natural mode, and the proportions of modes near to the ninth the tunnel side walls (see sketch in Fig. 4), the air in the were also large.

cavity behind the panel, or the air puffing in and out of the cracks along the panel side edges.

Effect of Mach Number Eflect of Air Cavity behind the Panel The thickness ratio t / l and the panel flutter parameter Panels on aircraft and missiles generally have some kind of have been computed for an aluminum panel a t sea level a space behind them which can have an effect on the flutter with I/w = 2, edges clamped, and for several Mach num- stability of the panel. Such an effect was reported in NASA bers from 1.1 to 2.0. The results are given in Table 2. The T N D-8278 on the flutter of a Fiberglas sandwich panel with thickness ratio to prevent flutter is virtually constant for a plastic-foam core. The panel had a length of 33.38 in.

M of 1.2 to 1.5, is 5y0 higher a t M = 2.0, and is 6% lower a t and a width of 20.31 in. Most of the tests were made with Af = 1.1. This trend for low supersonic Mach numbers is in cavity depth of 1.5 in. behind the panel, and flutter was ob- sharp contrast to the result for two-dimensional panels for tained over a h/Iach number range from 1.76 to 2.87. When which a great increase in thickness ratio is predicted to he the cavity was filled with layers of plywood so that the back required for Mach numbers less than about 2’/2. The ratios surface of the panel touched the plywood, no flutter was ob- of flutter frequency to first natural frequency fell between tained even though the airstream dynamic pressure was 1.0 and 1.1 for this panel and Mach number range.

doubled. But when the cavity depth was made 0.5 in., Reference 7 (Lock and Fung) reports the results of panel flutter occurred a t a dynamic pressure 40% lower than with flutter experiments a t low supersonic Mach numbers from the 1.5-in. cavity depth.

1.16 to 1.45. The panels tested were square or nearly so Since the flutter frequency was much lower than the lowest (Z/w = 0.95, 1.00, and 1.06), were clamped a t their leading acoustic-resonance frequency of the cavity, a simplified study at their trailing edges, edges, approximately simply supported was made on the basis that the only action of the air in the and free on their side edges to allow the flutter motion t o cavity is as an air spring that resists panel deflection in any mode that would compress the air but does not affect modes not involving compression. In the analysis the odd-num- Table 2 Clamped-edge aluminum panels at sea level bered modes (1, 3, 5, etc.) inXroIve compression, and the even- numbered modes (2, 4, etc.) do not. The air-spring effect M t / I (RE / a )l I 3 i t / 1 ) is largest on mode 1, much less on mode 3, and essentially 1.1 0.00292 0.212 negligible on higher modes. The analytically determined 1.2 0.0031 1 0.241 effect for a progressively decreased cavity depth primarily is 1.3 0.00312 0.251 to force the first-mode frequency upward toward the un- 1.4 0.0031 1 0.248 affected second-mode frequency. For sufficiently small 1.5 0.00311 0.247 cavity depth, the first-mode frequency could coincide with 2.0 0.00327 0.248 or even go above the second-mode frequency. One finding ~~ AUGUST 1963 lS0l FLUTTER OF FLAT RECTANGULAR PANELS P Fig. 4 Comparison . 0 07 of analytical and ex- periniental t / l as .0°6 functions of M for a square brass panel with clampcd lead- ing edge and simply supported trailing ,004 ‘IRST-MODE FREQUENCY INCREASING --t edge. OR CAVITY DEPTH DECREASING- .003LL , \ , Fig. 5 Qualitative effcct on t/Z and panel flutter param- 1.1 1.2 13 1.4 1 . 5 MACH NUMBER eter of an increase of the panel first-mode frequency, such as could be caused by a decrease of the cavity depth behind the panel.

of the analysiq was that determination of the natural fre- quencies of the panel in place over the cavity should be meas- would tend to bulge the panel. A panel flutter parameter, ured with the air density equal to (or over a range of density icc!-.’k?g) that i ~ ? the Civity rliiring the flutter experiment. involving panel stiffness and stream dynamic pressure, is plotted as a function of panel length-to-width ratio for both This is true because the test density is often a small fraction clamped-edge and simply supported panels and is compared of the density of the outside atmosphere (where the still-air with the experimental flutter envelope from Ref. 6. In panel modes and frequencies usually are measured), and be- terms of panel thickness to prevent flutter, the analytical cause the incremental effect of the cavity air spring is propor- results fall moderately below the experimental envelope, tional to the cavity air density.

which result is to be expected because the envelope was es- Since experimental frequencies were not available from tablished largely by the more severe condition of marginal Ref. 8 for the air densities of the flutter tests, flutter results thermal buckling.

have not been computed as a function of cavity depth as a The effect of variable Mach number was studied for a variable. Instead, the panel flutter parameter has been clamped-edge aluminum panel a t sea level with a length-to- obtained as a function of an assumed variable first-mode width ratio of 2. The thickness ratio was essentially con- frequency. The result is shown qualitatively in Fig. 5 as stant for 1 1 6 = 1.2, 1.3, 1.4, and 1.5, was 5’% higher a t M = the panel flutter parameter vs the first-mode frequency.

’

As the first-mode frequency increases, the flutter pnrametcr 2.0, and was 6% lower at M = 1.1. Another variable Mach number study was made to compare with a series of experi- increases at an accelerating rate, reache? a maximum value ’ in the vicinity of w, = wp, and then decreases sharply with ments in a wind tunnel on square and nearly square panels at Mach numbers from 1.16 to 1.45. Analytical results wl. It seems clear that the same trend further increase of agreed well with the experimental for the thickness ratio would be obtained for decreasing cavity depth.

but not for the flutter frequency.

Finally, there are undoubtedly other effects of the cavity a cavity behind the panel was analyzed quali- The effect of which would become important for sufficiently small cavity tatively on the basis of the cavity acting solely as an air depths. These are the virtual inertia and the frictional re- spring to resist panel modes that would compress the air, sistance of the air being pushed back and forth in the cavity thereby raising those modal frequencies. If the ratio of by panel modes higher than the first. There could also be cavity air density to cavity depth is high enough, the first- acoustic resonance effects a t sufficiently high frequencies.

mode frequency can be raised to equal or even exceed the The effect of the virtual inertia of the cavity air would be to second-mode frequency. An equality of first- and second- lower natural-mode frequencies. In Ref. 8 i t is reported mode frequencies is generally a very unfavorable factor and that decreasing the cavity depth from 1.5 to 0.5 in. lowered tends strongly to produce flutter.

the frequencies of modes 2, 3, and 4 by appreciable amounts, but the first-mode frequency could not be found. These fre- quency measurements were made at essentially sea-level air References density for which the cavity effects are proportionately larger than for the much lower air density of the flutter test.

1 Stocker, J. E., “A comprehensive review of theoretical and experinient,al panel flutter investigations,” North American Aviation Inc., Columbus Div., Rept. NA61H-444 (September Concluding Remarks 15,1961).

A panel flutter analysis has been developed for which the 2 Fung, Y. C. B., “A siimmary of the theories and experiments on panel flutter,” Air Force Office Sci. Res. TN 60-224, Guggen- generalized aerodynamic forces are computed by the “brute heim Aeronmit. Lab., Calif. Inst. Tech. (May 1960).

force’’ technique of dividing the panel into a larger number of 3 Luke, Y. L. and St. John, A., “Supersonic panel flutter,” boxes and computing and employing the aerodynamic in- Wright Air Dev. Center TR 57-252, Midwest Research Inst.

fluence of each box on each other box. Such a technique (July 1957).

can be resorted to where the aerodynamic forces are not Zeijdel, E. F. E., “Large deflection panel flutter,” Air Force obtainable from simple or elegant closed-form mathematical Office Sci. Res. TN 1952, Midwest Research Inst. (January 1962).

expressions. An important area of application of this 6 Watkins, C. E., “T!iree-dimensional supersonic theory,” technique is to finite panels a t lorn supersoriic Mach nuni- =IGARD Manual ot1 Aeroelasticity, edit,ed by W. P. Jones, Vol.

bers. Either experimental or analytical mode shapes and 2, Chap. 5 (available through NASA, Washington, D. C.).

8 Kordes, E. E., Tuovila, W. J., and Guy, L. D., “Flutter frequencies can be used. Thus, this type of analysis can be research 011 skiu panei~,” NASA TN 1)-451 (1960).

applied to any flat or nearly flat panel, whether unstressed Luck, M. 11. and Fung, Y. C., “Comparative experimental or stressed (as by thermal expansion), whether of isotropic and theoretical studies of the flutter of flat panels in a low super- or anisotropic stiffness, and even to a small-amplitudr flutter sonic flow,” .4ir Force Office Sci. Res. TN 670, Guggenheim superimposed on a buckled deflection.

Aeronaut. Lab., Calif. Inst. Tech. (May 1961).

Results are presented only for flat unstrrsscbd isotropic Tuovila, 1%’. J. and Presnell, J. G., Jr., “Supersonic panel rectangular panels with side edges parallel to the air-stream flutter test results for flat fibre-glass sandwich panels with direction and for a condition of no pressure difference that foamed cures,” NASA TN D-827 (1961).