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- - -A S A T E C H N IC A L

0 0404Pnz c 4c/Ie z

N O T E N A S A T N D -, 4 9 90-c*

LOAN COPY: RETURN TOAFWL (WLIL-2)K l R n A N D AFB, N M EX

EFFECTS OF S T R U C T U R A L DAMPINGON FLUTTER OF STRESSED PANELSby CharZes P , ShoreLangley Research Center a 1-Langley Station, Hampton, Va*N A T I O N A L A E R O N A U T I CS A N D S PA CE A D M I N I S T R A T I O N W A S H I N G T O N , D . C. J A N U A R Y 1969

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TECH LIBRARY KAFB,"I

0131944

EF FE CT S O F STRUCTURAL DAMPING ONFLU TTE R OF STRESSED PANELS

By Charles P . ShoreLangley Research Center

Langley Station, Hampton, V a .

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION ~

For sa le by the C lear inghouse fa r Feder a l Sc ien t i f i c and Techn ic a l In fo rmat ionSpr ing f ie ld , V i rg in ia 22151 - CFSTl p r i c e $3.00

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EF FE CTS OF STRUCTURAL DAMPING O NFLUTT ER OF STRESSED PANELS*

By Charles P. ShoreLangley Research Center

SUMMARYThe flutter of st re ss ed orthotropic panels with edge rotational restr ain t is investi

gated theoretically. A modal solution which utilizes two-dimensional quasi-steady aer odynamic theory and includes both str uct ura l and aerodynamic damping is presented. Theinvestigation w a s conducted in an attempt to explain the existing discre panci es betweentheoretical and experimental result s for str esse d panels, in particular for panels st res sednear buckling, the most critical flutt er condition. Struc tural damping is represented inthe present analysis in a manner consistent with the represent ation f or a Kelvin-Voigtviscoelastic body. Numerical re sul ts a r e presented to indicate that such a representationof str uct ura l damping elim inate s the physically untenable re su lt s that have been obtainedfo r certa in values of the input pa ra me te rs in previous analyses . Additionally, rep res entation of struc tural damping in thi s manner r esu lts in a reasonable correlation betweentheory and experiment over the e nti re rang e of stress from ze ro to buckling.

INTRODUCTIONRecent stu die s of panel flut ter (refs. 1 and 2 ) have shown that inclusion of the effec ts

of edge rotational res tr ai nt c an improve the agreem ent between theor etica l and exper imental flutter res ult s for st re ss ed isotropic panels. However, many instances remainwhere large discrepancies exist between theory and experiment for such panels. Theorywhich neglects damping predicts ze ro dynamic pre ssu re required f or flutter whenever themidplane load has caused two panel vibration frequencies t o be equal. This conditionocc urs for panels with a length-width ra tio gre ate r than 1 fo r many combinations of s t r e s sratio and edge rotational rest rai nt. On the othe r hand, experim ental flut ter boundariesdisplay none of the anomalous behavior associated with coincidence of the theoretical

* A par t of the inform ation present ed herein was included in a thesi s entitled "Flutterof St re ssed Pan el s Including Effects of Edge Rotational Restr ain t and Damping" submittedin part ial fulfillment of th e req uirem ents fo r the degree of Ma ste r of Science in Engineeri ng Mechanics, Virginia Polytechnic Institute, Blacksburg, Virginia, March 1967.

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frequencies. In fact, most experi mental data show that the minimum value of dynamicpressure for f lut ter occurs at or near the transition point, which is the inter section of thepanel prebuckled boundary and the postbuckled boundary. For examples of this behavior,see references 2 and 3 .

Inclusion of aero dynamic and str uc tu ral damping in the flutt er theory can removezero-dynamic-pressure flutter points. (See refs. 4 and 5 . ) However, the effects of aerodynamic and viscous s tru ctu ral damping (employed in ref s. 6 and 7) decr ease withincreasing compressive stress and vanish at the transition point (ref. 5 ) . Reference 8points out that viscous-ty pe damping is not a real isti c approximation of the damping char act eri sti cs of st ru ct ur al mat eri als and should, the refo re, be applied with caution. Anotherformulation for stru ctu ral damping frequently utilized in aer oelasti c stud ies is the f requency independent line ar hystereti c for m. Although the mechan isms resp onsib le fo rdamping in stru ctu res are usually nonlinear, fo r sma ll amounts of damping the li nearmodel often provides a good approx imation (ref. 9). In references 5 and 1 0 to 14 a linearhysteretic structural damping coefficient is, in effect, used to modify both the bending andmembr ane loading te r ms of the flutte r equation. However, the effect of st ruc tur al dampingwhen employed in thi s manner a lso decrea ses with an increase in compressiv e st re ss andvanishes at the transition point (refs. 4 and 5). Hence, this approach predicts the sam eanomalous behavior as the theory that ignores damping. However, vibration equationsderived on the basis of viscoelastic theory utilizing stre ss- str ain relations for a Kelvin-Voigt body (ref. 1 5 ) indicate that linear hysteretic structural damping should modify onlythose te rm s of the equations associa ted with bending. Such a procedure is believed tolead to the co rre ct fo rm of th e equations and is the basis fo r the flutter analysis developedin the present paper. However, in ord er to compare present re sul ts with resu lts fromprevious investigations, two sep arat e damping coefficients are introduced, one associ atedwith the membrane o r inplane fo rce s and the othe r asso ciated with panel bending. A solution fo r the flutte r of orthotro pic, flat, rect angu lar panels with ar bi tr ar y edge rotationalres t ra int is obtained by application of the Galerkin procedure. It is assumed that thelateral loading is given by two-dimensional q uasi-steady approximate aerody namic theory.

Results from the analysis a r e presented to indicate that the u se of t he two-dimensional quasi-steady aerodynamic theory yields satisfactory resu lts f or problems inwhich the effects of aerodynam ic damping are important. Additional resu lts ar e presentedto indicate that inclusion of str uct ural damping in flutter theor y is more realisticallyaccomplished by use of linear hyste retic stru ctu ral damping associate d with panel bendingonly. Such a procedure removes all physically untenable effects that otherwise occur andren de rs the value of the flutter p aram eter at the transi tion point relatively insensitive tovariations in length-width ratio, st re ss ratio, and edge rotational res trai nt, Finally, acomparison of resu lts fro m the present analysis with existing experimental resu lts is

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presen ted to verify the im proved correlat ion between theory and experiment for fl utter ofstre ssed panels over the entire stress range f ro m zer o to buckling.

SYMBOLS

Ars Fourier series coefficienta panel lengthB1, B2, B39 B4 coefficients defined by equations (17)b panel width

CO' c1,c 2 constants defined by equations (A2)fr ee- st rea m speed of sound

Dx panel bending s tiffn ess in x-direction

DY panel bending stiffn ess in y-direction

DXY panel twisting st iffnes s

D1, D12, D2 panel stiffness coefficients defined by equations (9 )E Young' s modulusFr stre amw ise deflection functionG S cro ss -st rea m deflection functiong st ruc tu ral damping coefficient

aerodynamic damping coefficient, -CgA Y W rgB bending st ruc tu ral damping coefficient

gM membrane structural damping coefficienti = @

3

C

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(11)mrp (12)mr (13)mr coefficients defined by equations (A3), (A4), and (A5)

(Jl), (J2),m (J2)mr (3) coefficients defined by equations (A12) to (A15)

coefficients defined by equations (A16) to (A19)

kX Nxb2nondimensional s t r e s s coefficient in x-direction, 2D1kx,cr buckling load with no airflowkx,T buckling load with airflow

kY Nyb2nondimensional st re ss coefficient in y-direction, 2D1kl,k2 constants

Lmr coefficient defined by equation (A6)

Lmr coefficient defined by equation (A20)*Lmr coefficient defined by equation (A21) 1 (x,y,t) aerodynamic loading ter m M Mach number NX inplane loading in x-dire ction, positive in compre ssio n NY inplane loading in y-direction, positive in comp ress ion P modulus of a complex number defined by equation (22) q free-s tream dynamic pressu re qX rotational rest rain t coefficient on x = 2 boundary, e XD1qY rotational restr aint coefficient on y =~t-boundary, bey2 D24

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tU W

X,Y ,=CY

PY

E

t imefree-s tream air velocitylateral deflection of panelCar tes ian coordinates of panelcomplex frequ ency coefficient, cp +i wcompressibility factor , d zpanel ma ss p er unit areaconstant dependent on deg ree of rotational re st rai nts t ra inmaximum s tr ain amplitudesp ring constant of rotat ional s pring supporting panel on boundary x =*ab 2and boundary y =f - , respectively2dynamic-pressure parameter dynamic-pressure parameter at trans ition point Poisson's rat io in x-direction and y-direction, respectively free-s tream air density stress real pa rt of complex frequency exponential coefficient complex eigenvalue (see eq. 18) real pa rt of complex eigenvalue imag inary pa rt of complex eigenvalue

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w

O r

Subscripts:

m , rS

circul ar frequencyreference frequency

number of half-waves in stre am wi se direct ionnumber of half-waves in cro ss -s tr ea m direction

ANALYSIS

The flat, rectangular, orthotropic panel configuration under consideration is shownin figure 1. The panel which is of length a and width b is exposed to super soni c flowover one surfa ce and is subjected to uniform inplane force intensities Nx and Ny whicha r e positive in compression. The panel is supported such that the la te ral deflection onall edges is zer o. Additionally, the panel edges are assume d to be elastically restrainedfr om ro tations by r esto rin g moments which a r e of equal strength on opposite edges anda r e proportional to th e edge angle of rotation.

Differential Equation and Boundary ConditionsThe small -defl ectio n equilib rium equation fo r motion of a n orthotropic panel in the

pre sen ce of inplane ten sile or compressive loads and supe rsoni c flow may be written inthe following fo rm (ref . 16):

1

+ a2w +Ny2w + y 2% =Z(x,y,t)Nx ay2 a t2The panel bending stiffnesses in the x- and y-di rectio ns ar e denoted by Dx and Dy,respectively; Dxy is the panel twisting stiffness, and y is the panel ma ss per unitar ea . The lat era l loading induced by the supersonic flow is ass um ed to be given by two-dimensional quasi-steady aerodynamic theory so that (re f. 17)

-2q aw aw2 (x,y,t) =--- pc at axwhere q =i p U 2 is the free- stre am dynamic pres sur e and p =J M T is the6

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- -

compressibility factor. The first t e r m on the right-hand side of equation (2) correspondsto a stati c loading and the second te rm is a viscous loading correspo nding to the aerodynamic damping. Refer ence 5 has shown that fo r M >1.6 and a from 0 to 10, the usebof two-dimensional st at ic aerodynamic theor y (aerodynamic damping neglected) yieldsflut ter r esu lts in good agreem ent with those predicte d by mo re exact aerodynamic theory .

Fo r finite rotational res tra int and nondeflecting supports th e boundary conditions(ref. 16) a r e as follows:awx a2w Ox -= 0 and w =0 at1 - pXp y ax2 ax

X = + - 2DY 82, a w - O and w = O at

1 - PxPy ay2 eY a y -DY a2, aw - 0 and w =0 at1 - PxPy By2 + o y a y

where ex and ey a re the spring constants of the rotational sprin gs on the boundariesx =* g and y =f-b respectively.2 2

The frequency-independent, line ar, hyst ereti c formulation of st ru ct ur al damping isemployed in the pres ent analy sis. Such a formulation may be derived for a one-dimensional case as follows. Assume that the material behaves as that for a Kelvin-Voigt viscoelastic body. For such a materia l the s tr es s u is a function of st ra in ra te

as well as of s t r a in E; thus,a t

where k l represents the elastic modulus E and k2 is a coefficient of viscosity; bothk l and k2 a r e positive re al quantities.

For simple harmonic motion the s tr ai n may be writteni w t

E =Eoe (5)where EO is the maximum s tr ai n amplitude and w is real.

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The stress now becomes

where the ter m i w k2- represents a stress 90' out of phas e with the s tr ain . Fork lfrequency-independent, linear ,hysteretic s tructu ral damping, - is replaced by ak2k lstru ctu ral damping coefficient g which is independent of th e magnitude of the frequencyw . The stress may now be writte n

c =E(I +ig)EoeiwtThu s, inclusion of thi s type of s tru ctu ral damping can be accomplished in dynamic analyses by using (1+ig) to modify all te rm s which contain the e last ic modulus E. This procedure is most nearly correct for simpl e harmonic motion. Therefo re, although this procedure may not be strictly valid for all regions of the solution to the pre sent p roblem, itshould yield good res ul ts at the point of s ustain ed sim ple harm onic motion which isassu med to be the point of incipient flutt er for the panel. With this procedure, only thete rm s in equation (1) involving panel bending will be modified by a structu ral dampingcoefficient. In ref ere nc e 15 plate vibration equations utilizing str es s- st ra in relation ssim ila r to equation (4) also indicate that only those t e r m s of t he equations associated withbending should be modified by a stru ctu ral damping coefficient. However, in the presen tanalysis a structu ral damping coefficient associated with m embrane fo rce te rm s is alsoretained so that the present results can be compared with tho se of previous investigations.The damping is introduced by multiplying the bending and membran e for ce t er m s in equation (1)by (1 + igg ) and (1 + igM), respect ively , whe re gB and gM a r e the bending andmembr ane stru ctu ral damping coefficients. Such a procedure is equivalent to assumi ngcomplex bending stif fne sses and fo r gM # 0,complex st re ss resultants .

SolutionFollowing the introduction of the general ized aerod ynami c loading and the s tru ctu ral

n4Dxdamping and dividing by , equation (1)becomes1 - PxPy

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where

Nxb2kx =D 1DC

A solution to equation (8) fo r sus tained motion of th e panel may be obtained by application of the Galerkin technique. The lat er al deflection is a function of t he sp ati al coordinates x and y and ti me t and may be expressed in the for m

where a is of the fo rma = c p + i w

The deflection functions Fr@) and Gs(b> a re taken to be the ze ro- st re ss vibration-mode shapes for supported uniform be ams with equal elasti c end rotational res tra int s andhence satisfy all the boundary conditions (eqs. (3a) and (3b)). On the basi s of pa st expe rience for a clamped panel (ref. 3), us e of th e beam vibration modes should yield excel lentresul ts .

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--

Hedgepeth (ref. 18 ) has shown that for simple supports th er e is no stiffness couplingbetween cro ss- st rea m modes. In reference 19 it was indicated that su ch a coupling wasinsignificant for a fully clamped panel; therefore, coupling between c ro ss -s tr ea m modesis assumed to be small for elastic res traints and is neglected herein. The two-dimensional aerodynamic theory prec ludes any aerodynamic coupling of orthogonal cro ss -st rea m modes. Thus, a one-term expansion in the cr os s-s tre am direction correspondingt o the lowest beam vibration mode is used to determine the panel dynamic stability criterion. Substituting the as sumed deflection (eq. (14)) into equation (8) , multiplying by0 0, integrating over the a re a, and noting that for a nontrivial solution to equa-F m E G1 must equal zer o yields the followingion (8) the determinant of the coefficients of

stability criterion:

i I

2-3-4

A rl

3 4

-h L14 . .

. .

=o (16

. .

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-In the de terminant of the coeff icien ts of A,., the coefficients (Ji)mr, (Ki)mr, Lmr,Aand Lmr rep res ent integra l expres sions involving the str eam wis e deflection function

and the constants CO, C1, and C2 rep res ent integral expres sions involving the cr os s-st re am deflection function. Details of the procedu re and the integral express ions aregiven in the appendix. The coeff icien ts B1 to B4 are given as

B4 - k Y($&

and

The expressions fo r B1 and B2 contain the panel geometric and stiffness par ame ter sand the expres sions fo r B3 and B4 contain the midplane loading te rm s kx and ky.The coefficients B1, B2, and B4 a r e al so functions of the cr os s- st re am boundary conditions. (See the appendix.) The te rm @ in equation (16) as su me s the rol e of a complexeigenvalue which mus t be examined to determ ine the st abili ty of the panel and includesboth the panel frequencies and the aerodynamic damping coefficient.

The panel behavior is charac teriz ed by the variation of cp +i w with an increas ein aerodynamic load A . Instability of the panel o ccur s when cp becomes positive and,hence, the condition when cp =0 is taken to be the critical condition. Th is conditioncorresponds to sustained simple harmonic motion and represe nts incipient flutter fo r thepanel. The relations necessary to determine cp and w may be obtained by solving fora! from equation (18), as follows:

Equation (19) yields two ro ots f o r a! and, thus, two sets of (cp + i w ) . The roots for woccur as * the sa m e absolute quantity; however, examination of equations (6) and (7 )

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rev eals that introduction of positive str uct ura l damping in th e flutter equation impl ies thatthe algeb raic sign of the damping coefficient g and w be the sam e. Thus, for a positive damping coefficient g the solution is valid only fo r the root of a! corresponding topositive values of w . For cer tai n panel conditions if both roots f or a! are considered,a false instability (c p >0) is found to occur over a wide range of X including A =0, acompletely unreal istic resu lt. The anomalies encountered in considering the negativeroot fo r w have been noted by other authors in references 20 and 21.

Substituting equation (15) for a! into equation (19), equating the r ea l and imaginarypart s, and selecting the root correspondi ng to positive w yields

and- =-or 2

where

Equations (20) and (21) govern both the dynamic and sta ti c stabilit y of th e panel. Zerovalues of t he te r m under the radica l in equation (21) yield ze ro values of the frequencywhich indicate zer o stiff ness and hence lo ss of sta tic stability f or the panel. Since cp =0is sought as the critical condition for dynamic instability, equation (20) is used to determine the relation between gA and the re a l and imagi nary par ts of the eigenvalue $ atflu tte r. The aerodynamic damping coefficient gA is defined as a positive quantity;therefore, only positive values of +I in equation (20) can le ad to dynamic instability.The relat ion between gA and the re a l and imaginary par ts of the eigenvalue @ thusobtained is

As both the aerodynamic damping and the s tru ct ura l damping approach zero , the us eof equation (23) leads to values of A which approach the values requir ed for coalescenceof the two lowest natura l frequencies . In the prese nce of damping, flu tte r occur s at avalue of X corresponding to the value req uired t o cause cp to become positive. Forspecific flow conditions and panel material, the aerodynamic damping coefficient may beexpressed as a function of X by combining equations (11)to (13) and is given by

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Thus, t o obtain flutter boundaries a solution for the eigenvalue Q as a function ofthe aerodynamic loading par am et er X was programed fo r a digital computer and utilizedan eigenvalue routine for a sq ua re complex matrix. To obtain points of the flu tter bounda ry fo r a particular panel, it is necessa ry t o specify the panel geometric, stiffness, mid-plane loading, and structural damping characteristics as well as the degree of el ast icrotational rest rain t present at the panel boundaries. The aerody namic loading pa ram e te r X is then var ied until both equations (23) and (24) yield identical values of gA.Limitations of the computer progra m re stri cted the maximum number of s tre am wi sete rm s in the assum ed deflection (eq. (14)) to 24 . Hence, the program yields accuratere sul ts only f o r problems in which 24 te rm s a r e sufficient for convergence.

RESULTS AND DISCUSSIONData a r e presen ted t o evaluate the accuracy of the e ffects of aerodynamic damping

given herein by the two-dimensional quasi-steady approximate aerodynam ic theory ascompared with the effects given in reference 5 by the mo re exact three-dimensionalunsteady aerodynamic the ory , to estab lish the validity of as sociati ng line ar hyste reti cstructural damping with panel bending only, and to determine whether such a procedureyields a better correlation between theoretical and experimental panel flutter results.Although the analysis w a s developed in a general for m to include orthotropic panels,res ult s a r e presented for isotropic panels only. Comparison of represe ntative res ultsfrom the present analysis with results obtained from the exact analysis presented inreference 1 revealed that the present results were accurate for clamped panels having alength-width r at io l es s than about 8.

Damping Obtained From Aerodynamic TheoriesFlutter results for a simpl y su pported panel obtained fr om both two-dimensional

quasi-steady aerodynamic theory and three-dimensional linearized supersonic potentialflow theory presented in referen ce 14 and utilized in ref erence 5 a r e given in figure 2 fora panel with a length-width ratio of 4 . The flutter parameter X 1 l 3 is shown as a function of kx/kx,cr, the rati o of the midplane comp ressive load to the cri tic al value req uire df o r buckling with no airflow. The dash-l ine c urve repr ese nts t he flu tter boundary obtainedfr om two-dimensional stat ic aerodynamic theory (no damping). The numbers on this cu rveindicate the modes that coalesced for flutter. The circles repre sent flutter results fr omthe three-dimen sional potential flow theory taken fr om figure 9 of refe ren ce 5, which were

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calculated from a six mode solution f o r aluminum panels at se a level for M =3.0; thisvalue of Mach number corre sponds t o the value at which most published experimental datafo r st re ss ed panels have been obtained. The solid-line curve repres ents res ult s fro mtwo-dimensional aerodynamic theo-ry and was obtained by utilizing 10 modes i n the p re sent analysis. Structural damping was zero fo r all ca ses and for the present analysis aer odynamic damping is given by equa tion (24). The fact that the buckling load in the prese nceof airflow can be l ar ge r than that for no airflow cau ses th e cu rve s to extend beyond-X =1.0. The buckling point on the flu tt er boundary in the presence of air flow iskx,crdefined as the transiti on point. Fo r no damping, the boundary indicates that zer o valuesof the flutt er par am et er occur at values of kx/kx ,cr of approximately 0.58, 0.7'1, and0.89. When the aerodynamic damping gA is included in the calculations, the effect isthe removal of the zero-dy namic -pres sure points although the saw-toothed-like cha rac terof the boundary r emain s. As can be seen, the differences between the res ult s from thetwo-dimensional quasi-steady aerodynamic theory and the three-dimensional potentialflow theor y a r e slight and may be considered insignificant in view of the fact that the la tt e r res ults were obtained by a method which entailed a grea t deal of cr o s s plotting ofresul ts f rom a six mode solution, which were probably not as well converged as the present resul ts. Reference 5 shows that results from sta tic aerodynamic theory agree rea sonably well with those f ro m three-dimensional potential flow theory f or Mach numbersgr eat er than 1.6; therefore, it is assumed that resu lt s from two-dimensional quas i-steadyaerodynamic theory will also agre e f or M >1.6. Two-dimensional quasi -steady aerod ynamic theory does yield satisfactory resul ts, when compared with result s fr om three-dimensional potential flow theory, and provides a great ly simplified solution for manyaer oel ast ic problems in which the e ffect s of aerodynamic damping ar e important.

Effects of St ruc tur al DampingFigure 3 ill ust rat es the effects of str uct ura l damping on the flutte r boundary pre

sented in figure 2 . The curves re presen t re sul ts fro m the present analysis for values ofthe aerodynamic damping coefficient from equation (24) for aluminum-alloy panels at sealevel fo r M =3.0. The dot-dash-line curve fo r no str uct ura l damping is taken fr om figu re 2. The dash-line curve was obtained for a value of the bending and memb rane st ru ctu ra l damping coefficients equal to 0.01. The solid-line cur ve repres ent s re su lt s fo r avalue of the bending damping coeff icient of 0.01 and a value of the membrane dampingcoefficient of ze ro. As shown by the dash-line cur ve, st ructural damping with equalbending and mem bra ne damping coefficients tends to smooth out the saw-toothed-likeflu tte r boundary; however, thi s effect less ens as kx/kx,cr approaches the tra nsi tio npoint. Simi lar res ult s wer e obtained in reference 5 which used equal str uct ura l dampingcoefficients. Thus, this type of str uct ura l damping ap pe ar s to have little effect on the

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flut ter of panels st re ss ed to the point of buckling (tra nsit ion point). However, as shown bythe solid-line curve, when membrane damping is taken to be ze ro, structural dampingsmooth s out the boundary t o a great er degree and also raises the boundary at the point ofbuckling.

Since the effects of aerodynam ic damping and Structu ral damping with gB =gMappear to be sm all as kx/kx,cr approache s the tran siti on point (see fig. 3), it is of interest to examine a special c ase f or which the theory, damping neglected, predicts flutter atzero dynamic pressure at the panel buckling load. Such a condition is shown in fig ure 4wherein the flutter param eter X'I3s plotted as a function of kx/kx,cr fo r a rotationally rest rain ed panel with a length-width r at io of 3 .3 and a stress ra ti o of Ny-=1.NXFor this condition kx,c r is equivalent to kx,T, th e transition-poin t value of kx. Th elower solid-line curve re pre sen ts the flu tter boundary for no damping, and the dash-linecurve repr esen ts the flutter boundary for gB =gM =0 . 0 1 and va lues of gA obtainedfro m equation (24) fo r an assume d aluminum-alloy panel at Mach 3 and sea-level flowconditions. The upper solid-lin e curv e is the flutter boundary obtained for the sa meflow conditions but with gM =0. As was shown in figur e 3 , i f gB =gM, neither structural damping nor aerodynamic damping has any effect at the tran siti on point (bucklingpoint). Thus, for thi s special case (dash-line curve), theory predi cts the o ccurrence offlutter at the transitio n point for zer o dynamic pre ssu re. This anomaly completely disappe ars when stru ctur al damping is asso ciat ed with bending only. The tren d of the flut terboundary thus obtained is physically reasonable and is typical of existing experiment alboundaries. Therefore, it appears that stru ctural damping is more realistically represented by modifying only the bending t e r m s of the different ial equation through us e ofcomplex stiffness coefficients.

Effects of Edge Rotational Rest rain t, S tr es s Ratio, andLength-Width Ratio on Tran siti on Point F lutt er

The structural damping results are particularly significant when it is realized thatexisting experimental resu lts indicate that the most c riti cal flutter condition occu rs nea ro r at the buckling point of the panel. Th is condition as defined by experi ment is the transiti on point (inter sect ion of prebuckled panel boundary and postbuckled panel boundary)and, as indicated in figu re 4 , past attempts at predicting the minimum value have in manyinst ance s led to the physically unreasonable ze ro values of dynamic pr es su re . Additionally, theoretical r esu lts which neglect damping o r include str uct ura l damping in the fo rmgB =gM indicate that the tran siti on point (kx,cr =kx,T) is extremely sensitive to variations in edge rotational restraint, stress ratio, and length-width ratio. Ther efor e, res ult susing the present r epresentat ion of s tru ctu ral damping are presented concerning the effectof the se pa ram ete rs on transition point flutter.

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Figure 5 i l lustrates the effect of varying edge rotational re st ra in t fo r a panel havinga length-width ratio of 3 . 3 , a stress rat io of 1, and equal rotational springs on all edges.The flutter parameter at the tr ansition point is plotted as a function of the edgerotational rest rai nt coefficient 9,. The lower curve r epr ese nts res ult s for any value ofaerodynam ic damping and any equal values of bending and membra ne str uct ura l dampingsince damping included in th is manner has no effect on flutter. The upper curve repre se nt s resu lts f or any value of ae rodynamic damping and a value of bending st ru ct ur aldamping of 0.01. A decre ase in flutter resistance with an increa se in restraint is exhibited by both curve s. (This resul t agre es with ref. 1wherein it was found that, fo r nodamping, as edge rotational res train t inc reases, a str ess ed panel can become more susceptible to flut ter even though its resis tance to buckling increase s.) In fact, the lowercurve i n figure 5 exhibits, at a value of qx =40, a zero-dynamic-pressure transitionpoint. Not only does the prese nt analysis elimin ate th is anomaly, as shown by the uppercurve, it also predicts a less sever e decrease in flutter margin with an increase in edgerotational restraint.

The effects of varying stress ratio on transition point flut ter a r e illustrated in figu r e 6 fo r a panel having a length-width ratio of 4 and clamped edges. Again, the lowercurve r epre sents res ult s which ar e unaffected by aerodynamic damping and stru ctura ldamping when gB =gM, and the upper cu rve repr ese nts re su lt s obtained by associatingst ru ct ur al damping with bending only. Comparis on of the two curves r evea ls that asso ciating str uct ura l damping with bending only not only rem ove s the many zero-dynamicpr es su re flu tte r points (which occu r when the panel has an equal choice of buckling modes)but a lso gives a nearl y constant value of the trans itio n point flut ter par am ete r over a widerange of s tr es s ratio.

Figure 7 illustrates the effects of varying length-width rat io on the tran sitio n pointflutter for a panel with clamped e dges and a stress ra tio Ny/Nx of 1. The lower curvere pr es en ts r es ul ts unaffected by damping, and the upper cur ve is again representative ofres ult s associated with bending s truct ural damping only. Comparison of the two curve sreve als that associating st ruc tur al damping with bending only removes the ex treme sensi tivity of the trans itio n point to variations in length-width rat io.

The results of figures 5, 6, and 7 suggest that the apparent sensi tivity of c rit ica lflutter boundaries to edge rotational re stra int, stress ratio , and length-width rat io wasspurious and was associated largely with the anomalous ze ro values of crit ical dynamicpr es su re which, in turn , we re due to an unrealistic theoretical model of struc turaldamping.

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Compa rison of Theo ry and Experim entThe experiment al flutter boundary presented in refe renc e 2 or a panel with a length-

width rat io of 2.9 and an average edge rotational restra int coefficient qx of 44 is compared with theoreti cal boundaries obtained fro m the pres ent analysis i n figure 8. Onlythe portion corresponding to stresses below buckling is shown. Beyond buckling, theexperim ental boundary rises and sm al l deflection theory is not sufficient to handle thiscondition. Th e boundar ies are presented in te rm s of the flutter parame ter A1/3 andthe perce nt of midplane com pres sive load required f or buckling in the pr esen ce of airflow.Theoretical boundaries are represent ed by the solid-line curves; the lower curve is forzer o damping and the upper cur ve is bas ed on an es timate d bending damping coefficientgB of 0.01 and values of the aerodyn amic damping coefficient gA calculated fr om th eexperimental tes t conditions. Reference 22 indicat es that the maximum value of m ate ria ldamping for an aluminum alloy sim ilar to that used in refer ence 2 is approximately 0.003.References 23 and 24 reveal that damping mechanisms present at panel boundaries mayincr ease the stru ctur al damping up to five tim es the value for m ater ial damping. It is ,the ref ore , be lieved that the es tima ted value of th e bending damping coefficient of 0.01 is arealistic value for the experimental data represented by the circular symbols.

Agreem ent between the expe riment al boundary and the the oret ical boundary fo r nodamping is reasona bly good fo r low values of midplane com pr ess ive stress but becomespoor in the regio n of the transi tion point. Aerodynamic damping and str uct ura l dampingassociated with bending only improves the agreement at the transition point, but the theoryremains conservative. The valu es of qx for t he points shown range f rom about 27 to 03o r fully clamped. Therefore, it is believed that the experim ental boundary is not trulyrepresentative of a boundary for q, =44 and that some scatter should be expected.Thus, a need exis ts fo r experimental flutter boundaries wherein the variation of q, isnot so large to provide dat a fo r a mo re adequate evaluation of th e prese nt the ory.

CONCLUSIONSThe flutter of s tr es se d panels with elastic edge rotational re str ain t was investigated

theoret ically. A modal solution which includes both st ru ct ur al and aerodyn amic dampingwas presented for the supersonic flutter of flat orthotropic panels. The solution utilizestwo-dimensional quasi-steady aerodynamics and is valid for panels with nondeflectingedges. The theoreti cal res ult s and the comparison of a theoretic al boundary fr om thepresent analysis with an experim ental boundary pres ente d i n NASA TN D-3498 evealedthat t he following conclusions c an be made:

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1. Two-dimensional quasi-steady aerodynamic theory yields satisfactory r esu ltswhen compared with three-dimensional potential flow theory for M >1.6 and may provide greatly simplified solutions to aeroel asti c problem s wherein the effects of aerod ynamic damping a r e important.

2 . Inclusion of str uc tur al damping in a sma ll deflection plate theory is more realistically accomplished by modifying only the bending t e r m s of the diff erential equationthrough th e us e of complex bending stif fnes s coefficients; such a representation removesall physically untenable results that otherwise occur and provides reasonable agreementbetween theoretical and experimental panel flutter r esul ts over the entire st re ss rangefr om zer o to buckling.

3 . Represe ntat ion of st ru ct ur al damping by us e of complex bending sti ffn ess coefficients rend ers the f lutter parame ter at the transitio n point (transition fr om flat to buckledpanel) relatively insensitive to variations in edge rotational rest rain t, s t r e s s ratio, andlength -w idth rat io.The ability to predict experimental results with reasonable accuracy over the entirest re ss range makes it possible to consider the flutter design of s tr es se d panels on a rational analytical basis rather than continuing to rely on empirical boundaries which must beperiodically revised as new experimental data a r e obtained.Langley Research Center,

National Aeronautics and Space Administration,Langley Station, Hampton, Va. , September 23, 1968,

126-14-02-22-23.

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APPENDIX

SOLUTION TO FLUTTER EQUATIONA solution to th e flu tter equation (eq. (8)) may be obtained by application of the

Galerkin technique. Substitution of the a ssu med deflection given by equation (14) intoequation (8), multiplying by Fm@)G1$, and integrating over the area results in thefollowing set of equations for the amplitude coefficients Ar1:

1

(m =1, 2, 3, . . ., k) ( A l )where

=r if m and r odd=r if m and r even

for m f r(A31

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APPENDIXfor m =r i f m and r odd

kKZ),, for m # r if m and r even

1 0 for m # r

0 for m +r = E ve n112 -Lmr = - Fm(f)Fr'($)d(s) = Lm, fo r m odd112

Lmr f or m evenThe pri mes and sup erscr ipt Roman num eral denote differentiation with resp ect to a orI - 6- The integ rals i n equations (A3) to (A6) a r e evaluated in te r m s of the ex pressions onthe right-hand side s of thes e equations by us e of t he following deflection functions f o r theze ro s t r es s vibrations of be ams with equal end rotational re str ain t ( see ref. 2 5 ) :

Symmetrical modesx - co s 6 jFjG) - COS 26 j i$ - cash G j cosh $ (j = 1 , 3, 5 . . .) (A7)

Asymmetrical modes(j =2 , 4 , 6 . . .) (A8)

where 6j is a constant dependent on the degree of rotational restr aint . The expressionsfo r determining 6j from reference 25 in the terminology of the present analysis a r e asfollows :

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APPENDIXSymmetrical modes

Asymmetrical modes

where-qx -

The inte gral expre ssion s i n equations (A2) may be evaluated by us e of equations (A7) to(A9), tha t is , replacing with Y-, replacing qx with qy =2,b D2 and taking j =1.Exp ressio ns f or the in tegral s i n equations (A3) to (A6) are given as follows:

cos26=26m2( - 9+26 COS 6m(COS 6 t m h 6 - s i n 6m)cash 6m

- cos 6 COS 6,(6m tanh 6 - 6 tanh 6JC O S ~ ~ , cos 6 s i n 6 cos26,

6m 6m

sin26, + l +26 s i n 6,(sin 6m coth 6 +cos 6m>-26, (inh 6m21

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APPENDIX

J

- 86,26,2 cos & cos 6 tanh 6 + sin 6, s in 6 coth 6,Lmr =6,4 - 6 4 sin 6, CO S 6 - 26,6, 6 +6 2

/cos 6, s in 6 +sin 6, cos 6 coth 6, tanh 6

L

6, cos 6 +cos 6, sin 6 tanh 6, coth2 26 - 6m

For a nontrivial solution to equation (Al) the dete rminan t of the coef ficien ts of A r 1mus t be zero. The determinant is given in the main text (eq. (16)).

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REFERENCES1. Erickson, Lar ry L.: Supersonic Flu tte r of Flat Rectangular Orthotropic Pane ls

Ela sti cal ly Res trained Again st Edge Rotation. NASA TN D-3500, 1966.2. Shideler, John L.; Dixon, Sidney C.; and Shore, C har les P.: Flutter at Mach 3 of

Thermally Str essed Panels and Com parison With Theory f or Panel s With EdgeRotational Res tra int . NASA T N D-3498, 1966.

3. Dixon, Sidney C.: Application of Transtability Concept to Flutter of Finite Panels and Experimen tal Resu lt s. NASA TN D-1948, 1963.

4. Guy, Lawrence D.; and Dixon, Sidney C.: A Cr it ical Review of Experiment and Theoryfo r Flut ter of Aerodynamically Heated Panel s. Symposium on Dynamics of MannedLifting Pla netary Entry , S. M. Scala, A. C. Harr ison, and M. Roge rs, eds., JohnWiley & Sons, Inc. , c.1963, pp. 568-595.

5. Dixon, Sidney C.: Comparison of Panel Flut ter Resul ts F ro m Approximate Aerodynamic Theory With Result s Fr om Exact Inviscid Theory and Experiment. NASATN D-3649, 1966.

6. Dowell, Ea rl : Flu tte r of Multibay Pan els at High Super sonic Speeds. AIAA J., vol. 2,no. 10, Oct. 1964, pp. 1805-1814.

7. Dugundji, John: The ore tic al Considerations of Panel Flut te r at High Supersonic MachNumbers. AIAA J., vol. 4, no. 7, July 1966, pp. 1257-1266.

8. Lazan, B. J.: Energy Dissipation Mechanisms In Structures, With Particular Reference to M ateri al Damping. Structural Damping, Jerome E. Ruzicka, ed., Amer.SOC.Mech. Eng., c.1959, pp. 1-34.

9. Lazan, B. J .: Damping Studies in Mater ials Science and Ma teria ls Engineering.Int ernal Frict ion , Damping, and Cyclic Plast icity , Spec. Tech. Publ. No. 378,Ame r. SOC.Tes tin g Mater., c.1965, pp. 1-20.

10. Nelson, Herb ert C.; and Cunningham, Herber t J.: Theor etical Investigation of Flutt erof Two-Dimensional Flat Pan els With One Surface Exposed to Supersonic Potenti alFlow. NACA Rep. 1280, 1956. (Supersedes NACA TN 3465.)

11. Dowell, E. H.; and V o s s , H. M.: Theor etica l and Exper imen tal Panel Flu tte r Studiesin the Mach Number Range 1.0 to 5.0. AIAA J . , vol. 3, no. 12, Dec. 1965,pp. 2292-2304.

12. Johns , D. J.; and Pa rk s, P. C.: Effect of Struc tural Damping on Pan el Flu tter. Aircraf t Eng., vol. XXXII, no. 380, Oct. 1960, pp. 304-308.

13. Kobett, D. R.; and Zeydel , E . F. E.: Resea rch on Panel Flutter. NASA TN D-2227,1963.

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14. Cunningham, H erb ert J.: Flutter Analysis of Flat Rectangular Panels Based on Thr ee-Dimens ional Supersonic Unsteady Potent ial Flow. NASA TR R-256, 1967.

15. Herr mann, George: The Influence of Initial St re ss on the Dynamic Behaviour of Elastic and Viscoelast ic Plates. Publ. Int. Ass. Bridge Struct. Eng., vol. 16, 1956,pp. 275-294.

16. Libove, Charles; and Batdorf, S. B.: A General Small-Deflection The ory fo r FlatSandwich Plates. NACA Rep. 899, 1948. (Supers edes NACA TN 1526.)

17. Houbolt, John Cornel ius : A Study of Several Aerother moelastic Pr obl ems of AircraftStru ctur es in High-speed Flight. Pro m. Nr. 2760, Swiss Fed. Inst. Technol.(Zurich), 1958.

18. Hedgepeth, John M.: Flu tter of Rectangular Simply Supported Panels at High Supersonic Speeds. J . Aeronaut. Sci. , vol. 24, no. 8, Aug. 1957, pp. 563-573, 586.

19. Kett er, D. J.: Flut ter of Flat, Rectangular, Ortho tropic Panels. AIAA J . , vol. 5,no. 1, Ja n. 1967, pp. 116-124.

20. Fr ae ij s de Veubeke, B. M.: Influence of Int ern al Damping on Air cra ft Resonance.Structural Aspects. Pt. I of AGARD Manual on Aeroe last icit y, ch. 3, W. P. Jones,ed.

21. Scanlan, R. H.; and Mendelson, A.: St ructu ral Damping. AIAA J. (Tech. Notes Comments), vol. 1, no. 4, Apr. 1963, pp. 938-939.

22. Granick, Neal; and Stern , Jesse E.: Ma ter ial Damping of Aluminum by a Resonant-Dwell Technique. NASA TN D-2893, 1965.

23. Mentel, T. J.; and Schultz, R. L.: Viscoela stic Support Damping of Built-In Cir cul arPl at es . ASD-TDR-63-648, U.S. Air Fo rc e, Oct. 1963.

24. Ungar, E ri c E.: Energy Dissipation at Structural Joints; Mechanisms and Magnitudes.FDL-TDR-64-98, U.S. Ai r For ce , Aug. 1964.

25. Weeks, George E.; and Shi dele r, John L.: Effect of Edge Loadings on the Vibration ofRectangular Plates With Various Boundary Conditions. NASA TN D-2815, 1965.

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_ .J U

Figure 1.- Coordinate system of orthotropic panel and direction of airflow.

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T w o - d i m e n s i o n a l t h e o r y ; gA= 0.00546 h2/ 3, g B= gM= 0

16

12

h3ha

4

0 . I . 2 .3 .4 . 5 . 6 . 7 .8 .9 1.0 I . I

k x

Figure 2.- Three- dimensional potential flow theory and two-dimensional quasi-steady aerodynamic theory for a fiat simply supported aluminum-allo y panel subjectedto midplane compressive load. M =3.0; =4.0; Ny =0. The numbers on the dash-line curve indicate the modes that coalesced for flutter.NX

c .-

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16

12

8

4

0

g = 0.01, g = 0B M l

-.- gB = gM= 0

T r a n s i t io n p o i n t s

No f l u t t e r

. I . 2 .3 . 4 . 5 . 6 . 7 . 8 . 9 1.0 1 . 1kX

kx, c rFigure 3.- Effects of str uctu ral damping on the flutter boundary for a simply supported aluminum-alloy panel at sea level. M =3.0; 4 =4.0; 2= 0.b NX

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20

1612

8

4

0 . 2 .4 . 6 . 8 I . o

kx, c rFigure 4.- Effects of structural and aerodynamic damping on flutter of a flat rotationally restrained panel subjected to midplane load. a =3.3; 3 = 1;

b NXex = By; qx =40.

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i

12

I O

8 F I u t t e r

6 4

2

0 . 2 .4 . 6 . 8 I . o . 8 . 6 .4 . 2 050

50 q xFigure 5.- Effects of edge rotational restraint and damping on transition point flutter. =3.3; 9=1; Bx =By .NX

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W0

r

I

...

1 No f l u t t e rI

I

0 . 6 . % I . o I . 2 I . 4 I . 6NS t r e s s r a t i o , YN X

F i g u r e 6.- Effects of s t ress r a t io and damp ing on t r a n s i t i o n p o i n t f l u t t e r f o r a p a n e l h a v i n g 2 = 4 and clamped edges.b

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.

8

6

4 No f l u t t e r

2

0 I 2 3 4

bNFigure 7.- Effects of length-width ratio and damping on transition point flutter for a panel w ith clamped edges and -Y = 1NX

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wN

E x pe r i me n t ( r e f . 2)20

gA c a l c u l a t e d f r o m16 F l u t t e r t e s t c o n d i t i o ns ,

gB = 0.01, gM= 012

8

4 No f l u t t e r

0 2 0 40 6 0 8 0 I O0P e r c e n t b u ck 1 ing l o a d

NY -igure 8.- Flat panel portion of experimental boundary of reference 2 and theoretical boundaries of present analysis. f = 2.9; - - 1; Ox = B y ;NXaverage value of qx of 44.

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