Principles of Aeroelasticity

2Br

(4-21)

Here the function F(I) is specialized for a stream with velocity U and ambient speed of sound a"", but the restricti on is nonessential. Furthermore, the co nstant factor in Eg . 4-21 may be dro pped, since only extreme values of the integral are sought; hence, the variational principle reads

'i"lIf[ (l",~ , "

2

+

U y-12

_

a", _ 8t

i

grad rp' grad rpJ;IY - l dV dt = 0 (4-n)

Only", is to be varied here. After some manipu lation, one fi nds that th e Euler-Lagrange equation is tantamount t o

{a",~ + y

2 1 U~

- (y -

1)[~; + {q. qJ }V 2 rp

_ 8=4> _ 2.. (q. q ) _ 81

2

a,

grad '1"

grad (0' 0) ~ 0 (4-23) 2

Eq uation 4-23 is the result we seck. Unfortunately, it is of the third degree in 1fJ. Therefore , exact mathematical solutions to compressible flow problems can be found only in tbe most elementary situations, such as one-dimensional motion in channels. Equation 4--16 shows that the coefficient in braces of the firs t tenn is just the local speed of sound squared. Hence an equivalent form ror Eq. 4--23 is V2rp _

.!..[a2p2 + 2..(q oq) + q . grad (0'0)] =0 a2

81

(4-24)

2

81

in Ref. 4- 1, Garrick has pointed out a physical interpretation by rewriting Eq. 4-24 as y2,p = ..!.. D,2rp (4-25a) (It Ott Here the opel"d.tor

D'=.!+ q

DI

at

' 'grad

(4-25 b)

AEII.OIl YNAI\1I(; U I' IWATO llS

77

where the sub~erip t ,; means th:!t 'I, is to be treated ns n constant with respect to th e o perations iJ/iJl and grad . I~ uution 4-25a states that the local field , observed in u coordinate system moving along with the fluid p:
,

["JIl[" " " "

v

]"'-'

dVdl=O (4--21)

-'--....!E...- U...!L-igrad p'·grad p ' ")' - I iJl iJx

A useful form of the Eu lerian differcn tial eq uation for Eq. 4-21 is

=

M2((' + I)O'/" a2,/,' + (r - j)'!f:[o~p' + Oi-rp'] z u

iJx iJx2

U

oy~

iJx

ik

+ 1[~!:.L + (hI" aZp'l + (r - I)?!t. Virp' U ay a'l' fJy a" Ox aJ u at + .l[arp' i"Pp' + a'l" iJ2tp' + iJ'I" aZrp' J 2

u1 ax ax a/

+ ~l grad 'l"

. grad

ay ayal

ik

az al

n grad 'P' . grad 'P'l)

(4-28)

Here we bave placed the terms linear in 'l", which form the basis of smaUperturbation subsonic and supersonic flow theory, on the left-hand side. All term s on the right are of the seeond degree except the last, which is of the third. T he linearized equation is evidently

(4-29)

78

J'RlNC II'U :S 0 1;

AI,IIOELASTICln'

In some applications involvin[; a<.:<.:cicraled or <.:urved nighl, it i8 1l1Ore convenient to write Eq. 4--29 in coord inates fixed to the nuid at rest. The result is the wave eq uation ofaeoustics, (4- 30)

The complete Eq. 4--28 can be solved by an iteration proced ure involving expansion in some small parameter, such as thickness ratio; and the first approximation is a solution of Eq. 4--29. All but the high Mach-number aerodynamic operators listed in Sees. 4-2 through 4-8 represent fully linearized solu tions. It is significan t for aeroelastic applications, however, that second-degree theory yields airload expressions which are still linear in the variables describing the motion, with coefficients determined by the wing thic kne ss distributinn, body shape, etc. This fact follows from considerat ions of sym metry. Second-degree theory has important practical implicatiOns for M > I, and it is capable of accounting for the presence of weak shocks in the supersonic flow. Compatible with the second-degree term s in Eq. 4--27 is the following approximate pressure formula, found by binomial expansion of Eq. 4-17:

c,~

_[l"

+1-" u2 al

uax

+M '[.!. arp' + _1

u a",

+(lti+(~i+ ('til u! aq,'J '

u· at

(4-3 1)

'x

(4-32)

The fully linearized versio n,

c• ~ e

u

= - p

2 ap' 2

u at

(4-33)

for coordinates at rest in the ai r, is suitable for computing first-order airloads on nearly pla ne wings. When treating bodies of revolution ami similar slender shapes, a consistent approx imation requires that certain second-degreeterms be retained (see Ward, Ref. 4-12, among other sources, for dctails). In particular problems, boundary conditions must be speCified on the body surface, at infinity, and sometimes on the wake or th e rcmaimle r of a plane of symmetry. T he infinity condition in volves the un iform (or time-dependent) stream U, with disturbances decaying and prop.1 g~\t i ng

A lmO[)YNAMIC O I 'EItATOItS

79

outward. Without viscosity. only the tluid velocity component 0,011 normal to th e instantaneous surface is lixed by the body's motion. If the body equation rC
2,

t) = 0

(4-34)

the surface condition to be satisfied on F = 0 is

-DF Dt

~

of - + (UI. + grad '1"). grad F=0 01

(4-35)

Fassumes more explicit forms for particula r shapes. The normal vector n to the body surface is often very nearly at right angles to the x-axis, permitting Eq. 4-34 to be written II

= n(x, y, I)

n = n(x, 0, t)

0'

(4-36)

Here 11 is a direction in the y.z-plane and f) is the polar angle in that plane. When one of Eqs. 4--36 is substituted into Eq. 4--35, there results

'Orp' = an an at

+ (u + '0'1" )"' + ax ax

(0'1"

oy

or alp' te rm)

(4-37)

iJfj

The linearized version of Eq. 4-37,

'0'1" = chi

'On

01

+ U~ ax

(4-38)

is adequate for many purposes. A common example of Eq. 4-38 involves a thin lifting surCace whose mean poSition is placed as close as possible to the :l;-y-plane (Fig. 4-1). Here" EO 2, and th e linearized boundary condition is split into parts for upper and lower surCaces:

0'1" (x, y, 0+, I) = Ozu + u ozu iJz 01 ax ) (..,) in ",ion R. (4-39" b) 0'1/ (x, y, 0-, ,) = oz[. + U OZL fh 01 ax Specification on the upper and lower sides of the ;>;-y-pIane at == ±O is permissible by virtue of a Taylor expansion in 2 and also by virtue oC the fact that the slender wing is customarily represented by singularities distributed on this plane. Any linear problem can be separated into a flow symmetrical in z (thickness effect) and one antisymmetrical in ~ (camber or angle of attack). This is done by writing

+ z.{;>;, y, t) -z,(;>;, y) + 24(;>;' y, r)

Zu = z,(x, y) ZL =

(4-40a, b)

11(1

J'IILN (:II'U'.S 01' AEIIOI(LASTlClT\'

, u

• •

u

• •

Section A_A Fig. 4-1. Thin, almost plane wing performing small unsteady motions in a .!ream U, showing aerodynamic load t.p. per unit area. A physical interpretation of this separation is attempted in Fig. 4-2. The latter (Ioad-distriblltion) problem is more important in aeroelastici ty. Its boundary condition reads or;' (x, y, 0, t) = oZa +

QZ

O{

_ w.(x, y, t)

(4-41 )

Equivalent conditions are readily constructed for bodies of revo lution, simple wing·body combinations, and more general slender shapes. In connection with the calculation of aerodynamic operators for plane wings from the linearized Eqs. 4-29 and 4-41, Flax (Ref. 4-13) has given a variationaL principle that merits ftlrther study. His theorem involves solutions for the same planform in forward and reversed main streams. It is entirely different from principles for the complete fl ow field , such us Eqs. 4-9 and 4-22, and consists of intcgrals over the x-y-plane of Fig. 4- 1. During this introductory review of aerody nam ic theory, we have

Ali ltOIlVNAM IC OI'I, '{ATORS

81

'0

cmphasizcd thc variational approach, not so much in an aUcmpt be differen t as to point out that i,s possibilities may havc been neglected in acroelastic applications. Very few examples of variational analyses of unsteady, compressible How are known to thc authors (sec Fyfe and Klotter, Rcf. 4-162, and Zartarian, Ref. 4-84). In a series of papers (Ref. 4-14 is typical), Wang and co-authors calculated some two-dimensional. steady, subsonic flows; and oue can imagine 'hat other invcstigators were discouroged by tbe algebraic complexity of their work. An important by-product of Wang's research was to clarify the manner of handling fields which extend outward to infinity. He thus iden,ified a serious error in an earlier development of similar type and discovered how to avoid infinite values of integrals like that in Eq_ 4-9. In the theory of elasticity, one very fruitful procedure has been to approximate the Lagrange density of the variational principle by a series of functi ons that satisfy certain of the boundary conditions and to solve numerically for a set of undetermined coefficients. An analogous technique seem to hold promise for time-dependent fluid motions in finite domains, such as those produccd by transient displacements of one- and two-dimensional pistons (Ref. 4-84). Waag's suggestion (Ref. 4-14) of using a conformal transformation to map the boundary into a simple geometrical shape would be belpful in the latter class of probems, and the small-perturbation concept is certain to yicld great simplifications. Even broader potential benefits are inherent in auother, somewhat more visionary scheme: the exact or approximate t(eatment or aeroelastic situations based on a single or composite variational theorem, describing both the solid and gaseous (or liquid) phases. One interesting effort in this direction is Riparbe!li's paper (Ref. 4-15). A second illustration of what cau be done is found in Miles' study of fuel sloshing il]side a flexible cylindrical tank (Ref. 4-16), wherein Hamilton's principle and Lagrange's equations arc sh.own to furnish an efficient mathematical representation of a practically important type of hydroeJastic phcnomenon.

u

• •

FIR. ~_l. Cross ,celi on of a Ihin wing or airfo il in" unirorm flow or velocily V • • howing Ihe dc(:o mpo.ilioll in lO" ,ym""elrical sh:opc al lero incidence and a ca mbered, inClined IIlcan line whid, i, pc ,millc,l by lincari''''li"".

S2

PRINCIPLES OF AEH.O .. LASTlCITV

4-2

GENERAL FORMS OF THE OPERATORS

The archetypal aerodynamic operator used in setting up Ihc equations of aeroelasticity is given by Eq. 3--6 as (4--42)

q. being a generalized coordinate defining the motion or displacement of the elastic system, and Q..,l being some sort of airload generated by,/,. For ol1e- and two-dimensional systcms whose deformations occur primarily in a single coordinate direction (i.e., norma! to some reference line or surface in th e structure), a convenient specification of Q..,l is the pressure difference ~Pa = (PL - Pu) between the lower and upper surfaces. In Figs. 4-1 and 4-2, for example, ~P,•.(~. y, /) is the aerodynamic force in the positive z-dircction per unit area of the x-y-plane, as it would also be if the wing wefe replaced by a body or wing-body combination. Since a constant z-displacement of the mcan-plane of a rught vehicle causes no loading, bilt a change in its angle of attack does, the best way of choosing q, is to equate it to the local streamwise incidence rx(x, y) in steady flow and to the dimensionless upwash w.(x, y, I)/U, defined by Eq. 4-41, for unsteady flow. (Note that w.}U red uces to (-00:) when the time dependence disappears from the boundary condition, which explains the minus sign in Eq. 4-43)· -rx(x, y) z;; aZA(x, y), (s teady flow)

ox

q, =

[ wix, y, t)

i!!

U

(i. + .!. ~) ,(" y ,) (jxual

~'"

(4-43)

(unsteady flow)

With these substitutions, we reformulate Eq. 4-42 as follows:

~;a = s/(\~)

~. = sr l(~;a)

(4-44a)

(4-44b)

Here (4-45)

is the dynamiC pressure, introduced to make ~P. dimensionless . .# and its inverse are linear operators, according to either linearized or sceonddegree ac rodynamic theory, with the thickness distribution 2z,(;r,.'I) • The structural sY l1lbol

II'

;s used in plac.: of •• in olher ch"ptcrs o r this book .

AlmOD VNi\ MI C O I'ElIi\TOI.!S

8)

(c r. Eq . 4- '10) appearin G li S a kn o wn. timc-indepcndcnt modifier in the Ill iler ca~e . II is casy 10 h : ,rmoni ~.e Gqs. 4-44 with th e definitions of o perat o r,; uscd in Chaps. 2 amI 5. Thus , ifil is desired to signify the actual linear den..'£ tion 11'. o ne replaces sI(· .. ) by

th e timc derivative wou ld be absent in steady-state problems and would be . 10 iw . . . . wrlttcn - -, (-. -) = - (. .. ) 11\ cases of simple harmomc molton . U ,

U

n ecau~e

of their special significance in theories of the strip (twodilllcilsionni flow) and lifting-line type, information is also furnished prese ntly abo ut aerodynamic opcrators that yield the lift per unit span (fo rce normal to the flight direction, positivc upward)

J"

L = _. t::.p~ dx

(4-46)

(l nd pitChing moment per unit span (positive in a nQ5e-up sense about a spallwisc axis at x = ba) M, = -

J" - b

tl.pa(x - ba] dx

(4-47)

The local chord of th e lifting surface under consideration is c = lb. (l IHI th e leadi ng and trailing edges are located at x = -b and z = b, respectively. Land M. arc related to two coordinates which completely tt cscribc the (small) motion of a chord wise-rigid airfoil, as follows: 11(1) = disp lacement, positive downward. of the axis at '" = ba (4-48)

0:(1) =

rot~tion,

position leading-edge upward , about the axis at '" = ba

(4-49)

" hcse coord inates arc illustrated in Fig. 6- 5. Thc lift and pitchin G momcnt per unit span are often presented ill the rO I'1Il of dimcnsionless coefficients L c, =(4-50) 'Ie

e

.,

_M _,

..

IJ~ -

(4-51)

8~

I'KI NC Il'U~

OF AIlIIO I" .•ASTlCITV

On occasion, Land M. arc also used to denote lolal lift or pilehing moment on a wing, lail, or an en lire night vehicle. Then their eoetlieients are distinguished by capital lettcrs, as follows: L CL = -

(4-52)

qS

C.II =

M

....:.:..:.L

qScu

(4--53)

Sand cR are the plan area (or other specified area) and some suitably chosen reference chordlength, respectively. Tn Sees, 4-3 through 4-8, the various operators are listed without any effort to elaborate often long derivations which intervene between the theoretical fundamentals and their final forms . Selected references are cited, in which the reader can find all important details. The listing proceeds from cases of two- and three·dimensional steady flow (Sees. 4-3 and 4--4) to simple harmonic and transient motions of airfoils (Sees. 4--5 and 4--6), and finally to the complicated cases of three·dimensional un· steady flow (Sees. 4-7 and 4-8), where the current catalogue of operators is by no means complete. In each section the compilation goes fro~ low subsonic to hypersonic speeds. d itself is displayed whenever available; otherwise, d- I is given. Afterward, lift and moment operators arc presented, along with information on loads due to a trailing edge flap in a few situations. Most of the simple harmonic operators arise from integro-diffcrential equations, containing a so-called kernel/wlclion K, These functions, which have been made dimensionless below, resemble structural influence functions because, in a loose sense, they represent the load per unit area at point (x, y) due to a un it impulse of sinusoidal upwash IV. concentrated at point (e, 1J). When only ,#- 1 is available, the kernel function describes the upwash at (x, y) due to unit concentrated load at (f, 1J). Using the symbol UI for the circular frequency of sinusoidal motion, we can say that each K depends on four dimensionless quantities: Y-1J

yo = - b

k

~

M =

(4--5411 , h)

wb - = the reduced frequency

(4-55)

~

(4--56)

U

".

= the flight Mach number

ll ere " is :III :lrhilr:lry rd~r~llrr len gtll. II is takcn t" ellll1ll tile sell1idlOnl In Cll ses "f tw u- dimcn siunal l1uw. when . 0 1' co urse. Ihe dependence or K un II ab" disappears. II" til e rresellcc 0 1" Ihe win g Irailing cdge mu,l bc accoun ted for during tli e ,krivalillll 01" .,1 , as is Irue for subsonic U. K may no longer be deter",ined o nly by .t" • .'/" but a lso by the absolu te locations of the points (.n, !ll ami (t . 'i), It is then necessary to employ additional dimensionless Yll riahlc s or the fonll (4- 57a)

Ii. ij

=

Y~1}

(4- 57b)

Some remarKs arc in order about the question of experimental connrma tio n. The rormulas given in Secs. 4-3 through 4-11 are purely theoretlelli. alth o ugh adjustlllents to make th e m agree better with wind-tunnel d nll, ;He orlen possible. Thus, steady-state measured lift coefficients, mo ment coefficients, and lift-curve slopes are available for an e normous vuri e ty 01" airfoils in a wide range of speeds. Aerodynamic operators for uli sleady flow arc sometimes corrected w ith an overall factor that assures Hoo d correlation in the limit or vanishing time dependence (e.g., w = 0), e~ l l\:dally when strip methods arc employed on a surface where properly Illey nrc no t vlllid . Occa.ioJl,ally, eve n a pha~e shift is applied as a function

or "'.

n e~ardil1g

the accuracy of um·teady calculations, however, the authors lira aware of few instances where reliable measurements show any .,IX'IIjil"a1l1 di sagreement, in the ligh t of probable instrumental errors, with 1\ Ihenry whic h reasonably models the true experimental situation. One Impo rlHllt e ~ecpt i on involves the data of Greidanus, van de VOOJ"Cn, and DOI'Sh (Rrr. 4- 17) ror oscillating airfoi ls in incompressible flow at the hi gher reduced frequenc ies. Fortunately. th e parametric range where these In vcsti gal o rs di sco vered la rge apparellt differences is not oflen encountered In [lfIl Cli ee. A comprehensive review of unsteady airload measurements Ilrlo r to 1956, with many refere nces. including such topics as wind-tunnel Willi elred s at subsonic speeds, will be found in the AGA RD report by M olync ux ( Ref. 4-25). Unlldj ustcd theory usunl ly proves uns:,tisfactory for controls with aerouynullli c balance and ror s mall tra ilin.g edge flaps or tabs, but here the Innuc no,;e 0 1" viSCOSity is o hviously dominant. A,\othcr situation requiring . pacl nl tre:l llllen! aris es when the mea n an gle of attack is so large that vlb rll iio lis ca rry a s url;,ce into the stall range. The papers by Rainey (Illlr. 4 2(,) and I !alrlllan cl al. (Rd. 4-27) contai n extensive data and

H6

1'll INCII ' U .;s OF A lmOICLASTI<;ITY

d iscussions Oflhi); subjeci lind of Lhe phenomenon or siull Outtcr, which is pcrhaps somewhat slighted in the prc,cl1t book. II is a conlinual da n!;cr on heavily loaded rotors, propellers, and rolnting machinery. With sueh qualifications as those just mentioned, however, the usc of measured aerodynamic operators for predicting dynamic aerocla:;lic behavior of primary Hfting surfaces cannot, with in the current state of the art, be expected to yield results significantly more reliable than intelligently applied theory,

4- 3 AIRFOILS I N TWO-DIM ENSIO NAL STEADY FLOW (a) S ubsoni c How (M

< L)

d=~JI

: Jl ~.!..{"-)dt

l+ ", J- I,J~xo

lI/J

(4-58)

Here the Cauchy principal va lue of the integral must be takcn becausc of the singularity in l /Xu at ( = i. Another quantity, appearing for th e first time, that occurs commonly in compressible How formulas is th e Prandll-G l(luert jac/or

M'I

(4-59)

Equation 4- 58 and its supersonic counterpart are well-known results of thin -ai rfoil theory, which is discussed in such books as Ref, 4-7, Rcf. 4-4, Glaue rt (Ref. 4-18), Pope ( Ref. 4-19), Kll~the and Schelzer (Re[ 4-20), and Ferri (Ref. 4-21). Forces and moments per unit span can be calculated by direct ir1\Cgration of tJ.p.(:r.), as in Eqs. 4-46 and 4-47. For example,

)1I -+ ((1I'.«() d( -4 b J' [JI _(2 + a ~1w.(f) d( {l ,J t={J u

L = -4 Q bJ' ~

M. =

Q

- I

U

(4-60)

2

(4- 61)

- I

Mathematical operations in Eqs. 4-58,4-60, and 4-61 arc fac ili tated by transforming to an angle va riable 0, defined by l = cos 8. The subsonic airfoil has one a ~is about which M . is independent of changes in angle of attack, ealled the nerOdyl!amiC C('llIer (A.c.). This is the quarter-chord line, as can bc seen from the vanishing of M . whcn w. is a co nstant and a = - to The force system is generally decomposed into a moment about the A.c., Iff.w, whieh for fixed q and M is a constanl determined by the camber, and a lift acting at the A.C., which dcpends 011 the di fference tJ.Gl between the angle of allack and its zero-lin value. Fr0111

A ."~ ItOI)VNAMt C

Ot'I':ItATOIt$

H7

E(IS. 4- 50, 4-51, 4-60. and 4- 61. the cocllicicnls corresponding to these

qllantities ;,re C",AO

~

M AC = qc!

~

fJ

f'

[--.1\ - e + ~2 JT+1lw'(l) d$ (4-62) i'"'=7J u oe, 21T (4-63a) e, = -'oc - I

8.C/. =

where

"

p

(4-63b)

In static aeroelastie analyses, the lift-curve slope ocdort. is widely used as an aerodynamic operator, connecting th e Change in incidence with the incremental force produced thereby. It is often detennined from tests. The subsonic value 21rIP usuaJ1y comes out somewhat higber than what is measured, but the l iP variation with Mach number agrees well for thin airfoils up to the critical M where sonic flow appears locally. When the airfoil is fitted with a control surface, such as the trailing edge flap shown in Fig. 6-1. Eq. 4-58 is still us eful for predicting the loadings. Without writing out all the forms explicitly, it can be stated that a flap rot.'ltion 01 (positive trailiog-edge downward) produces increments in running lift and MA C that can be written

t:.c, =

:~' or

(4-<4)

(4-65) An aerodynamic moment H" positive in the same sense as or' is generated about the flap hingeline due both to 0, .'Ind the angle.of.attack Change 6«. Assuming linearity and omitting any contribution of airfoil camber, the hinge moment is expressed as follows: _ H ,-qc t

A] ,[oCA. +oCA -e, -0' iM co:

(4-66)

where cf is the chordlength of the flap behind tbe hingeline. To obtain loads on the full span of a rigid lifting surface whose aspeet ratio is so large that threc·dimensional effects are negligible, all the formulas (4-62) through (4-66) are directly applicable if the coefficients are written in eapital letters. The reference area involved in their definitions is then S (Sf in the case of Eq. 4-66), replacing the area per unit span c ~ 2h.

(b) Transonic How (M ::::: I)

The theory of steady, two-dimensional transonic now is esscntially nonlinear, so that there exist no simple general forms for thc aerodyn~lI\1ic operators_ A recent book by Guderley (Ref. 4-22) contains a thorou gh treatment of this difficult analytical problem. (e) S upersonic How (M

>

J, Mb <{ [*)

d= - ~( ...)

(4-67)

Equation 4-67 is the fully lineari:/:ed result and e::<:hibits a simple pointfunction relationship, under which the local loading on any area clemenl is fixed entirely by the local incidence. It constitutes the basis of the wellknown Ackeret formulas for supersonic airfoils, of which

ae, -

4

(4-68)

~-

P JI fJ

'"

c".de = -1

- 1

w. (i) d'" - - oc U

(4-69)

are typical. The A.C. is at midchord (a = 0). Flap and hinge momen! derivatives corresponding to those in Eqs. 4-64, 4-65, and 4-66 are easily worked out or can be fouod in books on supersonic wing theory, such as Ref. 4-21. In contrast to the case of subsonic flow, the airfoil thickness distribution :a,(x) (see Fig. 4-2) often affects the A.C. location and chordwise load distribution to an important degree for sllch aeroelastic phenomena as divergence and flutter. Busemann's second-order supersonic theory or shock-expansion theory (Ref. 4-21, Chap. 7) provide the necessa ry corrections. They are also given in very simple form for M greater than roughly 2.5 (strictly, for ~.f2 » I, M~ .( I) by so-called piston theory (see Lighthill, Ref. 4-23, and Landahl, Ref. 4-24). The corresponding aerodynamic operator

d=_~[l+(Y~l)M~~](-")

(4- 70)

is suggested by Lighthill to be accurate enough for practical purposes CV\: 11 up to Mo= L • In connection with snbscction headings. some indicati,,", will be t:ivcn of order·of_ magnitud~

limilalions on Ihe importanl paromt:ters in Ihe range considered, altho,,~h these arc nol intended to be malhcmalieally complete. J he..., denO!cs thicknes. mti o, but it may also be ampliludc_to -chord rati o in cases of unstcady I11ol ion.

,u mon VNAM IC Ol' lmATOIIS

119

Whcn applicd to any :lirfoil with a closed lrai ling edge (z,( - b) = : ,(b) = 0). neithe r Buscmann's formu la !lOT c'l' 4- 70 causes any alteralion in lhe lift-curve slope, Eq. 4--68. Ho wcver, Eq. 4- 70 sh ifts the A.C. forward to the point
.JC

=_(H')MA. 8

and gives CmAC = -'-[Z4(+I) Mb

+ z.(-I)

_f'

Zii)di]

- l

+Y+ l rA",(z.(+n- Z4(-1)) 2

L4b2

(4-71)

b2

b

+fl - t

idz,dzadi] dxdx

(4- 72)

Here dz./dx has been substituted for w.jU, and A .. is the cross-sectional area of the profile, given by the chord wise integral of 2z,(x). (d) Hypersonic flow ( M2 )- I, MtJ = 0(1) or greater) Numerous approximate procedu res exist for calculating hypersonic airloads, most of them in nonlinear forms which do not easily permit the ex traction of relations resembling Eqs. 4--44. An excellent sununary, re leva nt to both two- and three-dimensional confi gurations, appears in a paper by l ees (Ref. 4-28). Pcrhaps most commonly employed in the United States is thc shock-expansion technique (see, for example, Egge rs, Syvertson, aod Kraus, Ref. 4-29). For our purposes, it serves as an iIlustratiolt to set down a few formulas from Ncwtonian theory, which has bccn described as th e limit when M ___ <Xl and y ___ I of more rigorous hypersoniC formulations. The basic physical idea (cr. Refs. 4-30 and 4-163) is that all fluid which impacts the front of a body remai ns within a layer of infinitesimal thickness adjacent to the surface as it flows by, so that the entire component of momentum norma l to the surface is transmitted as a pressure force . Sinc e the total momentum flux in a stream tube of unit cross section is 2q, it follows that a su rface clement whose inward normal makes an angle Ii to the direction of U should experience a pressure p - p", = 2q cos~ ii,

(4- 73)

Equation 4-73 has been refined in various ways. For instance, when the surface is curved, it ought to be corrected by a small centrifugal term ( Ref. 4-30) proportional to the curvature and negative where the body is convex. lees sugges ts (Rcf. 4--28) finding the correct stagnation pressure Po III Ihe nose, co rrespondin g to a coefficient Cpo < 2, and estimating P

90

I 'R INO I'US OF

ArmO I~ I .A S'l'l O T\'

locally by combining the known C,•• with Ncwton's

arpro~inm l i o l \ .

c. "" cos~ iJ

-

c -

It is generally assumed that nearly vacuum condilions cxist in th e Icc ,,],

hypersonic bodies (li > 11/2), where thc flow is probably separated. Newtonian theory achieves its grealest accuracy in regions wh cre (/ is small, and it therefore is especially useful for blunt noses and lead ing edges. When applicd to the forward portion of the airfoil considcred in Ih i ~ section, whether blunt or sharp, it yields the following non linear fOTUlldll for steady-state load per un it x-y-area :

IIp. = 2 q

(4- 75)

Eq uation 4-75 can be linearized for small mean·line slope tk./dx, regardl ess of the magnitude of the thickness 2:0" and gives the aerodynamic opera lor

(4- 76)

e:

The only terms resulting from the expansion of Eq. 4-75 which do nol vanish when z, is set equal to zero are

o( J) ,that is, 0«(;(2) and hi ghcr. 4

This result implies the "sine-squared law" of lin that is sometimes a ~so · dated with Newton's name. More detail abou t hypersonic theory appears in Secs. 4-5 and 4-6, which treat unsteady motion of airfoils. 4-4 LIFTING SURFACES AND OTHER CONFIGURATIONS IN THREE-DIM.ENSIONAL STEADY FLOW (II) Subsonic flow (M

< I)

For almost-plane surfaces of arbitrary pla nform ,\lid aspect mlio, Iho most compact statement of the operator is Multhopp' s ( Ref. 4-3 1). sf- I =

8~

ff '.

K (xu, !In; M )(- · ·j

.It d~

(4- 77u)

A I'; IUIIU'NAM I(' tW l-:II AT O llS

where Ihe kernel fUIlClioll

.... 1

rcatl~

K(·'tu• flo; IH) =

~[ I

(4-77b)

!In

fJ

is given by Eq. 4-59. As in all such problems, spe<:ial co nsideration must be given to the singularity of the kernel function at Yo = O. as discussed in Appendix I of Ref. 4-31. Although Eqs. 4--77 cannot be inverted in closed form.'" the li terature is replete with approximate methods for finding the load distri bution correspond ing to a given r.t('l', y), the majority of the m biased 10 favor a particular shape of wing or aspect.ratio range. With high-speed digital computers now gencrally availa ble. most such schemes would sccm to be obsolete except at the extremcs of aspect ratio. The needs of any organization which must ca rry out extcnsive airload calculations are best served by a general program which reduces Eq. 4--77a to an algebraic syslem,t (4--78) The two column matrices in EG. 4-78 contain the local mean-line slopes and dimensionless pressure differences at a preassigned set of points over the area R".. During the subseGucnt discussio n of oscillatory loading (Sec. 4-7). methods and references arc furnished on the algebraic re formulation of that integral eGuation, by direct nu.merical integration or series substitution ; and steady-flow fo rms like Eq. 4-78 can be obtained therefrom by setting the reduced frcq uency k = O. This solution procedure also includes the specification of Kutta's condition of smooth flowolT (t::.P. "" 0) from the trailing edge. On nearly all wings and tai ls, '" can be separated into symmetrical and antisymmetrical parts with respect to the veh icle's plane of symmetry (i.e., even and odd functions of y. respectively). For each of these. Eq. 4-78 is recast into a system involving points that cover just the right half 11 ~ 0 of R. (Fig. 4-1). The notation [.d'l and [.#~l is used for the aerodynamic matrices reduced in this way. III allY event, EG. 4- 78 or its eq uivalent is a rea l system of linear, simultaneous equations; and routine means now exist for inverting to obtain

{tl;_} =

[ .w]{",}

(4-79)

• The resullS whieh have be.n obla ined by ~erje , .olulion of Laplacc', equatioIl for ci rcular and elliptical pJanform! in incompress ible now might be regarded exception. to thi, Slaleme nt. Referencc 4-35 and sever~1 papers referred to therein comain the detai l, _ t Note tbat in Ihe roouction to matri. form' in Sc<:_ 4-4, Ihe known steady·state "pw",h dist 'ihtU ion "'.1 U i, everywhere replaced by its ncgaliv·e. the angle a("'. y)_

3,

even whcn (rcasollably weil-eolld it iol1 ed) matrices or order 100 or 1110n; are encountcrcd. In so mc cases, values or oc rn~\y nol be give n althe sa ,ne statioos where the ilp~/q are required; it is lhen on ly nceessnry 10 illtro _ duce a standard two-way interpolatio n matrix, tra nsferring the ava ilaht o a.'s to the pressure stations, When the re arc slope discont inu ities, such a~ occur at the Icading edge of a flap or control surrace, th ey are handled either by fefioing the network in Eg. 4-78 or, more rigorously, by approximating !:J.p,,}q with a series that includes the correct logarithmic singu la rit y along the discontinuity line. In the important special circumstance when the aspect 'dtio is la rge enough, Eqs. 4-77 can be simplified by choosing some elementary rcpresentation of thc chordwise pressure va riation and concentrating O l! more accurate computation of the spanwise load distribution. Sophisticated examples of Ihis technique are those where various orders of chordwise moments of ilp./q are made the unknowns. such as Rei ssner's L-andM method for rectangular wings (Ref. 4-32) and M ulthopp's OWl! .. lifting·surface" theory in Ref. 4-31. These yield algebraic equal ions, of which (4-1:\0) is a typ ical example, Here ot1 and "1! are the streamwise slopes at a pair of points along the chord of each spanwise station for which the lift and moment coefficients arc being found. Alternatively, as in Rer. 4- 32, oc1 and "1! may be weighted chordwise integrals of w./U, Additional a." and higher moments, such as hinge moments or integrals like 11lft(Y) =

f

L"oro

(4- 81)

xft ilp.(x, y) d x

might also appear. Even with the two unknowns displayed in Eq. 4- HO, very accura te results are obtained for aspect ratios in excess of IY:! or 2. In the past most airload estimates were based on lifting-lit,e reduclimls of Eqs. 4-77, wherein r,(y) remains as the single quantity to be de lermined. Beca use of their special significance, we shall review two vcrsions, first limiting ourselves to incompressible flow and then showing lhe simple extension to arbitrary subsonic M . Prand tl's classical eyuati on for straight wings, which can be derived either by a weighted chord wise integrat ion of the exaet equation (Reissner, Ref. 4-32) or by the fa mil i" .. physical reasoning (e,g., Ref. 4-18), reads ot(y) =

lY ) + c" 11 .!!.. (ccbl») . d>j -",-)ce I.." (, oe,ell

'"

8,...j- ,t/II

( !I

II)

(4- 112)

Here / and ell arc the semis p.\l1 mId lhe cho rd at a rcferenc e stMion, respectively; oc,/iJ« is the tw o-dimensional lift-curve slo pe, taken from experiment or ass igncd the thcoretical value 2,,; and «(y) is Ihe sectional angle of attack, measurcd from zero-lift attitude (cf. Eq.4-63b). Equation 4--82 is most conveniently solved by a series substitution based on Gauss' quadrature formula (M UltlIOPP, Ref. 4--33). Making the transformation (4--83) 1)=/ C050, y=/eos.p and choosing the Gaussian stations

IP. =

""

m+l

,

(/I

= I, 2, . . . ,II!:

m odd)

(4- 84)

as collocation points, Multhopp approximates Eq. 4-82 by

{.} ~ [drtl

(4--85)

H ere the square aerodynamic matrix reads - h"

-h,.

-h"

[.,,,''j_' .. c~ 41

(4-86)

-b., where c. is the chord at station /I and bi ; are sim ple numerical coefficients tabulated in Ref, 4-33 for several value s of m. Inversion of Eq. 4-85 is facilitated by the vanishing of hll when Ii - jl is an even nonze ro integer, Reference 4-33 deseribes the reduction of Eq, 4-86 to [d' ]- l, of order {m + 1){2, when the wing is symmetrically loaded, and to [da]-l, of order (m - 1){2, when the wing is antisymmetrically loaded. For a lift ing surface whose line of aerodynamic centers has all appreciabl e sweep angle A (positive for swcepback), Weissinger has derived a useful integral equation of lifting-line type in Ref. 4-34 :

94

j'RINClI'UiS OF AllRO I'.LASTICIT Y

where rJ.(II) is taken in a eross sed ion paral/elto the nighl direction. The dimensionless, nonsingular kernel L is a function of the four '1ualltitics indicated. Formulas and tables are given in Ref. 4-34 or in NACA publications dealing with this theory (e.g., Rcrs. 4-36 and 4-37). When th e Multhopp-Gauss substituti on is made in Eq. 4-87, another algebruic systcm resembling Eq. 4-85 emerges, {rJ.} =

[drt:}

(4-88)

Here the bur signifies generajimtion to arbitrary sweep angle, and the square matrix, in Weissinger's notation, reads

(b\\ + f.eu) (:'C" - b.. )

(f,g" - b,,) (b" + feu)

(4-H9)

The b;/ are the same coefficients appearing in Eq. 4-86, whereas the g,j are algebraic sums containing va lues of the L-function. For symmelrical\oading, De Young and Harper (Ref. 4-36) rewrite Eqs.4-88 (4-90)

giving complete listings of the ele me nts a.n of the (m + t)/2-order matrix. Reference 4-37 deals similarly with the antisymmetrical case. To summarize the foregoing, the lifting-line concept envisions an aerodynamic operator, (4-91) relating dimens ionless lift per unit y-distance to the angle from zero-lirt incidence, which may include the effects of camber, deflected flaps. · ant! the like. No information is provided about section A.C. location, • The deL.

~ap

OC'j O oct.C, measures,

and iM - aJ

""

contribution to:l is lL b. whereb is the actual nap

.

. , ,ero-l1 ft

on a two-dlmcnsion~l basis. Ihc rot'lllOn of the

direction per un it change in b.

rot~ti o"

AICIWIH'NAMU ; () I' I':IIATOII S

9S

(· .... w, or I.cro-lifl an g l~: so t h~sc must be presumed equal to their twodimc nsiollal v; dlle~. ""...tv shoull l Ix: ~orrcc!cd for sweep angle, as shown in SIX:. 4--4« '); otherwise Eqs. 4~62 and 4-63b apply at subsollic speeds. 110th the lifting-line and lifting-surface theories just prescnted are valid for arbi trary subsonic M, in virtue of the Prandtl-Glauert transformation for planar systems. A very dear presentation will be found in Scars' article (Ref. 4-38). The key idea is that a change of variable such as ",x x,y,z =p,y,z

(4-92)

will transform the steady counterpart of the linearized differential equation (4-29) into Laplace's equation, so that every problem of compressible flow has an incompressible equivale nt. For example, when lifting-line methods are used on a giveo wing of chord c(y) and sweep A at Mach number M, exactly the same lift per unit span L(y) will be found on a counterpart in a stream of incompressible fluid with the same density p~ and speed U. The new win g has the chord dimensions of its planform increased by l iP, so that c'(y') = c(y)

>0'

p

1 tan A' = - tan A

p

(4-93)

(4-94)

but the angle of attack and slope distribution of the mean surface z~ remain unaltered . References 4-31, 4-36, and 4-37 illustrate the practical application of this uscful result. At the low end of the aspect-ratio range He the very slender, pointed lifting surfaces (Fig. 4-3), to which R. T. Jones' adaptation (Ref. 4-39) of Munk's airship theory applies. As shown, for example, on pp. 244-248 of Ref. 1- 2, thc aerodynamic operator may be written in the direct form

Here sex) is the local semispan, as pictured in the figure. Equation 4-95 can Ix: integratcd analytical1y in several interesting rraclk:~d cases, of wh ich the most ilnportant are those when spanwise

%

I' IUNCU'LES 01'

AE~OIlI,ASTICITV

,

,

LJ

rX'

-~X!

~l

A

A

2s(x!

I

"

I

PlO" view

'"

'"

fi,. 4-3. Various configurations which can be lreated by slender-body theory. Typical .hapes of s(lil.nwise cross sections are illustrated in (b) from top to bullorn: wing of zero thickness, elliptical thickness distribution, midwing_body combination, body of revolution.

deformations are ne gligible and the operator reduces to

\V~/U

(or "') is a function of:l only. Theil

(4-96) Another integration of Eq. 4-96 with respect to y, across the span at station x , yields the following relation between ..(x) and th e lirt per unit of chord distance L(x): d

L(x) = 2q -

d.

[S(x)o:(x)]

(4-97 )

The quantity p",S(x) = p", m2(:l) is called the virlual mass of th e plane spanwise eross section and is seen toequa l thc mass of Huid contained in a circular cylinder based on the local span as diameter. Other conclusions that can be drawn from Eq. 4-96 are that the total lift depends only on the angle of attack at the trailing edge and that the lin-curvc slope of any

AtmOIJYNAl\lIC Ot'EllATOtlS

slender planform lV ith its maXUl1llm Sp:1I1 at the trailin g 4- 3, is

~dgc , ,IS

?7

in Fig. (4--98)

A{ denotes aspect ralio by the standard definition- wingspan-squared

divided by plan area. All of Eqs. 4-95-4-98 are essentially independent of M . It is of interest to complete thi s discussion of lining stlrfaees by setting down Diederich's semi· empirical formula (Ref. 4-40) for the lift-curve slope of any straight or swept wing at any subsonic Mach number, ( " ') cos A'

_'~_l. = -""""~;--_

(i;l.'"

A

7f"

+

~AJ + [(i;l.'O;AT

(4-99)

7f"fiAt

Here A' is the sweep angle of the equivalent incompressible planform, Eq . 4-94. Equation 4-99 reduces to Eq. 4-98 for slender wings, and measured data in Ref. 4-40 indicate excellent accuracy for aspect ratios above 1.5 and M up to 0.7. Bodies and wing·body combinations having cross·sectional shapes, of which the last two in Fig, 4-3(b) are typical, constitute a second major category where aerodynamic operators are often needed for aeroelastic analyses. Nearly aU aircraft and missile configurations can be built up by associating them together with almost· plane lifting surfaces. Except for interference effects, whose treatment unfortunately falls beyond the limited scope of the present chapter, wing and slender· body load data therefore fill most of the important requirements. Regarding the body itself, the basic operator is one relating lift per unit length L{x) to the local inclination cx(x) = -dz./dx between the mean line and the stream direction. Provided the fineness ratio is large enough and the nose is not too blunt, the aforementioned Munk-Jones theory yields this operator in a compact form identical with E'l ' 4-97, except that the virltlal mass p""S(x) assumes different mathematical expressions, depending on the cross-sectional shape. For instance, considering the lower three sections illustrated in Fig. 4-3(b), we have Se x ) = 7f"st(x),

(cllipse)

S(x) = ""[5"('1') - R'(x)

(4--100)

+ R'(X)] , .~Z(x)

(mid wing-body combination) (4-101)

\Ill

I' IH NC II'U'.S OF AI' 1101lI.ASTICITV

(lnd (4- 102) 5(x) = ,.,.R'(x), (body of revolution) Lateral loads due to sideslip angle arc found in a si mila r fashion. T he physical intcrpretation of the Munk-Jones formulas in terms of unsteady, two-dimens ional, incompressible fl uid motion in planes normal to tbe flight direction is well-known and will not be repeated here. Bryson's paper (Ref. 4-41) contains an exceHcnl presentation of its app lication to static and dynamic problems of various configurations, along with a list of other key references. Calculation of the subsonic loading on bodies, includ iflg the extension to highe r-order eIfCCL~ of fi neness ratio, is also discussed in Refs. 4-42 a nd 4-43. T he only shapes for which exact three-dimensional solutions exist are rigid ellipsoids in incompressible flow (Ref. 4-11), and for them the only aerodynamic operators in convenient linear forms are those giving total lifts and moments. For exa mpl e, the pitching moment exerted on a prola te ellipsoid of revolution at a small, steady incidence oc turns out to be a pure couple, (4-103) Here k, and k z arc inertia coefficients, expressed in Ref. 4--11 as elliptical integrals with the fineness ratio as argument. Formulas like Eq. 4-103 are useful for analyzing deformations of wings with external stores, When such bodies are relatively slender, the results arc quite insensitive to M ach number changes thro ughout the subsonic range, but often interference cannot be neglected. M.

(b)

Tra~onic

=

2qV[k2 - k,Joc

fl ow (M eE 1)

No explicit equations need be written for transonic aerodynamic operators, because whatever lillear ones arc valid constitute direct extrapolations of their subsonic counterparts. In the case of three-dimensional lifting surfaces, we conclude from the Prandtl-Glauert transformation as M .... I tbat the equivalent incompressible flow occurs about a wing of vanishing aspect ratio, to wh ich R. T. Jones' theory (i.e., Eqs. 4-95, 4-96, etc., when the tra iling edge is cut ofT normal to x) presumably applies. Equation 4- 99, for instance, yields a limiting lift·curve slope of ,.,.M/2, independent of sweep angle ; and this result is quile well confirmed by wind -tunnel tests near M = 1. Heaslet, Lomax, and Sprciter (Ref. 4-44) give one of the better discussions oftranson;c wing theory. An analytical approach to the problem (Ref. 4-45) suggests as onc order-of-magnitude test for the valid ity of steady-state linearized theory the following inequality: AVb[ln ;R01/3]! I (4- 104)

«

obeing Ihe maximum thickness ratio of the surface.

AEUOI)VNAl\lJC

OI'ElMTOUS

Y9

This is p~rh;lps too reslrictivcly wrillcn here, since, proper ly speaking, th e aspeCI ralio;l1 in (4- 104) should be replaced by a more direct (and smaller) measure of the span -to-<:hord pro portions, such as 21/c"",,. When At and ,j arc so large that linearized res ults are invalidated ncar M = I, an essentially nonlinear mathematical situation arises, and the comments made in Sec. 4- 3(b) apply. For slender bodies and wing-body combinations flying transonically, results such as Eqs. 4-97, 4- 100, 4-101, and 4-102 arc probably more accurate than they are either abovc or below sonic speed. An illuminating physical explanation of this fact appears in a 1956 paper by R. T. Jones (Ref. ~6) . (c) S upersonic flow (M > I, M,j

" If .,"

,r;I = - --:

n o'

where

-< 1)

-

K(xo, Yo; M)(- .. ) d; dij,

(4- \05a)

(4-lOSb)

P takes

on its supersonic interpretation of v'M2 I. Equations 4-105 apply, witho ut qualification, only to lifting surfaces with simple planforms, that is, those having all supcl"5onic leadi ng and traill ng edges. The region of integration R~' is then the ups/ream zone of inJfuence of point (x, 0, which is contained between the lead, ng edge and the forward-going Mach lines 1) = fj ± (x - ~)IP through (x, f) (scc Refs. 4-2 1 and 4-47 for examples). For wings with subsonic edges, no' consists of a portion of the projected planform and a portion of the disturbed diaphragm region adjacent to the \eading, wingtip, and possibly trailing edges. Ovcr the diapbragm, wJU is unknown but can be determined, during the process of integrating Eq. 4-105a, from the auxil iary condition that IIp. = 0 there. Many techniques, such as doublet superposition, conical, and generalized conical flow theories, have been developed for calculating the load ings on particular supersonic planforms with particular slopc distributions. Most of these have limited utility for the aeroelastician, who must deal with quite genera l deformation shapes. A full account, with many references, will be found in the article by Heaslet and Lomax (Ref. 4-47). These authors also furnish the derivation of Eqs. 4-IOS . .By an approximation procedure, whcrcby ",(x, y) is assumed to be coastant or to have some simple known variation in each of a large

100

I'IH NCII'U/'s OF AI, RO lll ,AS.... CITY

number of area element, dislributed ovcr th e disturbed :r.-!l-pl:inc. Eq. 4--I05a (for a lixeu set of point$ at Ihe centers of these "rcas) e~n be reduced to a syslem of algebraic equations,

(;:'1 ~

[""l{·)

(4- 79)

This result is analogous to the method of aerodynamic influence coefficients for oscillatory motion, which is described in morc detail in Sec. 4--7(c), Accordingly, we do not dwell on it here, except to point oullha1 the matrices {6.p.fq} and {o:} contain elements from both tbe planform and diaphnlgm regions. These can be ordered in such a way that, tak.in g advantage of the many zeros in Cd], the unknnwn « = -wJU for each dk.phragm element is computed in succession. The necessity of inverting any matrices is avoided. The remarks of Sec. 4-4(a) on the use of symmetry to break down [sI] to smaller square matrices [d' ] and ld"], covering only the right half of the wing, are equally applicable at supersonic speeds. Flaps and controls present no special difficulties, since a discontinuity in « across a line which is Jess swept than the Mach lines causes no pressure singularity. Running lift and moments are found by appropriate chord integrations of Eqs. 4-105a. For example, c,(y) = -

~

If

KTk{Xo,

YQ; M)I1.(( , ij) d( dij

(4-106)

R . TS

where subscript TE means th at the :i coordinate of the trailing edge at station iJ is to be substituted into the function so labeled, If cither « is iadependent of or KTk.U. can be suitably averaged across the chord, the ! integration is eliminated from Eg. 4- 106; and a relation is obtained which is reminiscent of the subsonic lifting_line operator, Eq. 4--91. A suitable approximation in the spanwise integral finally yields a form like Eq. 4-85, except that no inversion need be carried out. These observations have formed the basis of supersonic lifting-line the ories, of which Ref. 4-48 is typical. The familiar concept of spa nwise induction is not so meaningful at M > I, however, because the incidence of a given section of a supersonic lifting surface can influence only thc loading of immediately adjacent sections and not th e whole planform unless !R is small or M is very close above unity. Indeed, the effective three-di mensionality of the flow over any wing or tai l is measured, for a given sweep angle, by the parameter pAl" When A is small and fl At exceeds 4 or 5, strip theory yields much better accuTllcy than might hi: expected in a similar case at subsonic speeds, especially if thc loading is

t

AllROIlVNAMIC OI'I!I(ATOII,s

101

rounded oil' 10 ~e ro iUlhe small regiom afrecled by the tips. Furthermore, slrip lheory is e~acl for detcrmining ccrtain properties of sOl1\e simple planforms (cr. 4-49). Away from the aforementioned lip regions, the second-order in!luence of thickness on the load distribution can be found exactly when M2 I by means of Landahl's "inverse Rayleigh-Janse n method" (Ref. 4-24). If Landahl's terms of orders 61M, 61M3, and 61 arc retained, the sleady-state aerodynamic operator is as follows:

»

(4-107)

Here iiL~fi) is the local dimensionless coordinate of the leading edge, and w./U has been written explicitly in order to show its functional dependence under the integral sign. When 6 11M3, the last term on the right of Eq. 4-107 can be neglected, and the piston theory operalor, Eq. 4-70, is all that remains. This fact has led to the proposal that three-dimensional linearized theory might be roughly corrected for thickness effect by simply applying the over-all fuetor

»

to Eq. 4- I05u. Although promising, this scheme has as yet no real theoretical or experimental justification. As long as M () I, slender, pointed bodies and wing-body combinations in supersonic !low can be treated by Munk-Jones theory (see, for example, Heaslet and Lomax, Ref. 4- 47). This statement applies only to the normal loads, which are of principal interest in aeroelasticity; but we can state that Eqs. 4-97, 4-100, 4-[01, and 4-\02 supply the necessary operators. A not-50-slender body th.eory, giving higher-order effects of 6 and At, has been presented by Adams and Sears (Ref. 4--50).

«

(d) Hypersonic Ilow (M2 }> I, M

{j _

0(\) or greater)

The comments made in Sec. 4-3(d) about hypersonic aerodynamic operators need not be repeated. Three-dimensional configurations may be separated into those which are wing-like, having iR >- 6 and all leading and trailing cdges swept well ahead of the Mach lines, and tbose which arc essentially slender bodies. For the former category, the flow is nearly two-dimensional in y-~-planes. Equa tions 4--75 and 4--76 constitute high

102

PRINC II'LFS OF AEROIlLAST rc rrV

Mach-numher limits on the operators; whcreas iEI thc lower hypersonic range, = 0(1), we may take lighthill's suggestio n ( Ref. 4-23) and employ Eq. 4--70 as an approximat ion. Slender shapes- especially some of the complicated arr,wgemenls of wings, dorsal and ventral fins, end-plates, and central bodies that curren tly pass for re-eotry gliders-
M"

»

(e) Strip ,"col")' for swept win gs This section is concluded by restnting the well-known method whereby strip theory may be extended to an infinite swept wing (Fig. 4-4), all of whose stations are performing the same steady or unsteady motion. By resol ving the stream U into components normal and parallel to the swept span, it can be reasoned (e.g., See. 7- 3 of Ref. 1-2) that LA=LeosA (M.h = M. cos A

(4--108) (4- 109)

Here Land M. are th e lift and pitChing moment per unit of spMwise distance on a straight wing whose sections move identically to section AA. L{I. and (M, )... are swept-wing load. per unit length normal to U, and (M. ).'\. is taken also about an axis normal to U.

Fig. 4-4. Infinite swept wing in sirea m U, show ing nolali"" "",t secli ()EI A_A.

In terms of coclliciclIls, we may wril e

'" '"

D.c, = -

cos 1\ D.oc

(C,....W)A = c .. AccosA

(4--110) (4--111)

where iJc,/iJl1. alld c...,w refer to the corresponding straight wing flying at a Mach number M = U cos A/a..,. The incidence 6", is always measured in a plane containing U. Formulas referring to sections taken perpendicular to the swept y-axis can be worked out by a process of vector decompositioo, since only small angles of attack are permissible within the framework of linearized theory. When the motion is oscillatory, the reduced frequency k = (J)c/2U is the same on the swept and eq uivalent straight wings. For strip theories applicable to large aspect-ratio swept wings in three dimensions,sce Ref. 4--52, Sec. 7- 3 of Ref. 1-2, or the more rigorous later developments of van de Vooren and Eckbaus (Refs. 4-53 and 4-54).

4-5 AIRFOIlS IN TWO-DrMENS[QNAL UNSTEADY FLOW; SIMPLE HARMONIC MOTION

Mos t of the aerodynamic operators take on less complicated forms and some can be written only when the unsteady motion consists of a simpLe harmonic oscillation, which has gone on at circular frequency w long enough to assure the disappearance of transients. In view of the assumed linearity, we specify that the displacement of the mean line or mean surface of a wing or body is given by the real part of the complex function (4--112)

is a function of;t' aLone in two-dimensional problems. The tbickness 2<:, is usually regarded as independent of time, since structural deformations equivalent to thickness variations arc not often observed except on thin shells with widely spaced supports. It follows, from the interchangeability of the operation of taking the real part with other linear mathcmatical operators, that all the dependent variables can be recast in complex notation similar to Eq. 4--112. Thus the linearized boundary condition, Eq . 4-41, becomes, after canceling the common factor z~

e'·,

~: (;r, y, 0) = (u aOx + iW)Zi;t', y) == wix, y), (for ( x, y) in region Ra)

(4--113)

On tw o-dimensional airfoils. of course. R" is merely the strip ( - b <: :J; <: b) . The loading per uni t area of the :t-N-plane lIlay be written 6.p.(z, y, I) _ 6.jJ,,(z, y)e'"",

(4- 114)

and el - may be dropped from the definition of the aerodynamic operator. Eq. 4--44a, leaving (4-115) As a general rule throughout Sees. 4-5, 4--7, and dsewherc in the book

where sinusoidal motion is discussed, the bar over a dependent variable identifies it as a complex amplitude, from which the actual physical variable may be calculated by multiplying with el - and extracting the real part. For instance, the running lift, the pitching moment, and the two coordinates specifying the displacement of the wing in Fig. 6- 5 would become

L = Ie''''

(4-116)

M = /11

,

(4--117 )

h( t) = he''''

(4--118)

,~..,

f

..(t) =

(4--119)

(ie '"

We now proceed to list and discuss the aerodynamic operators, using the same system of classification as for steady flow. One new parameter, the reduced frequency defined by .Eq, 4-55, appears repeatedly throu gh the sinusoidal cases. (a) Incompressible How (/IP

d =

« 1)

J l If+? (...) -.lH -,-.11=1

~ 1(1 _ C(k)] tT"=i

~\

df

J' [ fI=i fl"+1 I + j - , -.lI+i.Jl=1~ _ ik In 2

(1 - x( + J I I

i(

(4--120)

.11

where H(2)(k) C(k) , - H~2)(k ) iH~2\k)

+

(4-12 1)

is ea11ed Theodorsen's function , H~2) being the Hankel function of Ihe second kind and order n (Ref. 2-4, page 624). E<juation 4--120 constitutes the inversion of a vortex-sheet inlegml equation, with the KuHn condition applied at the trailing edge during the process (Schwarl, Ref. 4-55).

A l(01)YNAM IC OI'EIl.ATOMS

105

AJdi lional rdefcnces and Jcrivalions from various points of view appea r in Sec. 5~ 6 of Ref. 1- 2 and in I~cf. 4-1, lhe laller including Kiissner's convenien l Fourier series sol ution. Thc incompressible operator may be thought of as the sum of a noncirculatory part,

_ 4 $.vc--

1r

f' [,11./1 -_ i' ~1

-i:

•

In

if

-I Xo

(~:~:f~~+'-'~~:~~:~~~:~~.) J<..

-)d( (4- 122)

which satisfles the instantaneous boundary conditions but gives zero total ci rc ulation around the airfoil, and a circulatory part (d - siNO)' which ta kes care of the Kutta condition and the infl uence of the vortex wake. d,vo can be generalized to arbit rary small motion by replacing ik with

.!!. ~

U ",

aod induJing the timc Jependence in 6p. and w•. The noncirculatory

loads involve derivatives of the displacements up to second order, with constant coefficients which are generally ide ntified as virtuol masses and virtual inertias. This concept of virtual mass is associated wit h the infinite speed of sound, however, and cannot be extcnded to compressible flow, where "memory" effects are present because of both the wake vortices and the finite rate of propugati on of di sturbunccs. Extensive tables are avai lable relating oscillatory lift, pitching momen t, and hinge moment to rigid.body motion of the airfoil and fl ap rotation. The symbol ism of Smilg and Wasserman (Ref. 4-56), whose table is com monly used in the United Slates, looks as follows:·

1.. = A1. =

-1rp
(! + a)LA]ii + L, 5}

1rp",b4W~1\ [MA - . (i + a)Lh] ~b + [M . + n + a)2L~]ii + [M,

-

(i + a)L,] oJ

(l

(4- 123)

+ a)(L. + Mh) (4-124)

(4-125) T he flap is here assumed to have zero aerOdynamic balance, and the airfoil rotates about the moment axis x = ba. LA' L~, M A, and M ~ are dimensionless functions of k only, wllile the r emuining coefficients L ft , etc., ure • To ndapl Eqs. 4- 123 -4~ 1 25 10 give l o tallo~ds on a two_dimensionsl wing of plan arc" S, replace one or the raCtors b ou tside each set or braces by S/2.

106

1'ltlNC1I'U'.S OF AEItO EJ. AST1 C1TY

tabula ted versus k and the frac1iona l-ehord location of the Hap leading edge. In Ref. 4-56, ope rators arc also listed for an ae rodynamically balanced nap and for a tab hing ed nt its nose. More complete tab les for trailing edge co ntrols and tabs appear in Ref. 4-57. To give an indication of the simplc math ematical forms in wh ich the incompressible aerodynamic operators occur, we reproduce here the expression for the running lift L(l) generated by the translational and pitching oscillations : L= ""p",b![ii

+ Uri -

bal:l.]

+ 2""p", UbC(k)[h + Ua. + be! -

a)i] (4- 126)

The noncirculatory terms are those contained in the firs t bracket. (b) Subsonic compressible fiot>' (M sf - I

=..!. JI 817 j

< I)

K (xu; k, M)(···) d$

(4-121a)

- I

where K(xu; k, M) =

1Tke~jk~·kr.",~') [; M ~71 Hiv(Mk

;QI)

(4-l21 b)

It is significant that this kernel function, given first by Possio (Ref. 4-58), can be divided by k and otherwise modified so as to produce a function of only two variables, M and kXn. The earlier solutions of Eqs. 4-127 were accomplished by collocation, taking account of the Kutta condition and of singularities in both the kernel and t:..p./q. Feuis' method (Ref. 4-59) is the most emcient; tables based on it were issued for the wing-nap combination at M = 0.7 (Ref. 4-60) in th e nota tion of Eqs. 4-123-4-125. Other similar tables have been published by Luke (Ref. 4-61) and T ur ner and Rabino witz (Ref. 4-62). What amounts to 'an inversion of Eqs. 4- 127 can be nchievcd by rcwriting the two-dimensional par tin l dilTere ntial equatio n in elliptic cylinder coordinates and solving it by infinite series of Mathieu functions. Amo ngsevera l ve rsions of this approach, we mention the work of Ti m mall, van de Vooren, and Greidanus ( Ref. 4-63) : Rcissller (Rcf. 4- 64): and

..umonYNAM IC O I'ElI ATOIIS

t07

Willi;lrllS ( Ref. 4- 65). Tables based 0 11 Ref. 4-63 arc given for Ihe wingflap wi th several flap-cbord ratios at M = 0, 0.35, 0.6, 0.7, and 0.8 in two Du tch repor ts ( Refs. 4-66 and 4-67). Finally, Jordan has used an interpolation procedure 10 compute derivatives for the airfoil without Ilap at !If = 0.8, 0.9, and 0.95 (Ref. 4--68). Unfortunately but naturally, there is complete notation al dis agreement among the various tabulations, so we must refer the reader to the indi vidual papers for information on this ]XJint.

» II - MI) 4( 0_ + ik) IX KC xo; k, M = ox

(c) T ransonic flow (M :::::: I, k

d

=

IK ..) d(

(4- 128£1)

- I

where (4--128b)

This form of the kernel function, whicb is directly related to th e disco ntin uity 6.[.: of the velocity potential, follows from the work of Nelson and Berman (Ref. 4--69). The earliest research on the problem is reported, however, by Rott (e.g., Ref. 4--70). Practical applications for particular upwash distributions rCCJuire tbat Eq. 4-l28a be integrated with various powers of ~ multiplying K, a process in which Fresnel integrals occur repeatedly. The full development is carried out in Ref. 4-69, where tables of lift, pitching moment, a nd hinge moment on the wing-flap combination without aerodynamic balance are presented. T he notation is that originally used for supersoniC coefficients (Ga rri ck and Rubinow, Ref. 4-7 1):

T. =

4p",b"w2 {[L I

+

fJ. = - 4Poob 4w'{ [M I

iLJ

~ + [~ + iL4]i + [4; + iLeJ Jj

(4--129)

+ iMJ ~ + [M3 + iM4Ji + [M~ + i MJ J} (4--130)

n=

- 4P oo b<w'{ [N I

+ iN 2] ~ + [N, + iN.]i + [N + iNc) J} 5

(4-131) 1·lere LI and La arc functions of k only; but the remaining coefficients depend on lhc axis location G, on t he dimensionless coordinate of the hin gelinc. or 011 bot h. For purposes of tabul ation, their dependence on G

108

I' IUNCII'LI!S 01'

A I;HOI~ LAS'I'K'ITV

is easily separated oul. Thus, those associated wilh ver tical translation and pitching are rewritten in Re f. 4-7 1 as follows:

+ iL4] = [L:a + i1:.] - [a + I](LI + iL,,] (4-132) [Ml + iMJ = [M; + 1M;] - [a + IJ[L, + iL,,] (4- 133) [Ma + iM.] = [M; + iMJ - [0 + 1J X {[M; + L:a + i(Mz + 11:.)] - [a + IJ[L, + iL...t]} [4

(4-134)

+

(Note that ill Ref. 4-71 the symbol:to = (a 1)/2 is used for the axis location.) The tabulation of forces and moments associated with the nap is simplified by certain relations which exist between them and those for the full airfoil. In view of the essential no nlinearity of steady, two-dimensional trd.nsonie flow, the foregoing results become meaningless as k __ O. By parallel physical and mathematical reasonin g, Landahl establishes the condition k II - MI for validity of the linearized theory(Ref. 4-72). In p ractice, results of flutter calculations and the like at M = I appear co nsistent with subsonic and supersonic counterpa rts when k exceeds 0, [-0 , 15, Landahl suggests that , in the neighborhOOd of sonic speed, the correct linearized differentia l equation to use is Eq. 4-29 with the (I - M 2 )Q'Atp'/a:il term completely omitted, On this bas is, it can be sho wn that the kernel function for M near but not exactly equal to unity reads

»

e-,(bo/V K (xo;k, M ~ l )=-

I ' M ,,21tlkxo

(4-135)

Another interesting examination of the limi ti ng behavior of the subsonic and supersonic solutions as M --->- I has been given hy Jordan (Ref, 4-68), One furt her important question concerns the influence of shocks adjacent to the airfoil surfaces a t very high subsonic speeds. Th ese discontinuities are not , accounted for in the co nventiona l linearized solution, but they are known to dominate the control·surface instability called aileron buzz. Preliminary approximate analyses of oscillatory transonic airfoil motions with shocks were published in 1958 by Coupry and Piazzoli (Ref. 4-73) and in 1959 by Eckhaus (Ref. 4-74), th e former pape r including some suggestiv~ corre lations with measured unsteady now properties, (d) Supersonic flow

(M > 1, Mb < I )

As in the transonic case, either the direct o r inverse kernel is available at supersonic speeds, the fo rmer being more useful for computationa l

t\t:II O IWNt\I\1I C O I'UIATOIlS

I~

purpo,cs:

sf' =

4(.! ox + ik)f'- 1 K(zo: k, MK ") dE

where K(x' k M) = _ 0',

!e-;(},~l~I~)J

(kM"o)

pUp'

(4-1300)

(4-J36b)

Again, this is a fully linearized form, and it is related to the velocity potential discontinuity across the airfoil. A straightforward derivation of Eqs. 4-136 is given by Garrick and R ubinow (Ref. 4-71). Applications require the evaluation of integrals containing K multiplied by various powers of l. These can, in turn, be expressed as combinations offunctions of the following type:

f~ =

J: lI~e-'(!kM'~I/)Ju ekiJ":lI)

dll

(4-137)

which are extensively tabulated by H uckel (Ref. 4-75) for}. = 0, I, 2,···.11. For the rigid airfoil with aerodynamically unbalanced flap, the tables of Refs. 4-71 and 4-76 provide coefficients in the notation of Eqs. 4--1294-134 up to Iff = 5. Tables utilizing the British symbolism were published by Jordan (Ref. 4-68) and more rcrently, for airfoil and flap up to M = 10/3, by Minhinnick a.nd Woodcock (Ref. 4-77). Reference 4-77 gives a useful comparison between British, French, American, and German notations. It should also be mentioned that Luke (Ref. 4-6 1) has tabulated data for the unflapped airfoil in the form of Eqs. 4-123 and 4--124. In Sec. 4- 3(c), we have a lready called attention to the effects ofprofiJe thickness in Shifting the aerodynamic center forward. Thickness may also have important influences on the aerodynamic stiffness and damping of oscillating wings, and second-order approximate theories for supersonic speeds appeared as early as 1948 (W. P. Jones, Ref. 4-79). Van Dyke (Ref. 4-78) gives similar results for pitChing and trans lation of a rig id airfoil, expanded in series up to the third power of rcduced freque ncy. His theory has been systematized by Rodd en and Revel! (Ref. 4- 80) for high-speed machine computation, with an approximate tip correction and sweep effect [in the mann er of Sec. 4--4(e)] included. Even at low superso nic M, calcuilltions for typical thickness ratios show large changes of damping-in-pitch from that predicted by linearized theory. As the Mach number increases, Re f. 4--80 points out that van D ykc's .,crodynamic opcrators approach the results of high-M expansions such as Landahl's (Ref. 4-24) and second-order piston theory. These latter forms arc sulllcie'llly gcner:.1 and elel11Clltary to be set down here. For instance,

rro

l'I!IN(:I r'u tS 011 Al'.rIO I':L"S'I' rCI'I'V

when terms of ord erso/ M. ,)/M". and ,I" ;lre retai ncd lutllc tw o-dimensional, sinusoidal case from Ref. 4-24, we obtain d= -

~[1 +

(y; I)M~~ (._.) _1.....E.: M" ax"

( f [x-~]t,-;k(£-i)(-.-)dt (4-13S) )-1

The first operator on the right is the point-function result of piston theory, which may be employed whenever Q )- 11M3 and M 2 '> 1 but M Q -< I. Landahl gives some computed values of aerodynamic Jift and momen t coefficients for the rigid airfoil in Ref. 4-24. Furthennore, a listing of nll eighteen dimensionless piston-theory coefficients in the notation of Eqs. 4-129-4-134 will be found in Rd. 4-81. Those not involving the flap rotation are listed below: L,=M,=O (4-139) L. ~ _1Mk

(4-140) (4-141)

I: = M ,=_I__ !

2

Mk

(,+I)A. 4k 2b1

M'- _1__ 3Mk'

(4-143)

M'=.....i.. _ (,

'3Mk

(4-142)

I) 4k +

M.

b3

(4-144)

Equations 4-1 39-4-1 44 require both leading and trailing edges to be sharp-pointed, but the extension \0 a blunt trailing edge is trivial. The quantities A.. and M .. are the area of the cross section and the first moment of t his area about the leading edge, respectively. (e) Hypersonic flow (M!

» I, M

" = 0(1) or greater)

The comments made in the first paragraph of Sec. 4-3(d) regarding appro)(imate hypersonic theories apply equally well to the unsteady motion of airfoils, except that considerably less work has been published

AllII.OI)YNAM IC OI'IlItATOItS

III

10 (\;111' on lime-dcf'l\'ml!;nl pro bkms. One al1cmpl at a slIrvcy appears in lhc fXlper by Mo rgan, Runyan, and H uckel (Ref. 4- 82). We cal l atten tion here to Ihree approximations which seem, among Ihem, 10 cover th e hypersonic rangc. For thc lowcr values of M"" satisfactory results can be obtained from va rious orders of piston theory (Ref. 4-83). All such formulas arc based on tbe assumption of isentropic flow, so that aerodynamic operators carried beyond the second order (the bracketed lerm on the right of Eg. 4-138) have no rigorous theoretical foundation. This order is accurate, however, only up to roughly M", == l As suggested by Ughthill (Ref. 4-23) and elaborated in Ref. 4-83, useful operators, both linear and nonlinear, can be obtained by higher-degree binomial expansions of the expression

W]2'' -1

p [ y _ 1 - = 1+ Pro 2 a~

(4-145)

Equation 4-145 gives the instantaneous isentropic pressure p on the face of a piston moving with velocity I<{t) into a perfect gas which is confined in a one-dimensional channel and has undisturbed properties identified by the subSCript 00. As one typical illustration, consider the following operator:

d = - -4 [ 1+ M

(r 2-') M -d~'''''''' (- .. ) d

(4-146)

'"' >- 2W and retaining linear Equation 4-146 is derived by assuming dx U terms in tbe binomial expansion of those parts of Eg. 4--145 that involve the motion but not the thickness distribution ef th e airfoil. This is the most exact linear form that can be extracted from Eq. 4-145; beyond it, effects of (w./Ur being to appear. has intermediate values, say 1 or 2, some adaptation of the When Eggers and Syvertson shock.expansion procedure (Ref. 4-29) is indicated. In their report, these authors suggest that unsteady airfoil motioll can be handled by a stepwise process, in which individual fluid particles are traced along the wing surface and the time history of pressure computed for each. Since this method is unwieldy to apply and difficult to linearize, Zarlarian has proposed (Ref. 4-84) scheme for combining van Dyke's hypersonic similarity (Ref. 4-51) with the shock-expansion. Reference 4-51 demonstrates, when the velocity disturbances arc not very large compared with U, that ench ?/-z-fluid slab moves past a thin airfoil without appreCiable distortion in the x-direction (i.e., U v, w). On this basis, flow past the

M"

a

«

III

I'R INCU'U\S OF AI!ROI!I.AS1·ICJTY

oscillating su rface is equivalent to ;( series of sleady flows pasl dislo rl ed surfaces, each one deri ved from the boullliary cond itions experienced by one particular sla b. The steady flows are computed as in Ref. 4-29. taking advantage of lhe well-known fact that the oblique shock strengths at the leading edge (assumed sharp) arc determined by the instantaneous effective turning angles on the upper and lower surfaces. Zartarian's method involves con~iderable algebraic manipulation, and the aerodynamic operators differ from one configuration to an other, so that no inclusive fonn can be set down here. To give some indication of the relative accuracies of two versions of piston theory and the shock expansion method, however, Table 4-1 shows the real and imaginary parIs of the dimensionless mOment due to pitChing oscillation about the leading edge (cf. Eqs. 4-130 and 4-134) for a 12 %-thick symmetrical double-wedge airfoil at M = 10. TABLE 4-t

Dime n,i o nless mom e nt due to pitching about th e tead ing edge ofa !l X_thi ck symmetrical doubte_wedge airfoil at M _ 10 (M
Coefficient kOM.'

kM: C.P. location (X chord behind L.E.)

M:

Second-order piston theory

Third-order piston theory

Shock expansion

0.0259 - 0.01 49

0.1174 0.1072

0.0971 0.076

30.7

28.6

12.9

Negative corresponds to positive workpercycledonebytheairstrcam on the oscillation, but the pitching instability predicted by se<:ond-order piston th eory for the ai rfoil about its leading edge is seen not to exist in the Light of the more precise calculation. It is worth noting, in this connection, that on physical grounds the piston-theory mod el should never yield negative dam ping on a single-degrec-of-freedom oscillation, since the normal velocity of any point on the airfoil surface prod uces an opposing change in pressure; this can be seen, for example, from the fact tha t th e linearized M~ (Ref. 4-83) is positive irrespective of pitCh-axis loca tion . For M6 :> 1, Newtonian theory (Ref. 4-30) supplies the limiting forms of the aerodynamic operators to be used on blunt noses and forward parts of airfoils, where the outward norma l is tilted towar~ the oncoming airstream. Rearward portions are essen tiaHy in vacuum. Section 4-3(d) previously discussed th e basic physical concept, an d it can be adapted to unsteady flow simply by observing that there are additional re lative velocities between the impinging fluid and the bounda ry produced by

AI::ROnVNAM IC OI'lmA"'OIIS

11 3

$mall time-de pcndent motio ns. T!lU.~, when the uppcr and lower surfaces arc described by zu(r,. I) :lnd Z I.(:I;, I), respcetivcly. (Fig. 4-J) the pressure difference is given witho ut linearit~ltion by

ilp~ = 2 q

1

+

[.!.U D~I'J' D,

[t~r 1+ [.!. DzuJ' U D,

(4--141)

Here DjD/ is the substantial derivative, which has the following forms (arbitrary motion) (4--t48)

(simple harmonic motion) When Eg. 4-147 is linearized under the assumption of small w.jU, although not necessarily small thickness slope dz,/dx, one obtains the linear operator (4--149)

Equations 4-147 and 4-149 do not contain the centrifugal term associated with curved surfaces; it is often small but ean be introduced following Refs. 4-30 and 4--82. Hayes and Probstein (Ref. 4-163, Sec. 3.7) have refined unsteady Newtonian theory to adjust in a more rational fashion for the pressure change that occurs through the thin shock layer. Their result is difficult to apply, however, since it calls for information on momentum flux in the layer which is not ordinarily available. By comparing Eg. 4--149 with Eg. 4--76-or Eg. 4--146 with its steadyfl ow counterpart- we see that all the hypersonic aerodynamic operators for which we are able to supply simple, general e~pressions aTe quasisteady. That is, the unsteadiness of the motion enters only through the time dependence of the boundary values w,JU. Although this fact reflects, in part, our incomplete knowledge of the theory, there is a strong presumption that quasi-steady airloads can be used in the majority of aeroclastic analyses at these speeds. We may e~pect many calculations to be facilitated, as compared to previous experience with subsonic and supersonic flow. The need to distinguish simple harmonic from other types of

114

I' RINell'LES 01' AEr{OEr, ASTlCrrv

motion disappears, and slich operators as Eqs. 4--146 ami 4- 149 apply beyond the sinusoidal restriction made in the present section. 4--6 AIRFOILS IN TWO-DIMENSIONAL UNSTEADY ).' LOW; TRANSIENT MOTION, GUST LOADING, AND ACCELERATED FLIGHT When a thin airfoil executes arbitrary small motions z.(x, /) normal to its chord-plane in a two-dimensional stream of constant velocity V, aerodynamic operators resembling Eq. 4--44 can be derived for some ranges of the flight Mach number. * Few similar results are available for three· dimensional wings, however, and in most cases the easiest way to calculate loads caused by ma neuvers and gusts seems to be by superposition of sinusoidal or indicial quantities. Current methods of analyzing transient aeroelastic problems either start out from some elementary input, such as a step function, OJ rely on a statistical formulation which, in turn, requires information only on simple harmonic operators lcf. Secs. 6--4(c), 7--6(c), etc.}. Running or total lifts and pitching moments are usually needed, as are sometimes higher-order generalized forces; but the details of the pressure distribution have Httle intrinsic interest. For these reasons, we follow the lead of the published literature and focus our attention here on resultant loads produced by certain stcp inputs. In the present section, the chordwise-rigid airfoil is considered ; consequently the only loads to be determined are the lift L(r) and pitching moment MvCl) per unit span. The manner in which these can be related to the displacements h(l) and 0:(1) is set forth, among other places, in the general papers of Heaslet, Lomax, et a1. (Refs. 4--129 and 4--130) or in Sec.. 6-5 of Ref. 1- 2. As a first step, we define two indicial motions commencing at I = 0: a vertical translation ho and an angular velocity qo about some fi;w;.ed a;w;.is, such as the leading edge. These are pictured in Fig. 4--5, from which we S~ that q(J involves maintaining the chordlinc tangent to the flight path at the chosen axis, so that the angle of attack remains zero Ihere. In effect, this procedure means that we make ,L temporary transrormation rrom the wi/ld-zu/llud axes, in which hand" are measured, to the body axis system used in dynamic stability. This is done in order to prevent the lift and moment generated by the ang Ular motion from diverging toward infinity, as they would if we started with a • Specifically. d has been given fur sonic and supersonic (linearized theory), high s upe ... onic (expansio n in 11M, Ref. 4-24), and hypc ... onic (Newtonian theory) . pt.",d •. The incompressible-flow operator may be constructed by comb in ing formuta. due to Neumark (Ref. 4-127) and von Karman and Scars (Ref. 4-128).

A I'; I{O"VNAMI C 0 l'ImATOl1 8

,. 0

li S

-

,., ""sui.. velocity q" _ CQn.tant

'"

fig. 4-5. Wing. perform ing the two indicial mot ions rererred to in the text. (,,) Indicial vertical translation. (b) Indicial pitching about the leading edge.

step var iation of follows

<:t .

The load~ associated with

,

no and % are written as

Lr!:s) = 21rq(2b) --.!'. (5)

(4-150)

M. r<s) "" 21T{/(2b)2....!' rl>nr(s)

(4-151)

,

U

U

Lis) = 41Tq(2b)

(q~b) rp.(s)

M. ,(s) = 41Tq(2b)2 (q~b) rp M,(S) Here

u,

.~= -

b

(4-t52) (4-153)

(4-154)

116

I' IIIN CII 'U~

OF AllItOI! I.A5"I'ICITV

is the di sta nce in sc michords travel lcd aftcr the slart oflhc IIUl11ClIV CT . Tu obtain total lift and pitching mOlllCll1 011 a (wo-d imcnsioll;'\ I lin iIl[; ~ u rilice of plan area S, one of the factors (2b) shou ld be rcp(;lccd by S in each Df Eqs. 4-150-4-153. (The same is tr uc of many subscqucnt formulas.) T he dim ensionless indicial functions
L(s) = 2".q(2b)([«(0)

+

t

dd(f

+ h~>}r(s)

[««(f) + h~)Jrp(.~ - u) dU)

+ 4rrq(2b)(!!...a(O)'J'js) + f ' !!... di(u) 'J'.(s oU dO"

U

M .js) = 2;rQ(2b)2( [,,(0)

+

h(u)] op.u(S - O")dO"j U

+ 4rrQ(2b)'(.£.i(0)'f'M'(S) + J'.£. d6.(u) '1'.11 . (5 U

(4- t56)

+ h~)J 'f'M(S)

J'd(f~ [««(f) + o

0") dO"j

aU dO"

-

a)duj (4- 157 )

Here the superscript dot denotes the derivative with respect to physic'l l time; replacing it in terms of an .1'- or u-derivative requires an olher multiplication by U/b. The chordwise axis whose displacemcnt is II allli the moment axis are bo th supposed to coincide with the one used ill defini ng an gular velocity qo. Transfer of th e loading to another axis em' be accomplished in th e usual way. A seco nd fundamental transie nt aerodynamic problem arises when Ihe airfoil meets atmospheric turbulence or a blast fron t. which c~!uscs; ' limedepend ent change in an gle of attac k. Onc typica l situation of lhis sorl

.... Ello rH'N .... MrC O I·ICIl .... TOIIS

11 1

•

• Gust or bI ••t i ront

Fig. 4-6. Approaching gust Or blast front observed in a CO
+ Ua)/].

is illustrated in Fig. 4-6. As usual, the airfoil itself has a flight speed U; but the front envelops the chord wit h an effective x.velocity U + UQ • lt is convenient to set I = 0 as the instant when the fro nt meets the leading edge x = -b. At any time thereafter. the normal z·compotlent IVa of gust or blast velocity has exactly the same aerodynamic effect as a chordwise deformation which produces an upwash (4-158) Moreover. since atmospheric motion occurs slowly relative to the envelopment rate, it is logical to assume that W II is fi xed to t he front and. (;Onscguently,a function of the combination (x - (U + UO)I] only. Natllrally, Wa U; otherwise catastrophic accidents would bcfall aircraft much more commonly. By analogy with ko and qo, wc can imagine a step gust of constant intensity \\'0' such that

«

wol.x,

r) =

11'0

I (. -

"

U

+ VI'

)

(4-159)

T he ru nning lift and moment associated with II'" are written

L(s) = 27Tq(2b) 11'0 tp(.v)

(4- 160)

M,(s) = 21Tq(2b)" ~o 'fI,l/(s)

(4-16 1)

U

II~

1'II INCII ' U .;s O F AmlOELr\ST ICITV

These il1 di<:i,,1 rUlictiol1~ ~(\,) and V',lI(.I) ;lrc par;tmctric;tlly dC(lCndclll on Mach number and th e ratio UI/IU. In NA C A literaturc. 'I~.\') is oficlI rcplaced by '"($)

k(s) :::; -' -

(4-16 2)

,(00)

arranged to have a unit asymptote like k .(s) ; k(s) is referred to ask"(,I') for a natural gust whe n Va = O. Clearly it follows from Eqs. 4-160 ;11,,1 4-161 that the loads due to an arbitrary II 'G' for fixed val ues of Va",1 VclV, are

Lr;(s) =

21Tq(2b){w~O) >p{s) + Ed: [\I'~O")]>p{s -

0")

dO"}

(4-16:\1

(4- 1{,4)

IV{,"{O") is defined here as (he /lormal velocity wh ich s/rike.! the feading !'dl'" at time t = hO"/V. Un ti l recently, the gust problem cente red around natural turbuicnce embedded in the atmosphere, for which the effective rate of convcclioll relative to the flying vehicle is U. and Va = O. T herefore, mo~t o r Ih e widely available litera ture cooccrns the calculation of >p{s) and If.llx) r,,1' this panicular case. On the othe r hand, the design or many mi litary aircraft today involves a study of the response to nuclea r blasts. T hc.' c may approach from any directi on, with fronts which propagate ;It Ih e speed of a shock wave. T he contribution Va to the chordwisc envd"l'ment rate, in two-dim ensional cases, is equal to the propagation spec,1 divided by the cosine of the angle between th e x-direction and the directi"'l toward which the front proceeds. For an act ual airplane in flight, there may also be a spanwise compone nt of frontal motion, so that the gcotllet ry of envelopment is a little more complicated. It is eviden t, nevcrthclc~s, that U o can have values throughout mos t of the range between * + w and -co and that the blast may come from ahead (Ur; > -V)or behind ( VI; < - V). Th is situa tion is fully discussed by Drisch ler and Dicdcridl (Ref. 4-131), who also poin t o ut th at, according to linea rized theory, only those components of blast dis turbance which are parallcl to the z-dircctioTl (i.e., wd produce appreciable loads. An additional source of biast 10;,dinB is the overpressurc behind the shock., which may produce high s trcssc~ locally in wing or fuselage skins; discu ssion of thi s clrcct exceeds the sWI''' of what we are II ble to cover here (see Ludl oO'. Ref. 4- 132) . • Note that. at both these timits. tit" indid"t r"ncti(}IIs 'r(,') " " .. I'M(S ) "'''SI identicat wi th 'p(.) a nd 'P .. (,,). rcsrccti vcly.

b"~",,,~

AEuonVNAM I C O I'I1H.ATOIIS

•• ,

In co nnectio n wilh Ihe SllbjCd of gusl s, wc mcntion onc othcr formation Ihal plays a major role in the stal istical Iheory of turbulence and elsewhere. T his is Ih c so-called sill"SQidal gus/- a simple harmonic disturbance cmbedded in the atmosphere and described (in complex notation) by "'dr, I) = wGei"I'-(~IU)l (4-165) Arter a wing or airfoil has proeecded through this gust for a sufficient length of time, the lift and moment settle down to the forms given by Eqs. 4- 116-4--117 and may be calculated by the methods of Sec. 4-5. In anticipation of future needs, we shall reproduce below a few expressions for airloads associated with Eq. 4-165. Tbe fonowi ng notation is adopted: La :;::

M (J

= -

Lae;'" R G~~;"I

= q(2b) wa el .., CLO U --

q(2b)!

w(J el '" U

C .11(1

(4-166) (4- 161)

T he remainder of this section is devoted largely to identifying sources where formulas and curves of the various indicial functions may be found. Certain equations arc written out, either when they have unusual in terest or fin special needs for applications in Chaps. 6 through 9. Section 4-6(e) treats briefly the problem of varying forward speed U(l). Beyond the references already cited, there are several general papers worthy of the attention of the reader who is attracted by the linearized th eory of unsteady airfoi l motion. Among others, we single out the following : for incompressible flow, Kussner (Ref. 4-133) and Fraeys de Veube ke (Ref. 4-134); for all flight speeds, Timman (Ref. 4-135) and Rott (Ref. 4-136); for applications of Rotl's moving-source method, Ordway (Ref. 4-131). (a) Jneompressible now (M"

« 1)

Only when the fluid density is constant can a separation be made between circulatory and noncirculatory airloads, as pointed out in Sec. 4-5(a). Mo rcover, when th e airfoil performs chordwise-rigid motions, the running circulatory lift depends only on the upwash at th e l-chord station, (4-168) 1I'14' = - [h + UI1. + b(l - a)li] (cf. Eq. 4-126); and the lift associated with the function C(k) always acts at the quarter-chord point. As a consequence, we need define only a single indicial function
120

l'IIINC II'I .ES OF AI.;IIOI( I,AS'!·IClT\'

and 4- 157, we di scove r th e foll owillg sim ple foriliulas (for deri vali ons, sce Sees. 5- 6 and 5- 7 of Ref. [- 2 or th e references given therein): L(s) _ 1Tp.,b'[Ji

+ U~ -

+ 21Tp", Ub

'l

d

-

,do

bll1tJ

[/I(a)

(4- 170)

+ UO:(o') + b(!

- a ):i(O')] 'l(~ - a ) dO'

M.{s) = 1Tp", b~[baJi - Ub(t - a}i _ b\t

+

(4- 171)

+ £I2.w,J + 21Tp .,Ub 2n + a) ( • .E.. [h(a)

Jo dO'

Uo:(u)

+ b(!

- £I):i(U)] '({5 - a) du

Here, as previously, dots indicate differentiation with respect to physical time; and resultant loads can be found by replacing b with S/2. The function ,,(5) == k,(s) was determined by Wagner (Ref. 4--138) in one of the earliest con tributions to unsteady wing theory. Its caleulation, using Fourier integral methods, is discussed by Garrick (Ref. 4-139) and, us ing Laplace transforms, by Sears (Ref. 4-140). Two convenient curve· fi ttin gs are O30

'1'<s):::::: I -

>0'

0.165e - D.r>I~

- 0.335e - .

(4--172a)

op(s) ~ s+2

(4--172b)

,+4

the former more accurate in the intermediate range of s and the latter showing the co rrect asymptotic be havior. For the natural gust (Uo = 0), the lift can also be proved to act at the quarter chord, so that only !f{s) is req uired. This is usually called Kussner's function, and numerical values appear in the paper by Sears (Ref. 4--140). Curve-fittings similar to Eqs. 4--172 are 1 - 0.500e- O.130' V<5)~ 5' + S ( s· + 2.82.5 0 .8

-

O.500e- ·

(4--173a, b)

+

Both 'tICs) and 'PM(S) for positive and negative Va are extensively plotted in Ref.4--131. Further information on the blast-wave problem will be found in a thesis by Hobbs, of whil; h Ref. 4--141 is a precis, and in a paper by Mi[es (Ref. 4--142). The sinusoidal gust loads for inl;ompressible flow, in th e notat ion of Eqs. 4--166 and 4-167, read as follows : CLO(k , M = 0) = 21T{C(k) [ J o(k) - iJ,(k)] CMo(k, M = 0) =

[~+

lJCdk,

These are also a development of Ref. 4--140.

+ J,(k)}

M = 0)

(4-174) (4-- 175)

AlmOIl\'N AM 1C OI' I': ltATOltS

12 1

( b) Subsoni c nov; (M < 1)

SOllie formulas (tlld calculations on all six of the indicial fultctions for subso nic flight are givcn in Ref. 4-130. The most complete work, however, is that which has been done by Mazelsky and Drischler using Fourierintegral superposition of sinusoidal airloads. 1n R efs. 4-143, 4-144, and 4-145 appear numerical va lues and exponential curve-fittings at M = 0.5, 0.6, and 0.7 on op(s), '1'M(S), rp.(s), rp .lT.(.r) (a)[is at the quarter-chord line), and 'f'{.!) (for Va = 0). At certain other values of Va' Krasnoff ( Ref. 4-145) reports 1jl{.!) and g:(.!). Finally, Drischler has published Cw(k, M) for a sinusoidal gust (Ref. 4- 147). (c) Transoni c How (M ~ 1) Indicia! functions for the sonic case M = 1 can eas ily bc developed by taking the limit as M ...... I of the supersonic formulas . Tn Sec. 6-8 of Ref. 1- 2, for instance,
r;', (fOrO

'/{S) =

,

~ s ~ 1)

l :~ {COS-1[1 - ~J + 2Js - I}, (for 1~ s)

(4-176)

As would be expected from the essential nonlinearity of steady, tw odimensiona l transonic flow, all these functions blow up more or less rapidly as $ -+ 00. This behavior does not seem to preclude their use in genUinely unsteady situations. At the lower end of the time scale, they constitute a reasonably continuous transition from their subsonic to supersonic counterparts. (d) S upersonic Row (M> I, flf6

«0

Fully linearized supersonic theory permits a relatively simple statement of the gencral aerodynamic operator, Wllich we set down first:

h(x - () , l' , 2 -- -";0;--;-'0; a",(M + 1)'

b(x
f) 1)

(4-177b)

In view of Eqs. 4-177, all the indicial functions may be computed by straightforward but lengthy integrations, This has been don e by Chang

122

J'HINC ll'LES OF Al(II Ogl,ASTl C ITY

(Ref. 4-14M) and by Lomax el al. ( R er~. 4- 3. 4- 129. aud 4- 130). Reference 4-130 presents rp. 'PM. 'PO' 'iI.llq (for an ax is al the ICalJin g edge). V'. and V'.u (for Va - 0). Actu ally, Ihe natural gus t functions were Iir:;1 derived by Biot (Ref. 4-149). The formulas are too long to be reproduced in full here. Each of the indicia I functions reaches its steady-state value (e.g .• If> = 2{7tfJ) after £ = 2M/(M - 1), which represents the time Te<juircd for all unsteady effects of starting to be swept 01T the trailing edge. Regarding loads due to blast waves in supersonic fiight, Ref. 4-131 plots Y' and 'I'M for M = 2 and many values of Ua/V. The sinusoidal gust functions, takcn from Sec. 6-6 of Ref. 1-2, arc the following: 1k

Cw(k. M CMo(k . M

>

1) =

>

1) =

4e Ii 10

Ii'" [(20 + I)fo -

(4-178) (4-179)

fJ]

where a is the dimensionless location of the moment axis and II is the function defined by Eq. 4-137. Because of their simplicity, the indicial functions according 10 the high supersonic theory ofLandah.1(Ref. 4-24) are noteworthy. The general aerodynamic operator, retaining as before the terms of order {J{M, 6/ M S , and 02, is·

(r + I)Md',] w,

,,~ _ ~[I + M

2

dx

U

"'f' ["" - ,,-et~w. ( " b(' -D)d'

2- -M3 aXl

U'

- 1

U

¢

(4-180)

Typical of the results which we can work out from Eq. 4-180 are the following:

2 [ -1+ -4M, 2] ' 17M (4-18 1)


1-] - 2 [ 1+2M2 .,.M '

.!... .,.M

[I + _2M1_z] + (, 217+ I) M b

z,(x

.3.. [ I +_1_]

17M

2

2M" '

<,

= b(s _ 1)),

2 <, (4- 182)

• We remark. that Eq. 4_180 may be a dapted to wings of ronitc s~m. for regio", IIninnuenccd by cut·ofl"tip.<. by adding o'/og' 10 the o'/ai' acling on the integ,al ;",,1 including the dependcJJC<: of .... on 9.

AEuonVNA~ttC

Ot ' tmAT O t! S

t23

Hcre the seco nd-orde r clrcets or thidn c;;s arc
.M

rw~ , '1'<.1) = \

O s s :-:;;; 2

(4-184)

.~ ,

The problem of calculating indicial functions for hypersonic flight speeds bas not been discussed in the literature. When Mo = 0(1), we may expect equations I ike (4-181) and (4-182) to furnish reasonable approximations, which can probably be improved by using an operator such as Eq. 4-146. The reader will have no difficulty in deriving corresponding forms from other quasi-steady hyperso nic theories, such as the various versions of Newtonian theory described in Sec. 4-3(d). (e) Unsteady

eff~ts

of longitudinal ae<:ekration

A subject of increasing interest is thal of a lifting surface which under-

goes rapid forward ae<:eleration or deceleration while simultaneously pe rforming lateral oscillations or some other unsteady motion. The general theoretical procedure for linearized treatment of the two-dimensional case was laid out by Lomax et at. (Ref. 4-130) and by Krasilshehikova (Ref. 4-151). A more recent discussion by Krasilshehikova (Ref. 4-152) deals with finite wings but is still somewhat remote from numerical applications. The acceleration problem is also discussed in one of the later chapters of Miles' monograph (Ref. 4-2), with emphasis on transonic drag calculation. When t he flow can be regarded as incompressible, as during the launching of rockets from the ground or the catapulting of aircraft, the work of Ref. 4-J50 may be useful. We present here a brief review of the two-dimensional theory. Consider a coordinate system at rest with respect to the gas at infinity, the wing moving at speed U(t) in the negative x-d irection . The bou ndary condition of flow tangency must be satisfied along different segments of the x-axis at different instants: a ",'

"

(x, 0, I)

=

wJx, /)

(on R.)

(4-\85)

124

l'IUNCII'U;.s 0 1' AmIO£LAS Tl CIT V M ..

-V i

I I I R. glOfl R.

!w. gi • • n hc: ,.!

"

Fig. 4-1. Steady-flow analog for a tw
Region R. is shown in Fig. 4---7 for the case when the leading edge coincides with the origin at I = 0. A condition of zero pressure

op' (:I', 0, r) =

a,

°

(4---186)

applies on the remainder of the x-ax is (or X-I-plane). Since small perturbations are assumed, the velocity potcntial is governed by the wave equation of acoustics, which reads as follows under the transformation I l = G.} : o!p' rPp' O"'/" (4---187) -+-~ ­

ax!

OZ!

01/

An exact mathematical analog for the problem defined by Eqs. 4--185 through 4---187 is a three-dimens ional wing with the same upwash W6 bnt fixed on the x.II-plane in a stream moving at M = v'2 in the positive redirection (Fig. 4---7), Il being regarded as a third spatial coordinate. The flow is then steady. In addition to Eq. 4---187, we have for boundary conditions

(4---188)

"d orp' = 0,

a"

(elsewhere)

(4---189)

A lmOlJ'o'NAl\ll C O I'EItATO I(S

115

T bis analogous flow can be solved by ae rodynamic influence coefficients, or by exact integration in simple case~, as in Ref. 4-130. The cffcrt of acccleralion A , as a correction to the airloads computed on the basis of instantaneous va lues of M(t) and k(/) = wb/U(t), is governed by the parameter Abfa~ 2. The correction turns out to be very small jf A bla ~! is small, say less than 0.05 (corresponding to accelenltions up to 200 g's on practical surfaces), and if M ::::. 1.5. At lower supersonic speeds the effect is somewhat larger. These conclusions are based on some unpublished calculations. A word of caution is in order here. The rapid accelerations of th e order of 200 g's imply a rate of change of Mach number of order 6/sec. T hus fOT slowly harmonically oscillating a.irfnils, the airforces arc far from harmonic, that is, 11'. will change with time in a nonharmonic manner. This is generally the case in practice where during one cycle M can change qu ite appreciably. Thus the solution of the harmonically oscillating wing is by no means the only one of interest to the ae roe!astician. However, the analog (Fig. 4-7) applies equal1y well to any arbitrary time,dependent motion. In such cases, the re sultant equations of motion of th e vehicle will be linear but will have time,dependent coefficients, a class of problems which we take up in Chap. 10. Moreover, the changes in ambient atmospheric properties during climbing or diving flight may have more significan t effects than those of forward acceleration. 4-7 LIFTING S URFACFS AND OTHER CO NFIGURATIONS IN TI-IREE-DIMENSIO NAL UNSTEADY FLOW; SIMPLE HARMONIC MOTION (a) Subsonic flow (M < 1)

d

- 1

=

g~ 11K(Xo, Yo; k, M)(- ..) d¢ di]

".

where Kix y. k M) = k 2e- 1kr• '0'

0'

,

(_1 - K (kl. I) klYol I

+ ~ [I,(kIYoil- L,.(k IYoIJJ 2klYul + N" J 1 +

I

Mk x"

(4-190b)

U

_ iMk lYol

+ Pe - '(Mkl ••lf#)

M {i(kYO)2

"'~e - 'kl. , h d"T _

"

+

(4-190a)

i

r·"" exp (iU- MJ 1.·+fi'(kYo)~])dl

M(kyo)' )o

+ .j(kx,,)~ + (t'-(kllo)2 cxp

M (k !Jo)2J( k.T,,)~

+ P'(kyuj'

p2

(I[bo - M ../(b:o)2 + P"'(kYo)2])) {i2

Here I[ and K[ arc modified Bessel functions of the I1rst order and the first and second kin ds, respectively, while L[ is a modified Struve function of the first order (see Watson, Ref. 4--85). Equation 4-190b relates to the lifting surface of Fig. 4-1 and is given in this form by Watkins, Runyan, and Woolston (Ref. 4-86). Note that K/k2 is a function of only three distinct quantities, kXo, k l !l~I , and M. Methods of numerical solution of the subsonic load distribution problem, which involve no further approximations in the representation of the physical system and are particularly suited to high-speed machine computation, are developed in R efs, 4--87 through 4-90. Further information on recent NASA research and some comparisons with experiments appear in Refs. 4-91 and 4-92. We describe briefly here the method of Hsu (Ref. 4-89), which we consider to be typical and equally as efficient as, although not superior to, the other numerical procedures. The solution proceeds by assuming an appropriate series of known functions with unknown coefficients to represent !:J.P_lf' These coefficients are related to the known upwash amplitudes at a number of points equal to the number of terms retained in the series, producing a set of linear algebraic equations. A suitable pressure formula, satisfying all edge conditions, is the following:

!:J.P. (£, ,.,)~8.,,!r. q . .J. hi;;)' '--; 7/ 1 [aoo + a~J!1 + aoo!/2 + ar:;;d + . '1./~ ~

+ [a lO + all~ + an,!! + ad]" +"

,}Jt -

+ ["20 + azl?! + az;.!l + a""if' + .. -)£•.11

£2

(4-191)

~. + ... J

Here ~ and ?l aTe coordinates chosen so as to transform the original wing (Fig. 4-8) into a square with sides at f' !l = ± I. They are related to the dimensionlcss physical coordinates by

e=

( -

-

t(llii) + ,,(1/»

! (e,(ii)

e/ij))

;

,

!/ = -,

e,

,

,,

y= ~

e,

(4-192a, b, c, d)

where = E,(1» and = E,(ij) d efine the locations of the leading and trailing edgcs. Substituting the kernel function from Eq. 4-19Ob and the pressure series from Eq. 4-191 into equation 4-19Oa, we are faced with numcrical intcgration problems that must be handled by a specia l tcchnique because of

AEIIO I)l' NAl\fI (.' OI'mtATOII S

121

th e high -order sin g ul:iri ties. Two disti ll c\ melhods have heen presenled by Watkins cl al. (Refs. 4--S6. 4- S7, lilld 4--S9). In Ref. 4-86 the authors divide the wingsp.1ll into a nu mber of stri ps (say 20). Within each strip the continuous c hordwise pressure dislribution is replaced in terms of a number of equivalent concentrated loads (say 4, at the ~-, 1-,:-, and ~ -chord points), following the "vortex-lattice" scheme. Hsu's scheme is based o n Chcbyshtlf's numerical integration formula. Although derived in a different way, this procedure is very similar to Mullhopp's (Ref. 4--31). The optimum chordwise and spanwise upwash collocation stations are arrived at by the condition that the pressure distribution given thereby will furnish the best possible approximation, in the sense that the total lift, the moment about th e midchord, and the second and higher moments are exact for certain special cases. The accuracy achieved depends on the number of stations used and the complexity of the mode of wing vibratio n. After properly accounting for singul,!-rities, th e numerical integrations both spanwise and chord wise are performed in terms of point values of the integrand. The locations of the stations for these point values follow from the condition that, for a given number of integration stations, the accuracy of the integration shall be a maximum under th e circumstances. The stations turn out to be located, a!Xording to the ChebyshefT formulas, in sueh a way that pressure and upwash points are interdigitated and never fall on the same spanwise lines. Hence no singularity is ever encountered when evaluating the kernel function. Values can readily be generated by a subrouti ne programmed for the high·speed computer, so that tables of the kern el function would appear to be unn ecessary for most applications . T he transformation Eqs. 4-192 arc first writte n out for the given planform. After choosing the numbers of upwash stations and integration

Tr-----< '0

tlot-=:::::::::::==t~H"I"" I I~ _ l) ,

fiC. 4-8. Wing of arbitrary planfonn, showing dimensionless chordwi.5e and spanwise variables.

1211

l'Il INCII' U ($ 01' AlmO!';l,i\STICtTV

stations, their loca tions follow frOIl) ChclJy~hc!rs formulas. Thus, if there arc both In chordwise upwash collocation points and 11/ chordwi,c integration stations, they are located, respectively, at x,=-cos ( 2i. ) , 2m + 1

i = 1,2, ... , m

(4-193a)

, ,= - cos (2j - 1)7T) , j = 1,2, ... , In (4-193b) ( 2m + t F or R spanwise upwllsh collocation stations, there will be (R + I) integration stations:

y<=-cos(~),

-

R

~. = -cos

+

r = I , 2, . . . ,R

I

(2, - 1).) ( 2(R + 1)'

$

= 1,2, .•. , R

(4-194a)

+1

(4-194b)

The number of terms to be employed in the pressure expression (4-191) must equal the product mR; and it should be even or odd in the spanwise direction, depending on whether the motion is symmetrical or antisymmetrical. For the symmetrical case, the final algebraic system of eqUlItions

"41r

Uw(H)-4 U Z( p", • " ~ Yo - P.., (R • ([a oo

211"

+

1)(2m

R+l '"

~~( 1 + 1).::1 j~l

+ a..1/ + ano'1/ + .. '](1

+ [a lo -

ali"l."

-

-

~I)

+ a14,,}.4 + . . '](1 -

+ [a.o + at2~.a + a2A.,,]/ + .. ']yl

')

!l•

I;j)

- §/) +

.. -)

. (:]K(Xo, ~o; k, M)} + 4p",U!(R + 1)7T(1-i 12m+l 27T ([a oo + uQ'!Y/ + a... y,4 + .. '](1 - 9 + [alO + a 1.y} + al~~,4 + . . -] (1 -

el)

+ [ow": a~!J!,2 + o:'!1/ + ...] ' £/ (1 _

~12) + .. -)

MmOI>\' NAM IC O I'I(IIATOKS

~' l) 1=, £,1 + 4p", U~(R + 1)2".{raoo + ar.rl./// + U01l!/ + . .. J .

(e -itl!, - ~,)

1211

.[.JI

-

1)[1+

x, -

~/ + sin -l=, +~

+ [ U10 + alU!!'~ + uu !!.': + .. .J

,r;!;, L2 ..)1 - -xl + i51n - -x, + ~J 4 1

+ [alo + fJ't2!} / + a""!t/ + ...J . [-1(1 - ; ,'1"] + ..

j

(4--195)

In Eg. 4-l'i5, K(.xo, Yo; k, M) reprcs~nts the kernel function evaluated for ::to = (x, - (I) and Yo = (jj, - iJ.)· Without further elaboration. it is evident that Egs. 4--191 and 4--195 can be written at the mR upwash stations in the following matric forms:

(0:.) _[K.]H (~) -

[K,]{"l

(4--196)

(4--197)

There are mR clements in each column matrix. Square matrix [Kll depends on the planform geometry an d the dimensionless coordinates x" fi.; whereas [K!J requires, additionally, the specification of k and M. Elimination of the pressure series between Egs. 4--196 and 4--[97 yields the algebraic representation of the aerodynamic operator, (4--198)

where (4--199)

Although comple" algebra is involved in the inversion of [K2 1-I, ma ny successful, practical ca[eulations of the type described above have been carried out. The prescnce of a control surface can be included either by refining the network of collocation points and artificially smoothing the upwash discontinui ty at its leading edge or, more rigorously, by adding the

1311

l'IUNCII'I .E$ OF AliltO liLASTrClTl'

proper logarithmic singularity alon g this line to thc pressurc series. Full de tails of th e latter procedure have not ye t been published. Finally, it should be menti oned Ihal many aeroelastie app lica tions do not require a complete evaluation of thc load distribution according to Eq . 4-191, but onl y certain weighted integrals (generalized fon;cs) of il over the planform. Some delails of the latter procedure arc given by Hsu and Weatherill (Ref. 4-164). Reference 4-90 shows that the amplitude of total lift on a wing of constant chord performing a vertical translation oscillation of amplit ude "Ib is

-

"2 'r['[

L = 8". q

000

' "J '[ +"2' 'J + '[

+"2 +"8

16 ao•

aoz

+ '''J)h "2 b

(4-200)

Here I is the semis pan, and three chordwise and three span wise stations on the half-wing are used in the collocation process. One can imagine many interesting special cases of the general method for subsonic oscillatory load determination just outl illcd; indeed. they effectively encompass the profusion of approximate theories in the puhlished literature. Thus, when the fluid is incompressible, the kernel function reduces to K(:co, yo ;k, M = 0) = k

+

2e-;Ut(_'_ Kl(klYol) + ~[ltCkIYol) klYol 2klYoi kxo

( kYO)2J(kx of

+ (kyo)!

-

;J(kxof

e'ut

(kYo)2

_ 11 J k

(kyu)~

+ (ky,,)~

L1(kIYol)]

..

j.1

+ (kyo)" e'" dJ.)

(4-201)

G

There exist more th an a score of methods designed to simplify the solution of the integral equation when M = 0 alone, covering the fuH range of aspect ratios from fractional to those where spanwise lift and moment distributions arc the only quantities requi red. An exceHent summary of most of the incompressible-flow schemes appears in a report by Laidlaw (Ref. 4-93). A noteworthy example is that of W. P. Jones (Ref. 4-94), for which many applications haw bee n published (e.g., Refs. 4--95, 4- 96, 4-91), along wit h extensivc tables of upwash coefficients (Ref. 4-98). When the number m of chordw ise collocalion stations is sct cqu;1l 10 unity, we obtain a theory of lifting-li ne type. l'articlIlarly IIU!llcroU, arc the incompressible lifti ng-line app rox imnlion, bascd on various modilicalions of the oscillatin g vortex slleel. The cnrliesl of these is Cicalll·s

AWIOIlYNAMII ' OI 'E I( ATOII S

1.11

(HcI".4 'N), hut I'cl"h:II's 1Il"~t widc!y !I~cd in the Uniled States is the one by It cissller :lIld Slevelis (lters. 4- 1(0). TheIr lec!lIlI'llIe is well-systematized fur cll mp"I; lli o n ;1", 1 yic!d, Ihe I"rit,na te rcsu ll Ih"t the only cha nge in the ~ Iril ) tllcory :lcrodynamic opera tor, Eq. 4- 120, consists of an additive c()rredion '1f!7) to qk). Th i, funct ion, wh ich represents i! purely cireuIli lory dkd, is found from the givcn p lanform shape and mode of motion by Fourie r-se ries solut ion of a spa nwise integral equati on. Finite-span IU]justlllclIls ca n be obtai ned in th is way for th e two-dime nsion al coefficie nts I.•. L~.·· · in Eqs. 4- 123-4-125. On st raight wings, sllch lifting-line the ories are accum te down to aspect ra tios of 3 o r below when M"'l. T he slibsonic ke rnel function may be expanded in ascending powers of reduced frcquency k (Eq. 54 of Ref. 4-86 is an illustration), and onc is (hilS led to ano ther so rt of approx ima tion. Reference 4-101 is concerned with low-fre quency problems of dynamic stability and shows, for all Mm:h nu mbers, how important simplifications arc achieved by retai ning tenm o nly up to O(k). An extended Prandtl-Glauert compressibility correction has been worked out by Mil es (Ref. 4-1(2). The subsonic th eory of W. P. Jones (Ref, 4- 103) also falls in to this category, and lypical >:alcu lations are given by Lehrian (Ref. 4-104).

«

Turni ng to the proh le m of simple harmonic mo tion of sle nd er, poi nted wi ngs llnd bodies, we can rely in part on the discu ssion of Sec. 4-4(a), he>:,. use the M u nk-J oncs hypothesis of two-dimensional , inco mpressible flow in planes norma l to Lhe flight direction re tains considerable usefulness . In ReI". 4- 2, Mil es gives th e following restrictions on its validity: for planar surfaces, plH -( 1 and kM2~~ I; for hodies,· MJ -( I and k = 0(1). Assum ing the reduced freq uency small eno ugh tha t these are met, we list several operators which arc the unsteady cou nterpar ts of Eqs. 4-95-4-97. Tile nota tion for plan form and cross-sectional geometry is de fined in Fig. 4- 3. For alm ost- plane lifting su rfaces of local span 2.;-(x),

-<

4 J,b)

J) ~- -

.,.U

f)/

- . /r)

1 h, ( r( x) -17!1- -hl(x) - !l2.jS~(X) - '11 )

.1(x)ly

( ••• )

d'll

'"/1

(4-202) • •" be"er"l. Ihc ampl ilude ralio of li mc-dc!"'ndent body motion mWI be small ",I io ,\; for further dcl"i(, see Mi les' chaple r, "Slender Non-planar

rd" li" to) thid n,"". Und ies."'

132

I'K INC II'LES OF AEKOEIAS'l'ICITY

where DIDl is once more the substantial derivative (cr. Eq. 4-148), or rate of change following a nuid slab,

a

D

a

U-+ - , oX 0/

D, U

. -a + IIlI, a.

(arbitrary motion) (simple harmonic motion)

A special case of Eq. 4-202 when wafU is

indc~ndent

of y reads (4-203)

The lift amplitude per unit chord wise distance results from another integration across the span. Regardless of the cross-sectional shape, (4-204) Equation 4-204 has an obvious extension to more general unsteady motion. Expressions for the Yirtual ma$S p "" S(x) appropriate to wings, bodies, and wing-body combinations are copied from Sec. 4-4(a) as follows: (ellipse or plane wing)

Sex) = 1TS\X), Sex) = "IT[S"(X) - R\x)

+ RI(x)], 5\X)

(midwing-body combination) (4-101)

SeX) = "lTR"(x),

(body of revolution)

(4-102)

F urther information on Munk-Joncs theory and its applications will be found, among many other sources, in Ref. 4-2, Ref. 4-41, or Chaps. 5 and 7 of Ref. 1-2. When ! (according to Miles), slender wings can be analyzed by omitting x-derivatives but retain ing the (J'lrp'/or 2 term in the pOlential differential equation

k»

OZtp'

iPtp'

1 (frp'

oz"

• o/

w~,

-+-~ - -- - -. ay~ a 2 2 a 2

(4-205)

•

An inverse aerodynamic operator resembling Possio' s for two-dimensional, subsonic flow might be given in this case; but aU the available solutions are actually developed by separating Eg. 4-103 in elliptic-cylinder coordinates. This process leads to infin ite series of Mathieu functions. Examples of its app lication appear in Ihe p'lpcrs by Merbl a nd 1..
A I,: ltO IH'NAI\IIC O I 'EI{ ATOU S

133

( I~ ef.

4- 105). Ma ~dsky ( I{cr. 4- 1(6). and M ilne ( I~ e r. 4- 107). to wh ich the reader is directed lo r for ms of the operators assoc iated with particular planforms and moJcs or molion. (b) Tran!>Onic flow ( M :::::: J)

8~ j j

d -I =

K(x o. Yo; k, M = 1)(' . .) d f dij

(4-206a )

'4'

where

K(xQ, Yo; k, M = I ) =

k2e-i~~,(_I_ K 1(k1Yol) + ~ [/](kIYol) klYo l

- LlklYol) -

~] 1T

cxp - kxo - -"Y')] - " - -I) (G'{ kx 2

+-2

"y2

"

"

- ~ i

kyo

2klYoi

1w.,1 " ['2( "")] ) exp - ;.-~

d).

(4-206 b)

I-

R;

fntegmtion region consists of the disturbed upstream wne of infl uence of (x, ii), which at sonic speed is limited to ( ~ i or "'0 ~ O. There also exists a direct form of the aerodynamic operator in th e three-dimensiona l sonic case, which may, after appro:dmations reducing the upstream zone of influe nce to finite extent, form the basis of a numerical procedure like the one described below for supersonic flow. As ca n be worked out from Landahl's thesis (Ref. 4-1 65), the direct operator reads

.PI =4(:i

+

ik) JfK(X"g'YO;k, M = 1)(·· ·)d!d17

(4-206c)

no'

where

(4-206d )

k»

Under the restriction II - MI, Ref. 4-165 shows how Eq. 4-206d can be generalized to other transonic Mach numbers in the nieghborhood of unity. Equation 4-206h first appeared in Ref. 4-86. A process of numerical solution can be worked out, pa ralleling that for the suhsonic integra l equation and leading to a ma tric relation like Eq . 4-198. The only fully doc ume nted technique now in the litera t ure is thatof R unyan and Woolston (Ref. 4-87).

The field oftrnn sonie flow past slender. pointed configurations has scen interesting recent developments of practical impo rtance. To begin with, it can be stated that, for sufficiently small aspect ratio or fincness ralio, the Munk-Jones operators (Eqs. 4- 202, 4-203, and 4-204) may be used light through sonic to supersonic flight speeds. The exact form of the limiting conditions on 6, At, and k is qui te complicated- as are the conditions permitting linearization itself-and they are thorough ly investigated by Landa hl in Refs. 4- 72 and 4-45. A genend requirement fOf linearization in the case of a wing or wing-body having thickness ratio" is stated as (4--207) Equa tion 4-207 seems to imply that a wing without thickness (6 = 0) is susceptible of linear treatment for arbitrarily small reduced frequency . This is true, however, only as regards in-phase or stiffness derivatives such as oC1JolZ. and oeM/oa. Unless lR is vanishingly small, damping derivatives become infini te as I/nki. Nevertheless, linearized theory yields satisfactory results for damping at any k which might be encountered in practice. Extension of tbe linearized formulation beyond Munk-Jones to solutions of the more general transonic dilTerential equation

(4-2"') can be accomplished by a process of cxpansion in M. and k resembling that of Adams and Sears (Ref. 4-50). Details of this development arc almost enti rely the work of Landa hl, who has published data on total pitChing moment and lift of pointed wings in Ref. 4-108; on midwingbody combinations and bodies of revolution in Ref. 4-109; on rC!Ctang ular planforms and more general oscillatory motions in Ref. 4-110 and elsewhere. No incl usive statement of aerodynamic operators can be made here. It is significant, however, that Landahl finds damping derivatives which deviate appreciably from Munk-Jones theory even when At is quite small. In Rds, 4-108 and 4-109, for instance, he calls attention to a large influence of leading-edge curvature on fixed-allis damping In pitch; whereas calcu lations by Eg. 4-205 show none whatcver. (c) Supersonic How (M > I, M/j
d

=

4(:x

+ ik)

II II,

K{x(}> YQ; k, M){' . ')

d! ilij

(4-209a)

where

(4--209b)

R:

As discussed in Sec. 4-4(c), the region of integration consists of that portion of the projccted planform and disturbed diaphragm regio n contained between the Mach lines ii = fJ ± (i - ()fP going forward from (x,0 . Equations 4--209 represcnt the fully linearized result for almost-plane surfaces, and K is actually thc kernel function for the velocity potential discontinuity 1::.'1':. This form seems to have been given first by Garrick and Rubin ow (Ref. 4--111). The calculation of oscillatory supersonic airload. has been approached in two general ways : by analytic solution of the linearized dilTerential equation for particular planforms and modes of motion ; and by numerical integration of the operator.<'/, or its inverse, in more general cases. Miles' monograph (Ref. 4-2) furnishes the most complete account of solutions of the former class. He presents extended treatments of such cases as the wide delta with all supersonic edges, the quarter-infinite wing (rectangular wingtip without tip interaction), and tapered planforms with straight edges. Watkins ct al. have analyzed rigid-body motions of rcctangular surfaces in some detail by expanding the integral equation relating ",' and w. in ascending powers of the red uced frequency (Refs. 4--112, 4--113). The delta wing has also formed the subject of considerable work by NACA. Deltas with supersonic leading edges and related wings are treated, for example, in Refs. 4--114, 4--115, and 4--1 t6. The same planform with subsonic edges poses greater practical difficulties, the series solution of Watkins and Berman (Ref. 4--tl7) being regarded as the most useful for applications. Thein; and subsequent airload computations are generally limited to clastic deformations which are, at most, quadratic in the coordinates x and fj. A great virtue of such exact solutions is that they serve as standards for evaluating app roximate methods. For the shapes to which tbey apply, a possibili ty also exists of building up tables from which generalized forces for actual flutter or dynamic response calculations can be constructed by curve-fitting the normal modes of the given structure. When supersonic expansions in k are terminated at the first power, airloads for low-frequency motion can be obtained with relative case by an extension of Evvard's theorem ( Ref. 4--118). Two reports by Watson (Refs. 4-119 and 4-120) are more recent ex.amples of this technique, which is also exploited in Ref. 4-101 and elsewhere.

136

l'flJ NCII'LEIS OF IIEItOIlLII ST ICITY

Fill" -4.'. Grid of boxes overlaid on a typical supersonic planform. Considering the diverse planforrns and aspect ratios of modern lifting surfaces, it is fOItunate that nu merical procedures like that of Pines and collaborators (Ref. 4-12 1) overcome nearly all limitations inherent in the analytical solutions. Except for some recent work on the inverse operator ,#- ,(. ..) (Ref. 4-122), all such schemes start from the aerodynamic influence coefficient, which is defined as the velocity potential (or pressure) developed at a point on a wing due to constan t am plitude of sinusoidal upwash IV.j U over an elementary area influencing that point. A grid of such areas or boxes is superimposed on the planform, as in Fig. 4-9. For disturbed areas of the :r.y-plane adjacent to a subsonic edge, the diaphragm concept is introduced by placing boxes there and requiring zero potential or pressure discontinuity. Details of box shapes and fineness of grid networks, which strongly affect the numerical accuracy, are discussed in two classified rep orts (Refs. 4-123 and 4-124). With an eye to high·speed computation, the whole airload de termination is reduced to a large aggregate of elementary, repetitive operations. Returning momentarily to the dimensional coordinates, the potential amplitude at point (x, y) on the upper wing surface is

WM ,

x 00' ( -..., (x

ulf

Aml.OOYNM1IC Ol'mIA'I'OItS

137

As i,l Fig. 4-9. the diswrbed arc~ is "pi it iuto red,lnguk,r boxes of ehordwise dimension ",. wilh diago nals paral ic! 10 the M ~e h lincs. If over each box, w. is madc co nsta nt and cquaL to its valuc at the central point (x" 11"), thcn the potcnti~ 1 discOlltilluity in the ccnter of box (n, m) is

x

(AIf.f.,~)

",p[_,wM\•• _<)],,,(w'",-/(z._,)'_(1'(Y._.l')

== - ~ I I

u~

Up .../(:c~ )il.(:C"

y.){R""

e ' ,

+

!)t

)

dedI)

P(Ym

1)2

(4-211)

iI",,}

The quantity in braces is the influence coefficient. In dimensionless fonn, C'.i>' is a function of M , = wb1 M 2 /Upt, and of the relative positions v = (/I - 1'), fi = (m - Il) of the recc iving and influencing boxes (Fig. 4-9). Although exact formulas are available (Rcf. 4-123), it is usually adequate to appro ximate C"" by replacing (e, 1)} in the numerator with its central value. There thcn result

"1

(

-1

+ 2M2 + 1 (k,)! _ 48

M

8M~

+ 24M! + 3(k,¥) 15,360

+iMI~(~) _ 2~2:3(~)} C." , = R•• , - + 11;;. , =

M}

(forv=p=O),

(for v 2. p,

ji

2. 2),

(fOTj= I,P_O).

and

c- - = ' .- ~

C,,, ."

{4-212a}

13H

"IUN CII'LI>S 0 '" ,U mOELA ST IClT\'

where cosh -1(2Y

+

I) - cosh - 1(2;; - I) + (v

-'-J

+ D['" - 2 cos- I 2,;+ 1

(forY~2,jj= O),

-Cii-t{".- 2COS- '2,,1 1J - (v - t) cosh _,2'; + 1 2f-1

(p.

fl. ,~

+ O[COSh - 12V +

+ (Y + !) cos- ,lf -

1 (forv =1'';2; 2), 2,:;+1

1_cosh- , 2ft2,; -+ 11J

2,1l+ 1

=

,,' I.!"

-

.) [ cos h - 1 2,; \l 2/1

A

+11

2 cos"-1 '; -

2ft

:J

H v + !)[cos- 1 2fi - 1 _ cos-l 2!l + IJ 2,;+1

2v+l

(4--2 12b)

The computation represented by Eqs. 4-212 is normally relegated to a subprogram in high-speed computers. Equation 4-21 1 is complete for finding potential distributions on wings with all supersonic edges. When diaphragm regions are present, the systematic calculation of the un- known w4 on diaphragm boxes from the condi tion 6 rp; = rp' = 0 is described in Ref. 4-1 23. Once I"~ is everywhere kn own, the tlrp: determination proceeds as before. In nearly all dyn amic analyses, the final aerodynamic step is to compute generalized forces or weighted integrals of the pressure distribution. A typical such force is described by the equation

Q;;

=

II

ap;(x, y)f;(x, y) dx dy

(4-213)

H.

wbere/l is the j th deReetion and 6p; is the pressure difference due to mode f,. When velocity potentia! influence coefficients are adopted, a single integration by parts permits QII to be expressed as a summation involving the predetermined tlrpc'(x n, y",) at the centers of wing boxes and at points

AlmQIH'NAMI C OI'I,-:IIATOII$

139

a long (he (rH iii ng edge. This su I11ma (ioll is based o n nu Illerica I evalua (ion, using stations ,,( ~"(; n(crs of th e boxes, o f (he foll owin g :

QIj = - :""U

II

6'1'/ (z, Y>[ikfAZ, y) - b afJ~~ Y)] dre dy

R.

-

p~UJ.

6«1;'(z" y)fAz" y) dy (4-214)

. ~.

since (4-215) and where the subscript t refers to points on the trailing edge. Reduction of the foregoing operations to matrix notation in a manner similar to previous sections is an elementary process, wltich will be omitted here for brevity. As just described, the method of aerodynamic influence coefficients does not rigorously account for the presence of the singularity in 6p. at a subsonic leading edge or that in w. adjacent to a side edge or subsonic leading edge. Reference 4-123 discusses modifications to the basic theory for these purposes. When 6Vo' is used in place of t:..p. as the principal unknown, however, only its slope is infinite at the Icading edge; and an increase in the number of boxes covering the planfonn usually prOVides an accurate enough way of dealing with the Singularities in practice. Flap-type control surfaces present no particular problem, so long as their leading and trailing edges are supersonic. With. regard to th.e second-order effect of wing thickness on oscillatory supersonic airloads, the comments made in Sec. 4-4(c) concerning threedimensional steady flow retain their validity. No rigorous theory is available in the lower range of M. When M2 )0 1 and the area concerned is outside the influence of a cut-off wingtip, Landahl's results (Ref. 4-24) may be used. Including terms of order dIM, 61 Ma, and
Reference should also be made to the approximate adaptation of van Dyke's theory to wings of finite span by Rodden and Revell (Ref. 4-80).

140

1'lUNell ' L ES OF AEIIOELASTICITY

»

As in steady now, Eq. 4-216 suggests (provided (j If Ma) tbat linC;lri~-cd theory might be roughly corrected for thickness by applying the overall factor

"'J

[ I + (' +2 ') M ox

to Eq. 4-209a.

When MlJ is small enough and k is also limited, the Munk-Jones operators (4-202), (4-203), and (4-204) may be applied to slender, pointed wings, bodies, and wing-body combinations in supersnnic fiight. Linearited theory is still permissible for higher frequencies and aspect ratios, but then the additional terms beside (cp'1",ay2 + a2rp'loz2) must be included in the governing differentia l equation. The various solutions of this type do not lend themselves to concise statements of ae rodynamic operators, but we shall give a few references. The topic of low-aspectratio planar lifting surfaces is discuss ed at length in several chapters of Miles' monograph (Ref. 4--2). Especially useful for large k are the Mathieu-function series, of which Merbl and Landahl's is typical (Ref. 4--105). Wing-hody combinations and bodies of revolution have been analyzed by the method of expansion in 1; and At (Refs. 4-125). These papers deal mainly with tota l lift and moment d ue to rigid-body oscillations, although the seeond provides some information on chord wise deformations where IVa is reprcsented by a polynomial in x. In the damping derivat ives, Refs. 4-125 find large deviations from Munk-Jones theory cven at relatively low values of {JiR. In fact, this would appear to be a field deserving much further investigation. There are as yet unpublished studies, for example, which indicate that sometimes the non linear effects of thickness on supersonic wing-bod ies may be as great as those due to higher-degree aspectratio terms in such expanded linearized solutions. Finally, it is essential that the transition be carried out to the nonlinear theory of van Dyke (Ref. 4-51) for slender bodies in hypersonic flow. A step in this direction is the teChnique of perturbing the steady-state characteristics solution (Holt, Ref. 4-126). (d) Hypersonic fl ow (M2 )- I, MlJ

=

0(1) or greater)

To avoid repetition, the reader is referred to Sees. 4-4(d) and 4-5(e) for discussions of this topic and literature citations. In parrieuJar, it appears that the comment regard ing the quasi-steady nature of acrodynamiC operatnrs, made in Sec. 4-5(e), may be extended to three-dimensional flows. For hypersonic lifting. sulfaces which arc neither too slende r nor too highly swept back, strip theory is valid; and res ults such as Eqs. 4-146, 4-147, and 4-149 are useful without modiliwtion.

t.I, tIOO \ 'NII.J', lI C O I' t'.MATOIiS

4--8

141

OTHER 1 ' I{onU~ M S OF UNSTEADY MOTION IN THREE DIMENSIONS

Any systematic presentation of time-dependent aerody namic operators for finite wings and bodies is hampered by the rathe r incomplete development of the theory in this area. Indicial functions like those defined in the introduction of Se<:. 4-6, along with certain generalized forms which are associated with elastic deflections, have been labo riou sly calculated on a few specific planforms in supersonic or incompressible flow; but there is little more in the literature. Except for setting down one or two generally interesting formulas, we th erefore confine ourselves in this section to listing some of the important references. From the standpoint of the aeroelastician faced with the routine analysis of transient phenomena where unsteady effects are significant, we can make some hopeful sta teme nts to ameliorate the apparently unpromising prospect of inadeq uate aerodynam ic tools. Iu the past, strip theory, usual!y with a steady-state tip correction, has been employed for lifting su rfaces in combination with slender-body tbeory for fuselages, external stores, missiles, and the like. There now exists a considerable body of experience on the determination and interpretation of dynamic loads and stresses in this way; by no means should this be hastily abandoned, except possibly on the moderate to low aspect-ratio surfaces, at subsonic, transonic, and low supe rsonic speeds, whose plate-like deformations render the strip concept wholly invalid. In high superso nic and hypersonic flow, two-dimensional aerodynamic methods have very broad applicability, as reviewed elsewhere in this chapter. But of even more direct conce rn is the validity of quasi-steady hypersonic theory, both two- and three-dimensional, for time-depcndent motions which would certainly demand the full unsteady treatmen t at lower fl ight velocities. Intensified experimental and analytical research is called for to delineate more precisely the limits of such approximations. There is .a growing tendency to try to unify the study of transient behavior of flight vehicles on a foundation of simple harmonic inputs. This is a development to be encouraged. Sinusoidal aerodynamic operators, even for quite general three-dimensional configurations, are now vcry well systematized, the majority of them having already been programmed for high-speed digital computers. The usc of sinusoidal responses in statistical treatme nt of t he gust problem is an excellent example of this sort. Another is the evaluation of maneuvering pe rformance by sinusoidally forcing various elements in thc control system. Even where such inpu ts ca nnot be assumed directly (fo r instance, when complying with

142

l 'II INOI ' t .ES 01 1 A lmOELASTI ClTV

current military and civilian gust and maneuver ~pcl: i!i cali ons). Fourier's superposition principle can prove userul in an approximale way. Thus. on a stable aircraft, graded gusts of var ious lengths ca n be applied in a periodic fashion, wi th the rundamental period long enough to allow thc motion to settle down. Then a Fouricr series rcpresentation of the gust inp ut permits the loads, stresses. or disp lacements to be synthesized from sinusoidal gust responses at a discrete set of wavelengths. In cases involving three-dimensional flow, when unsteady effects must be included, superpositi on may prove to be the only practicable approach here. To review the li terat ure on indicia l functions for finite wi ngs, we first consider incompressible linea rized theory. The angle-of-attack and natural-gust lift fUnctions rp(s) and op(s) [kl(s) and k!(s) in NACA notati on] were first estimated by R. T. Jones (Ref. 4--153), using elliptical planforms with JR = 3 and 6 as examples. W. P. Jones presented an operational app roach to the same problem in Ref. 4--154. His computations were on
S being the plan area. On a cambered wing, ~ in Eq. 4--217 is rep laced by the average chordwise slope <:>:(x, y) of the mean surface . • We remark. in passing, that .ome so...:allcd theories for calculating indicial function. do liUle more than fair a s mooth curve betwC(:n L(O) and the final steady.state value L(co).

AEII:OIlYNAMIC OI'EHATOIIS

J4.l

The princip
(4--218)

is accounted for by the following alteration in Eq. 4-205: L(x,

I) =

_ p",U o2 [1

+

1

Vo[1

u

+ JJI( [dS a;~ +

+ fJ

dx ax

[dSa;. dx

at

_ P S(x)dfaz. "'. dl ax

+ 2S(

Sex) Q2~.] ax

)

iPz_] +

x OXOI

S( x )

UG~[l + fJ2

ifza) 012

(4-219)

Here z.(x, /) represents the time· dependent displacement of the centerline. The virtual mass p",S(x) in Eq. 4-219 can be found from Eq. 4-100,

144

I'R INC II' LES

O~'

AllROELASTIC IT V

4-101, 4--102, or the appropriate formula fo r the cross-scclional shnpe under consideration. REFERENCFS 4-1. 4-2.

4-3.

4-4.

4-5.

4---6. 4-7.

4-8. 4-9 . 4-10 . 4-11. 4-12. 4-13.

4_14. 4-15.

Garricl:, I. E., Nonmady Wi/W Characur;stics, Section F , Vol. VIl, of High Spwi A~rodynamicr alUl Itt Propulsion, Princeton University Pr=, Princcton, N.J., 1957. Miles, J . W., T/r~ Pot~ntial Th~ory 01 Ul!$t~ady S"P~rsonic £1ow, Cambridge Monographs on Mechanic. and Applied Mathemati"" , Cambridge Unive!"!)ity, 1958. (Adapted from a monograph prepared for Air Researeh and Development Command, USAF, in 1955 _) Heaslet, M. A., and H. Lomax. Supersonic and Transonic Small Perlurbation T/r~ory (see Chap. 6), Section D, Vol. VI of H(~h Speed A erodynamics and Jet Propulsion, Prinedon University Press, Prinq ton , N.J., 1954. Robinson, A., and 1. A. Laurrnann, W;'Ii' Theory, Chap. 5. Cambridge University Pre", Cambridge, 1956. Temple, G ., "Unsteady Motion," Chap. IX, Vol. I , pp. 325 - 374, of Modern D~"dop~nl' In Fluid Dynomics_High_Sp~(d Flo w, Clarendon Prt$$, Oxford, 1953 _ F ranl:l, F. L., and E. A. Karpovich, Gas Dynomics of Slender Bodies, Chap. IV, MoSCQw, 1948. (Transla ted from the R ussian by Inte!"SCience Publishers.) Liepmann, H. W., and A. Roshl:o, EftJ.rell/s 01 Go.dynamlc-" John Wiley and Sons, New Yorl:, 1957. Bateman, H., "Notes on a Differential Equation Whieh Occurs in the TwoDimensional Motion of a CompreSiiblc Fluid and the Associated Variational Problems," Proc. Royal So~. , London,(A}, Vol. 125. November 1929. pp. ~98 -618. Bateman, H., Par/iaf Dijferenliof Equa/ions of Mo/h~malicof Physics, Dover Publications, New York, 1944. Hargreaves, R., "A Pressure Integral as Kinetic Pote ntial,"' Philosoph/cui Mago._ 21M. Vol. XVI, September 1908, pp. 436--444. Lamb, Sir Horace, Hydrodynamics. 6th ed., Dover Publications. New Yo rk, 1945. Ward, G. N., Lin~arized Tkwry of St~ady High·Spe~d Flu"" Cambridge University Press, cambridge, 19~5. F lax, A. H., "Rever$C-Row and Varia tional Theorerm for Lifting Surfaces in Nonstationary Compressible Row," J. Aero. Sciences, Vol. 20, No.2, February 1953, pp. 120---126. Wang, C. T., "Varia tional Method in the Theory of Compressible Flu id," J. Mra. Scl~nces, Vol. IS. No. II, November 1948, pp . 67'_685. Riparbelli, C. , "A Principle of Ma~imum Power for Real Auids in Steady Motion," Readers' Forum, J. Aero. Sciencu, Vol. 23, No. 10, October 1956, pp. 971---972.

4- 16. Miles, J. W., "On the Sloshing of Liquid ina FlexibleTank." J. Appll~"Muha"ics, Vo l. 25, No.2, June 1958. pp. 277-283. 4-17. Greidanus, J. H.,A. I. van dcVooren, and H. Bergh, ExfX!rimentol Delerminoao" af the Auodynamic CoeJ!klenlJ of an Osdlla/i/W Wi/W I" Incompressible, Two_ Dimensional Flow, Parts I- IV, Nationaa l Luchtvaart laboratorium. Amsterdam. Reports F-IO I , 102. 103, and 104. 1952. 4_18. Glauert, H. , The Elements of Anafoil
AIlII.QIlVNAM I C Q I' lmATQII.S

145

4-19. I'ope . /I. .. Ik/Sj'~ lVi",,' "'''/ , fjrf"U 71,eory , McGraw_Hill Ilo<Jk Company, New York. 19~ 1. 4-20. K ucthc, A. M., and J. D. Scheller, Fo""II"lil>". of Aerollp'omks. lohn Wiley and Sons, New York. 19:10. 4-21. Ferri. A., Ele",~"', of AlOk F1ows, The Macmillan Cornpany, New York, 1949. 4-22. Gudcrley, K. G., Th~ri~ sehul/n,,'Mr Slriimunge". Springer-Verlag, Berlin, 1957. 4- 23. lighlhiH, M. I ., ··Oscillating Airfoils at High Mach Number," J. Aero. Scienas, Vol. 20, No.6, June 19~3, pp. 402--406. 4--24. Landahl, M. T., "Unsteady Flow Around Thin Wings at High Mach Numll<:rs," J. Aero. Scienu., Vol. 24, No. I, January 1957. Pl" 33-- 38, 4--25. Molyneux. W . G .. MM.urem~'" of Ihe Aerodynamic Forces on Oscillating A erofoils. Advi.ory Group for Aeronautical Research and Development Rcpon J5, April 1956. 4--26. Rainey, A. G. , Meamrement of Aerodynamic Forus for- Vorlou$ MMn Angles of Allock on on Airfoil Oscillall"l{ in Pilch ond on Two Finiu-Span Wi,\\". ascii/ali,\\" in Bendi"l{ will, Empha"is on Dumping ill Ihe Siall, NACA T edmical Note 3643, May 1956. 4--27. Halfman, R . L., H. C. Johnson, and S. M. Haley, Evalual/on of High-Angle-ofAlla"k Aerodynamlc-D~rIVIJII"e DaM oOid Sial/-Fluller Prediclian T~c""i'l"es, NACA Technical NOle 2533,1951. 4--28. Lees, L., Hypusonic Flow, I nslitute of the Aeronautica! Sciences P"'print No. 554, presentoo at Fifth I nternational Aeronautical Conference, June 1955. 4- 29. Egscrs. A. J. , Jr., C . A. Syvcruon. and S. Kraus, A SI~dy of Inviscid Flow about Airfoils m High Supersonie Spuds, NACA Report 1123, 1953. 4--3(}. lvey, H . R., E. B. Klunker, and E. N . Bowen, A Methodfor Delumlni"l{ the !,!uodyOlOmic Chara(lerlslifS pfTw", and Thrce-pimenli"nal Sh(lpe~ 01 Hypersonic Spuds, NACA Technical Nate 1613, 1948. 4--31. Multhopp, H ., Mellwdsfor Calculaling Ihe Uft Dis/ribulion of Wing$ (,Subsl>llif Lifling Surface 11oMry), Britiih A. R .C. Reports and Memoranda 2884, 1950. 4--32 . ReislO of Sweplback Wi,!!," NACA Technical Memorandum ]]20, 1947. 4--35. Van Spiegel, E., and R . Timmsn, Lineariud ACTl>dynamic Theory for Wings of Circular Plonform in Steady and Unsteady Incompressible Flow, Nationaa! Luchtvaartlaboratorium, Amsterdam, Report MP 134, 1956. 4- 36. De Young, I ., and C. W. Harper, Theoretical Symmetrical Span Load1nc 01 Subsonic Spuds for Winp Having Arbilrory Plan/orhl, NACA Report 921, 1948. 4-31. De Young, J. , Theoretkal Anlisymtmlric Span Loading for Wing. of Arbilrary Plunform ul Subs""ic SfUeds, NACA R e port 1056, 1951. 4- 38. Scars, W. R .. Small P~rlurboliQOO Thmry. Section C, Vol. VI of High Spted Aerody"omics ond Jel Prop"lsion. Princeton University Press, Princeton, N J., 1954. 4- 39. Jone., R. T., Pro{'erlies of Low-A'~CI-Ralio Poimui Wi"l{' al Spuds Below and AIw"" lile 51'",,11 of So""d, NACA Report 835, 1946. 4-40. Diederich, F. W .. A l'I""fur", Pa"''''el~r for Correlotlng CerMiI' Aerodynamic C/",,,,rlrri.rlic,. of S~'e{'t Wing.•. NACA Tcchnic~l Note 2335, 1951.

146

I'IUNCII'LI~

01" AlmOIll .AS1'ICiTV

4-41. IIry801l. A. E., "Stabi lity I).rivulivcs for n Sle"dor Missile with Application to Q Wi ng· Hody-Vertical_Tail Configuration," J. AcTO. ScI~IIC'$, Vol. 20, No, 5, May 1953, PP, 297-308, 4--42, Brown, C. E., A."'odynamic. of Bodies 01 Hig" Speros, Sect ion B, Vol. VII of Hig" Sp~d Aerodynamic. and lei Propulsion, Princeton University Press, Princeto n, N.J., 1957. 4-43 . Ferrari, e., InteroCI/"" Problems (see: Chaps. C, 15 through C, 17), Seclion C, Vol. VII of H lg" Speed A~r()(/yMmia and Jel Propubwn, Prino:ton University

Press, Princeton, N.J " 1 9~7. 4--44. Heaslet, M. A., H. Lomax, and J. R. Sprelter, Lineariud C0"'Pffulble·F1alf' 'J"Mory far Sonic FligM Sp ffl4 . NACA Report 956, 1950. 4--45. La"dahl, M. T., E. L. Moll!>-Christenscn, and H . Ashley, Partir/r irie Siudies af VisCOllf and Nonuiscaus Unsteady £fows, USAF Office of $cienlilic R= arch T«hnical Report No. 55·13, 1955. 4-46. Jones, R. T., " Some Recent ne""lopmcnts in the Aerodynamics of Wings for High Speeds." Zeil$chrifl for Flug"'is~nsclwften, 4 Jahrgang. Heft 8, August 1956. pp. 257-262. 4-47. Heaslet, M. A., and H. lomax, SuperJanic and Transonic Small Perlurbollon '11Ieory, Section D. Vol. VI of High Speed Aerodynamics and Jel Propul.;"n, Princeton Unive ... ity Press, Princeton , N.J., 1954. 4--48. Mirel .. H., and R. C. Haefeli, "The Calculation of Supersonic Downwash Using Line Vortex Theory ," J. A(ro. Sdenct!~, Vol. 17, No. I, Ja nuary 1950, pp. 13-21. (This paper ~mphasju,. flow in the wake; see also references given herein.) 4-49. Walsh, J., O. Zartarian, and H. M. VOSll, "Generaliztd Aerodynamic Forces on the Della Wing with Supersonic Leading Edges," J. Aero. Sciences, Vol. 21, No. 11, November 1954, pp. 739-748. 4--50. Adams, M.e., and W. R, Sean, "Slender Body Theory-Reviewand Elctension," J. Arra. Sd"nCfls, Vol. 20, No.2, February 1953, pp. 85-98. 4-~1. Van Dyke:, M. D " A. Sludy of Hypnmn/c Small·Dlslurbance 'J"Mory, NACA Report J 194, 1954. 4--~2. Bannby, J. G., H. J. Cunningham, and T. E. Garrick, Sludy "f Effeci. "f Sweep on Ihe Fluller ofCantil-.r Wi'lf" NACA Report 1014, 1951. 4-S3 . Van de Vooren, A. I. , and W. Eckhaus, Sirip '11Ieory f"r Oumali'lf S wepl Wj'lf~ ill Incompressible flaw, Nalionaal Luchtvaartlaboratorium, Amsterdam, Report F 146, 1954. 4--54. Eclitau. , W., Strip Thtory for Osc/llo./ln8 Swept WUW1 in Camprus/ble Subsonic Flow, Nationaal Luchtvaartlaboratori um, AmS(crdam, Report F 159, 1955. 4--55. Schwarz, L., Ikru:hnu,,!: der Druckvtrteilung eilft r harlrWTli"'" .iell Nrformen
,,.,.

4--~6.

4--57.

4--58. 4-59.

4-60.

Smilg, B., and L. S. Wasserman, Appficalion afTh ffe- Dimens/O/iai Fluller Theory 10 Aircrafl SlruClur~z, Air Force Technical Report 4798, 1942. Wasserman, L S., W, I. Mykytow, and I. N . Spielberg, Tab F1Ullu n.eary and AppUcatiom, Air Force Technical Report 5153, 1944. Possin, c., "L'Anone Aerodinamica ~ul Proftlo Oscillank in un Fluido Comprcs.ibile a Velociul.lposonora," L'AuOIt!cniaJ, 1. XVIII, fase. 4, April 1938. Fettis, H . E" An Approximtlle Melhod for Ihe Co/cliialion of Nan.lolronory Air Forces 01 Subsonic Spuds, USAF Wright Air Development Center Technical Report 52- 56, 1952. Fetti" H. E., Tables of lift and Moment Cotjfichnts fa r on 0.r:i11a1i"K

AllRODYNAM IC O J'''MA'I'O MS

147

WI'IX-Allum, Cmnbl,!(JlloII III 7\vt>-D"tI~IIslona/ Suhsonlc How, Air Force Report 6688 (with .upplemcntory pabcs), 1951 . 4-61. Luke, Y. L., Toble50fCo~fficlcnlSfor CompruJibl~ Flult~r Ca/ca/al/CN", Air Force Technical Report 6200, 1950. 4-62. Turner, M. J., and S. Rabinowil>:, A~raJyllUmlc Coefficienls for on 03cil/ali"K Airfoil wilh HI"8~d Flap, wilh Tables for .. Mach Nllmb~r of 0·7, NACA T~nical Note 2213, 1950. 4-63 . Timman, R., A. I. van deVoon:n, a nd J. H . Gll'idanus," Aerodynamic Coef!Icients or an Osdllatin b Airfoil in Two-Dimensional Subsonic F low," J. Aero. Sci~nces, Vol. 18, No. 12, December 19~1 , pp. 797-.802. (Sec other ref=nces giVCIl hen:in.) 4-64. Reissner, E.., Oil Ihe Apph"coliotl of MUlhit " F/IIIctimlJ in Ihe Theory of Subsonic Comp,eu/bk FlGl" Posl OScillating Aj'foils, NACA Technical Note 2363, 195 1. 4-65. Williams, D. E. , On Ihe lmesral EqualiGl1J ofTwo-Dimeruional Suhsonk Flml~r Derit>Olive Theo'y, British A.R .C. Reports and Memoranda 3051, 1955. 4-66. de Jaber, E. M.., Tables of Ih~ A uodY""mic Aileroll-CotfficknlJ fa, all Oscillali'W Wills-A ilt,oll System In a SlIbJonlc, CompreJJiiJIe Row, Nationsal Luchtvaart. laborator;um, Amsterdam, Report F 155, 1954. 4-67. Anon., Tables of AuD
1950.

4-77. Minhinnick, I . '1'. , a nd D . L. Woodcock, Tabl~of Aerodynamic Flutter Deri.at/~. for Thin Wings mill Conlrol Surf""'" {n Tw",Dimerutonal SUJU'rJonie Flaw, Royal Air<:raft Establ i.runent Report No. Structures 228, October 1957. 4-18. Van Dyke, M.. D., SUfHrsonic Row Pun Oscil/atmc Airfoils lnc/llding Nonlinear ThlcJcneu Effecls, NACA Report 1183, 1953. 4-79. Jones , W. P., ~ 111ft",,""" of TIl/dRess/Chord Rallo 011 Soptrs()nic Derirx!li~es

148

4-80.

4-81.

~2.

4-83.

4-84.

4-85. 4-86.

4-$7.

4-88.

4-89.

4-90.

4--91.

4-92.

4--93.

4--94.

PRINCIPLFS

m'

AEROELAST ICITV

for Oscil/a,;ng AJrfoi/s, British Aeronautical Rcse~h Cou ncil Reports and Memoranda 2679,1948. Rodden, W. P., and J. D. Revell, Oscil/owry A~rodynamie C""Jlickms for a Unifi~ Supersonk-/fYfN'r$Onic Strip T&wr, Including tlt~ EJf~CIS ofSwup, Thicku.s and Fin/It Span, North American Aviation, Report Na-57-1549, December 1957, Ashley, H ., and G. Zartman, Supersonic Flull~r Tr~nds as R~veal~ by Pis/on Thmr, (Alcuia/imu, USAF Wright Air Development Center Technical Ileport 58-74, January 1958. Morgan, H. G., H. L. Runyan, and V. Huci<el, ''Theoretical ConsideratiollS of Flutter at High Mach Numbers," J . .Aero. Sdenus, Vol. 25, No.6, June 1958, pp. )7[-381. Ashley, H., and G. Zanwan, "Piston Theoty-A New Aerodynamic Too! for the Aeroclastician;' J. Aero. Sciences, Vol. 23, No. 12, D ecember 1956, pp. 11091118. Zartarian, G., Utu/eody Airloads on l'oinl~ Airfoils and Slent~rimenta/ Pr~ss"re Dillribut/olls ,,/I Low, Aspecl-Ratio Wing. Oscillollng inan Ineomp,~s.rl"/e Flow, Technical Report S 1_2, M.I .T. Aeroe1astic and Structures Ret.ea rch Laboratory, 1954. Jones, W. P., The Calculalkm of Aerodynamic Dcril... llvc CO<'Jlidelll.r fi" W/'~~,r of AnyP/onformin Non-uniform MOlioll, British A.R.C . Re portoHIlt! Mcmonllldn 2470,1946.

AERODYNAMIC OPERATORS 4-95. 4-96.

4-97.

4-98. 4--99.

4-]00. 4-101.

4-102. 4-103. 4-104. 4-105.

149

Uhrian, D. E., AuodyllQmlc Co~fJicJtJIIs lor an Oscillating ~/la Wi,!!" British A.R.C. Reporu and Memoranda 2841, 19~ 1. Woodcock, D . L., Aerodynamic D" I""tiKs lor Two Cropp(d Dell(l Wings and One Arrowhead Wi,!:\, Oscilluting In Disior/ion Modes, British R.A.E!. Report No. Structures 201, ApriJ1956. Woodcock, D. L., Calculated Aerodynamic Forces on a Sweptback Untupem! Wing Oscilluting ill Illcomp,e.. .lble Flow. British R.A.E. Report No. Structures 211, December 1956. Anonymous, Dawnwosh T(lb/es lor tM Colculolioll 0/ A erodYllamic Fo,ces on Oscillating Winc>, British A.R.C. Reports and Memoranda 2956, 1952. Cicala, P. , ComjMrlson 0/ Theory with Experiment in t~ Phenomenan of Wi,!:\, Flmw, NACA Technica1 Memorandum 887, 1939. Reissner, E. , and J. E. Stevens, Effect of FiJlite SjMn 011 the Air/ood Dist,ibu!ioIU fo,OscillalUW Wi'Ws, I and 11, NACA Technical Notes ll94 and ll95, 1947. J. B. Rea Company, Tnc., Aerodastlc/ty in Stability WId Conlrol, Chapter 5~ USAF Wright Air Dcvelopmont Center Technical Report 55-173, March 1951, Miles, J. W., "On the Compressibility Correction ror Subsonic Umte.ldy Flow," J. Aero. Sciences, Readers' Forum, Vol. ]7, No.3. March 1950, pp. ]81-]&2. Jones. W. p .. O.clllati"lJ Wi'Ws In Cl1I11pn Mibie Sulmmjc Plow, British A.R.C. Reports and Memoranda 2855,1951. Lehrian, D. E., Ca/culatlon (If Stability De,iMtllll's for Oscillali'W Wi,!:\,J, BritiSh A.R.C. Reports and Memoranda 2922, 19~3 , Merbt, H., and M. T. Landahl. Av O
4-106. Mazelslr.y, B., ''Theoretit::ll A~rodynamic Properties of Vanishing Aspect Ralio HarmonicaUy Oscillating Rigid Airfoils in a Compressible Medlum," J. Aero, Sciences. Va!. 23, No.7, July 1956, pp. 639-652. 4-101. Milne, R. D., 7M Unsteady Aerodynamic Forces on Deforming, Lew_A.pea. Ratio Wi/WI und Slentkr WUW-Body CornbinaliolU, Report No. 94, The Collego of Aeronautics, Cranfield, July 1955. 4-108 •.Landahl, M. T., 7M Flow Around Oscillating LA ....Aspect-Ratio Witrgs or T,anumic Speed, Swedish K.T.H. Aero Techn ical Note 40, 1954. 4-109. Landahl, M. T ., Forces amI Momenl$ an Oscil/atUW Slender Wing_Body Combl. Milam al Sonic Speed, USAF Office of Scientific Research Technical Note 56-109, 1956. 4-110. Landahl, M. T., The Plow Around Oscillating LQ ....lfsp«t_Ratfo Winss WId Wi'W. Body C"mblnOIiDnS at TrotuO"i~ Sped•• Proc. Ninth lntemational Con. gress of Applied Mechanics. Brussels, August 19~6. 4-11 L Gamek, I. E., and S. 1. Rubinow, '"f"Montkal Study of Air Foras on WI Ooei/-, lating (J/" Steady ThIn Wing In a S"I'erWllic Main StrMm, NACA Report 812, 1947. 4-112. Watkins. C. E., effectofA'jUct Ratio on she AI, ForceJond MomtmsofHoNnonlcally 05Cil/Oling Thin R eclangu/or Winss in Supersollic PO/ml!al Flow, NACA Report 1028, 1951. 4-11J. Nelson, H. C., R. A. Rainey, and C. E. Watkins, Lift and Momenl Coefficients Expn",lcd to Ihe Sevellth POwer of Frt'luency fa, Oscil/aling Rectangular Wings /11 S''I'ersOlllc Flow nnd Applied to a Specific FlUller ProbMm, NACA Technical Noto 3016, April 1954.

ISO

1')l:INCII'LES 01' AmWI' !..ASTICITV

4-! !4. Nelson, H. C .• Lift a"'/ M"'''''lIt aJl O,-rll/ol/IW 1f1""Sular a",/ R~/"I~d Wil\\,s willi SlIfH'rsolli~ EtIg~.<, NACA Tec hnica! Note 2494, 195! . 4-115. Cunningham, H. J ., Tala/lifl and Pitching MOlllent 01/ Thill Arrowheod Willgs OscillOling ill Supusollic Pounlial Row, NACA Technical Not~ 3433, 1955. 4--] 16. Cunningham, H. J., Lift ond Mrmoen/ On Thin Arrowll""d Wings wilh Suptrsonic E
AEHOIlYNAMIC O I'EHATOHS

4- 132. 4-133. 4-114.

4-\35.

4-116.

4-137. 4-1 38.

1~ 1

of SI""p·I£.Wed Tmvdl'w G"",., ...ith Al'plima"" Iv I'enc","iou 'if W.uk 81uJl W",'n. NACA Tc"hnica l Note 3956. May 1957. Ludlolf, H. F .• On AUOIlynamirJ of Blasts, Vol. HI. Advancu in Applied New Yorl<, 1953. pp. 109- 144. MUNm/cs, Academ ic KOn ner. H. G .• Zusomm .'!fa.utUkr /k, ich, «ber dm IM/aIiMii,." Auf/rid """ R Ugeln. Lufif.hrlforschung, .ad. 13, NT. 12. December 1936. p p. 410-424. Fraey. de Veubeke, Aerodynamlqw. IlIJtut;o"",,;,. rks profils mince. diformabl•• , Bulletin du Service Techn ique de I'Aeronautiquc. Brussels, .Belgium, No. 25, 195J. T lIllItlan, R., La tMorie Ms projils mlncu en tcoulement non &/oli
Pre".

192~.

4-139. Garrick, 1. E., " On Some Fourier Transform. in the Theory of Nonstationary Flows," Prouedings of /he Fifth In/unational Congress for Appli~d M~clwniCJ, John Wiley and Son., New Yor .... 1939. pp. 590-593. (Out of print.) 4-140. Sean. W , R., "Operational Method. in the Theory of Airfoils in Nonuniform Motion," J. Fronklin Imtllme, Vol. 230, 1940, pp. 95-1 11 . 4-141. Hobbs, N. P., "The Transient Downwa sh Resulti ng froro the Encounter of on Airfoil with a Moving Gust Field,"' J. A ero. ScienuJ, Vol. 24, No. 10. October 1957, pp. 731-740, 754. 4-1 42 . Miles, J. W .• "The Aerodynamic Force On an Airfoil in a Moving G ust,"' J. Aero. Sclt nus, Vol. 23, No. ll , November 1956, pp. ICl44-lOW. 4-143. Mazelsky, B. , Numerical lktermUtotion of Indicial Lift of 0 Two-Dimem{ono/ Sinking Alrfoo"! 01 S ubsoniC M ad, NUmMr$from O jcillutory Lift Cot!Jficients wilh Co/cu/ati
151

PRINCU'LES 0 11 AERQELASTICI1'Y

4-149. Biot, M. A., Louds "",, Super5,,"lc Wille Striking a Sharp-etiged Gust, Cornell Aeronautical Laboratory Report SA-247-S-7, 1'148. 4-150 . Ashley, H., J. Dugundji, and D. O. Neil.on, "Two Methods for Predicting Air Loads on a Wing in Accelerated Motion," J. Auo. Scknct's, Vol. 19, No.8, August 1952, pp. 543-~~2. 4-151 . Krasilshchikova, E_ A., Unsuady MOlio" of a Profile in a Compressible Fluid, Doklady, A. N. USSR, Vol. 94. No.), 1954, pp. 397--400. 4- 152_ KOl$iishchikova, E. A" Unsu(1dy Mat/a"af(1 P;"lu-Spa" Wing I" R Comprtssible Medium, Ook/ady, A_ N . USSR, VoL 97, No.~, Scptember-October 1957_ 4_153. Jon~ R. T., The Unstw dy Lifl a.f a Wing of Finile Aspect Ral/a, NACA Report 681,1940. 4-154. Jones, W. P., AtrO, New York, January 1962.

5 STRUCTURAL OPERATORS

5-1

INTRODUcnON

Since the early 1930's, when aluminum aUoys were first used, strong and rigid full-cantilever lifting surfaces have undergone substantial changes

in their geometry and method of constl"uction. Whereas a lifting surface

invariably had the geI1erai appearance of a sie!lder beam for many years, more recent trends in very high-speed military flight vehicles have been toward low-aspect-ratio plate-like surfaces. On the other hand, civilian jet transport aircraft retain exceptionally slender sweptback wing beams. We can observe the nature of some of these trends by studying the curves shown by Figs. 5--1 and 5-2. Figure 5--.1 illustrates the trend of wing thickness ratio in fighter and transport aircraft since 1920. T he emphasis on drag reduction is evidenced by a continuous lowering of thickness ratio from about 0.2 in 1930 to something on the order of 0,05 in 1959. Accompanying the changes in thickness ratio, we find interesting changes in aspect ratio. In Fig. 5-2, the trend of wing structural aspe<:t ratio is plotted, being defined as the ratio of the square of the structural span to the wing area. The teon structural span denotes the total span of the wing measured along the beam axis, as illustrated by the inset of Fig. 5-2. In Fig. 5-2 we observe th at lifting surfaces have had a continual upward trend in structural aspect ratio in the case of tra nsport aircraft, with the swept wing jet transports exhibiting higher values than ever before. In the case of fighter aircraft, however, we find that lirting surfaces tended to a higher degree of slenderness in the chordwisc bending direction unti l th e advent of supersonic fighters, which

'"

1501

I'NI NC II'U!S 0 11 AliNOlll.AfI"n CITY

1910

1920

19X1

1940

1950

'" '

I'la. 5-I. Wing thickneos ratio.

I'

"@:j..l0 .~

,

I> , g

• •• , ••

1900

1910

1920

1930

I'MII

'u,

1950

1960

Fir. 5-2. Win!: structurat aspect ratio.

STRUCl'URAI. OI'lmATORS

I ~~

ma rked a reversal of the trend , This reve rsal evidently res ulted from both aeroclastie and ae rodynamic reasons, It seems evident from these tre nds that ae roclasticians must deal no t only with one-dimensional structures or slender beams, but also with twodimensional structures or lini ng surfaces, Whereas there has heretofore been comparati vely lillie interest in the theory of bending and stretching of shells, this theory must now be employed as a basis for predicting the behavior of thin lifting surfaces . In the case of missiles and rock.ets, lifting surfaces are freq ue ntly omitted entirely, stability and CQntrol being provided by gi mbalcd engines and obliquely mounted jet nozzles. In t hese cases, the aeroelastician may be required to consider the vehicle as a lifting body or as a three-dimensional structure. Thus modern applications of aeroerasticity can run the gamut from one- to two- to three-d imensional structures. The effects of aerodynamic heating on structu ral behavior, he retofore neglected in aeroclastic analyses, now also assume importance. There nrc two principal effects. The first derives from the deterioration of the mechanical properties of materials at elevated temper
5-2 ONE-DIMENSION AL STRUcrURES

We take up first, very briefly, the one-dimensional structure- that is, a structure whose state of deformatio n can be adequately descri bed hy a set of functions of a single space coordinate.

156

l' IUNCII'U,:.s 01' .U ;MOELAS'I'ICITV

(a) Slender

1lJ1.~wcpt

wing

leI us consider a type of slender beam in which we can make the simplifying assumptions nonnally employed in engineering beam theory. First, the beam is penniued complete freedom to warp when torque loads are applied. This leads to the St. Venan! solution of the torsion problem and permits the simplification tbat bending and twisting are separate uncoupled actio ns. Second, we assume that plane sections remain plane during bending. This allows application of the well-known engineering bending theory. The beam may taper in the spanwise or y-direction and the properties or the cross section may vary with y. The reference axes are illustrated by Fig. 5-3. It is convenient in the present discussion to assume that the y-axis is placed so that it pierces each cross section at the centroid of the effective normal stress-carrying area and that the:l:- and z-axes are oriented so that the x.y- and y-z-planes pass through the principal bending axes. Lateral beam deflections. The lateral deflection of a beam, th at is, the deftection in the :-direction, according to the engineering theory, is made up of a bending deflection, W HI and a shearing deflection, WS' The former can be represented by (5-1) where Mj).) is the bending moment distribution about the x-axis and EI is the bending stiffness (Ref. 5--4). The positive directions of M~ and )II are as illustrated by Fig. 5-3. Differentiation of Eq. 5-1 twice with respect to y produces the following wen-known relation between beam bending moment and curvature: .f!wu = M"

dif

El

(5- 2)

The shearing deflection, ws. is given by Ws=

V(A) di.. • GK

l

¥

(5-3)

where V(A) is the shear distribution along the beam positive upward, and OK is the shearing rigidity (Ref. 5--4). Differentiating Eq. 5-3 with respect to y gives a relation between the shear and the first derivative or the shearing deflection, as follows:

d)lis dy

_

....!:::. GK

(5--4)

STItUcrUIt.... L OI·I(It .... TOItS

1!!7

,

,

---~---7

, ,, ,

/

W

t ,/

/

, ,,

v

Fig. 5-3. Axis system for .lender beam.

Making use of the fact that (5-5) where Z is the intensity of lateral loading. we can write the following relation for the lateral deflection of a slender beam: (5-6)

where fI'B is a structural operator for bending displacement defined by

g'._~(EI~) dy'

dy!

and fl's is a structural operator for shearing displacement given by fl's -=

~(GK~) dy

dy

The inverse structural operators for slender beams are obtained by integration of Eq. 5-6. We have already seen the form of the operator fI'B-1 in connection with Eqs. 3-23 through 3~30.

I!!M

I'RI NCII'I..I~

0 1' ,H ;ROEI.ASTI Cll'V

The Iw iSI in g dc nection of 11 bea m, accord ing to St. Venant SOIUlioll of the torsion problem, can be represe nted by TorJiUl1fi1 bealll r11j/eelioll.f.

6(y) =

f M.(d1

dJ.

(5-7)

where M .(J.) is the distribution of applied twisting moment about the y axis, positive as ill ustrated by Fig. 5--3, and GJ is the torsional stiffness (Ref. 5-4). We can derive by differentiation of Eq. 5-7 the fonn of m.

= .9'(6)

(>-1\)

where m. = dM./dy is the applied torsional moment per unit length and .9' is a structural operator defined by

The inverse structural operator for twisting can be expressed as

(5-9) where

S"'-'( ) = and where

f l·co.

C(Y,1/)(

l C( Y,1/=oG}' )

(. d'

C(Y,1/) =

Jo GJ'

) d1/

('I ;;:-: y)

(y 21/)

(b) Slender swept wings

In aeroeiastic analyses of slender swept wings, it is freq uently necessary

to think in terms of linear and angUlar displacements of streamwise segments of the wing, that is, segments which are parallel to the airstream. We refer to Fig. 5-4, which illustrates a slender swept wing of large aspa:t ratio together with the ax is system and other notation which will be employed. The deformation of a swept wing can be described in terms of linear and angular displacements of stIeamwisc segments due to the running loads and torques on these segments. In the case of slender swept wings of large aspect ratio, an effective root can be assumed as shown by Fig.~. Thus an elastic axis can be postulated; and the y reference axis can be loca ted along the elastic axis. If y is a true elastic axis, we can employ slender beam theory to compute the wing deflections. We may,

STRUCTUR ... L 0 l' EIIATOR5

159

Ellteli.., r~

Act",,1 root

Fig. 5-4. Sknder swept wing.

for example. visualize four distinct types of structural operators for a slender swept wing, which are expressible in the foHowing inverted forms;

w = f/'•• -l(Z)

(5-10)

8 .,. [/'" - \m.)

(5- 11)

o= [/'.,-tcZ)

(5-12)

w .. :;I'" - '(m. )

(5-13)

I n Eq. 5-10 we have an inverse structural operator which transforms forces, Z, on streamwise segments into vertical displacements, w, of segments at the y axis.

..

[f' -'(

(5-14)

where C"(y, 1]) =

,1- ' (cos\ G

1

C" (

Y,1]

)

~

EI

1

+

11J«>sA

['rotlA

d~ GX '

0

s.

";'O&A

o

(1J ;;::: Y)

(-'L _ ,) cos A

0

+

,)(-' - ,) dX cosA

EJ

(2- _,) cos A

d.l

dX

-GK '

and where EI and GK are slilIness properties of the beam on sections normal to the elastic axis.

160

"II:INCII'I ,I'S 0 1' A"II:OBI.AS'I'ICn'Y

The illverse 51ruetuml operator .9'~I- I( ) transforms pitching n'lOmcnts, III., on strcamwisc segments into pitching displacements, 0, of the segments. Y

where

..

-'(

) =

f:

(5-15)

cH(y, 1/)(

f"~ (cosGJ'A + Sill.EI'A) dX' .EI'A)dX ' C"( ) = " ~ ( cos'A + sm y,1/ ~ GJ f C"(

y,'1

)=

(1J :2 y)

0

The inverse structural operator .9'In-'( ) transforms forces, Z, on streamwise segments into pitChing displacements, 0, of the segments, ) =

f

(5-16)

c"(Y,1/X

where

c"(y, '1) = -sin A

f

~'_A

0

(-" -1) dX cos A EI

'

(-" -1) dl. rosA EI Finally, the inverse operatorS":o - I( ) transforms pitching moments on streamwise segments into vertical displacements, w, of segments at the ii'·a~is.

)=

L'

C" (y, '1)(

(5-17)

wbere we obtain the influence function C'~(y, TJl from C'''(y, '1), using the reciprocal relation, in the following way: (5- \ 8)

Elastic axis, In Parts (a) and (b) above, we have outlined the structural operators for computing bending and twisting displacements of beams with Straight elastic axes, In order to employ t hese operators in II given example it is necessary to know the location of the elastic axis, The elastic axis is located by drawing a spanwise line through the shear centers of the various cross sections of the beam. The shear center of each cross section is computed by establishing the point in the plane of the section al which a shear force can be applied to the section without producing a rate or twist

S1'ItUC]'U ItAI. O I'EItATOItS

161

at the section. Met hods for compu ting the shear cen ter location of multicell beams may be found, for example, in Ref. 5--4. Tests of full-scale wings and of models arc frequently employed as means for dete rmining elastic axis locations. 5-2 TWO-DIMENSIONAL STRUcrURES

As a basis for theoretical studies of lifting surfaces, we shall employ the theory of the bend iog and stretching of shell surfaces of variable thickness subjected to loads and temperature gradients. In some instances, the entire lifting surface may be regarded as a normally loaded shell surface. In other instances of built.up lifting surfaces or fuselage and missile bodies, the structure may be regarded as an assemblage of she!! elements. In either case. II knowledge of shell theory is an important prerequisite to an understanding of structural behavior. (a) Shell geometry

Let us confine our attention to thin shells in which the thickness h is small compared to the radius of cur vat ure and other dimensions. We define a reference surface as the middle surface of the shell. Shell geometry is defined by specifying the form of the middle surface and the thickness at each poi nt. Referring to Fig. 5-5, we may slate that any surface may be defined by a

h -Ct

"-?V'h= CI

,

J

, FIe. 5-5. Curvilinear coordinates on a surface.

posit ion vIX:tor r relative to a tixed origin "0." The vector is:! func tion of two independent parameters. ;, and ~2' as follows: (5- 19) The Cartesian representation orthe surface may be stated as (5-20)

and if we eliminate ofl and

f~,

we obtain the surface equation

F(:/:. Y. z) ... ()

(5-21 )

When a relation between [ 1 and~! is derived, !>ay,g(!" ~2) = 0, we obtain a eurve on the surface known as a pa rametric curve. A surface can be defined by a doubly infinite set of such parametric curves. The parameters ~l and t~ constitute a system of curvilinear coordinates, as illustrated by Fig. 5-5; and the position of any point on the surface may be defined by the values of tl and ! , at that point. In the vast majority of practical a pplications of shell theory, it is assumed that t he shell surface is described by a cu rvilinear coordinate system which lies along lines of curvat ure of the surface (Ref. 5-2), Lines of curvature of a surface are oriented a long the directions of principal cu rvature. These are the directions along which the normal curvatures to the surface exhibit principal or maximum and minimum values. Directions of principal curva ture may be shown 10 he orthogonal directions (Ref. 5-3). Referring to Fig. 5- 6, we form a right hand coordinate system on the undeformed midd le su rface such that the unit vectors corresponding to the curvilinear coordinates ;1 and f! and in the direction of the nor mal to the surface are represented by II' 12, and n, respectively. The radii of curvature in the directions of tl and t1 are represented by R, and R2 , respectively: and these radii are taken positive when the ce nters of curvature lie in the positive direction of n. By definition, the unit vectors t(

a,

1 _, t( __

(i=1,2)

(5-22)

rt., a~, where

are the so called "first fundamenta l magnitudes" of t he su rface. * There • According to ditfer<:ntial geometey, lin cleme nt of deflned by

~n:

d. on the

shell surface is

STK ucr UKAL OI'''KATOKS

/~/ft~

163

__

---;Middle surface

R,

Before deform.ii....

Middie surface

, ",",

deformation

,

;

Fig. 5-6. Middle surface geometry.

arc certain relations among the quantities defined above which are specified by the geometry of the middle surface. There are, first, t he matrix. relations

, 'I,

0

t, t,

~

'I,

~ n

~

R,

t,

0

I,

0

0

n

0

-1 -OIXz IX I eel

0

I,

I '., --

0

- .!.!. R,

I,

.,

0

0Cz a,~

n

,

-1 -oal

_ .!. ik t a,o;3

a. I O'1

0

"'

R,

., R, 0

~ n

(5-23a)

(5- 23b)

I~

I'RINCII'I.!!S 0 1' AI·:.ml!LA S'Tl CITV

!lnd Ihe conditions of Codazzi and Gauss represented, rcspecti vcly, by (5- 240)

(5-24b) (b) Stress resultants and couples Following the usual procedures of sbell tbeory, we formulate the equations in terms of stress resultants and couples. The stress resultants and couples pertaining to shell th eory may be defined with reference to Fig. 5-7, which illustrates an clement of a shell of length d~t, width del' and thickness h. The orthogonal curvilinear coordinates !l and ~I arc on the midplane reference surface an d lie along the directions of principal curvature. The coordinate axis Cis perpendic ular to the reference surface. The edges of the element are acted upon by the normal and shear stresses <1n> a;., T 11 , T'I' T ,V and TIt' as illustrated by Fig. 5-7. Assuming that the thickness II is small-compared to the principal radii of curvature and tha t the reference surface is a middle surface, we may state that a macroscopic description of the statics of the shell is provided by the three tangential stress resultants in the reference surface, · NIl

N.

_f~'2

=

NIl =

- A"

J-M''" f""

<111

d{

(5-25)

<1211

dC

(5- 26)

Tn d{,-kll N n

and the three stress couples

au' d,

¥ll =JUI1 -A,2

-f""

TIl dC

(5- 27)

- lfl

(5- 28) (5-29) ( 5- 30)

Since T lJ = "'I!' it is evident that Nil = NIl and Alu = Mw a nd that Eqs. 5-25 through 5- 30 represent. therefore, six independent quantities. • In writing th= e<Juation. we limit ournclves 10 Love's Rrst approximation by

seu in g]

+ URi '"

I.

STIt UcrUItAI. OI'I·:ItATO It!;

Fla:. 5-7.

165

Sliff.ne
The positive di rections of the stresses are taken according to the usual convention of theory of elasticity (Ref. 5-5); and the positive directions of the stress resultants and cou ples are, therefore, as illuslrated by Fig. 5-7. (c)

~train.di$placement

relations

We assume at the outset that not only is the shell thin, tha t is, the thickness is small in comparison to the radii of curvature and the principal dimensio ns, but also that the transverse shear and normal strains are zero. Our concern, then, is with two normal strains and one shear strain in la minates parallel to the midplane rererence mrface. The normal strains in the f, and f. dire<:tions in lami nates parallel to the rererence surface are given by the ve<:lor fonn (Ref. 5-3) ( n

"'"

1 (ar au t all ou) r1.~~ ae. · atK + '2 aen . at. '

(n =I,2)

(5- 31)

where u is the displacement vector of a point, defined by u = lI,t ,

+ 112'2+ wn

(5-32)

and where 111' Il:!, and ware components of the displacement vector. The radius vector r in Eq. 5-3 1 is now a ve<:tor of position to a point in a laminate which is parallel to but not necessarily in the reference surface.

I 'H I NCII' U~

166

011 M :HO£LA!i.- ' ·ICITV

The shear strain in lamimltes fXlra ll ci to the reference surfllce is given by

(Ref. 5-3)

, (Or au au ar au au)

Y'2

=

(5- 33)

<Xl~ ai, . a~2 + a~l . ai. + a~l . aE.

Equations 5-31 and 5-33 apply in the case of large displacements and they may be specialized to small displacements by neglecting the final terms On the right-hand side of each equation. In reducing these equations to scalar form , we adopt the Euler-Bernoulli·Navier hypothesis that the normal to the undeformed reference surface is rotated without extension into the normal to the deformed refercnce surface. After substituting Eqs. 5- 22 and 5-32 into Eqs. 5-3 1 a nd 5-33 and making use of Eqs. 5-23 and 5--24, together with the Euler-Bernoulli-Navier hypothesis, there may be obtained the following results for the strains in laminates parallel to the reference surface: (5- 34) "'ft = i " - 'JC~, (n=1,2) (5- 35) Yu = rll - 2,l
+ Hint + lu! + /n~ 121 + l(lu' + i02' + 132~ (II. + '~I) + (lllIJ~ + 1211.. + /3,/3,) 2 alru + _,_ G"', 133

il = III

i3 =

r.. = Kl

=

<XI a~,

"'. =

Cl,"'. a~2

.!. al w + _1_ 07". In

<x.

a~.

OXICl.

'. , _ .!. alsi _ _,_ '

where

III

-

"'I "~1 "

121.=.!.a v _~~

a[l "'lCl! aE. 131 = ~ +.!.aw "'I

OX I a[,

(5-37)

GE,

'" I

"'l~ G~,

=..! ou + --..£... 0(1,_ ~ "1 aEl OXl,,> 0~2 Rl

RI

(5--36)

+.!. olal "" "~2

""

I n

_ _ 1_oCl, I ,,31 "I"> "~g

=..!.~_..!...o~ ~ a E~ ""~ O~l lop uib: 2

/22=- ~

ae.

OXl"'l af)

In =~+Jow R.

w

+----

Cl 2 aE~

R~

STR UCI' URAL OI'ERATOIIS

167

In Eqs. 5-36, we have retained the non linear terms in the reference surface strains; but we have retained only linear terms in the formulation of the curvatures in Eqs. 5- 37. This combination of te rms is especially usefuL in applications to aeroelastic problems of shells where small vibrations are executed about pOSitions of eqUilibrium in which the static membrane strains may be large. (d) li:quations of state of an elastic shell element

Three sets of natural variables describe the state of an element of an elastic body : stress, strain, and temperature. Equations expressing a relation among these three are called equations of state. The equations of stale of a shell element are those equations which relate the stress resultants and co.uples wit h the strains in the plane of the reference surface, the curvatures, and the temperature. Let us assume at the outset that the present discussion is concerned with orthogonally aeolotropic elastic shell elements. [n this case, we may represent th e state equations by the following matrix forms: {NnN2~} = [KJ{itil

} -

{T1T2}

{MnMu} = -[Dj{J< 1J<2} -

{1\T1}

(5-38)

{N12M12} = [Ej{YUKlt} where

[K]_

' [KUK21 Kj K [£]~[K" o

0]

- D~

I n these equations [K j, [D j, and [El represent symmetrical arrays of influence coefficients for isothennal straining. Each coefficient may vary with its location on the surface of the she[1. Tand Tare functions of the tempera ture distribution, The T reflect the temperature 2 , O. . distridistribution over t he reference surface and the T the temperature bution throughou t the thickness. T(e b ez• 0 is an increment of temperature above a reference absolute temperature To for a state of zero stress and strain. The total absolute temperature, denoted by T .. is the sum of To and T. The elastic coefficients in Eqs. 5-38 may take on a variety of forms ifthe

ne,. e

-

-

168

I'RINCII'LES OJ!

AllROELA~" I C I'rv

shell clement were ort hogonally stiffened along the directions of the lines of.principnl curvature. We shall not attempt to list these forms here, but will merely state explicit values for the coefficients when the shell is elastic, homogeneous, and isotropic. They are Ku "" K,

Ku = Ky,

Dn = D,

DJ'}, = Dv,

K~

= Gh,

(5- 39)

Do. '"" D(l - v)

wbere K = Ehf(l - vll) and D = E!i3/12( 1 - vB). terms are

The thermoelastic

(5-40)

In Eqs. 5-39 and 5-40, E, G, v, and a' are Young's modulus, the modulus of rigidity, Poisson's ratio, and the coefficient of thermal expansion, respectively. (e) Equations of static equilibrium and boundary conditions of shell theory In Sees. 5-2(c) and Cd) above, we have reco rded the strain.displacement relations and the stress-strain relations of sheil theory. The mathematical statement of our problem is completed when we add the equilibrium equations and the boundary conditions. The equilibrium equations and boundary conditions may be for med by the direct summation of forces with refe rence to a sketch of an element of the shell or, a lternatively, by means of tbe principle of minimum potentiaL energy. We shall make use of the latter approach. The principle of minimum potential energy is a special case of Hamil ton's principle applied to a statics problem. We assume that the shell is loaded statically by forces Xl> X~, and X, per unit of shell area and in tbe directions of the axes fl' ~1l' and ~, respectively. It is subjected to a temperature distribution T (~l' g, variable over the surface and throughout the thick ness. Furthermore, the edge boundary conditions are divided into two categories. Over cc::rtain edges, designa ted by Cl • the edge forces and momenl s per unit length are prescribed; and over certain other edges, designated by Cz. tbe edge geometrical CODstraints are prescribed.

'z.

STKUcrUKAL O I'ffKATOKS

169

The va riation of the potential energy of the shell is expressible as (cf. Eq. 2-46) 6(U - W)

=

II(N .5i + N~.5i1 + Nn"YlI- Mn o'iKll , u 1

MjH;o'iK2;2. - 2M I1 "'''IJ

X":l
-II, (Xl /}U + X I t1(] + X,o'iW)iX1
FiC.

s-a.

Force boundary conditions on shell

edge.

170

I' IUN CII'IJ';S OF AI!ItOI'.LASTlCITY

~'ppnrclll,

then, thHtJor a thin shell- Ih:,t is, a shell in which 'fRI and 'f R2 may be neglected in comparison with one---we have

Sf.. =

f~ / I{ d~ - '/E

g . = DhI + MI.. m

(5-42)

M. =Mz.I-M1, m where / and m are direction cosines of the no rmal, ~, with reference to the local unit vectors t,. and t l . The results obtained from Eg. 5-4 l are dependent on the nature of the strain.displacement relations which are introduced. We shall adopt the strain·displacement relations of Eqs. 5-36 and 5-31. This leads to the following set of nonlinear thin shell equations (5-43) and boundary conditions (5-44) which we shall designate as Wasruzu's equations:

"

oe

{exa[ Nll(I

"

+ In) + N is /n ]} + oe

l

{ocl[NazI12

+ [Nu ll! + N a(1 + 1:rJ] 0011 0$2

_ (N u 131

[ N2!(1

+ /2'.1) + N1.2 11l] Oa.. 0$1

+ N Is 131) 01101 2 R,

00l2

1["( M) M +"(M\ - RI Oel exa n 22 0$1 0$1 011 121

,

+ Nn(I + In)]}

2

+ XI0I10lz =

0

0$2 { I[N2!(1 + In> + Nul' l]} OI

, +

0$1 {exa[Nlll~1

+ [NZi\I12 + N1il + Ill)] OOIS 0$1

[Nn(l

+

t MJ + OGt 0$. 13 + N u (1 +

Ill)

In)]}

+ NIsIlJ) 0011 oEI

(5-43a,b)

STIWcrUKA I, OI'J, KATOKS

a {<X:J[N u I + Nlzl~} + -a {«I[N2~1311 + N 0';:1 ofz -

31

+ [Nn(l +

In>

+ Nil/a] OCl~ + R,

[N!f.(1

I2131]}

+

i311 )

+ NulzJ OCIOC.

+ ~ (JJ a (oc2M ti) + ~ (ocIMn) + Mu OOCI _ 0';:1 OCI La!1

0';:1

J71

0';:2

~

Mrz VoctJ) 0';:1

The equations of linear shell theory arc obtainable from Eqs. 5-43 and 5-44 by putting III = 1'2 = I'll = 1~2 = in = I;». = o. (0 Formulation of the shell problem

The formulation of the shell problem, within the framework of the assumptions we have been making, is established by the equations stated above. In particular, we have six strain-displacement relations (5--36) and (5-37), six equations of state (5-38), and three equilibrium equations (5-43), making a total of fifteen equations. The unknown quantities which are defined by these equations are also fifteen in number, comprising six stress resultants and couples (Nu • N'l.1' N, •• Mil> M'I!/.. and M I !), six reference surface strains ( il> "2' Yu. "'I' "'2' and I(u,), and three displacements (u. 1.>, and w). The problem of static shell theory is one of solving these equations, subject to the force boundary conditions on the edge C" given by Eqs.

112

I'RI NCII>L1~

0 1'

AERO£LA!>~I'I CI1'V

5-44, nnd Ihe displacement Dol/mlnry conditions on the edge latter boundary conditions may be slaled simply by II =

u,

v ~ ii,

W=

w,

ow ow

C~ .

The

(5-45)

where the bar denotes a specified quantity, and differentiation with respect to "I' denotes differentiation in the direction of tbe normal drawn outward on the edge in the middle snrface (cf. Fig. 5-8). Subsequent paragraphs of the present section are devoted to the appli. cations of shell theory to structural configurations of interest in aero-elasticity, and to the derivation of structural operators which derive from shell theory. (g) Stiffened shallow shell tbeory

In the previous sections we have o utlined a ge neral theory applicable to shells of arbitrary curvature with cu rvilinear coordinates arranged in the directions of lines of curvature. Very frequently in aeroelasticity we are confronted with applications involving shells of small curvature. In these applications we may employ the shallow· shell theory of Marguerre (Ref. 5--7), a special case of the theory described above with a simpler and more tractable mathematical form. We refer to Fig. 5--9, which illustrates a segment of a shallow shell oriented with respect to the rectangular co-ordinates (x, y, z). The equation for the reference surface of the shell is taken as z = z(x , y) (5-46)

z

,

}----- y

FIC.~.

Segment of a shaUow mell.

STRUC'I'URAL OI'''RATOHS

173

A sha llow shell is quali tative ly one for which the slope of the reference surface is small at all points. A rul e for the latitude of interpretation of the wo rd "small" cannot be stated precisely; however, Reissner (Ref. 5-8) has suggested that shallow shell theory will be more than ,accurate enough as long as t he slope is less than i and often accurate enough for practical purposes up to t. In the case of shallow shel1 theory we may employ a rectangu lar coordinate system instead of the system of curviJjoear coordinates of the more general theory. We put .!'l = '1'" .!'~ = y, and ~ = ~ = I so that an element of arc on the reference surface is given by (5-47) and a radius vector to the reference surface is defined by r = :ci

+ yj + z(z, y)k

(5--48)

Quantitatively, we may say that a shallow shell is one for which

(a,)'

1 + iJz

"':i

l +

I,

(aiJy,)'

"""l,

The unit tangent and normal vectors to th e deformed surface are approximated by t~ = i

a, + aW)k + (oz oz

. (ikiJ!I awl k iJy

t.= J+ - +.~

(5-SO)

oz +ow) j+k _("ih: +aw)._ ( oy iJ'I' oy

Shallow shell theory may be formulated in precisely the same way as the more general shell thcory of the previous paragraphS by altering the nature ofthe assumed strain-displacement rel ations. In shallow shell theory these arc assumed to be (Ref. S-7) ( = iJlI

+ oz all' + .!(aw)'

( _ QV

+ oz all' +

~

iJz

• - iJy jiQ= ou

ay

.--"w c

ox"

ozoz oy iJy

+

20z

!('w)' 2 iJy

ov + in ow az az ay

(S- S1a)

+ dz ay

all'

a",

+ iJw all' az ay

. -a'-w " - a",oy

(S- Slb)

174

I'IUN CII'I.I;$ 01' AI;JIOEI.ASTICITY

An csscntinl feature of lhe reduction of the strain-displacement relations to the relatively simple forms stated above is the assumption that bending displacements arc significantly larger than stretching displaccments (Ref. 5-7). The stress-strain relations of an ort hogonall y stiffened shallow shell are identical to those of Eqs. 5-38. We shall formulate the equilibrium equations and boundary conditions by means of the principle of minimum potential energy as we did for the more general shell by Eq. 5-41. For a rectangular sballow shell element with dimensions 2a and 2h, and loaded by a traction of intensity Z (cc. Fig. 5-9), we have b(U - W)

~fa

-a

( . (N"",Oi.,+N... U.+N~6(.. -Mn OI<~

L~

" f -S' I -

f:. f.

Zowdxdy

Rn-clu

aw aw ' d. + N... ov+ f7.05w-ii?o'i--!i?.zOay ax - .

Nuo'iu

+ fI.~{j1J + f7

-a

- 6

2M~.fJl<:n)dxdy -

- M.. cll<. -

z05w -

tlu oow ax

~1n/jWI· ay-_dy == 0 (5-52)

Employing the strain-displacement relations of Eqs. 5-51, we obtain from the variational condition (5-52) the differential equations of equilibrium aN u ih:

+ aNzl" =0

aN~.

+ oNn =

ox

oy

oy

(5-53a)

0

iJ2Mz~ + 202Mzv + iJ2M.." +

or

.! [ N .£. (~+ w) + N n .£. (z + w)] ax ay ay1 ox u ax oy + i[N oy •• .£. oy (z + w) + N z • .£. ox (z + w)] + Z = 0 (5- 53b)

and the following foIX:e boundary conditions: (a) On edges x = ±a:

fin = N"", Vz

+ at? ----2'! = oy

N~"

a

-(z ax

Nla = N ... ,

fil"" = M,.,.

a oM aM + w) + N z• - (z + w) + -----2 + 2----E: ay

ox

iJy

STKUCI'UKA1, OI'EHATORS

175

(b) On edges 11 = ±h:

fi.. = N q Ii': •

+ oM.~ = N -.! (H ~· Ox

ax

= Nn ,

fl••

,

w)

M ••

= Mn

+ N ..?. (, + w) + aM"" nay

oy

+

2 OM~.

ax

(c) At the corners

When a corner is free and unrestrained, so that dw is arbitrary, the quantities M .. must be considered in formulating the boundary conditions. Equatiorrs $-53a and 5-53b may be reduced. to a single equilibrium equation by introducing a stress functio n F defined as follows:

,'F

N

__

"

N

oy"

... =

iJ2F a",~

,

N

"F 2, =--axoy

(5-54)

By means of Eqs. 5--54, we ensure the satisfaction of Eqs. 5-53a; aud then Eq. 5-53b becomes

O!Mu O:l't

+ 2 iJ2Mn + iJ2M... + Z = ax oy

+ 2 cPF

(jy'

_ iJ2F iJ2w _ aSP. O"w

oZ= ayt

iJ2w _ OJF fPz _ rJp. rPz

ox ay 0:1' oy or oy

0..1 ax'

all

+2

o",t

(fJF . ~ (5- 55)

ox oy

0:1:

ay

The three equations (5-5Ia), describing the relations between strains and displacements in the plane of the reference surface, can be combined into a single compatibility equation

",w)' ( ox oy

if'w if'w

ex' a1l (j2 ifw

"'",w

a;r;2

ayl

ih

a"w

- - z -+2- - -

all ax"

ax ay ax ay

(5- 56)

Equatio ns 5--55 and 5-56 thus represent in two equations all of the equilibrium equations and strain-displacement relations of shallow shell theory el[cept $-51b. When Eqs. 5-55, 5-51b, and 5-56 are supplemented by the equations of state of anelementoftheshell, we obtain a complete set. F or example, if we employ Eqs. 5-38, we obtain a mathematical statement of the problem of orthogonally aeolotropic elastic shallow shells with lateral loadi ng and with arbitrary temperature variation. We proceed by

176

1')l:INCU'L1!S 011 Almom.ASTICITY

introducing Eqs. 5- 54, 5- 5 la, a nd 5- S fb into Eqs. 5- 38 and reducing them to the forms (5- 51)

(a'. "w)

~T ~) (M "" Mn ) - - [ Doz~ ) -00/ - - (T ~ •

o D"

(5- 58)

1

o

"w "F) (ozoyoxoy

(5-59)

When Eqs. 5-57, 5-58, and 5-59 are substituted into Eqs. 5-55 and 5-56, we obtain coupled equilibrium and compatibility differential equations in terms of the dependent variables IV and F. The eqUilibrium equation is

(5-60)

and the compatibility equation is

" ("F F) + oz"( 1 "F) or K.. ow + Kz • a'oy! oy Ku az oy a' (a'F " (K.S~ + Kn T.) + 011 Kn oxa + K~~ "') at! - - or

f

" ( .'w 02W 02W aaz (Pw - ay2 (It.c~T~ + K~.T.) + oxoy - or oy'l - ail a1l f'J7. 02w

07.

ijIlw

--'-+2 -- ' - -

oy2 or

ax ay oz ay

(5-61)

where Ko are elements of the {KJI matri ){. Equations 5-60 and 5-61 constitute two nonlinear partial differential equations in the unknown quantities wand F which apply to elastic orthogonally aeolotropic shallow shells with lateral loading and with temperature gradients over the surface and through the thickness. Because

S" 'IlUc'''l'UIlAL. Ol' llRATQRS

177

of their similarity to the von Kurm{tn equations for the bending and stretching of fla t p lates, there arc numerous known tcchniques of solution, and a comparatively simple approach is thus provided for the anaLysis of shells which can be regarded as shallow, A linear shallow shell theory is obtained by negle~ting the squared and product terms involving IV on the right-hand side of Eq, 5-61, (b) Stiffened nat plate theory

Membrane and fiat plate theories with and without Icmperature gradients can be derived as special cases of Eqs, 5--60 and 5--61, A theory for the bending and stretching of stiffened fiat plates of variable thickness is of great interest in aeroelasticity since it is fundam::ntal to the treatment of lifting surfaces, By placing the quantity z equal to a constant in Eqs, 5--60 and 5--61, we obtain the equilibrium equation

(D i¥w D i¥w) + 4 02 (D O'W) + 0' (D 02 D azw) oZJ """ aZJ + '"" or o~ay ~ uzoy or/' oi! + ay1 = z _ O"T" _ oar. + atF utII' _ 2 a"F i¥w + i¥F iT'll' (5--62) a~z ay~ u:r:" or u~ oy a~ iJy iJy' a~' 0'

-

11'

U

n

~

and the compatibility equation

(5-63)

The simultaneous Eqs, 5--62 and 5-63 can be simplified, if it is assumed that the lateral deflections are small. In this case, the bending and stretching actions are uncoupled. This uncoupEng is accomplished mathematically by neglecting the last two tenns in II' on the right-hand side of Eq. 5--63, Thus, fo r a given temperature distribution, Eg. 5-63, together with the appropriate boundary conditions, can be solved for the stress function, F, By means of Eqs. 5- 54, the stress function can be transformed into expLicit initial stress resultants, N~), N~l, and N~). Under these circumstances we may rewrite Eq. 5--62 in the form .9"(11') = Z _

o~T~ _ (jilT,

ao?

all'

(5-64)

where f/' is a self-adjoint lil1car structural opcnltor deli ned by

"(" ") + 4 o:t""( o.o? Dn 0:t"1 + Dx. oy! oy D~ OZ") oy

f/' =

+ ayll "(D a" 0 or + D.. ay" 1

)

U

N(O) QI. U

or -

2Nml a" ~. oz oy

-

101

N ..

if oy2

In Eq. 5--64, we have shown .9" as a differential structural operator. It is also possible to represent the inverted form of V in terms of an influence function q:t", y; ~,'I/). The latter represents the deflcction of the plate at the point (z, y) due to a unit force applied at the point (~, '1/). When the boundary conditions on the plate arc homogeneous, the inverted form of JI' is

)-If,

(5-65)

C(z, y; E, '1/)(

Except at the source point (~, 7]), the fu nction C(:t", y; f, 1]) and its derivatives with respect to x and y up to the order of f/' are continuous. The function C(z, y; E, '1/) satisfies the differential equation v(e) = 6(x, y; E,7])

(5- 66)

and the prescribed homogcneous boundary conditions. The function 6(:t", y; Eo 1]) is the Dirac delta function. This is a function which has the unique property that it is zero everywhere except at x = E, y = 1/, where it is infinite. In addition, for any assumed function g(x, y) the Dirac delta function has th e integral properties

If ,

6(:t", Yi E, '1/1 dx dy = I ,

If

g(z, y) 6(x, y;

~,'1) dz dy =

g(f,1/) (5-67)

"

if (Eo '1/) is incl uded in the region of integration. If not, the function has the properties that

II H

b(x, y; f, 1/) dx dy = 0,

If

g(z, y) 8(x, y;

~,'I/) dzdy =

0

(5-68)

S

(I) Solution of the ptate equations In the case of lifting surfaces of irregular shape or thickness and of plates with complex boundary conditions where precise results are desired, it is obviously necessary to solve Eqs. 5--ti2 and 5-63 or Eq. 5- 64 by approximate numerical methods such as those illustrated in Chap. 3. Space does

S I"HUC rultAL OJ'EItATOII S

179

• T(y)

,/ ,"

- -,

Fi,l. 5-10. Slender plate !lUbjccted to a ehordwise temperature gradient and a pure Iwisting to"lue. not permit us to include such solutions in detail here, so we shall confine our remarks in this section to a single relatively simple illustration of general interest, The effective torsional $Iiffne$$ of thin wing$. Let us consider how the plate equations may be applied to compute a relationship between the applied torque, M t, and the twist per unit length, 8, of a long fiat plate with a chord wise temperature gradient. Figure 5-10 illustrates the axis system and some of the notation which will be employed. We assume that a pure couple, M I, is applied at th.e end and that th.e plate is subjected to a chordwise distribution of temperature, T{JI), as illustrated by Fig. 5-10, T(y) is taken as an even function of y. For a solid uniform plate, the appropriate differential equations derived from Eqs. 5-62 and 5-63 are 2 DV4w = iJ2F iJ2w _ 2 a F a'w + "ifF o"w (5-69)

oy' ax'

oxoy oxoy

ar oy!

The boundary conditio ns are the following: Fory = ±b, N".. =

N~. =

Mn =

V~ + aM u

'x

"" 0

(5-7 1)

l'IUNCI I'U~

1841

For:.: =

0 1'

A I~ItOEL AST I C ITV

±a,

(a)f.N",~dY=O. (b)f~N_YdY =O (c) M j = 4bD(1 - v)O

+0

f.

(5- 72) Nnlf'dy

Boundary conditions (5- 71) indicate that the boundary conditions are to be satisfied exactly along the free edges, and (5-72) that they are satisfied in an average way along the loaded edges. We assume that the chordwise temperature variation is described by T6J) zoo T... + tJ.TgfJJ), where T.... is the midchord temperature and g(y) is a function of y as yet unspecified . Since the plate is long and narrow, we assume that the stresses are invariant in the z-direction; and we put F = F(y). The form of the deflection shape is ass umed to be the same as the St. Venant torsional solution It' = Ozy, where 0 is the twist rate. When these functions arc put into Eq. 5-69, we find that it is satisfied identically ; and when they are put into Eq. 5-70, we obtain

, il'g -iJ'F = -£'1. 'I.uT -

all

a.,,"

+ E h"

(5-73)

Integrating Eq. 5-73 twice yields (5- 74)

The constants of integration, Ao and AI' arc evaluated by boundary conditions (5-72a and h). T he following result for the self-equilibraliog spanwise stress resultant is obtained:

Ehe' 6

N".,=_ (3 y2

_

btl

+ .'EMT [f' g(y)dy 2b

-b

J (5-75)

2bg(y)

When Eg. 5-75 is substituted into boundary condition (5-72c), we obtain the following relationship: M , = 9'(0)

where 9'( :?(

(5-76)

) is a nonlinear structural operator having the form of t::.T 4 b' )=4bD( I -,.) [ 1 - + - ( 1 +7)-( t::. Tor 15 h2

)

S·I·RUcrURM. OI'I!RATORS

till

and where

bh'

• T"

~ 3«'(1 --[""'i ----"'------"-" ' g(y)----' + v) ' g(y)y= dy - - 'J'----dy J -~

(5-77)

3 -.

the temperature difference required to buckle the plate in torsion. It is interesting to examine the physical significance of the various terms in Eqs. 5-75, 5-76, and 5-77. In Eq. 5-75 we observe that the first term, proportional to the square oftbe twist rate, gives the contribution of twist rate to the midplane stress resultant, N_. The second term, proportional to t::..r, gives the contribution of the temperature difference. In the structural operator of Eq. 5-76, the importance of the nonli near term in fl is evident if we observe that when this term is neglected, the torsional rigidity vanishes for t::..T= 6.T". The finiteness of t he twist rate has, tberefore, a profound influence on th e torsional rigidity und er these circumstances. Finally, in Eq. 5- 77, we observe that, if g (y) is a function such that it bas bigher values at the edges of the plate than at the center, then t::..T" is positive. For this type of temperature distributi on, we see by Eq. 5-75 that the stress resultant N= is compressive in the vicinity of the plate edges and tensile in the center, It can be reasoned that, for this type of stress distribution, a mechani.m exists for producing torsional instability. When the plate is twisted through a small angle, the effect of the compressive stresses near the free edges is to produce twisting moments which tend further to increase the twist angle. When th ese twisting moments exceed tile elastic restoring moments in the plate, a torsional instability is produced. The te mperature gradient required to produce such a torsional instability is given by Eq. 5-77. IS

(j) Thc

cir~uhn

cylindrical shell

Since fuselages and rocket and missile bodies are often of circular cross section, aeroclasticians are frequently concerned with studies of the dc, fonnation properties of circular cylindrical sllells. The derivation of the appropriate equilibrium equations for circular cylindrical shells rests upon the three basic equili brium equations (5---43). These equations are reduced to circular cylindrical equations by introducing the notation illustrated by Fig. 5-11 and by putting

1112

1'llINCI1'I.,;s 0[1 AliIIOELASTlCI'I'V

Fill. 5_1 I,

Cjrculll~

cylinder coordinate system.

This substitution yields the following nonlinear equations of equilibrillm for a circular cylindrical shell.

~ [N= R(1+ a,) ax + Nb a' oliJ

ih:

+ ali -a [,"" N.~--+N.~ (a")J 1+R 00 ox +XR=O

a [N

a;c

~~

(I a, w)J

R -a, + N,R 1+ - - - ox ~ RoO R

+ ~ [No(1 + .! a, _ ~) + N RoO

Of)

R

a'J _ [N"R (~+.! awl RoO

"'ox

(5-78)

+N aWJ_!(OM9B+2RoM~)+YR = O 6~

ox

R

00

ax

:x [N~~R ~: + Nh(V + :;)] + :0 [N,,(~ + ~ :;)

These equations are especially useful when they arc cast in terms of the dependent displacemcnt variables II, v, and IV and then linearized. This is accomplished for an isotropic elastic and homogeneous shell by introducing into Eqs. 5- 78 the stress·strain relations obtained by combining

Eqs. 5- 38 uml 5- 39. Hnd lincHrizing the resu lt. Wc obillin by this means thc cquilibrium equat ions 02u+ I _~ o2u+I+~

ox2

2R2 OO!

GJ'lv _!... ow

2R oX ao

R

o:t

h~

+ 12R2 FI(u, V. 11') + 1+ 1' iFu 2R oX of:}

1

a"v

-- ~+ - -+

R' of:} ,

i-vo!v 2 o~

I _va Eh

X = 0

- R! -low -of:}

hI

+ -12R!- Fa(lI, v. w) +

1-1'~

Ell

Z= 0

where we define in this section )

"( and where Flu, D, 11') = 0

a"v

1 (PI'

F~(II, v, w) = 2(1 - v) ar + R! aat

+ (2 -

Q3w

1')

1

a"w

o:t! of:} + R'I. ort'

(5- 80)

a3v

1 ih F.jll, v, 11') = -(2 - v) ox2 of:} - R - ' a,>

The temperature depe ndent terms in Eqs. 5- 79 have been omitted for brevity. Numerous versionS ,of the linearized circular cyli nder equations (5-79) have been published in the literature by several investigators of shell theory (Refs. 5-7 tbrough 5-15). All of these versions may be cast into the form given by Eqs. 5-79 in which the differences are reflected in the functions F1(1I, v, w), FZ(II, v, w). and F,(II, v, 11'), and all otber terms are identical. The explicit values of FlII, v, IV), Fill, v, 11') , and £3(11, v, IV) given by Eq. 5-80 are those of Washizu (Ref. 5-6) and Goldenveizer (Ref. 5-13). In Table 5-1, the values of these functions for six of the more familiar the ories of circular cylindrical shells are given. The simplest of these is Donnell's theory (Ref. 5-10). in which the functions Fiu, v, 11'), FZ(II, v, IV) , and Fiu, v, w) are put equal to zero. Since these quantities

-•

TABLE 5-1 Te rm s fo r yulous Circul a r cylind r l..al shell theor ies

Ftv<, II, w) Donnell (Ref. 5-10)

FI..u, v, ...)

0

0

,

~.

Timoshcnko (Ref. 5-1)

F.J.u.

hi iI8

0

~.

~,

iJlo +(1 - .) hi

••

FIUgge

- """"2R ~:r.aOI + R ax"

(Ref. 5-11 )

1-.iPu

+ 2R" Washizu (Ref. ~)

",d

I iJlv + ~ 8B'

-2- -~zli!O

Qi)ldenvei7.er (Ref. 5- 13)

3 - ~ j1311' - ,- il:r1i!O

ari18

1 - ~

., , -. - R"

3

+ : 2 ( 1 -.)

~,1:1

iJ(JS -

1- .

+ 2:R

aft'

'"ar

, '"

2(1-.)-+- -

+ (2

R' «I'

~w

- .) hli18

,

ii ~,

II'

R"

.,

i)£I'

iIIu

, 'w

+~

iJ(JS

3-

~

~

>

'-

g ;:

~

- RI - -MI - RI - - R -ar

3- .~1I'

0

2

Q

-(2 - . ) - - - -

+""""2R hiO' -

1 - . iJlw

;;

;=

+ RI 00i

1 - ~ ~w 'w - - - +R2 R h MIl hi

w)

0

2 iJI...

Vlasoy (Ref. 5- 12)

II.

~D

~ ~zli18

"

w R" - R hi 3 - . iJlv iJlu hifJl - ~ hiM)

iJlu 1 iJlo - (2 - .) ~ - R" «I'

9<

STRUCl'URAL O I'ERATORS

185

are multiplied by lI'fr2R~ in Eqs. 5-79, it may be argued th at their conlribution is small and Ihat th ey may therefo re be negleeted. It can be shown that Donnell's Ihoory is equivalent to tha t which is obtained whe n th e shallow shell theory is applied to a circular cylindrical panel (cf, Ref, s-IS). When the tangential loading terms X and Yare put equal to ZtlO, each of the theories ofEqs. 5-79 and Table 5-l maybe reduced to a single eighthorder differential equation in the displacement function rf> (Ref. 5-12). It is then possible to express in the usual manner (5-8 1) where the displacement function rf> is connected with the displacements U, v, and w. The form. of the structural ope rator f/ depends on the circular cylinder shell theory which is employed (Ref. 5-13). For e:>::ample, for Donnell's theo ry, we have f/(

) _ _ Eh(R'~+ 1 h VS)( R~ az& 1 _ ,,2 12R2 2

)

(5-82)

The connecting relations between the d.isplacements u, v, and IV and the displacemen t fllnction rf> are different in each case. In tbe case of Don nell's equation, they are simply u

= R ~ (iPrf> ax 00:

v .. _

_ "R2 a>",)

or

z .!(a 4- + (2 + "')R~ at4-) ao ao z a",'

(5-83)

w = -V~rf>

In fact, by making use of Eqs. 5-83, it is evident th at a further simplification in Donnell's equation is possible whereby 4

l!.- VSw + Eh aw = .1 v'z RS

Other relations connecting 4> with in Ref. 5-12.

R2 II,

ax4

R'

(5-84)

v, and w may be found, for example,

5-3 HOMOGENEOUS ISOTROPIC ELASTIC SOLID Let us consider the nature of the elastic operators which are appropriate for homogeneous isotropic elastic solids. In Eqs. 2- 21, we have given the equations of equilibrium in compo nent form. These eq uations may be

1 ~6

I'IUNCIPLES 0 1' AEII OEl _ASTI CI 'I'Y

ahcrcd by cxpressing thc m in tcr ms of dispbcc lllCll ts mlhcr than stresses. We may, for cxample, substitute for the normal stress componcnts such ex pressions as (Ref. S-5) (1~ = i./:.. + 2G au (5- 85)

a,

and for the shear stress components such expressions as

,

~

".

c(aw + a,\

where A. and G are defined by A. =

(1 +

G=

ay

ad

E~ v){ l

2~)

(5- 86)

E

2(\ + ~) and to.. is the cubical dilation defined by

to.. ...

all + av + aw a", ay az

(5-87)

When expressions such as (S-85) and (5- 86) are substituted into Eqs. 2-21, we obtai n thc Navier equations

(). +

azv a~w ) + ax ay + ax az

a'u G) ( a:x?

2 azu a v (fw )
(fu

().+G) ( - -

2

av -aW)' + --+ 2

axaz ay az

+ G'\' zu + X =

pa~

+GV'v +Y_pa.

(5- 88)

2

az

+GV'w+Z=pa ,

It is, of course, necessary to adjoin initial conditions a nd conditions at the boundary to Eqs. 5-88. When the surface displacements are given, the displaceme nts 11, v, and IV are prescribed at the surface. When the surface tractions F", F., and F, are prescribed, the bounda ry conditions arc given by

c[au au . -ov n-J. + ow ] - +-n'I+ _ no k on ax ax a:r F, = A.d n.j + G[av + aU n . j + aV noj + aJ<' n.k] on oy ay ay awan. au . aw ] F.=A.to..n.k+G [-+-n'I+-n·J+-n" on Oz az az " n o l' F ~=".."

+

(5-89)

where n is a unit vector norma l to the surface in the outward direction.

STItUC I' UItAL O I'I';ltATO IIS

E
e.~ p ress~d

(A + G)V(V . q)

1/11

in Ihe morc co mp;lct vcctor form of

+ GV~II +

R = pH

(5- 90)

where q, R, and a arc Ihe displacemcnl, body force, and acceleration vectors, respectively ; or thcy may be wrillcn in the followi ng operalof form: R - pa = j}(q) (5- 91) where j} (

) is a tensor differential operator with the form of ) = - [(A

+ G)grad div + GV2] C

)

(5-92)

in the case of a homogeneous, isotfopic material.

5-4

THREE-DI MENSIONAL ELASTIC STRUcrURE

Finally, "We take up the question of defining operators approprialc to Ihe elastic deformation of three-dimensional structures in general. Since such structures vary widely in their geometry and generaL arrangement, we shall be conlent to state onLy the general form of these operators. The tensor operator inverse to the operator !/. ( ) of Eq. 5-91 relates the displacement vector and the body force vectors by q = §l- I{R - pal where (j'>-l (

(5-93)

) is expressible in the .integral form of (5-94)

The quantity defined by

r

is a second order tensor of flexibility influence functions CUCx,

r =

y, ~ ;~, 1),

Oil

+ CU(x, y, z; ~,'I/, Oij + C~'(x, y,~;~, '1/, Oik + C>Z(:z:, y,~; E, '1/, Oji + CV"(:z:, y, z; E, 1], Oij + C' '(x, y, z; E, 1], Ojk + C~Cx, y, z; E,1), Old + C'"(x, y, z; E, '1/, Okj + C"Cx, y, z ; $, 1], {)kk

where, by Maxwell's law of reciprocal detlections, cn(x, y, z; ~,'I,

0

= CW~U, 'I. ~; x, y. z)

y, z; f, 'I,

~)

= CU(~, 'I, ~; x, y, z)

C~'(x,

O '(x, y. Z; ~, 'I/,

0=

c·¥(~, 'I, ~; x, y, =)

(5-95)

18!l

l' IU N(: Il' U/'s 0 1' AEHO m .•AS'I'ICITV

We sec, for e,;ample, that the quantity C'·(x, y,:; ~. r], C) represents the denection of the structure in the x-direction at the point (x, y, 2) due to a unit force in the y-direction at the poin t (~, '/, '). Such influence functions may be compu ted, or in very comple,; cases, obtained by experiment. For e,;ample, in the special case of a homogeneo us, isotropic, elastic solid, the influence functions cu, 0", and CU arc: computed by applyi ng Eqs. 5-88 in the following way:

+ G) ( or + ox oy + ox QZ + GV!C" + 6(:1:, y, z; E, fI, 0 = a QtC'~

(:I

Q'JC""

(flC'~)

+ G)((flCu + 02cu + 02C%2) + GV2C'~ _ 0 ox oy O!lt oy oz (l + G) (iPC + ;pc·· + Q2C + GV 2c = a 2 (.i

u

(5--96)

U

)

u

ox oz

oy OZ 02 where Q(x, y, z;~, 1), C) is the Dirac delta function (cr. Eqs. 5-67 and 5- 68). REFERENCFS 5- 1. Tim9Sne nko.S., andS. Woinowsk.y _Kricgcr, "TMMyofPlausand Shells, McGrawHill 11001: Compa ny, New York, 1959. 5- 2. Girkman, K .. F1ach~nlrag_rk~, SFringer_Verlag, Vienna, 1956. 5-3. Wang, C. T., Applied Elasticity. McGraw-Hi li Book Company, New York, 19535-4. Peery, D. J ., Ajrcrofl Siructuru, MCGraw_Hill Book Company, New York., 1950. S-S. Timoshenko, S. , and J. N. Goodier. Theory of Elaj·,idty, McGraw_Hill Book: Company, New York:, 1951. 5-6. Washizu, K., Som~ CorujderatjOJI.'1 all Shdl Theory, Aerodast ic and Structures Research Laboratory Report 1001 , M.l.T., October 1960. 5--1. Marguerrc. K. , " Zur Theorie der gek.rnmmlcn Plalle grosser Formanderung," Prac. Fifth Internat_Congress of ApDlicd Mechanics. pp. 93- 101 , 1938. 5-8. Rei"ncr, Eric, "On Some Aspect s of the Theory of Thin Elastic Shells," J. B<JJlon Sac. Civil Engineers, Vol . XLlT, ND. 2, pp. 100-133, April 1955. 5- 9. Love, A. E. H., A Treal;se Oil Ihe Malhema/i~al Theory of Elasticity, 4th ed., Dover Publicatio ns, New York:, N.Y. , 1'M4. 5--10. Donnell, I... H ., Siabilily ofThi,,-WallcdTubes V"der Tors;on. NACA Repor1419, 1933. 5--11. FI!lggc, W., SWfjk und Dynam;k der Schulen. Julius Spri ng~r, Berlin, 1934. 5- 12. Vlasov, V. S., Basic DifJ~rentjol E1uoli(ms ill Gmerol ~My 0/ E/OJI;C Shdls, NACA TM 1241, 1955. 5- 13. Goldenw:iztr, A. L., Teor;o uprug/kh IOllkikh obolochtk (Theory of Thjll Elo.tic Shells), Gosizdat tekhnikoteoretich lileralury, Moscow, 1953. 5_14. No\tOzh;!ov, Y. Y.• The Theory of Thjn SMI/s, edited by J. R. M. Radok. P. Noordholf Ltd .• The Netherla nds. 1959. 5- 15. Shulman, Ycchid, Some Dynamic 0",1 A~r~laSljc Probf~ms of Platt alld Sildl Sirucmrn, Sc.D. The,is, M.LT., 1959.

6 THE TYPICAL SECTION

6--1

INTRODUcnON

The typical section (Fig. 2- 3 and Sec. 2-7) was devised during the [930's by such aeroelastic pioneers as Theodorscn and Garrick (Refs. 6--1 and 6--2), seeking a system suitable for elementary examination of the flutter problem. They suggested that the dynamics of an actual wing might be simulated by choosing the properties of the typical section to match those at a station 70-75% of the distance from the centerline to the tip. Subsequent experience has confirmed their judgment in situations where the aspect ratio is large, the sweep is sma.li, and the sectional characteristics vary smoothly across tbe span.

Quantitative justification exists for this approach, because any twodegree-or-freedom lumped parameter approximation to a lifting surface which behaves elastically as a beam-rod will obey equations of motion in the same form as Eqs. 2-7&:1 and b, and for unswept cases the correspondence is exact even in the relations between the airloads and the displacement coordinates. Th is is true, for example, of the semi-rigid wing used in early British investigations (e.g., Ref. 6--3), which is a structure artificially constrained 50 that its spanwi5e distributions of twist, flexure, etc., are independent of the manncr of loading. There is no essential difference when either normal coordinates or arbitrarily selected modes of bending and torsion are adopted for setting up the analysis, as in Chap. 3, except that by the former method the clastic and inertia coupling terms are eliminated while the dependent variab les lose part of their simple physical significance. In every case, the various coefficients of Eqs. 2-76a and b

'"

1')0

l'IIiNCII'I ,ES 01' AI·:" OI':I.ASTI CITV

;\re replaced with weighted sp:lnwisc integrals of sectiona l properties lind airloads; it can be said thll t there is always 51ll/lcsystem like the one in Fig. 2- 3 whieh is aeroelastically eq ui vale nl to the true canti lever willg thus represented. The presence of an aerodynamic control surface just adds another degree of freedom or modifies the system inputs, as will be seen in subsequent seelions. For further clarification of this equivalence, we can correlate the eltamples which follow aga inst parallel treatments of the restrained beam-rod in Chap. 7. Surely the best justification for devoting a full chapter to the typicaL section is the way in which it permits a swift, orderly, uncluttered presentation of the majority of classicaL aeroelastic problems. A few essential features are lost, such as the effects of three-dimensional flow and of rigidbody degrees of freedom. But the need for extensive approltimation, some of it difficult to motivate o.r evaluate quantitatively, is wholly eliminated. The system will now be examined on its own merits, leaving to the reader much of the task of interpreting results as they mu-minate what may take place on more complicated structures. We assume the linearity of all the aeroelastic operators.

6-2 STATIC PROBLEMS

Figure 6--1 reproduces the typical section, as adapted for the discussion of steady-state aeroelasticity with no time-dependent motion. A trailing edge flap has been added 10 the system of Fig. 2- 3, although it is merely intended to be: representative of any of the several devices currently in use for modifying lift on a wing. The e.G. position is nol involved unless the weight has to be: taken into accoun t, but here the aerodynamic centert he altis about wbich pitching moment in steady fl ow is independent of angle of attack- has special interest; the A.e. is located a distance e ahead of the elastic axis. Bending spring K~ and its associated linear deflection, h, are not pictured in Fig. 6--1, since the equilibrium of angular displacements abo ut the elastic axis primarily governs the static aeroelastic behavior. The bending degree of freedom rnay be described as uncoupled from a; and h, because a pure vertical translation causes no lift or moment. Th llS the rotations are determined first; and, if desired, h can afterward be calculated from the steady-state version of Eq. 2- 76a, (6-1) seen in Chap. 7, the dccoupling no longer occurs on a swept wing, wbere a bending slope gives rise to aerodynamic incidence.

At; will be

TI m l'Yl'l CAL S ECfION

191

In lilly llc roela s!ic sys tem, the applied loads which piny the role of inpu ts may be divided into (I) external loads o f nonaerodynamic origin and (2) ai rloads due to the geome try of the structure of airflow. T he former category embraces gravitational forces, certain inertial effects ofmaneuvers, forces due to va rious types of sha kers, or eve n such things as the pull on a wire attached to the typical section. The aerodynamic load s arise gencrally from changes in angle of attack due to control deflections, maneuvers, gust or blast encounters, a turbulent wake acting on a tail, and the like. T he rigid-body angle of attack and camber associated with trimmed flight constitute suc h inputs for the steady-state aeroelastic problem. Nonaerodynamic external loads will not be considered in this section, since they have little practical interest and their analysis is trivi al. (a) Treatment of the torsional degree of freedom and divergence

As a first illustration, we take the system in Fig. 6-1 with the flap locked out, /j = o. T he angle of attack fro m zero lift includes a "rigid" part, «0, and a twist, a" in the torsion spring:

O(= «o+ a,

(6-2)

---

u

Fia:. 6-1. Typical section airfoil, with trailing edge flap, elastically suspended in an airs~am.

l'IUNCII'I.I'8 011 AI~N.OI",ASTICIT"

In

The struetunll opernlor relnting the to rque

7~

to (t. is the spring constant, (6--3)

Equations 4-63 and 4-62 furnish the aerodyaamic operators for lift and moment about the elastic axis,

ac

,.

L = C LqS = qS ---l! (~

+ !.t, )

(6-4)

M.=Le+MAC = qs [e As before, g.

O~L {.xl} + !.t,) + CCMACJ

(6--5)

c, and S denote the dynamic pressure, chord, and plan area

of the typical section, respectively. Equating the elastic and aerodynamic torques from Eqs. 6--3 and 6- 5, we find the equilibrium angle of twist to b<

'S(e OCt,1Xo + CClIf,w)

Ilt, ""

oK,. -'--~'e·_;:-",,_---,1- qSeoC L K. OJ.

Both IXo and C.MA O (which is directly proportional 10 the percentage camber of the thin airfoil) are seen to play roles as inputs to a tinear system. With the other aerodynamic and structural parameters held constant, each one makes a contribution to (t. in proportion to its own magnitude. Bcaring in mind that oCdo!.t, CMAC. and e will usually be functions of the free-stream Mach number M , we can conclude a great deal from Eq. 6--6 about the influence ofllight conditions on wing distortion. The second term in the denominator is often small compared with unity, suggesting an expansion of the form

'c,

L J[ 1+ qSeJC - --+ J(t

<x. = ="S f e--'~+cC .lIA C K~

"" <x;"

Ja.

+ <x;11 + ...

K.

(6-7)

Here (6-8) is the twist that would be produced by the pitching moment input, if no further aerodynamic torque resulted from the deforma tion itself. !.t;~) is precisely the incrementa! twist caused by the moment from angleof-auack change <x~I), and so on. The terms beyond <x~1) in Eq. 6-7 are

TIm TYI'I CA I. SECI'/ON

193

a measure of the "aeroclaslic effect," thlll is, of the interaction between airl oads and deformations, which is the distingu ishing featllre of ac roelasticity. Evidently,

'.

_-,:1;,-",-

(6-9)

,- :-I>-I_qseoc 1• K.

00(

may be: greater or less than unity, depending on whether the A.C. lies ahead or behind the elastic axis (e positive or negative). There is an eigenvalue problem associated with the twisting wing. This can be observed from the blowing up of the solutions (6-6) and (6-9)

qSeoCL

at a cerrain value of the parameter - K~

~;

alternatively, it is inferred

<10(

by causing the inputs CXo and CM~C to vanish in the moment-balance equation and Doting that there is oDe condition where

[1 - qseacLJIX. = 0 K. act.

(6-10)

may have a nonzero solution of arbitrary amplitude. This condition is known as tors ional divergence. It occurs at the dynamic pressure

K, q/J =

(6-11)

oC

s.--L

"

and comes about physically because the rate of change of pitching moment, M ., with angle of twist increases with increasing q until it becomes just equal to the constant K" and overpowers the torsional spring. The series (6-7) also diverges at this point. Divergence-type instability is not ordinarily an important practical problem on straight wings. since flu tter is likely to occur at a lower speed; bu t qD provides a useful measure of the general stiffness level of the structure. As might be expected, Eq . 6-11 shows the eigeovalue to be independe nt of both initial incidence an d ca mber. Except at the lower subsonic-speeds, Eq. 6-11 is ooly an implicit expression for divergence

.

.

act.

dynamIC pressure and Mach number, SlOce both e an d OIX depend on M D. However, q =

i

poeM!, so that an e:o: plidt solution can be devel oped

ei ther subsonically, where oeLla'/. '"""-' 1/'1/1'- .'l-f ~ and (' is fixed, or supersonically after similar sllb~litU( ions . The taHa ca~e has li ttle interest, because the A.C. is near m:dchord . e is usually . negative. and M n is, the refore, imaginary.

194

I'RINCII'UcS 0 11 AlmOHLASTl CITV

A resul t of some signifie~lIlee is derivable rrom Eq. 6- 9 ror any range of flight speed and alt itude around M/) wherein varia tions of e oCjjoa. can bc neglected:

:U,'J == CI-~(qC/CqC-;) -

" CI-C("M~\;;/CMOD-;;')

~

(6- 12)

For the third member of Eq. 6-12 to be valid, of course, fl ight must be at constant altitude; otherwise M t /MJ)! must be replaced by

PmM2/(P a,)DM IJ2 (b) Effects of a control sulface or spoiler Let it first be assumed thai the flap in Fig. 6--1, simulatin g any trailing edge control surface, is given a rigid rOlation (i unaffected by any aerodynamic torques exerted on its own hinge. Since pitching moments about the elastic axis due to initial incidence and camber have effects which arc additive linearly to those discussed here, they are omitted. According to Eqs. 4-64 and 4-65, the aerodynamic operators relating pitching moment to Iwist and flap angle are then M . = Le

==

+ M.f a

qs[e(a~:. ". + O~t b) + c aCa~'w 0]

(6--\3)

Equating moments between Eqs. 6--3 and 6-13, one calculates the ratio of elastic deformation to flap aogle

cac,u.w a;5+; ao oCr.

01,

-

o

~

-"-;;----'-c;;;"K~_OCL

qSe

(6-14)

a"

and thence the total lift which is produced on the typical section by deflocting the fl ap,

qs[ac + qSc oc l.ac,lIAoj, f•

L

=

ali

K. 001 00 l _qSeaCL

(6--15)

a"

K.

Equation 6--15 can be compared with the lift

sacLo

L' =

q

"

(6--16)

that would be developed by the Slime nap on an infinitely rigid wing. Their ratio reads I

+ qSc OC MAO / " K.

L

ad

06

(6-l7a)

where (6-11b)

is the wing angle of attack giving the same lift as unit flap deflection on a rigid wing. L/L' normally comes out less than unity, because aCM
,.

q~ =

K. Sc (_

ad

aC;;ltJ)

(6-J8)

Like Eq. 6-11, this is an implicit formula for q1l (or M R) whenever compressibility effects on a~la6 and aCM.,JC/a6 must be accounted for. The phenomenon of reveoal is usually associated with ailerons or elevon. mounted, as they oftcn are, ncar the tips of wings having considerable torsional flexibility. Several aucraft have actually flown at q's where the antisymmetric lift distribution caused by moving the ailerons, and therefore the rolling moment, became opposite in sense to that caUed for by the conlrol-column displacement (i.e., L/U became negative, at q > qR). While not necessarily causing structural failure, this condition is obviously undesirable. It is aggravated by sweeping back the wings, in which case the designer often finds himself forced to adopt spoilers or some other lateral con trol device. Horizontal and ,'ertical tail surfaces do not experience complete reversal but only a partial loss of effectiveness. This is because fuselage vertical bending and fuselage torsion are more influentia l thaniocal surface twisting in altering the control forces due to deflecting the elevator and [udder, respectively.

t96

I' IUN CWU!S 01'

A~~ ItOI! LAS'flCI1'Y

When the lIerodynamic dc rivntives in Eq. 6- 17(1 can be assumed constant, the following simple relation analogous to Eq. 6-12 is obtained:

(The third member again requires constant-altitude flight.) Equation 6-19 is plotted in Fig. 6-2 for several values of qrJ9n. It reduces to the compact linear form L q (6--20) - """ I - L' -

q1l

whenever both 911 and q are small compared to 90. On the other hand, an interesting anomaly appears in the less commo ncircumstancethat9R = qD. The control then remains fully effective (LIP = 1) right up to the catastrophic onset of the divergence-reversal condition; this phenomenon has a simple physical explanation. Reversal is not an eigenvalue problem in the same sense as divergence. Indeed, the characteristic dynamic pressure associated with the rigidly deflected flap is still 90. as may be seen from the vanishing of the denominator in Eq. 6-14 or 6--l7a. 30

'"

~

qR/'lD _1.1

~ t:'"

10

o -w

-

~" '\ '\ ,,\ "-

~0.7

1\

0.'

/ \

u

'\

\

'\ \

- 2.0

o

~

0.6

q/q"

/'

0.8

10

12

I.'

FiB. 6-2. Flap . n"'cliHn","~ I..IL' vs. dimensionless fl ight dynamic pressu,"" ,,!r{D (or 5eY<'rnl vnlue:; of (/>< " ,IiI) of reversal to diY<'rgencc dynamic pressures. Aerodynamic prope rtics ",~ 'Issumcd :r){lcpendcn( of Mach num\xr.

1'1 m 1'YI'ICA l . S):Cl'lQN

197

u

Fig. 6-1. Typical spoiler configuration, showing extended and

~tracted

positions.

The foregoing analysis can readily be adapted to other control configurations on the typical section by making appropriate insertions for aCJa6 and acM ..wla6. For example, a leading-edge flap has both these aerodynamic derivatives positive if 0 is defined to be the angle of nose-up rotation. Therefore, such a device would have its effectiveness enhanced by increasing dynamic pressure (Eq. 6-l7a) and is not subject to reversal. Most spoilers are highly nonlinear, since the incremental aerodynamic loads are not directly proportional to the control deflection. Thus the typical arrangement shown in Fig. 6-3 is recessed and has no effect at all except when O. is positive. A given value of O. produces (1) a negative contribution t:.CL(o.) to the 11ft coefficient and (2) a small moment 6C1-u.d/IJ, which can be negative or positive, depending on the configuration. although the latter sign is more common. Under these circumstances, Eg. 6--15 would be replaced by

(6--21)

With positive 6C.1F,H•.(".), there is a remote possibility that the second term in the numerator might overpower the first and bring on rcvcrsa l. In practice this eventuality turns out to be much leSS li kely than with a tra iling edge flap . (e) The influence of flexibility in the controls

Moving a step closer to actual condition> on aircraft, kt there be attached to the flap hinge a spring with torsional co nstant K~. II is so arranged that, in the ab sence of any aerodynamic moment, the flap would assume an angle "u. The spring torque tending to rotate the fl ap back toward neutral is then (6-22) " being the eq uilibrium flap p05ition. Tn the case of ailerons, for instance, 66 would be an an gk proportional to the turning of the pilot's control

1911

1'Jl:1 N<':I I'I .J,;S OF Alm Ol(LASTlC1TV

wiled or the latern l disphK'CIllCIlI of the ~tick, li nd K~ becomcs a gross ly simplificd rc prcscntlltion ofeablc flc xibili ty nndJor stilfncss of an actunlor and back-up structure. According to Eq. 4-66, the re is an aerodynamic operator giving the noselip hinge moment due to the incid ~nccs of the wing and flap:

H, -- qS ,C, [iJCJl ooc <1. + oC adu

{J]

(6--23)

Here Sf and Cf are suitably defmed flap area and chord, respectively. Since H, and H I must be num erically equal, Eqs. 6-22 and 6--23 can be rea rran ged to yidd the following re la tion among th e angles:· (6- 24) A second equatio n is provided by the balance of torques about the E.A., Eq. 6-1 4, which is here rewri tten

L _ K. ]., [aC art. qSe

+

[OC/: + ~ OC'\lAU]J = 0 a{J e all

(6--25 )

T he simultaneous Eqs. 6-24 and 6-25 furnis h both the twist, oc" and the result~nt flap angle, 15, as functions of the control input angle, 00. Carrying out this solution, we can find t he tota l lif\ on the elastically-mounted section,

L~qS [OC.Loc + OC /' o] iJ,,

'

iJo

(6-26)

• Observe thM. when the wing tors ional stiffness is high (ct. ~ 0), the negative sign of aCn /i16 causcsd from Eq . 6-24 to be smaller than the desired Ikflcct,on d, . This illll.<-

trates the phennmenon of blowbaci<, wh ich ea.n interfere severely with the ",,[,$factory operation of any aerodynamicaUy unbalanced control surface on a high-speed a'rr;;raft.

'1'1 IE 'I''iI ' ICA I.

S I~:C nON

199

'C

Dividillg by the "rigid" li n L' = qS ao" ,)0, we derive an expression ror control etTectiveness, 1 + qSc ilC.llA.C L

L'

jiJ

l1

~ c--"--=",----o,,K~.CO-,'::;'-'--":c'- - - - - -

[I _

qSI' OC I.] K«

il
[I _qSh ilC II] K.

(6-27)

0.'5

_ [aC L+ :. oc MAO](,5,) (qS6) (JC H

il
ao e ilo K. K, This more realistic system is seen also to display loss of effectiveness and reversal. In fact, the dynamic pressure qH' obtained from the vanishing of th e numerator in Eq. 6-27, is stiU give n by E<:j. 6-IS except in the unlikely event that the denominator goes to zero first. Equation 6-27 takes on a more meaningful form when the aerody namic properties of the typical section are independent of flight speed and altitude. Let us assume this for the moment and define the following three constanlS: (1)

qD. '"

K.

(6-2Sa)

oC

,.

Se~

This is the dive rgence dynamic pressure for an infi nitely stitT hingeline spring, KJ __ OJ . (2) qDf

~

K,

(6--28b)

s,c,( _il~;!)

Since oCHloo is negative in the large majority of cases, this is a positive constant proportional to K.. If the h inge moment derivative did happen to have an unstabie, positive sense, (-qm) would be the dynamic pressure for torsional dive rgence of the uncoupled lIap. ilC H (3)

A =

OII

OC,lUO

ocr.

+£

'C " , H

"

"

(6--28c)

,.

oC L

This is a purely aerodynamic constant, whose magnitude is normally less than unity. Its sign is opposite to tha t of the "floating tendency" OcfI/aCII OII 06 because tbe quantity in brackcts is negative so long as

q~

< qI),

In tcrills of Eqs. 6- 28, Eq. 6- 27 can be rewritten

(6-29)

Several interesting conclusions about the behavior of the two-degree- offreedom system can be drawn from a study of Eq. 6-29. The eigenvalue oftbe typical section is modified by elastically mounting the flap_ Divergence now coincides with the vanishing of the denominator in Eq_ 6-26, 6-27, or 6-29. Although this is quadratic in q, it usually bas only a single positive root. Retaining the restrictions· that underlie Eq _ 6-29, we can easily show this Toot to be

\ ( qDJ"In:!·n~p = qn.

+

[1 - qm] ) [1- ~r+ 4[1 q/J,

qRt

2[1 _ A]

A]lJJH. qDo (6-30)

T he condition described by Eq. 6-30 is called torsion-aileron divergence. Genera!!y it will be found to occur at a dynamic pressure below qD. when A is negative, because the flap then tends to float parallel to the wind and generates a nose-up moment, which assists in overpowering the spring K~. Conversely, the dynamic pressure exceedsqD, when A is positive. Two special cases of Eq. 6-30 possess more than trivial in terest: (!) The unrestrained flap, qDt = 0, for which

(6-3 1) (2) The case A = 0, which corresponds either to zero floating tendency or to qD. = qR' This yields (6-32) for evident physical reasons. Figure 6-4 plots Eq. 6-30 for several values of the constant A . • NOle that Eq. 6-30 can be extended 10 eover Ihe condition when the aerodynamic properties depend on the flight Mach number by rcsubstituti ng Eqs. 6-28. HOWOfCr, (q,,) ...ln/l.ft.p is then given only implicitly, as discussed above.

1'1W TVI' ICA I, S I,;CnON

101

"" 5""

"i'-

"-

A _ 0.5

5"' 0.25

0 -0.5

I

0.' 5

'.0

5 0

o

0.5

25

'.5

' .0

f ig. 6-4. Dimensionless dynamic pr<:!:lure of torsion-3ileron div<:rgcnce plotted vs. the stiffness-rauo parameter '/"'/'/"0 for live va lues of the constant A defIned in Eq. 6-2k A(rodynami( properti~ are assumed i1'ldepM.dent of Mach number.

6-3 THE SYSTEM UNDER P RESCRIBED TIME-DEPENDENT INP UTS As an elementary introduction to the subject of forced motion, we

e)[amine the bending-torsion typical section driven by a prescribed e)[ternal force, F(/), in the absence of aerodynamic loading. Either the wind-tunnel test section may be assumed evacuated of air or the airspeed, U, may be set equal to zero; in the latter case, the inertial pro perties of the wing must include certain small virtual i"erlias of the air, whose presence does not alter the analysis significantly. Figure 6-S shows the driving force to be ac ting at a distance d ahead of the E.A. In the notation of Sec. 2-2, the governing equations evidently read mh

+

K"h

+ S.ii =

s.h + I.'l. + K~oc

=

-F

(6-33a)

Fd

(6- 33b)

101

l'IU NC IP I.I"-S 01' AE HOEI.ASTI CITY

FiS. 6-.5. The typic.l section driv<: n by a p rescribed external force F(t), acting at a distance d ahead of the clastic axis.

The spring constants in Eqs. 6--33 are often replaced in terms of natural frequencies of uncoupled bending and torsional oscillation, defined by

JK.lm

(6-34a)

W.'" J K~I/.

(6-34b)

WA

=

So modified, Eqs. 6-33 will be employed as follows in this section:

+ w.~h] + S~ii = -F SJ + i.(ii + co/ex] "" Ed

m[li

(6-35a) (6-35b)

Three basic types of force input will now be considered, along with associated methods for finding the system response to more general inputs. The reader will recognize these techniques as the fundamental material of any modem book on vibrations in linear systems, but they must be introduced in preparation for the analysis of more particularly aeroelastic problems. (a) Sinusoidal forcing; the mechanical admittance method

We assume first that a simple harmonic driving force of circular frequency co hRS been acting for a sufficient time to ensure the disappearance of any transients associated with its application. (Damping is neglected in Eqs. 6-35 but must be present to some degree in Rny passive system.) Linearity permits the complex representation If"" to be used throughout; so that a force which, in physical actuality, is the real part of (6-36)

T il E TV l'le AL SEc.-nON

is kn ow n 10 generate rea l parIs of

/1('Y/!Wllt'tll

or

SI
203

responses which arc the (6-37)

>0'

(6-38)

Generally F, ii, and Ii would be complex numbers, to allow for phase differences; they may be regarded as real here, since the sinusoidal response of an undamped system is always either in phase or 1800 out of phase with the inp ut. Combination of Eqs. 6- 35 through 6-38 yields the matrix equation

[

"'b[W" + (iw)'] S.b(iwl

S,(iw), I.[w. t + (iw)z:J

l(~) ~ (-F) Ii

(6-39)

Fd

where the scmichord b = e/2 has bee n introduced as a reference length for reasons that will become clearer in Sees. 6--4 and6-5. Thesimultaneous system (6-39) is then solved for dimens ionless ratios of outputs to input, one such pair being the following:

(&-40)

(6--4\)

Equations 6-40 and 6--41 contain two convenient dimensionless parameters, S, (6--42) x~ = mb the distance in semichords by which the CG. lies behind Ihe E.A., and (6--43)

2(1.1

l'N INCII'U!S 01' AENOEI-ASTlC ITV

•• 0

'0

\:

0

0

----

- 1.0

0

'\

0

- 4.0 0

0.2

0.'

"'"0

M

0.'

••

-,.

' .2

'A

"0

Fig. 6-6. Mechanical admittance HOI relating translational d isplacement to applied fore<: on til<: typ ical section in vacuum, plotted VlI. frequency mtio wlw~ for the following :sct ofparamtters:
the radius of gyration of the typical section about its EA., also expressed in semichords. The symbol H(iw) is frequently termed the mechanical admit/ance-the complex amplitude of sinusoidal response per unit sinusoidal in put-of the linear system, generalizing the concept of admittance used so commonly in electrical work. The impedance Z(iw) is the inverse of H Ow}. In Egs. 6-40 and 6-41 the admittances arc given subscripts suggesting the output and input which they relate; such identification is even more important

when trenting complicated systems having many inputs and ou tputs, but n unique functio n of iw exists for each pair. For illustrative pu rposes, the adm ittance Hu.(iw) given by Eq. 6-40 is plotted versus the frequency ratio wlw~ in Fig. 6-6 for a typical set of the four section parameters w~/w«. dlb, X« , r" which govem its behavior. Dam ping being absent, the response is seen to become infinite and changc sign at each of the two abscissas w,/w", and w.Jw". These mark the eigenvalues of the typical section in vacuum, equal to the fOOts of the denominator polynomial in Eq. 6-40 or 6-41. They represent physically the natural frequencies of vibration of the co upled bending-torsi on system and can be expressed in the attractively symmetrical forms

(641)

-Equation 6----44 shows that the natural frequencies (but not, however, the associated norm al mode shapes) depend only on the two parameters w.lw, and (6-45)

(Here 'CG and XCG are the radius of gyration about th e e.G. and the C.G.-A.C. distance, respectively.) Since x/lr / according to Eq. 6-45 must have a value between 0 nnd I, Eg. 6----44 always yields real, positive frC<Juencics. The presence of any dissipative force out of phase with the displacements in Eqs. 6-35 will eliminate the singular resonance peaks from Fig. 6---6. But then, as will be elabo rated in Sec. 6-4, the admittance becomes a complex function of iw and must be illustrated by plolling both its real and imaginary parts versus frequency. More commonly, cu rves of amplitude and phase angle or complex polaTS witb freq uency as a parameter are used. Any advanced text on litlear systems (e.g., Gardner and Barnes, Ref. 6-4, or Vol. 2 of Dra per, McKay, and Lees, Ref. 6-5) demonstrates how the admittance function ll, with iw replaced by a general complex argument, contains all the information about system characteristics needed to calc ulate the response to any arbitrary input, £(/). In particu lar, the Fourier series and Fourier itltcgral methods rely direc tly on a knowledge

206

" IIl NCII' I,CS 0 11 IIlmOELIISTl ClTV

of lI(iw) . The former melhod is limited 10 periodically rcpea ling inputs, all of which arc known to be represe nta ble by the series F(/) =

. 2:

- - '"

c.c;ft""

(6-46a)

Here Wo - 271"/To, To being the fundamental period . The complex constant c" is computed by techniques of harmonic analysis, c. =

J..IT' F(t)e- I .,..! lil T, ,

(6-46b)

Evidently c_" is the conjugate of Cft' which permits Eq. 6-460 to be reduced to the more fami liar, but less compact, real Fourier series containing cos IIWo/ and sin n%/. The system's response to the nth harmonic of the input spec/rum is Jl(inwo)c.eI-ol, by the definition of mechanical admittance. However, one basic consequence of the property of linearity is that outputs can be superimposed to find the effect of a combined input. It follows that the bending and torsional responses to the force described by Eq. 6-460 are (6-47)

(6-48) Convergence of all these series may safely be ass umed, since 00 difficulty occurs in practice with physically realizable inputs. There is no question of taking real parts in Eqs. 6-47 and 6-48; the vanishing of imaginaries is easily proved from the conjugate character of c. and c_m and of H(iw) and H( - iw). Conceptually it is a short step from the Fourier series or /ine spec/rum of a periodic function to the Fourier integral or continuous spectrum of an aperiodic one. Indeed, many books furnish proofs tbat, whenever the integral of IF(I)I overall time eltists, we can find a representation of the form F(t) = { "'",G(iw)e'''' dw

(6-4911)

where G{im) is the spectral deruily of F(I) and may be obtained from the formula G{iw)

f"" F(I) e- I",' dt 271" -..,

<= - '

(6-49b)

For real or complex w, Eq. 6-49b yields what is carred the FIlI/fier Iratl.r[orlll of F(I). The extended theory of such transforms appears in Sneddon's book (Ref. 6-6), while Campbell and Fos ter have published an excellent table (Ref. 6--7). By obvious elttension of the reason ing leading to Eqs. 6-47 and 6-48, the typical se<;tion responses to the force described by Eq. 6-49a are h(l) = _1_

b

fOO

Hu{iw)G(lw)e'W' dw

(6-50)

f "" H.il.iw)G( iw)e""" dw

(6-51)

K.b - ""


- <0>

Integrals s uch as those in Egs. 6-50 and 6-5 1 will exist if the one in Eg. 6-49a does , for norma lly H (iw) = O( I/lwl) or higher at the limits. However, because of difficulties with both the convergence of Eq. 6-49 and evaluation of integrals, relations like 6-50 and 6-51 are rarely useful for direct solution of practical problems. One exception involves random inputs, described under (e) below. When the mechanical admittance is known, these relations are often employed to find the response to some elementary input, such as a step or impulse function. The lalter then forms the basis for analyzing more complicated inputs, as will now be discussed. (b) Transient foreing; sol ulwn by Laplace transfonna tion

Returning to Eqs. 6--35, we consider a force F(t ) initially applied to the typical section at 1 = O. At this starting instan t the system is either quiescent o r in a known state, characterized by a set of initial conditioIlS on h, h, a, and Gi. The concept of Laplace transformation, defined in the case of F(r) by"

.!l'{F(/)}

=

F(p)

=

f'

e-PIF(t) dt

(6-52)

proves valuable when computing responses. For the complete theo{y and for derivations of several fonnulas that will be exploited below, the reader is referred to Ref. 6-4, Churchill (Ref. 6-8), Doetsch (Ref. 6-9), or another of the excellent books on the subject. In this section we concentrate on zero initial conditions, since motion • Observe that, unless a dimensionless I is employed, thi~ transformation adds the dimension of time 10 any quantily on which it operates. The new dimension is dropped in the process of j""""iOlt Or return to the rul time domain. p has dimensions of inverse time. Similar co""""nts apply to Fourier tran:;formation.

l08

I' RIN CII' I.I':S 0 11 AI': ltO"LASTICITV

at I = 0 complicates the (malysis without contributi ng anyth ing of significance. Under these circumstances, th e transforma tio n of Eqs. 6-35 becomes

[

+ w.ol S.P~

S.P' ](I(P»)

,,[ p'

I«[l

+ w/]

(-I'{P»)

.j(p) =

(6-53)

F(p)d

(The argument p of each dependent variable is always written explicitly, to distinguish from the complex amplitudes of simple harmonic quantities, which were also identified by the bar.) In terms of previously defined dimensionless quantities, Eqs. 6-53 have the algebraic solutions

[I + wp"]' +~:I:. p. z I'{p) br'w • • • h(p){b = K.b [1+p'][ p'] p' ,. w· w· [H -p']br." d (ru)'. +....!.p' I'{p) w. r/ w." .j(p) "" [Hp'-][ 1 +wp']- -
(6-54)

<' -I+-- ~

«

Wh

W. •

-

l

-....!

3

h

r'w'~' ••

<

(6-55)

r~wtw' • • h

There is a manifest parallelism with Eqs. 6-40 and 6-41, illustrating the general result that mechanical admittances can be found by replacing p with iw in the Laplace transform of the corresponding transient inputoutput relationsllip (under zero initial conditions). Moreover, the roots of the denominator polynomial in p. which are ±iw! and ±iw., according to Eqs. 6-44, reveal by their pure :imaginary character that the typical section is undamped and neutrally stable. The bracketed coefficients on the right-band sides of Eqs. 6--54 and 6---55 are closely connected with the responses to two elemen tary types of input; (1) The step function

(6---56) (a dimensionless quantity). The Lap!ace transform of this function is 1 2"{I(t)} =P

(6-57)

so that each of the two responses has as its transform the prod uct of Ifp by the corresponding bracket.

1'1II; TV I'I CAI , SllCl'[ON

(2) 71/C

IIlIil

lOll

imp/II,re fill/c//oll ur Dirac de/Ill f'(1) ... J(t)

K.b

(6--58a)

6(1) being defined to have an infin ite peak, with unit area un derneath, centered arou nd I = 0 : (6-58b)

eXt) = 0,

(6--58c)

(dimens io ns of inverse time), Slnce the Laplace transfonn is just unity, Z{Cl (t )} = I (6--59) the brackets in Eqs, 6-54 and 6-55 are themselves the transformed respon ses, The name indicial admittance is customarily applied to the step-function response, its symbol being Adt), where i and j identify the output and input, respectively, By means of expansions in partial fractions, by direct application of transform tables, or by resort to tbe general inversion formu la fo r the Laplace transformation (Ref, 6--4, etc.), we can find the indicial admittances of the typical section:

(6-60)

(6-{i I)

110

I'JUNCII',"".s 01'

i\ElW~:Li\S'I' JCITY

Eat'il of these equations is arrangclI with the so-called pfrmmwlIl part (particular solution) first, followed by the Irallsielll or homogcncou .~ solution. The uni, impulse responses h,!I) arc determined either by im'erting the foregoingp-functions without the I Ip factors, or by observing that since their transforms differ only by the derivative operator p from the A,,{r), he,) = dA d /) "

(6-62)

dl

The resulting forms are

(6-63)

w.~ . \ - "- ' [ l+:r. -b - , · smw,l J d ro, W,

(<>-64)

The indicial and unit impulse responses of the bending degree of freedom are plotted in Fig. 6-7 for the same set of parameters as Fig. 6-6. In practical cases damping would, of course, bring about the gradual decay of these motions. In view of an important transform known as the convolulion or fa/lUng 'heorem, (6-65)

we can find three very useful expressions fo r the response, starling from rest at I = 0, to an arbitrary time-dependent input. Using the torsional degree of freedom as an example, these so-called Duhamel integrals read

_ _1_ ('F(r)A.F'(t _ r) d.,.

K"b

Jo

_ _ I_[F(O)A,,~I)

Kkb

+ ('F'(r)A.p(1 In

- T)dr]

(6-66)

The elementary physical interpretation of these forms is discussed in Appendix C of Ref. \- 2, among o ther places. In sitllations where the

0

-0.5

"

,,' , ,

'\

- 1.0

~

- 1.5

;

£

1'-.. - 2.0

*-bt +0.5

J

0

'--

\ k1r ~,

r

Fi e.

~7.

\' 1

.

Indicial admittance A..,(/) and unit imp ulse response - h•.{t) for bend Ing

w,

degree of fr=:lom of t~ typical sc:ction in vacuum (Fig. 6-5), sho"ing supcrpOlii(ion of the w, and w. contribut ions. D imensionless parame ter,: are d/b .. O. "'~ _ 0.2, ' • .,. 0.5, w.I"'~

.. 0.5.

driving force is not known in analytical for m, numerical, graphical, or analogue evaluations of Duhamel's in tegrals offer co nvenient, systematic techniques for obtaining responses. FinaUy, we point out a connection between the mechanical and indicial admittances, which is especially useful when the sinusoidal behavior of

111

J'RI NC II'I.~:S

Oil

A~;ROll I.As·n CIT\'

Ihe syslem is readily available. As shown in books on Fo urier integrals, tire step function can be wrillen 1(1)

f'"

1 e''''-.-- dm 27r -'" IW

z: -

(6--67)

where convergence is ensured by requiring the integration path to make a small semicircular loop below the origin. Therefore, analogously with Eqs. 6-50 and 6-51, the step function response must be / -_ -[ A " ()

f'"' H,tiw)e''''' d w

(6--68) 2.,. - '" iw Again in £q. 6--68 we must be careful of the pole at W = 0, but th is inconvenience can be avoided by adding and subtracting in the integrand: A "r ()

- H;'{O)f '" e;"'d _ - w 21T

~

_.,

H,,{O) \ (t)

iw

+ -2[... f -'"""[H,,{iw) -;(0 Hq(O)] e''''dW

+.!. f'"

2.,. - .,

[H,liw)

~

H,,{O)]e'''' dw

(6--69a)

,00

By means of the substitution H(iw) = R(w) + iT(oo) and full exploitation of conjugate properties in Eq. 6-69a. an extensive simplification is possible, resulting in A Jt) = !R,,{O)

+ 1. i"[R,/CO) sin wi + Tdoo) cos WI] 1T

Q

co

dn.>,

(I 2 0+)

W

(6--69b)

Equation 6-62 leads us to the even simpler result (I 2 0+) (6-70) The last two formulas are helpful for both analytical evaluations and numerical approximations. (c) Random forcing

We close this section with a short introduction to the effects of ralldom iI/pUIS. F(r) is said to be random when it varies irregularly and unpredictably. so that it can be defined only by certain probabilities or average values and manipulated only by the too ls of statistical theory. A full exposition of the subject of probability (c~. Ref. 6-10. fo r instance) is beyond our scope and unnecessary for our purposes. As a matter of fact, cenain semi-intuitive concepts prove wholly adequate for aeroelastic applications, alo ng with a few formulas relating them, whose rigorous

proofs, under the actual condi ti ons where they are used,

c~ceed

th e

3uthol'S' mathematical com prehe nsio n. F(I) can be fully defined by giving its probability density, the fraction of

observations which result in an arbitrarily chose n sample exceeding any given numerical value; this is often assumed to have a si mple exponential form known as the Gaussiall or IIQrmal distribu[joll (Refs. 6-10, 6-11). There arc, however, several other properties that are easier to measure or estimate and which are usually adequate for structural design purposes, etc. One of these is the 11th moment or 11th mean value F- = lim _1 JT F\I ) dt T-",

2T

(6-71)

- T

2T is an interval of observation, which in practice shou ld be chosen long enough [or the averaged F- to settle down to a clearly defined value. A slalionary random process is one having P' independent of where the time origin is chosen; all random variables treated in this book are assllmed stationary. Wh ere possible, the origin of F is selected so as to make the first moment or arithmetic mean F = O. The second moment or meansquare vallie -F2 = lim -1 F "(I) dt (6-72)

JT

2T - T provides a convenient measure of intensity and plays a central role in applications. The degree to which the magnitude of F at any instant depends on its preceding history is estimated by the autoco rrelation function 'P, whose double subscript is supposed to ide ntify the integrand, T~",

+ T) d/ (6--73) 2T - :r 9"FrtT) is an even function, because the stationary random statisticaL properties should he iodependent of direction in time; it falls to zero as T _ co because widely-separated observations are unrelated; and rpF.reCO) is obviously equal to the mean-square value. The Fourier tmnsfo rm of twice the autoco rrelation fllnction proves to have valuable mathematical properties and a reasonably dear physical interpretation: rpn{r) = lim _1 JT F(/)F(/ T ~«,


J"

'i'

'fF.reC")e -'''' d .. = -

- ~

r.

'fF,..{") cos ro .. dT

(6-74)

0

As shown in Appendi ... C of Ref. 1-2, among other places, we are led to !llF.reCW) by discovering that F(I) itself has a divergent transform, whereas the transform of the prodllct of F by itself (in the sense of lfF,..{T» is welldefined. $F,..{W) is proportional to the square of the magnitude of the

214

I'IU NC IPU!.S 01 ' AmtOELAS'J'I<; ITV

spectral componcnl of F hav ing rrcquency wa nd is therefore called power spcctral dellsity, This interpretation is clarified by inverting Eq.

J

6-74,

PF~7") =! '" eI" '$l<',p(w) dw

(6-75)

2 -,

and selting 7" = 0 10 obtain j"l

=!2 JO) $ ]; ....(w) dw = Jr'"$.b',p(w) dw

(6-76)

o

- 00

Thus the lotal power or intensity is, in a sense, the integrated sum of all its spectral components. The wallc analyzer is a standard item of electronic eqUipment dcsigued to make a direct determination of IIln {w) for any suitably random electrical signal. To illustrate the utili ty of the power spectral concept, let us consider a stationary random force applied to the typica l section, The outputs h aod « will also be random aud will possess autocorrelation functions (h.~,
Eq.6-66

«(I) = _1_

J' F(T)h~~t

_ ..) d.. (6-77) - '" where the lower limit is now - rc because the process must have been going on a long time. The unit impulse response must vanish when its argument is negative, 50 the upper limit in Eq. 6-77 may be replaced by co. Making the change of variable t' = t - T, K~b

a.(I) = -

, J"

F (t - t')h,...{t') dt'

Kh b -'" Now lei us compute the autocorrelation function of the output, 'P,.(")'" lim

1- JT :x(t),,(1 + 7") dt

T ~", 2T

- ~.

JT- T J" SOl F(I -

+ .. -

t'')

~ (K~b)2 1 J" J'" ".~f')ll.Af")llim 1- JT F(I T- ", 2T -T

I')

=

I

lim _,

(K~b)Z T_..,2 T

x

hd{I')h.~.(t")

- 0>

x

~

(6-78)

F(I

1 (Khbl

+T

-0>

t')F(t

-.,

dt' dt" dt

_'"

-

I") dl} dt' dt"

J" J'" 11..p(I')h.~I")tpl<'...{T + - '" - '"

t' - J") dt' dl·

(6-79)

The substitution in the last line is justified by mak ing «( - ,') into a new integration variable and noting that tile va lue of (l'Tw is unaffected by a shift of the range of integration wherein it is calculated. Equatio n 6-19 has some direct interest, but a much simpler relation is uncovered when the autocorrelation functions are replaced by their Fourier transforms (cf. Eq.6--75).

1

f'" (/)...(w)e"·" dO)

2 -.

~! 2 =

!

J. J. Jo> 4lFF{(~) -00

-0>

-'"

e''''h.,.,(I')e,.,,'h.p(t'')e-·'''· d,' dt' dw

(KAb)

J'" ¢JFJI.,~) ,I[J'" h.~t')e·"'· [J"" l

2 _., (KAb)

e,..

dt']

-0>

h.F.(t")e - ''''''

dt~]) dm

-'"

(6-80)

Each of the two integrals in brackets expresses the torsional response to a sinusoidal force of unit amplitude which has acted for a very long time, during which period the transien t would die out in the real physical system. Hence they are related to the mechanical admittance,

J"

iluA:t")e- '''' dl~ =

-Ol

=<

r"" e-i"l·h.~t") dt"

In

e-'' 'f . ,ei"'II~~t

- ..) dT =

H~~Jw)

(6-81a)

Equation 6-8la can also be derived by constructing the inverse of Eq. 6-68 and integrating by parts with respect to time. Similarly,

f'.,h~t')ei"l' dl' =

(6-8 1&)

H.p( - iw)

We substitute Eqs. 6-81 into Eq. 6-80,

{ "'GO>
e" '" dw

(6-82)

and observe that the equality of the two sides for aU values of .. is equivalent 10

Ib .{w) = IH~iW)1211n-(w) = I1,)F~UJ)/(K~W • (K/l.b)t IZ.~iUJW

(6-83)

Stated verbally, Eq. 6-83 asserts that the power spectral densities differ only by a factor which is the square of the magnitude of the mechanical

216

I' IU NCII'LES 0 11 AIlROUASTICITY

u

0_75

~ Jll If/{

0.50

~

0.25

0

.I ..;

I

, ../

II

II \

I~ 1'--. \V

~-

0.'

0.'

"

1\ L~

"

FII.6-8. Dimensionless input and output power .pectral densities ror torsional motion produced by applyin!; a random force to the typical section, as related by Eq. 6-83. System I"'ramct.rs aI1: d/b = 0, "'~ _ 0.2, r« _ 0.5, wJf1J~ = 0.5.

admittance. A similar relation can be written for any input-output pair. In Fig. 6---8, we illustrate Eq. 6--83 by presenting a hypothetical r.pF~ro)f(K,b)Z and showing how the spectrum is modified by passage through the two-degree-of-freedom system. Incidentally, Eq. 6----83 also applies when a simple harmonic rather than random input is involved. Then the spectral densities must be replaced by the squares of the amplitudes of a and E, as can be seen by multiplying Eq. 6-41 into its complex conjugate equation. When studying tbe linear system under random forcing, itis also possible to detennine cross-correlation funclions such as 'I'.~7)

= lim -l

T ~ ", 2T

IT-T(X(I + 7)F(I) dt

(6-84)

1'111'. TYI'ICAL Sf.C1·lON

211

The Fou rier transform of tp~ ", would be the cfoss-spectral dellsity (a complex quantity) !pol·(iw) = ! '1'.z;.(T)e-'''· dT (6-85)

f<

•

-<

Several useful re lations containiog these functions can be derived, such as

md

4l.,y( iw) = H.,y(iw) ¢lFp(w) K~b (Khh'!

(6-86a)

¢I_(w) = H.,y(-jw) q,.£~w)

(6-86b)

An excellent treatmcnt of engineering applications of all the foregoing material and much more on random processes will be found in the book by Laning and Battin (Ref. 6-11).

6-4 FORCED MOTION IN THE PRESENCE OF EXTERNAL LOADS DEPENDING ON THE MOTION Now we turn on the air in the wind tunncl which contains the typical section and expose the system to another assortment of time_varying inputs. The wing may be thought of as moving about a certain constant mean pOSition, determined as in Sec. 6-2 by the equilibrium of static loads due to initial incidence, camber, and weight. F ollowing standard practice in dynamic studies, the displacements 11(1) and aft) dealt with here are superimposed upon their static values ; time-independent terms have already been subtracted out of the equations of motion treated below. Three types of external load may act on the typical section: (I) the force F(t) pictured in Fig. 6-5; (2) aerodynamic lifts and moments due to gust-like disturbances in tbe airstream, identified by the superscript D; and (3) aerodynamic effects of the motions h(l) and <x(I) themselves, identified by the superscript M. In keeping with the general pattern of . linearity, the last are presumed to be linearly related to the displacements and/or their time derivatives of any order. With the foregoing conventions we may write the equations of motion

+ WAlk] + Sp. = s.Ji + [.[it + w.ta:] =

m[Ji

IP(t) _ LM(/)

(6-87a)

+ M.D(I) + MwM (/)

(6-87b)

- F(I) -

Fd

For the purpose of organizing a series of illustrative problems, we adopt the same order as in Sec. 6-3. The various concepts and techniques introduced there- in connection then with situations of negligible practical importance-will prove their worth as we analyze some mOle realistic aeroelastic phenomena.

2111

l'IUNClI'U';'' ' 01' AlmOEl ,AS'I'ICJTV

(a) Sinusoidul rorcing Two examples of steady-slale, simple harmonic rc~ponse will be investigated. This type of motion has special significance because, as is almost too well-known by aeroelasticians, the required acrodynamic theory has beeu developed and tabulated most thoroughly. First, we re-examine the oscillating concentrated force of amplitude P, described by Eq. 6--36. Once again Eqs. 6-37 and 6--38 defme the outputs, but now hand ii are unqucstionably complex constants. LD and M/ afe zero. The amplitudes of VII and kfMM receive linear contributions from h and Ii, as set for th in Chap. 4, but the coefficicnts in these relations are complex, transcendental functions of the Mach number /If and reduced frequency k = rvb/V. The following, conventional symbolism is used in this cbapter, consolidating Eqs. 4-123, 4-129, 4-124, and 4-130;

-~p"YSW2{L.i + [L~ l.;1l

=

(i- +a)Lh]i},

O:s:.:M < 1 (6-88)

i

2p"YSm2 {[L,. + iL..J + [L, + iLJi}, :: P"YSw 2 /[M h 2 \

M:lf =

-

I :s:.:

M

n- + a)L.] ~b

+ [M~ - (i + uXL~ + MJ + (! + U)2L~]lil, O :s:.: M < 1 l -2p"YSW"{[M 1

+

iMJ

~ + [Ma + iMJIi}'

1 :s:.: M

(6-89) Here a is the distance in scmichords by which the E.A. lies aft of the midcbord line. L _, L~, M~, and M ", are complex numbers independent of a. The supersoniccoefficients(L t + 14,), (~ + iLj.), etc., have been separated in to real and imaginary parts, but all of them except L1 and 4 do vary with the E.A. location (in a simple linear or quadratic fashion). This general choice of notation is neither hallowed nor especially desirable, but it has the advantage of broad usage throughout the United States and will continue to have such for some time to come. Explicit mathematical expressions for the aerodynamic coefficients will be reproduced only occasionally in this and subsequent chapters. The two-dimens ional forms arc presented in Chap. 4 and are fully tabulated. Equations 6--88 and 6---S9 are substituted, along with tbe otber simple hannonic quantities, into Eqs. 6--87, and the factor e'''' is canceled. Rearranging the homogeneous po rtions to the left-hand sides and resorting to matrix algebra, we obtain

'nu; n 'I'leAL

s

'

.

1

~

;::

I

" ••,

i-::: I -<:>

-

~

•

';j

~

'0

+ "

~

,

~

+ <:!: •

•

~

N

~

, "• ,

•

~

~

~

+

, "' + •, • • s• ~

N

I

~

I

-s ~ ~

'0

+•

~

~

•s If' ~

•

~

N

+

"',

SO

~

~

E

~

•

~

~. ~

-

~

•

:§

+• is

-, If' ~

•

~

N

+

•s

~

~.

I

,

~

S~:CJ'lON

2 1'

110

I' IUNCU'I.ES Of! AEHOlcI.AS·I'ICITY

Supersonic notation is used in Eqs. 6- 90, but the reader will have no difficulty in writing out the Su bsoll ie (."Ounlerp~rt of the square mat rix. TO'identify the individual aeroclastie operators on the left side, one need only separate the matrix into the sum of structural, inertial, and aerodynamic parts. For example, tbe matrie structural operator is

[9'J

~

mow.' [

0] [K. 0]

l.w~~

=

0

K.

(6-91)

Equation 6-91 holds for any motion of the typical section, but the other two operators are uniquely sinusoidal. In terms of dimensionless quantities which parallel those appearing in Eqg, 6-40 and 6-4 1, Eqs. 6-90 can be solved for the mechanical admittances modified by the presence oflhe ai rstream:

(6-92)

_(

m 2p",bS

)(~'r!I-(. m )[I+ (w')'J +[L,+IL,l) IW b\ 2p",bS IW

+( m )(w'II_( m )'+[M, +IMJ) _ __________ c"eo• b"S~~IW~\~-"2"•."b~S'--·-----------' a

(6-93)

where

" ~ ((2,:bS) [1 + (::)1- [L, + lLJ) x

((,,:bS) •.'[1 +

(:JJ -

[M,

+ lMJ)

_1_( t 2p",b S ),.+[L,+IL.J)(-( 2p", bS )'.+[M,+IM,]) m

m

(6-93a)

'n'm TVt'tCAL St,:CJ'tON

11t

The reduction to Eqs. 6-40 and 6-41 is read il y carried out after setting all aerodynamic terms equal to zero. A new pammc(er appears explicitly in Eqs. 6-92 and 6- 93; the mass ralio m/2p .,bS, which is a measu re of the relative inertias of the wing and a representative volume of air surrounding it; the symbol p is often used for this ratio, or for the same combination with 2 replaced bY1r/2. Altogether,Jour additional dimensionless quantities enter the picture when the wind is turned on. These are m/2p.,bS and three geometrical aerodynamic parameters M, k, and the E.A. location, Q. In a given experimental situation (e.g., the measurement of airloads by driving a two-dimensional wing in a tunnel), the ambient state of gas in the test section furnishes p., and the speed of sound VyRT "' . Knowledge of airspeed U then yields M; the driving frequency w determines k; and the inertial and geometrical properties of tbe model fix m/2p..,bS and the remaining parameters. In Fig. 6-9, the bending response is plotted versus w/w,,- at subsonic, sonic, and supersonic Mach numbers for a representative system. Here it is necessary to show both tne amplitude ratio and the phase angle rp. by which the (downward) displacement leads the (upwa rd) driving force, that is, (6-94) The static valuc of this angle is, of course, 180". We observe two effects of the airflow. It provides damping, rounding off the infinite resonance peaks of the system in vacuum; this damping is very small in the case of the peak near W., however, because the wiog mass is so large compared to the effective inertia of the air (m/2p",bS = 100). The second effect is a shift of the resonance peak locations, panicularly noticeab le in wI' as functioos of Mach number and dynamic pressure. In the present example the peaks move apart at supersonic speeds, but more commonly tbey move toward one another. The implications of such behavior with regard to stability is taken up in Sec. 6--6. For some purposes it is convenient to present the mechanical admittance in complex polar form. Figure 6-10 shows such a graph for the case M = 2. It bears Iepeating that, in principle, the availability of H.1" and HoP' implies the power to calculate response to any time-dcpendent F(l) at any airspeed. For a periodic but not sinusoidal force, the Fourier series Eqs. 6-47 and 6-48 are valid. Good convcrgence should be obtained whenever F(/) is not 100 jagged and the system is sufficiently stable. This technique is used in practice even for aperiodic inpu ts by app lying the same force repeatedly, but at long enough intervals to allow the decay of transients. Alternatively, the aperiodic input may be handled by a

222

1' II INC II'I£<; 011 AICI!O I·:I.AST IC I'I'\'

' .0

,1 " i

1.0

I

0.' f-- - - -+-- --".LO , ~-----+------~--~

,.,01-- -

I

M.

o wl_~

Fig. 6-9. Mechanical adminance H~lw, M ) relating translational displacement to appl itd force On the typica l section in an air.rucam at various Mach numbc1'$. The following sct of parameters was used ;n Eq. 6-92: dlb _ 0, z~ - 0, ~. - 0.5, w,Jw" ~ 0.5, m/2pd>S - 100, g = -0.2 (E.A. at 40 % chord), w~blg~ _ 1 (nceded 10 fix k). Note that the lack of inertia coupli ng implies "" _ w. and ro, = woo'

TIIH ·1'''I·,e .... L SEct'l ON

Fie.

~IO,

W

Complex polar plOI of 11., from VIf:. 6--9 for lhe: casc M _ 1.

combination of formulas like 6-69h and 6--66. As will be seen soon, however, il is somelimes easier 10 de t.e rmine indicia] admitlances directly rather than by numerical integration of sinusoidal data. As a second example. suppose that the concentrated force is removed. but that somehow we are ahle to genemte in the airstream a sinusoidal gust with a vertical velocity distribution given by the real part of

(6-,,) Since:t: is a down~mcoordinate fixed relative to the test.section walls. this gust is embedded in the moving gas and. as seen by an observer on the wing. varies simple harmonically with circular frequency {d. Chapter 4 discusses th e theory of two-dimensio nal airloads d ue to the sinusoidal gust and w ows that the dimensionless lift and moment are functions of Macb number and reduced frequency based on w. Hence we can write the

124

I'IUNCII'U>S 01' AI'.ROIll,ASTI CITY

disturbance loading LIJ(I) = qS

~ ei"' CLG(k, M)

M/(I) = qS(2b)

(6-96a)

wa ei"'ClHaCk, M)

(6-97a)

U

By reference to Eqs. 4-174, 4-175, 4-178, and 4-179, these coefficients can be cast in the following forms appropriate, for instance, to low subsonic or supersonic speeds;

2'1t{C(k)[JO(k) - iJ 1(k)] CLaCk,M)= (

,4e'" "M

2

io(M,k),

+

iJ1(k)},

M::::::O

M>l

1

[~ + ~J Cro
,jM2

(6-96b)

(6-97b) M> 1

1

Here the origin of x is assumed to 00' atmidchord; C(k) andf"are functions discussed in Chap. 4. It is of interest that the entire incompresSible-flow lift acts at the i-chord A.C. (a _ -i), regardless of the frequency. After the disappearance of transients, it is clear that the foregoing gust will produce a steady.state response of frequency ill. Once again Eqs. 6-37 and 6-38 apply, and the airloads caused by the motion are given by Eqs. 6-88 and 6-89. The appropriate substitutions in the equations of motion, including a choice of supenonic coefficients, lead to Eqs. 6-98, as shown on page 225. The solution of Eqs. 6-98 will nnt be written out, but it is obvious that there are two mechanical admjttances for the gust input, H.aCiw, M) =

njb "'ajU

H. a(iw, M) = - "-

>Natu

(6-99)

(6-100)

Except for the absence of the moment arm djb, each of these depends on the same set of parameters as the admittances }/~F and HoE in Eqs. 6-92 and 6-93. Plots like Figs. 6-9 and 6-10 could be constructed and would have the same general forms, since the only essential distinction between the concentrated force and tbe gust loading is tbat the latter is distributed along the wing chord. The tecbniques outlined in Secs. 6-3(a)

1'm ; Tv ,'le AI, SECI'ION

~,.c.

r;;;

0',

.-.., :l:

~

-

':j +

.s

~

~

"~

•

~

:;;

.. -. !.:J • + >: £ .; ~

,.•

,j'1~

+

'"n

~

~

N

" -:;r .N

+

~I

+

.r

~

•

~

~

•

~

N

+

" I

I

"

~• ~

" + >:•

~

"a "~

'" • ~

N

+

".a

"a

,•

I

~

~

- -, ~

",'

I

"

~

N,

~

225

226

I'R INCII'I,ES O li AEIl OEtAS'I'ICITV

and (b) also proviJc a basis f<J r calCllhlting re sponse to (lilY sort of gust structure embedded in the airstream, once the mechanical adm ittances are known. (b) Transient forcing

In view of the complexity of the aerodynamic lift and moment expressions (d. Chap. 4 or Sec. 6--5 of Ref. 1-2), the problem of arbitrary time-dependent forcing will not be worked through for the ful! range of Mach numbers. As an introductory example, we return to the concentrated force of Fig. 6--5, initiating t he motion from rest at t = 0, but we restrict the airspeed, U, so tha t incompressible-flow theory is valid. Under tbis limitation, the aerodynamic operators relating LJI(t) and M/f(l) to the displacements assume the rc!atively simple forms of Eqs. 4-170 aod 4-17 1: etc/) = ~ p"" bS[J.

+ "p"" US M. M(t) =

i

+ U
f.

bali]

,

d

- [h(o) • d,

+

(6--101) U(;t(a)

+ bet

p", bS[ baJ. - Ub(l - ali - b"\}

+ 1Tp"" UbS[i

-

a]

'i

d

- [heal

• d,

- a)
+ (1 )<J.]

+ Ua(a) + b(} -

x q>(s - 0") da

a)i(O")] (6-102)

Here s = UI/b is a dimensionless time measured in semichordlengths traveled after the start of motion; dots, however, indicate derivatives with respeet to physical time. 11'(.1) is the indicial admittance (of unit asymptote) which describes the circulatory lift build.up after a sudden cbange of incidence. This incompressible case is facilitated by the fact that only one such function appears, whe reas generally four separate indicial admittances are needed for lift and moment due to vertical translation and pitching. Turning to Eqs. 6-87, we set LD and Mv D equal to zero, insert Eqs. 6-101 and 6--102, and take the Laplace transform with respect to the dimensionless variable s, viz., F{frj =

L'" e- i • F(s) ds

(6- 103)

using the tilde to distinguish from the dim ensional p. The resulting matrix equation reads

~

.~

<5

-

~ ~ I

~

'"

~

I

""'.

~~

~

-

". a ~.

+

,~

,~

~

~ • +

'"= !!>

~

,~

•

~

-" ,

~

N

+,

~

I

,,10, ",' ~ ~

-;;;a

• +

~

,~

~

~

'"X

+ '"W + Sl~

~

~

•

•

,~

+ ~ 0; • '"W +

-'"'"

W

'" - ''"" 1:''"" '" 1:

"' '", ''"X ~

~

+

,~

'"

" I" ~

+

N

~

~

..s ~

-"-

~ •

~

+

'"'", ~

~

N

•

~

'",

~

I

" I" ~

~~

~

•I

~

w

N

+

llII

I'RINCII'LI'-S 0 1' AlmOm,ASTlCI'I'Y

In lenns of k~ = wa.b/V and other previously defined parameters, Eqs. ~104 have the following transformed solutions:

(6--105)

(6-106)

The common denominator determinant is

.~([

2m +1+ 2 7Tp", bS

x ([ 2m r.B 7Tp .,bS

+ [n -

.vi]t'+k.'(W')' 2m ) w. 7Tp..,bS

+ (l +

a) - (I

2 u )

+ (2a B -

+ 2a)<j;(j)Jp +

_ ([ 2m ". _ , _ (1 "'p", bS

!)45(p)]r

2m r /k. 2 ) "'p",bS

+ 2,)~p)]r)

X ([ 2mx. _ u + (1 - 2a)f(ji)]p

+ [l + 2if{ji)Jji) (6-107) 7Tp""bS Obviously Eqs. 6--105 aod 6--106 cannot be inverted except by numerical means, even when F(s) has an elementary functional form. But the process is reducible to manageable proportions if one introduces an exponential approximation for
O.165~.ou:lt

- 0.335c"-3<

(4-172a)

is the best known. This hns the tmnsrorm (6--108)

and thererore leads to a ratio or polyn-omials in p when inserted into Eqs. 6-105 through 6--107. Inversion is accomplished by the laborious procedure orfactoring the tenth-degree denominator and applying the method or partial fractions (Chap. VI, Ref. 6--4) to construct the corresponding series of time functions . A step input F(j) "'-' 11ft can thus be made to generate a pair of indicial admittances for hand oc, whose usefulness in determining the response to more general transient forcing is implicit in Eq.6-66. We now examine a problem where the calculations are mueb less unwieldy. The torsional degree or freedom is suppressed by means of an infinitely rigid spring K"" so that IX. "'" O. Regardless or the range or flight Mach number, we can then write (Eq. 4--156) for zero initial conditions J.:V(s) = 21TQSJ.' "'.[II(a)]p,(s - a) da oda b

(6-109)

The subscript· c, meaning "compressible," on the dimensionless indicial function distinguishes it from its incompressible counterpart in Eqs. 6--101 and 6-102. As an input, we create a transient gust structure Wo(1 - 3;fU) in the air stream, such that Wo(l) (or wJs)] denotes the illstantaneous upward velocity striking the leading edge at time t. According to Eq. 4--163, tbe gustinduced lift is LJs) = 27fqS r' !!.-[wa(O")]'P«S - q) da o da U

J

(6-110)

assuming the first encounter to occur at S = I = 0 and waCO) = O. 'P,(s) is the indicial function for a step or sharp-edged gust. Specializing Eq. 6--87a to the single degree of freedom, we get

+ 21TqSl'~ [h«(I'")]p,(s Qd~

b

a) da

= -27fQS1' ~[wG(a)] 'P.(s _

oda

U

a) da (6--111)

Clearly, the Laplace transform of the solution reads

h(ji) =

-piP.(P)wrlPJIV

b [m + if.(ji)Jr + (7fp",Sb m )k.' 1Tp.,Sb

(6--112)

:Lln

1'lil NC lI'l ,ES 011 M-:HOELASTICITY

where kA= m.b/V. Exponential approximations analogous to Eqs. 6- 1ill! permit fai rly straightforward inversions of Eq. 6-112. An especially interesting ease is that of high supersonic speeD. T hen the only aerodynamic force due to the motion is a lift directly opposite to the velocity h, which has the effect of putting a little "viscous" damping inlo Ihe system. By Eqs. 4-1 83 and 4-184, 'I'.(s) = -

2

nM

'I',(s) =

(6-113)

,

rrM

2

(6-114)

nM ' whe re the influence of airfoil thickness is neglected in Eq. 6-114 during tne interval up to $ = 2 while the gust front is enveloping the chord. Since
(6-[ 15)

Since k,.1 = wht and tbe radical in the argument of the sine is very nearly equal to unity, Eq. 6--115 shows that the typ ical section wi ll execute damped vibrations at close to its natural bending frequency. Inverting Eq. 6-112 by a double application of the convolution theorem, we obtai n

(6- 116)

TIlE 'fYI'KAL tmCI'ION

In IXl rl iclilar, for the response to a II(~) :;:::; A IS)

b

11'0

~Gl.U

= _

~h:Jrp-cdgcd

23 1

gust wa(,I') = 1t'0 1(,,'),

11'"

U

Xf'" [-(~)(~) J'''' (k.C' - .))1 - (;;:J).,(')d' • J("'p",Sb mk. j (6-11 7) 1 ",~M2

Equation 6-117 can be integrated without difficulty when Eq. 6-114 is inserted. The very slowly decaying response to such a unit step at AI = 5 is plotted in Fig. 6-11 for a typical section havi ng k. = 0.1 at a density ratio of 20. (c) Random forcing To illustrate the principles set forth in Sec. 6-3(c), we adopt the same physical system as in the last example (i.e., a. = 0) but expose it to a gust which is a stationary random function of time. In the buffeting problem, for instance, such a gust would represe nt the partially separated wake of a second lifting surface located upstream. Alternatively, it might be generated by a coarse screen or lattice hung across the test section. Espccially in the latter case, the air motion would resemble so-called homogeneoUJ isotropic IIIrbulence, in which all three fluctuating velocity

components, Ii, v, and w, are present and have similar statistical properties. If these components are generally small compared with V, as they must be to assure linearity, the u and v disturbances produce second-order airloads compared with IV = IVa and are neglected. We assume that the turbulence is embedded in the airstream and does not change appreciably in time during the interval needed to traverse one chordlength, so that IVa = WG(I - ;). Finally, tbe turbulent structure is taken to be two-

dimensional, witb no variation of 11'0 in the spanwise direction; this is an unrealistic assumption, unless the wingspan is very small compared to the typical scale of turbulent eddies. Und ex the prescribed conditions, Li.epmann (Ref. 6-12) suggests that the following autocorrelation function and power spectral density represent rather well tbe turbu lence which might be encountered:

,

'PoMT} = ~ e· u /~[l - (U{2).}T], (T:2 0)

(6-118)

(6-119)

1/ I'..

I I I

I I I I I

ti.,, I I

,Y ,,

,,

I

Ii\

,I

I

•

,

,, ,

,

-..i

I

I

I

,, ,, ,, I

,, ,, ,t I

I

..-"

,, ,

I

• ,I

•

1:),

I

,,

o0

I

a,

•

,

d

~ I

d I

,,

,

I

nit: TVt'tCA t,

S~crt ON

lJl

Here IVOI is the mearl-squnre value: and). is the imegraJ scale a/turbulence, which is n rough measure of the average size of eddies. In the earth's atmosphere, ). might bc of the order of 500 to 1,000 feet, whereas windtun ncl turbulence has a seale eq ual to or smaller than the vertical dimension of the test section. The reader can eonfirm that oa<w) satisfies Eq. 6---76. Equation 6---83 furnishes the most convenient way of calculating statistical properties of the output, here chosen to be the bending displacement h(t). The mechanical admittance relating IVa to II is found fro m the first member of Eq. 6-98, which reads, in the absence of twist,

{mb(w~e _

(

2

)

+ 2p..,bt Sw2 [Ll + iLJ} ~ = -q SC.ulk, M)

wa

(6---120)

U

Solving and reducing to dimensionless form, we obtain

(6-121)

By analogy with Eq. 6-83, the power spectral density of the bending motion

;,

T he mean-square displacement is -h' -

b'

L" •

... (w) dw

(6-123)

and the autocorrelation function, if desired, can be calculated from a formula resembling Eq. 6-75. The high-supersonic speed range again serves as a simple nu merical

:l1~

l'IlINClr'I.I' S or' ,\EliOELAS'I' rCIT\'

•

../'!

-

I\--

~ • • t~w) Wu (!l

'\

'" ,

,.,

V~.." Wu tv

,,

,

. t ..lw)

,

h~_

•

Fig. 6- r2, Input and Output pov.'C' spectmJ densities for bending motion of the ty pi'" r section exposed to iwtropic turbulenc". S]"tcm parameters are M _ 5, k, _ I, ",/2r",Sb - 4(}, and Alb _ 1.

example, Piston theory, with thicknc~s effects neglected, gives the following formulas for the aerodynamic coefficients: Ll = 0 L! = I fMk

C

(k M)=8sink

lfl

'

Mk

1

(6- 124)

J

(The vanishin g of CLG for k = mr occ urs because, according to piston theory, the sinusoidal gust produces no net lift when the chord is an in tegral multiple of its wavele ngth. ) A fter some algebra, we obtain

1+ 3(¥r

(l+en si n" k

x

M'(

m

4p", Sb

)'k'[4(k2_ k.~)' + (4PooSb)"kZ] HIM

(6-125)

Figure 6-12 ill ustrates the two power spectral dcnsities for a particular set of system parameters. Some closely related examples, based on incompressi ble-flow aerodynamic coefficients, will be fo und worked out in Sec. 10-7 of Ref. 1-2.

'1'111·: 'l'VI'tCAI. SIlCI'ION (>-5

ZJ5

EIGENVALUES; J)YNAMIC AEROELASTlC INSTABILITY (FLUITER) IN VARIOUS )'L1GHT SI'EED RANG.ES

The reader has no doubt already observed that the system consisting of a typical section in an airstream possesses dynamic eigenvalues. Mathematically they consist of values of the complex variable p which ea use the determinant of the SCJuare matrix in Eq. 6---104-0r its compressible-flow coun terpart- to vanish. Because r{!(p) and simi lar transformed indicia! admittances arc transcendental fUnct ions, there exists in theory an infinity of such roots. This should not surprise us, for the combined system, with air included, really has an infi nit e number of degrees of freedom. When computed in practice, the set of roots normally contains just two complexconjugate pairs, the remainder being rea l and negative. These pairs are the ones of principal interest to the ae roci asticia n, since they represent coupled, damped (or divergent) oscillations of thc wing acted on by airload s. In more complicated sy~ te ms, we nearly always discover co njugate pairs up to the number of degrees of freedom of the clastic body; and the vibrations associated with them ale called aaoe:lostic modes. In the case of the Iypical section, the two modes can be traced continuously as functions of airspeed down to U = 0, where they coalesce with thc coupk'd natural modes whose frequencies arc given by E'ls. 6--44. The most logical way of studying the dynamiC aeroelastic stability of a structure, and in partic ular of detcrmining its margin of safety for any given flight condition, would seem 10 be 10 calculatc a roolS locus of pas a function of airspeed and altimde. In enginCl:ring practice, however, this has not bcen the customary approach, for the very sou nd reason thaI more data are available on airloads resulting from simple harmonic motion. What is actually done is to eSlablish a srabilil)' bmmdurv, a eurvc or surface along which certain preassigned system parameters (cithe r natural or artificially introduced for the purpose) vary in such a way that stability is neutral, that is, p = ik for one of the roolS. In [he case or [he typIcal section, for instance, maximum use would be made of the available aerodynamic operators by assuming sinusoidal motion in advancc and tillding roots of the determina nt of the left -hand side in Eq. 6--90. Altemativcly, one might apply the Nyquist criterion of stability (Ref. 6-5) 10 a series of rcsponse curves like the one in Fig. 6- 10, as has been suggested by Dugundji (Ref. 6- 13; see also Chap. 10 of Ref. I-I). For OUf discussion oflhe simple bending-torsion system, we propose to lake up the less familiar roots locus first. It merit~ much more consideration for applications than it has received in the past, and some of the alternative methods may engender positively misleading conclusions.

l36

l'IH NCII'L1,;$ 01' AI!ROIll. ASTleITV

There nrc IWO airspeed regimes in which Ihe properties of the nonsinusoidal aeroelastic modes can be ascertained relatively easily: the low subsonic, where incompressible operators are valid, and the high supersonic. In the low subsonic case, the characterist ic equation to be solved is stated (6--126)

t:.(fr) = 0

where t:. is tbe typical section determinant, made dimensionless by division with a nonzero factor and expanded as in Eq. 6-107. Each complexconjugate pair of roots of Eq. 6--1 26 has the fonn

ft

= (P,R

± iw)! U

=

+Pp± ik

(6-127)

subscript R denoting the real part. T he damping rapo of the associated mode, in Ihe parlance of vibration engineering. is given by (6-128) Provided no roots of Eq. 6--126 lie to the right of the imaginary axis in the complex p-plane, the linear system is known to be stable with respect to all small motions. One or more positive real parts-in particular, a negative value of {- imply instability. This situation is generaUy agreed to be inacceptable on a vehicle in flight, despite nonlinear effects which might limit the amplitude of the actual os-cillation. The critical jluller -condition is defined to occur at the lowest speed U 1/", for fixed free-stream pressure and density, at which the damping ratio of any dynam.ic ae roelastic mode passes through zero. Both conjugate pairs of eigenvalues have ass ociated with them eigenfunctions of the form (6--129a) (6-129b)

when the complex representation is used for the vibration. Each of these is found, in the standard way, by substituting its eigenvalue in to one or the other of the homogeneous Eqs. 6-104, obtained by setting F(j) = 0, and solving for the amplitude ratio and phase angle between the two degrees of freedom. Since either <'t.o OJ ho may be specified arbitrarily, the mode shape is completely defined by

a.J: and '1'~.

This information

has only minor interest in stability studies, and modes are often not even calculated. Their variation with airspeed can throw some light on the

,• •

o

I•

...,

0

'1'1m TVI' ICA I, SECt'ION

-- -- -- ---',

''%

~-, ,

"~

~ -_ ..... '4»

-, ""

•

......~~

-

...... ~J)

.,

o••

T~i""

~

" , ,I '~ -' ,I , ,, ,, ,

0

02

••

0.'

I

[

... -

,.

+ 0.10

, \i I \

0

o

branch (r""f',.j l

Bending b... nch ("')

0." o.2

237

\ \

+0.20

+ 0.30

..

,I '" I I

l.o~./ u,

1.2

-"-

Fie. 6-ll. Dimensionless frequencies <J)I<J)~ and damping rotios C of the aeroela:;(lc modes of (he typical section, plotted ",", airspeed parameter I'k~ . UfbUJ~ for incompressible How. System paromtters are "'~ _ 0.024, r ~ = 0.62, wJwa = 0.227, m{2p",hS _ 38, a _ -0.36. (Adapted from Ref. 6-14.)

physical nature of flutter, however, an interpretation we shaH attempt in Sec. 6-6. According to Eqs. 6-107 and 6-126, the aforementioned properties of the acroelastic modes in incompressible flow depend on six dimensionless parameters characterizing the Iypical section and ai rstream: wJw~, x'" ra.' a, 2m/"'p",bS, and ka. == mjJ/U. T he first four of these descri be the wing and its su pporting mechanism. The mass ratio· (2m/"'p",bS) couptes it to the fluid state through the re lative densities. Finally l/k~ is a dimensionless speed that may be regarded as the independent variable parameter in a stability investigation. Another suitable combination for this purpose would be some multiple of "'p",bS/2mk,.2r/, which is proportional to dynamic pressure q divided by K~// and is tberefore a ratio bctweeo aerodynamic and structural stiffnesses. As an illustration of how the eigenvalues vary with speed, we have adapted some calculations of GolaDd and Lu ke (Ref. 6-14) in Figs. 6-13 and 6- 14. Allhough the objective of these authors was to analyze tbe • The reader will recall thai, in defining rna.. ratio, the 2f" i. often replaced by 1/2 or some other faCior.

238

I'UINCII'U:" 0 1' ,l.EIIOI':L,l.S',',CITY

'.00

.."

--""::.a~d'ngl - - .!."'h

- ", (t)

,I

- ,- "

':'-< Torsi"'.....

bra"ct,''''''

.,,

...."" ~

•

'" B'~dir; br.ndl (",)

•

br.""h (",I

•

i:1

•

'0J ,, -- • • /11-

/'

/

., ,.. ,., ,.. ,., ...

, •

""

Fig, 6-1 .., Dimensionless frequencies wfwa ~ nd dampi ng mtios ~ of the acroelast>c modes of the typical section, plotted V$. airsp=:! parameter' I /k~ '" Ulbw~ for incompressible How. System parameters are "'a _ 0.024,'« _ 0.62, OlJ(JJ~ _ 0.909, m/2 p ",bS _ 38, U _ - 0.36. (Adapted from Ref. 6---14.)

stability of a uniform wing with concentrated masses at its midspan and tips., their system is mathematically equivalent to a typical section with certain values ofthe parameters listed in the previous paragraph. We bave computed th ese as accurately as the original data (Ref. 6---1.5) permit, and they are: enumerated in the figu re captions, The scheme employed by GoJaud and Luke for finding tbe frequency and dampi ng ratio is based on Eq. 6-126, combined with the approximation (6--108) for ij(ft). They devised an iteration procedure:, startin g with an assumed value of the root fi in ij{ft), factoring the quartic polynomiaJJeft after thus replacing f(pl, using the r oots so obtained to rcfine-the first approximation, and repeating the steps to cOllvergence. The final results are inexact, mainly· because of the small difference between Eq. 6-108 and its transcendental counterpart, which is (6--1 30) • As described in Ref. 6---14, this calculation appears to destroy the conjugate property of the roots by int roducing compleJ; cocffieielliS in to the characteristic polynomial through the approximation (6-108). T he damping ratiM arc generany quite .maU, however, so the additional inaccuracy i. probably negligibl e.

'1'1 iii

TYI'ICAI. SECI'ION

239

K" being Ihe modified Bessel functi on of the second kind . No significant error is suspected in the present case. Figure 6- 13 shows an c!lample qualitatively resembling the largeraspect-ratio ca nti lcver wings and tail s.urfaces whose bending frequencies are ralher small fractions of the torsional. The aeroclastic mode marked "torsion branch," the one which merges as U -+ 0 with the higherfrequency coupled natural mode W:! involving mostly rotational vibration, e!lhibits a lower damping ratio than the "bending branch" at all airspeeds and proves to be the one which becomes unstable at a critical condition U,. = 1.153. bw. Figure 6-14 presents a case of nearly equal uncoupled frequencies such as might occur on an all-movable control surface, where "'~ refers 10 oscil1ation as a rigid body against whatever torsional restraint is attacbed to the 3)[is. Here the "bending" aeroelastic mode shows the instability. In both figures , the dampi ng. which is purely aerodynamic in origin, is seen to increase steadily with airspeed up to a point ro ughly 15 % below critical. Then one of the modes reveals its tendency 10 turn unstable, while thc othcr damping ratio simultaneo usly increases very rapidly. The frequencic"s of the acroelastic modes approach each other gradually but do not become equal at nutter, as has sometimes been hypothesized. In fact, further work on this qu estio n indicates that, had th e calculations been carried to higher val Ue:> of U/bw", they would converge onto tbe same horizontal asymptote from above and below. In Fig. 6-15 is given the roots locus itsclf, on the complex p-plane, of the data from Fig. 6-13. Here tbe instability of the "torsion branch" is clearly indicated by its crossing the imaginary axis from negati ve to positive PR' Incidentally, for any point on such a locus, the angle between the vertical and a line to tbat point from the origin has a sine equal· to ,; this angle is often treated as a meas.ure of th e degree of stability. The reader will note that this system has such a small value of static unbalance x~ that "'1 and "'2 essentially coincide with "'. and w~, respectively. It is also of interest to observe the damping ratio of the critical mode changing quite lapidly with U/b"'~ in the neighborhood of neutral stability, suggesting Ihat the onset of flutter would occur violently and any small excess of airspeed beyond U}<" would cause certain destruction. This effect is morc pronounced in Fig. 6--13 than Fig. 6--14, the latter wing being very lightly damped at all speeds below flutter. At higllsupersonic Mach numbers(M ~» I, MtJ <; I), the piston theory • This oomtruction requires, of course, th at the horizontal alld vertical scales be equal, which is not the case in Fig. 6-1~ wh~re the abscissa has been expanded for darity.

240

l'fUNClI'LI',S 0 1' ARROELAS1'ICITV

02 0

0.'

I

,

0'

2 Toroion

'00'

0."

t'---

0.""

cBendjng ~,

0 -0.020

-

0.Ql5

0.010

'.

-

000'

o

0.00>

fir. 6-15. Locus of coojugate roots of the charactecistic equation ArP) = 0 of tile typical section, plotted on the upper half of the cornplc~ plane of p _ p~ + ik, for the same parameters as in Fig. 6-13. l/k" varies along each e urve. (Note that horizontal and vertical scales are unequal.)

aerodynamic operators discussed in &:cs. 4-5(d) and 4-6(d) are applicable. and the aeroelastic modes of the typical section can once more be det.er~ mined with relative case. It is a simple matter to adapt Eqs. 4-139 through 4-144 to arbitrary time-dependent motion, substitute for LM(/) and M~M(/) in Eqs. 6-86, and arrive at the following dimensionless, homogeneous relations:

+((2p..,bS m ),'k'--'-[' +(Y+ ')MA .]).~ O • M 4 2b! ~

(6-131b)

Tim TYPICAl,

S~:Cfl ON

141

Here s = UI/b is Ihe lime variable from Sec. 6-4(b). Olherquunlilies have been defi!lcd in this chaptcr. except for A .. and M .,. which arc thc crosssectional area of the wing profile and its area moment about its leading edge, respectively. (The leading and trailing edges are assumed to be sharp-pointed.) Whe n tbe Laplace transformation is applied to Eqs. 6-131, we again get a characteristic equation like 6-126. Only a quartic in p now has to be factored for the exact eigenvalues, however, because all the coefficients in Eqs. 6--13 1 are constants. The process is sufficiently direct that it need not be elaborated. Clearly the eigenvalues and mode shapes are functions of the same basic set of dimensionless parameters as for incompressible flow, plus the Mach number M. When thickness effects are accounted for, two profile parameters MA.Jb 2 and MM../!? must be added; each of these will be directly proportional to Mr5 for a given geometry. Recently Zisfein and Frueh presented an illuminating study of typical section fiutter based on piston theory (Ref. 6--16). Figures 6--16 and 6-17 are replotted in dimensionless form from their report. The values of the pa rameters needed, after adopting their approximation that , -< I, are listed in the captions. In both these examples, the aeroelastic mode corresponding to the "bending branch" goes unstable. The resemblance between the damping cur ves and tbose of Fig. 6--14 indicates that, for this simplified system, the dimensionless character of the modes is rather insensitive to the range of M wherein th ey are calculated. Although the large number of pa rameters makes it dangerous to attempt broad generalizations, experience does show that the form of these curves is principally controlled by the mechanical properties of the struct ure. Thus we find, in two-degree-of-freedom cases involving a trailing edge flap, that stability reappears at an airspeed higher than the minimum Up. This leaves a finite range of fiutter, something which is not observed in the bending-torsion case. Wben there are three degrees of freedom, often two aeroelastic modes become critical atd ilferent speeds. Even more complicated behavior is naturally observed on the elaborate structures met in practice, especially when large concentrated masses are attached to a lifting surface. A multitude of examples will be found in the pUblished literature, and some are presented in subsequent chapters. Zisfein and Frueh discover two interesting properties of curves of frequency versus speed like those in Figs. 6--16 and 6--17. At the lower end of the llk~ -scale, small C being assumed, tbe two branches merge with a so-called base CIIrve formed by dropping first-derivative terms (i.e., eliminating all damping) from tbe equations of motion (6--131). When the C.G.

242

1'lliNClI 'l,ES 0 1' i\ I'.IIOm,i\STI CITY -0 2 - 0.

,

/ nd;n~ bfa ndl

,

,

/

Torsi"" bra

!~

+ 0.2

I I

"-

+0.3

,.,

-

0

• ,

,I

'""on b'~"clt

.

~ >:)(Of\~1 ", " I-" I .

I

o.'

- - - BaIt cu" e ~

,

o

1

0

I ,

" "

3~ /~O "L

"

"' "

Fi a:. 6-16. Dimensionless frequencic5 w/wa and damping ratios { of th~ aeroclastic modes of the typical section, plotted vs. I lk~ '"' Ulbw~ For high sUJl(:l"Sonic M . S)"'tem pa rameters are "'~ _ 0.2, r~ _ 0.5, w.lw" _ OA, m/2p",bS - 50, Q . . -0.2, thickness

ratio

~ .. 0, a",lbw" '" 2.325.

6-16.)

( Note that M = :" 2

j:::.J

(Adapted from Ref.

a

is at midchord ("'a = - a), Ref. 6-1 6 derives a biquadratic relation for this base curve,

Equation 6-1 J2 al so assumes Ihickncs. nllio /) = 0: it call be cxlclldcd (0 other e.G. location:; by replacing the factor a in tho de nominator with

a[1 - ;~:(l + :;)]. On the other hand, at high speeds beyond UFlbw... , both wjw,,-brancbcs are asymptotic 10 tile

sin"glc''--'7"ChC'' --" , __---;CC,

J I

Woo _

+

(1+:3)(1_:'::-) "

_ __ _

-

1

---'(~';-"",.·"I'~.'-')----

w«~(l

w."

+ x_) a

This expression simplifies considerably to 0)

_

J

w t +W2

h • (6-134) '" 2(1 _ r . 2(r.2) for the case x~ = -a treated in Ref. (6-16). Properties of the typical-section aeroelastic modes can, in principle, be computed for Macll numbers intermediate between the low subsonic of Figs. 6-13 through 6- 15 and the high supersonic of Figs. 6-16 and 6-17. No exact results of thi, sort have been published. As discussed in Sec. 4--6, four indicial functions would be n:qulred for lhecharacteristic equation in place of thc single rp(s) needed at 111 = O. These functions are known only numerically when AI < I, while for supersonic speeds they reach fixed limits within a finite time and sometimes have multiple extrema, thus rendering curve-fittings like Eq. 3-172a unsuitable or, at least, very difficult to achieve accurately. The best prospcct for future calculations on systems or all kinds probably lies in thc use oftlie analogue computer, with aerodynamic elements designed to approximate the vario us indicial or mechanical admittances by electronic circuitry. To find boundaries of ncutral stability for the typical se<:tion by means of sinusoidal aerodynamic ope rators, we must in some way determine . roots of the complex, algebraic equation·

aUm) ~ IIzp",bS m [, + (w.1(",.,]_ (L, + ;L,l) IW} ("~ x I m ,.'[1 + ("'ofJ - (M, + ;M,lj Izp",bS 1 1( 0

_I "':t:. _ [4 + iLJ)1 \2p", bS

mx. _ [M! \2p",bS

+ jM!]).., 0

(6-135)

• Supersonic symbols are employed h.~e for the lift and moment coefficients. The correspondi ng notation for suhsonic speeds is discussed in Sec. 6-4 (a) and ebcwh.ere in the book.

244

I" UNCII'LES 0 11 A"RORLASTICITY

, ,

-0.

-0

I

/

• bY

~~,~'1.

0

, I

, ,

+0.

+0

I Torsion I branch

C\

I I

\

,, , ,

''%,;

,

;rh

br..

,

0

,, \

:

ik:' .0

~.

~,~~

~ :

• ,o

• ____ B.,e curve

,I

"", J "

~.

30

40

50

b{;,. - --

Fie. ~17. Dimensionles.s froquency ratios """''' and damping mtios { of the neroelastic modes of the typical sr:ction, ploUed vs. Ilk" iiii Ufbw" for high sUl"'rsonic M. System parameters are the same as in Fig. 6-16, except forlower radius of gyration, f " _ 0.354. (Adapted from Ref. 6-16.)

which is obtained from Eg. 6-93a, the dimensionless denominator determinant of the homogeneous form of Eqs. 6-90. With airloads acting, the system is not conservative. Equation 6---135 constitutes an elementary example of a complex eigenvalue problem, a topic closely associated with the name of H. Wielandt (e.g., Ref. 3-8). This characteristic eq uation is essentially a pair of real, implicit relations among the eight parameters k, M, ",{w", ",.fro". :1:", ' ", a, and (m/2p ",bS); A. .,/b2 and M .. f1J3 must be added to this list when thickness effects are

TilE TVI'I CAL SECI'I ON

14S

introduced at supersonic speed. The quantities ())/())~, ()),J(J)~, !l:~, r., and (m/2p",bS) appear explicitly in Eg. 6- 135. so that the linear or quad ratic depen dence of the real and imaginary pa rts of the equation upon each of them may be inferred by inspection. The E.A. location a enters linearly into [~ + iL4! and [M1 + iM z!, and quadratically into [M J + iM.!; hence (6-135) is also quadratic in a. Closed-form solutions can be developo:d for any pair out of these six just mentioned. In contrast, all the aerodynamic coefficients are ~anscendental fUnctions of k and M, except under the Simplification of piston the ory. This mathematical situation is either convenient or very inconvenient, depending on whether one is making a parametric study of flutter or wishes to ascertain the critical speed of a particular wing in a particular flight condition. For the former case, k and M may be selected in advance, along with all but two of the other parameters, whereupon these two follow by elementary algebra from Eg. 6-135. A common instance involves picking density ratio m/2p""bS and frequency ratio wJw as the dependent parameters. With other quantities fixed, the imaginary part of Eq. 6-135 furnishes a linear relation between them, which is used to eliminate one from tae rea l pa rt of Eg. 6--1 35, leaving a quad ratic equation for the other. This process migh t be repo:aled at several values of k and A.{, yielding a neutral-stability surface on a threedimensional plot of critical M F versus m/2p",bS and a", /bw" (whic h eq uals w/w"Mk). Such a stability boundary has several uses. For example, a test of a certain wing model in a va riable Mach-number wind tunnel is described by a line, which may penetratc th e surface at one or more flutter points. The effect of varying altitude in the standard atmosphere on bending-torsion flutter is shown by a line, which is the intersection between the stability bounda ry and a cylindrical surface. The shape of this surface follows from the variation of p", with a", = v' yRTgc, in the atmosphere. and the surface generators are parallel to the M-axis. A few results of such parametric studies and citations to many more in tbe literature are described in Subsecs. (a), (b), and (c) that follow presently. To compute the critical speed and frequency of a typical section with given properties in air at an assigned the rmodynamic state (p '" anda "" given), trial -and-error or implicit solution is unavoidable. Thc quantit ies k and M are the unknowns, with wjJ/a",. the mass ratio, and the structural and inertial parameters specified. Unfortu nately. tbe complicated relationship between the aerodynamic coefficients k and M demands that these two be selected before Eq. 6--]35 can be solved. Two of tbe other quantities must therefore be aUowed to deviate artifiCially from their correct values, while a sufficient number of k -M combinations are tried to permit double interpolation to the tr ue critical point. Since a curve of M F versus altitude

is of'ten useful to the designer (at least those portions of it which do not turn out to be below sea level!), themostJogicaJchoiceofartificialunknowns would seem to be the density ratio and frequency ratio wJw, leading to a stability surface like the one mentioned two paragraphs above. Past practice in the United States has leaned, however, toward the employment of a fictitious structural damping coefficient. This procedure is treated further during the discussion of distributed-properly structures in Sec. 7-7, where more infonnation is also supplied about the influence of internaL friction on stability. (1/.) ElTects of some of the parameters In incompressible lIow

This case constitutes a major simplification, because M is eliminated as an argument of the transcendental airloads. The many incompressibleflow studies that have been published retain more than historical importance, because of the light they throw on how aeroelastic stability is affected by the mechanical parameters at all airspeeds. Significant examples whose results may be applied to the typical section are the following: the extensive figures ofTheodorsen and Garrick (Ref. 6-2), giving stability boundaries for bending-torsion, bending-flap, and torsion-flap fl utter; a second paper by these authors (Ref. 6--17) on the same system with all three degrees of freedom coupled ; Duncan and Griffith (Ref. 6--18); Duncan and Lyon (Re f. 6--19); van de Vooren and Greidanus (Refs. 6-20 and 6-21); and Falkner's work (Ref. 6-22) on the bending-flap case. We shall present here a few curves ada pted from Ref. 6--2. As in other early papers, Tbeodor&<:n and Garrick cboo&<: Uj..jbw" as the ordinate for their stability boundaries. On a plot of this quantity versus a second dimensionless parameter, with all other system properties held fixed, the area below the curve is identified as stable. Tills means that any combination of airspeed (for M2 -( I), semichord, and uncoupled torsional frequency which ensures U < UF puts the wing in a flutter-free region, and conversely. Some writers now use bwJUj.' as an incompressible-flow ordinate; this inversion simply m akes stable conditions faU above the boundary, but it is more consistent with various ordinates proportional to bwJa." which arc widely used in subsonic and supersonic studies. Figure 6-18 iUustrates the influence of frequency ratio w,Jw" for four values of the static unbalance. A little shading is added on the unstable side of each curve, as in other figures below. The most significant feat ure herc-one which is very famiiiar to aeroel asticians and appears repeated ly on such plotS- iS the tendency for instability to be much more prono unced in the neighborhood of wJw" = I, espc:.:iaUy for moderately aft C.G. locations. When pointing out this so-caned "resonance" phenomenon.

, Fig. &-18. Stability boundaries for the typical section in incompressible Row, shown on a plotofdimensionlcss fl utter speed U,{bw", \'S. frequency ratiowJw", for four values of static u nbala nce. O ther 5ystern parameters are (2mI"p",bS) '" 20, Q = -0.3, r", ... 0.5. (Reproduoxd from Ref. 6-2.)

we must add that reasonable amou nts of internal friction, such as might occur in actual wing structures, tend to ameliorate, although not to eliminate, the more severe stability minima. Reference 6-2 and otbers demonstrate this, and they also show the much less dramatic stabilizing effects of structural damping at lower frequency ratios. The curious dependence of U8lbw", on the relative density (2mfwP
148

I'RINCII'U;:S OJI AI,ROI!IAS'I'ICITV

Stable

Fig. 6-19. Stability boundaries for the typical section in incompressible flow, sho,,?\ illl a plot of dimensionless flutler speed UAbw,. vs. relative density (2ml,.,,,,bS) for four values of stati,;: unbalance. Other syilem parameters ~ w.I()j~ _ 0.707, " _ - 0.3, r,. _ 0.5. (Adapted from Ref. 6-2.)

we reproduce Figs. 6-20 and 6- 21 from Ref. 6-2; the abscissa is (i + a + X,.), the dimensionless distance by which the e.G. lies behind the steady-now aerodynamic center (A.e.) on the i-chord axis. Figure 6-20 contains points corresponding to four different E.A. positions. The fact tha t they fall very nearly on the same line proves, for the low values of W~/illa involved, that the instability is governed by the C.O.-A.c. offset. In fact, Theodorsen and Garrick suggest the empirical formula

'!L_ J( billa =

2m ) "'p",bS (l

'.'

+ 2(a + :c~)]

(6-136)

which is also plotted on the figure. Using quasi-steady aerodynamic operators, Dugundji (Ref. 6-26) has provided a firmer theoretical foundation for Eq. 6-136 and also stated more precisely the circumstances under whic h it may be expected to hold; in particular, it may prove to be unconservative when the C.G. fa lls behind the 5S or 60 % chord axis. Figure 6-21 refers to a higher frequency ratio. A little examination will show that these curves, which correspond to six E.A. locations between the ~ - and i-chord lines, would draw much closer together if replotted ve rsus :c. alone. Here it is the static unbalance itself tha t governs the instability. Larger positive values of :Ca are unfavorable, wit h stability

nm TV l'leAL SECJ'ION

149

I

(J+a+x .. ) -

Fig. 6-20. Stability boundary for the typical section in incomprwoible flow, shown on a plot of dimensionless flutter speed U,/I:>w« vs. dimensionless A.C.--c.G, offset (I + II + x,,) for four values of E.A. location. Other system parametelll aT!) m,Jw« - O. (2mf"p",bS) _ IO,~« _ 0.5. (Adapted from Ref. 6-2.)

I

~Il

(

+a+%",) _ _

Fig. 6-21. Stability boundarios for the typicu,J section in incompressible flow, .hown on a plot of dimensionl= flutter speed U,II:>w" I'S. dimensionless A,C.oC.G. offset (i + a + "'..J for six valur:s of E.A. location. Other system paramctelll are m,Jw" = 0.707, (2m/"P«i>S) _ 10, r .. _ 0.5. (Adapted from Ref. 6-2.)

250

1'lliNClI'l.ES 01' AI'.ROI,;!.ASnCI'I'V

luinima occurring ncar x. = 0.2. It is a simple illustration of thc principlc of mass-balancing that a e.G. ahcad of thc E.A. supprcsses fluttcr completely, alleasl for the forward E.A. positions. (b) Subsonic lind supersonic compressible flow Pammetric studies which arc applicable to typical section flutter in compressible flow arc more restricted both in number and scope than those published for the low-Mach-number regime. Four important e ~amples, all of them utilizing fully linearized aerodynamic operators, are the following: Garrick (Ref. 6--27) on the bending-torsion case at M = 0.7; Nelson and Berman (Ref. 6-28), who emphasize sonic speed but give a few curves for M's between 0 and 5; Garrick and Ru binow (Ref. 6-29) on the bending-torsion case in supersonic flow; and Woolston and Huckel (Ref. 6-30) on the bending-flap and torsion-flap cases between M = 0 and 2. Several other reports have been prepared, for limited distribution, by various industrial organizations. but tllese generally deal with finite wings and other systems having distributed properties. Mach number is the primary independent variable used for presenting most parametric data. The practice of showing stability boundaries on a plot of UFlbw~ versus M is regarded as undesirable, however, for tbe airspeed occurs in both ordinate and the abscissa. Following its introduction by Regier of NASA, the aforementioned ratio bwafa"" usually multiplied by some power of the relative density such as V2mj7Tp.,bS. is currently gaining wide acceptance as an ordinate. Figure 6---22, modifkd from Fig. 5 of Ref. 6-28, is such a graph. Here the boundaries are presented for a wing havi ng a low frequency ratio and an E.A. at midchord. The wide range of relative densities represented by tlle four curves might be achieved either by varying the air density in a wind tunnel or wi th a series of different models. The stable region lies above each boundary in Fig. 6---22, in the sense that a combi na tion of wing characteristics and ambient state of the gas (i.e., test-section condition or altitude in the standard atmosphere) which yields a numerically higher ordinate, at a given M, assures stability. Thus, the torsional stiffness necessary to avoid iluller is measured by the value of w~ required to bring a wing of given mass and geometry at least up to the boundary. Flight at a fixed ambient state corresponds to a certain horizontal straight line; no Huller is expected if this line faUs entirc:1y within the stable zone for all Mach numbers of interest. The common tendency of lifting surfaces to be especially critical from the aeroelastic standpoint in the transonic range is typified by tlJe sharp peale near M = I in the curve for the highest relative density. The behavior of the boundaries at tbe higher supersonic Mach nu mbers can be

., Stable

,

/~.IOO D

1-\\.

,,

"

"

,

,

JAI-

D

,., Unst.b50

,, , "

"

"

"

M_

"

"

Fir. 6-11. Stability bOlmdarie$ (or the typical section in compressible flow, shown on a plot of

bw~ ~ II", ,J;;:p:bS

>'$.

Mach number M for four values of relative density. Other

system pnameters are w.JfJ)~ ... 0, II

...

0, x" _ 0.2, r~ ... 0.5. (Adapted from Ref. 6-28.)

misleading, because of the destabilizing influence of profile thickness, which is not accounted for here but is di scussed under Subsec. (c) presently. An incidental feature that renders plots like Fig. 6--22 helpfu l to designers is that flight ofa parricular wing at constant dynamic pressureq is described by a straight line through the origin of coordinates. The reader will have no trouble convincing himself that the slope of such a line is inversely proportional to Vq. Figure 6-23, also adapted from Ref. 6-28, shows the inftucnce of frequency ralio (J),J{J)~ al five preassigDed values of M between 0.7 and S.

6

'\

,

Stable

~M _ 5,O

• ,

,

VI

~

'\ 1.0~

"-

00

0.'

\,\28 ~' ~

Unstable

1

J?!£

.,.. 2.0

o.

~

~

1.6

fig. 6-23. Stability boundaries for the typical section in compressible flow, !ihown on

a plot of -hw. a~

- -

~p~bS

'/1;,

rrcquc[J\;Y

.•• nhO - for five values of Mach number M. w~

Other system parameters arc (lm"rp.,bS) _ 10, a _ 0, r", = 0,2,

r~

.. 0.5.

As in Fig. 6--18, the undesirability of ()J.JfJ)~ ~ L is evident especially from the high peak when M = 5. (c) High supersonic speeds; the importance of thickness

The high-M simplification of piston theory has naturally produced a spate of analytical flutter studies, of which Refs. 6--31 through 6--34 are representative. Chawla (Ref. 6-31) and Morgan, Runyan, and Huckel (Ref. 6--32) treat bending-torsion instability of the typical section. The extensive tables of Weatheri!l and Zartarian (Ref. 6-33) cover all three binary cases ass ociated with translation, rotation. and the trailing-edge flap, but most of their data also refer to the bending-torsion type; only a single profile shape-the symmetrical double wedge-is considered. Reference 6- 34 undertakes to interpret the results of Ref. 6-33 and present them graphically. Elementary expressions for the stability boundary in many forms can be derived by manipulating the characteristic determinant of Eqs. 6-131, after it has been specialized to simple harmonic

nm

TV I' ICAI. SECI'ION

lSJ

motion through the substitution dIlls ... ik. The wing analyzed in Refs. 6- 33 and 6- 34 serves us a sllitable illustration; for the double·wedge prolile of thickness ratio o'i , the cross-se<:tional area is 2b zo'i, and M

A

.. =Mo'i

2b'

MM"=2Mo b'

(6- 137)

(6-138)

We then obtain the following fonnulas for dimensionless critical speed and frequency ratio; Up

hw. = M(2e:bS)

JX x

MCP:bS)l [-, - (r; 'H x [(::jx -IJ+ ".} + [r; 1 M oT- ~

(6- 139)

where

- (w.)' -

x~

ro

With thickness effects included, the eigenvalues are now seen to be functions of only six distinct parameters 1IJ~/w~, x«, r~, 11, Mo, and M(m!2p ",bS), the Mach number and relative de nsity having coalesced. While this is a great convenience, it renders inefficient Regier's parameter, which was used as the ordinate of Figs. 6-22 and 6-23. There is, however,

~4

I'IU NCII'U:S 0 11 AEHOm..AS'I'I CITV

a possibility of constructing another ra lio with many of the same properties by inverting Eq. 6- 139 and mu ltiplying it through by M(mI2p~bS). This process yields

(6-141)

A considerable qu antity of helpful information can be deduced by closely examining Eqs. 6--141, 6--139, or any of the simplified versions that are found by specializing the system. Since much of this is done in Refs. 6--31 and 6--34, we content ourselves he re with reproducing a few figures that ex hibit Significant behavior of the bending-torsion ty pical section. When looking these over, the reader should bear in mind the limitation of piston theory to M > 2.S or thereabouts for unswept wings; in Ref. 6-32 some discussion appears on how results like these can be extended to lower supersonic Mach numbers, using van Dyke's theory. On the plots taken from Ref. 6--34, the profile thickness is introduced through the parameter M..2p",bS/m), which is just the quotient of M6 and M(mj2p ",bS) and is independent of the flight speed. Figure 6- 24 shows the influence of the hasic Mach-number-density parameter at three values of thickness ratio. Inc reasing 0 is defin itely destabilizing to this system, whose properties are quite characte ristic of the small, slraight wings of certain high.speed vehicles (except that the CO. lies somewhat farther aft than is normal). The ordinate va ries in a roughly parabolic fashion with M(m/2p",bS), ascan a lso be seen by studying Eq. 6-141. This fact means that an airpla ne or missile accelerating to higher M at a filled altitude can be expected ultimately to encounter aeroelastic instability, even though its surfaces are stiff enough 10 survive in the dangerous transonic range. Figure 6--25 presents the effects offreq uency ratio, including the familiar peaking near whlw~ = I. A new phenomenon here is the crossing over of the boundaries for the three different thickness ratios just to the right of th eir maxima. While the larger 6 is unfavorable at w.lw .. < 1, it has a small stabilizing tendency in the higher range, something of interest to the designer of all-movable controls, whose l')~'S are relatively low.

TIn: 'r'\ 'r'leA t,

~;r.:CI'ION

25$

ro' r---,---,---,---,----,---,

I---~

~V

St~b l~

, "'+--

'''''''' -----/-+--Un.table

"'1--+- -1r---+ - --t--+--'k,-----,,*oo,----,"'~---''''~----.'''~----''''!o----"'''' Fig,

~H.

Stabi lity boundaries for the typical section wi th double-wedge pror.tc

.

'•• (m)

(m)

- - VI. M - - . for aocording to pISton theory. shown on a plot of a., 2p.,bS 2p",bS Ih= values of Ihid:nesl_r~t io parameter. Other sy:stcm parametcl'!l are w.I"'~ .. 0.7, ,, _ O,,,~ = 0.2, r~ _ O.~. (Adapted from Ref. 6-34.)

Figures 6-26 and 6-27 bear a close rcsemblance to Figs. 6- 20 and 6- 21, in that thcy summarize the roles of the relative chordwise locations of A.C., E.A., and CG. for whlw~ - 0 and for th e less common hi ghe r w.lw~ = 0.7. respectively. Exactly as in incompressible flow, Fig. 6-26 presents ascability boundary which varies in direct proportion to the square rool of the A.C.-C.O. offset for a va riety of EA. locations; but here the influence of thickness va riations is also included. Since the A.C. is displaced forward by a distance proportional to Mo, tbis phenomenon is the principal explanation of the much greate r instability due to increased thickness ratio on wings which have both low w,Jw~ and CG:s in the vicinity of mid chord. Incidentally, this A.C-CG. offset can be shown to affcet flutte r in generally the same way al aU Mac h numbers from nea r zero

'",-,-----,---,---,

StatJ.It

Uostabl.

FiC'. 6-2.5. Stability

ing 10

boundar~.

pi~ton theory,

for Ihe typica l section with double-wedge profile accordg

shown on a plot of bw

" ""

(~)

vs. r"'quency ratio for ltorte

2p,.,bS

:~~, 0;" t~ic~~~i~la7::~~~~O~:; 'k~~~melers are M(ZP:bS) - 240, to high supersonic. Figure 6-27 suggests that the static unbalance x. tends to be the controlling factor at higher w,Jw~. In fact, Ref. 6-34 indicates a n extraordinary sensitivity of stability to this parameter when w.lw~ = I exactly, bllt the significance of this resull is questionable because no account is taken of the presence of structural friction. To emphasize the importance of thickness, we close Ihis section by copying Fig. 6--28 from Ref. 6-32. Here the stability boundary is plotted versus frequency ratio for four ai rfoils: a 4% thick symmetrical double wedge, a 4 % thick NACA 65AOO4 profile, a flat plate, and a single wedge with blunt trailing edge. Particularly at low w_fw~, the former two, which possess nearly coincident aerodynamic centers, have almost the same behavior and are mue h less stable than the latter two. The complete

" ,c ----r----,----,----,----, 50

" 0

.,I '

"

I

) '0

00

.I~

l>.

0.2

~(!6.!:!)

o-

0.1

[a+ ( _'_ ;

•

I

,~

UOI

----' ,~

0 Ull

.'1.:

..

V,

" , 30

}fC,'MOlnd

I J-v:

11:

0 -0.1

=

+ 0.1

21 )

)0 '

1

SO""

)

J;

-~ .

~e-l"')

" "I

OM"" 0.2

0'

-~

0.'

I) M6 + .. ,,11_ _ _

01 _0.1

f r

.., o

- 00

~o

_u

o

- II

00

,

- 0,)

:0:

~~

0

0.)

"

0.'

0.,

'0Fla:' 6-26. Stability boundary for 1M typio;al ~Iion wilh double-wed~e profile aocording \0 pilton theory. shown on a p lot of bwm (-"-) VI. dillllra (,om A.C. 10 e.O. iI .. 2p",bS in semkhord. for a variet)' of thickneu ratios and E.A. positions. Other

fJQJ",_ - 0, r~

paIllmc\ef$ arc /If (

M ) _ 240, 2p,.,bS - D.S. (AdajMcd from Ref, 6-34.) S)'Sltm

§

FI,. 6-17. Stability boundaries for 1M l)'pic~1 $eCtion with double·wedge profile according to piston theory. shown On

,,"0(")

a 1'101 of - - :;r--

.,.. w uic unbalance parameter

"1

.... -
system parameters are

M(2p~S)

_ O.S. (Adapred from R .... 6--34.)

-

:<

240, """..... _ 0.7, r,.

,

G i"

!l

25"

l'IU NCII'LI'-S 0 1' A,," OEI..AS TICITY

I

llJ

o~­ .

Fig. 6-28. Stabil ity boundaric:< accurding to piston t~ory for the typical section with four different profik shapes, shown on a plot of bw,.}o"" 'IS. frequency ratio. For the single wedS" and double wedge, ,j ~ 0.04. Othel" system parameters ar<: M _ 5, (m/2p",bS) _ 250, Q . . - 0.2, %~ _ 0. 2, r.. - 0.5. (Reproduced from Rer. 6-32.)

stability of the flat plate and single wedge at w,Jw~ = 0 is attribu ted to the fact that neither profile has any effect of thickness on A.C. position, the A.C.'s and c.G.'s falling togethe r at midc hord in both instances.

6"-6

THE PHYSICAL EXPLANATION OF FLlrnER

It has been said that the development of unsteady airload theory for OSCillating wings did more to promote misunderstandin g of the flutter phenomenon than any other factor. While this is an exaggeration, it is true that the insights which aeroelasticians have---{)r think they haveabout flutter are largely mathematical. An event of some importance is, therefore, the reappearance lately of papers such as those by Pines (Ref. 6-35), Dugu!,\dji (Ref. 6-26), and Crisp (Ref. 6-42), which try to contribute to the physical comprehension of this type of instability. One key step is the inspired return by these authors to quas i-steady aerodynamics when

TilE "'YI'ICAI.

S~x.. nON

1~\I

exa III i11 illg cases wit h 1110re tha n one degree of freedol11: Ihis represents a more respec ta ble approximation now thaI the success and accuracy of sueh theory at high supersonic speeds is recognized (cf. Rcf. 6- 16). Actually, quasi-steady theory was in common usc prior to 1935. The reader will find many references and interesting examples of its application in the classical study by Frazer and Duncan (Ref. 6- 36), and it lInderlies the discussions of flutter in more recent books by von Karman and Biot (Ref. I-II) and Rocard (Ref. 1-33). Because the reduced frequencies are so high, neither steady-state nor quasi-steady aerodynamic operators are sufficiently accurate for most qualltitatice flutter predictions during the final design process, in the same sense that they serve for calculating dynamic stability of aircraft. Whell seeking physical explanations, however, things are different. For this purpose, we should like to distinguish between systems with one degree of freedom and those with two or more, and we hazard the assertion that unsteady effects are essential to the understanding of the former, but not of the latter. Ld us try to clarify this poi nt by treati ng two elementary examples in detail. Consider, first, the purely rotational motion of the typical section (Fig. 2- 3 or 6- 5) obtained by restraining its axis with an infinitely stiff bending spring. Let the flow be incompressible. In the absence of any external driving force, the governing differential equation is 1,1:1.

+ K.p =

M,

(6- 142)

The pitching moment M. is given by Eq. 4-171 (wi th h = 0) for arbi trary smaU «(t), or by a complex function of k = wb/U when the 11«4 notation is used to describe a simple harmonic oscillation. In any event, the significant Features of the problem are preserved by writing Eq. 6- 142 in the approximate form [ 1"+::': 2

P h'S(!8 + a2)JI;I. - aM, Ii + [K - aMwJo:: = 0 ad "GO:: 00

(6-143)

Here l~ is augmented by the virtual moment of inertia of air accelerated with the wing, which is usually small and does not affect stability. The derivatives aM.lali and aM.lao:: are not constants but depend on the flow parameters and the time dependence of the rotation (although not its amplitude). If these parameters and the mode of motion are thought of as known, however, the derivatives take on fixed values; and Eq.6-143 describes a second-order, lumped-parameter system with a single degree of freedom.

hI tc rms of thc Laplacc-transform variublc p, Eg. 6- 143 would yield a characteristic polynomial aop2 + alp + O2, There are only two possible ways that instability can occur. Onc is by having thc coefficicnt of ox change fro m positive to ncgative, corresponding to the condition at s.: 0 in Routh's criterion (Ref. 6-4). This is just torsional divergence, the state of static instability where the negative "aerodynamic spring" about the E.A. overpowers K,.; it is fully discussed in Sec. 6-2(a). The second possibility is that of negative damping, wherein aM.loli changes from algebraically negative to positive (a l s.: 0 by Routh's criterion). It is the source of dynamic instability or flutter. Since no mechanical friction has been included, the negative damping must be entirely aerodynam ic in origin. It follows that the neutral stability boundary is defined, in part, by the necessary condition

Jm{M.l == 0

(6-144)

The circumstances under which Eg. 6-144 can be fulfilled in incompressible, potential flow have been anaLyzed by Smilg (Ref. 6-37) and Fuog (Ref. I - I), who interpret the work of Greidanus (Ref. 6-38). These authors find that t~e rotational axis must lie ahead of the ~-chord line (0 ::;;; -1) and the reduced frequency must be sufficiently small (k less than an absolute maximum of 0.0435). It is possible to explain qualitatively both the negative damping at low k and the existence of a minimum critical Hutter speed. Regarding the former qucstion, we imagine a slow, nearly sinusoidal variation of at, shown by Fig. 6-29 at four stages phased 90° apart. The principal part Lo of the lift due to the incremental angle of attack is in phase with ox and acts at the l-chord A.C., thus adding to the toC'OionaL spring K~ a further CQntribution proportional to the dynamic pressure q when a < -i. This inphase lift has associated with. it a ci rculation ro bound to the airfoil, whose sense is indicated in each panel of the figure. ro is Changing with time; and, because the total circulation in an incompressible, potential flow must remain constant, its alterations are reflected by t h.e continual appearance of countervortices of equal strength. These countervortices arc shed from the trailing edge and swept downstream at the airspeed U, giving rise to a wake vortex sheet, th e near portions of which arc pictured in Fig.6-29. Now, an out-of-phase loading at low k is caused by the angle of attack (or upwash) induced along the airfoil chord by these wake vortices. This upwash resembles a gust, embedded in the air and going downstream with speed U, since the wake itself moves in this way. Hence, as ~hown for gusts in Sec. 4-6(a) and in the references listed there, the additional lift ~ which it produces acts precisely at the !-chord. The induced upwash and L2 arc also sketched in Fig. 6-29 in the directions

Tim TV I'ICAL SEC I'ION

2lil

c1-77-'l-)-)-:t-0E-EWoke vortices

u

Lo~O

Llt

ro"'o I

!

E-(-E E-7C!-3-)-:t-7

I ! /bJa _ O,it< O

u

E-(-(- E -(-(- E -(- -:1-')-

Ie)

M.. imum negat". "

u "" I I I (d)o"O, it >O

Flit'. li-2t. Four positions of an airfoil oscilhlling about a fixed pitch axis in incom_ pWlsible flow at low reduced frequency •• howing the bound circulation r •. wake vortices, upwash. and lift L. induced by the wake. they would have allow red uced frequ ency, when the wake vortices mainly responsible for the upwash are those which left the trailing edge during the small fraction of a cycle just preceding the instant of observation. the remainder having been convected to large distances. The point of the discuss ion is as follows : when the rotation axis lies ahead of the i-chord. the moment due to L2 is in the same sense as the angular velocity 6; during more than half of eac h cycle o f oscillation and therefore does a net positive work per cycle on the wing. This transfer of energy is the origin of the negative damping. At higher values of k, the damping becomes

262

I'IUNCII'I.ES 01' AI.;IIOEI.ASTICITV

positivc for a complcx vllricty ofrClisons, among th cm thc facts that mo re cycles of wake affect the upwash, th e bound circula tion lags behind the angular position, and the center of pressure of the resultant lift begins to oscillate. The essence of the foregoing reasoning is the incl usion of unsteady effects. At least at subsonic speeds, quasi-steady theory is precisely what one arrives at when the influence of the wake is omitted entirely. To see how misleading are the results that one would obtain in the present example by overlooking the wake, we note that the incompressible quasisteady damping derivative is

= -rrp",UbIS[a(t _ ( o~.) a."

a)]

(6-145)

Equation 6--145 implies negative damping, independent of reduced frequency. about rotation axes between the i- and i-chord lines, at complete variance with the facts . It is of interest 10 me ntion here the anomalous low-speed flutter discovered hy Greidanus (Ref. 6--38), and Biot and Arnold (Ref. 6--39). Their discussions relate to bending-torsion instability with a vibration node near the i-c hord axis. In the limit, however, such motion is indistinguish. llble from single-degrce-of-freedom rotation about a = i- According to Eq. 4-171, the damping moment vanishes for just this particular value of a, so that the system is neutrally stable at all airspeeds. Returning to the discussion of forward axis lOcations, the onset of flutter is easy to understand in terms of the variation of oM.Ja& with k established above. At any values of U and q, the frequency w of the aeroeJastic mode is controlled primarily by the stiffness and inertia terms in Eq. 6-143. A good approximation is

I~+~p,ysH+ a~)

(6--146)

Since oM./o;t. is negative when a < --I and proportional to q, w starts from a value slightly below W u = -vi K,.fI.. Rt U = 0 and rises slowly as the speed is increased. As a consequence, k = wb/U falls steadily from its infinite still-air value. The damping stays positive unti[ k reaches the bounda ry wbere aM.Jali = Q, which defines the critical UFo At higher U and lowcr k, the oscillation becomes and remains divergent. IncidentaJ1y, a line of reasoning similar to the foregoing can be followed to prove that purely translational motion of this same system can never exhihit an instability, regardless of the Mach number range.

'rim TVI'I CA I, Sl<:cnON

263

As a second aW::mpl to visualizc the nutler mcehnnism, we take up the two.degree-of-frt:edom typ ica l section. For our aerodynamic operators, we adopt the same convention uscd in dynamic stability studies that the lift is determined by the instantaneQus angle of attack (I< + hlV) on a steady-state basis and acts at the aerodynamic center. The equations of motion for any Mach number are then the follo wing: (6-147a)

(6-147b)

All symbols in El:js. 6-147 have been defined previously, including e, which is the distance by wbich the E.A. lies behind the A.C.; thus, e = b[t + a] in subsonic flow, whereas according to piston the ory with thickness effects e = b [a

+

(Y; I)M~~J.

Avoiding unnecessary repetition, we merely observe that the eigenvalues and aeroelastic modes from Eqs. 6-147 are found by factoring a cbaracteristic polynomial of fourth degree in the Laplace transform varia ble p. For a given wing. the coefficients are constants with a parametric depend. cnce on q and possibly M. There are two oscillatory root pairs, each of the form p = Pn ± iw. Choosing a representative set of subsonic paramo eters, we plot in Figs. 6-30 through -6- 32 the dimensionless frequency and damping, amplitude ratio, and phase angle for each mode as functions ofreduced speed Ulbw~. The last two quantities are defined in Eqs. 6-129, the angle 'PA denoting the number of radians by which the downward dis· placement h leads the positive.nose-up rotation
164

,

+1

I'R INCII'I.I!:S Of' AEltOIlLASTICITV

Ttnion bronch

.>

(wI

---

+1 ,0

+0

• ,

--

•mlo

, ,

~ ,

-"

•0$

"'WM: ~.

--

(wlw,,)

Bo.o ing bnonch

•

+0 "

Q~

,

•O,W

0

0

,~

, ,

-0

-0

+0.12

,• , ,

,,

___ Torsion b.. rod> (P.. /w"J

., Bondi",

0

r'ncIl "

./

(P./OJ,,)

"'

"'

~

... u

0,'

, , , ,i ,

-

'''' -

0,"

U,

-

"'

>0

_ 0,17

FII. 6-30. Dimensionless frcquelX'y U)/U) ~ and damping pJw~ of the ae,oc)astie modes of the typical sect ion, estima ted using steady.,state aerodynamic operators and plotted \'S. reduced airspeed Ulbw~. System parameters arc "'~ .. 0.5. r~ _ O,~, w.lw~ - .O.S, (lm ''''p",bS) - 10,

~/b

- 0.4,


_ 2"..

Simultancously, w drops, because thc lift is applied at the i-chord aDd constitutes a negative "aerodynamic spring" on the torsional freedom with a "spring constant" proportional to the dynamic pressure. T be small advance in rp~ is attributable: to the lift due to A. Flutter sets in when the bending amplitude passes through zero, leaving a pure rotational oscillation around the E. A., on which no damping forces act according to the approximations underlying Eqs. 6-147. The origin of :nstability becomes dearer when we examine the energy

'1'1 m 1'Vt'ICAt, SliCn ON

16S

exchange belween Ihe nirstream and t he wing, The aerodynamic work: done on the system per cycle of oscilla tion is

w= -

f'·'· (_0

£l"{L}:Jt" (dh -d/) + dt

f.""'-0 :Jt" {M.}£l" (doc- d, ) dl

== W,.,+ W.

(6-148)

Here we must take real parts because of the nonlinearity of thi s operation on quantities expressed in the complex notation. If we neglect the small deviations from simple harmonic motion, it is easy to show that tbe torsi onal work W. is proportional to

Ib~ i qk cos 'f'1, and is done entirely

by the moment due to h, The lift work WA is proportional to

with the first term coming from the dephased oc-Jift and the second the pure damping experienced by any traostatio nal osciUatioo.

,,

,.,

,

,.o

0

• • • , 0 0

.1.,

0

\

r~d"l ",oc,

~ ,...., .~,

"

"

~I

!

• ••

0

I

0.4

~

l!

0

Oh

::'.--

r---"~ ,.

0.'

0

~ _ 0.87

FIe- 6-31. Dimensjonle5ll ampli tude ratio I "" !>,..I bel""",n the bending and torsional displacements of tbe aeroelasti<:: modes of t he system described in Fig. 6-30.

266

I'IU NC II'LES 0 1' AIt.1I.01C1,ASTI CITY

•

I I I

!

,•

I

I I

-,

~

0

.,,",

~ingbranth .If

Torsion

't

-,•

-.

i I I

Toroion brl~nch 0

r

02

0.'

-

0.'

~~

-•0.8 V, bw

1.0

0.87

FI.. 6-32. Phase angle '1'. by which the hending leads the lor.;ional displacement for the aerodastic modes of the system described in Fig. 6-30.

The two contributions to Ware ploUed in dimensionless form in Fig. 6-33. As the airspeed is first brought up from zero, both W~ and W extract energy and assure stability, since: the rotation about a rea rward axis generates both a damping force and moment due to h. These negative works are roughly proportional to U at first M

but they begin to drop off as the bending amplitude decreases and vanish simultaneously with it at Rutter. For U > U1<.' tbe effective rotation takes place about an axis ahead of the E.A., corresponding to a 1800 shift in <{JA. The II-moment then does positive work and dcstabilizes the motio n.

Tim TVI'le AL SI!C1'ION

267

A similnr ann lysis en n be made of the behavior of the bending aeroclastic mode. AI low speeds, this consists of rOlation about a line far forward of the leading edge. Thus the sta bility is controlled by the translational damping in W~, although th ere is a small positive work W~ due to the h-moment (see Fig. 6-34). This moment associated with the bending velocity also excites the torsional freedom, driving it at a frequency less Ihan w~ and therefo re producing a rotation in phase with the torque itself. One can reason that this reduces IhofbOCoI and causes the increasingly negative, or lagging, 'l'~. T hese phase and amplitude shifts are unable, however, completely to overpower the h-damping, so that this mode remains stable up 10 the flutter speed and beyond. Not every bending-torsion instability comes about in the manne r just described. In some cases, both the cbanges in mode shape and the variations of 'P_ with airspeed play an important role. When unsteady aerndynamic effects are accounted for, we find tbat W~ and W~ do not simultaneously vanish at the critical UFlbw~ but become equal in magrutude and opposite in sign. A qualitative description of one example of this type, in which the lower-frequency aeroelastic mode develops flutter because 'Pit. becomes positive and the lorsionalliftdoes work on the bending. basbeen attempted by Kassner andFingado(Ref.6-47; see also Ref. 6-48),

:~"

-0.0

-0.0 2

,

- 0.0

""

./

w. _"u,i",,,ta.1

"

/ w,

/

~Mbll ..",,, ••

0.2

D.8l

~o

~~ ..

-0"

"

~_

~!i",:j".i

/

'-

0.'

0.'

-"--

"'0

FIe. 6-Jl. Aerodynamic work done on the system of Fig. 6-30 during !he cycle of o!Cillation commc:ncing at f '" 0, ptoued ..... rcduDCd airopeed for the auoelastic mode associated with torsionol frequency w, .

2611

I'R INCII'LHi Oil A(;ROlCLASTICITV

0."

w.

nob'w,,'I;oI" 0

I, I,

\

'i

-OM

w

-- :"·'<-"'11'-1' -0.08

- 0.1

,

~")0 "-

w. -0.1

,

-0.20

0

.-.b ....

,,· I~l i

[\ ,,

"

0.'

.1L_

0.'

O~

)

'.0

", "'. Fig. 6-]4. Aerodynamic work done on the syslem of Fig. 6--30 during the cycle of oscillation <.:Ornrmndng at t _ O. ploued vs. reduced airspeed for the ucroela.stic mode anociated with bending frequency "',.

The reade r will find it an interesting exercise to undertake a physical explanation, along the foregoing lines, of flutter invoLving coupLing between bending and rotation of a trailing edge flap. This reasoning proves somewhat less complicated, and, if careful attention is given to the effects of the aerodynamic hinge moments, one can demonstrate the stabilization attainable by mass.balancing the flap . A significant conclusion which follows from aU such discussions is that changes of the acroelastic mode shapes have a major influence on how and where the instability of a multi.degree·of.freedom system sets in. We also believe that some light can be thrown on studies such as Ref. 6-49, wherein the alterations in the bending.torsion stability boundary are calculated which result fro m certain arbitrary modifications of various aerodynamic terms in the flutter equations. It is found, for instance, that the in·phase component of lift due to the torsional motion is often the airload 10 which stability is most sensitive. This discovery is not surprising in view of the example of the present section, where this lift causes the modal modification that leads to negative aerOdynamic damping. Another verification is implied in the explanation given by Ref. 6-4";'.

'rm: TVI'ICAL

S~:cnON

169

(a) Instability lind lhe mcrglng of naturlll frequcncies Adopting even more ex treme simplilications than those made in the foregoing discussion , Rocard (Ref, 1- 33, Sec, 119) and Pines (Ref, 6-35) have suggested that flutte r of a multi-degree-of-freedom system can be explained by discarding aerodynamic energy diss ipation entirely, Although we feel that something essential is lost through eliminating the effects of continuous variations with airspeed of the phase angle !Ph> which is one consequenceof this approximation, their conception is neverth eless intensely interesting. Consider the equations of motion that are obtained by dropping the terms involving hfU fro m t he right-hand sides of Eqs, 6-147. Only derivatives of even orders remain, and the resu lting characteristic polynomial must he a biquadratic, floP' + a~ + 04' In practice, the two r-roots lurn out to be either real and negative, corre sponding to undamped vibrations p = ± iw u P = ±iw" or complex conjugates. In fact, when plotted versus Ufbw~, the p-roots form the "base curve" shown in Fig, 6-30 or Fig. 6-16. In Fig, 6-35 we sketch a typical roots locus on the upper half of the complex plane. Evidently the instability is associated with a merging of the frequencies of the aeroelastic modes (in Rocard's French, auto-oscillation par confusion de deux jrequences propres), This coalesce nce must occ ur at the critical speed UF' because it follows from the realness of the coefficients 00, a2 , a4 that, at any U > UF , there arc two modes of equal frequency, one decaying and the other divergent. Instability by merging of frequ encies may also appear in an undamped system wit h more than two degrees of freedom. For example, exactty this phenomenon underlies Hedgepeth's very successful approx imate scheme

u.. u, ---j"---u>u, '" .. "'I (u :O)

"

Fla. 6-35. LOCtls of roots of the characteristic polynomial for the typical section wit hout aerodynamic damping; airspoed U increases in the direction of the arrows, (Lower ha lf of the locus is symmetrical with respect to the p..-uis.)

27U

I'RINCII'I. ES OF AlmOEI.AS'n CITV

for prcllicting panel nuttcr at the higher supersonic Mach numbers (Ref. 6--40; see also Chap, 8 of Ho ubolt's thesis, Ref. 6-43), It is, of course. necessary that the system be unconservative and capable of drawing on external energy sources. The lack of energy conservation is always betrayed by asymmetric coupling between the degrees of freedom, such as can be seen in the aerodynamic terms of Eqs. 6-147. Indeed, when a system is conservative and the pammeter which symmetrically couples two degrees of freedom is enlarged, the natuml frequencies associated with them are forced apart and can never merge. This principle is demonstrated for free vibrations of the typical section by reference to the kinetic and potential energies, which are (6-149)

"od U -jK I< h'+

~K • •a.~

(6-150)

When the structure is vibrating at either natural frequency W I' with modal amplitudes hOi and 11..01' the equality between the maximum values of T and U leads to the following expression for Wi~: f>.)2

,

Imagining

S~

tK.h o,'

=

!mh~,z

+ IK.Clo,2

+ tIcl1..o,2 + SA/hOi'

(1=1,2)

(6-151)

to be incre ased, we find for the rate of change of frequency

o(rull as....

h -11..0,

iK~hol

+

~K.a.o;2

·' Uml!o,~ + 1'.11..0/ + S.Clo1hol]2

(6-152)

It is easily proved for an initially positive S" tbat Clo1 and hgh corresponding

to the lower '"' ' are both. of tbe same sign, wbereas ~ and hM arc opposite. Therefore, added coupling tends to depress Wl and increase W2'· (b) Pines' approximate rules for bending-torsion flutter

Many useful things can be learned from close examination of the idea of instability by freq uency coalescence. For instance, Rocard is able to show that adding certain types of damping is always destabilizing, in the sense that it reduces the critical valne of the coupling parameter (Ref. 1- 33, Sec. 120; sec also Salaiin, Ref. 6-50). With such damping present, the frequencies do not merge by the time the stability boundary is reached. Putting the II-terms back into Eqs. 6-147 provides an illustration of this • To complete the proof, we must point out that ~wi')/ah " and a(w,')/a"" vanish, so that the frequmC)' is stationary with respect to the small Chang.. in mode shape that 11.1.0 occur. This is a special case of some importa nt theorem. of Rayleigh (ReL 6-41) regarding the elfects of system changes on nalural frequencies. A partial account is given in Sec. 13-3 of Ref. \-2.

Till; TVI'ICAI- S llCl'lON

27 1

surprising theorem, and it thus explains why l1utter oceurs to the left of the noses of the base curves in Figs. 6--1 6, 6- 17, aud 6--30. Very unfortunately, space limitations prevcnt our reprodu ci ng Rocard's demonstration in full. We also warn the reader that it is by no means universally applicable in aeroelasticity. Thus, Hedgepeth points out (Chaps. 7 and 8, Ref. 6--43) that damping of aerodynamic origin at high supersonic Mach numbers can increase the critical speed for a skin panel. In Ref. 6--35, Pines deduces a set of approximate rules governing typical-section flutter by studying the characteristic polynomial of the undamped system. With. the "·terms omitted from Eqs. 6-147 one finds that the (dimensionless) characteristic equation reads

+(rohr[l_qSbOCL ~J=O w.

K.

oa.

(6--153)

b

It is easy to see that Sialic instability comes about with the vanishing of the bracketed factor in the constant term, * which yieldS the dynamic pressure qJ) of torsional divergence, as in Eq. 6--11. Dynamic iostabilitycoincides with themcrgingofthe natural frequencies, the lest for this situation involving negative or zero values of the discriminant of Eq. 6--153, that is,

(1+ (;;:l·-P.'~·['.+ilr

-4[1- z·:l(w"r[l _ qSbaCI'~l ;5; O r~

w.

K. 017. b

(6--154)

Pines identifies this as a limitation on the magnitude of the dynamic pressure parameter (6--155) Equation 6--154 constitutcs a quadratic expression in Q. Sufficient conditions for instability are that Q be Tcal. positive, and lie between the roots of the quadratiC, (6--1560) • This is j ust Routh'sconditlon at ~ O. Act ually. the ab:se "e<:of 0 , and a, meanS that the system is never beller than neutrally stable.

I'RINCII'U;S 0 1' ,U :ROELASl'ICI'I'V

:t12

whcte

Equations 6-156 imply a range of dynamic pressure within which flutter might be observed. Furthermore, the coeflkient of Q~ and the constant term in Eq. 6-154 are both positive, so that Ql and Qt, if real, must have the same sign. This sign will be positive provided the coefficient of the first power of Q is negative, which leads to the condition - -, +2-.....!.. ,(wv'[ "'J 2 0 w " b b

01'+-+ ~

b

2

x

r.t

(6-157)

Additionally, Ql and Q 2 will be real on ly if the quantity under the radical in Eq. 6-156b is positive, that is,

[

( )'(

)J;" .

e ex. x. x. +--e -w. 1+ --,

b

b w.

b r.

(6-158)

Both of the inequalities (6-157) and (6-158) must be satisfied before dynamic instability can occ ur. These combined requirements generate the following conclusions (Ref. 6-35): (1) No flutter exists for e.G. positions forward of the E.A. (x~ < 0). (2) Flutter at some speed is possible for all frequency ratios if the

A.c. lies between the E.A. and the e.G.

Ix~l>

i).

(3) If t he. A.C. is forward of the E.A. (~ only if

>

(z~ > 0, ~ <

°

and

0,x~ > 0),flutter is possible (6-159)

'1'1 III 'I'V I'leAI. SEc.-n ON

(4) If th e A.C. is aft of the CG. possible only if '

(

x~

,

> 0, - < 0, b

flutter is

,

-%.

+ b

w, > ,-,.,--_-'-"" ()

(6-16&)

1;1[1+~:',J •

w.

~I > z~),

273

(6- 160b) It is not a difficult exercise to prove that these four statements follow

from Eqs. 6-157 and 6-158, if account is taken of Eq. 6-45. Reference 6-35 illustrates them pictorially and makes some other interesting deductions. One case of practical importance is that of small (w.JruJ~. The limiting zero value of frequency ratio mu,t be approached with care, because this removes an elastic constraint from the system and changes the total order of the differential equations. It is clear, however, that Eq. 6-l56b can be approximated by

(6- 161)

For instability,

['I'" + ~J must be positive along with

'1'"

itself, which means

that the CG. must fall behind the A.C. Flutter occurs only in the neighborhood of the dynam ic pressure parameter Q~ qSboCL=

K"

oa

1

['I'. + ~J

(6- 162)

When solved for the speed, Eq. 6-162 yields

u bw. =

J(

2m ) p", bS

'.'

a~L[ '1'" + ~J

(6-163)

which is reminiscent of the empirical formula (6-136) and shows the same effects of the various parameters. Since th e flutter frequency can also be shown to vanish, Pines correctly identifies the limiting instability at

1701

l'II.INCU'U,:g 01' AEII.OEI..ASTICITV

= 0 as nonoscilla tory dynamic divergcnce involving both tmnsilitionlii acceleration and rotation about the CG. lt must be relilized that all the results just stated lire based on rather extreme assumptions and cannot be accepted quantita tively. They are, Wh

however, very helpful when interp reting parametric data such as those of Refs. 6-2 and &-34. Moreover, ReI. 6-35 points out how these ideas can be carried over to assist in dealing with much more complicated syste ms and states, "the problem of the flutter design enginee r is tn establish methods which will prevent regions of frequency coincidence from occurring within the flight regime." Based on considerable experience, Pines recommends procedures for assuring this prevention.

(c) Necessary conditions for InstabiUty based on aerodynamic energy Input As with any sort of instability, supc:rcritical flutter exists because the total mechanical energy of the lifting surface is, on the average, increasing with time. When dissipative forces within the slructl.lre are negligible and the system is linear, this test is eq uivalen t to the statement that the work per cycle of oscillation done by the aerodynamic loads on t he surface is positive, as has already been discussed above in connection with the typical section. In the early works of Frazer and Duncan (Ref. 6-36), this idea was employed extensively for studying the stability of a particular type of semirigid wing with aileron. It has been elaborated in a number of more recent publications, among them those of Garrick (Ref. 6--44). Greidanus (Ref. 6-38 et al. ), Rott (Ref. 6-45), and Duncan (Ref. 6-46, a very general and elegant investigation). Crisp (Ref. 6-42) in troduces an ulTimate stabilily crilerion, or relationship be tween mode shape and reduced frequency, which must be fulfilled in a given configuration before flutter can exist. Some of Crisp's results regarding energy transfer to a vibrating wing, wh ich are esse ntially found ed on the use of quasi-steady aerodynamic operators, contri bute to the physical understanding we are seeking and will be reviewed here. Consider a linear system whose motion can be described in te rms ofa set of generalh ized coordinates q,(t) (e.g., ql = band 92 = a for the typical section). Crisp observes that, when the airloads involve only q;, q;, and if; multiplied by constant factors (this comprehends virtual inertias as well as the conventional quasi.steady effects). the equations of motion can be written in the matric notation

[Aj(q,)

+ [BI{q,) +

[C]{q,)

~

0

(6-164)

'1'1 m TVI'ICAI .. SECl'ION

17$

The same matrices o f coefficients appearing in Eqs. 6-164 form the basis of the following three quantities: T

~

IU, -.J[A]{;,)

(6-165) (6-166)

F= iL4i~{B){(M U = ! L q;~{C){ql}

(6--167)

'r and U are generalized kinetic and potential energies, including aerodynamic inertias and stiffnesses, respectively. F is similar to the dissipation junction of Rayleigh (Ref. 6--41). [AJ, [B), and [CJ are symmetrical when dealing with an elcmenlary mechanical system, but thc addition of the aerodynamic couplings spoils this symmetry in the case of the last two. Each of them can be decomposed into symmetric and skew-symmetric po rtions, for example,

[B)

=

{Btl

+

IB~J

(6-168)

Moreover, only {BII and {Ctl contribute to Fand U, in view of the properly of skew-symmetric matrices exemplified by the equation L q, -.-I[ CtJ(q,} = O. Under conditions of uniform flow at infinity and in the absence of heat transfer, all three matrices are made up of conslanls; but Ref. 6-42 considers the possibility that they migh t be functions of time, Ihus allowing, on a quasi-steady basis, for variations in Jl.ight speed and altitude or for aerothermoelastic modifications to the flexibility of the structure. It can then be shown that the set of Eqs. 6-164 reads {A(t)J(ii,}

+ ({A(t)] + {B(t )]){<j.} + {C(t»){q.} =

0

(6--169)

the dot over a matrix meaning that the time derivative of each element is implied. The total energy of the system· is (T + U) = E. By taking the derivative of this quantity, using Eq. 6---169 to eliminate {iii}' and applying the aforementioned property to drop out some skew-symmetric matrices, Crisp derives the following expression for the instantaneous rate at which ·work is being done by Ihe alr:

~~ ~

- L;, -.J([BJ

+ ! [AJ){;')

- L q,-.-I[CJ{ql}

+ I Lql~ [C1J{q,}

(6--170)

The first term on the right ofEq. 6-170 is a generalUed dissipation function, composed of the sum of - 2£ and an obvious effoct of the rate of change of • NOle that E il; nOI c:
inertia,

it is ncarly always negative and thererore stabilizing. T he s~cond term, con taining the skew-symmetric part or the aerodynamic stiffness, is the one which has the major controlling inftuence on stability. Its contribution to dE/dl may be positive or nega tive, depending on the mode shape and dynamic pressure; but in any event [Cal is a large quantity in aeroelastic systems. For instance, on the typical sectio n as described by Eqs. 6-147,

,.

~ qSOCL

0

(CJ

~

,.

_i qSoC1,

(6-171)

0

The significant role of [C2 1 is even clearer in uniform flight, when the square matrices are independent of time and Eq. 6-170 reduces to

dE"" -(2F

d,

+ L Ij;-.J[CJ{q,})

(6-172)

Several interesting conclusions regarding the influence of system properties on stability are deduced in Ref. 6-42 from Eqs. 6-170 and 6-172. T he reaqer will find it instructive 10 examine more closely the application to th e typical section, including the consequences of rapid variations in ambient gas state and airspeed, such as might be found in certain intermittent wind tunnels. For a structure in vib ration at a critical flutter boundary, one can state a necessary condition for equilibrium by requiring the net energy input per cycle to vanish: E

' · blO>

1' - 6

= -

I.20'"(2F + L Ij,-.J [CJ{q,}) dt

= 0

9

(6-173)

The generalized coordinates here have the forms

q, - Q. sin (WI

+ f/';),

(i=12 " · ··n) ,

(6-174)

Reference 6-42 shows that Eq. 6-1 73 is reducible to the dimensionless relation(6-175)

where R j - QJ QR' )"'J = «({" -
c;p

• Rigorou5 unsteady aerodynamic operators can b<: employed in Eq. 6-173.

TIm , 'VI'I CAL SIlCfION

177

of the symmetric pliTt of IB] and the skew-symmetric part of [CJ, respceth'ely. When binary flutter is being analyzed (/I = 2), Eq. 6-175 can be given the elementary graphical interpretation that all possible flutter conditions fall on a circle whose size and location depend on the aforementioned matTi:\ elements. Flutter may be prevented, in theory, by reducing the radius of this circle to zero. This faci forms the basis of one of Crisp's stability criteria, and in certain systems no flutter is possible at reduced frequencies greater than some limiting value. Other criteria can be constructed by examining the energy input to individual degrees of freedom. Applications of some of the foregoing ideas to Dutter of lowaspect-ratio wings will be found in Ref. 7-34.

REFERENCES 6--1. ibcodors<:n, T., Gtneral ThnJry of AuodyMmlc IlIJlabilily and 1M M echanism of FlUII~r, NACA Report 4%, 1935. 6--2. ibeodorsen, T., and I . E . Gamel<, MechtmiJm of RUlur, a 'J"MortliCtlI and Mpt,imtnlalfn",migallon "fthe Rutter Pr<Jbltm, NACA Report 685, 1940. 6--3. Roxbee-Cox, H., and A. G . Pugsley, Stability "fSlalic Equilibrium of Ekmicand Atr"dynomlc ACliollS on a Wing, Britisb AR .C. Reports and Memoranda 1059, 1934. 6--4. Gardner, M. F., and 1. L. Barnes, TraIlS/tillS in Lintar SYJlemS, John Wiley and Sons, New York. 1942. 6--5. Draper. C. S., W. McKay, and S. Lee!l, In.trumtnt Ene/nurlns, Vol, 2, McGraw_ HiD Book Company, New York, 1953. 6--6. Sneddon. I . N ., Fo~ritr TraMform., McGraw-Hill Book Company, New York, 1951. 6-7. CampbeU, G. A., and R. M. Foster, Fourier [nltf(ra/s/or Practical Applicllii/JIU, Tedtllieal Publication, Bell Telepbone S),stem, 193\. 6--$. Churchill , R. V., Modern OporaliollOl Morh ~marics /n E,,!{iMeri"IJ, McG raw-Hill Book Company, New York, t944. 6-9. Doetsch, G., Thtorit und Anwtnd,,'W de," Laplact·TrtJ.lIJ!o,mat;on, Dov<:r Publication!, New York, 1944. ·6--10. Feller, W., An lmroducliOJ! 10 Probability neory and TtJ Applica/iIJl1', Jabn Wiley and Sons, New York, 19.50. 6--11. Laning, J. H.,and R. H . Battin, RandomPro<:esJeJill AUlomatit; Con/rol, McGrawHill Book Company, New York, 1956. 6-12. Liepmann, H. W., "On the Application of Statistical Concepts to tbe Buffeting Problem," J. AHO. Scitnct., Vol. 19, No. 12, Dec. 1952, pp. 79J--800, 822. 6-13. -Dugundji , J., ,oA Nyquist Approacb to Flutter:' J. Aero. Soiences, Readen' Forum. Val. 19, No.6, June t952, pp. 422-423. 6--14. Goland, M., and Y. L. Luke, "A Study of tbe Bcnding-To~ion Aeroelastic Modes for Aircraft Wings ," J. 14"'0. Sciencts, Yol. 16, No.7, July 1~9, pp. 389396. . 6--15. Goland. M., and Y. L. Luke. "The: FluUerora Uniform Wing witb Tip Weigbts," J. Applied M~ chanlC$, Vol. 15, No. I, Marcl\I 948, pp. lJ-20.

27M

I'H: INCJI'LES 0 1' AlmOl;U.ST ICIT V

6- 16. Zisfcin. M. LI .. and F. J. Fruch. A SlIIdy "f Veltwl'y-"''''1'''·'or)~ 0111111'1118 ReI",i,m5hip. for Wi"J ",,,II',,,,c/ 8i,mry Sys, ..ms ill 1I(~h Supersonic nu",. USA F Office or Scientific Research Technical No!o 59_969, Oc!ober 1959. 6-17. Thcodorscn, T., and r. E. Garrick. F/mtu Ca/culoli""s ill Thr~~ Otgrus af Frudom, NACA Report 741, 1942. 6-18. Duncan. W. J., and C. L. T. Griffith, .,.,,~ Influence of Wing Taper on the Fluller of Camllroer Wi".;'>, British A..R.C Rcports and Memoranda 1869, 1939. 6-19. Duncan. W. J., and H. M. Lyon . Calculated F/e;,;ural-Torsional flutrer Charac/eristics of So"", TYI'IMI Call/lllmtr Wings, British A. R.C. Rcportli and Memoranda 1782, 1937. 6-20. van de Vooren, A. I.. and J. H. O~idanus , Diagrams of Critical Flwltrr Speed for Wing. of a Cu tain Standard Ty/M, Nationaal Luchtvaartlaboratorium, Amsterdam, Report V. 1297, 1946. 6-21. van de Vooren, A. I., Diagralm of flulter, Dioergencc and Ailer"" Rcw of V"r/alio,,~ In S~u(ru/ Parameter. '"e/uding Fluid DelISi'! on Ihe F/uller Spud of Lighl Uniform Camih..,~r Wings, NACA Technica l Note 2.558,1951. 6-24 . Henry, C. J., Jr., J. Dugundji, and H. Ashley, Aeroe/aslic Srability of Lif,ing Surfac~s in High.DeMlly FluiM, USAF Office of Scientific R.esearch Technical No!e 58·626, June 1958. 6-25. Abramson, H. N. , and W. H. Chu, A Discu..iOlr of the Fluller of Submerged Hydrofoil.. Southwest Research Ins!itute Technical Report I, Contract No. Nonr 2470 (00). August 1958. 6-26. Dugundji , J., ·'Effect of QuaSi-Steady Airforecs on Incompressible BendingTorsion Fluuer,' · J. Aero. Scienre~, Vol. 25, No.2, February 1958, pp. 119--121. 6-27. Garrick, I. E., IkRdil!~- Torsioll Flulter Cu/cu/ali()llS Modified by Sub.oorlir Compressibilily Correc/lon •. NACA Report 836, 1948. 6-28. Nelson. H. C., and 1. H. Berman, Calc/uO/loM Oil /hc Force. and Mom'nI.for all Osclllaling Wing_Ai/eroll Comblna/ion in Two- Dirn~lISiollal Potelllial Flow 01 SOllic Spoed, NACA Report 1128, 19B. 6-29. Garrick, I. E., and S. !. Rubinow, Fblller alld Osrillali"!r Airfora Ca/cuIOliOliS for all Airfoil in a Two-Dlmensionol S,,~rsonic Flow, NACA Report 846, 1946. 6-30. WooI
Till·: 'I'V I'r CAI , SEcn ON

T~'Ch /lical Report M"y 19SH. I'ines. S" "An £kmcntary f.xplan~tion of the Fluller Meehanism," I'roc. N31. Specia lists Meeting on Dynamics a nd Acroelas ticity, Insti tute of the Aero· nautical Sciences, Ft. Worth, T",a., November 1958, pp. 52- 58. Frazer, R. A., and W.l. Duncan, Th~ Fluller of Auopla~ e Wi"Cs, British A.R.C. Reports and Memoranda 1155, 1928 . Smilg, R, " The Instability of Pitching Oscillations of an Airfoil in Subsonic Incompressi ble i><Jtcnlial Flow," J , A~ro. SciMcts, Vol. 16. No. 11, Nov.:mbeT 1949, pp. 691-696. Grcidanus, J . R., " Low·Speed Fl utter," J. Aero. Sciences, Reade..' Forum, Vol. 16, No.2, February 1949, pp. 127- 128. Biot, M. A., and L. Arnold, "Low Speed Flutter and Its Physical Interpretation," J. Mro. Sci~nces, Vol. 15. No. 4, .... pril1948, pp. 232-236. Hedgepeth, J. M., " FluUer of Recta ngular Simply Supported Panels at High Supersonic Spcods," J. (lero. Sciencer, Vol. 24, No. 8, August 1957, pp. 563-573, 586. Strutt, J. W. (Baron Rayleigh), TIle TIIto.y of Sound, Vol. I, 2nd cd., Dover Pu blications, New York, 1945. Crisp, J. D. c., "The Equation of Energy Balance for Fluttering SystclIl'l with Some Applications in t he Supersonic Regime," J. AuolSpau Scimcu, Vol. 26, No. 11, November 1959, pp. 703-716, 738. Houbolt, J. C., A Study of S~ra/ Aerolhermotlarlk Problem. of Aircraft Slruelurcs In Hlgh,Spe<'
11,.",y Cil/rulllllm,s, USAf' Wrigh t Air Dc'dopment Center S~·74.

6- 35.

6- 36. 6---37.

6-38 . 6-39.

6-40. 6-41. 6-42.

6-43.

6-44. 6-45 , 6-46, 6-47. 6-48.

6-49. 6--50. 6-51.

279

7 ONE-DIMENSIONAL STRUCTURES

7-1 INTRODUCTION For the purposes of this book, we define a one-dimensional structure to be any linear elastic system whose state of deformation can be adequately specified by a set of functions of a single space coordinate. Because of the presence of the additional independent variable time, we are still

forced to deal with partial differe ntial or integnll equations when analyzing dynamic problems. But the conceptual and mathematical simplification that is accomplished by the one-dimensional approach is hard to overemphasize; and for some years the practice of aeroelasticity was indistinguishable from "beamoiogy," as it is irreverently called. The many configurations which can be treated in thi s waY-----1ilraight or swept wing and tail surfaces of large aspect ratio, elongated fuselage" struts, booms, and the like-will retain their importance simply because of the high flexibility whicn is associated with their Shapes. More recently rockets, guided missiles, and slender, winged aircraft such as fe,entry gliders have been added to the long list of approximately one-dimensional vehicles. Consider an aggregate of elastic material clustered around a straight or curved line. If straight, let this line coincide with the (spanwise) y-ax is of a rectangular Cartesian system; otherwise, let s denote distance along it measured from some reference point. Provided that cross sections of the structure taken norma! to the coordillate line do not change thei r shapes or warp appreciably, any instantaneous small deformation from the un, strained position is fuHy specified by three orthogonal displacement components u(y), t{y), ..~y), [or u(s),·· -J of points on the line, plus the

'"

ON"-l)[l\mNSIONAI. Sl'HucruRr.s

1111

twist O(y) of thc cross sections. 0 is tnkcn positive in a ri ght-hllnd ~c n ~e abou t th e direction of increasing y or J. The axial strctching /.{ y ) is neglcctcd in aeronaut ical work. Whcn dealing with lifting surfaces 0 1" small thic kness rutio, thc rcsistance to bending in thc chordwisc di rection is usually so great that only thc normal displ;lcement n~lI) and the twist are significant. It is with such wing-Ii ke structures that we shall be mainly concerned in this chapter, although chord wise bending can easily be included when setting up eq uations of motion in terms of gencralized coordioates. Both bendiog directions have compardble flexibility on fuselage and missile structures, but the two are usually uncoupled sinl'e thcy coincide with inertial and clastic principal a)(es. From the discussion of aer(}Clastic equations in Chap_3, we sec thm any system of loads acting on a one-dimensional wing-like configuration C;1I1 be reduced to a running force F,(Y, t) in the direction of IV and a running torq ue TJ,y, t) about the coordinate line. Each of tlJese represents an aerodynamic, inertial, gravitational, etc. load per unil d;Jlal1Ce al01lg !I (or J). When studying the deformations caused by such forces and torques, it is convenient to distinguiSh between those situations where there is a straight elastic a)(is and all other cases. The concept of elastic axis, or linc of shear centers, is a familiar one, fully treated in such places as Chap. I and Appendi)( I of Fung (Ref. I-t). All that need be said here is that, when the axis is straight, bending and tor~ion are uncoupled in the sense thut loads on the a)(is produce no twisting and couples about it produce no linear displacement. A line structure possessing this property is often _called a beam-rod or simply a beam. The elastic operators of the beam~rod were reviewed in Chap. 5. Its motion is most conveniently described by the differential equations o f dynamic torsion and ftexure, which are derived from considering thc equilibrium of the segment of length dy pictured in Fig. 7-1. Here the displacement w of the segment's center from its unstrained position is composed of portions rx due to bending strains and P due to shearing strains: (7- 1) w('1, I) = «(y, t) + P(y, I) With the arbitrary infinitesimal dy cancelled, the conditions of force and moment balance read, respectively, .. F ml\'=

o

#-/.1.=

ay

as,

• +0,

(7- 2)

S, +-oM,

(7- 3)

ay

For thc moment, the e.G.'s are assumed to lie along the E.A, ; m(y) is the mass per unit length; #(11) dy is the mass moment of inertia (rolary

1M2

l'IIiN CII'U:S 01' AlmOISLAS TlCITV

Pure sh"",

P"r~

d e l1~ o~on

bendme deflection

+

External for~ F,ly,l/dy -.

fie. 1_1. Normal forces and moments acting on a segment

d.~

of a be3m·rod. A ll

qUllntitie5 an: ~hown in their posilive senses. inenia) of the segment about an ::t"-axls through its center; S, and M . are the elastic shear and bending moment, Ih~ subscript e being used to

distinguish them from aerodynamic symbolS. Dots indicate partial differentiations with respect to t. In view of the elastic operators CODneding So and M. with the corresponding deflections,

ap

= 5,

GK

a,

EI

;PG'. = M ,y' •

(7-4) (7- 5)

Eqs. 7- 2 and 7- 3 can be transformed to the following force-deflection relations:

il'

"oj "J- +1I1W..= • -'["ay ay2 - ay "0] - "-_0 '" GK -,~ + '[ ay oy or ay

oyt

[E 1 -

F

E1 -

(7-6)

(7-7)

ON"·I)IJ\.mNSIONAI. !>,'RUCI"U MES

l8J

The bending [IOU shcMing stiO"ncsscs Ef(!J) and GK(.'I) conform with the notation of Chap. 5. When the properties of the beam vary along its span, Eqs. 7- 1, 7-6, and 7-7 must be solved simultaneously fo r w, 0';, and (J. subject to geometrical boundary conditions on p, Gl., and orx/o-y, or elastic boundary conditions on S, and M . at Ihe ends. For dynamic problems, initial condition s on 11, .., p, and p must be suppl ied at t = O. The case of a uniform beam is a fa ir approximation to some aircraft structures and serves even better for promoting the understanding of mo re complex problems. The coefficients EI. GK, m, and p are defined to be constants. It then proves possible, by successively eliminating oc. p, M ., and S, amongEqs. 7-4 through 7- 7, to construct a partial differentia l equation for the single variable w('I, t):

il'w

EI - -

ay'

.. ! tm .... -GK- +,J(PW - +mw+ - w~ oy~ GK

[mEl

F

(7--8)

•

Similar equations can be derived for IX and p, except for different terms on the right. Equations 7- 6 through 7-8 have been derived principally for the purpose of discussing the importance of shear deformation and rotary inertia in aeroelastic applications. Let us temporarily defi ne dimensionless variables ,;; = w/I. ii = y/I, and i = wt, where f is the beam's length and w a circular frequency at which it might be oscillating. Us.ing these quantities, the left-hand member of &j. 7-8 can be recast as follows:

"I Eil'w .. ,m .'. ) I - - [mEl - -+1'Ja', +ml1'+ EI iJy" GK or GK

-

O"lw =

ml'w~

afl + EI

102,;;

aF -

[EI GKf"

'J + ml'

O"lw arr 012

2 fJ.W

+ GK

"'014w)

(7-9)

During the oscillation which we envision, the four derivatives with respect to fj and l all maintain the same order of magnitude as IV itself. Hence the .sizes of their coefficients measure, at least qualitatively, their importance in determining the characteristics of the molion. The first two terms on the right form the basis of what we shall call slender beam theory. They evidently predominate over the effects of shearing when

~ «l

(7-100)

GKe

and over those of rotary inertia when ,",w

• -< I

GK

(7-lOb, c)

184

I'IUNCII'LES 0 11 AlmOl; .....s'l'I c l'ry

In essence, the inequalities (7- 10(/) and (7- 10},) require lhat the ratio of depth to length of the beam be small. Equation 7- lOc calls for a frequency whic h is not too large, that is, for motions confined to the lower end of the vibration spectrum. Now, if we leave aside low-aspect-ratio plates or shells and occasional impulsive inputs, the latter condition is generally met in aeroelastic problems. On the other hand, the slenderness condition is no longer fulfilled so universally today as it used to be. In our forthcoming applications of the bending differential equation, we shall be neglecting shear deformation, not because it is always inSignificant but because it complicates the mathematical manipulations. In any event, shear, rotary inertia, and several other effects are automatically accounted for in certain alternative techniques of solution, such as those based on integral equations or modal superp
for the examples of this chapter. Confirmation of the validity of the limitations (7-10) is provided by analyses offree vibrations of beams which do include shearing and rotary inertia. References 7-1 and 7-2 are two excellent papers on the subject. Although the techniques are straightforward, not much has been done toward work ing out exact solutions of Eq. 7- 8 for cases of forced motion. (Leonard does provide some useful examples in Ref. 7-2.) The reader might find it an interesting exercise to try to extend some of the uniform beam remits given below, particularly to the study of shear deformation of swept wings under aerodynamiC loading. Turning to torsional motion of the beam-rod, we first generalize in the following way the elastic operator which determines the internal torque T, associated with the twist 9: T - GJ

.-

ae _ ..£. [CI aZe] ay ay ar/

(7-12)

GJ(y) is the familiar torsional stiffness that is characteristic ofsolid sections,

closed structural boxes, and the like. The second term on the right is contributed by differential bending, which is the resistance to twisting provided by upward and downward bending of a pair (or more) of spanwise longerons or spars acting in opposition, as described, for example, in Sec. 12-3 of Ref. 1-2. a(y) represents the second moment of the bending stiffnesses of all such spars; in the Ci\se of front and rear spars separated a

ONt'~ DtMI>NStONAL

SnWC)'UMII.5

285

distance D and having individual stiifncsscs 1::1, and 1£1" rcspectively, a

~ "D 7'"","";:,,,", El. + El,

(7-13)

Equilibrating the applied and inertia couples on a segment dy of the beam-rod, we find (7- 14)

where T~ is the running external torque and I. the moment of inertia per unit length about the E.A. When the internal torque is eliminated between Eqs. 7- 12 and 7- 14, -if' all

["'J '[ ayt iJy

"J

a - - - GJ-

ay

+11}=T.

•

•

(7-15)

When analyzing aeroelaslic problems by means of differential equations in this chapter. we shall generally neglect the differential bending tenn in Eq. 7-15 for the same reasons that we choose to emphasize slender beams. Moreover, most of the torsion material in current wing structures is furnished by thick metal skins, which lend to minimize the differential bending parameter (JIGJP. (analogo us to EIIGK1 2 in Eq. 7-9). Its greatest interest may be in connection with dynamic model wings having multispar, sectional construction, as discussed in Ref. 1-2. It· is presumed that the reader is familiar with the uncoupled freeoscillation characteristics of slender, uniform beam-rods under various combinations Qf boundary conditions, since these are covered in so many books on elasticity and vibrations. The torsional mode shapes are, of course, simple sine curves. Valuable information on bending eigenvalues and modes will be found in two University of TeJlas publications (Refs. 7- 3 and 7-4). For some applications, especially those concerning nonuniform or nonslender beam-rods, it is more convenient 10 cast the equations of motion in integral form. We then need two fleJlibility influence functions [cf. Sec. 5-2(a)]: C"(y, 1), the normal deflection w at station y due to unit concentrated normal force at station 1); and C"'(y, 1), the twist at y due to unit concentrated torque abou t the elastic aJlis at IJ. These functions imply within themselves aU structural boundary conditions, and they lend themoelves to eJlperimental determination in this and more oomplicated situ.ations. Since the net eJlternalload and torque on a strip

2116

I'RI Nell ' LES 01' IIlCMOI,;I,ASTI CITY

+

+

III) at station 1/ are ( - m('I)i"(II, I) I'~(II' Il{ Ih) :II1U (-I.b)(J(I/o1) T. (I), I)J til), thc integral relations equ ivalent to Eqs. 7- 15 and 7- 1 I (or

possibly to Eqs. 7-1 , 7- 6, 7- 7) are w(y, I) ""

f

CU( y, 1'/)[ - m('l

)i"~1), /) + F.(1), I» 1/1)

(7- 16) (7-17)

The structure here extends from y ,.. 0 to Y "" I. Two important assumptions arc implicit in Eqs. 7-16 and 7-17. First, the beam-rod is fuJly restrained and has no rigid-body degrees of freedom; th e removal of this constraint is taken up in Chap. 9. Sel;ond, the chordwise centers of gravity of an sections lie on the E.A.; if there is actually a static unba lance S.('l) per unit span about this axis, the terms s.(,M(1), /) and S.(1)i';{'1/, I), respectively, should be added inside the integrand brackets or Egs. 7-16 and 7- 17. Motions of a one-dimensional structure without an E.A. can be conceived or in terms of integral equations which resemble Egs. 7-16 and 7-17. We consider the situation illustrated in Fig. 7- 2, where there is a curved structural reference /ine so chosen that sections normal to it experience negligible warping or in-plane deformation during bending and twisting. The displacements are w(s, I) and 8(s, I), the latter representable by a vector locally tangent to the line. Since it is no longer true that forces produce only pure linear deflections and torques, pure twist, we need four influence functions: C"(s, a) and ca"(s, a), defined analogously to

Fill. 7-1. Onc:-dimensional struct uTC with coupling belw""n bending and twist, showing the runni ng 1I1O
ONf:'-O IMENSIONAL STRUCl·URI·:S

2lI7

C" (lI. '}) ami <:""(11. 'I) CXCl::pt that torques arc Vt'CtOI"l' tangen t to the str uctural refcrcm,:e linc; C'"(s. a), the normal deflection at s due to unit

concentrated torque at u; and C"'(s, a), the twist at s due to unit concen'trated normal force al a. A new type ofloading often found on curved or swept wings is the distributed eltternal bending moment, a couple with its vector normal to the s-altis. We use symbol Mis, /) for such a moment per unit length. positive in a sense to compress the upper fibers . It is not difficult (cr. Sec. 4--4 of Ref. I-L) to treat M l as a limit of equal and opposite Z-forces, and thus to show that oC"joa and o~joa are the influence functions giving the normaL deflection and twist, respectively, due to unit concentrated bending moment at u. As in Ref. I- lor 1- 2, we can synthesize integral equations for w(s, /) and 8(.1, tJ from the information just set down. For the sake of brevity, let the inertia forces and moments be temporarily included in the defini tions of F" M l ' and the torque Til abou t the s-altis. The desired equations will then read w(s, /) = L'C"(s. a)F.(a, t) du

+ f' C"(s, a)T,(u, \) du + ('ac"es, a) M i.lI, t) da Oe5, /) =

l'

Jo

Jo au

(7-1 8)

C"(s, u)F.(a, t) du

+ (, C"(s, a)T,(u, /) da +l'oC"(S, u) M/...u, /)da'

Jo

Q

oa

(7-19)

We close this seclion with a few words about aerodynamic operators. The equations of motion of one-dimensional structures contain the airloads in the form of running lifts and moments, which are chord wise integrals of the overall load distribution. It is, therefore, natural to consider two successive degrees of refinement in the theoretical teChnique of relating lift and moment to the displacements. The simpler of tbese is strip theory, wherein the local loads at a particular spanwise station are assumed to depend only on the local motion or angle of attack the re, with perhaps a time-independent multiplying factor to approltimate the effects oftbree-dimensional flow. The more refined method is lifting-line theory, wherein spanwise induction between stations is accounted for. On very slender configurations which are one--
288

I'RIN CU'U'S 01' Af.:ROELAS1'ICITV

equation approach, we shall not go beyond strip theory, for spanWise induction is much more compatible with integral forms of the structural and aerodynamic operators.

7-2 STATIC PROBLEMS OF LARGEASPECT-RATIO STRAIGHT WINGS In this sectio n we study the cantilever lifting surface of Jarge aspect ratio which behaves structurally as a beam-rod with an unswept E.A. This system has, in COlIU11on with the typical se<:tion of Sec. 6-2, a fully decoupled bend ing freedom..The complete aerodynamic loading can be determined from the torsional equilibrium, after which bending may be calculated, if desired, with the already known spanwise lift distri bution as an applied force. The concentration here is on the torsion, since it constitutes a true aeroelastic problem. Lacking space for exhaustive coverage of the available methods of solution, we examine represen tative cases of both symmetrical and antisymmetricalloading. The system with its steady-state loads per unit span is pictured in Fig. 7-3. Allowance is made for an upward normal acceleration gN(y), which produces a running downward force mNg along the line of C.O.'s, located at a distance d(y) ahead of the E.A.

(a) Symmetrical load distribution and divergeIK:e Consider first a uniform cantilever wing or tail. Adopting the differential equati on approach and neglecting differential bending, we rewrite Eq. 7-15 (7- 20)

of EI~$tic

"""7

~ e rodynamic

,

centers

Fi •• 7-3. A st raight, cantilever lifting surface subjr:<:ted to steady-state aerodynamic loods and an upward maneuvering acceleratio"gNCl/).

ONI~"I)lMENSJON"'L

STRUCTU RES

m

The boundary conditions arc 0(0)

= 0

and

dO(I) =

0

dy

(7-2 10, b)

in terms of the dimension less spanwise variable fJ = till. If the undeformed surface has an initial angle of attack""" measured from the zerolift attitudes of individual sections, the resultant incidence is (tlQ + 6). Section 4-3 furnishes the strip-theory aerodynamic ope rators for run ning lift and moment about the A.C., L(y) = qcci = qc VCI (tto

M..I.d..fi)

= qc'c~o

'"

+ 8)

(7-22)

(7- 23)

Lower-case symbols are employed for aerodynamic coefficients per unit span; in this immediate example «0, ocdoCl., c..,tc, and N are assumed constants, but they may in other applications be dependen t on y, allowing for variable camber, aerodynamic twist, or a spanwise factor to adjust approximately for t hree-dimensional effects. Combining Eqs. 7- 20, 7-22, and 7-23 yields

If 0 +

dY'

,1,'ll!) ..,

K

(7-24)

where

,.d K

~

mNgdl" GJ

-

qcP [ ,'" - "'0 GJ 0'"

+ cC""'cJ

(7- 26)

The homogeneous and particular solutions to Eq. 7- 24 add up to O(jj) = A sin ,1,fj

+ B cos,1,fi + ~

(7-27)

When applying boundary conditions to determine the free constants A and B in Eq. 7-27, imagine that the surface is a half-wing encaslre at iJ = 0 in the body of an aircraft or missile. A symmetricaillight condition is ass umed. There are then two ways of looking at the problem. First, the rigid angle of attack <Xo might be a given constant K becomes, in turn, a known qu antity, and Eqs. 7- 21 are evidently satisfied by 8(Y) =

~ [1-

tan}. sin},y - cos Ay]

(7-28)

290

I'NI NC II'L.ES 011 AER O ELAS'I'ICITY

The IOlallift on two halves of the wing is 2qcll' c1 d y= 2cl iJc1l'( CL~-'" qSo sat(o

+ 0) d-,

"" 2e1 Ilan Aac, + [I _ Ian A] [mNgd _ S

aoc

).

0:

0

A

qce

: e

c"."w]l

(7- 29)

(Note that the factor mu ltiplying the braces approaches un ity if the body width is negligible compared to the wingspan. This result also reduces to what would be expected in the absence of elastic twist when the dynamic pressure is small and tan )./A -:::=. 1.) Equations 7-28 and 7-29 may be used directly for any situation where N is given, such as a wind-tunnel test of a flexible model with a preset root angle 17.0 (N = I because of the weight). For an aircraft in free flight one would have to solve for N by equating the total lift from Eg. 7-29 to NW, W being the vehicle weight;as se t forth in Sec. 8-3 of Ref. 1- 2. As a second interpretation of Eqs. 7-27 and 7-28, we migh t consider a typica l structural design ~ndition, where the total willg lift CLqS is the prescribed quantity. Now N is known, because, to an excellent degree of approximation on most large-aspect-ratio airplanes, CLqS = NW. The unknown quantity is ~, for which we solve by multiplying Eq. 7- 29 by qS and equating the result to NW,

i.

(7-0 = - .

NW

Ian l 2qcl iJc,fiJrI.

+[1 tan) 'JfLqcemNgd iJc,fiJrI.

(7-30)

The first term on the right here is the rigid-wing value when )./tan J. = 1. The elastic deformation can be computed by inserting Eq. 7-30 into Eg. 7-26 and using this value of K in Eq. 7-28. It is interes ting to examine the modification of spanwise lift distribution due to the fle xibility. For an untwisted. rigid surface pulling N g's, one has

,

NW (7- 31) 2qcl Some algebraic manipulation yields the r unn ing lift on the flexible wing at the same load factor

c, = - -

_"._lcOSAti+ 1 • , . ~ . ,,-slnl.y c' tan).

,

dl • + [2mg We

2qc'1 NWe

]

- . c,..lC

[I

-

,.,~ I cos ).fJ] Sin y tan }.

(7- 32)

ONI'!-OIr.mNSIONAL SJ'RUCI'UR[S

2')

Plots of Eq. 7- 32 for large-llspcct-mtio surfm;cs in subsonic flight show appreciable distortion even at rather low q. For example, on an uncambered wing with mg dll We ::::::: 0, the overload ratio at the tip is

"'I ' C,'

i- l=

(7- 33)

sin.1.

This rises from I to «/2 before the structure becomes unstable. The mention of instability calls to mind that there is an eigenvalue problem associated with Eqs. 7-24 and 7-21. This can be seen from the blowing up of solutions like Eqs. 7-28 and 7-29 at certain critical values of the parameter I.. But perhaps the most important fact is"that, when the forcing terms on the right-hand side of Eq. 7- 24 are equated to zero, we obtain an elementary example of a Sturm-Liouville problem (cf. See. 6-3 of Ref. 7-5). It can be proved that there is a denumerable infinity of distinct, re al, positive eigenvalues, and, that the corresponding eigenfunctions form a complete set, that is, they can be used to represent to an arbitrary degree of accuracy any sufficiently well-behaved function in the interval 0 ::;; iJ :::; I, by an elltension of the familiar concept of Fourier series. For twisting of the unifonn beam-rod, the characteristic values and functions are

I. u = (2n 'P~(fj) ... sin

+ l)~)

2•

.

( n=012···) , ,

(7- J4a, b)

A"Y

Each such combination represents a dynamic pressure and mode of deformation in which the wing (starting from zero initial incidence, camber, and normal load factor N) can assume an equilibrated twisted position of arbitrary amplitude. The fundamental eigenvalue Ao = ,,/2 has the most direct physical interest. It corrcsponds to the lowest dynamic pressure of torsio na l divergence, which Eq. 7-25 shows to be ,.,.2

qD =

G1

4-" c"C-:;"",,",,,"

(7-35)

Significant comparisons can be made with Eq. 6-11 for the typical section. Once more Eq. 7-35 is merely an implicit fonnula un less e and oc,/rJrzare indepeodent of ffight Mach number. The inverse proportionality to [2 reveals the major influence of wingspan or aspect ratio, but even this difference from Eg. 6-11 disappears if we regard GJII as an effective torsional spring stiffness K1 and also replace Ie by the wing area S. In common with buckling of columns and flutter, the divergence problem has a special cmphasis on the lowest eigenvalue. It is therefore

1\12

I'IUNCII'LES 011 AElWI,;I,ASTICITY

wotlh mentioning, in passing, the utility of Rayleigh's energy method for estimating Ag for syslems eithcr with constant or variable spanwise properties. The twisting wing is neutrally stable in any mode B(y) provided the potential strain energy stored in the elastic deformalion just equals Ihe work done by the defonning aerodynamic torques. According to strip theory, these torques .§Tow in direct proportion to the local 8; so onc obtains the following fonnula:

(de) a, B dy=qv]' iceJ. 'lGJdy 1

2

dy

(1- 36)

DOC(

g

If the divergence mode is known, it can be substituted for 0(y) in Eq. 1-36, leading to the exact q". However, any reasonable function which satisfies the structural boundary conditions will yield a close apprOXimate qJ), whether or not 0(y) can be found analytically. At worst, the two simple integralS in Eq, 7- 36 have to be evaluated numerically. The estimated dynamic pressure always exceeds Ihe exact, since the true potential energy is a minimum relative to any apprOXimation which applies artificial conslraint~. If we use the unifo rm cantilever as an example, but replace the true mode sin (l1fJ/2) with 0 = [2y - Y"], Rayleigh's method proceeds as foll ows:

ai' " e'df[2 -

qD~1-cc 2

aCt:

GJ =

•

y

D

2y]2dy

'" oe, £"[2 - -'J' d a- •• y - y y

m

5 OJ

= --2f

,, -oc, a.

(7- 37)

This res ult is greater than Eq. 7-34 by only 1.4 %. Incidentally. improved aerodynamic theory of the lifting-line type can be introduced by replacing

::' 0 with c,G in the right-hand member of Eq. 7- 36 and solving for clj)) corresponding to the preassigned 0(y) before evaluating the work

integral, A well-known by-product of Sturm-Liouville theory is [he powerful way in which [he eigenfunctions of the homogeneous differential equation are ada pted to solving nonhomogeneous problems. We illustrate this techn ique by returning to Eq. 7-24_ ).t must still be constant, but K may

ON~';.J)tf.mNS t ONA t,

S·t·UlICI'UUES

19J

be a function of y, thus allowing for :lcrodynamic twist, variations in CG. position, I:Olll:cntratcd masses like cxternal stores, or variable normal load factor. The solution O(j}) is construded by assuming that both it and K0J1 can be expanded in Fourier series of the functions ,!,"(j}) from Eq. 7- 34b, • (7- 38) Ocy) = 2' a,,'!'iY)

""' K (y) =

•

2' A"rp,,(fj)

(7- 39)

.",

Since K is treated as a given quantity, the constants An are efficiently calculated by resort to the orthogonality or the eigenfunctions,

f' o

'F"(j})rp,,,(il) dfJ =

i'

sin (2n

0

+

.

1) fry sin (2m 2

+

.

I) fry tiy 2

(1--40)

Wi th Eq. 1-40 established, Eq. 7-39 is multiplied through bya particular <}'",(9) and integrated over the span to produce A ...

= 2

f

K(fJ )'1'",(il) tiff,

(m=O, I,2,"')

(7-4\)

To determ ine the constants Q. in Eq. 7- 38, we insert both series into Eq. 7- 24 and recall tbat (f,,(Y) satisfies the homogeneous differential equation when A = AK , that is,

d' •• d"'"

.v

+ ~2", = . .,.-

(AI _ A 3)m .. .,. ..

(7-42)

These ste ps lead to the relation (7-43)

.and to the final solution

~ O(y) = -

2 '" fK(i)J 'Fiff)dfj ,. ,,2' _0

1\

("

,.

~

_ 1)

_ 'Fi y)

(7--44)

Good convergence is assured, in practice, by the fact that the denominator quantities A" 2/}.2 increase from term to term in proportion to the squares of the odd integers I, 9, 25, 49, . . '. For any realizable il.ight condition, }. < 10; the blowing up ora(y) as torsi nnal divergence is approached shows dearly in Eq. 7--44.

19~

t'Rt NCl I'U;.s O t' AEROEt .•ASTt Ct'l"V

One othcr class of straight, c~ ntiJcvcr wings for which Sturm-Liouville methods have genuine practical merit consists of the generali zed tapered planforms first identified by H. Rcissne r (for a fuJI treatment, see the paper by F. B. Hildebrand and E. Reissncr, Ref. 7-6). The chord, A.e.- E.A. offset, and torsional stiffness are described by the functions

[1 _ XY)" _

c(j}) -= e(fi) _ CR eu

Yl"

GJ(i/) = [1 - xii] " = Y1" GJ R

(7-45) (7-46)

Here subscript R refers to the wing-r oot section, and Y1 is a dimensionless variable running from (1 - X) at the tip to 1 at the root. With the steadystate loads illustrated in Fig. 7- 3, the differential equation (7-15) can be written in terms of Yl for the system under consideration,

~(y"dO)+ dY1

1

(7-47)

dY1

The modified parameters are ).,/ = qC llCRtt OC,

(7-48)

GJR Ol1. _mNgdf K( l!l1j -

GJlI

-

qcf[ ac, C l1. 0 GJR all.

+ cC ...-I O]

(7-49)

The section lift-curve slope must be a constant. As long as Z < I, the boundary conditions are Eqs. 7- 21. When developi ng solutio ns for Eg. 7--47, it is convenient to distinguish between two sub-cases.

*- 2(r, + 1).

Under this r estriction, any extensive treatment of Bessel functions (e.g., Chap. 2 of von Karman and Biot, Ref. 7-7) shows that the homogeneous solution reads (I) Y2

/I11(Y1) = Yi" [ AJ, e

R " X

yf') + BY, eRoS yt")]

(7- 50)

,

where (7-51a, b) J, and Y, are Bessel functions of th e first and second \d nds aod order v; except when v is an integer, Y, =" J _ •. Thc eigenvalues and eigenfunctions

are established by app lying conditions (7- 21), which process yields transcendental relations whose roots must be determined for each distinct

ONE-IUMENSIONAt, STII UCI'UMffi

2~

combination of),,, )'2' and x. Once these characteristic quant ities arc available, K/Jh) and O(YI) may be expanded as in Eqs. 7-38 and 7- 39, leading to systematic and rapidly convergent series. When computing the coefficients A., we ma ke usc of the orthogonality of eigenfunctions with respect to tbe weight YI~l. (2) )'3 = 2()'1 + I). In this important special case, the transformation " I = In Yl leads to the following homogeneous solution, which also satisfies the boundary condition at the fOot : (7- 52) where (7- 53) Substituting Eq. 7- 52 into Eq. 7-21b, we get a transcendental characteristic equation tan (" In (I _ X) =

2#

I

+ 2)'l

(7-54)

whence the sequence of positive eigenvalues .l.11~ can be computed by numerical or graphical means. The eigenfunctions 'P/Yl ) then follow from Eqs. 7- 52 and 7- 53. The nonhomogeneous solution is finally found as in the first sub-case. Particular examples of torsional divergence win be found worked out in Ref. 7- 6 and Sec. 8-3 of Ref. 1-2. Reissner's results have received less attention in cngineering applications than thcy deserve. The reader will be able to find an excellent approximation in the form of Eqs. 7-45 and 7-46 for almost any beam-rod cantilever which has no disrontinuities. Sub-case (2) has been elaborated because it encompasses )'1 = I, Yt = 4. an exact description of a slraighttapered surface with geometrically similar cross sections at all spanwise stations, fabricated from the same structural material. (b) An elementary example of aerodynamic beating effects The study of uniform wings by means of the torsional differential equation provides an opportuni ty for us to illustrate the role of aerodynamic heating in aeroclasticity. Although they are taken up to some extent in Chaps. 2 and 5, the full presentation of boundary-layer heat transfer, temperature distribution in solid structures, and generation of thermal stresses would require another book larger than the present one. Accnrdingly, we must accept without proof certain essential results and emphasize those cousequences of the thermal environment wh ich have

2%

1' llINe ll'U.s 01' AI;1I.0I!l, ASTICITV

primary importance for the acroelastician, namely, the losses of stiffness due to thermal stresses and to deterioration of the elastic constants. Budiansky and Mayers (Ref. 7- 8) have analyzed an unswept lifting surface of constant cross section, which has a large enough aspect ratio to permit the examination of aerodynamic heating and its effects on a two-dimensional basis. If· a particular section, of area A, has an instantaneous distribution of temperature T throughout the material, this will give rise to a varying spanwisc normal stress
If

T~ "" ~:
(7-55)

A

r is the distance from the E.A. to the area element dA at which (fw acts. Unless there is a net spanwise tensile force, which would be present only in whirling beams such as rotors or propeller blades, the integral of a~ over A equals zero, and T~ is a pure couple. To get the total internal torque T. we must add the familiar contribution GJ UOliJy of the shear stresses, tha t is,

(7- 56) Since this relation replaces Eq. 7-12, we see that there is an effective torsiol/al stiffneS5 GJen. = 1

GJ

+ J... fJa

,

GJ·

,. dA

(7- 57)

to be inserted in place of GJ in all of the beam-rod formulas . The foregoing development presupposes that the shear modulus G is unaffected by heating. Should there be appreciable reductions of G in t he hot zones, the shear equilibrium of the cross section must be reexamined to find a reduced value of the coefficient GJ for Eqs. 7-56 and 7- 57. This adjustment is usually much smaller than the one· associated with thermal stress, because strength losses force the designer to turn to different, mo re heat-resistan t materials before temperatures are reached where E and G drop significantly. Moreover, the elastic constants which control dynamic phenomena, such as vibrations and transient response, turn out to be nearer their "cold"' values than those which apparently

ONI!:-I) IMENS IONAL STRUCTURES

297

must be used when dea ling with static problems, because the lalter are always 3ffectcd by II eerlain IImolint of high-temperature creep (Ref. 7-9). It is also of interest that self-equilibrated spanwise thermal stresses do not alter the bending stiffness at all, within the framework of slender beam-rod theory. The implications of Eq. 7-57 are clarified by studying what bappens following a rapid, large increase or decrease in the forward speed of the vehicle mounting the lifting surface. If we adopt the assumptions of Budiansky and Mayers that recovery temperature and heat-transfer coefficient are independent of location along the airfoil chord and that radiative heat loss is negligible, the entire structure starts the maneuver at the uniform adiabatic wall temperalure

(Ta ..)., = To> [l

+ t).1' ~

1 Mol]

(7- 58)

associated with the initial flight Mach number Mo and ambient temperature T "'. The recovery factor 1J. is a number of order 0.8-0.9 which adjusts for the drop of T... below stagnation temperature achieved by convection of some heat away in the viscous boundary layer. If the vehicle accelerates, heat is fed in at a rate roughly proportional to the defect of instantaneous skin temperature below a new T... appropriate to the higher B.ight Mach number. Since there is relatively little time for this heat to be condu cted chordwise along the slender airfoil sC(;tion, it penetrates largely through the depth of the wing at the place ",here it is received. T therefare increases faster at the leading and trailing edges, where there is relatively small heat capacity compared with the thicker center section. Trying to expand excessively in the spanwise direction, these edges find themselves restrained by the cooler center and develop compressive thermal stresses (negative u.). A simultaneolls positive u. appears around midchord to preserve equilibrium. Since r is larger whereu. is negative, the integraL in Eqs. 7- 55 through 7- 57 has a minus value, and the effective torsional stiffness is reduced. Severe, sudden heat inputs, especially to structures Jiaving thick or solid cross sections, may even lead to negative GJorr• and thermal buckling. By the converse of the foregoing reasoning, deceleration produces a torsional stiffening and is favorable from the standpoint of elastic and aeroelastic stability. (n either event; the modification to GJ is a transient effect, which disappears once thermal equilibrium is attained. In their app lications of the theory. Budiansky and Mayers adopt the general assumptions discussed above, going so far as to neglect chordwise heal conduction completely and to treat T as constant through the thickness at any point on the wi ng. For solid cross sections, the temperature

is then calculated from a simple first-order differential eq uation and is related to the thermal stress as follows: fl.

=

£'J.1,i.!..A Jebor
- (T.Jg]z, dx

+ [T - (T~ ..k.)f\

(7- 59)

where '1.1: is the coefficient of linear lhermal expansion and z,(x) the local depth of the section. Based on Eq. 7- 59. effective torsional stiffness can be determined in terms of tabulated functions of a modified time variable when the profile has a simple analytical shape. Most numerical results in Ref. 7-8 refer to a 3 % thick symmetrical double-wedge airfoiL with a 3-ft chord, fabricated of solid steel. Figure 7-4 reproduces their calculations for a sudden acceleration from low flight speed to a final Mach number M, at an altitude of 50,000 ft in the standard atmosphere. For M, = 3, there is a peak loss of all but 22.5 %of the torsional stiffness of the cold wing, occurring about I t min after the speed-u p. An appreciable interval of thermal buckling appears when M , = 4. Reference 7-8 investigates the relieving effects of finite ae<:eleration and of the presence of a free wingtip where fl. = O. The numbers show, for the particular solid wing considered, that the minimum value of transient GI. rr . is relatively insensitive to the rate at which the velocity is increased over a range that might be regarded as including any full-power maneuver of a supersonic fighte r or missile (e.g., for a time of less than 5 min to reach M t = 3). As measured by the reduction in fundamental torsional frequency squared. w}, on a uniform, free-free wing of span B, Budiansky and Mayers find only very small deviations from two-dimensional results when Ric exceeds roughly 2. Thus one may conclude that simplified estimates of GJ. rr. arc adequate at least for preliminary design purposes on unswept lifting surfaces having continuous, thick-skinned structures, but that stability margins wi]] always contain an element of conservatism. The effe>:tive torsional stiffness may be inserted directly into formulas li ke Eqs. 7-28. 7- 32, and 7-35 for static aeroelastic analyses. Since high supersonic speed is involved. we must take care to include the infl uence of profile shape and thickness on the AC. location and c.... c. Piston theory will normally yield aerodynamic operators of sufficient accuracy. For example. it gives the following lift-cufl'e slope and A.C.-E.A. offset in the case of a symmetrical doub le-wedge airfoil of thickness ratio b:

---

oe,

4

0:.

M

,-~["+(';l)M'J

(7-QO) (7-61)

ONI·:-I)IM ENS IONAI. STII UC'I'UII ES

299

Hcrc II is thc distance in scnlichords by which the E.A. lies art of the midchord line. When Egs. 7- 60 and 7-61 arc substituted into Eq. 7- 35, we obtain the implicit divergence formu la (S, = lr: is semispan area) ,,!MpGJ,tr.

(1- 62)

This can be solved as a quad ratic for MD' Typically, the E.A. of such a wing would be close to midchord (a = 0); and the Mach number then becomes M

-" S,

D -

J

y(y

GJetr. + J)p",d

(7- 63)

Equation 7-63 reveals another undesirable effect of airfoil thickness at high speeds: forward displacement of the A.c., occurring in direct proportion to Md, increases the moment arm of the lift and augments torsional instability. It is clear from Fig. 7-4 bow thermal effects can temporarily reduce the rigidity below a safe level and cause dangerous deformations or divergence on a surface which otherwise possesses a perfectly adequate margin. Typical calculations of this sort will be found I.0

, 1\ ,

,,-

I

,

o.

0

o

iL1.08"

Altitude 50,000 It

"-

M,= 2

1

•

~

.,/ V

d

/, /

\

/

I.

V ,

j

/ •

,

,

Fig. 7-4. Time history of effcctive torsional S1iffness for the double_wedge steel wing showo, following a sudden acceleration \0 Mr. fraken fro m Budiansky and Mayers, Ref. 1- 8.)

JOO

I'RINC Il'I.ES 01' AI<:RO"I.AS1'ICll'V

in Ref. 7-11. On the other hand, a decelerated maneuver might be able to pasS successfu lly through a zone of "cold" instability. Since GJ.ff. is a lime-dependent quantity, its variation might conceivably gene·rate dynamic responses on a loaded wing in steady fiight undergoing thermal transients. The characteristic times of temperature rise and fall in the structure are so long compared with a typical period of vibration, however, that a quasi-static approach always seems to be justilied in aeronautical practice. We rocall, nevertheless, that quenching following heat treatment sometimes causes metal objects to sing. More relined torsional stiffness calculations of the sort discussed above will be found in Refs. 7- 10 through 7-13, along with further aeroelastic applications. One especially signilican t error may arise from the neglect of mid plane stretching where the thermal stresses become so large that the simplified theory predicts buckling. When this stretChing and initial imperfoctions are considered in a large-deflection theory, GJdr. will always remain greater than zero, and the curve in Fig. 7-4 for M, = 4 is therefore unrealistic. (c) Antisymmetrical load distribution by integral equation methOds; control effocth·eness In Sec. 7-2(a) we analyzed the effects of torsional deformation on a surface attached either to a fixed cantilever mount or to an airborne vehicle executing a symmetrical maneuver. We now look at antisymmetrical maneuvers and, in view of the general restriction to one-dimensio na l structures, consider only loadings produced by a fl apped control of the aileron or elevon type. Linear theory permits the portions of the total load distribu tion which are even and odd functions of the spanwise variable y to be examined separately. Accordingly, the lifts and mome nts needed for the present treatment arise from only three effects: control surface rotation d(y), antisymmetrical twist I.l(y), and angle of attack changes due to motions in the lateral degrees of freedom of thc vehicle. Angles are taken to be positive when in the nose-up sense on the right wing or stabilizer. Among the latcral freedoms, only the rolling velocity p (positive in a sense to swing the right wing upwa rd) will be included, because aileron effectiveness is normally measured with yaw and sideslip trimmed to zero while performing some sort of helical motion about a fixed longitudinal axis. This situation is achieved exactly in wi nd-tu nnel tests where the model is constrained to roll about a spindle, as in the experiments described by Refs. 7-1 4 and 7-15. A more complete discussion of aeroelastic effects on lateral stab ility and control of free-flying aircraft appears in Chap. 9.

ONt',.t)tMI!NStONA t, Sl'MUcrUMES

30t

The system shown in Fig, 7- 3, with its straight E. A.. Mill serves our purposes except for the added freedom in roll. To provide va riety in the exposition of methods of soluti on, however, we set out from the intcgral equation (7-17) and from lifting-li ne ae rody nam ic operators. The intra· duction of approximate integration met hods will then lead us to a ma trix formu lation of the sort common ly used in the aircraft industry for nonuniform one-dimensional structures. It is consistent with the lifting-line approach to write the an tisymmetrical ang!e of attack distribution o:(y) = 8( y)

+ a", b(y) _ py " u

(7- 64)

where p>AU is the (small) angle of attack produced by rolling and 0((/06, which may also vary across the span, is the wing incidence equivalent to unit control rotation (cf. Eq. 6-J7b). As discussed in Sec. 4-4, the running lift is given in linear operator form as follows: L( y) = ely) = d qe = e '( ) I II

I

8(y)

1

Pill

+ 0CJ. b(y) 06

+ oel(Y).o + oc,(y) oi)" o(~)

U

pi

(7--65)

U

OR is the an gle at some reference station, and this representation is satisfac tory as long as eithcr b is a constant or 6(y) is known in advance; if Ihe control surface is permitted to twist under th e action of ae rodynamic hinge moments, the te rm
OC,(y)IO(~)

can

be determined by measurement or theory independently of the aeroelastic ana lysis. The,lift in Eq. 7- 65 acts along the line of aerodynamic ccnters. The only antisymmetrical pitChing moment accompanying it is that due to 6, which can be written M .H;(y) = ( qcZ

-

.1) _ oc",-, c (y) 6 a
," .~ L1 Y -

(6-66)

the righ t-han d member being usefu l when twisting of the contra! surface is neg lected. A tin a! contribu tion to the torque in Eq. 7-\7 might come from the ine rtia force (-mpy) caused by roll accelc:ration, ac ting with its

3111

I'R1 NCIf'U~

0 1' AlmOEl.ASTICITY

arm d about the E.A. Such a time-dependent maneuver can properly be considered if its rise time is long compared to the fundament,l! structural frequency; elastic deformation and airloads may then be calculated OQa quasi-static basis, as is sometimes also true of longitudinal maneuvers. Making the appropriate substitutions into Eq. 1-17 Md dropping 0, we obtain (J(y) = q LCU(y , l1)eCC,' d7J

+ f.(y)

(1- 61a)

where

In these relations, as in Eq. 7-64, Y = 0 may be regarded as the vehicle centerline i[the torsional influence function and the aerodynamic operators are suitably defined. No exact sol ution of Eqs. 7-61 or their differential counterpart can be expected unl ess the geometrical and structural prnperties are given as ele mentary functions. Hence we turn to quadratllIC. As discussed in Sec. 3- 5(a) or any book on numerical processes (Ref. 7- 16 is an exceUent example), the introduction of Simpson's rule, the trapezoida l rul e, Gauss' formula, or any equivalent device permits a typical integral to be written for a set of spanwise stations y/ on the right wing, (7-68) where

Cw-J

is a diagona l matrix of weighting numbe rs, and the

definitions of the other matrices are obvious. To construct algebraic ' equati ons in terms of the single unknown quantity c,'(y,), we use any standard approximate scheme for solving the lifting-li ne aerodynamic problem to recast the requ i~d terms in Eq. 7-65, (7- 69) Here CR is the chord at any convenient reference station. If possible, Eq. 7- 69 involves the same locations y, choscn for Eq. 7-6&, taking advantage of antisymmetry to work on only half the wingspan; otherwise, an interpolating matrix must be employed to make the two sets compatible.

ON I~.I)IMI~NS IO NA I_

S'I'Hu c rUHES

31)3

Inverling Eq. 7- 69 lUld cllrrying out a series of operalions similar 10 Eq. 7-611, we find il po~sible to rewrite Eqs. 7-67

[d"J-,(ee,' j =qcl/[f:] (eel'j + {f. } ell

{/"l

~ q[ ~,*:'j £]

d"

+

(7- 70a)

eu

+

(,(~)) ~)

q[F](";,"j '" ~ [G] [YJ(m)p

(1- 10')

The three square matrices un the right- hand sides are abbreviations for the following:

[EJ ~ [d'JLWJ~'J [FJ ~ [C"JCWJ~"J [GJ~ [d'JCWJ CdJ

(7- 70c)

(7- 7Od) (7- 70e)

Treating tlR' p, and p for the mnment as known coefficients, the antisymmetrical load distributions are found by elementary mat ric manipulations,

(1-11) (7-72)

The tntal rolling moment ex.erted by the lifting surfare is i{(

= 2q

fee,,,

dy

~ 2qcllLH ..J{:~}

(7- 73)

where t he row matrix. resulting from span wise numerical integration is (7-74)

Two maneuvers of practical importance to the handling qualities of aircraft will serve to illustrate how the foregoing solution is used. The first of these is the response to a sudden fixed displacement 60 of the ailerons, (7- 75)

3(14

I' IU NC II'I .ES OF AIl UOIiLASTl ClTY

Wit h all but the rolling degree of freedom suppressed, th is motion IS governed by the differential eq uation

i"p

= //1

(7-76)

and the initial condition P(O) = O. I", is the mass moment of inertia of the vehicle about its roll-axis, a quantity whieh is practically una.ITected by wing twist. When Eqs. 7- 70b, 7-71, 7-72, and 7-73 are combined with Eq. 7- 77, and the roles of input and output are specifically identified, we obtain the following : (7- 77) Here the constant coefficients are the rather elaborate expressions A,

~ 2qc ~ + LH -.J(f"'·l-' -

q'"fEl)-'[GJ~,J(mI

(7-78")

R

B,

~ - LH -.J(L,:J H(f""l' - q'"fElt'[EJ~oJ)(q(~)) (7- 7Sb)

C1 =

qL H .J([ JI'.'] - l

- qCR[E])- l [F){OC;t'}

+ LH -.J(L,:J + q((",T' - q,,(Elt'[ EJ[,J)(::') (7- 7ac)

Equation 7-78a is solved by an exponential expression typical of firstorder systems,

We observe how the motion starts out wit h a maximum roll acceleration f(0) = C1 "0

A,

(7-S0)

and speeds up smoothly until the moment due to rolling velocity (related to the damping in roll) just balances the effect of the ailerons. The second maneuver to be considered is steady rolling, which is described by the asymptotic form of Eq. 7- 79. (7- 81)

ON";" IlIM~NStONAL

Sl'ItUCI·UtU:.s

When divided by bu, Eq. 7- 8J yields the aileron effectiveness

3115

a(~) / ali

or rolling power of the elastic wing. After substituting Eqs. 7- 78b and 7-78e into Eq. 7-81, we can divide numerator and denominato r by the wing area S and the ratio tjen' We then recognize terms in C, and H, which are, respectively, the aileron derivative and damping in roll of the rigid wing:

c 1$ -_ qS(21) 1 2LH ' ''~1('''1 q -lL'\J ali

(7- 82)

c I, --

(7-83)

1 2 LH -.J",l( "') L '::J a(~)

qS(2IJ q

With these defined, the aileron effectiveness becomes

,(~)

" (7-84a)

From the corresponding rigid-wing performance

('(~))' ~ fJlJ

_ C"

C,.

(7-84b)

we can construct a ratio somewhat analogous to Eq. 6--17a for the typical section. One significance of results like Eq. 7-84a is that rigid-wing derivatives are available from the wind tunnel or from accurate aerodynamic calculations, and the aeroelastic corrections can often be made on a modified strip-theory basis with considerable economy and little loss of over-all precision. Since C,. is negative, Eq. 7- 84a yields a numerically positive result that decreases with increasing dynamic pressure because of the influence of the principal numerator term, which is the one containing the As in the case of the typical section treated negative quantity

aCm.JclalJ.

306

1'lilNC lI'l.ES OF ,..mWELI\S"I'ICITV

in Sec. 6--2(b), an aileron reversal condition

q = Q1l. g iven implicitly by the equation Sic

'R

o(p')/O/j = 0 is reached a\ U

'.

(7- 85)

qR=-LH-.J( [ d"] -l_qRCR[E]) 1

X([FJ(";;W) + [EJ~'J(;:')) Since the aerodynamic quantities vary with flight Mach number, 11 IS us ually easiest to find q1l by calculating al leron effectiveness over the speed range of interest and obse rving if and where it vanishes. While discussing reversal, it is natural to bring up the eigenvalue problem for the integral equation of the elastic wing. We note, for c;.;amp!e, that Eq. 7-70u has a nonze ro homogeneous solution of the divergence type at the dynamic pressure qD for which the determinant

(7-86)

When [d a ] is independ ent of Mach number, Eq. 7- 86 expands into a polynomial in qlJ' whose degree is one less than the number of spanwise stations and each of whose real positive roots identifies a possible antisymmetrical divergence. Only the lowest root is practically important; and even it serves mainly to measure the level of rigidity, because 9R < qD on straight wings with trailing·edge controls. This is a good example of a p roblem where the technique of matri x iteration can be efficiently applied. We rewrite Eq. 7-7Oa withf~ = 0 as follows:

{c:J = qJ.c,,[d~][E]{c:J

(7- 87)

A trial solution for c,(y,) is inserted on the right, and a preliminary estimate for qD is found by requiring the two sides of Eq. 7--87 to be equal, either at a chosen spanwise station or in some average sense. The result of the calculation e< ') (

c~

,

[s1"4][EJ(cc"j

is chosen for the second approximation to

'R

and the process is repeated until convergence occurs in the

sequence of qD values. The final column of lift coefficients also yields the divergence mode of twist throu gh the operation {O} =

[.#·l-l{:~·}.

If desired, the aerodynamic matrix can be adjusted after each step to reflect the improved estimate of Mach number. This procedure is much

morc ellicient thall n'ctor ing a polynomia l, whcn 11 rcalistic numocr of statiollS is cmployed . It can oc rigorously proved to convergc onto the lo west positivc eigenvalue if the problem is self-adjoint, tha t is, if the square matrix is symmetrical (Ref. 1-11). This condition is fulfilled fo r straight wings when strip theory is used, and satisfactory convergence occurs in practice even with much more complicated aerodynamic operators. Incidentally, tJle matrix solution for symmetrical divergence resembles the forego ing, except that a diffe rent aerodynamic matrix [dO] and possibly another influence function eM appropriate to the other type of wing twist must be used. In some simple examples, the symmetrical qv has turned out slightly lower than the anlisymmetrical, because the evenly twisted finite wing is a mo re efficient generator of lift and torque. Another iteration technique can be adopted to compute reversal from Eq.1- 85. A trial number is substituted for qR in the denominator on the right, the calculation being repeated until the two sides of the equation and the Mach number of the aerodYllamic quantities all agree (Ref. 1- 18). An extended presentation of various methods for finding the antisymmetrical load distribution and aileron effectiveness of straight wings will be found in Sec. 8-3 of Ref. 1-2. It is pointed out, for example, that many of the foregoing results can be closely approximated by assuming the E.A. and the line of A.C:s to coincide, so that elastic terms containing

oe,/oo and oc,/a(~)

may be discarded and only the deformation due to

aileron twisting moment oc,..J.clab retained. Strip-theory solutions are de ri ved for the uniform straight lifting surface, which parallel those developed in Sec. 1-2(a). Using the notation defined therein, the aileron effectiveness can be expressed as follows (Eq. 8-110 of Ref. 1-2):

.(~) ~

"

cOSA' _ IJoe, + [cos i., _ 1_ 2 - ).l !J~ OC"'.HJ [ cos). oli cos A 2 Ii 00

."

(1-88)

' o,[tan). _ 1J

,

The control surface envisioned in Eq. 1-88 is one of constant chord, extending from y = " to the tip y = I. )., is the dynamic pressure parameter (1-89) associated with the inboard end. Reve rsal occurs at the vanishing of the

303

I' IU NC II'LES 011 II lm OELASl'lCITY

,

2.

•,

.

""

2

\ ,., ~'F

,.•

- 0.4

'"

--- --- r-- - -- --- --- --

, ,.

Lim it dive spffi

,.. M -_

"

:\ 1\ 1

" " " "

Fi r. 7-5. Ail~ron ~lJecti""ness ploU~d V~. fl ight Mach number at .5<'a level for the straight wing of a sub,onic monoplane. (Taken from Otap. S of Bisplinghoff. Ashley, and Halfman, Ref. 1-2.) numerator in Eq. 7- 88, but the presence of transcendental fIJnctions of ). and ~I preve nts the wriling of an explicit formula for gR' even when the aerodynamic derivatives ar~ independent of M . As a warning, it must be stated thai uncorrcx:ted strip theory is less accurate fo r the quantitative estimation of aileron effects than for sym· metrical loadings. Except at high supen;onic and hypen;onic speeds, the aerodynamic induction is much stro nger in conncction with angle·of. attack discontinuities like those at the control·surface extremities than it is when the spanwise incidence variation is continuous. Special care must therefore be taken in evaluating Oc ... Acf(j{), ocJo6, and C w Despite the difficulty of determi ning absolute values of 0 (~) /06 and

q", it turns out on most straight wings that the linear variation of effective· ness with q/qR found for the typical section in Sec. 6- 2(b) is a close approxi· mation of the true behavior. T his point is well illustrated by Example 8-4 of Ref. 1- 2, from which we reproduce Fig. 7-5. The system considered is a tapered wing of semispan I = 500 in. (M = 6.15), root and tip chords 225 in. and 100 in., respectively. The A.C. is at 25% chord be hind the leading edge, and there is a straighL E.A. normal to the flight direction at 35% chord. The spanwise stations HI are chosen, in connection wi th a

ONI(-OlMEN SIONA I. STIWC1'U IUCS

309

7-point Gauss qua drature fOr nlll[;I. 10 be c{[ually spaced in an angle variab le defined bY .'1 = fc os 0, so lhat on the se misfh1 n· 46 1.94 353.55

m,

(7- 90)

m,

(7- 9 Ia)

191.34

o Other necessary matrices of wing properties are 10.95 13.66

C·] ~

17.72 22.50

109.515

C,]

136.612

m,

~

(7- 9Ib)

177.165

225

[C" ] =

424.3

424.3

424.3

0

424.3

186.6 186.6

0

424.3

186.6

0

0

0

78.45 0

x 10- 10 rad /i n.-Ib

(7- 91 c)

0

There is an aileron lying between II = 370 in. and I z = 487 in., which is assumed to have constant sectional deri vatives 8"-1805 = 0.4 and oc,..J.c/86 = - 0.45. The data in Fig. 7-5 were computed from Eq. 7- 84a for sea-level flight, using strip theory throughout, neglecting the twist due to the A.c'-E.A. offset, and adjusting aU aerodynamic coefficients by a factor I/VI MZ for the (subsonic) influence of compressibility. A generally parabnlic varia tion of effectiveness with M is evident. correspo nding to * To be precise. the~ Gaussian (or Multhopp) stations are used in Ref. 1- 2 for divergence and syrrunctricalloading calculations, while a different set of y, is employed in Example 8-4, co ncentrating on lhe portion of the span cover<:d by the aileron. Equations 1- 90 and 1-91 serve, however, 10 give a description of the system adequate for our purposes here.

310

I'IUN CII'I.ES 0 11 AElWEI.AS·I'ICITY

the aforementioned linear dcpendence on If/If/{. A hypothetical limit dive speed of 480 knots at sea level is indicated on the figure. At this limit 70% of the rolling power has a lready been lost to aeroelastic effects. Reversal occurs at 534 knots. In Examples B-1 and 8-3 of Ref. 1- 2, the symmetrical influence of twist on load distribution and the divergence eigenvalue are computed for this same wing from the integral equation. To permit comparison with Un' we list the following estimates or sea-level UD, found by using three different forms of the aerodynamic operator: 13 17 knots, by strip theory with ae,fiJl•. = 5.5; 1583 knots, for symmetrical divergence by lifting-line theory; 1659 knots. for antisymmelrical divergence by lifting-line theory. No attempt was made at a Mach-number correction, since these speeds exceed that of sound and are therefore artificial. In actuality, this system would never encounter torsional instability, because the A.C. would be shifted behind the E.A. in supersonic flight. With the background provided by this relatively brief treatment of th e integral-equation-matrix approach t o the aeroelastic analysis of straight, one-dimensional structures, the reader will be able to fill many gaps left open because of space limitations. For instance, if the trailing-edge control surface is replaced with a spoiler, we would c.ommence by substituting Ilc,(d,) and Ilcm<,C
ON I,.I)IMIlNSIONAL STIt UCTUU.f.:S

7~3

311

SJ'ATIC PROIJLEMS OF STRAIGHT AND SWEPT WINGS SOLV ED BY APPROXIMATE METHODS

Turning to lifting surfaces without clearly defined elastic axes or with appreciable sweep, we are confronted by an extensive literature, con taining a bewildering array of techniques for predicting their static aeroeiastic properties. The objective of what foUows is merely to illustrate a few of the more powerful and broadly useful methods, furnishing as we go some information on the influeuce of Ihe principal parameters. Appearing at the dose of World War ll, the swept wing offered to the ae roelastician his first real novelty since the cantilever monoplane. It is characterized by a structural aspect ratio (ratio of the swept span to the chordwise dimension taken normal to the swept allis) which exceeds the ae rodyn amic aspect ratio in rough propo rtion 10 1/cos2 A. A denotes t he sweep angle, positive for sweepback. Thus t he deformations are generally larger, but not necessarily more undesirable, than those of comparable straighl wings. Indeed, the most interesting feature of sweep is the ae rodynamic coupling between flexure and torsion. It is easy to see that a smaU upward bending slope ow/ofj, fj denoting a coo rd inate meas ured out along the swept str uctural reference axis, is equivalent to an incidence - sin A ~w/ofj in a plane parallel to the flight direction. A positive lift increment on a Aell ible, sweptback wing usually unloads the tip region and displaces the center of pressure inboa rd and forward. Such a wing is incapable of diverging, since added torque due to positive twisting is more th an counteracted by the negative effect of the bending slope. Even 5 or 10 degrees of sweep angle ca n cause qn to go to infinity, the asymptotic value depe nding mai nly on the GJ/El ratio. This observation led Hill in England to the concept of the aero-isoclinic wing (Ref. 7-19), which has a combination of parameters that makes qo just infinite ; the spanwise load distribution is unaffected by aeroelasticity, with bending and torsion cancelling each. other identically. Sweepback obviously de tracts from lateral control effectiveness, because upward bending and nose-down twisti ng both work against a positively deflected aileron. As a result, most large·aspcct.ratio swcpt wings are fitted with spoilers. Sweptforward surfaces are fine from the control standpoint, but th eir very low divergence speeds seem 10 have con tributed heavi ly toward their abandonment in practice. The often predominant influence of bending is illuminated by the elementary example of a uniform, slend"r swept wing whose line of A.C.'s coincides wi th its E.A. In the absence of camber, there is no torsional deformation, and th e flellllral differ ential eq uation (7-11), for the

312

]'ltl NCII'U:S 0 ]' M :It OELASTIC]TV

steady-st,lte de lleetion

LI~!i)

LI nd er aerodynamic loading, reads

E1 d'", = [ If

dY'

+ C] cos A

(7-92)

Here L is lift force per Ur,!t distance normal /0 the flight direction, as it would be given by a theory based on ehordwise sections take n this way. Superscripts , and e refer, respectively, to loads due to initial incidence of th e rigid airfoil and t o those added by bending itself. We assu me that U (!il "" qcc{CiI) is k::nown and that U (y) can be estimated on a slrip basis from the aforementioned angle of attack due to bending slope, J!(y) = qCik'coS A[_ dw SinAl aoc dfi

(7- 93)

The derivative oc,Iooc.represents l[fi-c urve slope for a straight wi ng whose profile is the same as a section normal to the y-axis, with cos A providing the adjustment for sweep effect discussed in Sec. 3--4(e). Cantilever boundary conditions require wa nd dwldy to vanish at the root y = 0, while the second and th ird de rivatives aTC zero at the free tip fj = 1. Solu tion of Eqs. 7-92 and 7- 93 is facilitated by adopting

r ==

.,

dw

(7-9411)

d-

as the principal unknown and introducing dimension less quantities, -

1

_

qrP sin A cos 2 A

1]=

"~

-7fj

(7-94b)

Ef

oe,

-arl.

h = qcP cos A

(7-94c) (7- 94d)

EI

a and h are constants on the uniform wing.

In these terms, the differential equation and remaining boundary conditions are

,pr dir r (\) = 0

,

ar =

dr(O) = dii

-hct(if)

°'

tfr (O) = 0 dij!

(7- 95) (7-96a, b, c)

The \V = 0 root condition is applied whe n integrating l' to find w. Our prinCipal interest here is in the homogeneo us problem obtained by equa ting t he right-hand side of Eq. 7-95 to zero. Despite its simple

ONI'A>lMENSIONAI, STM Uc)'UMl':S

313

appearance, this is no longer II Sturm-Liou~il1e problem, because of the prese nce of the third deri~at i ~e, and it wo uld be incorrect to conclude that the eigenvalues a arc all rcal and positive. The homogeneous solution has the form (7-97) r = Ale'" + A 2e" 'l + A3e"'~ where

r, are the roots of the equation , 3_ii=O

(7-98)

When Eg. 7-97 is substituted into the three conditions (7-96), the requirement for nonzero coefficients A, turns out to be the following lranscendental equation: (7-99) After the three well-known roots r~ = {Iii, ' t = 1;(- 1

'3 =

+ tV3) {la,

H -1 - i-v3) {Iii are inserted into Eg. 7-99, it simplifies to

e(3l·) ~i + 2 cas (~j

{fa)

= 0

(7-100)

Equation 7-100 is not satisfied by any real, positive ii, because the exponential term is always too large Ie) be cancelled by the cosine; but there is a denumerably infinite sequence of negati~e eigenvalues. The first of these, (7- lOla) iio = -6.33 must be found by trial and error, while the rest are closely approximated by the vanishing of tbe cosine term, _ (211 + 1)31r3 G" = 3.)3 (n .., 1,2,3, ...) (7- 101b) By reference to Eq. 7-94c, we note the physical interpretation of these results that only sweptforward wings can undergo pure bending divergence. Corresponding to au, the dynamic pressure of the fundamental mode is 6.33E[

q n = .d';;'CI,C;C"CACIC'CO='~'A.-c'>"J"'C, 6.33EI

7c,"o","ACC;'-,f, "'. - c".,7,7;"- A -- ' 1

1

(A

< 0)

(7- 102)

where i = c cos A is the chord measured normal to the E.A. FOr the sweptback case, the bending divergence speed is always imaginary. By way of application, the last member ofEq. 7-102 refers to a cantilever wing of fixed planform dimensions, wbich can be Jotated abollt its root

3 14

l'IUN CU'Lf..5 01'

AI:!: " OEl,A~mC rrV

Line of A. C.'S of C. G.'s

JV

, Fir.1..(;. Swept wing of arbi!n..), planform and Iliffn=. showing aerodynamic moments acting on a section dy parallel 10 the ~nterlinc.

from A = 0° to 90°. Neglecting variations of ocJ8a; with Mach number, qD approaches infinity as I/l sin AI when the sweep is brought back toward zero. It is also infinite when the wing is swung all the way forward, because of the theoretically complete loss of aerodynamic efficiency. Between these limits qD has a minimum value of 12.66Ellcl3 ocd8a; at A ... _45°. Nonhomogeneous solutions of Eq. 7-80 describe the bending of the pre[oaded wing, either sweptbaek or sweptforward. They are found by adding Eq. 7-97 to the appropriate particular solution. For arbitrary e{(1/j, the latter is best developed fro m the standard integral formulas (pp. 52<)....530 of Ref. 7-5). If e{ is constant, the particular solution is just be/Iii, and we can easily work out the complete result by applying Eqs. 7-96:

(7-103)

In the analytical presentations comprising the remainder of this section, we shal1 focus on one-dimensional struct ures ha ving swept," rectilinear reference axes but with arbitrary spanwise variations of all properties. For use with modern dig ital computers, the most convenient mathematical approach relies on numerical approximation of the integral equations. As pointed out in Sec. 7-2, aerodynamic operators of lifting-line type fit in naturally. We introduce the airloads in connectio n with strips parallel to the vehicle centerline (Fig. 7--6), whose cross sections are assumed to

ON ll -J)IMENSIONAI. STlWCWUES

315

renmin undefo rnlcd, be~:a u sc such 11 viewpoint is espcciully e0111patihle with Weissinger's theory of s[l.1nwisc loading in subso nic High!. This choice of system is not intended to renect critically upon the many published treatments based either 011 the beam-rod differential eq uations or on integral equations with secti ons normal to the reference axis. In fact we should li ke to call partkular attention to two examples of the former wh ich are just as use ful today. for the continuous structures to which they apply, as they were when proposed a decade ago. The first is t he method of bending-torsion divergence calculation of Diederich and Budiansky (Ref. 7-20). Using strip theory modified by an over-all aspect-ratio correction, these authors have solved the cantilever beam-rod equations for two impor tant cases: the wing with uniform properties, and the linearly tapered wing with El and GJ proportional to the fourth power of th e chord [cf. H. Reissner's solution in Sec. 7-2(a) with Y. = I, Yt = 4]. Their solutions show the divergence eige nvalues to be governed primarily by the ratio of two stiffness parameters qiRcRJ! cos" A ae,

aIX

Gil{

where R identifies prope rties of the root station. We reproduce Fig. 7-7, which plots both theoretical and experimental results on a uniform, sweptforward surface from Ref. 7- 20 and points up the controlling influence of A.

A second procedure deserving special mention is Diederich's numerical integration of the differential equations for a cantilever with arbitrary spanwise stiffoess distributio ns (Refs. 7-21 and 7- 22). Reference 7-22 contains charts and approximate formulas from which such. things as span wise load distribution, lift-curve slope, aerodynamic center location, and damping in ro U can be corrected for ae roelastic effects. In order to synthesize integral equations for the deformations of the surface in Fig. 7--6, we must carefully distinguish betwe<:n barred quantities, associated with cross sections taken perpendicular to the structural reference axis, and unbarred quantities, related to streamwise segments. Either y or fI is suitable as a spanwise coordinate, but we choose the former in conformity with our emphasis on sections normal to it. Then it is clear, fOf example, that the bending deflection 1I"(y) and twist O(Y) about th e axis combine to produce a (small) rotation O(y) of streamwise segments, as follows: dw 8(y) = O(y) cos A - dfi si n A

-

d.

= O(y) cos A - ~ sinAoosA

d,

(7- 104)

316

I'HIN CII'UlS 01'

A~: HOIl "'.STI CITV

"" "'"

..

V/I-'

I '"

- n_

~60 o

•

~> o Experiment

,

0

\

,,

0

0

0 -A (Degrees) _________

Fit:. 1_7. Measurai divergence dynamic pressure 'In on a unifOlm, sweplforward wing compa~d with modified .lrip-theory calculations a. a function of sweep angle A , . (Taken from Diederich and Budiansky, Ref. 7- 20.)

To modify the general equations (7-18 and 7- 19) into a system containing wand 0 as dependent variables, we let (7-105) stand for resultan t upward force per unit distance normal to the fiight direction, acting at y - 'I. The total aerodynamic moment on the strip d7J, which is the resultant of Tg d7J and M~ d7J , is called T.(7J) d71, and to it mu st be added the inertia torque -m(7J)N(7J)g d(7J) d'l. Thus we arc led to the following formulation : w(y) =

f

C"(y, 7)[L(7J) - mNg] d1)

+Lc·,(y,7/)[T,.(71) /.l(y) =

f

- mNgd] d7J

(7- 106)

c"(y, 7) [L(7J) - mNg] d7J

f

+ C~O(y, 7/)[7;.(71) -

mNgd] d71

(7- 107)

ONI>·D1MI·:NSIONIIL STRUCI"UIU:S

317

The defi nitions of the stru ctural influence fUnetiolls in Eqs. 7- 106 and 7- 107 are dear from the manner in which they appear. When lifting-line or streamwise.strip aerodynamic operators are employed, this general approach gives the impression of having decollpled bending from torsion, because L and T~ depend only on the spanwise distribution of fJ. This is somewhat misleading, since bending-slope contributions are impHeit in (!l.,~, aod fJ itself. Nevertheless, it is a great convenience to carry out aeroelastic calCulations with the single Eq. 7-107 and to determine the flexural defonnations afterward from Eq. 7-106. Any advantage is lost when we seek information on shears, bending moments, torques, and stresses. The assumption, implicit in the foregoing, that streamwise segments are wholly free from camber bending is fuLfilled best when the ribs are oriented parallel to the vehicle centerline. Since mostlarge-aspect-ratio swept wings are laid out with ribs normal to some swept axis, it is reassuring to know that experience proves their camber bending to have very little influence on aeroelastic phenomena. Any lifting surface which deforms in this way to a significant degree is p robably best treated by the methods of Chap. 8. (a) Symmetrical load distribution and clIvergence

For the symmetrically loaded case. N is a constant, and the angle of attack can be separa ted into portions due to twist and initial incidence of the undeformed surface, o:(y) = fJ(y) + r'J.()(y) (7-108) The generic lifting-line operator (cr. Eq. 4-88), relates less running force, L(y):;; ely) _ ,s:t"{O(y) q' = c,'(1/) + et(y)

IX

10 the dimension-

+ <Xo(y)} (7- 109)

This is presumed to act along some l ine of aerodynamic centers, whose location may be influenced by the finite span, and it is accompanied by a momen t about the A.C. Therefore, the combined torque must be Tiy) = e(y)L(JJ)

+ MAdJ;) (7- 110)

When both airload expressions are inserted into Eq. 1-107 and the terms describing the effect of flexibility are separated out, we obtain

B(y)

=

q

f

C(y, l'/)cc1' d7J

+ I.(y)

(1-11111)

318

I' IU NC II'U<:S 0 11

where

A~:ROEI.AST ICITY

L' C(y, 1'/)cc{ L' [C'«y,

I.(y) = q

- g

L

dlJ

+ q e"(y, 1/)C·C ..A c dlJ

1'/)

+ cH(y, 1'/) d(IJ)JmN dlJ

(7-111b)

The quantity (7- 112) is a modified torsional influence function, which plays a role analogous to that of cH on a straight wing with a true E.A. C is evidently not symmetrical under an interchange of 11 and 1'/ . We introduce numerical integrations with a weigh ting matrix and approximate Eqs. 7- 111 through a series of operations similar to those in Sec. 7-2(c). Assuming that the simultaneous equalions

CWJ

L: e,}

(7-113)

= [ d'J{oc}

defined by the form of the aerodynamic operator, can be set up at or transformed to the same stations used in the structural analysis, we finally obtain

[.rI'']-I{C:J =qcR[EJ{C:;"} + {I.}

(7-114a)

The various matrices abbreviated in Eq. 7- 114a aTe the following:

'I

{I.} =qCR(£J\j" c~ +q[F]{C...... cJ

-

Ng[G]{m}

[t] ~ ([c"] + [C"] ~..J)lWJ [F] ~ [C"]~'\llWJ [G] ~ ([c"] + [C"]~
(7-1I4b) (7-114c) (7- 114d) (7-114e)

As discussed in connection with the straight-wing differential equation in Sec. 7- 2(a), we can conceive of solving Eqs. 7- l l4 either for a given incidence at the vehicle's centerline or for a given norma! load factor N. The former case is quite straightforward because .:lo(Y), ce{(y), and C."AcW) are known quantities, the rigid load distribution often being available from wind-tunnel tests, if not theoretically. Equation 7- 1I4a is manipulated to give the elastic loading

{';J

=

([d'r l

-

qcR(EJ)-I{/,}

(7-115)

ONl," I)IMI~NSIONAI,

STKUCTUJU(S

319

whence the resulla nt lift on Ihe full span and ils symmetrical spanwise distribulion are (7- 116) (7-117) If Eqs. 7-115 through 7-1 17 refer to a free-flight condition rather than a restrained wing. the final tenn in {I,} contains the unknown factor N, which must be eliminated by equating total lift to N times the vehicle weight W. (This involves neglecting tbe tail lift, but it can be introduced by the techniques appropriate to unrestrained elastic aircraft presented in Chap 9.) The most direct way of handling the problem of free flight is to replace cc,' by (CCI - cc{) in Eq. 7- 1l4a and make the substitution

[. . TtJ-{..}

(7- 118)

The reSUlting relation is

[....~ - 'i"·) ~ q'R[d"') + {..} +q[F]{,~o} CIl

Ng[G]{m} (7-119)

'C N

Eliminating N by means of the balance condition qSCL = NW, we are led to th e following solution for the lift distribution:

t:) ~ ([. .

T'-q'n[E]

+ 2'!'R[G]{m}L ULwJt

x ({eto) +q[p]{cm.dc}) (7-120) Total lift and pitching moments are detennined by operations like Eq. 7-117. One useful application ofEqs. 7-115 and 7- 120 is to compute wing aero· elastic .effects on stability derivatives. Consider, for instance, the lift· and moment-curve slopes, whicb are found hy adding a fixed increment aeto to the rigid.wing angle of attack and are unaffected hy MA O' Two situations are possible: the process might be carried out on a restrained elastic model, as ifmounted on a wind·tunnel balance, in whieh ease N = 1 and does not influence the incremental load acc,; or it might be done for free flight, in which ease the airplane must be balanced, and Eg. 7-120 applies. The solution of the fonner problem results from replacing U,} by aClol]cR[E][d'l{I} in Eqs. 7- 115 and 7-116:

{~c:,} = (~;:"} + ~'Xo'1cd"[d'rl -

QCIl[E])-l[£][ d1{1}

(7-121)

3211

I'HIN CII'L ES Oi' A"HO I!I.M;'I'ICITY

TI1C changes in total lom.l coeflicicilts nrc

LWJ(""'u )

6e, _ 2,,, L U s

tlC 111

= _ 2cll tall A (MAC)S

(7-122)

C

L -.J f-W 1(tlec l) Y L 'oJ

(7- 123)

cR

Here pitching moment is taken about the y-axis in Fig. 7- 6,· for which. the lift at station y has an arm - :J; -= -y Ian A; (MAC ) is the mean aerodynamic chord, used as a reference length. The actual derivatives are

oCL = tlCL = (OC T.T iJa. drxo O~ J

2

+ 2q;1I L

1-'

C'u(j([ d']

- I -

qc

R[fJ) - '[ £][ d

']{1 }

.~

~1.)

oC M =tlC M = ( OC M\' iJa. D..rxo a~ J _

2~~I:~~A L y-' LW,J([d,]-1_qCR[£] )-I[£][d' ]{J} (7-124b)

where the rigid-wing contributions, which correspond to the first term on the right of Eq, 1-121, have been separately identified, The static stability parameter about this moment axis js

oCJf = oC Nloa. oC I, oC [Joa.

(7-124<:)

For th.e somewhat more realistic example of free flight, Eqs. 7-120, 7- 122, and 7-123 yield the following derivative expressions:

aCr. = 2cR L

a.

s

I -'LW~ ( [d ] - 1 O

qCR[£]

+ 2,;,,, [C]{m)L 1 -.J ['w~r{l)

(7- l2Sa)

L y ~LWJ ([NT' - ",,,[E] (7- l2Sb) • The moment ca n be transferred to any other span wi.., axis by the standard mcLbod.

ON~~I>IM.:NS t ONAL

STItU(;I'Ult t,s

32 t

The read er will h~lVC no uifiicu lly in rcorg:lI1i zi ng I:qs. 1- 125 so that the rigiu-win g ucrivativcs appear explicitly, if hc will first separatc out the clastic part from Eq. 1- 120 before carrying out the tota l lift and moment integrations. On a typical slreplback wing, the bending deformations und er load overpower the twist, tending to reduce the lift-curve slope and the static stability (Le., rendering oCMlorx and oC",/OCL more positive). Tbis fact is evident from Eqs. 1- 124, if we observe that the principal contributions to matrix [E] are from [c*'], whose elements are ob viously negative. The converse is true in the sweptforward case. It is interesting that the inertia loads resulting from a symmetrical fl ight maneuver partially counteract these static aeroelaslic effects, regardless of the sense of the sweep, as can be seen from the opposite signs of the terms containing [E] and [0] in Eqs. 7- 125. Pai and Sears s~m to be the first to have analyzed this phenomcnon quantitatively (Ref. 7-23). Nume rical examples computed by the integral-equation-matrix approach will be found in their paper, as well as in Sec. 8-4 of Ref. 1- 2. Nothing has been done in th e foregoing about the case of fl.ight at a prescribed value of the normal load factor. This situation can be handled, howevcr, by substituting the known N and an un known centerline angle of attack .xa(0) into Eqs. 7-(14. Eq uation 7-117 is then employed to compute "
322

1' lUN CII'UiS 01' M:HORLAS'I'ICITV

." . . .-6'L'lyl for rigid wing 1----- -----.0.1. U y ) for elaslic " ,ng . ............ ,

~"

'- "

~

,

"

"

,.,

,.,

y,,"y!l-_

Fig. 7-8. Spanwi&(: load distribution. with and without aeroelastic effects, on the right wing of the XB-47 in a 3$ puU-out al M _ 0,8 and 27,OOO-ft altitude. (Taken from Pian and Lin. Ref, 7-24.) provide fairly good agreement with the more precise computations mentioned above, The divergence problem for swept wings and one-dimensional lifting surfaces without clearly defined E.A.'s, which form tbe principal subject matter of this section, is commonly analyzed by "brute· force" numerical calculation of the eigenvalues of the integral equation. Although the distinction was not brought out in Sec, 7-2, there are actually two different types of symmetrica l bending-torsion divergence. The more familiar static divergence coincides with the blowing up of th e elastic loading or twist described by Eq. 7-115, that is, at the roots of the determinantal equation (7- 126) The lowest positive value of qR' if any, defines an aeroelastic stability boundary for a wing that is either restrained or mounted on a vehicle flown at a fix ed normal load faclor. In uncon trolled free flight, however, Eq. 7-120 applies, and the eigenvalues mus t be found from 1 [d']-1

- qj/(cR[l] -

2~R [G]{m}L W..J )1 - 0

(7- 127)

ONlo;. l)IMI~NSIONA"

STHuc n .l HES

323

This secolld i l1~tability, associated with uy namic press ure q'/' is sometimes called Ilymllllic dilJ<" gellre. lIs onset wou ld be accompanied by a looping maneuver which went into an ever-tightening spiral as the increasing structural deformations augmented the acceleration Ng. Both gR and gR' are likely to be negative on swept-back surfaces. But in the sweptforward case the problem is a severe and real one. As may be guessed from the opposite signs of the two terms in parentheses in Eq. 7-127, it usually turns out that q R' > gft. During the approach to dynamic divergence, the downward inertia forces tend, in part, to counteract the positive bending caused by upward airJoads when A < O. With high-speed computing machinery, there are many ways of determining the roots of Eqs. 7-126 and 1-127; these will not be discussed here. A trial-and-error procedure is generally indicated, since Machnumber effects must be induded in the aerodynamic matrix. Primarily because lEI and leI are not symmetrical, matrix iteration is, strictly speaking, not applicable. Nevertheless, satisfactory convergence will usually be obtained in cases of practical significance, such as sweptforward wings, where the eigenvalues are positive. Divergence a,nd loading of large-aspect-ratio swept wings have special theoretical interest because they involve the solution of adjoint differential and integral equations. General information on the theory will be found in Ref. 7-5 (Chap. 5 and Sec. 8.2). We mention here one simple illustration, which can be given a physical interpretation. In the case of inslability at a constant normal load factor, the homogeneous form of Eq. 7-1l1a applies,

l'

&(y) = q

C{y, 'I))cc,' d'l)

(7-128)

For the moment, suppose that we neglect twisting and adopt the striptheory aerodynamic operator, Eq. 4-110, to rewrite this in terms of the single variable O. O(y) = q

J'

a, cos A C ' (y, 1)cO d'l a, 0 -

!

9

(7-129)

Equation 7- 129 is self-adjoint only if C'(;j, I) = COt('1, y), but this is IIOt true of swept structures. The adjoint integral equation is, therefo re, obtained by interchanging the variables in the influence function, w(y) = q

a, cos Af.'• C'(1), y)cw d'l _I

a.

(7-t30a)

The reason for choosing the symbol w in Eq. 7- IJOa becomes apparent

324

I'IU NC II'U;S O il AmWE\.ASTICI1'V

whell we observe that. by the reciprocity re la tion for linear el:lstic systems, C"'(,/, y) = ~(y, 11), and

a, a.

w(y) = q - ' cos A

l'•

C ' (y, l1)cW dl1

(7-130b)

Since cce(y, 1/) represents the bending displacement at 11 due to unit ooncentrated stream wise torque at Tj, Eq. 7-1 3Ob describes the bending of a wing which is somehow loaded with distributed torques everywhere proportional to cwo Thus there is a meaning, albeit somewhat artificial, to the adjoint problem of Eq. 7- 129, which itself describes streamwise twisting of the same structure under normal forces proportional to c/J. As proved in books on integral equations, the eigenvalues of Eqs. 7-129 and 7-130 are eq ual, if real, and conjugate, if complex. Moreover, the eigenfUnctions corresponding to distinct eigenvalues are orthogonal with respect to the weighting factor c. That is, if 0,(y), 9/ and WiY), 9, are solutions of Eq. 7- 129 and Eq. 7- 130b, respectively, (7-13!)

This result has particular importance in connection wi th methods of approx.imate solution based on superposition of eigenfunctions, because it is definitely not true that tbe integralS of ({;\0 1 and CW,W1 vanish. All the forego ing ideas are readily extended to the more general integral equations of swept wings and to the matri x relations which are obtained by numerical integration thereof. For example, the adjoint of Eq. 7-128 has the kernel function C(,/, II), when aerodyna mic strip theory is used. The identity of eigenvalues and orthogonality of adjoint eigenfunctions (or eigenvectors) is preserved, but the possibility of simple phYSical interpretation is lost. The practical value of biorthogonal eigenjWlctions, as the sets of adjoi nt solutions are sometimes called (Refs. 7- 5, 7-26), is brought out when we introduce the idea of ge neralized coordinates for solving the nonhomogeneous integral eq uations. A variety of such procedures is discussed in Chap. 3, the two in most common use being those of Rayleigh-Ritz and Galerkin. In self-adjoint problems, Rayleigh- Ritz consists simply of substituting a series of approximating functio ns into the variational statement of the principle of mini mum potential energy. It can also be adapted to swept surfaces (cr. Flax, Ref. 7-25) but will lose its elementary physical interpretation . We illustrate here the Ga1erkin method, applying it to Eq. 7- 111 a.

ON I~· DIMIlNS I ON"' I.,

rcpre.~C!lIHli ons

Let us lISSllnJe the rollowing distributions:

0(1/) =

c,'(y)

STItUCl'UIII':S

ror the twiM lind elastic lift

•

,.L,0 ,(Y)ll,

(7- 132,,)

•

=

325

,.L,c,lY)ii,

(7- 132/))

Here the 0 ,(y) satisry the boundary conditions 0(0) = dO(l)/dy = O. Em;h member of the lift coefficient series is calculated from the corresponding 0;(y) by whatever aerodynamic operator has been chosen, c,,(y) = s.ot"{ 0,(y)}

(7- 133)

When substituted into Eq. 7-11la, Eqs. 7-132 yield

.t

i

(0,(y) - q

('(y, 1/)cc" d1/}ol' = /,(y)

(7- 134)

One version of Ga)erkin's scheme is to solve for the coordinates iii by requiring the two sides of Eq. 7-134 to be equal, in the mean, when weighted in turn with each of the assumed twist functions. Mu ltiplying by c(y)0 )(y) and integrating over the span, we obtain

1",A'Jol; =

..

where

Au =

Bj =

f f

U =I.2. · ·· ,n)

(7-i3S)

f f

(7-136)

Bi •

c0,0; dy - q

e0;

C(y. II)ee" d1} dy

(7- 137)

C0 ,/,dY

Both the twist and lift distributions are calculated by solving Eqs. 7- 135 for the oli' th en inserting these constan ts into the series 7-132a and 7-132b. A more efficient variant orthe foregoi ng is to choose for 0,W) t he eigen. functions of the homogeneous integral eq uation (7-128) and use the adjoint eigenfunctions Wiy) in the weighting process. It is then known that

Jf' C(y, 1/)[("1, d1"j = - 1

0,(y)

(7- 138)

4o,

o

qlJ, being the associated eigenvalue, and the coefficients A." in Eq. 7- 136 are replaced by

A;;

=

i'c8,W1 dy - q

= [I -

L'CWJR' C(y, 1/)cc" d'l dy

q:JfC0iWldY

(7- 139a)

316

I' IH NC U'I ,ES 0 1' AtmOm,ASl'ICI'I'Y

Hence, (7- 1J9b) (i = j)

l'

Sinee they have been deooup[ed by the orthogonality, Eqs. 7-135 can now be solved independently, B cWJ. dy (7- 140) ' ,,--~''-,-7C--­ qJ = JJ = j qn, 0 Thus one laborious step in the ma the matical process is avoided in eases where the biorthogonal eigenfunctions are known. On sweptforward wings, for which thegn, are us ually real and positive, Eqs, 7- \32 and 7- [40 indicate clearly the divergence of th e solution to infinity at each critical dynamic pressure. Sweptback wings often have only real, negative qD,'s, so that the bracketed factor in the den ominato r of Eq. 7- 140 represents an attenuation of the defonnation- the familiar inftuence of the bendi ng slope. Indeed, in some simple examples, such as a sweptback surface which bends without twisting, IqD,1can be proved to be the jth di vergence dynamic pressure of the same structure sweptforward at the angle (- A). For a more tho rough and rigorous treatme nt of the solution of sweptwing aeroelastic problems by means of biorthogona[ eigenfunctions, the mathematically-minded reader is du-ected 10 Seifert's article (Ref. 7- 26).

A [I _ .!LJfc0 W,dY

(b) Antisymmetrical load distribution; control effCl.:tivenes.<;

When we consider the lifting surface in Fig. 74;) to be rolling about the x-axis with angular velocily p (positive to swing the right tip upward), the purely antisy mmetricalload distribution arises from only three effects: antisymmetrieal twist rJ(y), deflection 6(y) of a lateral control device, and p itself. Therefore, the development followed in Sec. 7-2(c) can be adapted for our purposes here, with a sligltt redefinition of symbols. Equations 7- 64, 7-65, and 7-66 are unchanged, except that the aerodynamic operator .vi may have to include sweep effects. The normal load factor N(y) is determined by the rolling acceleration, according 10 (7- 141)

Ng = F.!!

Turni ng to Eq. 7-107, we make a series of substitutions similar to those which produced Eqs. 7-11 1 and are led to O(y) = q fC(Y, '1)cct d"'1

+!.(y)

(7-142a)

ONI;·I)I,\mNSIONAI , STKUC'I'UKES

where

I[ '" . '"

!..(y) = qJDr'~ '-\7!. 11 Cad

"u

+ Ca(~)

327

Pi] u d'l

+ q i'C"(Y, 1J)c' ac;:o tJ1/ d'l - P f (c''(Y. 1)

+ C"(y, 1)

d(1))]m 1) d'l

(7-J42b)

tJ R is the eonlrol rotation at some reference statio n, and C is given once more by Eg. 7- 112. Introducing a numerical integration formula with the weighting matrix

['W-J ' along with the antisymmetrical aerodynamic matrix

[d~l, we derive the following system of simultaneous equations for the r unning lift at the y.-stations ~

[dOr1(CC") =qcu[£'J(CC!,) + {I.} cn

(7-143a)

(;1<

T he column matriJl of "known" qua ntities on the right is evidently

{I. }

~ q[E1~,J ((~:,) '" + (,(~)) ~)

+q[FJ(",;C)', - [GJ~YJlmlp

(7- 143b)

wi th the three elastic-geometric matrices [£'], [F], and [0] still those defined in Egs. 7- 114c, d, e. By direct analogy with Sec. 7-2(c), a number of conclusions can now be reached without recourse to detailed derivations. The motion of the vehicle in response to a sudden deflection 6n = 6n l(t) of the roil controls is given by (7- 144) where AI' R aod c't are the constants from Egs. 7-78b,c,dwith bars added " matrices [da]. [E], [F ] , and [GJ. The asymptotic form of over the square Eq. 7-144 corresponds to the steady-state rollin g power of the elastic

31H

l'IH NCl J>I ,I'~"

OF AEItO I·;LAS·I'ICITV

swcpt wing and can be written

(7- 145)

Here C'6 and C'. are the control-surface derivative and damping in roll of the rigid wing. The dynamic pressure for reversal occurs at the van ishi ng of the numerator in Eg. 7-145, expressed by the implicit fonnu la

,, ~ - L H.J([""or' - q"R[E]J-' ( [E J~.J I;;') + [Fll"",,)) (7- 146) Antisymmetrical bending-torsion divergene<: is a constant load-factor phenomenon and therefore takes place at the fundamenta l positive eigenvalue of Eg. 7-126, [d' ] bein g replaced by (dO) and adjustments made to the structural infl uence coefficients for the effects of antisymmetrical loading, if they are required. To illustrate the severe deleterious influence of swccpback on the operation of trailing-edge controls, we reproduce the results of two calculations on the XB-47 airplane. Figure 7-9 shows the antisymmetrical portion of the load ing on the right wi ng, both flexible and infinitely rigid, at the start of a maneuver conSisting of a sudden lOa aileron deflection during a 2g pull-out. Flight Mach number is 0.72 at an altitude of 5,000 ft in the standard atmosphere, and the gross weight is 125,000 lb. Geometry of the controls and other detai ls of this calculation, wh ich is based on the same integral equation studied in the present section, appear in the paper by Pian and Lin (Ref. 7- 24), from which Fig. 7-9 is ad opted. Our second example, Fig. 7- 10, is taken from Sec. 8-4 of Ref. 1-2. It shows how the ai leron rolling power varies with M for the XB-47 flying at sea level and

.- ""' Ir

, I ".'I--y / V/ ' -~

M

1."1) '1 lor "l id

0

~-)()OO

o

--f:: 0.2

I

0.4

,-y/I -

"-- Llyll"

.1 "t~, ";01 08

0.6

' .0

Fig. 1_9. Ant;~ymmo:trka l load d istribution, with and without acroclastic effects, d ue 10 s udden HI· aileron ~/kct ion on the ri~h t wing or the XB-41 in a 2g p ull_out at M _ 0.72 and 500().ft altitude. (Taken rrom Pian and Lin, Ref. 1-24.)

indicates reversal at a speed not far fro m the maximum pe rformance of the bomber. Needlc:ss to say, resul ts like th ese were influential in leaditlg to the adoption of spoilers for the B-47 and B-52. Perhaps as undesirable as th e lo w reversal condition is the manner in which the effectiveness drops ofT more mpidly with M fM JI. than the parabolic depc:ndence obsc:rved on straight wings. A comparison between Figs. 7-5 and 7- 10 illuminates th is point. In addition to the publications cited above, the reader will find extensive information on antisymmetrical loading of swept wings in Re fs. 7~27 , 7- 28, and 7~29. 0

'"

I "'I" "'"

0' 0 0.' 0

l~

~

1

I

0

~O.l 0

o

0.'

0.'

0.'

0.'

"-

~

0.'

0.'

Fi,. 1_10, Aileron effectiveness as a function of Hight Mach number for the XB47 Hying at sea level. (Taken from E~ ample g·9 of Bisplinghoff, Ashley, and Ha lfman, Ref.1 -2 .)

330

,' IUN CII'U.;s 01<'

AERO~:L"STlCI1'V

7-4 CHORDWISE DEFORMA nONS AND VERY SLENDER CONFIGURATIONS Sections 7-2 and 7-3 were concerned with the relatively familiar and long-understood static aeroelastic propertics of large-aspcct-ratio wings and tails. To the list of one-dimensional problems there have been added more recently those where the deformation depends principally on the single cho rd wise coordinate:l:. The attainment of supersonic flight has augmen ted in a number of ways both the catalogue and the severity of suc h problems. For one thing, lifting surfaces of small to fractional aspect ratio are the optimum solu tion for many high -speed design requirements. The majority of these must be treated by two-dimensional structural methods,but some are slender enough so that the spanwise variations of their bending deflections are negligible. Most missiles also fall in this category of slender configurations. Another well-known consequence of supersonic Mach numbers is that the local aerodynamic loading often grows in nearly direci proportion to the local surface angle of attack or chord wise slope. This behavior, which differs entirely from the subsonic regime except when the aspect ratio is very low, implies the possibility of a new type of divergence. Moreover, we hardly need mention the enormously larger dynamic pressu res that usually go with supersonic speeds. We are now entering an era when rapidly accelerated missiles and entry vehicles will encounte r as great dynamic pressures as will probably ever be attained by controlled flying machines within the earth's atmosphere. Somewhat later, highperformance submarines will be setting similar records for liquid media, albeit not supersonic speeds. The .aeroelastic problems which arise, accompanied by intense aerodynam ic heating in the case of entry, may cause our earlier experiences to look like child's play. The differential and integral equations which govern the chordwise deformations differ from those presented in Sec. 7- 1 mai nly by a change of independent variable from y (or s or YJ to:l:. When analyz ing simple bending, for example, we overlook rotary inertia and shear deflections, rewriting Eq. 7- 11

" [ "w] or + m(:I:)iI'(:I:, /) =

o:r:! £I

F.(:r:, /)

(7- 147)

where Fi:r:, /) is the running external load and m{:I:) the mass per unit chordwise distance. There are two possible interpretations of Eq. 7- 147, corresponding to the two systems which we shall examine in Sees. 7-4(a)

ONt;'IlIMENSIONAI,

STRUcrUIU~

33 1

ami (b) below. Either we cun consider chordwise (camber) bendi ng of a unifo rm, relatively large aspect-ratio surface 011 a structurally one-d imensional (plane-strain) basis, or we may be dealing with a genuinely slender configuration. In the former instance, m(z) represents mass per unit area, and El(z) is the component of flexural rigidity of a plate resisting curvature about a spanwise axis. Antisymmetrical deformation of slender wings and bodies consists of torsion about the longitudinal axis and side bending. Torsion is subject to a differential equation resembling Eq. 7-1 5, with the differential bending term omitted,

,[ "Jaz

-

az

GJ. -

- 1/z)O(z,l) = -TJ.z,t)

(7- 148)

The meaning of these symbols is obvious. It should be noted that the torsional problem appears less important in practice than bending. In fact, there are situations involving high longitudinal accelerations (e.g., the anti-missile missile), where time-dependent axial defo rmation u(z, /) may be much more significa nt than torsion. The appropriate differential equation reads (7- 149)

where A(z) is the cross-sectional area of structural material and F. is a running longitudinal force. Equation 7-149 has homogeneous solutions of the longitudinal-vibration type, which usually occur at high frequencies compared to lateral bending but may be violently excited by certain impulsive loads, such as the starting thrust of a rocket engine. For missile structures composed of thin cylindrical skins, Eq. 7-149 is oversimplified, and the problem must be approached by means of shell theory. An integral equation fOf chordwise bending, in direct analogy with Eq. 7-16, can be written

w(z, t) = fencz, ~)Z(~, I) d~

(7-1 500:1)

where Z is the resultant running load, inertia forces included. Since slender configurations are usually encountered in the free-free condition, w(z, I) is then the elastic displacement relative to a reference surface tangent to the structure at a given station. wis not the absolute normal acceleration, because it also receives contributio ns from the rigid-bOdy accelerations in vertical translation and pitching. Sometimes the slope

332

I'Ml NCU'I .ES OF A- l' MOl' I.A-STl Cl'f"V

a., = -ow/a", represents a more useful variable. In such cases, the derivative of Eq. 7- ISOa with respect to x may be used.

"'lx,

r)

=

fC"(X, ':)Z(':, /) d;

(7- 150b)

with C"(x,

n=

- :x C"(x,':)

(7- 150c)

The remainder of th is section discusses static problems which illustrate what can he done with a one-dimensional chord wise model. The first of these relates to the curling up and divergence of thin leading edges in supersonic flow, emphasizing elementary profile shapes that are amenable to differen tial equation solutions, Next, both the differential and integralequation approaches arc exemplified in conne<:tion with the chordwise stability of slender vehicles. (a) Cbordwise bending at supersonic speeds We consider a thin, uniform structure with a chord wise cross section of the form shown in Fig. 7-1 1. The intention is to represent the forward half of the' profile of a straight wing in a supersonic airstream, replacing the connection to the rearward half by a cantilever base. The section is solid and has a slightly blunted leading edge, because, as we shall see, chordwise stabili ty is strongly affected by thc closeness of approach to mat hematical sharpness. There is a mean incidence "0 at midchord. The aspect ratio and structural arrangement are assumed to be such that the

---'- . ..

-J~ " '-

Fig. 1-11. Cross '-"'Ction of forward half of an a two-dimension al superson ic a irstr<:arn.

--

airfoi llln~rgoing

chordwioe bending in

ONI"'!)IMENSIONAI. S·I·ItUCr UItI",o;;

33.1

flow can be approximated Iwo-dimcnsionnlly und the mcan-surfacc bending onc-dimcnsionally according (0 Eq. 7- 147. All angles arc small, Eliminating timc-d<:pcndent quantities from Eq. 7-147, we obtain (7-151)

Here E1 = Et(! - v3) is a reduced modul us of elasticity for the assumed state of plane strain, v being Poisson's ratio. Since the equation refers to a unit spanwise segment, 1= ;Z,3 (7- 152) in dimensions of feet' per foot. For convenience in a case of linear chordwise taper to be treated below, the origin of z in Fig. 7-11 is placed at the vertex of a hypothetical point added onto the leading edge; the actual chord is c = 2b. We are able to include in the analysis the important second-order aerodynamic <:ifects of profil<: thickness, which yield a linear relation between upward force per unit area and the slope (dll'/dz - 0:0), contai ning a nonlinear factor dependent on the thickness distrib ution z,(x). Busemann's theory (Ref. 7-30) would provide tne rigo rous ly correct form of the operator; but an eX<:ellent approximation for all but the lowest supersonic ~peeds is fou nd by replacing M with

p - -';'f' -

1

(7-153)

in the denominator of the piston theory operator, Eq. 4--70. This step gives·

ap.(x)~ j[t + (r; I)M~~J [- ~: + o:oJ

(7-154)

Since w itself will not appear in the differential equation, we work in terms of the bending slope

and also employ a dimensionless independent variable (7-156) • It should he noted that tlte generat property of supersonic flow involving no upstream influence on aerodynamic load ing provides an excellent justification fo~ analyzing deformations of the forward hatf of the profilc wi thout regard for what occurs to the rear.

334

I'HtNCII'LES 01' AEHOf:l.AS·t'tCJTY

Inscrting Eqs. 7- 152 through 7- 156 into E(I. 7- 151, thc foll owing dimcn· sionless relation is obtained: (7- 157)

where (7-158)

and zlR = z,(l) is the semi-thickness at midchord. Boundary conditions for the free edge at i == a l and clamped midchord line at i = I read (7- 159)

Equations 7- 157 and 7-159 define a nonself-adjoint mathematical problem reminiscent of the pure bending of the swept wing, discussed in the introduction to Sec. 7-3. One achieves the exact analogue of a uniform sweptforward wing by spe¢iatizing to the trivial case of a flat-plate ai rfoil (ZI = Z,R, dzddx = 0) and selling the origin at the leading edge (a l = 0). Then, kJ becomes identical with the parameter (-a) in Eq. 7-95. From Eqs. 7- 101, the eigenvalues of the homogeneous form of Eg. 7-157 are known to be 6.33, (n = 0) (7- 160) kl • = ( (211 + 1 )3~ (~l 2 3 ... )

3.J3

'

n

,,'

Each of these corresponds to a possible mode of chordwise bending divergence of the fiat-slab wing. According to Eg. 7-158. k i • furnishes the lowest dynamic pressure at which static instability is possible qn =

In view of the relation q =

1.055PEl(~J

(7-16Ia)

~ p .,M2, this implicit representation can be

solved for the divergence Mach number (7-161 b)

where (7-161c)

ONI,.,OIMENSIONAI..

STH UCI'U IU~

335

(One limitation is K > 2, as it onJin:Lrily will be because of the magnitude of £ ._ The root with the minus sign in Eq. 7-161b is likely to be physically meaningless; it occurs ncar M /) = I because of the smallness of fJ in the de nominator of Eq. 7- 54, and Ihis is a range where the aerodynam ic theory is invalid.) Finally, Eq. 7- 103 gives US:, for the deformation of the plate under load at q < qt),

f el f" _ t, + 2e - 11 f' _ t , eos eif~) J3

. 'l

IX

(x) =

IX

{

.

2 - 1f"=I;: cos2J3 (3~ kl) ef" _ t , +e

-\

(7-162)

Equation 7-16 1a reveals the proportionality between dynamic pressure and elastic modulus which character.iz.es all divergence formulas for distributed-parameter systems. Somewhat more unusual is the ve ry strong dependence on the profile thickness ratio z,/b. In his exhaustive studies of the chordwise-deformation problem (e.g., Refs. 7-31 , 7':'32, and 7-33), Biot has solved Eqs. 7- 157 through 7-159 for a trunca ted, tapered wedge having

.

-

z, = ztJ/- = z,# b,

(7- 163)

In this case dzdd~ is a constant related to the thickness ratio of the profile, so that k\ is independent of z, The particular imegral of Eq. 7-157 is simply (7-164) "'" = -<10 and Biot therefore concentrates on the homogeneous solution and the related question of stability. Equation 7-163 leads to the following restatement of the eigenvalue problem: (7- 165a) (7-165b)

This differential equation is of the equidimensionai type, and we can easily show that it is solved by (7- 166) "'. = 1 CnX'"

ft -'.2."

where

inM

= z. - 1 and the

Zs

are the three roots of

z(ZZ _ I)

+ k,

_ 0

(7-167)

336

l'IUNCIl 'LBS 011 AEHOm.ASTICITY

Sin~ek,

>

0, Eq. 7- 167 must have at Icast onc rea l root, %1' which is Icss th an -I. %, can be plotted versus k, by substituting a series of val ues of 2\ into Eq. 7-167, and it varies from -I when kJ = 0 to - ~ when kJ is large. It fo llows from the conditions z, + 22 + Z3 = 0 and %,Z2 + 2 1%3 + 2M = - I tbat the other two roots may be written (7- 168a, b) ~

and ~ are real or complex. conjugate, depending on whether 2 1 is algebraically greater or less, respectively. than - 2/YJ ;;;;: - 1.l548. Through Eq. 7- 167, this condition -corrcspoQ.ds to k I being less or greater than 2/(3V3) = 0.3849. By applying the boundary conditions (7-J65b) to t he solution. Eq. 7-166, Biot (Ref. 7- 32) constructs the eigenvalue problem for kJ and proves that, when 0 < a 1 < 1, no l eal k t arc possible for real Zu and 2:1. Hence the fundamenta l k t, exceeds 0.3849 always. With the latter can· dition fulfilled, the characteristic equation becomes

Dzt['3 + i](.!.)3~L zt - l at

+

2 +32) ~m .

2

(V In-.,I)

+ D[2zt + 1] cos ( Din ;)

= 0 (7-169a)

where (7-l69b) Having found %, as a function of a l from Eqs. 7-169, we can compute the eigenvalues by substituting zt into Eq. 7-167. There is, of course, an infinite number of roots for each bluntness parameter at; but only the fundamental is of interest. Biot defines a second characteristic parameter, based on the a~tual semichord,

(7- 170) and gives the following table of kn, corresponding to the minimum diver· gence dynamic pressure. Critical dynamic pressure or Mach. number can be romp uted from Eq. 7-170 as in the case of the sla b with constant thickness.

ONI(" I)IMENSIONr\L STItUC)'UHES

331

TABLE 1-1 Critical value of the stability parameter for chordwlse divergence of a two·dimension ...1 $traight wedge, vs. bluntness parameter (I ,

"

kD

0 0.000016 0.0204 0.069 0.1055 0.117 0.241 0.460 0.582 0."" 0.712 0.823 0.981 1.000

0.3849 0.528 1.04 LSI 1.19 2.25 2.63 3.78 4.39 4.74 4.99 5.51 6.27 6.33

(b) Slender aircraft and rnbI;iles treated as one-dimensional structures Figure 7-12 depicts a generalized Hight vehicle of the type we shall examine next. Since dynamic phenomena are touched on in Sec. 7-7, we restrict ourselves here to the static case and drop terms containing iii from equations of motion such as Eqs. 7- 147 and 7-150. If the product of Mach number by the maximum local angle between the flight direction and planes tangent to the wing or fuselage surfaces is small enough compared to unity,· the slender. body aerodynamic operator (4-97) furnishes the required rnnning toad per unit x-distance. Differential equation (7- 147) then becomes

d'w]

d( [

dWJ)

(7-171) -d' [ EI(x) - 2 = 2q- S(.:t') exo - dxZ dx Jx dx The cross-sectional virtual mass p",S(x) can be taken from one of Eqs. 4-100 through 4-102 in most practical applications; for inslance, Eq. 4-101 yields S(x) = 1T[s2(.:t') - R2(x ) + R'{x)fs2(x)J

when symmetrical deformation of the midwing-body-of.revo[ution Fig. 7-12 is analyzed . • The great""t local slope usually occurs along the wing leading edge.

In

JJ8

I'HI"': CII'I.ES OF AI, ROEI.ASTl CI'n'

I 3-u

--

FiB. 7-11. Three views of a slender configurntion, con!iisting of a body of «volution centered ;n a thin wjng and undergoing small symmetrical bending deflections ".(.rl «Iuli,·. 10 a plan. tangent at .orne ref.rence station.

Among the 5Cveral published treatments of the static problem, tbe reader will fin d those by Dugundji and Crisp (Ref. 7-34) and Martin and Watkins (Ref. 7- 35) especiany illuminating. Reference 7-35 presents a scheme for numerical integration of Eq. 7-171 that can be adapted to arbitrary chordwi,e distributions of bending stiffness and to aerodynamic operators more genera! than slender-body theory. For fiat, triangular planfonns having I(:r) propnrtional to any power of:r, Martin and Watkins also furnish closed Bessel-function solutions to Eq. 7-1 71 witb "-0 = 0 and calculate chordwise-divergence eigenvalues corresponding to boundary conditions of clamping or pitch-spring restraint at the trailing edge :r = I. Because their procedure has ~rtain parallels to Biot's work discussed in Sec. 7-4{al, we shall not go over it in detail but rather turn to a modal analysis of divergence in free flight ( Ref. 7-34). As pointed out in Chap. 3, a ve ry natural choice of assumed shape (such as the y, in Eq. 3---103) to approximate the deformations ofa onedimensional or two-dimensional structure consists of its normal modes of vibration ill tYlCUQ. This is true even for static problems, because normal modes still satisfy the elastic boundary conditions and their orthogonality eliminates off-diagona! terms from the [.9"] matrix. The autbors of Ref. 7-34 consider both flutter and divergence of tbe slender, plane

ON.;.. mMI' NS IONA I. STR UC.1·U IWS

339

triangular (delta) wing which has uniform mass amI stiffness distributions. so that both II1(X) and 1(x) vary directly with x. Thus we can write (7-1720) (7- 172b)

e

= where subscript zero refers to the trailing-edge station, and Free vibration in the ith normal mode 'P,(e), defined according to

:1;/'.

(7-173) is governed by the homogeneous fonn of Eg. 7-147, which is easily reduced (7- 174) The characteristic parameter is

8;

=

wJ~vmolElo

(7-175)

Dugundji and Crisp solve Eg. 7-174 subject to the free-flight boundary conditions

tr..".

",

Moment """ ~ - ' = 0 Shear "'-' !!...(~ d~

If'de{l,)

(~ =

0, I)

(7- 176)

The re are, of co urse, two rigid-body degrees of freedom associated with the zero-frequency nonnal ized modes If' = 1, If! = [I - { n The elastic modes are determined by truncating a power-series solution, and each has the form 'P,(.:)

=

1+

Q:,jCi

+ ~es

+ ?'·lt~1! + ... + ~~ +?':s~~ +

~~ + ...

(7- 177) The ?': K-coefficients tabulated in Appendix D, Ref. 7-34, for i _ 3,4, and 5, show satisfactory convergence. Eigenvalues corresponding to them are 83 = 28.80992, 84 = 73.21273, Os = 137.4223. Orthogonality requires that (i "* j)

Given the

'{I;

(7- 178)

as computed above, we represent the total z"displaccment

)40

]'UINCII'I.ES OF' AEUOE I.A S Tl CI'I'V

rcla t.ive to:lIl inertial coonJinate sys tem in the manner alrcmly cxcm plitlcd by Eqs. 3-96, 3- 103, 3- 1 1J. and 3-127: (7- 179)

The q; are independent, dimensionless generalized coordinates. Therefore, Lagrange's equatio ns provide a description of the aerodynamically loaded system (ef. Sec. 2-6). By applying Eq. 7-178, together with the fac t that the maximum ki netic and potent ial energies are eqnal during free vibration, we calculate for the potential function U U

=

!moll!q/Wi~Jl,'I'/ d, .- 3

(7- 180)

0

The generalized force cxerted by the running lift on the ith degree of freedom is

Qj =

l~fL(
=

d,

(7-181)

where the operat or (4-97) has been substituted. In th e last member of Eq. 7-181, only loading dlle to the displacements themselves is shown, although an additio nal "disturhing" load can easily be added. From Eq. 4-100, SW =

1l's!m

= 1l'1~~tan2A'

(7- 182)

i\' bei ng th e semi-vertex angle of the del!a planfonn (the compleme nt of the conventional sweep angle i\). By substit uting Eqs. 7-180 through 7-1 82 into the general Lagrange equation (2-67) and eliminating dimensinnal factors, we deriv e the homogeneous equations

(7-1 83)

([5") - ["'D{,,) - 0 He re the individua l dements of the aerodynamic matrix are

-1'1/(1)'1,(1) + i f e"./(f)'1/Wd;

A;j =

while those of the structural matri x may be S

"

~

(

moM/ 4Q1l' tan -' ""

Elo) (

)1', 0

writt~n

(I) · (I) dl I{,

0,', ,) r'",'(' )d',

= (( qJ~41T tan- A 0,

Ifj

(i=I ,2

(7- 184)

Jo

or

if=) )

( i = j)

(7- 185)

TrunC(lling Eqs. 7- 183 (It i = 5, Dugundji and Crisp havc calculated the following numerical quantities:

0 0.75

Ajj =

s,, -_

- 0.85297

1.0504

- 1.2095

0

0

0.21096

0

0

0.10305

0

0

- 0.03450

0.19524

0.00279

0

0

-0.00697

- 0.06808

0.28415

El g

41rqlt tan 2 A'

-0.35281

0.45685

0.07130 -0.18634

0 0

0

0

0

0 0

0

0

0

0

0

0 0 0.02065e 3!

(7-1800)

(7-186b)

0 0

0

O.0l321O.!

0

0 0

0

0

0.00969(;/~2

The zeroes in Eq. 7- 186a can be interpreted as meaning that, although steady-state displacements of the elastic degrees of freedom generate forces tending to excite the rigid-body freedoms, the converse is not true. (This statement is obvious in the case of the translation 9'1' which causes no static airloads at all.) Regarding the phenomenon of divergence, it is seen to be controlled entirdy by the dastic freedoms, provided SQme mechanism is assumed to act which preserves static eq uilibrium or "trimmed flight," so that a dynamic divergence ofthe sort discussed in Sec. 7- 3(a) does not take place. Divergence can occur at each of the eigenvalues of the determinant of Eq. 7- 183, (7-187) 1[9"] - [sf)1 = 0 After the zero roots are cancelled, this evidently reduces itself to third

Efo(

0,'t A' ) . Reference 7- 34 order in some quantity proportional to -[4 q 411 tan solves the resulting cubic equation and gives three real eigenvalues, choosing to present them in the form of reduced divergence speeds U 1) 1% ..../ (mol211p"'fl tan 2 A')

=

q 1)1t 4rr tan2 A' Efo

= 0.494,

(}S2 0.54 1,

0.973

(7-188)

The lowest figure yields, of course, an upper bound on the speeds or dynamic pressures at which the wing can safely fly.

3.41

I'RIN CU'LHS 01" AI(HOELAS'rICITY

" ,----,- --r---i-----, >.01-- - 1---1---1 Divergenc:e

::;,

~ 0.6 f----f-'1,

-f--l----,

2, 0.• 1___J~--~L~~~!~~~

r

0.'

1---1---+_ +-+_ ---1

:; - FI,. 7-/3. Di"""gence and .outter ~haracte,islico1 of a free-.oying, fiat-plate della wing accord ing to slender-body aerodynamic theory. (Adapted from Rd. 7-34.)

Since elementary slender-body theory has bee n used, Eq. 7-188 shows that the dimensionless divergence properties of this particular configuration are independent of flight Mach number and are fixed by a single combination of system parameters. T hus, we can say that U/1wa is determined entirely by the mass ratio

(mo/'1Tp.,Ptan I N),

9oJ'/Elo

(4'1T tanIN/Os'). TO ,shed more light on the couplings involved, Dugundj i and Crisp have computed the influence on these eigenvalu~ of changing the frequency ratios wJ% and wJ~. For instance, Fig. 7-13 shows divergence s~d VS. w4 {roa, with the mode shapes rpM) kept unaltered and W5 adjusted according to Wi ~ 2.4461 W. _ 1.4461 (7-189) W3

or

by

Wa

(Equation 7-189 causes the three frequencies to spread in linear proportion as w4 is increased.) It is of interest that, just below the frequency ratio (wJwJ = 2.5412 which characterizes the actual flat-plate delta, the divergence speed jumps from a value associated with a mode that couples

ON"·I)IMI·:NSIONAL STltuCI' UJ{ES

343

princirxlily Ihe (I'~ and V'j degrees of freedom 10 II mueh higher value associa led with the V'6 Ilorm
Iw 3 J(mol2rrp,J). tan' A')

-

J

7.390

+ 2.495 (W .)1± ""a

J

(m.;',,)'

6,225

(w.)~ _45.~91 (w.)'+ 54.612 wa

/.('3

(7- 190)

Equation 7- 190 is plotted as the dashed curve on Fig. 1-13. A great deal more information on the delta wing as a one-dimensional structure will be found in Ref. 7- 34. Thus we have added on Fig. 7-13 a curve marked "flutter speed," which is an excellent approximation to their slender-b ody result for high mass ratios (molrrp",fI tan t A') > 25. At tbe true frequency ratio (Wl/WJ = 2.5412, the static instability is seen to occur at a slightly lower speed than the dynamic. In co ntrast, in Sec. 9-60 there are presented similar computations for flutter speed based on the pistontheory aerodynamic operator, whose use is limited to higher Mach numbe rs andMs inA' > I. As an indication of the success with which slender-body theory predicts divergence of triangul ar wings, we reproduce Fig. 7-14 from Marli n and o Experiment

low .SJItCI (000

th eo~

o

____ ~is~~~~:~_.~~<~"~-~-_-_~-~" S"

]0' Ape. ho ll angte,

IS"

20"

K

Fig. 7_14. Experimental and theoretica l divergence characteri$tics at Mach number 2.0 of fiat·plate delta wings cantilevered from the trailing edge. The pla(e thickness is r. (Reproduced from R~f. 7-35.)

344

I'RI NCII'U'S 01' AI·;ItOI(tASTlCITY

Walkins (Ref. 7- 35). These Ultla ren:r 10 a n:tl-plale delta su pported as a eantile.vcr from the rear. The writers suggest that one possible explanation for the experimental points' falling below theory at very low vertex angles is nonlinear aerodynamic effects; on such wings, it is well known that vortex separation from the leading edge above rather small angles of attack causes th e lift to build up more rap idly than estimated by slenderbody methods. Among other studies on the static aeroelasticity of low-aspect-ratio wings, we cite the papers by Hedgepeth and Waner (Ref. 7- 36) and Hancock (Ref. 7-37). These references go somewhat beyond the limitation to one-dimensional defonnations made in the present chapter. For example, Hedgepeth and Waner treat a rectangular planform supported along a chordwise line at midspan; th ey assume deflections which are quadratiC in the spanwise coordinate y, but make no restrictions on the :z·dependence. 7-5 PRE'SCRIBED TIME-DEPENDENT INPUTS

Superposition of normal free-vibration modes, which has been applied to a static problem in the preceding section, also serves as a very efficient method for finding the dy namic response to an input whose form can be prescribed independently of the system'S motion. These situations arise during the ground operation of aircraft, such as when conductio g vibration tests, braking or taxiing over rough surfaces, catapulting, and analyzing the effects of certain weapons. One might also mention the static firing ofrockc:t engines and tmnsient loads associated wit h miss ile laun ching, especial1y when this takes place from a hardened site. A related class of problems consists of those where the aerodynamics due to the response have a negligible influence ; landing loads an d vibrations induced in a missile or other flexible strocture exposed to the wi nd us ually fall into this category. In the present section, we illustrate the analysis of forced transients with examples involving a uniform cantilever beam and the landing impact of an airplane. (a) Time-dependent forcing or a uniform slender wing

The mass of any lifting surface is usually a rath er small fraction of tb e total mass of tbe fuse lage or missile body to which it is attached. Provided the uncoupled vibration frequencies of the body and the wing aspect ratio are both high enough, its bending motion can then often be simulated by that of a cantilever beam built in at some station y = 0 between the body's

ONE.I)IMllNS IONA I. S'I'HUCI'UHI':S

345

cCll lcrJillc ~lJ1d sidc wall.· Wc adopllhis approxi mation and a lso assume Ihallhe bending stiffness Ef and rUllllin,g mass 1/1 arc independent of span· wisc dblancc. Wilh the applicd load f~(y, /) regarded as the given input, Eq. 7- [ I becomes Ef

~; + I/1W

(7-191)

"'" Fiy, t)

The oonvenient transformation of variablts y = YII, l = (tIl!)'" Ellm, II' = wll, / "'" F,PjEf eliminates all coefficients from Eq. 1-19[, leaving (7- 192) The cantilever boundary condilions call for an encas/,'! root and zero shear and bending moment at the tip: (7- 193) Any modem book on vibrations deve lops the solutions 10 the homo· geneous (i.e.,J "", 0) form of Eqs. 1-[ 92 and 7-193. The normal mode shapes are composed of trigonomctric and hyperbolic functions in the combinations

Tiff) = ±Wcosh

J9li - cos J8 y) j

- ( 'ieh ft. cosh J8; where the eigenvalues

(J/

-

, in -10, ) (. . -Slllv . ,-, iY) . smh v,,-,Y + cos .J&,

J

(7-194)

and corresponding natural frequencies w. = (&.W)'" £llm

are roots of the characteristic equation 1 + cosh

vo, cos YO; = 0

(7-195)

Section 3-2 of Ref. 1- 2 lists 8l = (0.597.".l (J2 = (1.497T)"

0_(" -1 ,2 7T)',

(7- 196) (i

=

3,4,···)

• A similar structural co nfiguration occur. with a vertically cre<:tcd missile clamped firmly to its launching pad, as is done prior 10 firing Atlas.

34~

I'R I:\"C Il'I.ES OF AEROEI.ASl'lCITY

We have chosen the constant outside Ihe brackets in Eq. 7- 194 so as to make Ihe tip amplitude 'Ii(l) equal 10 unity; the sign should be tnken the same as that of sin \IF; and therefore alternates between plus and minus with increasing i. As shown, for example, on pp. 327 If. of Ref. 1-32, the orthogonality condition redu~s to

r' (-)

(-)d' Ih/(l)=i, Jolf'!/lf i!/ Y=lo, (i -;'-j)

(i = j)

(7- 197)

Two different schemes are in common use for computing the forced transient response from Eq. 7-192. The classical procedure, which need not be reproduced in detail, represents the entire deHection as discussed in S~.

2-5(c):

(7- 198)

When Eq. 7-198 is substituted, multiplication by If iii), integration across tbe span, and application of known propenies of the nonnul modes dj)' (e.g., d'r,

ifl

=

.,) ,If,

+ Olq I =

lead to

4 frO), i)'f ;C1i) dfi

= 4Q ;(/),

(j = I, 2, 3, .. .)

(7-199)

Dots here indicate dimensionless time·differentiation. The uncoupled equations (7-199) can readily be solved by Laplace transformation for the individual generalized coordinates. For instance, we might consider motion starting from rest at f _ 0, whence li"(g,O) = li"(g, 0) = 0

(7-200)

qlO) = 4;(0) = 0

(7- 201)

corresponding to The response to the jth generalized force tben becomes qli)=i fQ,(7)sinOP-T)dT

Returning

\0

""

(7-202)

Eq. 7-198 for the complete response,

"(y, 1) - 4

.I

;~l

'I';(Pl f l Qm sin (),(l- T) d... OJ 0

~ 4 f ' f.' [J('1' r) .I 'I',(fi)'I',{'1) sin 0,(1 - T)] dfJ dr o

0

;~ l

OJ

(7-203)

A less familiar technique, but one which has many pru ct ie:11:!!.h'ullhll:!es in aeronautical problems, is lhal origirm lly proposed by Williu ms (Ref. 7- 38). The dc Heetion is wrilten as

ro{ y, i) = ro-,(y, i )

+ Z 'f,{ y)r,(i)

(7- 204)

;- L

';.,(y, l) being the quasi-steady dispfacemelll due to the instantaneous load fly, f) under the constraint tha t all vibratory motion is suppressed. By direct integration of the static equation d' 11", -

-, =

d;

!(Y,I _ _)

(7- 205)

treating time as a parameter and employing the boundary condilions (7-193), we find (1-206)

(The reader must recognize the presence of four dummy integratio n variables in Eg. 7-206, each distinct from the upper limit of the outcr integral, which gives the actual dependence of Ii·, on the spanwise coordinate.) To derive the dynamic correction to "iy, i), we once more insert Eq. 7-204 into Eg. 7- 192, use Eg. 7-205 to cancel}against the fourth deri vative

d',

of"·,, and observe that drti = 0,21;.

..Z,

[f;

+ (I/r ;]'1..(Y) = -ii·,(y, i)

(7-207)

Tbus we see tbat th is correction may be interpreted as the beam's dynamic response to tbe inenia loading due to the actual time-dependence of ".,. Multiplication by
' 1+ fJl' l ""

- 4f,I-i.fJ,i)
:: -4R,(i),

(j = 1,2,3,' · ·)

(7-208)

Tbe expression for the "generalized force" on ' 1/) is capable of simplification. Thus, we can replace in terms of its fourth derivative and integrate by parts four times, using the struclUraJ boundary conditions

3411

I·III N(; II ' U~.·:;

01' AlmOEl.AS'I·ICIT\'

10 ciiminate intcgmted portions, as follows : R;(l)

==

f" "W,

O'l.iY) rly

f

~ _l l\V(~ i)If'Pl dt=_1 'at,V• (-)r ~y , d-' YO' d~' ,/" Y .1/ , ~ Y

10

(7-209)

JOY

In view of Eq. 7-205, however, R;(f) =

,J ~ f J(y, /)q>ifi) dfJ • •

= Q;Ci)jO :

(7-210)

Equations 7-:ro8 through 7-21 0 require initial co ndit ions. For instance, the system might be at rest at 1 = 0, as described by Eqs. 7- 200. If Eq. 7-204 is substituted into 7-200, then the usual multiplication by 'P~fi) and span wise integration lead to r;{O) = -4 RJ~0») ftO) = -4R;(O)

(7-2 11)

Tbe solution to Eqs. 7- 208 through 7-21 1 reads

rilJ "" -4R,-{l)

==

r

+ 40J

o - 4R ,(O) cos 0/ -

Rlr) sin Oil - or) d-r • Rl-r) cos O;(i - or) dr

41

(1- 212)

the latter form being preferable because of improved convergence and the vanish ing of R!(O) whe never th e force F, has no step funct ion at f:= O. Returning to Eq. 7-204 for the complete response, f;~ii,

f) = l;',(Y, i)

- 4I <,',cY)[R,(O) cos OJ + (I R,(T) cos

a,(i - .,.) dT]

{'iy, /) - 4fl(l), O) I [';;;(Y~'t,(l)

OJ] d,

J.

,kl

=

o

- 4

, ~ I

cos

,

C(' [All, T)l r/y); ,('1) cos 0,(£ , ml a;

oo J o

T)] dl) dT (7- 213)

\".(g, I) is, of co urs e, calculated by Eq. 7-206 and involves no summing of infinite series. If 1<'1,0) = 0, we can compare the series in Eqs. 7-213 and 7-203 and discover why good convergence of practical calculations is to b~ expected from the Williams solution, in view of the fast·growi ng factor (j ,2 in th e de nominat or.

When making vibra tion tests on II st nlclure, we npply distributed or concentrated sitll/SQida/ forces. Thcrcforc JIg, 1) is givcn by Ihe rCli I p,trl of

fif}, f) =

F(fj)e'wI

=

(7- 214)

F(fi)e'ut

0= wf'v'm/EI being the dimensionless driving frequcncy. (/\ conn:ntrated force is treated by including in F(fj) a sui table Dirac delta functi on centered on its point of action.) In this case, the alternative forms (7- 203) and (7-213) for the system response must be mod ified to yield the steady oscillation which is all that remai ns after the inpu t has acted for any lenglh of time. The reader will have no difficulty in showing that the two restill> now read, respectively,

r

'fly) Ii{i}, l)

= 4e'''.4 '. 1

F('1''fl'1) d1)

(7- 215)

0

«()/

()2) !'

'f,(1I)11F('1)'f,('1) d'l

'(·I)~W(·)eill+4eil>l~~)" y, , y ~ ( e' ('"

._1 ,

w.cv) =

ff

•

-

0')

=

(7- 216)

(7-217)

fflfnfJ)(d fj )'

In particular, a constant loading amplitude F(fj) W(fj)=FD

fJ,

Fo gives

Y' --y' + -Y'] [-24 6 4

(7- 218)

whereas a concentrated force Po at fj = fio, which is described by (7- 219) will be found to producc a somewha t more complicated W" reflecting the shear di~continuity at fJo' The faster converge ncc of the series in Eq. 7- 216 is again apparent. Other interesting ded uctions may be made from these sinl.lsiodal results. We note the famil iar resonance condition that occurs whenever th e driving frequ ency passes through any natural frequency of the beam. The theoretically infinite resonance peaks wou!d in reality be reduced 10 large finite size by the inevitable presence offriction in the structure. Such damping has been neglected in the foregoing development; although this is an acceptable approximation when analyzing many forced motions, it is wholly unsatisfactory for the resonance phenomenon. Equation 7- 215 reveals that lhe key requirement in an experiment designed to measure properties of one particular normal mode is to nave

3~O

"RtNCII'U:S Ot l AlmOm,AS1'tCI1'Y

the force-amplitudc distribution F{,) us nCllrly as possible orth ogo nal to nil other mode shapes ofthc systcm. This rcquirem cnt becomcs progressively more stringent the higher the order of the mode onc wishes to isolatc, because of the decreasing relative siu of the coefficients l/(Ojl - 02) whenever the driving frequency does not coincide exactly with the desired wJ' In this connection, many excellent discussions of vibration testing will be found in the literature; the papers by Mazet (Ref. 7-39) and Lewis and Wrisley (Ref. 7-40) are representative. The two procedures for transient-response calculation have as their counterparts the mode-displacement and mode-acceleratiOIJ methods for finding dynamic stresses in 3i=ft structures (cf. Sec. 10-3(b) of Ref. 1-2). The idea is that local stress, shear, bending moment, etc., may at any instant be written as linear sums of contributions from each normal mode. The advantage of the mode-acceleration scheme is that the principal part of the stress distribution is the static one, obtained from the instantanenus loading F.(y, t) without any reference to its time history whatever. This can be done by the t ried, well-developed rules of tbe stress analyst and can include allowance for types of structural nonlinearity that are not encompassed by ordinary modal superposition. Thus the modeacceleration design stress at a station y is expressible as u.(y, t) = a••(", t) -

AM

L1---' ._ w.i" q.

(7- 220)

where, as shown in Ref. 1-2, A/i ) denotes the maximum stress at y due to unit displacement of the ith nonnal mode. For a given accuracy in a practical computation, the series in Eq. 7-220 can be truncated much lower than it should be if the mode displacement method were being applied. Incidentally, an ingenious variant of mode acceleration, for situations with external loads due to the system's motion, bas been derived by Mar, Pian, and CaJligeros (Ref. 7-41). We close this discussion on transients in unifonn cantilevers by directing the reader to many other examples of this sort which win be found in Ref. 7- 2 and papers listed therein. Thus, Leonard tabulates the exact Williams solutions for cantilever and simply supported configurations and for freefree beams with a central coneenttated mass. Results are givcn for both the elementary beam and the ''Timoshenko beam" having finite shear Ilexibility and rotary inertia. The importance of these latter effects is assessed by means of illustrative computations, in which exact responses are compared with approximate numerical estimates found by the method of characteristics and by a stepwjsc--integration scheme adapted from Houbolt (Ref. 7-42).

ONE- mMttNSIONAI, S'rHUcrURI£S

351

(b) 'Illc IlInding problem When predicting 3ccelcrations and strcsses due to ground landing impact, dynamics engineers agrec that thc lift lind moment on the airplane can generally be ass umed constant during the critical phase. Hence one rather difficult part of the loading due to motion is eliminated from this type of transient analysis. What remains is not wholly straightforward. however, because of nonlinearities in the shock-absorbing strut, landinggear structure, and tire force-deflection characteristic. For a period of II few tenths of a second following contact, these dements build up a system of vertical and fiOrizontal forees which depend in a complicated way on the heights of the two landing-gear attachment points above the runway surface and on the rates of change of these heights. Simultaneously, the rest of the elastic vehicle vibrates in response to these forces. Both symmetrical and antisymmetrical modes are observed, except in the rare event that the angle of bank is exactly zero at touchdown. If the linear techniques of aeroelasticity are to be used to compute landing loads, it seems necessary to be able to find we time history of the gear forces independently of vibrations in the airframe. Fortunately, several investigations (e.g., Ramberg and McPherson, Ref. 7-43, Pian and Flomenhoft, Ref. 7-44) indicate that this is the case to an acceptable degree of accuracy. By adopting their approximation and considering only one-dimensionaL wing struct ures, we bring We problem within the

purview of the present section. Let us examine the vertical translation and wing bending produced by a pair of landi ng forces ~.(t) and F~ (t) acting upward along the elastic axis at distances Y = ±YG from the c~nterline. Typical time variations of F, and F, are given in Refs. 7-43, 7-44, and elsewhere; Fung (Ref. 7-45) has published illuminating data on their statistical properties. Regarded as portions of a distributed spa nwise load, they would be expressed (7- 221) FiJI, t) = F. (t) d(y - Ya) + F. ,(I) 6(y + Yo)

.

where the delta flLnctions contribute unit area to any integral passing through their respective stations, in the usual manner. Almost universally today, this problem is treated by superimposing normal modes. We therefore write the vertical position of the elastic axis relative to some inertial frame as \\(y, t) = ,(qo(l)

+ 'PJI)Y +,-, I rp,{y)q'{t)j

(7-222)

Here 21 is the wingspan and Y = y/l. The normal mode shapes rplff) of the free-free structure are presumed known; they are orthogonal with

3~2

I'IUNCII'LES 01' AElWm.AS1'ICI'I'Y

respect to the splUlwise distribution o f runnin g mass m(!i). Two rigid-body degrees of freedom arc allowed: (11 vertical translation, in which '%(1) denotes the position of the CG. of the ills{alJIGlleously deformed system; and (2) roU, in which 'P.(I) denotes the angle of bank about the fore-andaft principal axis. Pitching is excluded, as are chordwise deflections and fuselage bending, but it is an easy matter to add any of them to such an analysis whenever their importance is suspected. As outlined in the preceding subsection or in Chap. 3, we apply the Lagrange technique and develop the following equations of motion: Mriio=flF.(iJ,I)djj

~ ifo = Mj[qi +

2~w;q~

fl F,W,

t)!i

dy

+ w;~q;]

=fl

F .(ii, I)op,.(ii) dY.

(7- 223a)

(i

(7- 223b)

=

1, 2,3, ' ..) (7-223c)

Mo and T~ are the total mass and roll moment of inertia of the vehicle. M( is the generalized mass of the ith elastic. degree of freedom, M( =

If,

m(fi)'Pl(fi)dfi

(7- 224)

and is often made equal to Mo by appropriately choosing the reference amplitude of the correspondin g mode shape. [Note that the masses of nacelles, central body, etc .• must be included in m(i1).] For variety, we allow here in an approximate way for energy dissipation in the structu ral vibration. The symbol ~. represents tbe damping ratio of free oscillation in the ith normal mode. A visco us type of dissipation mechanism is hypothesized, and no consideration is given to generalized forces on other modes due to frictional forces caused by anyone of them (zero "damping coupling"). The peak response to certain landing loads may not occur until several cycles of significant higher modes have gone by. and in such a circumstance the natural decay of these modes might affect the design. It is nearly always conservative to omit damping effects. When Eq. 7-221 is substituted into Eqs. 7-223 and we take account of symmetry, a further separation into motions which are symmetrical or anti symmetrical with respect to the centerline occurs. The former are described by (7-225a)

(i for symm. modes) (7-225b)

whereas the latter obey (7- 2260)

M;[ql +

2~lw,'h

.

+ wl9,1

,

=[F, (/) - F, (t)] 'Plyo),

(i for antisymm. modes)

(7- 226b)

In Eq. 7-2260, the generalized force is the rolling moment about the airplane centerline, (7-227) Clearly there will be no antisymmetrieal response at aU in a landing with Fts = F.,:

The solution of the foregoing equations can be illustrated by reference to the symmetrical case. Typical initial conditions would involve a starting rate of descent Vo and undisturbed normal modes" 90(0) = 0,

(7- 228)

9;(0) = q'(O) = 0

(7-229)

Equations 7- 225a through 7-228 are solved for the rigid-body response (7- 230a)

(7- 230b)

Since the vertical velocity must drop to zero during landing, we observe that the total impulse

must ultimately settle down to equal the momentum change Movo. The vibratory responses from Eqs. 7- 225h through 7- 229 may be calculated by Laplace transformation or by the indicial admittance method [cf. Sec. 6-3(b)] . In general, they will contain terms with both sines and cosines of Wit, but a very satisfactory approximation, in view of the smallness of the ,,' reads

qtCt)"'" 'PM,(f)(J) r:"' - " ""('-')sin w,(t - T)[F.iT) ;00,

Jo

+

F • .<,.-)] dT,

(i for symm. mod es) (7-231)

The resultant motion is described by the symmetrical terms in Eq. 7-222.

JS4

I'IU NCU'U;;S 0 1' A)(ItOEI ,AS'I'/CITV

Local accelerations, bending monlcnts, stresses, and the like can bc computed efficiently by adjusting the mode acceleration procedure to include the small influence of damping. Since the various contributing modes are going at different frequencies, it may not be an easy task to determine the instant at which a particular one of these quantities reaches its peak value. One suggestion (Biot and Bisplinghoff, Ref. 7-46) is to add the maxima from the individual modes in scalar fashion, without regard for their phase differences; but this may be an oversimplification when the structure is a complex one. Limitations of space prevent more than this rather sketChy review of what is often a severe design problem, so we must call attention to the many thorough treatments now available in the literature (e.g., Refs. 7- 34 through 7-49). Two further points should be made. The first is that even greater complexity enters the picture when we deal with rough-water landing of seaplanes and flying boats. There is considerably mo re doubt as to whether the mutual interaction between vehicle motion and hull forces can be neglected, and a statistical approach to estimating loads is certainly indicated (Refs. 7-45, 7-50). Finally it mus t be observed that the fascination of the touchdown pheno menon sometimes obscures the fact that other aspects of ground operation, such as taxiing and instabilities in the braking system, often affect the design or landing-gear struct ure to an equal or greater extent (McBrearty, Ref. 7-5 1). 7-6

FORCING IN THE PRFSENCE OF EXTERNAL WADS

DEPENDENT ON THE MOTION

In connection with the forced , time-dependent behavior of systems with a finite number of degrees offreedom, Sees. 6-3 and 6-4 devoted considerable attention to the concepts of mechanical admittance, indicial admittance, and unit impulse response. It should be evident to the reader that these ideas can be carried over to more complex linear systems, including the continuous, one-dimensional variety covered in the present chapter. Moreover, techniques such as Fourier series, Fourier integralS, and Duhamel integrals for calculating the response to arbitrary inputs also retain their usefulness. Provided the spatial variations of tbe driving force are known in advance, the position coordinates play the role of parameters. Simple harmonic excitation of the uniform cantilever, as described by Eqs. 7- 215 through 7- 217, furnishes an illustration of this point. Thus, one may divide out the common time function ei<.l == tiel and normalize the force amplitude in some such manner as F(y) = Fo'tJ(Y)

(7-232)

ON~- IlIMI!NSIONAI..

STltuC I'UKES

3S!

Equation 7- 215 is then manipuillted to yicld the following admittance, relating bending amplitude as the oUlpullO Fo as inpul: W(y);;; HWJ.{O.ifl F.

=4I

;_1

f

q;,(fi)

0

$(1/) 9',(1/) d7]

(6.!

IJ2)

Similar admittance functions exisl for acceleration, shear, bending stress, and the like. With tbese introductory observations, we undertake to review the analysis of three representative situations wherein tbe vehicle is driven by a prescribed external agency, while its response is sigllificantly affected by airloads due to transien t defonnations of the lifting surfaces. The first is a case of steady-state sinusoidal forcing, and we have chosen the Holzer-MykJestad procedure of dividing the structure into a number of concentrated segments. The second example deals with discrete gust or blast loading of a swept wing and typifies the indicial approaeb. The third involves the (more realistic) statistical treatment of the effects of continuous atmospheric turbulence. (a) A straigbt wing under sinusoidal forcing; in-ftight ribratiOD

teo>ting; the Holzer-Myklestad approach It is a routine part of the testing of most large aircraft to excite the aeroeiastic modes and observe their frequency and damping characteristics throughout a series of fli ght conditio liS. The airborne excitation may take the .form of rapid control-surface displacements, impulsive loads caused by small sbaped charges, or sinusoidal forcing from an electro-mechanical, hydraulic, or aerodynamic shaker. Altbough the impulsive type of excitation is the more economical in time, experience seems to prove that the greatest amou nt of useful illformation-especially regarding tbe nearness to a criticalll.utter boundary----ean be had from carefully interpreted sinusoidal measurements over a range of frequencies... To provide an analytical description ofa flight vibration test, let us stUdy a straight-tapered wing with a rectilinear elastic axis normal to the body centerline. Following a procedure suggested by MykLestad (Ref. 1-9) in connection with free vibra tion and flutter prediction on beam-rod structures, we split the wing up into several bays and concentrate the mass • For c:onfirmation of this statement, the reader is referred 10 the consensus of papers c:ollccted in Ref. 7-52.

356

I'H INCII'U:S 0 1' All ROI, I.AS.... CITV

Body or Ju,elage cente,line

Side of

"""

m'+ I ./~+1

-S.+1

In"

,

I.

Force F .cls at di,tance d ahe.d of thi' POint

Fig. 7_15. Lumped-mass equivalent for the right half of a tap",ed wing with straight

elastic axis nonnal to the fuselage centerline.

m. and moment of inertia 1ft from each bay at a sUitably located ceDtfal point y~. As shown in Fig. 7-15, the center of gravity of each !ump lies a distance ·S. ahead of the E.A. It is connected to its neighbors by weightless flexures and torsion members. Each segment is assigned constant chord 2b.. and span B,. for ae rodynamic purposes; the total lift and pitching moment act at station Yn. We remark that the process of constructing this lumped equivalent for any given configuration is an art, beyond the scope of the present discussion. Such approximate representations have their greatest value today when analog computation is applied to aeroelastic problems. The wing of Fig. 7-15 is assumed to be driven by a pair of shakers, each acting d feel ahead of the E.A. and aligned with the outermost mass ml on its half of the span. The forces are simple harmonic and positive upward, (7- 234) Let w;e'~ (positive upward), ii,-e;o-'I, and O;e'O
I e;'" = 4p", b;3B,W2{_[L, + iLJ :; + [ L:. + iL,]O;k"" ,

M.I/'" =

_ 4p",b;~B;W2( -[MI + iM;] :: + [M3 + iMJo,Je''''' ,

(7-235)

(7-236)

ON I',. IlIM I!NS IONA I, STII.UCI'UII.ES

3~7

II is necessary here 10 usc strip theory, bet:ullse the HoI1.cr- MyklcSlud proL'Cdure makes no allowanec for
" SM = L",l{[m, - 4p ", bl8;{L:t

,-,

+

iL,,)]w,

(7- 237)

IJn

Tn

"-,

...

L w2{[ m, ,-, "

= l w~{[m;sl

4p""b?B,{L l

+

i4)]W,

+ 4p", b?B1(M 1 + iMt)]w(

'~1

+ [I,

- 4p",b,4B,(M a + iM,)]O,}

+

Fa

(7- 239)

The four bracketed factors appearing in Eqs. 7- 237 through 7- 239 are abbrrviated as follows:

+ iL!)] == Pi [m,s, + 4p", b?BtCL3 + iL,)] = P/ Em ,s, + 4p",b/ B;(M 1 + iMI )]:::: Q, [I, - 4p"" b/B,{M, + iM.)] :;;;; '2: Em, -

4p"" b/B,{L~

(7-240)

These quantities depend on the properties of the ith section, on the air density, and (through the dimensionless aerodynamic operators) on the E. A. location, flight Mach number, and reduced frequency whdU. All this information is available for a specified flight-test condition. Using Myklestad's notation, we ne-xt introduce five influence coefficients which eharacterile completely the flexibility of the structure between station y~ and station y~71 :

358

I'RII'ICII'LES 011 Almm: LASTICITY

VT• = Difference between angles of twist at Y.+! and 1'" when unit torque acts at Y... vl'". = Difference between bending slopes at lI"+l and y" when unit shear

(and no bending moment) acts at Y... v.'U". = Difference between bending slopes at Y..+l and 'y .. when unit bending couple (and no shear) acts at y". d F • = Bending displacement of V.. relative to Y.+! when Y"H is held as a cantilever and unit shear (but no bending moment) acts at y". dM • = Bending displacement of Y .. relative to Y,,+ l when Yo+l is held as a cantilever and unit bending couple (but no shear) acts at Y...

All of these are positive as defined.· They are illustrated pictorially in Ref. 1- 9, and their calculation from tbe known flexural, sbear, and torsional stiffness distributions of the wing is discussed. Because the stwcture is linear, the deflection and slope changes over the length I" can be expressed as follows:

=

•

a" - vF.wzI [P,w, + .P/ iia - uF.F ;_1

(7-241)

.-

= W.. - I"a,,+! - dF.W1.I [P;W; ;- 1

+ P/OJ

(7-242)

•

- 0" - vT.w"I[!J;w; ,., - !2:o;] - VT.Fd

(7-243)

Here Eqs. 7-237 through 7-240 have been combined into each third member . • Note that the structure nero not ~ assumed a simple beam-rod. It is also not difficult to account for the effect of rotary inertia by adding a co"""ntratcd bending couple at y. proportional to ~.

ON Ic. IlIM I~NSIONAL

STRUCl'URES

3.59

The computation proceeds inward rrom the tip station, where the slopes and delle<:tion have the still-unknown values

(7-244)

Tbe linear character of the problem also pennits these same quantities at any inboard station to be written

'. ~ I..• - I..J -I,.;; -I,-.F

)

~.. = -g:"ifi + g,} + g~.if - gF.F O~ = h••'1' - h,,3 + h."..", - hF.F

(7- 245)

where the amplitudeli eoefficients ! r., ... , hF • are independent of p, is, ;po and F. (It is evident tbat !., = g~, = h"" = I, whereas the remaining coefficients vanish for II = 1.) Equations 7-245 are substituted into Eqs. 7-241 through 7-243, replacing /I by /I + 1 where appropriate. We thus arrive at three relations of the form Ap

+ Bd + C;p +

DF = 0

(7- 246)

Since p, is, if, and Fare, as yet, arbitrary and independent, their coefficients must separately equal zero. Hence one is led to twelve equations among the amplitudes coefficients, which may be abbreviated as follows:

i.' +1 =1... + VF,G'#. + u},[.G.,: /,.+, =h. + VF.G", + V.lf.G,.'

I.,,+-, =!",. + VF.G"" + UM.G"" , .-. IF'+1 = IF. + VI'. + VM. .=L1I, + I n /." ' " - d.•.. .G. . . - dM • G... , -dF.'. G - d,y.G' go.""+, _,... +1' nfl •• , I. g- .,'+' ~,+1/. ". .. 1"'+ ' -dF.G". -duG' ,y.~ • -, gF.+, = gF. - In/FHI + dF• + d M, L II ,-, g-"".. - g-...

.

n"" •• =

h..,

+ vr.D...

h" " = h",

+ VT.n,.

h"'+1 = h",. + vT.il,,,, hF•• , = nF. + vr.d

(7- 247)

360

IIIUNCII 'Ll~

0 1' All It OIll .AS'l'ICI1'Y

The nine capital-leltcr factors in E'ls, 7- 247 stand ror

, G",. = ",";[[P,.g", - P,'h",,] ;- I , 0 •• = W2 ~(P;g~. - p/".J ;_ 1

,

G". = w~1 [P,g ... , + P/ liw] ;- 1

H•• =

•

w!1 [Q,g. , ' ~l

~:'I.J

,

w"l [ O,i6, - Q/h,,] ,, ' • 2 = w 1 [O,g,.; + ",'Ii",] .-,'-I = 1: I;G.;

R 6• =

R F• 0. :

G.: = G~.' =

(7- 248)

.-....

....

,-I

I;C:;••

11,0",

'-I

It is a simple task to show that the shear, moment, and torque ampli. tudes just to the left of y. can be expressed in terms of the tip motion as follows: (7- 249) S. = -G••if + 0 ••5 + G". if + Gp.F

M" "" -G".'I/> + G,.'3 + G....'ii> + GF;F where

T.. = -R". I/>

+ R•.J + R".if! + RJo.• F

,-, GF .' ""' I I,G",

(7- 250) (7-251)

(7-252)

; ~l

•

H". = d - w 2 2: [O;gp, + Q/h p,]

,- ,

For each flight altitude, Mach number, and forcing frequency. the computation implicit in Eqs. 7-247 and 7- 248 can be started at n = 2 and repeated for each successive spanwise station until the centerline n = b is reached. Reference \ - 9 offers many illustrations ofweU-orgaruzed

ONR.IlIMENSIONAI. STItUCI'UItI("<;

361

labnlar procedures fo r th is purpose. Since Ihe uiraaft is in Ihe free-free tlo ndition. III. ami ',represellt aile-half the mass and pitching moment of Inertia orthe centra l body. When (he excitation is sym mctrical at the two wingtips, the centerline boundary co nditions require the vanishing of bend ing slope i., shear S., and torque T•. In the foregoing nota tion, we therefore get the three equations

I . ,rf - /',5 - I N;; = I~·, F -C",rf + 0,,5 + O",,;p = - CF,F -R• .';; + B,,5 + R",Jp = - fip,F

(7- 253) (7- 254) (7-255)

These constitute a simulta neous system with complex coefficients, which is easily solved for the tip motion amplitudes .p, 8, and;p per unit amplitude of F. rpfF, 5/F, elc., are typical mechanical admittances or transfer functions for the wing; they depend on frequency w, flight altitude, and Mach number. Through the extensive collection of relations derived above. we can obtain a mechanical admittance connecting any other load, stress, or property of the motion to F. These computations are well adapted for high-speed digital machinery. One can also treat a ease of antisymmetrical forcing, in which the wingtip shakers are operated 180° out of phase. The calculation then proceeds inward along the right wing just as before, but the boundary conditions consist of (7- 256) w~ = fj~ = M~ = O These again provide the three relations needed to specify rpfF, 51F, and ;pjF. Other types of root and tip restraint are studied ill the various examples of Ref. 1-9. The majority of these deal with free vibration but are readily modified to include airloads. (b) The discrete gust problcm for a ,swept wing Disturbances due to nat ural or artificial nonuniformily of the wind and pressure fields are a frequent occurrence during regU lar operation of airpla nes and missiles. Although natural atmospheric turbulence can be fully described only by statistical means, it is a common design practice to compute the response to discrete gusts of preassigned shape and size. Such results are often very useful, when properly interpreted in the light of the designer's experience. Moreover, discrete-gust analyses apply directly to encou nters with blast waves from large explosions such as bomb bUThts. For these, considerable data on t he structure, decay, and propagation rates are now available, so that accurate load calculations are a fairly straightforward matter (Ref. 7- 53).

We cun gain a crude idea of the rel
g

Here g is the gravitational constant and WfSthe wing loading based on the area for which the lift-curve slope oC1.looc is defined. Equation 7-257 assumes that the vehicle's stfllcture is rigid, that gust envelopment takes place instantaneously, and that the incremental lift build-up is not delayed by unsteady flow effects. To improve upon these approximations, all of which are deviated from significantly in practice, aeroelasticians devised the concept of atrevialion jaClor. This is a number by which the result of Eq. 7- 257 or a similar formula for bending moment, stress, etc., is multiplied to yield the actual maximum of the quantity during the timedependent response. The Dame is somewhat misleading. because the factor often exceeds unity on account of dynamic overshoot. . Wings with appredable sweepback have a special interest, because their alleviation factors turn out smaller than those for straight surfaces with the same general stiffness level, aspect ratio, and wing loading. The reduction is due principally to the longer envelopment interval and to the unloading influence of negative angles of atlack produced by bending deformations, the latter being the more important effect. In this section, we sketch how we would go about computing the response and alleviation on a large-aspect-ratio swept wing meeting the elementary "one-minuscosine" gust, which form5 the basis of certain military specifications. Our modal approach is suggested by the work of Pian, Lin, and co-authors (Refs. 7-54 and 7- 55); but it also resemb les most other treatmenls in the recent literature. Consider the configuration of which the right half is pictured in Fig. 7-16. The vertical gust velocity which first strikes .the vertex of the leading edge at I = 0 is described by wa(sJ =

(iWo[ 1 - cos ;;0]. 0,

(2s a

(0 5: So

s

2s o ) (7-2 58)

s~)

where So == Ut fb ois a dimensionless time based on midspan semichord, and .fa is a dimension less half wavelength. Only bending deflections are permitted, through a set of mutually-orthogonal cantilever modes YlY) . Freedom in rigid-body translation is introduced by means of the vertical

ON.;. OIM .....NSIONAL STRUCtURES

363

coordinate '1o(r) at the centerline, bu t the pitching moment of inertia is so large as to suppress any rotation abou t y. Hence the upward displaceme nt of the clastic axis in Fig. 7- 16 becomes w(y, t) = qit)

.,

+!

(7-259)

y,{y)q.(J)

Only symmetrical modes are included because of the obvious symmetry of the input. Following the Lagrangian approach of Sees. 2-6 and 3-4, we derive for the instantaneous kinetic energy of the right half of the system

T = !M.,qo' + tjo!tj.i'm(y)y.(y) dy + i! g,1 P m(y)y!(y) dy ' _I

.~l

0

J0

(7-260)

Here m(y) is the mass per unit distance perpendicular to the centerline, and Mo the total mass. In terms of the natural frequencies w. of cantilever bending vibration, the potential energy is U =

!",iq,tw,' (' m(Y)yl(Y) dy '-1 Jo

(7-261)

All generalized forces can be built lip from the running lift L(};, I) dy on the typical chordwise strip, shown shaded in Fig. 7-16. In the manner of Sec. 6-4, L is separated into a portion La due to the gust and a JlQrtion LM caused by the dynamic response itself. Thus we find

(1,

== QoM + Qt ""

f

LM(y, t) dy

UAi"~

+

J:

Lc/..y, t) dy

(7-262)

U

FI,.7-14. Half span of a sw~ptbad, wing with rectilinear elastic ax is, encountering a two-dimensional gusl front normal 10 the /light direction. T he taper is linear.

364

I'IIlN CII'I.I' S OF "'''ltOl(LASTlCITV

ami Q, ;;;

Q/' + Q/' =

f

V"(U. f)y,(II) ,I!!

+

f

1.-0<11, I)y,(y) if!!,

(7- 263)

( ;=123' • • • '')

Applying Eqs. 2-67 and substituting

MQ qo'

(~r q." + (~J~l q /

f

(~) Z

m(y)y,-{y) d!!

f

So

for physical time t, we obtain

+ QoD

(7- 264)

(i = t, 2, 3, ...)

(7-265)

m(y)y f,.y) dy = Qo,li

f

+ q/ (~) Z

m(y)y,t(y) dy

primes denote differentiation with respect to st.. A unique aspect of th e present development comes in the form of the disturbing airloads Qol) and Q,/). According to Eq. 4-160, the aerodynamic operator which gives running lift d.ue to a unit step gust striking the \eading.edge of any strip of the wing at t = 0 is La = q(2/1)IlA

1 U !f(s)

(7- 266)

Here s = Ulfb, and 0A is a generalized lifl-curve slope chosen, in this example, so as to make the total wing lift approach some preassigned value as s ..... GO, 'JI'is) being defined such that !p(oo) = I . For instance, in incompressible flow over a swept wing of very large span and constant chord, 'f'($) would be Ktissner's function described in Sec. 4-6(a), and aA = 2.... cos A. The local semichord on the wing of Fig. 7- 16 is b(jj) = b(l[1 -

where il;;; y/l, ,s

(lYl

(7-267)

~ I -

l, and l is the taper ratio. A gust front which . ytan O meets the vertex at I=-O would begin to envelop station y at I = U Hence s in Eq. 7-266 must be replaced by a delayed dimensionless time variable u [t _y tanO] , ~ -'~-;---'U,---~ = .I G - py b l ,sy

where

P::: I tan Ojb o.

(7- 268)

Leading-edge sweep is called 0 10 distinguish it

ONI(.I) JMJ·;NS JONA J, STUUC rUU J,,'i

365

fro III A whcn lhc chn rd is not COI1.~tal1 l. Su bsliluti n£ for dyna mic pressure, locHI scmichord, and x in Ell. 7- 266, we get La =

poo UhoClj\[1

-

"iiJ1JI(su - (J~) 1

(7-269)

fJy

Ncglccting spanwisc induction effccts, t he build·up of total lift on ha lf the wi ng due to the step gust would therefore be

(LOlrOT = 'f'Lo<:y,so)df}

= PooUbrPj\l ("[1 - btiJ'f'(So -

Jo

E

PooUbouj\ltpo(So)

P~)

l - {)y

dj

(7- 270)

'f'iso) is an indicial function for gust loading of the finite p[anform; the upper limit YI is used to distinguish whether the gust front has reached the tip statio n y = I

y! =

(So/P. I,

(7- 271)

In the manner e:.:emplified by Eq. 4-163, it is now possible to write the generalized lift force Qo 1J associated with a n arbitrary gllst structure 1I'0(xo), which starts from zero at Xo = 0, as follows:

(7- 272) Here

IJ

is a dummy variable. replacing Xo in the co nvolution . Similarly.

(7-273) where (7-274) Equation 7-258 can be substituted directly when tha t particular gust model is employed.

366

1"UNCII'U:,s 011 IIEROIll .ASl'I CITY

In Refs. 7- 54 and 7- 55, V'o (Sg) lmd \l'1(SO) (for YI = if) are evalua ted in closed form, using the rational-fraction and exponential approxima tions to the incomprcssible flow Kiissntr function, Eqs. 4-173b and 4-173a, respectively. The two approaches have comparable accuf'.lcies in practice, but the former is simpler and much better suited for analytical work. As an illustration, Ref. 7- 54 gives
,, _

So

'110 _

+

Wr0.09984 In (0.32«(1 - So6») L(p + O.32,W (1(so + 0.32) 1.72

(ft

+ 2.50:1)3

So.'))]

I (2.5({J n peso + 2.S)

so]

50[(12 _ jJd «(1 _ 0.32.5) +0.312.8 2 fJ (fJ + O.326)~ 30[,82 -

+ 0.688 P

po «(1 -

2.56)

2

(fJ

+ 2.56)2

So] f1

(7- 275)

for the range 0 ~ So ~ {J prior to when the front reaches the wingtip. The generalized forces due to the motion follow a simllar pattern, originating with the modified Eq. 4-]50, (7-276) which represents running lift after unit step change of stream wise incidence: ats -= O. (Once: more op(s)is to be c1losen so that op(<>:l) = L) The bending response of the wing will commence at the instant the gust hits the vertex. Therefore, S = Utlb in Eq. 7-276, and s can be expressed in terms of the reference time parameter So by {7- 277} Referred to properties of the midspan station, J.jv

{)Y]tp( ' ••)

= q(2b,JaA[t -

1 - tly

(7-278)

The total lift on the half-wing resulting from the unit step of angle of attack is (LMhOT = If.ler dfj = q(2bo)a,..l (l[l _ o o

J

tlYJtp( . So

_) dfj IfJy

(7- 279)

ONE-OIMENSIONAI. STMUcrUIU;S

367

Hence, for arbitrary angle-or-attack history, rJ.(jJ, so), we get from Duhamel's pri nciple

[ I - vYJ'P .~ (L•" J'l"YI' = q(2b) 0 QA '!.- i10o:,,(fJ,0) -

ono"

(SO-o)d _ il d 0

loy

(1-280) The instantaneous angle on the flexible wing receives contributions from the bending slope and vertical velocity. Thus, introducing Eq. 7-259 and noting that il = WI£) cos A,

_)

low

. AOW -

"'y,t ( = - ---sJn U at

= _

iJy

10 _ ..!:.. I r;(f/YJ, _ sin A cos A I or, q, U

I

U i_I

(7-281)

' - IOf}

In terms of dimensionless time, _,_ )=_qo'_.!..~ (;n,_sinAcosA~or, () (1-282) "",Y,So b b~r,y,q, ,~~~q,so o

.- IVY

D. - I

Combining Eqs. 1-280 and 1-282, we get for the generalized force on tbe rigid-body degree of freedom Qoat = -2qQ AI

Jr"o J(lo [qO"(O) + ;I_ 1y,{y)qj(") + sinAcosA hOI

11~1

dr;q/(o)] dy

x[ 1 - OiiJ'I'(;o - 6~) dil do

(7-283)

In a similar fashion, the force on tbe ith bending degree of freedom must

re,d

Q,M = -2qQ I>-/1" fJ[qo"(O) a

Jo

+I

YI(g)qt(o)

I- I

+ sin A cos A bo I d~l q/(O)] I i _Idy

xr,(!l)[l -

"iJJ'I'(~ - ,,~) diJ do

(1-284)

When Eqs. 1-272, 7- 273, 7-283, and 1- 284 are substituted into Eqs. 7-264 and 7-265, we are able to construct the final set of equations of motion. These can be made dimensionless by division with (2qaf',lbO), whereupon tbe response to a given gust is found to depend in an important

36/1

l'IHNC II'U,:.s (II' AElWm,ASTICIT"

way on Ihe mass ratio (MQ fp",h"2/U,, ) anll the mass lIistribution m(fi)/p ",bo'

After the number and form of the modes arc selected, we are confronted with the problem of solving a linear set of integrodilferential equations with So as the independent. variable. This may be done, in principle, by means of Laplace transformation, but Ref. 7- 54 suggests the impracticability of such a procedure. Tn preference, these authors turn to stepwise schemes of numerical intcgration due to HOllbo!t (cf. Ref. 7-42). Without going into full details, we reproduce as Fig. 7-17 the results of a typical computation of th is type from Ref. 7-54. The actual mass distribution of the airplane is simplified to one concentrated mass mo"" 56 slugs at each wingtip plus a mass 2(Mu - mol = 3886.8 slugs at the 0

-

,.,

Tip . ccele,otion. Ktip - - _ _ Fu,elage .tX:eleroU 00. K

I

0

\

,.,

,•, '" .

0

, ,

o.

0

o

--/---,

•

•

-

Fig. 1_17. Time hi.tories of acceleration ratius at "" nterline and wingtip for a simplified .wcptback wing. Details of the configuration are listed in the text.

ONI;:"I)lMI' I"'SIOi"AI. STMUC I'U IIES

369

fu selllgc centerline. A single p.1mbolic bending mode rIO}) "'" .Ii 2 was used. Other signifkan t para me ters arc 1"",58ft

a,\ = 6

..1=0.418

WI

n=

c

36 40'

A"", 34 0

S = 1428

Altitude

= 7.55 radians/sec =

II,()(X) ft

E.A.S. = 460 mph ft~

50=15

bo "",8.7ft

Plotted in tbe figure versus J Q arc the acceleration ratios or alleviation factors K and K 1tp ' These are the ratios of the instantaneous centerline and wingtip accelerations, respectively, to the reference acceleration from Eq. 7-251 for a sharp-edged gust of amplitude equal to that of the oneminus-cosine gust used in the actual analysis. The peak value of K is 0.517, which clearly illustrates the ameliorative effect of sweep; however, the tip motion is quite violent in this casco Many other similar examples of swept-wing gust response will be found in Ref. 7- 56. (c) Dynamic Te!iponse to continuous turb ul em:e

To arrive at a more sophisticated formulation of the gust-loads problem than the foregoing, we must seek to generalize the ideas set forth in Sees. 6-3(c) and 6-4(c) to distributed-parameter systems. The foundations of such procedures arc given in a paper by Liepmann (Ref. 7-57) and elaborated by Diede rich (Ref. 7- 58), Ribner (Ref. 1-59), Foss and McCabe (Ref. 7-60), and a host of olher authors cited, inter alia, in the last reference. Perhaps the two most significant generaliZAtions concern the influences of finite wi ngspan and three-dimensionality of atmospheric turbulencc. First, we need a statistical desc riptio n of the turbulence itself. It has, of course, three Cartesian velocity components Ua , Va, w"" which are at any instant random functions of a position vector r = xl + yj + zk in a coordinate system filted relative to the mean motion of the air. Each component has a mean-square value; and they are funher specified by a correlation tensor, composed of nine averaged products like u(,{r;JuU(r2), ur;(r l )oofr2) , •••. In the most genera l scnse, the mean squares depend on position and direction, while the correlation tensor elements vary with the locations f, and r . of the individ ual points for which the observations are averaged. In the commonly ass umed homogeneouJ, isotropic turbulence, however, uri = orl "" woz "'" const., and the correlations depend only

370

I'IUN CII'I .ES OF ,u:ROELASTICI1'V

on relative location tor "" r, - rt ; moreove r, it can be shown that only two correlations are distinct. When lineariud theory is applied to the present problem, the component W(J normal to the plane of the lifting surface is th~ only one producing firstorder loads, and the history of these loads is determined by the variation of wa in th e x-y-plane. A partial statislical description of the dimension less walV is provided by the autocorrelatio n function 'l'arJ..T, YI - y,J, which generalizes the 'PaG(T) defined in Sec. H(c) to account for the spanwise y-dire<:tion. Suppose we adopt Liepmann's symbol (7- 285)

for the %-%-element of the correlation tenSOT. Then, if the turbulence embedded in the airstream V flowing past the flight vehicle undergoes no significant chan ge during the brief period required to move one chordlength, it is evident that "'GCCT, y, - Yz) =

RJ.z., - :r,., Yl -

y~, 0)

(7-286)

(7-287)

is the time interval corresponding to the correlalion distance (ZI - x 2) in the flight direction. Associated with 'Paa and R33 are power spectra, which play an important ro le in practical computations. To understand their definitions it is helpful to start from the concept of the power spectrum of the correlation tensor. Since th~r~ are three coordinate directions, we expect that there should be three wavelengths or frequency variables appearing in this spectrum. They are defined in terms of the wave numbers k" kz, and ka, giving the number of waves per unit X - , Yo, and z-distances, respectively, in a particular harmonic component of the turbulent structure. In a ' manner quit~ analogous to Eq. 6-75, we therefore define the z-z-element ¢l 33 of the power spectrum tensor by means of Raa(Xl -

XI'

y, - Y2' Z:t - z:) "'"

x ex p [i{kl:r, -

~)

L: L: L: 41~~kl' 1sJ,

+ k,.(y, -

Yz)

+ klh -

k,)

0}) dk1 dk,. dkJ (7-288)

The dependence of 4133 on kJ is of no interest in the present discussioll and

ONlr.. D1M~: NSIONAI. STMUCI'UMlS

31 1

may be inlegrllled out. Thus, Ihe combin ulion of Eqs. 7- 286 llnd 7- 288 yields 'fiOG(T, YI - Y.) =

L: f~o:;cI>GMkl' k~)

x exp {i[kl(:l:l where

WaMk1, k.)

t'",

EO

+ kt(Yl -

:1:2)

Y. )]} dk l dk2 (7- 289)

(7-290)

cD:d:k" k., k.;J dk3

The mean-square gust velocity is (7-291) Reference 7-57 suggests the following approximate formula for describing isotropiC atmospheric turbulence: ¢l:clk k k)_~wGt;sf h

2_

3

-

u! . \[1

.,.,.

2

+

kI +k/ ) (7- 292) ;.z(k12 + kit + k32»3

Here, as in Chap. 6, ). is the integral scale of turbulence. The integration in Eq. 7-290 leads to

cI> (k k) GO\.

I,

2

=

3

w(?,/

4:;;. u! A. \[1

k,'

+ ka l

+ i_\.k 1 + k1')]~ J

)

(7 293) -

(A second spanwise integratioo gives a result equivalent to Eq. 6-119.) Many studies (e.g., Ref. 7-62) suggest that, while Eqs. 7-292 and 7-293 may be inadequate for representing all tfle gustiness enoountered over a long history of:fljght operations, the superposition ofa few such structures with different intensities, scales, and durations will probably yield satisfactory design-load estimates. As in the case of two-dimensional flow, spectral techniques greatly reduce the actual computing labo r. We illustrate this pOint first with Liepmann's simple el
=J' J"' -I

-00

Wqf.T', Y) hu!.t - .,.', y) d.,.' dy U

(7-294)

The wingspan is 21. Diederich's discussion in Ref. 7-58 of tbe enct

J7Z

I'R INC II' I.ES 01' AlmOEl. A!S'I'lc nV

meaning of "w(i. 11) and similar response funct ions will be found he lpful: it rep resents the lift generated over the entire wing when a unit Dinie·delta function· of wGfU sweeps past the y·ax is at spanwise station y and time t = 0, The roles of t and x are interchangeable, since z = Xo + UI for any gust element, The upper time limit in Eq. 7- 294 may be replaced by + 00. The transformation l' = (I - 1") thereupon gives an equation pa ralleling 6--78, L(1)=f' f '" Wa (J - T.lI)h u ;{T,y)dTdy - , - '"

(7- 295)

U

Proceeding as in Eq. 6--79, it is easy to show that the autocorrelation fu nction for lift is related to that for the gust as follows: rpLL(-r)

= =

lim ..!...2T

T_oc

fT L(/)L(/ + T) dl - T

f, f ,L: L:

halTl' 1h)h/.O(T1, Y.)

x rpad.T + Tl

-

T!> 'ft. - Y'I) dT, dT2 dy, dy! (7- 296)

The powel' spectral density of the tift is defined, somewhat as before, to be the Fourier transfo rm of'Tu., 'l'L/,(T) =

f"",

(7- 297)

,/""<() /,f,(('J) d(<)

We introduce Eqs, 7-297 and 7- 289 into Eq, 7-296, no ting that it is possible to make the substitution (7- 298)

where kJ = wfU is the wave number conncrted with the frequency w of Sinusoidal-gust passage, After breaking up the six integrals on the right by appropriately distributing the five factors in the exponential, we are led

'"

J:./'''$L1'<w)

dO) =

J:/'"f:",

1,k l{[f, f..,

$(u;(k

z

"Lib, y,)

x exp [i(k, x, +

k~y,)] dr, cl y ,] , [f, L"'", hLd. T2' Y2)

x exp [ -

+ kill~)] dT. dYuJl dk2 elk,

i(k,Z2

(7- 299)

• ThaI is. unit ~olumc under a plot of '".IV vs . :<,11 when the volume integral is carried oUI over a very sma ll "'_II area in the vicinity of where the impulse is located,

ONI'; 'I)IM.:NSIONAI. STIIUcrURI'S

J1J

The reasoning which leads from Eq. 6- tlO to Eq. 6- 83 can now be ge nen,li~ed. It is not dimcult to prove that

HuJ..k l • kt ) = =

f, L"'}W(72' 1. f' i'" U -,

YI) exp {-i[k L"'3 + k1yJ} dT3 dY2

hL()(=-'

-0>

U

tI) exp {-i[kJ.:t: + k,y]} d:r. dy (7- 300)

represents the complex amplitude of the simple harmonic lift force produced when the wing flies through a steady sinusoidal gust having wave numbers kJ in the x-direction and ( - kg) in the y-direction. (The crests of these gust waves wouLd be oriented at an angle tan- l (k,Jk,) to the y-axis, and their wavelength is 21f,Yk lt

+ k, l.)

Moreover, the product

of the two double integrals in brackets in Eq. 7-299 is the square of the magnitude of this mechanical admittance, IH uJk1• k2112, Thus we come

to the conclusion that the spectra of the lift and the gust are connected by the relatively simple relation IIl LL(W)

= -1

f"

V -.

IIlG&
(7-301)

where k J = w/U. The mean-square lift is

13 =

L"'.,¢lLL(w) dw

= U

L:

f'", L"'",

IIlLL dk 1

\(l (:{l(k1• k 2) IH w
k~)I' dk

j

dk2

(7-302)

Several interesting specializations of Eq. 7- 302 are worked out in Ref. 7- 57. Probably the most valuable aspect of the foregoing development is its generality. In exactly the same sense that Eq. 6-83 relates any inputoutput pair in a lumped, linear system under stationary random forcing, so we can say that any time-dependen t characteristic of the random-gust response of a rigid or flexible flight vehide may be computed from the turbulence spectrum as in Eqs. 7- 301 and 7-302. For instance, Foss and McCabe (Ref. 7-60) write the mean-square stress at any point in the structure as

rr -

l: L:

WOC(k 1, k!) IHoQ(k j •

k~ll2 dk dk~ l

(7-303)

Their paper provides considerable detail on the practical problem of calculating ull and other properties of the response to the spectrum in Eq.

J7.j

l'IIiNCII' I.I~

or A"lwm..I\,.,,~n c rrv

7- 293. A ri gid airplane with freedo m in vertical trallslalioll and an .airplane with a large-as pcet-ratio wing permitted on ly bending deformations are used as examples. Peak stresses and accelerations frol11 discrete-gust analyses are compa red with statistical resu lts in the important special case where the scale A. is quite large relative to the wingspa n. It is found that similar conclusions with respect to the size of design gust loads are reached from both computations in many practical cases, but that th e discrete-gust approach may be seriously unconservative when the inherent aerodynamic damping on the airplane is lo w, as may occur in flight at very high altitudes. As mentioned above, there now exist numerous studies or the statistical treatment of loading due to gusts, buffeting, ground operation, and other repetitive but individually unpredictable phenomena. The case for the rationali ty of this approach and a strong plea for its continued development will be found in Ref. 7-61. Several excellent illustrations, incl uding comparisons with flight meas urements, have been published, among them Rcf. 7-63. It is a matter of distress to the present authors that space limitations preclude their providing a more extended treatment of this singularly important aspect of the influcnce of aeroelasticity on design. Finally, the remark made in Chaps. 4 and 6 bears re peating, that unsteady ae rOdynamic operators are much more fully developed for sinusoidal motioll than for other types of transients. This is even more true when three-dimensional flow and fini te-span effects are bc:ing considered, as in the foregOing section. It is exactly qua ntities like the HJ.{...(kJ , k a) and HGJ..k\. k2) in Eqs. 7- 301 through 7-303 that are the easiest to determine from the aerodynamic standpoint. 7-7 EIGENVALUES; FLUTTER OF ONE-DIMENSIO NAL STRUcrURES (III) The aeroelastic modes of

III

uniform straight wing

Most of the conclusions reached in Sees. 6-5 and 6-6 regarding the influence of various parameters 0 11 the aeroelastic modes of the typical section ean be carried over, at least qua litatively, to the bending-torsion flutter of unswept, one-dimensiona l lifting surfaces. In view of this fact and of the prominence which flu tter has received in the aeroelastic li ter· ature, we confine ourselves in th is chapter to summarizing a number of interesting special results. The first of these concerns the method (originally suggested by Goland, Ref. 7- 64, and refined by Runyan and Wat kins in Ref. 7-66) for solving exactly"the homogeneous differential eq uations of the uniform beam-rod in an airstream. So universal in practice is the usc of generalized coordinates for analyzing flutter that engineers may overlook

this "cxact solution," it, valuc as a stalld(tn.l of com paris 011 for approximate c.1.iculations, and even its direct applicability to some simple wing and tail deSigns. Consider a rectangular surface with straight elastic axis, whose stiffnesses EI, GJ, running inertias m, I", and running unbalance S~ (about the E.A., positive when e.G. is aft of E.A.) are independent of the spanwise coordinate y. Adapting the differential structural operators of the uniform beam and rod from Chap. 5, we derive for the general equations of motion

a'""

(j'o m fP - w- S -

+EI - ~L(,

a1l

~ ol!

0,2

02Q I. ali -

s.

fPw

' ,)

(7- 304a)

a~o

(7- 304b)

012 - GJ ay! = M i Y, I)

Representative of the boundary conditions which might be associated with Eqs. 7- 304 are those for the cantilever,

'1 = ow (0, /) = 'ay

w(O

0(0 ,) =

fPw (I, /) "'" ffw (I, I) = a8 (I, I) =

'a!l

all

ay

°

(7-305)

A characteristic.value problem is obtained when we replace the runnin g lift L and moment M . by means of the strip-theory equations (4-156) and (4--157), then substitute the motion coordinates 11', 8 themselves for (-h) and (:I. If we examine the case of unsteady flow at arbitrary flight Mach number, th e indieial functions 'P, 'P., etc., arc so complicated that there is no hope of solving Eqs. 7-304 and 7- 305 and in this way determining the frequencies, decay rates, and shapes oflbe coupled aeroelastic modes. Under tbe approximation of quasi-steady aerodynamics or of piston-theory operators (e.g., Eq. 4- 183), however, we can reduce Eqs. 7-304 to a linear set with constant coefficients. Although the total order is six in y and four in /, a solution can be worked out-for cxa'mple, by Laplace transforming on both the independent variables. It will consist of products of exponential functions of y and t, the complete set being denumerably infinite in correspondence with the infinite number of complex-frequency eigenvalues. The details have apparently Dever been carried through. A less unwieldy problem, and one in which the exact forms of the unsteady operators may be retained. consists in seeking the stability boundary where a particular mode is simple harmonic,

1.-'(yYooI

(7-306)

Q(y, /) = ii{y)eio,t

(7-307)

w(y, I)

=

I'IU NCU'U:S 01' AF.NOI!l.,A&1'ICI'n'

l71i

When the superson ic airload notation, Eqs. 4-1 29 and 4-130, is il1scrtcd into· Eqs. 7- 304 along with Eqs. 7- 306 and 7- 307, we can cancel the common facto r e'ooJ to obtain £1

~; -

[m - 4p.Y(LI +

iL,,)]w~II'

- [-S. GJ

d'iJ + [ - S. + 4p",b'(M I

dy'

+ T he pauern of Eqs. 7- 308

+ 4p", b"(La + iL.)]w20 = + iM. )](,iJ;'

[I. - 4p,.Y (M 3

i~

+ iM,)Jw">O"",

0

(7-308a)

0 (7-308b)

as follows:

dIll'

_

--ali' - ~{j=O

(7- 309a)

tPii + ylV + ()(J- =

(7- 309b)

dy'

-2

dy

0

Here .. ~ (m - 4p",b l ( L j + -i~)] w 2 Ef

(7-31O)

and the rema ining coefficients have similar expressions that are evident from the two sets of equations. The J;",l!Itilever boundary condi tions (7- 305) become 11-(0) = w'(O) = 0(0) = ,,,"(I) = )V' (f) = O'(!) = 0 (7- 3[1) Equations 7- 309 and 7- 311 may formally be solved by means of a series of terms like A "e"·1 (Ref. 7-64); but it is more efficient to adopt Laplace transformation on y,

h(p)

==

L'" e- "b(y)dy

(7- 312)

Applying this operation to Eqs, 7-309 and using the bou ndary conditions at y = 0 to eliminate three of the n:sidual terms arising from d4 1V{dy' and ,,20Idy2, we get (7-JJ 3a) r ~ (p) - rI.;{p) - PV(p) = pW! + W3 rU(p)

+ yw(p) +
(7-3 [3b)

The following abbreviations arc empl oyed on the righ t-h and sides: IV! IV,

= ,inO)

1

== 1t'~(O) I 0, == 0'(0) J

(7- 314)

ON I~. I)IMllNS I ONA I .

SrlWCl'U HI'$

377

The algebraic solulion of E<Js. 1- 313 rC:ids -w( ) = [p'

+ bp]W: + [II~ + bJ W + ($0 1 3

P

/l(p)

O( ) =

[pO - elJ0 , - pyW, - yW. P d(p) the denominator polynomial being d(p) =

rI' + lip'

-

elp2

+ ({Jy

(1- 31511) (1- 315b )

- Gel)

(1- 316)

The standard in version of Eqs. 1- 315 re9uires findi ng the rOOls of thc bicubic d(P) and then exprcssing 1;;(y) and O(y) as series of expone ntials in y by means of the familiar technique for rational fractions (Ref. 6- 4. Chap_ 6). To avoid the excessive labor thus encountered, Runyan and Watkins(Ref. 7- 66) proposed to take advantage orthe relationship between the power series expansion of a function in 11 and its transform in ( IJp) . They write

_'_ =2..1

Tn

p~ .. ~ O

d(p)

(1- 311)

rP

whence there is no difficulty in sbowing that Tn"" I T, = -6 T2"'Cl+~2 T., = -tP-&t-iJy T. = -IIT"_1 + xT. _ ~

(7-318)

+ (elO -

{Jy)T. _3 •

(n :;;:: 3)

Because of rapid convergence, very few of the T n need to be computed in practice. Once Eq. 7-3 17 is substituted into Eqs. 1- 3 15, all terms in both formulas are of the form A,./p'" and can be inverted through "" - '(-!

~

1-

p") - (m

I

,

",-I

I)!

(1- 319)

We are thus kd to something like the following :

+ hiy)Wa + h3(y)01 g,(y)Wz + giy) Wa + gb)0.

w(y) = h, (y)W2

(7- 320a)

O(y) =

(7-32Ob)

where, for example, T

h.(y) =

2... 'l.

TyhH

y 1 n + .:I "1. ~(2n ~",:--;" n _~ (2n + 2) ! + 4)!

(7- 321)

3111

l'IU NC II ' I,I,,"i 01' AI!RO I;I,ASTl CIT \'

The remaining polynomia ls ,LTC lisled in Rer. 7- 66; their 1;0Llverge m;e is obvious. The eigenvalues of Ihe proble m can be co nstructed by resort to the boundary conditions at the wingtip (the last three members of Eq. 7- 3 1I). They produce the homogeneous equations

0)

h;N(I) W3 + h! (I) W3 + h3:(1)9 1 = hi (I)Wt+ h z (l) W, + h3 (1)9 1 =0

(7-322)

gN)Wz + gz'(I)W3 + g3'(1)91 = 0 Clearly, if the solution is to be non trivial, the denominator determinant of Eq. 7-322 must vanish, ht(1)

hs"(f)

ha"(l)

ht"'(/)

h!"'(I)

h3"'(J) = 0

gN)

g2'(f)

ga'(/)

(7- 323)

Equation 7-323 is nothing but the flutter determinant. It is of interest to observe that its order is jusl one·hatf that of the original system of ordina ry differential equations, whereas when modal methods are employed the determinant has the same order as the preselected number of generalized coordinates. In terms of the flutter eigenvalues, Eq. 7-323 has a much higher degree, controlled by where the se ries (7- 317) is truncated. Indeed. factoriz.ation is generally impossible, since the T~ (through 11:, p, )I, and tI) are complex, transcendental functions of the reduced frequency k ;;: lub/V. A tria l and error process must be used in practice to ascertain flutter speeds and freq uencies. Given a set of wing parameters, flight Mach number, and ambient density, for instance, one can compute the real and imaginary parts !!I" and J m of the determinant (1-323) for several values of w and k. Plotting !it" and Jm versu s w for each k, we find a set of w-k combinations at which !!Ie and Jm vanish separately. A cross plot of {/t" = 0 and.fm = 0 on the w.k·plane finally yields one or more intersections, each corresponding to a speed and frequency of neutral stability. If desired, the aeroelastic mode shape at flutter can be expressed as two complex power series in y by substituting into Eqs. 7-320 the values of Tn calculated from th e known eigenvalues. It shoutd be evident to the reader that the generali ty of these results can be increased, when making parametric studies and the like, by recasting Eqs. 7- 309 and 7-31 1 in dimensionless te rms before proceeding. A very important extension of the exact solution has been presented (Refs. 7-65, 7- 66), which accounts for a concentrated mass attached at an

nrbitrllry spa nwise ami chord wise loc(Ltioll 011 the uniform wing. Such lL mllSS causcs d isconl in ui lies in I he vi h ratory shear II I1U torque, bu t these II re efficiently handled by tile L
Flutter prediction by superposition of normal vibration modes or artificially defmed generalized coordinates is the familiar procedure used regularly in the United States at least since the publication of the SmilgWasserman report (Ref. 4--56), and probably longer in certai n European countries such as Great Britain (cf. Refs. 6- 19, 7--68). Here we describe its application to computing the prima ry bending-torsion stability boundary for an essentially unswept (IA I <. 15°) lifting surface with onedimensional structural properties. Techniques for setting up the equations of motion are reviewed in Sec. 3- 5, whence we may take Eq. 3--JJ8, for example, and specialiJ:c it to the case where all loads and defl~tions occur only in the z-dire<:tion. It is generally agreed that the "camber-bending" type of chordwise de fo rmation (Ref. 7- 69) has little influence on the flutter of moderate-to large-aspectratio straight wings; therefore, we deal with modes ylt", y) which are no more than linear in the streamwise "Variable z. As in foregoing sections, the entire defte<:tion consists of a flexure w(y, t) and twist O(y, t). There are two common ways of defining the generalized coordinates: on a structure with a true clastic a)(is, it is aerodynamically convenient to adopt "uncoupled" mod es of pure bending and torsion; whereas, for more complicated elastic configurations, the orthogonal normal modes are more suitable. Taking up the former first, we choose a total of n coordinates

380

I'IUNCII'U'S 0 1' AliHO I".ASTlC I'rl'

q,(I), ,of them associated with bending modes I .. ,(y), the rest with torsion

l o/Y): w(y, I) = O(y, I) =

•

,.L, j",(y)q,(t) 2:•

f . ,(y)q.(l)

(7-324) (7-325)

/mr +l

Since theifunctions are dimensionless, the q, have dimensions of length for I ,.;: i :=:;; , but are angles for, < i. The instantaneous kinetic energy of one wing, lying between a root at 11 = 0 and tip at y = I, is easily shown '" b,

(7-326) From Eq. 7-326 we can immediately dedu~e the form of each term appear· ing in Eq. 3- 1 [8. The running inertial properties m, S~, Ia have the same definitions as in the preceding subsection, but they vary with y and may even contain delta functions to represent a fuselage, nacelles, or other concentrated masses. Since y is supposed to be an E.A., it follows that the potential energies of flexural and torsional deformation can be separately computed and added. Furthermore, it is customary to eliminate aU elastic coupling by assuming a pseudo-orthogonality among the shapes '", , . . " ) 'O' '. and 10><, ," ', )' .1 . ' so that each of their contributions to U is )'1 independent. Then, as set forth on pp. 559- 560 of Ref. 1- 2, a series of natural frequencies w,. . and W6. can be invented by imagining a coostrained free vibration ;'n each ~ode. These considerations lead to

Since the only external loads are lift L(y , /) and moment Miy, I), the generalized forces on the various degrees of freedom are

f l'

L(y, 1)/.)y) dy,

(1 :=:;; i :=:;; r)

(7- 328) M .(y, 1)/. ,(y) dy,

(r+l :=:;; i :=:;; n)

ONfI.. UlMENSIONAI. STHUcrUHI,s

)lI1

Defo re writing down the cyuatiolls of mOlion, we introduce st ru ctura l friction in thc custo mary fa shiol1 . Since this phenomenon may havc a stro ng influence on the location of the stability boundary or even suppress the instability altogether in certa in critical cases, its inclusion is much more important when treating flutter than when treating forced response. As discussed in Ref. 1-2, friction can be satisfactorily appro ltimated with small forces, opposite in phase to the velocity but distributed in direct proportion to the elastic restoring force due to each mode of bending, and with a similarly distributed torque on eac h twisting mode. One thus represents a damping effect which is observed \0 destroy an amount of mechanical energy per cycle of "Vibration that is proportional to the square of the amplitude but independent of frequency. All this is accomplished by multiplying each frequency 00",,2 or w" t in Eq. 7- 327 with a complex constant (I + ig..,l or (I + ig6,J. The g"factors are positive, with values in the range 0.005-0.05, depending on the st ructural mate rial, design, and fabrication ; they can often be estimated from free-"Vibration tests. Gi"Ven Eg. 7-326, Eqs. 7-328, and Eq. 7- 327 with the added structural damping, one applies Lagrange's procedure or Eg. 3--1J8 to obtain the following:

, [' mf..,!." dy - ' -L",+1 J'S.!.,J"" dy

Zij,

ij,

,• • - 0

0

+qjw..: [1

+

(1 $. j $. r)

ig..Jfmf..,2c/y = fL(y,')f.., dY,

(7- 329)

'J.' S./.,/. , dy + Z" J' I.f,,!., dy

- Z ij, <- I

ijj

0

+ Q1w,,![1 +

' ~ .+1

0

ig,Jf1.f.,2dY = i'M.(Y, t)f., dy, (r

+

I 5. j $. n)

(7- 330)

As a rule, the integrals of mJ..,/.., and 1..1,,/&, for i "'" j are made to van ish by means of the same pseudo-o rthogonalization of the artificial modes that is involved in Eg. 7-327. Since neutral stability is under investigation, we next assume each generalized coordinate to have the form ij~ ... t. It is then possible to arrive at formulas for the running lift and moment, based on any of the linear strip-theory aerodynamic operators from Sec. 4-5 or th e three-dimensional operators fro m Sec. 4-7. Inas much as this sort of computation has been fully deveJopedelsewhere, Jet us merely point out that, in the homogeneo us

3112

I"UNCU'LES 0(1 ,UmOItLASTJ CITV

case, each of lhe gener31ilcd rorees can ullimalcly be redu ced to the fOflll (l '5. j ~ r)

(r+1 '5. j '5. lI) (7- 331 )

Here b /{ is the semichord at some reference station such as the wing rool, and the QH are dimensionless unit generalized forces, each depending on wing geometry, Mach number, reduced frequency wbRiU, and the two mode shapes that are identified by its subscripts.· When Eqs. 7- 331 are introduced into Eqs. 7- 329 and 7-330 and the common factor e''' ' is cancelled, we can divide the bending equations by 4p~wzbIl3f and the torsional equations by 4p gc(,)lb R4f to obtain the foHowing:

" ·11'm PR - f.:/,.,dY + Q"I\

1: -b'R

, m}

o mR

(1 '5. j '5. r)

(7-332)

(7- 333) Properties of the reference station are marked with subscripts' R'; thus, Jill

=

-

Inn

4p", b n z

(7-334)

As before,:r:~ and r« are the static unbalance and radius of gyration about

the E.A., made dimensionless with respect to local scmichord b . • Note that Q,; '1' Q", as

i!;

easily proved by example.

ON~:-I)JMt;NSIONAI.

STItUCr UItI,;s

3113

The Outler determinant has o rd~r 1/ nnd is formed from Ihe coelJicients of the ijdhl/ and ih. For a givell configuration, ambient air dens ity. and fli ght Mach number, this complex determinant amou nts to two real equations fo r speed V". and frequency ' UFo There are, at most, It pairs of physically significant roots, the one of grea test practical interest being that which yields the lowest speed. For illustrative purposes, consider a cantilever with the calculation based on a single bending mode f ..(Y) and a single torsion mode fJ..yl. The generalized coordinates are q, and q•. respectively. If we then neglect span wise induction, it is an easy matter to express the generalized aerodynamic forces in supersonic notation as follows: (Jll = -

J:(b~)h-] + iLJf..!dY

=1: (:j~[L3 + iLJfJ , dy 1221 f: (:~J[MI + iMJf..!, dy -I: (:}[M + iM.Jf. (Ju

(7- 335)

=

(Ju =

1

a

dfj

If desired, the dimensionless coefficients in Eqs. 7-335 may be multiplied by overall functions of Y or otherwise manipulated to approximate three:dimensional aerodynamic effects. The determinantal flutter equation finally reads

("R[' - (:;)'(:;)'e1 + ;g.)]f.'::'/-'dY

-f.'(:~h, + ;["J /.' dyl -f.' (:j'[M, + ;MJ/,' dyl = 0 (7-336) For flutter calculation based on true normal modes, ",lx, y), the y-axis is simply a convenient structural rererence line and need not play any special role in dccoupling bending and torsion. No ioogcrcao the potential

384

I'RINCII'LFS OF AEROI!.LASTI CITV

cncrgy be cast in such a simple form as Eq. 7- 327 except by taking advantage of the general orthogonality of the '1'/5. We write the normal displacement of the wing as w(:I:, y, t) =

.

,-L

'P.!..:I:, y)¢it)

= e;a'

..

,L

'P/..:I:, y)t;

(7- 337)

Here the 'Pj are usuaUy no more than linear in x. The flutter equations are (j =- 1,2,3,"')

(7-338)

where (7-339)

M/ = II p(x, y)rp/1A y) dx dy planform

..d

EM j

_

(7- 340)

II6j.illf(:I:, Y)'Plx, y) dx dy

pl..,{orm

The only coupling among the degrees of freedom is aerodynamic and arises because the pressure difference 6j.i"(x, y)e ;'" is linearly related to all of the 'l l' By analogy with Eq. 7-331, we can say (7-341 )

Evidently the flutter detcrminant is {M,lw' - w,'} Q,,'

0.'

+ Q,,'}

Q,' {M.rm' - ",,' ]

Q.t

Q,,'

+ Q..']

Q••

(M.lw' - w,']

+ Q..')

-. (7- ]42)

As in the case of Eq. 7-336,- this can be regarded as the determinant of a sum of inertial, structural, and aerodynamic matrices. The order is equal to the number of modes needed to assure an accurate solution, whieh may vary from two on simple cantilevers to more than twenty on large, flexible airplanes with clasticaUy mounted external stores. As a rule, a division can be made bc:tween modes which are symmetrical and antisymmetrical with respect to the vehicle's midplane, for no coupling exists between tbese. Structural friction is readily introduced into Eq. 7-342 by replacing each ml with ml[l + iC/]. Also it isgood practice to render thefluttcr cquations dimensionless, thus reducing the amount of computation, whenevcr making

ON&-OIMt:NStONAL Sl'MUCJ1JKF.s

lIS

parametrie studies or elltensivc eonfiguration cha nges. If strip theory or its equivalent is used for the aerod ynamic operators, we lind that the normal-mode scheme is the more laborious. This is bocause each shape function ,/,,(%, y) has both bending and torsion contributions, so that each of the Qi/ contains all of the coefficients~ , ~,· - - ,M•. By contrast, Eqs. 7- 335 show how these coefficients appear only two at a time when artificially decoupled modes are employed. The wide availability-of highspeed computers has tended lately to minimize this distinction as a factor in deciding the best way to proceed with a particular flutter analysis. Another question which has lost some of its erstwhile significance is how to go about the mathematical determioation of the eigenvalues of the flutter determinant. We have already reviewed this to some extent in Sec. 6-5. All that needs to be added is that the trial-and-error nature of the computation, which results even in the simplest cases from the transcendental dependence of the aerodynamic forces on reduced frequency add Mach number, is reinforced when modal methods and Jarger numbers of gene ralized coordinates are used. Among the va riou s techniques of solution, the U.g-method perhaps merits special mention bocause it repl aces the problem of finding pairs of real ro ots of two simultaneous equations by the somewhat easier calculation of complex roots of a single polynomial with complex coefficients. The latter factorization can be done in closed mathematical form for polynomials up to the fourth degree, which correspond to systems with four generalized coordinates. This advantage is gained only at the price of a certain artificiality: the structural damping coefficients of all the degrees of freedom are assumed equal to a single constant g, which is then adopted as one of the unkn owns. Under these assumptions, one observes from Eq. 7- 336 or any other dimensionless modal Hutter determinant tha,

(1- 343) occurs linearly in each term of the principal diagonal. (The reference frequency WR would be Wo in Eq. 7- 336.) Furthennore, these are the only locations where wand g appear explicitly in the equation. If the C{)nfiguration, density, Mach number, and reduced frequency wb ll !U are picked in advance, the determinant expands into a polynomial equation. Special procedures have been developed (e.g., Ref. 7-70) for getting th e roots directly from the determinant when its order is too large to allow C{)nvenient expansion. T he output of a U'g-calculation is generally displayed as a plot of artificial structural damping versus airspeed with the actual UF occurring at the point where g equals the true

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