Principles of Aeroelasticity

PRINCIPLES OF AEROELASTICITY

PRINCIPLES OF AEROELASTICITY RAYMOND L. BISPLlNGHOFF CHANCELLOR UN IVER$rrV OF MI SSOURI

ROLLA

HOLT ASHLEY PROFESSOR, AERONAUTICS AND ASTRONAUTICS STANFORD UNIVERSITY

DOVER PUBLI CATIONS, INC. NE W YORK

C"I'),ight CD I%~ 1» R,,)",olltl L lli'I'Hnghotf 11011 ,\,hley, ,\11 right< ,'"",,-,,·,1 ,,",kr Pall """,,'ieall and I III~l'lIa' 'Oil" 1 (".01')' 'gh, COlll'cn' '''"'_

~Il'!

P"l>lish ",t ' " C"",d" h)" G""c,'.1 I'"hl ;,hill& Com· I""'}", Ltd .. ."\0 t.~,", ;tl ~o",l. non Mills. "l'Otonlo. Onlario. l'"bli,hl~1 ill Ihe lini(cd Kinb"lolII b)" eomlahlc "111 eo"'I',,"y. 1.1 01 .

Thi' n",'", editioll . r"-,, , pub!i,he.1 ill 1975. j, " " lI11al"-i,!!.,,,,!. content; this is the case when surrounding atmospheric fluid interacts with a flexible structure, giving rise to aeroelastic phenomena. There are other examples, such as the force of gravity or the impingement

lN1'ltODUCI'ION

13

5'1S~~;:3co..".. Sdt X-" I$

FIt:, 1_7.

~

~l kI

IUght vehicle oonccived as a coUection of interacting forces surrounded by

eQvironmental fields. of solar radiation OD the very cold skin and solar batteries of a space station, when the action is, for practical purposes, unidirectional. 1-3 A NEW SCHEME OF PRESENTATION In this introductory chapter, we are purposely refraining from presenting a list of verbal or quantitative descriptions of typical problems in aero-

elasticity. Chapter 6, in effect, amounts to such a breakdown, based on one very simple two-degree-of-freedom system. Pursuant to the point of view expressed in the previous section, we are laying out the book along the following lines. First, we give general methods of constructing the static and dynamic equations. The concept of aeroelastic operators, suggested by Fung (Ref. 1- 1, Chap. II), has proved very helpful here. Chapters 2 and 3 deal with the laws of mechanics for heated elastic solids and with associated ways of finding inertial operators for the equations of motion. Chapter 4 reviews the forms of aerodynamic operators which describe extemalloads on bodies and lifting surfaces in various ranges of

I"

I'MINCII'LI!S 01' AHHOKl.ASTICITY

night s~d. Chapter 5 discusses struetuml optrutors, including the innuence of temperature variations throughout the solid. In these three presentations, we appeal as much as possible 10 Ihe va riation al principles, which furnish compact statements ofllie fundamentals and sometimes also provide natural means of approximately solving practical problems. We admit that this approach is more successful in connection with elasticity than with fluid dynamics, but it docs add an element of unity that deserves to be emphasized. Particularly in the field of aerodynamic theory, we no longer feel there is a need for a full logical development of the tools in a book on aeroelasticity. An inordinate amount of space is consumed thereby, and such prodigality is better supplanted by jUdicious references to the excellent literature now available. Accord ingly, parts of Chaps. 4 and 5 an: no more than catalogues of operators for subsequent use. Chapter 3 completes tbe introductory material with further details on techniques of setting up and solving the mon: familiar equations, such as energy methods, nonnal coordinates, and other superposition schemes. There is no extensive review of the Subject of mechanical vibrations. Although this subject certainly falls within the province of the aeroelastician, the profusion of excellent treatises already available relieves us of this task(see,forexample: Den Hartog, Ref. 1-31; Timoshenko, Ref. 1-32; Rocard, Ref. 1-33; and also Refs. 1- 9 and 1- 11). Chapters 6 through 9 form the heart of our survey of the current state of linear aeroelastic theory. The primary classification is by the numbtr of independent space variables required to describe the physical system. It proceeds from simplified cases which have only a small, finite number of degrees of freedom, to one-dimensional systems (line structures), and finally to two-dimensional systems (plate- and shell-like structun:s). This would seem to cover all situations of major interest in aeroelasticity, so Chap. 9 combines some of the previous results by treating the unrestrained elastic vehicle in flight. Chapter 10 takes up the increasingly important but relatively unfamiliar (to theoretical aeroelasticians) subject of systems which must be represented by nonlinear equations or by equations with time-varying coefficients. For preparing the bulk of this final chapter, the authors are deeply indebted to two colleagues, Garabed Zartarian and Eugene J. Brunelle, Jr. Witbin every category, it is possible only to describe a few typical structun:s and particular problems which cbaracterize them. Each chapter starts with steadyand quasi.steadyphenomenaand then goes on todynamic phenomena. A practice is adopted of examining forced displacements and foreed motions first, so that the homogeneous parts of the equations thus obtained will afterward apply directly to eigenvalue problems such as

INTRODUCTION

15

divergence and nutter. As long as the limitation to linear mathematical representations is preserved, this constitutes a consistent, efficient proced ure. But we must always bear in mind the warning, so well expressed by Fung (Ref. I-I, Introduction), that the re is a "very important distinction between the response and stability problems, in regard to the justification of the linearization process." When examining the eigenvalues and eigenfunctions which determine stability, the ampli tude is for the most part of little interest, and it is logical to consider infinitesimal departures from static equilibrium. In analyzing gust loads, control elf~tiveness and tbe like, however, the degree of finiteness of stresses, accelerations, and displacements becomes important. Linear theory has severe weaknesses in more such situations than are now r«:ognized, and an urgent subj«:t for future research is the clearer quantitatiVe establishment of its limitations. The contents of this book are essentially theoretical, not because of any prejudices on the part of the authoTS but because of the way in which the topic was assigned them. Where possible, the t«:hniques outlined in what follows are chosen because favorable comparisons with measured data exist. Enormous advances have recently been made both in the experimental procedures of aeroelasticity and in the related instrumentation. Another book migh t be written entirely about dynamic modeling, and yet another on full-scale vibration and flight testing. The aeroelastician also has available to him several valuable additional tools, of which sled testing, rocket-model testing, and 1he direct measurement of unsteady airloads are illustrations. Another important area which tb e authors have been abte to touch on only in passing is that of digital and analog computation methods in aeroetasticity. The viewpoint of this book is somewhat colored by the authors' experience with high-speed digital machinery, as some readers will observe from the cboice of examples. Nevertheless, the direct analogic representafion of aeroelastic systems by electrical networks is perhaps the most promising technique that can be singled out for elltel\.Sive exploitation in the future.

REFERENCE'S F~g, Y. C., An in/r{X/ucllon to Ike Theory (Jf ArrIM:Wsriciry. 10hn Wiley and Sons, New York. 19S5. 1-2. Bisplingnolf, R. L., H. Ashley, and R. L. Halfman, Aeroelosrlcily, AddisonWesley Publishing Company. Cambridge, Ma!.!t" 19S5. 1-3. Scanlan. R. H. ,and R. R05enbaum, Inrroductlo~ /(J Ike Study of AirCrtlfl Vibrofi(J" tllld Fluller, The Macmillan Company, New Yorl
oq, + oq, =

Q,

(i=l, " ' ,n)

(2--67)

These are known as Lagrange's equatiom of motioll for the meclJ.anical system. In the terminology of the calculus of variations (Refs. 2- 5 to 2-7), they are the Euler equations of the variational principle. Su~b equations exist wbenever tbe state of tne system can be represented in terms of a set of discrete coordinates. Lagrange's equations are extraordinarily valuable when dealing with neroelastic phenomena and possess far wider practical utility than the broader principle from which they spring. In an efficient, weD-organized fashion, they generate both the exact equations for lumped parameter

36

I'KINCII'Llts 011 AmWELAS"IC ...."

systems and approximate representations, to aay desired degree of accuracy, for continuous or distributed paramcter sys tems. They form thc basis of the uscful approx ima tion technique, known as the Ray leighRitz method. A typical Eq. 2- 67 can be considered as a gcneralization of Newton's second law of motion by placing oU/oq, on the right-hand side. The force system th en consists of two parts: the -Qu/aq, terms describe any conservative generalized forces that have been included in the potential energy, with minus signs to account for U being the negative of the scalar potential function; and the Q, terms eJ[press all remaining forces, particularly the unconse rvative ones. The acceleration terms are also divided into din:ct linear and angular "generalized inertia forces," contained in dldt(oT/Qlli), and nonlinear inertial effects -OT/Oq" involving products of velocities aad going by such names as centrifugal, Coriolis, or gyroscopic forces. It is not a difficult matter to broaden Hamilton's principle and Lagrange's equations to cover electrical systems, coupled electromechanical systems, or situations with ILeat conversion. All these matters possess at least subsidiary interest in the field of aeroelasticity. Then: are certain n:strictions on the comprchensive validity of what has just been presented, of which at leas t two deserve formal mention. The first concerns the physical constraints placed on the motion: it is required that the system be holonomic. Mathematically, th e equations applying these constraints to the system coordinates must be algebraic. In a nonholonomk system, the same equations constitute relations among differentials which cannot be rendered exact and tberefore integrated for arbitrary time intervals. This distinction is fully discussed by authors like Fox (Ref. 2-6) and Lanczos (Ref. 2-7). Of greater practical interest is the requirement in Egs. 2--43 and 2-46 that conditions be specified at both time 10 and 11' This is Dot characteristic of actual dynamic problems, whero inirial conditions are given at, say, to = 0 and the motion is then to be calculated for times t theroafter. No difficulty is encountered when Lagrange's equations can be constructed, for these are differential equations which may, in principle, be integrated from instant to instant. But th e question of how to handle the upper limit 11 during direct application of Hamilton's principle is a more subtle one.

2- 7 A SIMPLE APPLICATION OF LAGRANGE'S EQUATION In a host of texts on dynamics and vibrations the single-degroe-of-frccdom mass on a linear spring (Fig. 2-2), or its mechanical or electrical analog, forms a first elementary illustration of how the laws of motion nrc

MATliUMATICAI. I'OUNUATIONS OF ARiWICLASTICITY

Fi,. 2-1. force.

General~d

37

single degree of freedom linear oscillator acted on by IlII external

applied to systems with and without external forces. The reader's eJ{perience can perhaps be counted on for supplying the details, and he may find it an interesting exercise to try to adapt Hamilton's principle andLagrange's equations to this case. A logical aeroelastic counterpart to the simple linear oscillator is shown in Fig. 2- 3. It is the so-called typical sec/ion wing, a rigid airfoil section suspended in an airstream, and permitted degrees of freedom in bending

Und. flected pos~lon

01

airfoil centerline

, ~irstream ~ocity

01

U

lunneilloor

, translation

FI,.2-3. Two-d imensional airfoil suspended in an aintream by bending and torsion

springs,

38

PMINCII'LI'.S 0 1' A"Mom .....STICI'l'¥

and lorsion by suitable suspension frolll two sets of springs. Both th e motion and the airflow pase ehe wing arc arranged to be two-dimensional, that is, independent of the spanwise coordinate. Aerodynamically, this is done by mounting it wa\1-to-wall across a two-dimensional stream between plane, parallel sidewalls. With all the flexibility in "massless" springs, this b; a true lumped parumeter system. Only two generalized coordinates are needed to specify the instantaneous position fully; natural choices for these are the downward vertical displacement h of the line of attachment of the springs (elastic oxis) and the Jeading-edge-up angular rotation tt about this line. Both are measured from the unstrained positions of the corresponding springs, and gravity b; neglected. Let it be assumed that ox and hIe are small, c being the wing chord, and also that the thickness ratio (ratio of m:uinlllm airfoil depth to c) is less than about 10%. oX b; a chordwise coordinate taken positive aft from tbe elastic axis, and z is normal to the chord in the plane of motion. The scalar equivalents of the transformation eq uation (2-58) then read

v = 0,

u = 0,

w=-h-xox

(2-68)

provided that the cubes of sman quantities are negligible. If the bending and torsion springs have total stilfnesses kA and k", respectively, the potential energy of strain is (2--69) This is just the energy in Eqs. 2- 67 associated with the conservative internal forces or the system. Equation 2-69 contains no cross-product term (Ba = 0 in Eq. 2~1), so the system is said to have no sialic and elastic caupling between the degrees of freedom. In a similar way, the kinetic energy can be written as 'T

=

i

r

Jthord

WZ dm = ! (

J eI\Ord

(h

+ ~ijl dm = ~mlz' + S.hii. + !I.ii.t (2- 70)

where

m = total mass, S. = m:c,. = static unbalance of wing about its elastic axis, I" = mass moment of inertia of wing about its elastic a){is. In the absence of air, the wing constitutes a conservative syste m. Choosing ql = hand q2 = 1.<ed wing surface.

where we have assumed linear operators and where IV is the lateral displaccment of the wing surface and Zp is the net lateral disturbance ]lrc~surc. When the wing surface is divided into a network of n points. IL~ illustrated by Fig. 3- 1, Eq. 3- 79 can be satisfied at each point. Then Its form changes to that of a matrix equation, as follows: ([9') - [JaI"] - [.9J) {w} = {Zn}

(3-80)

In Eq. 3-£0 the square matrices [9'], [.#], and [J] are n x /I matrices of structural, aerodynamic, and inertial influence coefficients, respectively. The column matrices {w} and {ZD} are n x 1 matrices of the deflection and Ihe disturbance force. respectively. When the matrix fonn of Eq. 3- 80 is formed, explicit solutions which depend on the exact nature of the problem cun bc obtained. By using the inverted form of the matrix of structural inl1ucllce coefficients, we can rewrite Eq. 3- 80 in a more suitable form for unu lysis: (3-81)

whcre [9'] - 1 is a matrix reciprocal to [SI' ]. In a dynamic aeroeiaslic problem, Eq. 3- 81 represents a set of simultaneous ordinary differential equations with the independent variable lime; but in a static problem, it represents a set of simultaneous algebraic equations. Let us look in some detail at the formation of Eqs. 3- 80 and 3- 81 by IIpplying them to the static torsional equilibrium of a slender straight wing as illustrated by Fig. 3- 2. We make the fund amental assumption that the wing is a slender beam with rigid chord wise cross sections. The

/U , 4 - -- -,- - - -./ Fi g. 3-1. Slende r straight wing.

elastic twist angle, fl, can be expressed in terms of an integral equation of static torsional equilibrium as foHows; 8(y) =

J~ C'~(y, 71Xecc + C2C",A.o)q d1J t

(3- 82)

We can regard the total angle of attack distribution, ~(y), as a superposition of a rigid angle, Gt'(y), and the elastic twist, O(y): Gtw) = Gt'(Y)

+ U(y)

(3--83)

and the correspo nding local lift coefficient distribution, c,(y) , a superposition of elY) = c{(y) + c!"Vi) (3 - 84) where c{(y) is the local lift coefficient distribution resulting from rigid twist, ,,'(y), and c,'(y) is the local lift coefficient distribution resulting from elastic twist, O(y). Substituting Eq. 3-84 in to Eq. 3- 82 yields O(y) = q

f

Cee(y, 1J)ecc,'d'i

+ fey)

(3-85)

where fey) = q fc:H(y, l1Xecc,' +

2 c,yOC )

d"l

In those cases where aerodynamic strip theory is admissible, we ca n put oo«(y) = c,(y) and obtain the following simplified version of Eq . 3-85: O(y) = q where

l' ~(y,

"I)a,c"'() dl)

+ f(y)

(3- 86)

AmH JI·:I. ASTI ( · 1':()lIAT IONS AN I I 'l'lmm SOI.UTIONS

57

By Illc:on.' "rl''!' .l XI,. We c:ln ""'lIputC the :I ngle " i" twisl ,It a linite number oi" points. say is given by fJ(y;)

II.

", = fJ C J 0

as sbown hy I,' ig. 3 2. The clastic twi st :lIthe ilh station

fO

(!/ •. "I)I/oed} (/11

+ iCy;),

( i = 0 , I " 2 .. . , u)

(3-87)

An approximate numerical evaluation nftbe definite integral can be carried oul which leads to a linear combination of the ordinates of the loading diagram, as foHows:

•

0, = q L C:J' W/ aoceft)J + /;'

(i = 0, I, 2, . . " /I)

,.,

(3- 88)

G/

where are the influence coefficients a ssociated with the n points, and WJ arc weighting numbers ( Ref. 3- 1) wh ich depend on the method of numerical integration tha t is empl oyed . Equation 3-88 becomes ill matrix for m,

I') ~ q[ c+ I~) (3- 94)

Silllpson's rule yields an cxact result if the function which is being approxi. mllted is a po ly nomial of th e third degree or !ess. T he weighting matrix for Simpson's rule is I 0 0 0 0 0

C'w-J

0 4 0

0 0

0

,

0 0 2

0 0

0

0 0 0

4 0

0

)

0 0 0

0

o

0

0 0

(3-95)

2

I

Olher forms oflhe weighting matrix arc useful in aeroeJasticity, but space docs not permit us to delve further into this matter. The reader is referred, for example, to Ref. 3- 2 for a discus sion of Lagrangian interpolation functions and to Ref. 3-3 for Multhopp's quadrature formula. (b) Collocation with generalized coordinates aod assu med modes

In the coilocation method described in the previous sectio n we discussed the de formation of the structure in terms of discrete deflections at sele1'",10/, /'roMr",. 1/1 Engl/l"uitlJ', O. V"o Nostrand Co., Ncw York. 1937. 3- 6. Flax, A. J.I., "Acroc\aSlic Problems at SUpersonic Speeds." Pro are made at altitudes whe re the molecular structure of the gas is not directly sensible, so that it behaves like a continuous medium. Its motion is then described by the space and time distributions of six scalar variables: density p, pressure p, temperature T, and three components of the velocity vector· q = ui

+

tlj

+

wk

(4-1)

The former arc re lated by an equation of state which, for atmospheric flight conditions, is the perfect gas law p = RpT[1

+ "1

(4--2)

Here "(1', T) is an effective mass fraclion of diatomic molecules in the dissociated condition (cL Liepmann and Roshko, Ref. 4-7, Chap. I), and R is the "lLndissodated" gas constant. Dissociation can be neglected at sea-level pressures when ambient temperatlLre T is less than about 4OOO° F, but dissociation gains greater importance with increasing altltude. Combinations of even lower p and higher T can lead also to ionization reactions of the gas atoms, but these are not expected to have any significa nt cffect on the aeroelastic stability or loading of currently envisioned vehic les. Both dissociation and ionization must often be considered, however, in the viscous theory of heat transfer at high speeds, along with relaxation times of the various molecular cbanges which vitiate the usefulnc~s of conventional equilibrium thermodynamics. As readily provcd. Eg. 4--2 with" = 0 implies that the specific internal cncrgy, e, and cnthalpy. 11 = e + (pJp), are functions of T only. From • To co"ron " wilh lirmly cslablished acrooym"nic prac\ice, we ","ploy the same .)'mool s here r",· velocily component.< as arc lISed c1sewhcn:.: in lhe book for clastic (II~JlIa"~"I(;"I" No co"rusi o" sho"ld arise. b.:cm,sc the nOI~liotial uhcrraliun i. confined 10 a lii I). 4.

72

1'ltlNCII 'I,I'S 01' A lmOlcI.As .... CITY

this il follows that Ihe specific heats, 1'. = (all/o 1')" :111 _ 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:;-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 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, spej -",-)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- 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-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 }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 COp{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 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

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)

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 principllif Lifling Surface 11oMry), Britiih A. R .C. Reports and Memoranda 2884, 1950. 4--32 . Reis-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 AuDay,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 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 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":lL1~

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 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 ) 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 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
(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 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

=
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 CroNatu

(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

+ Uc 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}:)(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'$.

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-seO

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

(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,

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

i).

(3) If t he. A.C. is forward of the E.A. (~ only if

>

(z~ > 0, ~
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:
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 - (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: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"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·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 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.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 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) "'" = -
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 Ilormifi) 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 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 + 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("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 '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)

,/""
39~

ballistic missiles :ITC not aClivllled until alicf clearing Ille pud, becuuse Ihey would be unsta ble on the ground. These [ower pre-launch frequencies can also muke the slructure liable to resonale with the periodic air forces due 10 its own vortex wake in a wind (Fung, Ref. 7- 78), as tall smokestacks are prone to do. Special measures, such as spoilers or protective fairings, have had 10 be used in certain cases. After launching, the clastic missile is subject to a kind of gust excitation that differs from conventional atmospheric turbulem:e in horizonta l flight. Even on esse ntially turbulence-free days, the variation of wind speed with height produces a time-dependent crosswind veloci ty. More experimental evidence is needed, especially at the shorter wavelengths, on statistical properties both of this wind shear and of th e horizontal component of conventional turbul ence. The assumption that such turbulence is isotropic in all three of its components does not always seem justified, particularly near the ground and in stable layers of the higher atmosphere. Turning to dynamic instability of tbe vehicle in flight, we note that this problem is related to that of "feedback coupli ng," which bas often been discussed in connection with flutter of airplanes (see, for instance, Ref. 7-76). Although other papers undoubtedly exist, the authors have been able to find only one unclassified report (Edelen, Ref. 7- 77) which contains a full analytical trea tment. Motion of the missile relative to inertial space is usually sensed by a system of linear accelerometers, plus gyroscopes arranged to measure angular velocity about each. of three axes fixed to the structure. The possibility of coupling exists because bending vibrations can produce signals in these _instrumen ts. If thes e signals come through at frequencies below th e cut-off frequencies of the various electronic and mechanical clements of the control systems, they will cause spurious corrective forces and moments to be applied to the veh.icle. The structural feedback loops thus established may produce undesirable waste of power or even a vibrational instability of the missile, which is akin to ftutter but is probably better described as "dynamic aeroclastic instability." The various loops arc illustrated schematically by the block diagram in Fig. 7-20. This refers to lateral and angular motion in one plane by a missile which is assumed to be effectively roll-stabilized. Since each system differs in many details from every other one, the situation can only be represented by typical examples. A rather general "fix" which is often suggested, however, consists of placing the accelerometers at the nodes and the rate gyros at the loops of the vibration mode that is expected to be most strongly coupled. The difficulty with this scheme, which seems to invalidate it for many practical applications, is that often more than one mode is involved, and the node and loop positions of different mode shapes

I·IU NCII'I.I~

396

-

l.,,,..

oc",.,,'

comm."" (Input, A.... ""

..1ocity

--=< -

"""mano (\riP",)

•

~

0 11 AEROELAS'r'CITV

-..... -,

- -

~,.

,"""10'

Con"" lIO'i'ion

r-

Ri ....

d,..,.,.,k.

..... ' .. ..... .. ,L.... otion

01·"··,,,>1

Ri~d

bod)'

(OUlp

-~

""

r-~'••

.... '"'1~ o---~I"'''' .~,

lMH' ~ ..

OCCOI.,.,ion

• •

- ocoaI.,_ LI .... '

-

• •

••

Fi,l. 7_20. Block diagram of missile control system with a~roelastic r~edback. never coincide. It would appear better in most cases to adopt tbe more sophisticat~ approacb of including important elastic degrees of freedom in the analysis by which the control system is designed. Unfortunately, this may the n create requirements for more elaborate computation in ffight and the use of more than one sensing element in different positions along the length of the missile. The foregoing discussion applies regardless of whether the control actions are produced by aerodynamic surfaces, auxiliary vernier rockets, or swiveling of the principal rockets. (The transfer function relating timedependent control motion to the forces and moments generated thereby is more difficult to determine in the first case.) As a simple example of how the equations of motion are set up, we consider the dynamics in a single plane of a roll-stabilized missile with a single, gimballed rocket engine whose angular position is adjusted in response to signals generated by one rate gyro located at x. on the axis (Fig. 7- 21). This represents a considerable idealization of the control system of the Vanguard. The rocket thrust-line is rotated through an angle~. relative to the lowest slope of the deflected centerline. T his,j. is related to the local absolute slope 8# of the centerline at the gyro station by the transfer function of the control system and actuator dynamics ~,=

A{O.}

(7-353)

A is a time-dependent control system operator, whose La place transfonn can usually be expressed as the ratio of two finite polynomials in the transform variables.

ON t,.J)lM ..:NStONAL S, 'HUCl'UHES

397

For convenience, all structure aunching the engine and its actuator to the missile is assumed infinitely stiff, .md inertia forces connected with the angular accderation J. are ignored. The absolute displacement \i.~x, I) of the cc:nterline is conveniently represented by the summation (7- 354)

of two rigid-body modes plus the DOrmal modes rp;(x) of vibratio n of the elastic structure. The extemal force system in the z-directioo consists of an ae rodynam ic force L(x, t) per unit length plus the concc:ntrated force (small angles assumed) T[dr

+ OW(I" ax

I)l ~

T [ A{O. }

+ OJi.t) +!U/) drp'{x')l

••,

dx

(7-355)

Finally, 8.(/) = 8Ji.1)

+1 ~ll) drpb. )

(7-356)

dx

j ~1

It is easily shown that the equations of motion of the system are as follows: (i) Lateral translation:

·· = T[ u' , + ow(:t"., MlotWR o:t"

I)l +1.

L( :t", I) d:t"

(7-357)

I)l +1.

I J.J..:t", I) d:t"

(7- 358)

bv 401

I'R INC II 'I.ES 011 IIEROELIISTlCITV

7-66. Runy"n, I-I. L. . "nd C. E. Watk ins, PlulI~r of" U"ifl"'" WillS wllh "" Arbllrarlly Placed M",", AeeMdilw 10 " DiffN"~IIII"I'&JlIliIIOl' Analysis "lUI" Comp"rl$o" wllh ExperimelU. NACA Report 966, 19s(). 7_67. Woolston, D. S. , and H. L. Runyan, Appraisal of a Melhod of Auller Analysis Baud on ChIMeR Mode. by CO"'f'l"isOll wilk Exl"riment for Cases of Large Mass CouplinK, NACA Technical Note 1902, 1949. 7--(jE. Duncan. W. J., and C. L. T. Griffith. TIle l"./fu. nce of Wi"g Toper on Ike Flutter ofConll/~e, Wi'Ws, British ARC Reports and Memoranda 1869. 1939. 7-69. Spielberg, I. N., The Two·Oj_nsion,,1 Iilromp"ssible Auodynamic Coeffidenu for Osdllatory Clwnges iii Ai'foil Camber, U.S. Air Force Technical Note WCNS 52.7,1952. 7-70. 1.q>pert, E. L., Jr., "An Applicatlon of IBM Machines to the Solution of the Flutter Determinant," J. A.ero. Sciencts, Vol. 14, No.3, March 1947, pp. 171-174. 7- 71. Ash ley, H., W. J. Mykytow, and J. R'. Martuccelli, " Prediction of Lifting Surface: Flutter at Supersonic Speed. ," Proc. Second Intemat. Congress of the Aero. Sciences, ZUrich, September 1960. 7-72. Molyrw.:Ull, W. G., The Flu/tu of Swep/ and UrIS"'ept Wi,!!' wilh Fixed-Roo/ CondillDnS, British A.R.C. Repom and Memmanda 2796, 1950. 7-73. Barmby, J. G ., H. I. Cunningllam, and T. E, Garrick, Study of Elfec/sofSwup on the FIullu o/CantileJJ- II M~, M~ » I, and M 6 ]YJ2 (Chordwise)

'I'WO-I)IMt:NS IONAI. STRUCT URI';S

4 11

where (/,. /1" oc" lind oc, lire given by

fl,

oc,

= 1,875 1

(l3 = 4.7300

=

0.7341

oc3 = 0.9825

The natural frequencies corresponding to these modes are the experimentally determined frequencies of the MW- 2 wing given in Ref. 8- 2 as

w,

"), = 67 cps,

=

265 cps

(8--\ 9)

The mass di stribution is assumed uniform, an assumption which is nearly satisfied by the MW-2 wi ng. The biconvex profile cross seI.ASTICIT Y

In the casc oflhe NAC A MW- 2 wing, the divergence dcterminant (8- 27)

reduces to - 2.720

,

,

-3. 2M

- 4.2UI

...)+ !.J31}lMk.' u;(• ,

....

(u;-)

....

_2.1s(

-1.44M

.- .

~O

(u;-) + J.40A. I-'Mk,

(8-28)

where kR = k!, and y _ 1.4. When the determina nt is expanded, the re are obtained the three roots

!!.... )1 (hm.~) b%fl2b

=- 00 0.646 0.527A

,

,3

(8-29)

Since the determinant is of triang ular form, the divergence speeds are evidently those of the individ uaL bending, torsion, and chord-wise modes. The boundaries represented by these divergence speeds are superimposed upo n the flutter boundaries in Fig. 8-3. It is evident that the divergence boundary.occurs above the flutter boundary over most of the pract ical range except for a narrow region in the vicinity of A3 = 1.8.

8-4 SURFACE SKIN PANElS (11) Introduction

Other two-dimensional aeroeiastic problems of interest arc those concerned with the behavior of surface skin panels that have one side exposed to an airstream and the other side to still air. Dynamic aeroelastic instability, or panel flutter, is the principal problem ofthi. group, although static aeroelastic instability and dy namic response to aerodynamic excitation are matters of some interest. (b) Aeroelastic equations Let us consider a thin panel which is located on the surface of a flight vehicle and has one side exposed to a supersonic airstream and the other side to still air, as illustrated by Fig. 8- 4. The initial geometry of the surface may be arbitrary, and the panel may be subjected to midplane stresses arising from internal pressurization or from thermal effects . We shall assume that the smal11ateral deflection, lI', constitutes the sole dependent variable of the pro blem.

'i'W().I)IMI'.NSI ONA I. S'I'II.UC I'UIUlS

417

Ilased upon shallow shell theury [cr. Sec. 5- 2(g)], the following operato r equation governs the phenomenoll of skin panel aeroelasticity when th e surface has an initial midplane shape spedficd by z = z(x , y ) and when the panel is homogeneous, isotropic, and of constant thickness, and has a temperature distribution T(x, y ) constant throughout its thickness: (J2P &2 (.9'-'#-..1")(w) - Z/)+ ( - - -

oyZ or

(J2F

OZ

(J2) (,)

(J2F

2-- - - +-- -

&x oy ax oy

ax' 011

(8- 30)

where the stress function F is governed by the compatibility equation

"w)1 2

J. ,( 0 "'0'''' Eh V F - -I1..V T + th; oy 011' -

ax'

o·Z OZw ox. 011'

o~z 0' w Ch OI W - - -- +2----

011 or

a", oy ax oy

The structural and inertial operators are

a2 F o' + 2 a2F . ~ _ (J2F .!.-J{ oy' ax' ax ify axoy axl ay'

.9'(

) _ [DV4 _

J{

)--[m."J( ,,' )

)

(8- 31) (8-32)

We shall employ the aerodynamic operator "'{

)=_

2q

uJ Ail _

[ui.+M2 - 20J{ 1

ox

M'

101

)

(&-33)

which reduces to the case of piston theory when y'"M""-'1 -+ M and [(M! - 2)/(M 2 - 1)] -+ 1. &juation 8- 33 represen ts a first-order approx.imation to the aerodynamic theo ry in which we neglect the influence of three-dimensional aerodynamic effects and limit our results to Mach numbers beyond approximately 1.6.

,

lip/x. y,t)

>

"'/X.,.. I)

, "J

Fir. 1-4.

Surface skin panel.

41M

I'RINCII'U.s Of AI'1l.01; LA-'>·' ·ICITV

Combining Egs. H- 30, H-3 1. H- 32, and H- 33, we obtuin the following equilibrium aeroelastic equation of s kin panels:

(8- 34) where the compatibili ty equation in the stress function F is the same as that of Eq. 8-30.

8-5

FLAT PANElS

We treat first some of those aeroelastic problems which concern fiat panels. Equation 8- 34 is rewritten by ass uming tha t w(:t',1/) represents small perturbations from an initialLy B.at and initially stressed condition characterized by the stress resultant:; N~I, N~I, and N~). This results in the ro llowing differential equation :

il'w

o~

(,)~ o'W

(,)40 ' W

+ 2 b 0~2 il-ql + b aq' + R~~ OIW

oW

+ Rn -OrjI + , o~ -+'

02W 02W of 2 + 2R ,.. of 01/

M! _ 2 a oW M2

mo'l40t w

-- + -= 1U 01 D 01 3

a' -Zf) (8- 35) D

where we have made use of Eqs. 5-54 and introduced the nondimensional parameters

w __w

(8- 36)

,

and where R."" R,.., and R... are regarded as constants. It will be observed that R= and R.n, represe nt compressive stress resultants.

TWO· I) IMI' NSIONAI. STItUCTUltflli

(a) IlIslllbilil y of

pand.~ .

4.9

!'anel III.lter

At first. we shal l focus our nllentio n on th e problems of panel instability. Experiments have shown conclusively that flutter of surface skin panels can exist ( Refs. 8-4 and 8-5), and J()rdan (Ref. 8-6) has suggested that such flulter contri buted to ea rly G erman V-2 rocket failures during World War II. !'anel flutter diffe rs from the more conventional lifting surface flutter in at least two important respects: first, it is entirely a supers onic phenomenon ; and second, structural nonlinearities associated with the lateral deformations of panels tend strongly to limit the flutter amplitudes. The latter limitation often causes the modes of structural failure to be those peculiar to fatigue rather th an explosive fract ure of the skin surface. Theoretical attacks on the panel flutter proble m have been diverse and have led on ocrasions to different conclusions. The widest atten9_on has been given to th e relatively simple problem of a flat, simply supp O. (b) Dynamic stability when a1¥3 - 00032 - a~a;' > O. When these results are investigated, we find that no dynamic instability condition is possible ill the stable buckled range. For dynamic instability to occur for ilT/fiT., less than 2.5 and 6.06 for the simply supported and damped panels, respectively, the panel must be unbuckled. Thus the complete stability boundaries of tne pane! may be obtai ned by combining Figs. 8-9 and S---12 as illustrated by Fig. 8- 13. It will be observed that the value of ~,.,-2 in Fig. S---9 has the same significance as t.TjtlT.. in Fig. 8-12.

TWO·l) t M t,:NS tONAL

~"m UCt 'UIlES

433

(iii) III~,/(ibilily of n'ell/llglllar 1/(11 plllwll of jilli!e mper:l ratio. In the sections directly abovc, wc havc COI1(;c ntratoo aHcntion on scmi-infinitc panels with the air blowin g in the direct ion of the finite dimension. We sball discuss in tbe present section the flutter of finite.aspect-ratio panels. It will be shown that the solutions presented above for semi·infinite panels apply also to finite.aspe- a:l. Neglecting the damping term proportional to aw!at and the shear tc: nn proportional to N.-.. solutions are obtained by putting W(.;,1], /) -= Wn(;) cos n1Njeu"c

Fa, 1],

/)

= Fn'{I;) cos n1Nje j ""

(8- 85)

where we introd uce the assu mption of simply supported edges. When these quantities are substituted into Eqs. 8-83 and 8- 84, there is derived for • Here we represtl(mlc Problems of Aircraft SlrW:luru, Mitleilung auS dem l nSliWt rur F lugleugstatik und Lekhtbau, Nr , 5, E.T.H., Zurich, 1958.

8- 1S. Dug'mdj i. Jollli. 1'",,,.1 I'/UI/
,

Fig. 9_1. Axis system for unrestra ined veh icle.

451

I'IU NCU'!.&S 0 1' AmlOm.. AS'I'I CI'I'V

opcmlor of Egs. 5- 93 llnd 5- 94. III the former Cllse, we have the vecto r differentill! equlltion of C(luilibriulll

+ pa =

.9'{q)

0

(9-40 )

where a is the acceleration of an arbitrary particle according to

• ~"dt (d.,) ~ "- (d •• + d1 + dq) dt at dt dl dt and where the boundlll"Y conditions are those already stated by Eqs. 5-89. Tbe vector i is a position vector from the center of gravity to a particle in the un.'ltrained vehicle, and the vector q is an elastic deforma ti on vector. These vectors relate to the position vector r through the relation f = f + q. In making use of thc third vector equation of equilibrium, it is important to fC(:ognizc tha t, in the majority of cases, the vehicle with which we are deali ng cannot be considered a simple homogeneous, iso· tropic body. It will be desirable, therefo re, to construct this equation in the most general terms, Equations 5-93 and 5-94 provide a basis for a more general approach in which the elastic infl.ucnce coefficients can be left in unspecified terms for later definition. These equations yield a vector integral equation, as follows: q - qo -

lCV x qJ x i' =

-

Iff, ['.

Iff

+

"

ap dY

r . (F.l!

+ F.1t) b(r -

r,) dV (9-4b)

Equations 9- 40 and 9-4b provide alternative forms of the required third vector equation of equilibrium. Initial conditions must be adjoined to both Eqs. 9-4a and 9-4b. H owever, boundary conditions need be stated explicitly only for differentia l equation (9- 4a) since tbey are implicit in th e integral equation (9-4b). In Eq. 9-4b , we have employed the influence function tensor r which describes deformations of the vehicle with respect to an :r·y·z·system of axes with a point in the structure which we designate as Po located at the center of gravity of the undeformed airplane. In other words, we assume the vehicle clamped at this point Po and we evaluate the various influence functions which make up r by computing the deformations at (x, y,~) due to unit loads at (e, 1/, ,). The center of gravity of the deformed structure, however, does not remain contiguous to the original point PQ' The latter moves linearly with respect to the center of gravity by an amou nt represented by the vector qOo and lines through the point rotate angularly by an amount represented by the vector !V ~ q•. Therefore, since q represents a defonnation with respoct to an altis system with its origin at the center of gravity of the deformed

TII ~

UNUES'I'RA INI'.I> VI!IUCLE

453

vehicle ami since I' represents a tensor of innucilce coemcients Ille,lsurcd with respect to an a~is system clamped at Pit< we must introd uce on the left-hand side of Eg. 9-4b th e diffcrclIce represented by q - qo-l(Vx qo) x i'

In the second termon the right-hand side of Eq. 9-4b, we have incorporated the effect of Ihe Sl.Irface tractions FII + FM within a volume integral involving the same influence function tensor, f. This is made possiblc in a formal sense by introducing the Dirac del ta function d(r - I ,) with the property that

Iff,

g(x, y,~) 6(1 - r.) dx dy dz = 0

(9-5a )

provided the surface, S, is not included in the volume of integration and

Iff

g(x, y,

~)6(r -

I,) d;l; dy dz =

v

If

g(x, y, z) d;l; dy dz

(9-5 b)

,

providing S is included in thc volume of integration. Equations 9-2, 9-3, and 9-4 form the bases for mathematical analyses of the aeroelastic behavior of unrestrained three-dimensional vehieles. The latter portions of the present chapter are devoted to applications of these equations to aeroelastic phenomena of practical interest.

9-2

FREE VIBRATIONS IN A VACUUM

A fundamental questio n which arises in thc dynamics of an unrestrained vehicle is that of defining ils natural modes a nd frequencies of free vibrations. In addition to being important physieal parameters of the system which can he measu red experimentally, these quantities are useful in constructing solutions of practical aeroelastic prohlems, In the absence of external forces t hc :r;..y-z-axis system remains inert, and the equations of free vibrations, derived from Eqs. 9-2, 9- 3, and 9-4b, are the following:

III ~;

p dV=O

v

Iff(i+q)X~~PdV - O

,

q - qo _!(V xq,,)xi' = -

ffff .~; pdY v

(9- 6)

"54

I'R1NCII'I.I\S 01'

A~ROKLAS'I'ICI1'Y

We may take, as a solution of these equations, the product of a spatial function and a time function q(.x, y, z, I) =

~(.x,

(9- 7)

y, z)T(I)

where o) =

W"l\f'-l-IfJprx IIIr'(e, 11, t;) pde d1} d~

(9-1 S)

y

where G is a second-order influence function leDwr having the form G(%,

y, Zj ~,11, 0

rex, y, z;~, 7j, ';;)

=

- ~ Ilf,

rer, s, I;

e, 7j, ')p(r, $, 1) dr ds dl + i'(x, y, z)

)( {'Y -l. [SHier, s, /) x r(r, s, I;" 7j, ')p{r, 5, /) dr ds dlJ} , (9- 19)

It is easily shown by substitution into Eg. 9- 18 that two different mode shapes, denoted by JPdV",;

(9-33 )

v The orthogonality condition, Eq, 9- 20, reduces Eq. 9- 3] to

U =!.-2 ;_, ~ M,w,",;'

(9- 34)

Hamilton's principle is stated by Eq. 2-46 in the form

,j\,. - U) dt =j'\)W dt I,

I,

If we regard 10',1'w dt, and ,; (i = I, 2, . , . , co) as the degrees offreedom,

the virtual work q uantity, 6W =

II(F / } + ,

"W, is

~ro' dS +

F,II).

If,

r x (FD

+ ;~, IFF /) + F 'U ), 4>; 6;, dS ,

,

+ F.l!)·"1'w dt dS

(i = 1, 2, . . . ,

0:»

(9- 35)

I ntroducing Eqs. 9-30, 9-]4, and 9-35 into Eq. 2-46, carryi ng out the variations, and integrating by parts, we obtain

r {[M~~:' -ff(FD+

FM)dSJ ·6ro'

+ ~t('Y'W)

s

-ffr

x(F D

+

fw

Since drD" d

+ F .lI)dSJ

(9- ]6)

. , { w dt

D+ F M) '4> dSJ Ii';,} = 0 [Mi/ + MjroNI - ff(F .,

dl, and

j

6;j are perfectly arbitrary, Eq . 9-36 can be satisfied

only if the sq uare-bracketed te rms wi thin the integra nd arc put individually equal to zero. This procedure leads, the n, to the three equa tions previously recorded as (9- 25), (9- 26), and (9- 28).

460

9-4

1'lil NC II'I...s 0 11 AlmOlll .ASTICITl·

EQUATIONS Of' MOTION IN SCALA lt FORM

In the previous sections, we have developed the equations of motion of the unrestrained vehicle in vector form. In order to carry out analyses and numerical computation it will be necessa ry to reduce the equations to their component scalar form. We assume that the linear velocity vector of the center of gravity, that is the origin "0," is represented by dr n' = U=i d,

+ U..J + U,k

(9- 37)

where U~, U., and U, are the magnitudes of the lineaf velocity vector components along the body axes, a.~ illustrated by Fig. 9- 1. In additio n, the angular velocity vector of the hody axis system is given by (9- 38)

w =pi + qj+rk

where p, q, and r afe the magnitudes of the angu lar velocity vector com· ponents along the body axes, as illustrated also by Fig. 9- 1. In trod ucing Eqs. 9- 37 and 9- 38 into Eq. 9-2 and making use of d; - _ Ij_qk

d,

dl""'_ri+pk d, dk d,

.

= ql -

(9- 39)

.

PJ

we obtain the scalar_equations M(U~ -

M(U.

+

V.'

+ U,q) = Ps (9----40)

V.r - V,p)'" p.

M(U, - VA

+ V,p) =

P,

where P", p., and p . are components along the body axes of the force vector p =JJ(FD + F·If)dS

(9-41 )

,

Equation 9- 3 can be expanded into component form by introducing r = i'

+ q.

'"

This yields, by making use of the facts thnt - = w d'

~

i'

,x. + w

dq and - = .. rtf

vI

~

.

q. the followlIlg res ult :·

:JJJ[;: ~(W xi) +;: K(Wx q) + q x (w xi) + i' x ~ c

+ q x~~ + q x W x q]pdv=fJr X (FD+F·U)dS

,

(9-42)

For simplification of tile present discussion, we shall assume tha t the elastic deformations are sufficiently sma ll so that tenn s involving their products may be neglectcdalld that t he moment of mom entum is un affected by deformationa l c.hanges in geometry. Thus, we neglect the second, third, fifth, and sixth terms on the left-hand side ofEq. 9-42 in comparison wi th the first and fourth term s. Expanding the vector triple product of the first term, Eg. 9--42 becomes

:, ['I" '

w

+

Iff

Fx

~ p dV] =

v

If

r x

(FD

+

F JI) dS

(9-43)

,

where '¥ is the inertial tensor. Introducing into Eg. 9-43 the qua ntities r _xi+ yj+ zk q -ul+ vJ +wk

(9-44)

w=pi+qJ+rk and assuming for brevi ty that the product of ine rtia tenns aTe zero, that is, the body axes are principal ine rtiaL axes, we obtain the following compon ent scalar forms: InP

+ (I" -

I n )q r

+ III [yi,;

- 2ii

+ ("'W

- 21l)r

+ (",Ii

-

yll)q ] p dV =

L~

v I ...ti

+ (1~~ - l .. )pr + Iff[zii-x;v+(yW - zli)r + (yu - xu)p]

pdV= L.

v 1,/

+ (I,.

- I=)p q

+

Iff

[:r:ii - yii

+ (zu -

yoi!)q

+

(zu -

xw)p] p dV = L.

" • The symbol i'Jjdt ;5 em ployed to denote a partial differentiation in vectors i, j . and k are held fixed.

(9--45) wh ;~h

the unit

462

I' HI NCII ' U !S 0 1' A1,ItOll I.ASTI Cr I'V

where vector

L~,

L" and L, are components along the body axes of the moment L=

Iff,.

r

~

(FIJ

+

(9-46)

F"'I) dV

In add ition to the si:l; equations (9-40 tbrough 9-45), we require three more equations obtained by putting Eq. 9--4b in compo nent form. These are the following:

,_ '. _'[.('" _'w.) 2 ax _,('" _"')] ay (1z

=

IIf,.

(1",

{C=(x, y, z;~, ,/, m(p.,D + F /I)b(x - x.) -

pa~]

+ Cn(x, y,z;~, 1), O[ (F/' + F. U)6{y -

Y.) - po,]

+ C'''(x, y,:;~, 1/, {)[(F ,D + F;If)6{z -

z,) - pa.J}d~ d1) d{

,_'. _'[.("'_ "') _.('w. _"')] 2axoy oyoz =

IIf,. {C'~(:t,

y, Z; $, 1),m(F ~ 1) + F ~ M)O(x - x.) -

p(/~]

+ C" (x, y, z;~, 1), {)[(Ft + F,M)b(y - y.) - po, ] + C"(x, y, z; e, 1/, N(F ,II + F ;lI)6(z - z.) - pa .]} de d'ld{

w_w, _2aya: ![,('w,_,••) _.('"ozox _'w,)] =

IfI

{C",(z, y, %)i

+ 4>. /:r, y, z)j + 4>,/;r;,!/, z)k

(9-50)

When the substitutions of Eqs. 9-49 and 9--50 are made, Eqs. 9-40 remain unChanged, and Eqs. 9-45 and 9-47 reduce to th e followi ng :

+ (I.. - l••)qr= L~ I ,/i. + (Iu - I .. ) pr = L" 1,/ + (I,. - 1".J pq = L, J=p

M / i··j

+ M ;W /' I i

-

-0

I!,/

+ I:! /

-.'I[

'

(J- 1,2, "',n

(9--5 1)

(9-52)

T he ge neralized mass is (cf. Eq. 3-117a)

M,

=

JfJ(1)~/ + "'~, 2+ ., + F,I'4>,, )dS

"

(9- 54)

5!M= ff(F"M.pr; + F..·'1"o, + F: lI4>.,)dS

,

We recognize Eqs. ~ and 9-5 1 as comprising the six Euler equations of motion of a rigid body that is free in space. The angular position of such a body is described by the Eulerian angles 8, 4>, and 'P. as Sh OWl1 by Fig. 9- 1. In fact, when the angula r velocities p, q, and r are known functions of time, the angular positioll may be computed by solving the follow ing equations for 8,.p, and 'P {Ref. 9-1): {j sin 'P-¢sin /l cos1Jl=P {j cos 'P

+ if, sin (J cos 'P = q

ri>cos 8+1fo 9- 5

(9- 55)

=r

THE EQUATIONS OF SMALL DISTURBED MOTION FROM STEADY RECTJLlNEAR FLIGHT

The complex nature of the equations of motion of an elastic vehicle free 10 undergo large angles of r otation is evident from the preceding sections of this chapter, T hese equations may, of course, be solved numeri, call)' to obtain tne responses of all elastic vehicle subjected to an)' flight condition. However, for purposes of discussion here, it will be desirable to introduce some sim plifications. One possible simplification is that of linea rization to represent the case of small disturbed motion from steady, rectiliriear flight. Let us suppose that the vehicle is climbing with steady forward velocity U, makin g a path angle with the horizontal of Yo, as shown by Fig. 9- 2, before it is subjected to a disturbance. The :N'plane of the vehicle is the plane of symmetr)', and the velocity vector U is assumed collinear with the :t'axis. If we define Eulerian angles 4>, 0, and 1JI as rotation angles of the :t,y,z. coordinate system from the direction of rectilinear flight, the assumption of small disturbances assures th'a t 4>, lJ, and 'P are small; and we can put p - (x, y) T(I)

(9-6Id)

and by combining Eqs. 9-6Ia, 9-61b, and 9-6lc in a manner similar to that employed in obtaining Eq. 9:" 18 of Sa;. 9-2, we can derive the homogeneous integral equation (x, y) = w 2

where

JJ

G(:I:, y;

~.1j)#.~, 1J)m(~, '1) d~ d'l

(9-61e)

s

-J.JC(r,s;~''7ll

,

LA{

+ys+xrlm(r,s)drds* I",

I~

is the influence function of the unrestrained airplane. Equation 9-61e is satisfied by an infinite number of pairs of deformation mode shapes ,{:I:, y) and frequencies W i' The same remarks conceming orthogonality of mode shapes in Sec. 9-2 apply also to the natural m,ode shapes of the elastic plate derived from Eg. 9-6le. The generalized forces of the unrestrained elastic plate are computed by referring to the forces which act during the disturbance. We assume that • This form of the influence function of a free elastic plato: may al.o be derived directly by breaking Eq. 9- 19 down into il5 component form .

46"

I'RINCII'LI~

011

AHROI~",. S'I' I C rrY

a pressure 6pl!(z, y, I), wilh urb-itTnry spa tiul and time dependencc, is applied by an atmospheric disturbance. As 11 result, there are disturbed displacements, velocities, and acce!er.ltions; and acrodynnmic pressures de noted by 6p''I1(x, 1/, t) are brought into play. The generalized force component producing the disturbance IS an explicit function of time defined by

EP(t) =

If,

6]1\x, y, r),p;(x, y) dx dy

(9-62)

The component resulting from the disturbed motion, 'B.;'J., serves not only to damp tne motion, but also introduces coupling among the nonnal coordinate s SI.i1(E I ,

. • .

t

,E,,; l " .. ,

~ft ; ¥l' ... , ~~) =

If,

6p.'ll4>ix, y) dx dy (9--63)

Thus, it is evident that although the norma! coordinates uncouple the system elastically and dynamically, E JM terms may provido very strong aerodynamic coupling.

9-6 EXAMPLES OF THE DISTURBED MOTION OF UNItESTRAJNED ELASTIC VEHICLES In previous.sections of the present chapter we have derived the equations of motion which are appropriate 10 a three-dimensional unrestrained aeroc:1astic system. Following the practice of Chaps. 6, 7, and 8, it would be possible to derive in detail various aeroelastic phenomena in connection with these equations. However, lack of space precludes the detailed trealment which is presented, for example, in Chap. 6; and we shall restrict our attention to three examples in which the rigid-body degrees of freedom are present. (a) Dynamic re.'lponse to a discrete atmospheric disturban(e Let us take up first the application of Eq. 9-59 to the problem of the small dis turbed motion of an elastic Vehicle which is in steady level rectilinear flight and subjected to a discrete atmospheric disturbance. We shall make use or the theory presented in Sec. 6-3 pertaining to the typical section. The disturbanctl generalized force is assumed to be expressible as E/(t) = CJ(I)

(~)

nm UNR£S1'RAIN HI) VmUCLE

469

whcre /(1) is a nondimensional function representing the time variation of the disturbing force. The differen tial equations, based on Eqs. 9-59, 3re M!~I

+ Mjw/€; =

E /V([t, . . . ,[~ ;

elo .. . , en; ~l' .. . , e.)

+CJ(t),

(j= 1.,2, ···,n) (9-65)

with initial conditions $,-{O) - t;(C) = C. In Eqs. 9-65, [ I' ~.. and $3 represent the small distu.rbed plunging, rolling, and pilching motions from steady rectilinear flight; E.,"', represent the disturbed quantities which are to be superimposed upon the quantities correspond ing to steady level flight prior to the onset of the disturbance. Of the vario us methods of solution of transient problems mentioned in Chap. 6, the Laplace transform (cf. Sec. 6-3b) is selected for the present illustration. }, defined by Eq. 6--52, applied to Eq. The Lap lacian operator .2'{ 9- 65 yields

e,.

MFl/..p)

+ M;wNlp)

- 2'{Ef

+

CJ(t)},

(j = 1, 2,' . . , n) (9-
=.!... {Klk, K)}

(9-101)

4U

+ !Cdk)

and ""

lS

the Kronecker

delta. The solution of Eq. 9-101 is obtained by matri x inversion as (9- 102) where (blsl = lalJl-1 ; and th.c solution for each normal coordinate is, therefore, (9- 103) The quantities l , are transfer functions of the system with respect to displacements of the normal modes. It is evident that when the matrix {a;;] is singular, a condition of dynamic aeroelastic instability or flutter exists. Transfer functions may be constructed for any output quantity that is desired. For example, suppose that our principal interest is in stresses. The steady.state stress response, in terms of its transfer fu nction r i x, y, k, K), is 0(>:, y, t) - rG(x. y, k, K)e'1 (9- 104) The stress transfer function compu ted by means oftbe mode-displacement method has tbe form r~( x, l/, k, K) =

I; _"4 Alx, y)§,(k, K) (9- 105)

4HO

PltlNCII'LIlS OF M:ItOELASTlCITY

It is evident that in computing th e tr:lnsfcr fUllctions it is necessary to invert the matrix [a,!l as many times as necessary to detine the variations of b;;(k) with k. T he matrix la;!} is of order /I x n with complex coefficients, and its inversion is equivalent to the inversion of a 2/1 x 211 matrix with real coefficients , When the transfer function 1'. has been computed for the stress at a given location (x, y), the frequency spectrum of the time variation of the stress is found from

'"

(l>~(,,»., ~ fJIf~(kl' k2)1 2 .u

" Ii\w I~ , '",-

"

-0.1

-" -"

liM _ 50

"

-

"

"

-OJ

_D.l

,,' Q,W

o0

'" ;, M _ WO "MoOe 3 O Mode 4 • r.locj. 5

• o. 1.11

" ...,"';'~M 0

"

'"

~ " . . ·10.

.

- ",

,,' Fig. ' _9. U-g plot for delta wing (piston theory).

'"

i

•, \'!II

"", • ..;';/,1

,., ,

,

1-'1

" "

,.. "

0'

"M~

! "

,.j

"'-,V'i;M

..

4!16

I'RINC II'LES OF AEItOELAS'I'ICITY

seen to be mode 5: whereas. when pM is increased, mode 4 becomes unstable. The nature of some of t he U-g-cu rvcs is a fac lor worthy of consideration. At very high values of 17M, Ihe unstable branch lends to double back on itself after crossing the zero damping line. Such behavior indicates thaI an increase: in damping of the system will d~rease its stability. This somewhat anomalous behavior has also been observed in flutter analyses of swept wings and in panel flutter.

REFERENCES 9--1. 9--2. 9-3.

9-4,

9- 5.

9"'-;.

9_1.

9-8.

9-9. 9_10.

9- 11.

9--12 .

9--13.

Whittaker, E. T" Ana/yllra! Dynamics. Cambridge University Press, 1937. Hildebrand. F. G., Ad",mced Ca/cu,,,.• fo. Enginttr~. Prentice- Hall, New Yo rk, 1949. Willi.m" D., Dynllmic Lud~ in A ~rup/anes Under Gi""n }"'pu/silH! Load wilh Parllea"'r R~fumu to l4nding oml Gust Loads on a lArge Flying ihJat. Great Britain Royal Aircraft E.tablishmc"t Report. SM E 3309 a nd 3316, 1945. Mar, J. W., T. H. H . Pian, and J. M. Call igeros, "A Note o n Methods (or the Deter mination orTransient Stresses," J. Aero. &i~nC~$, Readers' Forum, Vol. 2), No. 1, January 1956. Kirsch,.A. A., J. M. Calligero •• and K . A. Foss, Effect;' of Structural R exibilily "n GU'I Loading of Alr~'''fr, WADe Technical Report 54-592, Part 2. August 1955. Liepmann, H, W. , "Ext~nsion of the: Statistical Approach to Buffeting and Gust R.esponse of Wings of Fi ni t~ Span," J. Aero. Sdencts, Vol. 22, No.3, March 195~. Fo~ , K. A., and W. L. McCabe, Gu,·t Loadint: of Rigid and Flexible Aircraft ill ContinlLl)ll$ Atmospheric Turbulellce, WADe Technica l Report 57_104, January 1958_ Liepmann, H. W., "An Approach to the Buffet ing Problem from Turbulence CoMideration,; ' J. Aero. Sciences, Vol. 19, No. 12, December 1952. Bispl ingholT. R. L.. T. H. H. Pian, and K. A. FO$S, Resp(Jns~ "f Elastic A/rNufl 10 Contilluou, Turhwlenct, AGARD R~port No. 117. April- May 19~7. Bisp li nghoff, R. L., "Some Strvetural and Acr".,l""tic Consjd~ ratjons of 1righ Speed Flight," J. Auo. Sc/~lIre. O

1

q~ A{COS ['11 + f'( J-R + 32RtJ~R) dTJ)

+p Gm~ :i:;Hl· +

R

(to-I6b)

r H{cos [q> + {weT) dT]} {exp [ -

if !~:~ dT]} (10-27)

Garber proceeds by improving the estimate of w2 with a sequence of values .. 3. , , R W,w, (i= 1, 2,··-) (10-28) 00/+ 1 = --+ --" 2w, 4w,

496

PRII\'CIPLES OF AEROELASTlClTY

The initial

W,'

=

WI'

is given by

-R~ -(:aY- :J2:)'

WI

=

= .j-R·

(10--29)

It is of interest to note that, if we adopt for w the initial value

WI' Garber's result is exactly that of Eq. 10--16b with the 1~2-tcrm omitted. So far we have considered the homogeneous solution of Eq. 10--4. If the system is subjected to a sp«:ified applied force /(1), we must add to the transient solution the particular integral, which is easily derivable once two linearly independent homogeneous solutions, say q = ql' q., arc known. The total solution is found by the method of variation of parameters (cf. Ref. 10--6, Sec. 1-9):

q(t) =

f ~~~~

I [ql('1 )qz{t) - q.(1))qlt)] W - [q l(1), qt 0)

Moreover, since I, is arbitrary, we can state that the system is stable in the neighborhood of time I as long as 1 da. a 1(1) > - (10--45) 2 d, when a~(tl) > O. A closer examination of this derivation reveals that the parameters aa(t) and all) cannot vary too rapid ly, lest the coefficients v, and v. get too la rge. Also, the assertion regarding stability is valid for the displacement q and not necessarily for the q- or tj-responses. To determine the stability of the q-response, we can differentiate Eq. 10--39 and obtain (for N = 2) (0

a~(t) +

(/) 1

).

I P

+ ( 02 + a, ) . + = al(t) + 1 P P

a (I)"' + a (/)' 2 PIP

+

P

= 0

where P = q. This is of the same form as Eq. 10--34 with modified a.(I) and ail), and the test will then be _ 1 da. al(l)

> --

2 " Applied to the system of Eq. 10--4, that is, to

the condition of stability from Eq. 10-45 is b(t) c(1)

1 d (a(l))

>"2 dr

c(t) ,

( 10-46)

0'

Since it is assumed that a. = ale> 0, Eq. 10--46 is equivalent to (10-47)

which is in agreement with Sonine's theorem (cf. Ref. 10--12). For comparison, the discussion following Eq. 10- 18 requires Q(t»O,

I ,R: b b lR or --+->0, or -> - - 2R

a

a

2R

(10-48)

500

I'RIN CII'U:S 0,., AI!:ROt;LASTICIT'V

Conditions (10--47) and ( 10--48) afe equivalent. if Ii

c

R

"

,

R

--+ - = -

( 10--49)

This equality is satisfied in the WKB approJtimation, where it is assumed .. ot

The foregoing can be eJttend ed to higher-order systems, and Routh's criterion (e.g., Ref. 10--13) can be used to test the stability. Without furnishing details, we summarize the results from Ref. 10--12. When N > 2 the eJtpression correspo nding to Eq. 10-40 is

,

..! ,

A Hq( ~ 1

For a third-order system

+q = 0

(10-50)

(10-51) and for a fOMfth-order system we have

(10-52)

where terms contammg products of derivatives have been neglected in Eqs. 10--51 and 10--52. In the vicinity of some I = 11' Rou th's criterion is applied to Eq. 10-50 with the "con stant" coefficients indicated by Eq s . 10-51 or 10-52.

10-5 SOME AERONAUTICAL APPLlCAT[QNS Systems described by Eq. 10--4 are not uncommon and will soon be more frequent ly encou ntered in tbe aeronautical field, particularly in connection with the aeroelasticity and dynamic stabi lity of roc kets ami other unmanned vehicles. We review two studies of this sort bere. As a first eJtample, consider the free bending vibration of a stender bc(lm. which might represent an elongated missile. If t he mass per unit length In and the stiffness EI are functions of time as well as the spanwisc loca lio n x, the pertinent equation of motion for the lateral deflect io n q( ,I'. I) is

SYSTEMS WITH TIME-VARYING COEFFICIENTS

501

given by (cf. Refs. 10-7, 10-14)

mq +nl(j+

:~(Ef~;)

(10-53)

=/(x, t)

where fix, I) is a forci ng function. He re rotary-inertia effects and shear deformations have been neglef initial conditions, this equation may be solved graphically in general [for instance, by the metho d of isoclines, see Ku (Ref. 10-1S)] to yield :i as a function of x. Once th e relation :i = i(x) is established, an integration of Eq. 10-89b then yields X as a function of time. T he process must be repeated for each new pair of initial conditions. If the results are plotted as :i versus x in the phase plane (f-x-plane), we obtain a series of curves, called solution curves or trajectories, which, when assigned directions of increasing time, £ive us a qualitative description of the system behavior. Associated with the differential equation (10- 90) are singular points in the phase plane where the slopes of the curves are not unique. These situations arise when both t he numerator and denominator of Eq. IO-S9 vanish simultaneously. The character of these singularities, wh ich play important roles in determ ining the nature of tile responses, can be studied by the use of Poincare's criteria (cf. Refs. 10-3, IO-IS). The phase-plane method of solution just described, apart from being somewhat tedious in numerical details, is indeed one of the most important tools for the treatment of nonlinear problems. It is capable of extension to higher-order syste ms or to systems of more than one degree of freedom • F or convenience. the problem is stated for a two-degree-of.freedom system. Its oountcrparts for one and several degree, of freedom a re quite obvious .

SYSTEMS WITH TIME-VARYiNG COEFFICIENTS

511

at the expense of considerably more involved calculations and geometrical interpretations. For instance, for a third-order system, we must consider a three-dimensional phase space and three-dimensional trajectories. These matters are fully discussed by Ku in Ref. 10-18. A sometimes more fruitful alternative approach is the approximate solution developed by Krylnff and Bngoliuboff (Ref. 10- 19). It is strictly applicable only to systems with weak nonlinearities, small linear dampings, and couplings. * To make this restriction quantitative, we state that the aggregate of terms represented by F, and F~ in Eqs. 10-870, b are to be of the form where}l- is a small constant parameter indicative of the extent of the nonlinearity, and w, a nd w. arc the "linear" uncoupled natural frequencies. Consider once more the single-degree-of-freedom case, for which we have (10-91) As !-' --+ 0, its soluti ons are given by '" .., a sin (w,t

x=

+ rp) (l0-92a, b)

wla cos (w,1

+ '1')

For weak nonJinearities, it is logical to assume that the form of the solutions is the same as in Eqs. 10-92a, b, but with a and 'P as slowly varying functions of I. We may briefly summarize the steps as follows: the differentiated form of Eq. 10-920 [with a = a(1), 'f = '({t)l, when compared with Eq. 10-92b yields the condition (10--93)

x

Furthermore, if the expressions for (obtained by differentiation of Eq. 10-92b),:t: (Eq. 10-92h), and X(Eq. 10-92a) are substituted into Eq. 10-91, we obtain a second condition

+ '1') + /If[a sin (W,I + 'P), aWl cos (w,1 + 'P)] = (10--94) From Eqs. 10-93 and 10-94, the expressions for a and P arc found to be 0= - J:!..... f[o sin (w,1 + rp), aWl cos (w,t + rp)] cos (w,t + rp) w,

ow, cos (w,t

+ rp) -

awlP sin (W,I

°

(10--95a , b)

¢ =....!!.... i[o sin (w,1

,w,

+ rp), aWl cos (W,l +

rp)] sin (w, /

• See the comments in the co ncluding paragraph or this section.

+ rp)

512

PRINCIPLFS OF AEROELASTICITY

As a and rp are assumed to be slowly varying functions of time, in th e sense that they do not change appreciably during one cycle, it can be shown by an averaging process over the cycle that (to first order) da = _ .!!... gl(a) WI dt

(10-96a, b)

drp = .J::.... gz(a) dt

aWl

Here

(10--97)

'0'

,..J."

g2(a) = -I

f(a sin

!p,

aWl cos

!p)

sin 'P dIP

(10--98)

Once the functions gla) and g2(a) are determined, Eqs. 10--900, b can be integrated to obtain a and rp as functions of time, and the solution wi!l become X(I) ~X(t) = a(l) sin [4111 + 'l(t)} (10-99) Let us apply this method to the vibration of a mass attached to a cubic spring (Ouffing's problem), i.e., i Wlz:j; pz' = 0 (10-100) Successively we have,

+

+

,..1" Lb{a

{a 3 sin 3 tp} cos tp d'P = 0

gl(a) = -I

I gia)=211 0

a = al

3

3

3a sin 3 1p}sin1pd>p=_ 8

(constant)

3aI2 . - -J.I --/+rpo w, 8

So that

x~alsin ([WI +

=3 aI2}+ 'Po)

(10--101)

The effective frequency is to first order in J.I

WI= [WI + .!!.... 3a/JI = w/ + ~ a/J.I WI

8

(10--102)

4

which is in ag reement wi th other methods. Note that the frequency dependent on the motion amplitude a l •

W

is

SYSTEMS WITH T IME-VARYING COEfFICIENTS

513

The above can be obtained by a slightly different, but closely related pnx:edure (cf. Shen, Ref. 10-20). If the disturbed motion becomes pedodic (not necessarily harmonic) eventually, we assume

•

x=~a ft sinnwl

(10--103)

,

Substitution of this expression into Eq. 10--100 yields

•

Z

."'

+ b. ) sin nwt =

(_n 2w 2 0 .

where h. is the nth harmonic component of W12X W12x

+ p.:rf' =

•

~ b.

,

sin nwt

=

0

+ p.XJ; F( x )

(l0--104)

I.e., (10-105)

If the first harmonic is balanced, we have WI

=

h,fa,

(10-106)

Furthermore, if the first harmonic predominates, by Fourier analysis,

b,=3 r~F(x)sinwtd(wl)f':;3 r~F(alsinwl)sinwtd(wl) 1TJO

... 01[W 1Z

1TJ9

+ Ip.O,2]

(10--101)

whence, from Eq. 10--106, (10--108)

which coincides with Eq. 10--102. As stated earlier, the Kryloff and Bogoliuboff solution assumes weak nonlinearities and small damping and, when extended to a multi-degreeof-freedom system, it further requires weak coupli ng between the dependent variables. Although applied successfully to some "strong" nonlinear systems (see comments in Ref. 10--20), it has been shown to suffer considerably in accuracy when strong linear dampings are present. Bru nelle (Ref. 10-1) suggests a modification which accounts more accurately for linear dampings and couplings. For instance, Eq. 10--91 is recast into the form x+ 2fJi

+ w,!x + 4(i, x) =

0

Hence, the original starting points would be

x = ae - 6' sin (W 21 + '1') = oe- 6'{w t cos ( in Mechanical and Electrical Sy&lems, Pure and Applied Mathematics Series, V1. 11, Interscience Publishers, New York, (2nd printingJ l 954. 10-4. Collar, A. R., "On the Stability of Accelerated Motio n: Some Thoughts on Linear D iffe",ntial Equations with Variable Coefficients," A.ro. Quart., Vol. VIII , November 1957, pp. 309- 330. 10-5. T.ien, H. 5., Engilteu ing Cybernetics, McGraw-Hili Book Company, New York, 1954. 10-6. HIldebrand, F. B., Ad\Jaltcrd Calculus for Engine",s, P",ntice-Hall, Inc., New York,1949. \0- 7. Brune\1e, E. J., Transicnl ond Nan/jnt llr Efftcts on High Spud, Vibrolory 1"Iotrmoclastic Jrmabmry Phenomena, Part l-~orerical Considerations, WADD T R60-484, July 1960. \0-8. Squire, W.• "Approximate Solution of Linear Second Order Differential Equat ion.;' J .. Rayal Aero. Sociery, Vol. 63, No. 582, June 1959, pp. 36&-369. 10- 9. Reed, W, H., " Effects of a Time·Varying Test Environment on the Evaluation of Dynami. Stability with Applic"'tion to Flutt~r Testing," J, Aero/Space Srj~IIUS, Vol. 25, No.7, July 1958, pp . 435-443 . 10- 10. Garbe r, T. B., "On the Rotatio nal M otion of a Body Re·Entering the Atmosphere," 1. AerolSpac~ Sd~ncu, Vol. 26, No.7, July 1959, pp. 443-449 . 10-11. Duncan, W. J. , "Indidal Admittances for Linear Systems with Variable Coer_ fic ient~," J. Royal Aero. Society, Vol. 61, No, 553, January 1957, pp. 46-41. 10-12. Grensted, P. E. W., "Stability Criteria for Linear Equations wi th Time-Varying Coefficients," 1. Royal AerQ , Society, Vol. 60, No. 543, March 1956, pp. 205_208. 10-13. Gardner, M. F., and J. L. Barne., Tmnsiems In Linear Sysrem., Vol. I. , LumfUdCormam Syll.",S, John Wiley and Sons, New York, 1950. 10-14. Birnbaum, S" Belldlnc Vibration, of a Perforaled Grain Solid Propel/ant Ro~ht During Powered Flight, Institute of Aero.pace Science. Preprint No. 61-30, January 1961. 10-15, Bi.plinghoff, R. L., H, Ashley, and R. L. Halfman, Aeroelasridty, AddisonWesley Publi.hing Company, Camb ridge, Mas.., 1955. 10-16, Inee, E. L., Ordinary DijJi:rentiul Equalia"" Paperback Publication, Dover Publications, New York, 1956. 10-17. Minorsky, N" l"trodll~tio" /0 Non_Un""r MechaniCS, J. W. Edward., Ann Arbor, Mich., 1947. 10-18. Ku, Y. H " Analysi. of Control of Nonlinear Systems, Nonlimar Vibrurlons u",1 Oscillation$ of PhySical SyS1em., Th~ Ronald P,..,ss, New York, 1958. 10-19. Krylolf, N., and N. Bogoliuboff, Imraducrion /a Non-Linear MechQnj~s, Trans_ lated from Ru"ian by S. Lefschetz, Princeton University Press, 1943. 10- 20. Shen, S. F., "An Appro~imatc Analysis of Nonlinear Fluner Problem.," J. Au olSp