THE ROYAL AERONAUTICAL SOCIETY

Paper received June, 1946 THE EXPANDING DOMAIN OF AEROELASTICITY by A. R. COLLAR, M.A., D.Sc, F.R.Ae.S.

Professor Collar took his degree at Cambridge with honours in mathematics and physics; he also took an honours degree in physics at University of London. In 1929 he joined the staff of the Aerodynamics Department of the National Physical Laboratory and was for some time Secretary of the Airscrew Panel and the Fluid Motion Panel of the Aeronautical Research Committee. At the beginning of the war he was specially attached to the staff of the Royal Aircraft Establishment, Farnborough, to take charge of a special group investigating aeroplane flutter and vibration problems. In 1945 he was appointed Sir George White Professor of Aeronautical Engineering in the University .of Bristol. He was elected a Fellow of the Society in March, 1944.

SUMMARY. 1. INTRODUCTION. Y MEANS of a broad survey of the The branch of aeronautical science which Bdevelopments in aeroelastic science during has become known as aeroelasticity is of the past ten years and of comparable comparatively recent development. It was developments in associated fields, in so far first studied as such rather more than a as they involve elastic deformation, the decade ago, when Roxbee Cox and Pugsley, following their pioneer work on loss and present paper attempts to show how the reversal of aileron control and on the rolling individual subjects concerned are developing power of a monoplane, began a correlation strong interconnections. At one time the of this work with the flutter investigations of subjects were for the most part regarded as Frazer, Duncan and Collar. This correlation discrete entities; now they are developing led to the formulation and development of into component parts of an integrated whole stiffness criteria for aircraft wings and com­ —the dynamics of a deformable aeroplane. ponents: criteria which have combined an admirable simplicity with a remarkable It is suggested that in the future it may degree of success in the prevention ot be necessary to discontinue the present troubles involving elastic deformation of methods of treatment of the component sub­ aircraft. jects as separate, and to adopt a standard In 1937, Pugsley presented to the Royal method of attack on the problem treated as Aeronautical Society a statement1 of the pro­ a unified whole, especially when adequate gress of the subject to that date. Since then, computational aids become available. A no general review has been made, in spite of modified approach, involving an arrangement great developments and changes of emphasis, of the subjects in a frequency spectrum, is particularly during the past few years. The also discussed. Some speculations on the rapid advances in aeronautical science which elastic provisions which may be necessary in have been made during the war have resulted the future are included, in corresponding strides in aeroelasticity, and 613 A. R. COLLAR the subject has been expanded to such an In the case of wing-aileron flutter, aileron extent that it is now encroaching on fields mass-balance had provided the cure; for the which have previously been regarded as the other three problems, however, wing tor­ preserves of other subjects; and indeed, the sional stiffness was shown to be the property subjects involved may now be regarded as of prime importance. Pugsley's paper con­ beginning to merge into one another. cluded with some speculations as to which The implications of this trend of develop­ of the three phenomena, if any, would be the ment have an obvious interest, and it is determining factor for the elastic properties thought that a discussion of the existing state of wings. of affairs may be helpful at the present time. Accordingly, the following notes have been 3. THE FORCES INVOLVED. prepared to indicate how aeroelasticity is beginning to link up with other subjects, and As a preliminary to the discussion of the what may be the future methods of dealing expansion of the domain of aeroelasticity, it with this coalescence of subjects. is convenient here to consider what forces are involved in the subject. The name itself 2. THE ORIGINAL SCOPE OF defines by implication two of the forces, AEROELASTICITY. namely,' aerodynamic forces and elastic Although Pugsley's paper1 drew attention forces; and these appear alone in studies of to the possible extension of aeroelastic studies unaccelerated motion such as problems of to tailplanes, in fact the phenomena dealt steady rolling or aileron reversal. The flutter with referred exclusively to wings: and and buffeting fields, however, introduce a indeed, at the time, little attention had been third and equally important type, namely, paid to aircraft components other than wings. inertia forces. In addition, there may be Work on tail units had been carried out in certain other forces, for example, gravita­ flutter and buffeting investigations, but from tional forces which make contributions to the the point of view of stiffness wings presented total stiffnesses; but these are relatively the main problem. Pugsley illustrated his unimportant, and we shall restrict ourselves subject by a stiffness diagram in which the to the consideration of aerodynamic, elastic ordinate was a modified (non-dimensional) and inertia forces. wing torsional stiffness criterion and the abscissa, nominally, the wing density: in 4. THE " TRIANGLE OF FORCES " point of fact his independent variable might equally well have been taken as time, since AND PROBLEMS AKIN TO there had been a steady increase in wing AEROELASTICITY. density through the years, and he was It is illuminating to consider some of the regarding the future as a period of very high problems governed by the three types of wing density. force mentioned above; to do this, we may He showed how the criterion had changed make use of the diagram shown in Fig. 1. with increasing wing density, and how it had The three types of force are placed been modified by the influence of various separately at the vertices of the equilateral phenomena in turn: these phenomena, in " triangle of forces " AEI, where the initials chronological order of importance, were: — indicate the force concerned. We now write (a) Wing-aileron flutter. down the subjects originally within the (b) Reversal of aileron control. domain of aeroelasticity, bonding them to the (c) Wing flexure-torsion flutter. appropriate vertices as shown, and placing (d) Wing divergence. them within the triangle when they are 6H THE EXPANDING DOMAIN OF AEROELASTICITY

Fig. 1 aerodynamic forces F flutter elastic forces B buffeting inertia forces D divergence R reversal of control bonded to all three vertices. Thus flutter (F) of which a major component is the subject and buffeting (B) are located within the of stability and control (S); again, most triangle. On the other hand, reversal of loading or strength problems (L) of past control (R) and divergence (D) are outside years, including gust problems (G), have been the triangle, since they are unconnected with treated on the basis of interaction of aero­ inertia forces;* though we may draw a bond dynamic and inertia forces. In the inertia- between divergence and flutter. elastic field is a subject which, at the time of Inspection of Fig. 1, however, leads Pugsley's review) was rapidly growing in naturally to the question: in view of their importance—that of mechanical vibration obvious connections with aeroelasticity, what (V); and entering this field we find the of the subjects in the aerodynamic-inertia problem of the stresses induced by impact force field and in the inertia-elastic force loads (Z) which, originally treated as an field? We may in fact add to our diagram, inertia problem only, was acquiring a con­ as in Fig. 2, a number of subjects originally nection with elastic forces. belonging to these fields. In the first field is Fig. 2 shows clearly that the no-man's land the vast subject of rigid aeroplane dynamics, surrounding the territory of aeroelasticity was of very small extent; and any attempt at By divergence we imply here an infinitely slow expansion must involve intrusion into other divergence, such as occurs at a critical diver­ gence speed. fields of work. 615 A. R. COLLAR

In the next paragraph we attempt to define is needed therefore if we deal with the sub­ briefly the scope of the problems shown in jects in a very broad and general way, and Fig. 2 at the beginning of the past decade if in so doing we find that in our statement of development. considerable omissions and lacunae appear. 51. AEROELASTICITY. 5. DELIMITATION OF THE ORIGINAL We have already indicated, by reference to SCOPE OF THE PROBLEMS. Pugsley's work, what was the general scope If we were to attempt here to define any of this subject. It related almost exclusively precise limits to the subjects with which we to aeroplane wings, and -Jftid down, on a are concerned, we should have to include in theoretical-cum-statistical basis, standards of a single paragraph a very large part of the torsional stiffness for wings: standards of whole science of aerodynamics. No apology flexural stiffness were not found to be

B.

Fig. 2 A : aerodynamic forces F flutter E : elastic forces B buffeting 1 : inertia forces S stability and control D divergence R reversal of control C gusts L loading problems V mechanical vibration z impacts 616 THE EXPANDING DOMAIN OF AEROELASTICITY

required. It is true that standards of massive, so that in effect it had no bodily torsional stiffness for ailerons had been freedoms; and the components whose flutter tentatively proposed also; but the number of properties were to be studied were regarded criteria was very small. In addition, the as having only a small number of degrees bases on which the component subjects were of freedom and as being encastre" in the fixed built up were quite narrowly limited in the fuselage. manner indicated below. Fyven with the above simplifications, theoretical treatment was very complicated, 5.1.1. Flutter. and further assumptions were necessary, The degree of complexity of the theoretical particularly for the formulation of a simple treatment of any subject by any one method s'.iffness criterion. The aerodynamic actions, increases very rapidly with the number of for example, were treated by strip theory, forces involved and particularly with the making use of a fundamental set of constant* number of degrees of freedom which must be derivative coefficients proposed by Duncan introduced. In consequence, flutter, which and Collar in a research on airscrew blade 5 from the first involved all three types of flutter. force, and which was treated by the classical In the formulation of the stiffness criterion method of examination of the Routh stability it was also necessary to treat of an criteria, had of necessity to be restricted to " average " wing, in which the distribution problems involving very few degrees of of chord with span, the distribution of mass freedom: the maximum number was usually in the chord, and the position of the three. In addition, very considerable simpli­ flexural axis, were all maintained constant: fications had to be introduced: of these, the although in various researches the effects of most important was the concept of semi- these parameters were examined. Lastly, rigidity, in accordance with which the elastic no distinction was made between the stiffness body with its infinite number of degrees of appropriate to the semi-rigid representation freedom is replaced by a body having only of a wing and the stiffness of the actual wing a finite number. For wings, these degrees as found by test. of freedom were usually a single mode of displacement in torsion, another in flexure, 5.1.2. Divergence. and (rigid) aileron rotation. The problem of divergence can be treated On this basis, it was found that, up to ten either as a special case of the general flutter years ago, most flutter problems could be investigation or as a relatively simple pro­ adequately dealt with.2 It thus appeared that blem per se. The remarks on flutter there­ it was unnecessary to introduce bodily free­ fore apply also to divergence. There is, doms of the aeroplane as a whole: this result however, one additional limitation: a wing was largely inferred from the satisfactory approaching its divergence speed, if encastre solutions obtained on the restrictive assump­ in such a manner that its root incidence were tions made, though one early research of maintained constant, would change its lift Frazer and Duncan3 had examined the effect with increasing rapidity. Strictly, therefore, of rolling freedom on antisymmetrical flexure- a specification of the lift conditions, implying aileron flutter and had shown that little pitching of the fuselage, should be introduced; modification was required to the anti-flutter but this was generally neglected. recommendations. * It was of course realised that the aerodynamic Broadly speaking, therefore, it was found forces are strictly functions of the frequency to be possible to regard the fuselage—or at parameter : this dependence, for translational and pitching derivatives, had been investigated least the forward part—as rigid and infinitely by Duncan and Collar,4 617 A. R. COLLAR

Some attention had been given to wing- usually of an ad hoc nature; and by proper aileron divergence; but the pilot's reaction to positioning of the tail organs and by clean this phenomenon was generally ignored. At design the phenomenon can be avoided. the wing-aileron divergence speed, both wing Since the whole trend of modern develop­ and aileron are neutrally stable, and the ment has been towards cleaner designs, cases latter condition implies zero stick force. Thus of occurrence of buffeting are now rare; and a pilot will have warning, through the pro­ we shall not refer to the subject again. gressive lightening of his ailerons, of the approach of this phenomenon, and its critical 5.2. SUBJECTS ASSOCIATED WITH AERO­ .speed can be controlled by alteration of the ELASTICITY IN THE " TRIANGLE OF aileron hinge moment characteristics. More­ FORCES." over, since aileron travel is limited, the corresponding wing twist is also limited: The subjects of 5.1 have been treated very usually to an amount which would not imply broadly, and with an almost complete lack danger to the structure. In vie'w of these of detail. Even more broad and less detailed considerations, it is not surprising that wing- must be our attempt to delimit the scope, aileron divergence has not presented a pro­ ten years ago, of the subjects with which we blem of urgency in practice. are now concerned. They had been studied extensively for many years, and were in 5.1.3. Loss and Reversal of Control, consequence much more developed than Ten years ago this subject was restricted aeroelasticity; we shall therefore only attempt to loss and reversal of aileron control.6' ' to consider how they stood in relation to Here also the remarks given above on flutter aeroelasticity. apply almost in their entirety; although some investigations into methods of dealing with 5-2.1. Dynamics of a Rigid Aeroplane— aerodynamic loading8 and into the case of an Stability and Control. 9 elastic (as distinct from semi-rigid) wing The structure of this subject had been were made. erected on a foundation of aerodynamic and In view of the relatively simple nature of inertia forces only: it had not been found the calculations involved, the results were necessary to introduce elastic deformations. not normally presented in the form of a stiff­ It is true that the work of Gates showed an ness criterion, although a torsional stiffness awareness of the possible importance of criterion of the normal form was involved; elasticity—he had at an early date been con­ and hence parameters such as wing taper cerned with flutter as a problem in stability were taken into account in the calculations —but with the exception perhaps of control made. circuits, elasticity effects were generally ignored. The formal treatment of stability 5.1.4. Buffeting. and control, including the separation of The subject of tail buffeting falls within longitudinal stability from other aspects, was the domain of aeroelasticity, but is in many of course well established; and attention was ways a subject apart from the others. It is therefore mainly directed to the important concerned10 with the transient vibrations component details of the subject, and in resulting in the tail unit from aerodynamic particular to control surface hinge moments. impulses due to the eddying wing wake. The development of aeroelasticity re- Since these impulses are quite random in emphasised the question of elastic deforma­ character, there is no analytical theory tion; and in their work on the effect of associated with the phenomenon. Its cure is response on aileron hinge moments, Gates 618 THE EXPANDING DOMAIN OF AEROELASTICITY

and Irving11 introduced the appropriate of gust effects were not very advanced, aileron reversal speed correction. having been severely hampered by an almost It may be remarked that the exclusion of complete lack of knowledge of the structure elastic forces was in fact quite natural, for of gusts. But investigations of the loads due two reasons: first, the frequencies of oscil­ to gusts of arbitrary structure were being lation occurring in stability investigations made under the guidance of Bryant13 (for were on the average much lower than the wing loads) and Williams14 (principally for gravest natural frequency of elastic oscil­ tail loads). lation; secondly, there were very little data In the main these investigations related to on the elastic properties of aircraft. Such rigid aeroplanes, but in each case some information as existed was mainly of an ad attention was given to the effect of bending hoc nature, and Roxbee Cox's statistical freedom of the wing. Thus, although gust surveys were only just reaching a point at problems belonged principally to the aero­ which it was possible, through the medium dynamic-inertia force field, elastic effects of criteria, to fix the desirable elastic proper­ were beginning to appear in the theory: in ties of aircraft and to give guidance to the Fig. 2 this is indicated by a tentative (dotted) research worker. bond to the elastic force vertex.

5.2.2. Loading Problems. 5.2.4. Mechanical Vibration. In the determination of the aeroplane Ten years ago this subject was still in its strength needed to resist the aerodynamic and infancy. In the early days of flying, inertia loads applied, elastic deformations mechanical vibration does not appear to have were almost invariably ignored. For one presented a serious problem. Probably this thing, the number of loading cases to be was due to the small powers involved (which covered, and the lack of precise information imply also fairly small amplitudes of on the aerodynamic loads, implied margins of impressed force) and to the relatively high safety which were more than adequate to deal damping of the wooden construction then with variations in load due to elastic defor­ common—particularly since this involved mations. Again, for the main components many bracing wires, joints, and so forth. affected, strength was largely determined by With the coming of cantilever wings, with spar design, while stiffness provision was the replacement of wood by metal, and with made (if the intrinsic spar stiffness was steadily improving methods leading to a insufficient) by the addition of skin of such more integrated form of construction, the a gauge that modification to the calculation internal damping available had steadily of the strength characteristics was unneces­ decreased. At the same time, power plants sary. had developed greatly; so that, given a Thus, • although it had been foreseen by coincidence between an airframe frequency Pugsley and Roxbee Cox12 that ultimately and a powerful engine harmonic, unpleasant there must be some inter-relation between vibration was almost bound to develop. strength and stiffness, that stage had not been At the time we are considering, Constant's reached; although in one field—that of gust work15 on thresholds of unpleasantness in . effects—it was just being realised that vibration was underlining the existence of the structural elasticity must appreciably affect problem, while various facets were attracting the magnitude of the loads due to gusts. the attention of Carter and of Morris. But theoretical solutions' of the problem were very 5.2.3. Gust Problems. restricted. They usually involved some far- At the period we are considering, studies reaching assumptions—e.g., in engine-air- 619 A. R^ COLLAR

screw vibration, that the engine bearers were resulted in the developments concerned, attached to a rigid body and that the airscrew these same conditions have prevented the could be regarded as a rigid disc—which publication of most of the researches—indeed, rendered the solution of doubtful general many have not been fully reported in any applicability. form; and this adds considerable difficulty to •» At the same time, even these solutions the task of a reviewer. However, as we are were almost entirely restricted to the deter­ concerned in the main with general trends, mination of natural frequencies of vibration, we must regard the subjects touched on as for the very good reason that there was illustrative only, and not as component parts available very little reliable information on of a detailed picture. the forcing torque impulses or on the damp­ We may naturally enquire at this point ing; with the result that no serious attempts what are the physical reasons underlying the to forecast amplitudes of response were developments in our field. In a word, the possible. In view of these considerations it answer is speed. The remarkable develop­ was obviously not expedient to introduce air ments in power plants—first in reciprocating forces into the problem, and no attempt to engines, then in jet propulsion units—coupled do this had been made. with continued efforts in the direction of drag reduction, have resulted during the past 5.2.5. Impact Problems. decade in the first real excursion of aircraft Up to the time we are considering, landing into the much discussed but highly specu­ impacts had been treated on the basis of lative realm of compressibility effects. inertia loads only: that is to say, the aircraft (apart from its undercarriage) was treated as At the same time, the strength questions rigid, while aerodynamic forces were ignored posed by this speed increase have been in view of the low landing speeds involved. solved, partly by material developments, However, it was realised that while these partly by improved constructional methods, assumptions might be justifiable so long as but in the main by a steady movement wings were light and possessed a fairly high towards the ideal of the flying wing: i.e., by flexural frequency, the increasing weight of a steady process of transference of mass from wings required a reconsideration of the the rest of the aircraft to the wing, in order picture. A paper by Fairthorne16 drew to obtain the maximum relief from inertia attention to this and made a first examination loads. As a result, the elastic properties of of the effects of wing flexibility on landing aircraft, at one time a secondary or even a impacts. Aerodynamic forces, however, still last-minute design consideration, have in played no part in the theory. The bond to many cases come to the forefront as one of inertia forces, and the tentative bond to the principal factors in structural design. elastic forces, are indicated in Fig. 2. This growing importance of aerodynamic and elastic forces* implied in the design 6. DEVELOPMENTS IN THE SUB­ requirements for aeroelasticity,17 has been JECTS DURING THE PAST strongly underlined, not only by the develop­ DECADE. ments of aeroelastic theory itself, but also by We must here content ourselves with only the way in which the implications of elastic a brief summary of the main developments deformation (which introduces both aero­ which have occurred during the past decade, dynamic and elastic forces) have obtruded in so far as such developments are related to themselves on the notice of those concerned the science of aeroelasticity. While war with branches of theory in which such defor­ conditions provided the impetus which mations could previously be ignored.

620 THE EXPANDING DOMAIN OF AEROELASTICITY

In the next paragraphs (6.1 and 6.2) we laying down the physical parameters and the shall try to show how aeroelastic theory is stability condition, and then solving (by expanding into other fields, and how the means of Routh's criteria or their equivalents) subjects in these fields are expanding into for the corresponding critical speeds and aeroelasticity: we shall find there is some frequencies, one leaves free certain of the overlapping of our discussion, as there is in parameters and fixes the critical speed and the subjects concerned. frequency; the corresponding values of the parameters are then found (a process usually 6.1. EXPANSION IN THE AEROELASTIC involving only the solution of algebraic FIELD. equations). In this way a curve of critical In the main, the expansion of aero­ speed against the values of the parameters elasticity in recent years has been in the may be found, and the appropriate critical nature of the growth of existing subjects speed corresponding to the correct value of rather than of the birth of new subjects. the parameter is thus obtained. The majority of the subjects can still be com­ This inverse method of solution has now prehended within the fields of flutter, diver­ become standard procedure for problems gence, and loss of control: only one new field involving many degrees of freedom, and has of major importance, that of effects on aero­ indirectly been the means of widening greatly plane stability (which is, of course, closely the understanding of the flutter problem. bound up with control effects) has emerged. Under the next few headings some of the In consequence, the developments have been problems which have been treated in recent largely in the nature of studies of new years are described shortly. We shall not methods for dealing with the more compli­ refer here to flutter of wings not carrying cated problems posed by the rapidly acceler­ engines, since this is dealt with later ating aircraft of the period. (6.1.1.5) in relation to stiffness criteria; nor shall we refer to propeller flutter. The 6.1.1. Flutter. latter subject has been extensively studied We have already said that, in classical experimentally, but no adequate general methods of studying flutter, it was necessary theories have yet been proposed; and indeed to limit the number of degrees of freedom to the origin of flutter of metal blades remains about three. However, one study had been in doubt. made in 1936 by Duncan, Collar and Lyon18 of a problem in which six degrees of freedom 6.1.1.1. Elevator Flutter. had been treated. We shall here restrict our discussion to It is not easy to say precisely how com­ symmetrical elevator flutter. Purely as a plexity increases with number of degrees of matter of interest, however, it may be noted freedom: probably it is not far wrong to that aeroelastic science had its beginnings in suggest that the complexity is roughly a study of antisymmetrical elevator flutter, proportional to the factorial of the number made in 1916—thirty years ago—by Bairstow of degrees of freedom, if any one method is and Fage.19 Following the particular occur­ adhered to throughout. Thus an increase rence they studied, the port and starboard from three to six degrees of freedom halves of elevators were given stiff inter­ necessitated new methods, and these were connections, and cases of antisymmetrical found in the paper considered by the flutter have since been very rare. adoption of inverse methods of solution. Originally, studies of symmetrical elevator Broadly speaking, the change from the flutter involved two degrees of freedom only: classical method is as follows: instead of fuselage bending and elevator rotation. 621 A. R. COLLA K

Calculations on this basis had, however, interest for flutter investigators. It was always given numerical results of the wrong realised that the " mass-balancing " effect of order, even allowing for the lack of know­ engines, as normally disposed, must be con­ ledge of the appropriate aerodynamic forces. siderable; but no quantitative estimates of the In a study of the problem made under the effects had been made. Research into this direction of Jahn20 at the R.A.E., therefore, question has been vigorously prosecuted both the following additional degrees of freedom at the N.P,L.23 and at the R.A.E.," as well were successively introduced: pitching of the as by certain aircraft firms. The problem is aircraft, vertical translation of the aircraft, as yet far from final solution; but it is and wing bending. (In this particular case already apparent that a large number of two further freedoms, involving elasticities normal modes must be introduced if an present in the circuit, were also added.) adequate solution is to be obtained. Parallel with this investigation was an experi­ mental research conducted at the N.P.L. by 6.1.1.3. Antisymmetrical Wing Flexure- Scruton.21 Torsion Flutter. The results obtained in this way were very In this phenomenon, as in others, it has satisfactory, and design recommendations been found that it is necessary to introduce based on the studies were found to be not only the antisymmetrical normal vibration beneficial. From the motions occurring in modes, but also the relevant rigid body the critical condition, it was also found that freedoms—in this case freedom in roll. the problem could be treated rather more simply by considering the motion to be 6.1.1.4. Mass-Balancing and Flutter Pre­ built up from the normal modes of vibration vention Devices. of the aircraft as found in a resonance test. In addition to the studies of elevator flutter The implications of this latter finding are referred to in 6.1.1.1, cases of flutter of of great interest and importance. It will be ailerons, rudders and tabs have also been seen that the degrees of freedom involved are extensively studied. These studies, while not only the normal modes of vibration—it underlining the necessity of treating defor­ may be noted here that Duncan22 had sug­ mation of the aircraft as a whole as a con­ gested the use of normal modes of vibration stituent motion, have also yielded much as an approach to complex problems—but valuable information on mass-balance as a also the rigid body freedoms (which may in cure for control surface and tab flutter. a sense be regarded as normal modes of zero It has been shown, for example, that the frequency). virtual inertia of the air25 must be taken into Thus the investigation pointed to the fact consideration in the determination of the that an instability having its origin in the appropriate inertia characteristics. Another tail organs, involving as it does appreciable very important result relates to the position­ wing bending and bodily movement, can no ing of mass-balance weights. If it is desired longer be treated as a local affair, but must to achieve mass-balance by the use of masses be regarded as " elevator-aeroplane " flutter. not attached to the control surface, but Or, in more general terms still, the operated through a linkage, then the position phenomenon must be regarded as an aspect of the mass in relation to the nodes in the of the stability of a deformable aeroplane. vibration modes of the aircraft must be accurately known.26 6.1.1.2. Flutter of Wings Carrying For a balance mass operates as such in Engines. virtue of the accelerations imposed on it by This subject has always held considerable the vibration of the aircraft, and if it is 622 THE EXPANDING DO MAIN OF AEROELASTICITY positioned near a node this acceleration will have promise,29 and with careful attention be small; the mass is then practically inoper­ to details of design may in time be adopted ative as a balance. In fact, if it is. on the as alternatives to mass-balance. wrong side of the node it becomes an anti- balance mass. The node may be regarded 6.1.1.5. Researches on the Aerodynamic as a virtual hinge; and if the balance mass Forces of Flutter. is to play its proper part, it must be appro­ While, as indicated above, methods of priately positioned in relation to this virtual dealing with the flutter problem have steadily hinge and the control surface hinge. improved and extended, studies of the In the study of mass balancing of elevators relevant aerodynamic forces have also been and rudders in particular, it has been prosecuted. At the N.P.L. a number of necessary to consider as part of the practical investigations have been made into the problem the effects of the balance masses on validity of the constant strip theory derivative longitudinal and directional stability (in the coefficients by observations of the flutter classical sense). Thus there is, in a way, a characteristics of a series of straight-tapered dual linkage with stability and control wings; and some modifications have been matters; not only is the flutter problem proposed.30 At the same time Jones31 has becoming a problem in the stability of a made a number of detailed theoretical investi­ deformable aeroplane, but the curative gations of the aerodynamic characteristics of measures adopted must be considered in oscillating wings of finite aspect ratio and relation to the stability of the rigid aeroplane. varying planforms. Considerations similar to those mentioned Some attention has also been given to above apply also to the mass-balance of compressibility effects. In the subsonic field spring tabs. In this case the problem is com­ investigations by Frazer32 and by Jahn have plicated by the appearance of elastic coup­ indicated how increasing Mach number may lings, and if the effects of these are to be affect critical flutter speeds. In the super­ overcome, the tab must be treated as sonic field, the aerodynamic forces have been rotating about its own hinge and a virtual studied by Collar33 and by Temple and hinge fixed by the geometry of the system. Jahn,34 and wing flutter calculations for Here again, the balance mass must be supersonic conditions have been made. correctly positioned in1 relation to these two hinges; this consideration has led to the 6.1.2. Divergence. definition of a " limiting length of balance There has been no evidence that divergence arm "" and to great clarification of the represents a problem of acute practical problem of spring tab flutter.28 importance, and in consequence relatively Studies of methods of prevention of flutter little attention has been paid to this other than mass balance have also been phenomenon. It has not been entirely for­ prosecuted, but without very promising gotten, since there has been a general results. The introduction of motional forces, tendency for wing flexural axes to move aft, or damping, does not seem likely to provide and divergence speeds have therefore tended a practical solution for a variety of reasons: to become lower. In particular, studies have the damping required varies acutely with been made of the divergence speed of an 35 36 many of the parameters involved, such" as elastic wing by iterative methods. ' The height and control surface inertia, and in any result still emerges, however, that divergence case would require prohibitive effort from the speeds are sufficiently high not to present pilot if applied directly to the control surface. practical problems. Irreversible and quasi-irreversible units still If divergence does become a problem of 623 A. R. COLLAR practical importance then, as remarked in The results indicate the very great impor­ 5.1.2, it will become necessary to introduce tance of torsional stiffness for aircraft flying either a specification of the lift conditions or in the neighbourhood of the sonic speed, and —since the varying lift will probably be show further how inadequate may be the unknown—to treat the response of the air­ stiffness found by the conventional test craft as an integral part of the problem. In method as a measure of the effective stiffness this case inertia forces, usually excluded in of the elastic wing; since the deformation studies of divergence (since the inertia forces modes under a concentrated torque and under corresponding to the deformation are neg­ aerodynamic loading may be widely different. ligible) will reappear in the problem. While, therefore, it is apparent that the, Wing-aileron divergence has also been subject of aileron reversal and loss of aileron > studied;37 but here the remarks in 5.1.2 still control has progressed considerably, at the apply. same time it has been restricted to unacceler- Tailplane-elevator divergence speeds have ated conditions; and inertia forces have been received attention for the first time, in a excluded from the researches. This has study of loss of elevator control: this is dealt largely been due to the importance assigned with later in 6.1.3.2. to a high steady rate of roll as a criterion of aileron performance. 6.1.3. Loss and Reversal of Control. It is convenient to divide this topic under Nowadays rates of roll are becoming so two headings: loss of aileron control and loss high that there are signs of a change in this of elevator control. respect; and it is probable that aileron performance will in future be judged by the 6.1.3.1. Loss of Aileron Control. rapidity with which a vertical bank can be The question of loss of aileron control has reached, or by some similar criterion. Such had to be kept continuously under review a manoeuvre will involve' acceleration and throughout the past decade, since the wing deceleration in roll; and studies of loss of stiffness determined by the associated require­ control due to wing deformation may well ment has often been greater than that have to include these inertia forces in the required by the wing stiffness criterion future. (which has assumed more and more the role of a flutter preventive). 6.1.3.2. Loss of Elevator Control. Up to a certain stage the work on aileron With the increasing speeds of ten years reversal was ably summarised by Victory.38 ago, and in the absence of requirements for Since that time the increasing importance of the stiffnesses of fuselage and empennage, it compressibility effects, and the possibilities of was to be expected that aeroelastic effects on flight at supersonic speeds, rendered a new the tail organs would make themselves felt. investigation desirable; more particularly The first evidence of this was through effects since wing tapers and stiffness distributions on longitudinal stability, and though no were becoming very different from those of cases of actual reversal of elevator control the orthodox wing of ten years ago. have been reported, observed tail defor­ Accordingly, a study of the rolling power mations and a crop of associated troubles of an elastic wing was undertaken by Collar rendered an investigation of those effects and Broadbent,39 and the results were necessary. applied to flight at subsonic and supersonic The investigation, made by Collar and speeds:40 the aerodynamic forces being those Grinsted," treated tailplane twist as the main given by Glauert"- *2 in the subsonic range variable, but examined also the effects of and by Collar43 in the supersonic range. elevator twist and of fuselage flexibility under 624 THE EXPANDING DOMAIN OF AEROELASTIC! TY shear and moment loads. It was shown that in a somewhat desultory way over a number severe structural distortion might result unless of years: and appropriate criteria were the elevator reversal speed were of the order defined and proposed for adoption. of twice the maximum speed, and that in 6.1.4. Elastic Effects on Stability. consequence high standards of stiffness for It has been indicated above that this the fuselage and empennage were very problem has appeared for the first time desirable. The tailplane-elevator divergence during the last decade. It might have been speed was also studied—a phenomenon of a thought that the principal practical problem somewhat different character from wing- would have been one of lateral stability, aileron divergence. resulting from the changes in effective Finally, the investigation included a treat­ dihedral produced by wing bending; but in ment of the effect of tailplane twist on longi­ fact the effects of elastic distortion on longi­ tudinal static stability, which showed how if tudinal stability have proved much more the tailplane stiffness was inadequate a stable serious. aeroplane might at high speeds become more The previous problem had been envisaged stable or might develop instability, depending by Pugsley, and Bryant and Pugsley46 on tail setting, wing pitching moment and demonstrated how, with increasing wing so forth. loadings, the values of wing dihedral would This investigation was followed by a second have to be carefully watched if lateral study in the same field by Collar and instability were to be avoided. But further 45 Victory. The object here was to define, expansion of aeroelastic theory in this field in the form of stiffness criteria, desirable does not appear to have been required by standards of structural stiffness for tailplanes, experience. elevators and fuselages, and to examine how Regarding longitudinal stability, an early far these stiffnesses were interdependent in examination of the effects of wing torsion was their effects on elevator control. made by Pugsley.47 The question was approached theoretically Reference has already been made to the by a consideration of loss of control and extension of the aeroelastic field to include more particularly with a view to limitation loss of elevator control and to the study by within reasonable bounds of the structural Collar and Grinsted of the effects of tail and distortions due to application of elevator. fuselage distortion on longitudinal stability. By this means appropriate forms for the Further studies in this area are largely due criteria were derived, and the required order to Gates, Lyon, and their collaborators, and of magnitude indicated. are more in the nature of introduction of To supplement this a statistical survey* elastic effects into general stability studies; was also made of the experimental evidence and, as such, will be dealt with later. on all th6se stiffnesses—evidence accumulated 6.1.5. Developments in Stiffness Criteria. We have already remarked that, when * The word . " statistical " is used somewhat loosely here and elsewhere. In nearly all such Pugsley reviewed the subject of aeroelasticity, surveys of stiffness criteria ..thd data are so varied in nature and origin, and so scanty, that there was virtually only one stiffness it is not possible to obtain frequency distri­ criterion: and even that criterion had not butions in the strict statistical sense : the best that can be done is to weigh the data on a been accepted for current use except as basis of experience, common sense, and a recommended practice. This criterion laid knowledge of the origins of each item, and to take an arithmetic mean. To be honest, it is down, in very simple terms, the wing sym­ often difficult to avoid an element of casuistry metrical torsional stiffness, measured between in the process, since the data are often being used to test a theoretical result. the wing root and the mid-aileron section. 625 A. R. COLLAR

During the period under review, some the maximum diving speed by an adequate « changes have developed in this criterion and margin; while it has not been possible to raise . new proposals have been made. At the same the flexure-torsion flutter speed economically ] time, new criteria have appeared; it is con­ by any means except that of increasing the § venient to discuss these aspects under separate wing torsional stiffness. It follows that the I headings. divergence speed has in practice remained at \ a higher level than the flexure-torsion flutter ] 6.1.5.1. The Wing Stiffness Criteria. speed, and the latter has therefore defined j The developments in the mid-aileron tor­ the criterion. sional stiffness criterion have recently been Thus the mid-aileron wing torsional stiff- i 48 reviewed, and new proposals made; we ness criterion has assumed almost exclusively > shall here make only a brief prdcis of this the role of a preventive of flexure-torsion work. flutter. This criterion is remarkably simple First let us note that for conventional in form, since it involves as parameters only wings it has not been necessary to lay down the maximum equivalent airspeed, the mean any standard for bending stiffness: this in chord and span, and the torsional stiffness. spite of the fact that, partly as a result, the No account is taken of parameters which are ratio of bending to torsional stiffness has in fact known to affect the flutter speed decreased on the average.to about one-fifth_ appreciably, such as wing taper, bending or" less of the value it had ten years ago. stiffness or position of the flexural axis. It would seem probable that the design of Even the most important parameter—inertia the wing for strength leads almost auto­ distribution—is left out of account except in matically to adequate bending stiffness. This that different values of the criterion are laid view is supported by a survey of bending down for wings with and without wing stiffnesses, which shows smaller variations in engines. Finally, compressibility effects are a bending stiffness criterion, from one aero­ ignored. plane to another, than might have been The great simplicity of the criterion, and expected if there were no controlling factor. its uniform success in preventing trouble, On the other hand, two criteria for wing rendered any modification undesirable so long torsional stiffness have been current for some as no great weight penalty was incurred in its years, having been made mandatory shortly satisfaction (it must be remembered that a before the war: these are the " mid-aileron " certain standard of stiffness, of the same and the " equivalent tip " criteria. Only the order, is in any case required by aileron first of these has been important in practice: reversal considerations). This fortunate state satisfaction of the mid-aileron criterion has of affairs has continued until quite recently. almost invariably implied satisfaction of the In the past two or three years, however, equivalent tip criterion by a wide margin. an increasing tendency towards strong wing Regarding the mid-aileron criterion, we taper has produced wings in which the may revert to Pugsley's speculation, referred natural tendency is for the local wing to in 2, as to whether this would be defined torsional stiffness to fall off very rapidly by reversal of aileron control, wing flexure- towards the tip; and this and other factors torsion flutter, or wing divergence. The demanded that the form of the criterion be answer, so far as the past ten years are con­ reconsidered. To this end a number of cerned, is wing flexure-torsion flutter. investigations into the effects of the para­ Reversal of aileron control has had its own meters concerned have been made. criterion, in the sense that it is a requirement It has been demonstrated that the optimum that the aileron reversal speed shall exceed reference section for the measurement of the 626 THE EXPANDING DOMAIN OF AEROELASTICITY

torsional stiffness is a function both of the wing and tailplane torsional stiffness criteria modes of distortion assumed and of the taper, are identical in form, the tailplane criterion but that a section at about 0.7 span can is not determined by flexure-torsion fljitter safely be used. Investigations have also considerations, while the wing criterion is. been made of the variation of critical flutter The tailplane criterion is fixed largely by speed with taper, with assumed modes of considerations of longitudinal stability and twist and of bending, with wing density and control; it is noteworthy that the two inertia axis position, and with wing bending criteria have, however, nearly the same stiffness and flexural axis position. Account numerical values. On the whole, the tail- has also been taken of the difference between plane criterion is slightly more severe than dynamic and static stiffness. the wing criterion, and it is probably this All these investigations have recently been fact, together with the simpler structural reviewed,4* and the results have been ex­ problem, which has resulted in the practical tended and incorporated in a proposed new freedom of tailplanes from flexure-torsion criterion. This new proposal is an elabor­ flutter. ation of the existing criterion, additional Two tailplane criteria are defined, one for terms introducing the more important para­ tailplanes without outboard fins and rudders meters such as taper, inertia axis position and and a rather more severe criterion for tail- Mach number. planes carrying outboard fins and rudders. At present, a survey of values of the pro­ The fuselage stiffness in vertical flexure is posed new criterion achieved by existing also determined from considerations of longi­ aircraft is in progress. tudinal stability and control. Corresponding to vertical fuselage flexure 6-1.5.2. Additional Criteria. and tailplane twist are lateral fuselage Requirements now exist,17 in the form of flexure and fin twist. No theoretical investi­ criteria, for torsional stiffness of ailerons, gations into desirable values for the stiffnesses elevators and rudders. These criteria are all in this second case have been made; but a similar in form: this form being that derived lateral stiffness criterion for fuselages has from the theoretical studies of loss of elevator been defined, by analogy with the vertical control mentioned earlier.4*'4S The numeri­ stiffness criterion, and a value assigned to it cal values to be achieved are, however, on the basis of practical measurements. different, being based on practical results It has not been found practicable to define rather than on theoretical predictions. a criterion for fins, in view of their very wide In the case of ailerons two values are variations in shape and attachment—parti­ defined, one for ailerons with distributed cularly in the case of outboard fins. mass balance and the second for other The last major structural criterion is that ailerons; similarly, different values are for torsional stiffness of fuselages: it is this required for elevators with and without horn criterion which would enter into calculations balances. of antisymmetrical elevator flutter. In addition there is a requirement for the Finally, standards for the stiffnesses of flexural stiffness of the overhang of an control circuits have been laid down. These elevator beyond its outermost hinges. take the form of a requirement that the With the exception of wings probably the stretch produced by a reasonably big effort most important stiffnesses are those of the on the part of the pilot shall not exceed more tailplane in twist and of the fuselage in than a given percentage of the total available vertical flexure: and criteria have been movement. There is in addition a stiffness defined for these stiffnesses. Although the requirement for the aileron balance circuit, 627 i A. R. COLLAR designed to ensure that serious upfloat of the the appropriate stiffness criteria and the ailerons shall not occur. elevator reversal speed. It is in the field of longitudinal stability 6.2. EXPANSION IN FIELDS ALLIED TO that another important deformation effect has AEROELASTICITY. first become serious: that of panel distortion under load.53 In elevators, the distortion of In 6.1 we indicated how aeroelasticity has the panels of the structure under aerodynamic been, in effect, expanding into other fields. loads can produce sufficient change in the In the present paragraph we shall attempt to effective camber to result in quite abnormal show something of the reverse process. hinge moments; and the stick free stability 6.2.1. Elasticity in Problems of Stability can be adversely affected to a serious extent. and Control. To a lesser extent, panel deformation in ailerons can also be serious. We have already shown how elastic effects It may be remarked that the deflection of enter into both lateral and longitudinal con­ a panel is obviously not proportional to the trol. In lateral stability, however, elastic applied load; and thus panel deflections have effects do not appear to have presented a for the first time introduced serious elastic serious problem: such effects undoubtedly non-linearities into aeroelastic problems, and exist, since, for example, under normal we can no longer talk with accuracy in acceleration the wing dihedral must change terms of stiffnesses or stiffness criteria. How very appreciably. These problems of lateral this difficulty is to be met is a matter still stability have been kept in mind; but it is in under consideration. the field of longitudinal stability that the effects of elastic deformation are most pro­ 6.2.2. Elastic Effects in Loading Problems. nounced and can have serious consequences. In 5.2.2. we stated that up to ten years ago loading problems were treated on the The treatment of the problem of longi­ aerodynamic-inertia force basis, and that tudinal stability has been greatly simplified elastic deformations were in the main ignored. by the expositions by Gates of the funda­ In this area, the position has not changed mentals of longitudinal stability49 and of the radically, and we are not yet being driuen idea of manoeuvrability.50 By studying to a correlation between strength and stiff­ independently the static stability, stick fixed ness. Nevertheless, there are indications and free, and the manoeuvrability, stick that we are moving towards such a correla­ fixed and free, the important aspects of tion, and we shall try here to indicate the longitudinal stability of an aircraft are all roads by which this movement is taking covered. Without this subdivision, the place. introduction of elastic effects into longitudinal In the first place, both aeroelastic require­ stability investigations would have involved ments and aerodynamic requirements for much more arduous calculations and made smoothness of profile are demanding much the interpretation of the results much more greater skin thicknesses than were common difficult. > ten years ago. In consequence, the skin is As it is, Gates and Lyon51 and their rapidly becoming a main stress-bearing part collaborators have now proposed a theory in of the structure, and it is not impossible that which the effects on all aspects of longi­ we shall soon see spars disappearing from tudinal stability of elastic deformations in the wings in favour of all-skin designs. In this fuselage and empennage are taken into event the strength and stiffness will be much account;52 and following the earlier pre­ more closely tied together than is at present cedent44 have absorbed into stability theory the case. 628 THE EXPANDING DOMAIN OF AEROELASTICITY

Again, there is an increasing realisation of the leading edge, then a gust load would that the structural strength of an aircraft can cause a change in wing incidence near the no longer be adequately treated on the basis tips in the sense required to reduce the of static loads alone. Much attention has applied load there; and other examples been given, and more is required, to the might be given. problem of fatigue of built-up structures; Here, then, is a problem in which design and the oscillatory loads involved may be of for strength may be appreciably affected by aerodynamic origin as well as due to the design for stiffness; and, it must be mechanical vibration. added, such a design would have to be In this connection it may be remarked that watched carefully from other aeroelastic proposals are in hand for strength testing an viewpoints, particularly those of flutter and airframe under conditions designed to simu­ of stability. late those which would obtain during an 6.2.4. Mechanical Vibration. occurrence of flutter. In this field of non- In this field considerable advances, which static loads we may also note the work which owe much to the energetic researches directed has been done on the strength of structures by Morris,54 have been made in the past ten under repeated slow loading; although this years. We cannot enlarge on these advances has not resulted directly from elastic effects, here, since up to the present time the it has an obvious bearing on strength under researches have been confined to the inertia- oscillatory loads of aero-elastic origin. elastic field. We may, however, note that the stage has now been reached when the 6.2.3. Gusts. determination of the natural frequencies of Knowledge of the structure of gusts has a complete aircraft, and particularly those not increased to any appreciable extent frequencies which seem likely to be impor­ during the period under review, and in con­ tant from the vibration viewpoint, can be sequence calculations still have to be made predicted with fair confidence. for somewhat arbitrary gust cases. But the Thus the investigators have reached a methods of dealing with these cases have milestone in the road of their research; and considerably developed. the next stage must be the determination of This is best illustrated by reference to a amplitudes of response by the inclusion in calculation made by the Bristol Aeroplane the equations of motion of the forcing Company in relation to one of their aero­ impulses arising in the engine and of the air planes; in this calculation, the growth of the forces; since the latter provide both damping air forces according to Wagner's theory, the forces and modifications to the elastic stiff­ response of the aeroplane, and a large nesses. number of elastic freedoms for the structure, That mechanical vibration and flutter are were all taken into account in some detail— beginning to overlap is illustrated by the a complete aerodynamic-elastic-inertia force following occurrence. On a particular problem. aircraft the elevator trim tabs were not mass- At the same time, since gust cases are often balanced, since the backlash in the operating severe, attention is being devoted by many system was very small. On one mark of the investigators to the problem of cheating the aircraft, which had one particular type of gust. One way of doing this is to employ engine, no trouble occurred. But on a the elastic deformation of the wing to reduce second mark, with a different engine, very the adverse loadings. For example, if it unpleasant vibration developed, and was were possible by a structural artifice to obtain cured by reducing the backlash in the tab to an effective flexural centre near to or ahead a much smaller amount than had previously 629 A. R. COLLAR been thought necessary. It was evident, In this triangle we see how most of the therefore, that the different forcing impulses subjects have now become bonded—although from the second engine were producing a at present the bonds are often tentative—to tab nutter, within the limits of the tab back­ a third vertex, where previously only two lash, where previously none had existed. such bonds existed; moreover, certain of the subjects are becoming bonded together by 6.2-5. Impact Effects. obvious interconnections. Thus, for example, As in the case of mechanical vibration, the loss and reversal of elevator control is researches on this subject have been confined, obviously directly connected with longi­ up to the present time, to the inertia-elastic 55 56 tudinal stability; while the study of gusts field; - and some important effects, par­ has much in common with both flutter ticularly on the increase in local stress above studies/ and investigations of landing impacts, that given by the rigid aeroplane case, have and so on. been observed. Although increasing attention is being paid In short, it is evident that we are no longer to this subject it has still not yet reached the dealing with a series of subjects, each in its stage where even on the restricted inertia- own watertight compartment: there is a elastic force basis, accurate quantitative definite coalescence of the subjects into an predictions of the important local stress integrated whole, which may be defined as variations resulting from a landing impact the dynamics of a deformable aeroplane. And can be made. When this stage is reached it we are faced with the question: what are to will be expedient to introduce the air forces be our methods of treatment of this unified also; for, as speeds at touch down problem ? It is obviously not possible to steadily increase, these forces are becoming be dogmatic on this important question; but correspondingly more important, and may the writer would suggest the following provide a significant correction to the stress possible answers. calculations. 7.1. TREATMENT OF THE PROBLEM. 7. THE FUTURE. We have seen in 6 how the individual To the mathematician the description of studies with which we are concerned are the subject—dynamics of a deformable expanding; and we may revert to our aeroplane—suggests the formal solution at " triangle of forces " diagram to examine once. We select as generalised co-ordinates what is happening to the subjects there des­ quantities defining the translational and cribed. We may perhaps take the subject of angular freedoms of the aeroplane as a rigid aeroplane stability as an example. body; we add co-ordinates representing the In Fig. 2 this subject is bonded to aero­ control surface angles, tab angles, stick and dynamic and inertia forces only, and is pedal movements, and with these we include unconnected to the third vertex of the co-ordinates to represent the freedoms due to triangle—elastic forces. In consequence, it automatic controls; and finally we introduce lies outside the aeroelastic triangle. But, at a number of co-ordinates sufficient to des­ the present day, it has acquired a bond to the cribe the modes—normal or otherwise—of third vertex—not perhaps so strong as its elastic deformation of the aircraft. We then connection with aerodynamic and inertia derive, in terms of the inertia and elastic forces, but nevertheless sufficient to draw the properties of the aircraft and the air forces, subject within the general domain of aero- the complete Lagrangian equations of elasticity. We may, in fact, re-draw our motion, including in the appropriate equa­ triangle as shown in Fig. 3. tions forcing terms representing mechanical 630 THE EXPANDING DOMAIN OF AEROELASTICITY

Fig. 3 aerodynamic forces F flutter elastic forces B buffeting 1 : inertia forces S stability and control D divergence R reversal of control G : gusts L loading problems V mechanical vibration Z impacts #

vibration impulses, gusts, landing impacts, must not dismiss out of hand the possibility pilot's efforts, and so forth. of solving such a set of equations, at least In this way, we obtain a set of simul­ so long as they are linear in the unknown taneous differential equations, probably in co-ordinates or approximations to linearity twenty to thirty degrees of freedom, which can be safely made. For we have already describe completely the motion of the air­ remarked that inverse methods of solution craft and which can tell us whether serious have remarkable power; and by adoption of trouble is likely to arise or not. such methods of dealing with the equations From the practical viewpoint, of course, we may well be able to reduce the problem the labour of dealing with such a set of to that of evaluation of numbers of numerical equations is prohibitive at present. But we determinants of large order. 631 A. R. COLLAR

At present this process would be far too could prepare a table with a spectrum of arduous and would offer such great possi­ frequencies such as follows:— bilities—even probabilities—of error that no SUBJECT FREQUENCY attempt would be worth while. But in the Reversal of control last few years considerable strides have been Zero Divergence made in dealing with large numbers of Loading problems simultaneous equations by the use of elec-- Zero or low trical, mechanical, and electro-mechanical Stability machines;57' 5S while the growing use of Gusts Low or medium matrix algebra59 makes the formulation of the Impacts 1 Medium or problem, as well as its solution; much easier. Flutter J high It is thus not outside the bounds of Vibration High possibility that within a period of years we On inspection of this table we see that it may have available computational aids of may be possible to solve our unified problem such power that the solution of equations of by unified treatment, but that it is probably the kind we are considering may be a routine not necessary to take the whole group at process. once. Thus the connection between the first In the writer's view it is more probable and last subjects is negligible: we need not that the chief difficulty will be, not in the consider vibration questions in dealing with solution of the differential equations, but in reversal of control, and vice versa. But in the derivation of the equations themselves dealing with reversal of control we have a from the basic data: even at the present time, strong connection with stability studies: in in advanced flutter calculations, the solution stability studies we have a growing connec­ of the equations is the simpler half of the tion with flutter; and flutter and mechanical work. However, the formulation of the vibration are becoming closer: so that the equations may not be so much more complex tie between first and last exists. than at present, since in the majority of cases Thus in the future, given our computa­ the number of coupling terms will probably tional aids, it may well be that we shall write be relatively small. down for each subject a sufficient number of Meanwhile, we shall obviously continue to equations of motion to cover appropriate discuss our subjects as separate entities. The neighbouring subjects in the spectrum. In fact that the coupling terms in the equations studying reversal of control we may include of motion are few (or weak) has permitted divergence and stability: in studying us to regard the subjects as separate in past stability we may include reversal of control years, and there is no reason why we should and flutter: in studying flutter we may not continue to take -advantage of this fact include stability and mechanical vibration. (provided that we are careful to assess the Viewed from this angle, it appears that magnitude of those couplings we ignore) for the unification of the subject does not some time to come. But it may well be that necessarily involve a hopeless complexity but a reconsideration of programmes and offers rather the advantage that we may methods at this stage is worth while. study the various aspects in relation to their We may, for example, take the subjects associated subjects by a standard method of we have discussed and, since they are attack. becoming unified, try to relate them accord­ ing to some particular characteristic: one 7.2. STANDARDS OF ELASTICITY. such characteristic that suggests itself is We have speculated in the last paragraph frequency of oscillation. On this basis we on how our subject may be treated in the 632 THE EXPANDING DOMAIN OF AEROELASTICITY future; we may now perhaps be permitted matters as aileron reversal, longitudinal to speculate further on the methods of stability and flutter. And the bending and curing trouble as it arises. And since the torsional stiffnesses are almost indissolubly aerodynamic and inertia loads are likely still mixed in the theories—it is in fact no easy to be dictated mainly by other considerations, matter even to formulate unambiguous the principal control on aeroelastic and definitions of bending and torsional stiffness associated phenomena will continue to be for cranked wings. exercised through stiffness provisions. We If, therefore, we find our simple stiffness may make more use of inertia distributions criteria unsuitable or insufficient for design than in the past, but except for mass-balance purposes we may have recourse to a process of control surfaces to prevent flutter, varia­ of " stiffness stressing "—a study by stress tion in inertia distribution is likely to be a offices of deformations under various loading supplementary measure to the main process conditions, parallel to the studies of strength of stiffness variation. now made. In this way we may find a new In the first place, it would appear that the correlation between strength and stiffness simple stiffness criteria which have served us coming about. But in any case, it seems so well in the past are likely to require evident that for very high speed aircraft the elaboration. We have remarked that this utmost efforts should be made to ensure that elaboration has already been proposed for elastic deformations are minimised; and, the wing torsional stiffness criterion; although ceteris paribus, those aircraft having high it does not at present amount to more than structural stiffness are likely to be most saying that the familiar criterion is no longer successful. a constant, but is a function of all the important relevant parameters. 8. CONCLUSIONS. On a point of detail, it may be found to In what has gone before we have be desirable to square the criteria in their attempted to review in a very small compass present form and to introduce the standard the progress in the past decade of aero- air density to render the criteria non- elasticity and its allied subjects. The writer dimensional; if this were done the criteria is aware that, in attempting this task, he has would be in the nature of force or moment laid himself open to a charge of insularity; coefficients corresponding to unit increments and, indeed, some may regard the reference of displacement, and would define the list as parochial. required stiffnesses explicitly. As regards the latter charge we may, But we may lose even this elaborated form perhaps, be allowed to remark that during as we modify our designs to penetrate the past ten years the number of papers further and further into the sonic regions and presented to the Oscillation Sub-Committee beyond. For not only must we introduce of the A.R.C. alone is nearly six hundred; different functions for Mach number correc­ and almost without exception these papers tions at subsonic and supersonic speeds; we have been related to aeroelasticity in the are also likely to find that new criteria are wider sense implied in the present paper. involved and that there is such an essential The number of relevant communications interdependence between the criteria that we to other bodies is not known. But it is cannot define them separately. apparent that in any attempt to survey the ^ We are beginning to realise this in studies general trend of work in all areas, it is of sweptback wings for high speeds:60 wing necessary to make a very restricted choice bending stiffness becomes increasingly more of references; in the present case it is hoped important, as sweepback increases, in such that those quoted will themselves give the 633 A. R. COLLAR interested reader more extended references in duction for permission to refer to many their own areas. Reports which are as yet unpublished and to In the flutter field, a bibliography of work for the Ministry on which he was British work containing about two hundred personally engaged during the war years. and fifty references was recently prepared b}' Graham;61 other similar bibliographies in all REFERENCES. fields would be very helpful. NOTE.—It must be clearly realised that the list of references given here is by no means com­ The charge of insularity is well sustained plete; indeed, it is no more than a small and cannot be refuted. In explanation, how­ sample of the papers which might be con­ sulted. However, in most cases the works ever, we must point out that, except in the quoted themselves contain reference lists flutter field, countries other than Great covering their own areas reasonably fully. Britain have given little attention to aero­ 1. A. G. Pugsley—Control Surface and Wing Stability Problems. J.R.Ae.S. Nov., 1937. elastic matters until the past few years. 2. R. A. Frazer, W. J. Duncan—The Flutter of Only recently has it become the practice to aeroplane wings. R. & M. 1155. Aug., 1928. examine on a more or less routine basis the 3. R. A. Frazer, W. J. Duncan—The Flutter of effects of distortion on lateral and longi­ Monoplanes, Biplanes and Tail Units. tudinal control and stability. R. & M. 1255. Jan., 1981. 62 4. W. J. Duncan, A. R. Collar—Resistance Indeed, we have recently learned that Derivatives of Flutter Theory, Pt. I. German military aircraft suffered heavily at R. & M. 1500. Oct., 1932. 5. W. J. Duncan, A. R. Collar—The Present one period during the war as a result of the Position of the Investigation of Airscrew lack of attention paid to aeroelastic Flutter. R. & M. 1518. Dec, 1932. phenomena; and much concentrated effort 6. H. Roxbee Cox, A. G. Pugsley—Theory of Loss of Lateral Control Due to Wing had to be applied to the task of finding out Twisting. R. & M. 1506. Oct., 1932. what was going wrong and into the redesign 7. A. G. Pugsley, H. Roxbee Cox—The Aileron Power of a Monoplane. R. & M. 1640. which was found to be necessary in many Apl., 1934. cases. 8. A. G. Pugsley—The Aerodynamic Charac­ teristics of a Semi-Rigid Wing Relevant to Moreover, nowhere else has theory or the Problem of Loss of Lateral Control Due experiment thrown up a method of dealing to Wing Twisting. R. & M. 1490. June, 1932. with aeroelastic phenomena of a simplicity 9. A. G. Pugsley, G. R. Brooke—The Calcula­ comparable with that of the stiffness criteria tion by Successive Approximation of the Critical Reversal Speed. R. & M. 1508. used in Great Britain. Oct., 1932. 10. Staff of Aerodynamics Dept., N.P.L.—Two Thus it may fairly be claimed that in Reports on Tail Buffeting. R. & M. 1457. co-ordinated aeroelastic research, the most Oct., 1932. 11. S. B. Gates, H. B. Irving—An Analysis of consistent developments have taken place in Aileron Performance. R.A.E. Rept. Aero. Great Britain. This is not to say, of course, 1624; A.R.C. 4652. July 1940. 12. H. Roxbee Cox, A. G. Pugsley—Stability of that in individual areas we have not learned Static Equilibrium of Elastic and Aero­ much from other countries: the reverse is dynamic Actions on a Wing. R. & M. 1509. Oct., 1932. the case. To take only one example, in the 13. L. W. Bryant, I. M. W. Jones—Stressing of flutter field the present emphasis on normal Aeroplane Wings Due to Symmetrical Gusts. R. & M. 1690. Feb., 1936. modes of vibration as found from resonance 14. D. Williams, J. Hanson—Gust Loads on tests owes much to American influence. But Tails and Wings. R. & M. 1823. May, 1937. co-ordinated aeroelastic research belongs 15. H. Constant—Aircraft Vibration. R. & M. peculiarly to Great Britain; and, as the 1637. Oct., 1934. 16. R. A. Fairthorne—The Effects of Landing writer has attempted to show, a. larger co­ Shock on Wing and Undercarriage ordination may very soon be necessary. Deflexion. R. & M. 1877. Jan., 1939. 17. A. P. 970, Part V. The writer wishes to express his thanks to 18. W. J. Duncan, A. R. Collar, H. M. Lyon— Oscillations of Elastic Blades and Wings the Ministry of Supply and Aircraft Pro- in an Airstream. R. & M. 1716. Jan., 1936. 634 THE EXPANDING DOMAIN OF AEROELASTICITY

19. L. Bairstow, A. Fage—Torsional Vibrations 37. A. R. Collar, M. Victory—Reversal of on the Tail of an Aeroplane, Pt. II; Aileron Control, Wing Divergence, and Oscillations of the Tailplane and Body of Wing-Aileron Divergence for the Case of an Aeroplane in Flight. R. & M. 276: an Elastic" Wing: A Study of Recent July, 1916. American Methods of Calculation. R.A.E. 20. H. A. Jahn, G. H. L. Buxton, I. T. Min­ Rept. S.M.E. 3262; A.R.C. 7107. Aug., hinnick — Fuselage Vertical Bending — 1943. Elevator Flutter on the Typhoon. R.A.E. 38. M. Victory—The Calculation of Aileron Rept. S.M.E. 3286; A.R.C. 7744. Mar., Reversal Speed. R.A.E. Rept. S.M.E. 1944. 3279; A.R.C. 7486. Jan., 1944. 39. A. R. Collar, E. G. Broadbent—The Rolling 21. C. Scruton, G. Sim, W. T. Burke—Experi­ ments on Tail Flutter: Effect Of Fuselage Power of an Elastic Wing, Pt. I: Com­ Pitching Mobility on Symmetrical Elevator pressibility Effects Absent. R.A.E. Rept. Flutter. A.R.C. 6717. May, 1943. S.M.E. 3322; A.R.C. 8786. May, 1945. 40. E. G. Broadbent, A. R. Collar—The Rolling 22. W. J. Duncan—A Suggested Investigation Power of an Elastic Wing, Pt. II: Com­ on Wing Flutter. A.R.C. 4281. Nov., 1939. pressibility Effects and Results for Super­ 23. N. C. Lambourne, D. Weston—An Experi­ sonic Speeds. R.A.E. Rept. S.M.E. 3347; mental Investigation of the Effect of A.R.C. 9239. Oct., 1945. Localised Masses on the Flutter and 41. H. Glauert—Theoretical Relationships for Resonances of a Model Wing. A.R.C. 7604. an Aerofoil with Hinged Flap R. & M. Apl., 1944. 1095. Apl., 1927. 24. I. T. Minhinnick, J. Yarwood—Interim Note 42. H. Glauert—The Effect of Compressibility on on the Theoretical Effects of an Engine the Lift of an Aerofoil. R. & M. 1135. Mass on Wing Flutter. R.A.E. Tech. Note Sept., 1927. S.M.E. 143; A.R.C. 6720. May, 1943. 43. A. R. Collar—Theoretical Forces and 25. A. R. Collar—Note on Virtual Inertia and Moments on a Thin Aerofoil with Hinged Control Surface Flutter. R.A.E. Tech. Flap at Supersonic Speeds. R. & M 2004 Note S.M.E. 218; A.R.C. 7469, Feb., 1944. Nov., 1943. 44. A. R. Collar, F. Grinsted—The Effects of 26. A. R. Collar—Mass Balancing of Controls Structural Flexibility of Tailplane, Elevator and Flutter Prevention Devices. A.R.C. and Fuselage on Longitudinal Control and 8091. July, 1944. Stability. R. & M. 2010. Sept., 1942. 27. R. A. Frazer, W. P. Jones—Wing-Aileron- 45. A. R. Collar, M. Victory—Standards of Tab Flutter, Pts. I and II. A.R.C. 5668. Structural Stiffness for Aeroplane Tailplanes, Mar., 1942. Elevators and Fuselages. R.A.E. Rept. 28. A R. Collar—The Prevention of Flutter of S.M.E. 3251; A.R.C. 6818. June, 1943. Spring Tabs. R.A.E. Rept. S.M.E. 3249; 46. L. W. Bryant, A. G. Pugsley—The Lateral A.R.C. 6837. May, 1943. Stability of Highly Loaded Aeroplanes. 29. A. R. Collar—A Proposal for a Control R. & M. 1840. Sept., 1936. System Operated by an Aerodynamic Servo 47. A. G. Pugsley—Influence of Wing Elasticity and Avoiding Large Mass-Balance Weights. upon Longitudinal Stability. R. & M. R.A.E. Tech. Note S.M.E. 293; A.R.C. 1548. Jan., 1933. 8411. Dec, 1944. 48. A. R. Collar, E. G. Broadbent, E. B. 30. J. Williams—Note on the Fundamental Puttick—An Elaboration of the Criterion Coefficients for Flexure-Torsion Flutter. for Wing Torsional Stiffness. R.A.E. Rept. A.R.C. 8193, Nov., 1944. S.M.E. 3357; A.R.C. 9540. Jan., 1946. 49. S. B. Gates—Longitudinal Stability and 31. W. Prichard Jones—Aerodynamic Forces on Control. A Simple Exposition of the Wings in Non-Uniform Motion. A.R.C. Fundamentals Employed in this Aspect of 8928. Aug., 1945. Design. " Aircraft Engineering," Sept., 32. R. A. Frazer—Notes on the Influence of 1940. (Page 260.) Mach Number on Flutter and Divergence 50. S. B, Gates—A Proposal for an Elevator of a Tapered Wing. A.R.C. 4949, Feb., Manoeuvrability Criterion. R.A.E. Rept. 1941. Aero. 1740. Mar., 1942. 33. A. R. Collar—Resistance Derivatives of 51. S. B. Gates, H. M, Lyon—A Continuation of Flutter Theory, Part II. Results for Longitudinal Stability and Control Analysis. Supersonic Speeds. R.A.E. Rept. S.M.E. Pt. I: General Theory. R.A.E. Rept. 3278; A.R.C. 7470. Dec., 1943. Aero. 1912; A.R.C. 7687. Dec, 1944. 34. G. Temple, H. A. Jahn—Flutter at Super­ 52. H. M. Lyon, J. Ripley—A General Survey sonic Speeds, Pt. I: Mid-Chord Derivative of the Effects of Flexibility of the Fuselage Coefficients for a Thin Aerofoil at Zero Tail Unit, and Control Systems on Longi­ Incidence. R.A.E. Rept. S.M.E. 3314; tudinal Stability and Control. R.A.E. A.R.C. 8583. June, 1945. Rept. Aero. 2065; A.R.C. 9015. July, 1945. 35. A. G. Pugsley, G. A. Naylor—The Diver­ 53. M. B. Morgan, H. H. B. M. Thomas—Con­ gence Speed of an Elastic Wing. R. & M. trol Surface Design in Theory and Practice. 1815. Oct., 1937. J.R.Ae.S. Aug., 1945. 36. E. G. Broadbent—The Estimation of Wing 54. J. Morris—The Escalator Method in Divergence Speeds. R.A.E. Rept. S.M.E. Engineering Vibration Problems. Chapman 3355; A.R.C. 9493. Dec, 1945. & Hall, 1946. 635 A. R. COLLAR

55. D. Williams—Dynamic Loads in Aeroplanes 59. R. A. Frazer, W. J. Duncan, A. R. Collar— under given Impulsive Loads. A Simplified Elementary Matrices. Camb. University Method of Calculation. R.A.E. Rept. Press, 1938. S.M.E. 3369. Mar., 1945. 60. A. R. Collar—Some Problems in the Design 56. A. Robinson—Shock Transmission in Beams. of Sweptback Wings. A.R.C. 9264. Jan., R.A.E. Rept. S.M.E. 3319. Apl., 1945. 1946. 57. R. R. M. Mallock—An Electrical Calculating 61. M. E. K. Graham—A Bibliography of British Machine. Proc. Roy. Soc. A. 140, p. 457. Work on Flutter. R.A.E. Tech. Note Mar., 1933. S.M.E. 224; A.R.C. 7685. Apl., 1944. 58. R. J. Schmidt, R. A. Fairthorne—Some 62. A. R. Collar, H. A. Jahn, A. Robinson, Comments on the Use of Hollerith Machines E. G. Broadbent—German Aircraft Industry for Scientific Calculations. R.A.E. Rept. —Aeroelasticity. B.I.O.S. Rept. No. 12, A.D. 3153; A.R.C. 4721. June, 1940. Item No. 25. 1946.

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