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2、of ElementarySolutions_-.-. . . . _ . APPLICATION OF REVERSE FLOW THEOREM_-_. ._ APPLICATION OF RESULTS TO DYNAMIC STABILITY ANALYSIS.L.-. PART III-EFFECT OF NONLINEARITIES- _ _ _. _ - _ -_ _ _. _ _ _. _ 1: APPENDIX A-RESPONSE IN LIFT OF TWO-DIMENSIONAL TAIL TO TWO-DIMENSIONAL VORTEX SYSTEM-_- _ APP
3、ENDIX B-RESPONSE IN LIFT OF TWO-DIMENSIONAL, RECTAN- GULAR, AND WIDE TRIANGULAR TAILS TO TWO-DIMENSIONAL VORTEX SYSTEM-GUST ANALYSIS _ - _ APPENDIX C-RESPONSE IN LIFT OF APEX-FORWARD AND APEX- REARWARD WIDE TRIANGULAR TAILS TO TWO-DIMENSIONAL VORTEX SYSTEM-m _ _ _-_ _ _-_-_-_-_-_- APPENDIX D-BOUNDAR
4、Y CONDITIONS AT THE TAIL DUE TO PENE- TRATION OF VELOCITY FIELD OF TRAILING-VORTEX SYSTEM- REFERENCES-_-._- _-_-_-_-_- . page 1 1 2 2 8 10 12 12 14 14 16 17 19 19 19 20 20 20 21 22 23 23 25 26 29 29 30 30 32 32 33 34 37 39 40 41 42 43 III Provided by IHSNot for ResaleNo reproduction or networking pe
5、rmitted without license from IHS-,-,-. I.- ,.1 , -, REPORT 1188 ON THE USE OF THE INDICIAL FUNCTION CONCEPT IN THE ANALYSIS OF UNSTEADY MOTIONS OF WINGS AND WING-TAIL COMBINATIONS By MURRAY TOBAK ,- SUMMARY The coneem! of indicial aerodynamic functions is applied to the analysis of the short-period
6、pitching mode of aircraft. By the use qf simple physical relationships associated with the indicial-function concept, qualitative studies are made qf the separate e$ects on the damping in pitch of changes in Mach number, aspect ratio, plan-form shape, and frequency. The concept is .further shown to
7、be of value in depicting physically the induced e#ects on a tail surface which follows in the wake of a starting forward surface. Considerable e$ort is devoted to the deuelopment qf t.heoretical techniques whereby the transient response in t 2vo g-w PART I-ISOLATED WINGS APPLICATION OF INDICIAL FUNC
8、TIONS TO THE AERODYNAMIC THEORY OF UNSTEADY FLOWS One of the most useful tools in the study of unsteady flows is the concept of indicial aerodynamic functions, which may be defined briefly as the aerodynamic response of the airfoil as a function of time to an instantaneous change in one of the condi
9、tions determining the aerodynamic properties- of the airfoil in a steady flow. Theoretical aerodynamic indicial functions were first derived by Wagner (ref. 2) for the two-dimensional wing in incompressible flow. More recently, these results have been extended to cover the com- pressible case for bo
10、th subsonic and supersonic speeds (refs. 3 and 4). In addition, theoretical indicial functions have now been obtained for both wide and slender triangular wings and rectangular wings, all for supersonic speeds (refs. 4 to 6). The indicial function derives its usefulness primarily through the ease wi
11、th which it lends itself to the powerful and well-established methods of the operational calculus (refs. 7 to 9). With the use of these methods, the aerody- namic forces and moments caused by arbitrary motions of the airframe can be studied from a fundamental standpoint. Because of the wide range of
12、 applicability of this means of approach in unsteady flow analyses, a considerable portion of the succeeding discussion is devoted to the fundament.als involved. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- _ -_.- ._ - _._ - _ _ . _ _- - -._ ;- .
13、 - , USE OF INDICIAL FUNCTION. CONCEPT IN ANALYSIS OF UNSTEADY MOTIONS OF WINGS AND WING-TAIL COMBINATIONS 3 I DEFINITION OF COORDINATE SYSTEM In the succeeding analysis the stability system of axes is used. The origin of the coordinate system is placed in the airfoil so that the y axis which is per
14、pendicular to the vertical plane of symmetry is coincident with the axis of rotation of the airfoil; the positive branch of the x axis is pointed in the direction of flight; and the z axis lies in the vertical plane of J _ -symmetry, .,positive downward. _ The.-angle of attack a is measured as the a
15、ngle between the chord plane of the airfoil and the xy plane, and is shown as positive in figure 1. The / r-Flight path FIGURE l.-Definition of coordinate system. ,-Flight path Angle of pitch = 8 Angle of attack = 0 Here a and 0 are equal, so that the maneuver is defined by one variable, the time hi
16、story of either a! or 0. Let the angle of attack be cr and the angular velocity be p (q=dO/dt =da/dt). At any instant, the normal velocity at any point on the airfoil surface is composed of two parts, one due to the instantaneous angle of attack aV, the other due to the angular velocity at the same
17、instant -pa: (see fig. 3). These are two of the instantaneous boundary conditions of the unsteady flow. Solutions for the aerodynamic forces and moments which correspond to these boundary conditions may be derived by a number of methods involving various degrees of approxi- mation. In succeeding sec
18、tions, the use of the concept of indicial functions and the principle of superposition for this purpose will be illustrated and compared with other current widely used methods. CONCEPT OF INDICIAL FUNCTIONS In order to illustrate this concept, assume that the airfoil under consideration has been fly
19、ing a level path at zero angle of attack. At some time, which is designated time zero, the wing is caused to attain simultaneously a constant angle of Angle of pitch =0 Angle of attack= 0 FIGURE 2.-Maneuvers corresponding to purely (a) angle of pitch and (b) angle of attack varitaions. angle of pitc
20、h e is the angle between the chord plane of the airfoil and the horizontal plane (an arbitrary reference) and is also shown positive in figure 1. Forces are measured as positive upward, whereas pitching moments are positive when tending to increase the angle of pitch in the positive direc- tion. Whe
21、n the airspeed V, is constant, which corresponds to the condition under study, the translatory and angular motions of the airfoil with respect to any system of coordi- nates are defined if the time histories of the angle of attack (Y and the angle of pitch e and their derivatives are known. For purp
22、oses of clarit,y, two different harmonic motions of the aircraft are shown in figure 2, illustrating the difference between a flight path which involves a constant angle of attack and a varying angle of pitch and one which involves a constant angle of pitch and a varying angle of attack. Now conside
23、r the case of a wing executing harmonic rotary oscillations about the y axis while the origin of the coordi- nate system traverses a level path at constant velocity V,. This case corresponds to that of a wind-tunnel model mounted to permit single-degree-of-freedom rotary oscilla- tions, or to the sh
24、ort-period mode of an aircraft in flight when the plunging velocity of the center of gravity is zero. . LNormal velocity-qx due to angular FIGURE 3.-Unsteady flow boundary conditions at airfoil surface. attack LY and angular velocity q. The normal velocity of the flow next to the surface of the airf
25、oil therefore changes dis- continuously from zero to a pattern that is constant with time and identical in shape to the pattern shown previously in figure 3. The lift and pitching moment that result are of a transient character and attain their steady-state values corresponding to these new boundary
26、 conditions only after a significant interval of time has passed. It should be noted there exists an essential difference between the length of this time interval at subsonic and supersonic speeds. At super- sonic speeds, the vorticity shed into the airfoil wake cannot influence the flow about the a
27、irfoil but at subsonic speeds this Provided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 REPORT 1 IS%-NATIONAL ADVISORY COMMITTEEZFOR AERONAUTICS, influence exists for till time. The result is that the lift and moment reach steady-state values in a fini
28、te time at super- sonic speeds but approach these values asymptotically at subsonic speeds. In either case, however, the time responses in lift and moment to the step changes in (Y and q are termed “indicial functions.” Figure 4 illustrates typical subsonic and supersonic indicial lift responses to
29、a step change in the angle of attack. FIGURE S.-Typical indicial lift responses to step changes in angle of attack. It is obvious that the time history of the wing motion during a short-period oscillation may be broken down into an infinite number of infinitesimally small step changes in the angle o
30、f attack and step changes in the angular velocity. The summation of the indicial lift and moment for these steps then yields the total lift and moment at any prescribed time. In figure 5, the mechanics of the procedure are illustrated for an arbitrary angle-of-attack variation. Here, the given angle
31、-of-attack variation is replaced by a number of small step changes. Within each step the corresponding response in lift is shown plotted for convenience. It is then apparent that the total lift at time t is equal to the sum of the increments of lift in each step at time t. As indicated by the leader
32、s, however, it is clear that the increments of a FIGURE 5.-Illustration of superposition process. lift for the various steps at time t are equivalent to incre- ments in the first step at time C-t;. Alternatively, then, the total lift at time t can be written as CL(t.(t)=CL,(t)a(0)+) $ (t;)At; (1) 0
33、After a transformation of variables, t-ti=r and letting the increment of time approach zero, equation (1) can be re- written in a form of Duhamels integral (see, e. g., ref. 9) A similar procedure is carried out for the angular velocity variation, whereupon the total lift coefficient at the prescrib
34、ed time t becomes It should be pointed out that in this form equation (3) is applicable to the analysis of arbitrary motions, the only restriction being that the flight speed is constant. In the following sections, however, the application of equation (3) is restricted to harmonic motions having a s
35、ingle degree of freedom. The reasons for this restriction are two-fold: first, the motions of a statically stable aircraft in response to a disturbance are most generally of a harmonic nature; and second, such a restriction permits an assessment of the influence of the time rate of the airfoil motio
36、ns on the aerodynamic forces and moments. APPLICATION OF INDICIAL FUNCTIONS TO HARMONIC PITCHING OSCILLATIONS Consider first a pure sinusoida, pitching oscillation, the angle of attack being zero throughout the motion. The flight path for such a motion has been illustrated in figure 2 (4. In this ca
37、se, the angle of pitch is given by O(t)=O,ei”t where BO is the maximum amplitude of oscillation and w is the angular frequency. The angular velocity is, of course, q(t)=B=iweoei=iwe(t). Inserting the value for q(t) in equation (3) and performing t.he indicated operations, there results Fote in figur
38、e 6 that CL, (T) is equal to CLQ(t)-p2(), and that P*(T) approaches zero as 7 a.pproaches t. Replacing PrJ7) in equation (4) by this equality, For subsonic speeds, let t approach infinity. With this substitution, equation (5) thereby represents the lift coeffi- cient due t,o the harmonic pitching mo
39、tion after the transient loading subsequent to the start of the motion has reached a steady periodic variation. Then separating equation (5) into components in phase (real part) and out of phase Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- .- .-.
40、-. “, ,: , . . -. , “ USE OF INDICIAL FUNCTION CONCEPT IN ANALYSIS OF UNSTEADY MOTIONS OF WINGS AND WING-TAIL COMBINATIONS 5 FIGURE 6.-Definition of the function J(T). (imaginary part) with 8, there is obtained Introduce the nonclimcnsional parameters, 2 I, p=-r T number of half M. A. C. lengths tra
41、veled in time 7 k=cvt reduced frequency. In t,erms of these parameters, equation (6) becomes, for nfO 1 1 (8) I.P. ($-)=k CL Gh,k) (18b) Thus, after fixing pl, choosing a (small) value of k, and computing G(L, k), the finite areas corresponding to the terms - s “F, (cp) see, e. g., ref. 10) have poi
42、nted out that the above-mentioned theory overlooks important contributions to the aerodynamic forces and moments which, though still within the first order in fre- quency approximation, arise from time-dependent boundary conditions and must be calculated from unsteady-flow theory. It has been shown
43、by these authors that with proper inclusion in the equations of motion of these coefficients, the deteriora- tion of damping in the short-period mode actually occurring for aircraft flying at speeds near the speed of sound can be successfully predicted. The consequences of the assump- tions involved
44、 in the classical dynamic-stability theory will be more evident from a brief review of the equation of motion and boundary conditions for the single-degree-of- freedom rotary oscillations of a rigid wing flying at constant supersonic speed. At the very outset, the assumption is generally made that t
45、he aerodynamic reactions to the motion of the airframe depend only on the angular position and angular velocity and not upon angular accelerations or higher time derivatives. The equation of motion for the change in pitching moment following a displacement from an equilib- rium position is then writ
46、ten in the form of a power series in LY and dr: It should be rem.embered that for the rotary-oscillation case, the airfoil is subjected to changes in both angle of attack LY and angular velocity q, and that these motions produce normal velocity patterns at the airfoil surface which are different in
47、character. Thus, although for the single-degree- of-freedom case, ti ,-Starting sound waves from leading edge -J starting y.aves :-,:,A A from II r! t X FIGURE 12.-Relation of wing positiou to startiug souud waves for supersonic speed. At t/=0, the starting waves just cover the wing and the loading
48、is uniform as described previously. At t=ti, the starting waves have grown in radius and the wing has begun to emerge from their influence. On that portion of the wing which has emerged, region in figure 12, the loading has already reached its steady-state value. Notice that in this region the characteristic tip Mach cone has already formed. On the portion of the wing uninfluenced by the starting waves from the edges, region in figure 12, the loading is still uniform as at t=
