1、NATIONAL ADVISORY COMM E FOR AERONAUTICS TECHNICAL NOTE 3283 AERODYNAMIC FORCES, MOMENTS, AND STABILITY DERIVATIVES FOR SLENDER BODIES OF GENEFLAL CROSS SECTION By Alvin H. Sacks Ames Aeronautical Laboratory Moffett Field, Calif. Provided by IHSNot for ResaleNo reproduction or networking permitted w
2、ithout license from IHS-,-,-NACA TN 3283 TAB- OF CONTENTS Summary Introduction List of Important Symbols General Analysis Differential Equation and Pressure Relation Total Forces and Moments Reduction of the integrals The complex potential Stability Derivatives Relationships Among the Stability Deri
3、vatives Apparent Mass Applications of the Theory Wings with Thickness and Camber Plane Wing-Body Combination I Wing-Body-Vertical-Fin Combination Concluding Remarks Appendix A: Differentiation of a Contour Integral Appendix B: The Residue A1 of the Complex Potential References Tables Figures Provide
4、d by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE 3283 AEBODYNAMIC FORCES, MOmNTS, AND STABILITY DERIVATIVES FOR SLENDER BODIES OF GENERAL CROSS SECTION By Alvin H. Sacks SUMMARY The problem of determ
5、ining the total forces, moments, and stability derivatives for a slender body performing slow maneuvers in a compressible fluid is treated within the assumptions of slender-body theory. General expressions for the total forces (except drag) and moments are developed in terms of the geometry and moti
6、ons of the airplane, and formulas for the stability derivatives are derived in terms of the mapping functions of the cross sections. All components of the motion are treated simultaneously and second derivatives as well as first are obtained, with respect to both the motion components and their time
7、 rates of change, Coupling of the longri- tudinal and lateral motions is thus automatically included. A number of general relationships among the various stability derivatives are found which are independent of the configuration, so that, at most, only 35 of a total of 325 first and second derivativ
8、es need be calculated directly. Calculations of stability derivatives are carried out for two triangular wings with camber and thickness, one with a blunt trailing edge, and for two wing-body combinations, one having a plane wing and vertical .fin. The influence on the stability derivatives of the s
9、quared terms in the pressure relation is demonstrated, and the apparent mass concept as applied to slender-body theory is discussed at some length in the light of the present analysis. It is shown that the stability derivatives can be calculated by apparent mass although the general expressions for
10、the total forces and moments involve additional terms. INTRODUCTION Ever since R. T. Jones (ref. 1) in 1946 demonstrated the use of Munkls apparent mass concept of 1924 (ref. 2) for solving problems of slender wings in a compressible flow, an ever-increasing number of investigators have entered the
11、field of analysis now commonly known as Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TeJ 3283 slender-body theory. The stability derivatives of slender triangular wings were treated by Ribner (ref. 3) in 1947 following the pattern of Jones, a
12、nd in 1948 Spreiter (ref. 4) extended the latters result by means of conformal mapping to include certain wing-body combinations, Shortly thereafter, in 1949, Wards general analysis for slender pointed bodies in steady supersonic flow (ref. 5) was published. After the appearance of Wards analysis, a
13、 number c;f papers were written on various aspects of slender-body theory including extensions to subsonic flow and to “not-so-slender“ bodies (e.g., refs. 6 and 7), and in 1952 Phythian (ref. 8) developed an analysis zhich included time variations in forward velocity and angles of incidence. Althou
14、gh many papers (e.g., refs. 9 and 10) have been devoted to the calculation of various stability derivatives for specific configurations, it is only in the past few months that a report by Miles (ref. 11) has given the com- plete counterpart of Wards analysis for unsteady flow. The determination of s
15、tability derivatives has long been of concern to the engineer in connection with the dynamic behaior of airplanes, but the problem has assumed even greater proportions in the more recent slender configurations of missile design. The stability derivatives them- selves correspond to the coefficients o
16、f a Taylor expansion representing a particular component of force (say lift) or moment as a function of the airplane motions. The coefficient of any particular motion (say q) in the expansion is equal to the partial derivative of the force or moment component with respect to that motion. Ordinarily,
17、 stability derivatives are defined as these partial derivatives evaluated with all of the independent variables except a set to zero, so that the usual stability derivatives depend upon the initial angle of attack as well as on the configuration. In the present paper, however, all derivatives are ev
18、aluated with all of the independent variables (a, P, p, q, r, B = - - 2n dx . distance from airplane nose to pivot .point complex potential (p + iq length of airplane force in the z direction (approximately lift) rolling moment about the x axis reference length pitching moment about pivot point x =
19、cl yawing moment about pivot point x = cl angular rolling velocity about the x axis pressure angular pitching velocity about the y axis fluid speed relative to axes fixed in the body component of qr normal to body contour in plane x = const. (positive into the fluid) Provided by IHSNot for ResaleNo
20、reproduction or networking permitted without license from IHS-,-,-NACA TN 3283 qs component of qr tangential to body contour in plane x = const. (positive counterclockwise looking upstream) r angular yawing velocity about the z axis ro radius of transformed circle corresponding to airplane cross sec
21、tion S cross-sectional area s r reference area t time uo component of flight velocity along negative x axis o component of flight velocity along positive y axis V V, - r(x-cl) v1 speed of a point fixed in the xyz system of axes ur,vr,wr components of qr in the x,y,z directions W o component of fligh
22、t velocity along positive z axis W wo - q(x-c1) Y force in the y direction (side force) XYZ Cartesian coordinates fixed in the body (x rearward, y to starboard, z upward) a angle of attack (angle between arbitrarily chosen xy plane and flight direction) B angle of sideslip (angle between xz plane an
23、d flight direction) 8 angle between the positive y axis and the tangent to the body contour in a plane x = const. P fluid mass density V outward normal to the body contour in plane x = const. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 32
24、83 (c complex coordinate of centroid of cross-sectional area (YC + izc) u complex coordinate in transformed circle plane 9 velocity potential Ip,“yt partial derivatives of P with respect to x,y,z, and t 929 pt 92 2% velocity potentials for unit velocity of the cross section in the y,z directions 9 v
25、elocity potential associated with variations in shape and size of cross section with x 4f stream function qs stream function along the contour of the cross section Special Notations contour integral taken once round the body cross section in the positive (counterclockwise) sense Force coefficients:
26、Cy = I , etc. ( 1/59 puo2sr Moment coefficients: C, = M etc. (1/2)uosrr Stability derivatives: All derivatives are evaluated at * a=p=p=q=r=a=p=p=q=r=O Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3283 real part imaginary part derivative o
27、f ( ) with respect to time complex conjugate of ( ) GENERAL ANALYSIS The problem to be treated here is the determination of the aero- dynamic forces and moments (except drag) and the stability derivatives for a smooth slender airplane or missile of arbitrary cross section per- forming slow maneuvers
28、 in a compressible fluid. The configuration will be limited in that the base ( if any) of the fuselage and all wing trail- ing edges must lie in a plane essentially normal to the longitudinal body axis. Differential Equation and Pressure Relation The linearized differential equation for the velocity
29、 potential of unsteady motion of a compressible fluid is the well-known wave equation where the system of axes APE is fixed relative to the undisturbed fluid, co is the speed of sound in the undisturbed fluid, and T is time as measured in the kl system. Thus the velocity potential Q is express- ible
30、 as = (A,p,!,r). In general, the pressure relation associated with the velocity potential is given by (ref. 13, p. 19) P,= -m, - 1 q12 + f pld($) + const. P 2 where pl is pressure and ql is the magnitude of the fluid velocity expressible as2 q1 2=c+42+Q2 P 5 (3) 2he subscripts on p and q are used to
31、 distinguish them from the angular velocities of rolling and pitching. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3283 It will be convenient for the present problem to introduce a coordinate system xlyz which is fixed in the airplane. Th
32、e axes chosen for this purpose are shown in the sketch and comprise a Cartesian coordinate system endowed with the translational velocities Uo, Vo, Wo and the rotational velocities p, q, r of the airplane. (ote that this does not constitute a com- pletely right-hand system.) The xl axis passes throu
33、gh the airplane nose, and the ori- gin of the xlyz system is fixed at an arbitrary distance cl from the nose as shown in the sketch. Since it is the purpose of this paper to study only instantaneous forces and moments (i.e., no time histories), it will be sufficient to choose an instant of time such
34、 that the positions of the moving xlyz system and the stationary bk, system are just coincident. Thus, equations (1) and (2) will be expressed in the xlyz system only for this instant, designated T = 0. For this purpose a new function 9 is introduced such that Now, through the use of the transformat
35、ions relating the moving and stationary coordinates (see e.g., ref. 13, p. 12, and ref. 14, p. 79) one finds at T = t = 0 that and It can be seen from the sketch that the quantities (vo + pz - rxl) and (wo - py - qxl) are simply the velocity components in the y and z directions of a point fixed in t
36、he xlyz system. Note that in the corresponding x component (-uo + ry + qz) the products ry and qz are considered negligible compayed with Uo. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3283 With maneuvers , in planes the assumptions of s
37、lender bodies, small angles, and slow the differential equation (1) reduces to Laplaces equation xl = const. near the body; that is It follows also that the density P must be treated as a constant in the pressure relation (2) which now becomes 1 - (my2 + mz2) + const. 2 Here, for the sake of convent
38、ion, one can transfer the origin of the moving axes to the body nose by letting so that pitching and yawing rotations are still made about an arbitrary pivot point x = cl. Thus, introducing the notation equation (6) can be written This, then, is the pressure relation (referred to the moving body axe
39、s) upon which the calculations of the forces and moments will be based. It should be noted that a consistent application of the slenderness approx- imation requires the retention of the squared terms qy2 and PZ2. Thus slender-body theory is not a strictly linear theory although the differ- ential eq
40、uation (5) is certainly linear. This means that solutions of equation (5) for V (and hence the velocities) can be obtained by super- position, but t3e pressures cannot. Likewise, the forces and moments cannot be calculated by superpositioa except for those special cases in which the contribution of
41、the squared terms to the loading vanishes. Furthermore, when the airplane is performing combined maneuvers ( e . g . , simultaneous rolling and pitching), the squared terms may contribute additional forces and moments. These in fact give rise to the second- order stability derivatives that will be i
42、ncluded in the present analysis. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3283 Total Forces and Moments The analysis to be presented here for calculating the total forces (except drag) and moments on a slender configuration will be the
43、 counterpart of a method originally due to H. Blasius (ref. 15) for obtaining the forces and moments on a two-dimensional body of arbitrary shape immersed in a steady incompressible stream. This analysis, although not suited to the calculation of total drag, will nevertheless take proper account of
44、the local forces associated with leading-edge suction. Consider a lamina of the slender air- plane, cut parallel to the yz plane, of thickness dx as shown in the sketch. One can write immediately the differ- ential lift and side force on an ele- mental area in terms of the local pressure pl on the b
45、ody: and Now, by introducing the complex variable ( = y -I- iz, one can express the differential complex force as - where 5 is the complex conjugate of 51. In a similar fashion, the differential rolling moment about the x axis can be expressed as where a denotes the real part. Further, the different
46、ial yawing and pitching moments about the pivot point x = cl are given by Integration of equations (91, (lo), and (11) gives for the total forces and moments - - - 3he method of Blasius has been extended to two-dimensional unsteady incompressible flows by L. M. Milne-Thomson (ref. 16) and recourse w
47、ill be had to many of his techniques throughout the present analysis. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3283 and where the contour integrals are taken round the boundary of the airplane cross section in the positive ( counterclo
48、ckwise) sense. Before these integrations can be effected, the pressure pl must, of course, be expressed as a function of the complex variable 5. Toward this end, it will be convenient to introduce two new definitions pertaining to velocities in the fixed and moving coordinate systems. First, the square of the speed of a point fixed in the xyz system can be written as Second, it is noted that the square of the fluid speed relative to the xyz system is given by so that, neglecting Tx2 in comparison with my2 and TZ2, one can write equation (7) in the form This expression will now
