https://ntrs.nasa.gov/search.jsp?R=19930084194 2019-02-15T09:01:28+00:00Z /%#c/? Td . . . 33m NATIONALADVISORYCOMMITTEE FORAERONAUTICS TEcHmCAL NOTE 3350 LOAN COPY: I?ETIIRN pf~f~ (WLL.-I KIRTLAND AFB, N MEX THE LINEARIZED EQUATIONS OF MOTION UNDERLYING THE DYNAMIC3 STABI~Y OF AIRCRAFT, SPINNING PROJECTILES, AND SYMMETRICAL MISSILES By A. C. Charters Ames Aeronautical Laboratory MoffettField, Calif. Washington J~wy 1955 TECHLIBRARYKAFB,NM I;lll[llull[ c NATIONAL ADVISORY COMMITTEE FOR AERONAUTIC: DObb07g TECHNICAL Nm 3350 TEE LINEARIZED EQJATIONS OF MOTION UNDERLYING THE DYNAMIC STABILITY OF AIRCRAFT, SPINNING I?ROJECTILES, AND SmmlzlmIc!m mssms By A. C. Chsrters i3uMM4RY Linearized equations of moti.ohare derived for both conventional aircraft having mirror symnetry and spinning projectiles or missiles having rotational and mirror syrmnetry. The aerodynamic coefficients are introduced as a formal series expasion in the customary variables, Wth additional terms being included to account for the aerod~amic effects of spin. The requirements of symmetry are used to reduce the system of aerodynamic coefficientsand, in the projectile or missile ca6e, to clarify markedly the geometry. A common mathematical approach and staadard NACA nomenclature are used throughout. The equations for aircraft are compared with those for missiles and shortcomings in the currently accepted theories are pointed out. The dynamic-stabilityrequirements for spin-stabilizedprojectiles are dis- cussed briefly. The remits are applied to the analysis of flight-test data from the aerodynamics range. Relations are derived between the aerodynamic coeffi- cients and the const=ts of the equations of motion. A comparison is made with ballistic theory in current use and iE found to be satisfactory. INTRODUCTION The dynamic stability of aircraft is a subject that has been explored at great length. The theory of the motion of spinning projectiles has also been studied extensively. No further development of either sub~ect is needed per Be, at least in .sofar as first-order effects are concerned. On the other hand, mcdern trends in aeronautics and ballistics have brought the flight performance and even the physical appearance of aircraft and projectiles closer and closer together. Hence, a need now exists for a 2 NACA TN 3350 theory of motion that covers both cases simultaneouslywith a 6ingle nomen- ‘ clature and a common mathematical development. ~ In the present analysis, aircraft are distinguishedfrom projectiles and symmetricalmissiles as follows: 1. Aircraft are asmmed (a) to have a plane of mirror symmetry through the longitudinalaxis and (b) to fly only slightly disturbed from a steady-state equilibriumattitude so that all components of the angular velocity of the aircraft are small. Q Projectiles and symmetricalmissiles are assumed (a) to have not only a’~lane of mirror symnetry but also 90°rotational symmetry (or its equivalent)and (b) to fly similarly to airvaft except that the axial component of the angular velocity, the spin, may be large (with the restriction that the change in the spin must be 6mall). Despite these differences, the flight of both aircraft and projec- tiles takes place under closely similar circumstances. The analysis of the motion in each case is the classical analy8ie of a rigid body moving under the action of external forces. The differential equations of motion are derived in both cases from the vector equations relating the rates Of change of the linear and angular momenta to the external force and moment. : Furthermore, the conditions of flight prescribed are the “same,namely, both must fly in nesrly a straight line at nearly constant velocity and the inclination to the flight path must be 6ma11. The aerodynamic-force ?. system postulated is the same: The aerodynamic forces and moments are assumed to be linear functions of the velocity and the angular velocity. In both theories the equations of motion are linearized by the neglect of second-orderterms. One might well suppose that the dynamic stability of aircraft and projectiles had been treated by a common development. Such is far from the case. The dynamic stability of aircraft was first a.mlyzed by Manchester around 1900; the correspondinganalyais of the d~amic stability of spin- stabilized projectiles was made by Fowler and his’aflsociateain 1920 (see refs. 1 and2). Both men had very practical objectives in mind. Manchester was interested in the flying qualities of airplanes. Fowler wi8hed to find the design criteria for an artillery shell that would inmre a true flight to the target and a strike head on. Both attacked the problem inde- pendently and their divergence of interest is strongly reflected in the two developments of the theory, a divergencewhich has continued for all practical purposes to.the present time. A casual observer reading the literature on dynamic stability would ‘k be left with the impression that balli6tici&ns and aeronautical engineers were concernedwith entirely different problems. The nomenclature is different; the geometry does not appear at first glance to be related; the 5’ mathematical treatments differ radically. The casual observert~ judgment NACA TN 3350 3 M would not be so superficial after all, for it is only fair to say that the theory developed to.handle the dynamic stability of aircraft is not i’ adequate in its present form to predict the dynamic stability of spinning projectiles, and vice versa. For example, the aircraft equations do not describe the gyroEcopic nutation and precession of a spinning shell, and the projectile equations do not describe the phugoid oscillation of air- craft. Recently, R. E. Bolz and J. D. Nicolaidea have derived the dynamic stability of spinning projectiles and ~ymmetrical missiles in terms famil- iar to the aerodynamicist (see refs. 3 and 4). Although their derivations go a long ways toward joining the theories of Manchester and Fowler, both omit the force of gravity and contain certain other restrictionswith the result that a gap as yet remains between the case of the aircraft, flying with its weight balanced by lift, and the case of the gyroscopically stabilized projectile, flying with varying velocity and gpin. It is the purpose of this paper to bridge the gap, that is, to treat the two cases with strictly similar mathematical developments and with a common nomen- cIature. In addition, it is desired to apply the results of this theory to the analysis of experimental data obtained from a relatively new flight- iest facility, the aerodynamics range. The development of the theory presented herein follows closely the customary treatment of the dynamic stability of aircraft with the primary difference being the simultaneous treatment of longitudinal and lateral stability. The nomenclature conforms throughout to NACA standards. The other departures of importance from conventional aircraft theory are a rather’formal development of the aerodynamic force system and the thorough use of the coriditionof symmetry assumed for the body, as is done in the ballistic theory. In fact, it might be said that the present development is a welding together of these aspects of the conventional aerodynamic and ballistic theories which, in the author!s opinion, represent the most effective means of attacking the problem. SYMBOIS ao, al, constsmts in the x(t) equation a2, as } A any quantity (This symbol is usedin the development of general transforma- tions.) Al, A2 constants in the ~(t+) equation ho, bl constants in the A(x+) equation b2, b~ } 4 NACA TN 3350 . co, cl, constant6 in the ~(ti) equation C29 Cs } ‘“r’ force moment c aerodynamic coefficients: (p/2)V=%‘ (p/2)v%l c~ coefficient of aerodynamic asymmetry force CM coefficient of aerodynamic asymmetry moment do, dl, d2, da, constants in the 5(x+) equation d4 I U D operator?DA == ID operator, m=* Cm + CFa f. 2 f0, fl, f2) f= constant6 in the 7(x+) equation } F force, external, acting on b~y acceleration of gravity g H angular momentum of body i unit vector along X axis i d= I moment of inertia of body 3 unit vector along Y axia product of inertia of body Jxz k unit vector along Z axis “d . KX2 NACA TN 3350 Iy d Ky2 mZ2 4 Iz KZ2 m22 K1 Jxz % T z characteristiclength % c Zr Zr Zp ‘i z~ m masa Of body % mq 6 NACA TN 3350 * % Cm. a m. a 4Ky2 mq + X. + iv(K - mrp) Cmrp %p 8PKY2 v . %P % ‘6P 8pKy2 M linear momentum of body M moment; external acting on body ‘P 4K7 CnT 4KZ2 NACA TN 3350 % 4Kz2 C%p @K~ X ccmponent of = variation in p over measured trajectory Po constant component of angular velocity about which the angular velocity in flight varies (p = PO + p~) P(s) polynomial factor in constants dl, d~ of ~(x+) equation -. . !I Y component of U t Q(S) polynomial factor in constants f2) f3 of 7(x+) equation r Z component of E R(S) stability quartic F v’ 0 stability factor, — % s arc length along trajectory s characteristic area s independent variable in stability quadratic .. Sl> S2 roots of stability quartic, s1=a2, &=q . t time NACA m 3350 . t+ ‘2 Tl, T= u variation in velocity along trajectory v velocity of center of gravity of body v~ constant component of velocity about which the flight velocity varies (V = V. + u) angular velocity of body with respect to XYZ axes distance along ~ axis . . mg sin 70 pSV02 Cxo % -r % x. a %+ Xq %. ‘oD T c% %3 -E- c NACA TN 3372 9 . Cx PP ‘PP 8P2 %’ x coordinate axis x~ space-fixed coordinate axis Y distance along Y. axis Y+ Cy P Yp T ‘i . . Yp i Yr ‘$L Yap Cy. O,p Y&p @- cYqy Yqp 8f Y coordinate axis . To space-ftied coordinate axis z distance along & axis z+