Symmetric Missile Research Articles

Minimum Ballistic Factor Problem of Slender Axial Symmetric Missiles

The problem of determining the geometry of slender, axisymmetric missiles of minimum ballistic factor in hypersonic flow has been solved via the calculus of variations under the assumptions that the flow is Newtonian and the surface averaged skin-friction coefficient is constant. The study has been made for conditions of given length and diameter, given diameter and surfacearea, and given surface area and length. The earlier investigations/sup 8/ where only regular shapes were determined has been extended to cover those class of bodies which consist of regular shapes followed or preceded by zero slope shapes.

Spin-Yaw Lock-In of a Rotationally Symmetric Missile

Spin―yaw lock-in has usually been associated with small aerodynamic asymmetries that produce a trim angle that rotates with the missile. Recent trajectory calculations using wind-tunnel measurements ofthe aerodynamic moment acting on a rotationally symmetric missile predicted spin―yaw lock-in. Thus, an asymmetry-induced trim moment is not necessary for lock-in. For a side moment that varies with the roll angle between the angle-of-attack plane and a fixed plane on the missile, this paper shows that lock-in of the coning motion and the rolling motion can occur. The conditions on the aerodynamic moments for occurrence of lock-in are derived and appropriate stability criteria are obtained. A simple case of a hypersonic finned missile is considered and the existence of lock-in is demonstrated.

Flight Dynamics of Unguided Rockets with Free-Rolling Wrap Around Tail Fins

This paper presents a study of the dynamics of configurations with free rolling wrap around tail fins. The linearized theory of motion of symmetric missiles is extended to these configurations to define a dynamically equivalent fixed-wrap around fin tail rocket. The effect of asymmetries is introduced by forcing functions at body and tail rate frequencies. Finally Magnus effects and conditions at resonance and resonance lock-in are discussed. Nomenclature CD = drag force coefficient CLα = lift force coefficient slope CNα = normal force coefficient slope CNβ = side force coefficient slope CNq, CN = α� pitch yaw force derivatives CNpq, CN = β� p pitch yaw force derivatives q CN�, CN = q

Lateral Motion of Free-Rolling Tail Rockets in Free Flight

A study of the dynamics of free-rolling tail configurations is presented. The linearized theory of symmetric missiles motion is extended to free-rolling tail configurations to define a dynamically equivalent fixed-tail rocket. The asymmetries effect is introduced by small forcing functions at body and tail roll rate frequencies. Finally, Magnus effects and conditions at resonance and resonance lock-in are discussed.

Flight Motion of a Continuously Elastic Finned Missile

Abstract : The motion of elastic finned projectiles has been analyzed by various approximate theories. In this report the exact equations of small amplitude motion are derived for a symmetric missile. The aerodynamic and elastic symmetries are used to allow the use of complex variables to describe the lateral motion in a non-rotating coordinate system. Although the resulting equations are both ordinary and partial differential equations, frequencies and damping rates of free oscillations are obtained from an ordinary differential equation with boundary conditions. Equations for a permanently deformed bent missile are derived, and an ordinary differential equation for the forced motion of a bent missile is obtained. Sample calculations for a finned projectile with a fineness ratio of 20 show resonant motion at the aerodynamic frequency as well as at each elastic frequency. The nonlinear roll moment associated with a bent missile is computed and the location of possible spin-yaw lock-in is determined. The flight motion of an elastic missile is shown to be the sum of two elliptical motions: a low frequency pitching motion and a higher frequency flexing motion. The induced drag coefficients for both motions are computed as functions of the missile's elasticity.

Flight mechanics of an elastic symmetric missile

Flight mechanics of an elastic symmetric missile

Flight Mechanics of an Elastic Symmetric Missile

Abstract : The free-flight motion of an elastic missile is approximated with three bodies connected by two massless elastic cantilever beams. If the mass distribution of the three bodies is 1-2-1, the frequency of the symmetric oscillation of the outer bodies is within 5% of the classical frequency of the oscillation of a free-free beam. A second combined pitching antisymmetric flexing motion can occur with a frequency that is almost twice that of the symmetric flexing motion. As the beam stiffness is reduced, the symmetric flexing motion frequency approaches the rigid body aerodynamic zero-spin frequency, and the flight zero-spin aerodynamic frequency is considerably reduced. Moderate beam damping can cause dynamic instability for spins greater than the zero-spin aerodynamic frequency. Resonance mode amplification can occur when the spin is equal to the zero-spin aerodynamic frequency, but more importantly it can occur when the spin is equal to the two elastic flexing frequencies. Spin-yaw lock-in occurs at the lower elastic frequency.

Some special cases of spin-yaw lock-in

Abstract : A slightly asymmetric missile is a basically symmetric missile with a nonzero pitch moment at zero angle of attack. In flight this moment causes a trim angle that rotates with the missile and has a maximum when the spin is near resonance with the missile's natural pitch frequency. This report considers a roll moment that can be induced by this trim angle and can cause resonant lock- in spin. Simple expressions for this induced rolled moment are given and the existence and stability conditions for equilibrium spin are derived. The special case of the induced roll moment caused by a radial center of mass offset is considered and a number of different types of possible equilibrium spin combinations are shown. The possibility of resonant lock-in spin in the opposite sense to the expected steady-state spin is indicated. It is further shown that there are induced roll moments for which the design steady-state spin will not occur under any launch conditions. If several stable equilibrium spins are possible, the one that occurs in flight can be determined by the orientation of the initial pitch angular velocity. Finally the form of an induced pitch moment is given and its possible effect on the angular motion is discussed.

Symmetric Missile Dynamic Instabilities

_ / Magnus moment V */2PSlVpd / = axial and transverse moments of inertia = amplitudes of the two yaw model arms for a symmetric missile = trim angle magnification factor = reference length = mass -XYZ components of the missile's angular velocity = reference area = dynamic stability factor = 2c0/d0o gyroscopic stability factor = (op) /4C^ = time = magnitude of the missile velocity = missile-fixed axes, the X axis along the longitudinal axis of the missile, positive forward = nonspinning aeroballistic axes = angles of attack and sideslip in the missilefixed system = circular limit motion radius = polar angle of £ = K j / K j ( j = l , 2 ) = value of \j for a linear Magnus moment = the complex angle of attack in the missile-fixed system = 0 4ict = be i9

Angular Motion of Spinning Almost-Symmetric Missiles

Abstract : An almost symmetric missile is a missile whose zero-spin pitch and yaw frequencies are 'nearly' equal. The angular motion of a spinning almost symmetric missile can be described by five rotating modal vectors. Two of these vectors vanish when the frequencies are equal. Good approximations for modal frequencies and amplitudes are derived for the special cases of near zero spin and spins near pitch or yaw resonance as well as for spins away from these special regions.

Bounds for the critical load of certain elastic systems under follower forces

References 1 Murphy, C. H., Free Flight Motion of Symmetric Missiles, Ballistic Research Laboratories, Aberdeen Proving Ground, Md., 1216, AD 442757, July 1963. Nicolaides, J. D., 'Tree Flight Missile Dynamics, University of Notre Dame, unpublished notes. Gradshteyn, I. S. and Ryzhik, I. M., Tables of Integrals, Series and Products, Academic Press, New York, 1965. Ingram, C. W., A Computer Program for Integrating the 6-D of Freedom Equations of Motion of a Symmetrical Missile, University of Notre Dame, unpublished.

Response of a symmetric missile in a spin-varying environment

Response of a symmetric missile in a spin-varying environment

Nonlinear motion of a symmetric missile acted on by a double valued static moment.

The static moment of a symmetric missile is usually assumed to be single-valued although double-valued moments can occur. For example, lee side separation can occur at an angle of attack that is greater than that for reattachment and the moment for angles between these values can be multivalued. If a mathematically simple double-valued static moment consisting of two linear segments and two jump points is assumed (Fig. 1), an exact solution for planar motion and zero aerodynamic damping can be obtained. The effects of nonplanar motions and both linear and nonlinear damping moments are studied by the use of a quasilinear analysis. First regions in an amplitude plane are determined for which the multivalued moment is brought into play as well as regions for which the motion is only affected by the single-valued linear segments of the static moment curve. Next, quasilinear approximations are derived for frequencies and damping rates of the motion caused by this multivalued static moment coupled with one linear and two cubic damping moments. The accuracy of these approximations is determined by comparison with the exact solution for planar motion with no aerodynamic damping and is found to be better than 1% for most cases. The character of the planar singularity for linear damping is investigated and found to be a stable limit cycle.

Angular motion of a re-entering symmetric missile

For the case of cubic static-moment constant air density and no damping, the exact solution for the combined pitching and yawing motion of a symmetric missile can be expressed in terms of elliptic functions. A perturbation method that makes use of this exact solution is then developed to determine the effect of air-density variations and nonlinear aerodynamic Magnus and damping moments when the static moment is strongly nonlinear. Stability boundaries for initial conditions are computed, and the conditions for circular limit motions are derived. These circular limit motions have been experimentally observed and conditions for a possible nonlinear moment expansion have already been derived for a slightly nonlinear static moment by a quasilinear analysis. The various predictions of the approximate theory have been verified by numerical integration of the exact differential equations of motion. The great value of the methods developed lies in their ability to yield quickly the effect of various parameters and initial conditions on missile motion, thereby releasing the designer from the necessity of performing a large number of numerical integrations. The results of the theory, however, are limited to altitudes for which the density variations over a cycle of the motion is small (i.e., less than 10%).

Quasi-linear analysis of the nonlinear motion of a nonspinning symmetric missile

In der Arbeit wird mit Hilfe einer Erweiterung der quasillnearen Theorie vonKrylow undBogoljubow von einem System zweiter Ordnung auf ein solches vierter Ordnung die nichtlineare Schieb- und Giebewegung (Querbewegung) eines symmetrischen Flugkorpers untersucht. Dabei finden statt der linearisierten Frequenzen die exakten Verwendung. Das Verhalten der Losungen wird in der als Pseudo-Phasenebene auftretenden Amplitudenebene studiert. Zu einem gegebenen aerodynamischen Moment konnen damit die Anderungen des Maximums und des Minimums des Anstellwinkelbetrages bestimmt werden. Die Methode ist im besonderen fur den Fall eines nichtlinearen Dampfungsmomentes ausfuhrlich dargestellt.

Steady-State Circular Pitching and Yawing Motion of Symmetric Missiles

Steady-State Circular Pitching and Yawing Motion of Symmetric Missiles

The Prediction of Nonlinear Pitching and Yawing Motion of Symmetric Missiles

The Prediction of Nonlinear Pitching and Yawing Motion of Symmetric Missiles

Effect of Roll on Dynamic Instability of Symmetric Missiles

Effect of Roll on Dynamic Instability of Symmetric Missiles