Perturbation Equations of Agile Missile

Agile missiles execute severe maneuvers for target intercepts. Seldom do they experience steady flight conditions. To study their flight dynamics, the perturbation equations of unsteady flight are derived, and the aerodynamic forces and moments expanded into derivatives. The second-order derivatives disclose the strong cou­pling between the yaw, pitch, and roll channels.

A six-DoF air-to-air missile simulation, stripped of the flight control system, is used to investigate the aerodynamic and inertial couplings. We will see that the aerodynamically induced rolling moment couples into the pitch channel during a yaw maneuver, and the yaw channel is excited by rolling motions in the presence of a pitch maneuver.

7.5.2.1 Equations Of motion. Of particular interest is the intercept trajec­tory of a short-range air-to-air missile. Its speed is constantly changing during thrusting, and it is turning in response to maneuvering targets. This reference tra­jectory is characterized by the linear velocity components ur, vr. wr and angular velocity components pr, qr. rr in body coordinates. The perturbation variables consist of the linear velocities u, v. w and angular velocities p, q,r.

We start with the general perturbation equations of unsteady flight, derived earlier as Eqs. (7.38) and (7.39) and repeated here for convenience. The first equation governs the translational and the second the rotational perturbations:

Bp

+ m[Q. MBr[sv, BBp + m[sQBI]BP[vIBr]Br

= [efoBp + ШВр + ([T]BpBr – [E])[T]Br! lfgrY

d є со1 dt
KT"

[єта]Вр + [єт,]Вр

These are two sets of first-order differential equations, each having three state variables: u, v, w modeling the translational and p, q,r the rotational degrees of freedom. The body rates p, q,r couple into the translational equation by the term т[є£2ВІ]Вр[vBr]Br of the attitude equation, and the aerodynamic forces and moments impart additional cross coupling.

Unsteady reference flights are characterized by curved trajectories, i. e., their ref­erence angular velocity [шВг,]Вг is nonzero. Three terms of the perturbation equa­tions contain unsteady terms. The translational equation is affected by m[Q. Br’Br єу’вBr, a centrifugal force, and the attitude equations by the two terms

We are surprised by the roll rate p appearing in four equations. It couples through wr and ry into the translational equations and through r, and q, into the attitude equations.

It is a well-known phenomenon in missile dynamics that once the missile begins to roll, the yawing and pitching channels are adversely affected. Yet, what causes the missile to roll? We have to look for aerodynamic phenomena that induce a rolling moment on the missile.