Missile equations

Missiles have simple MOI tensors. Because of symmetry, the body axes coincide with the principal axes, and the tensor exhibits only diagonal elements. A particular important case is the missile with tetrago­nal symmetry, which manifests two equal principal MOI. (We exclude here the cruise missiles with their aircraft-like symmetries and attribute them to the aircraft equations in the next section.)

Equation (10.10) with a diagonal MOI tensor [IB]B is in component form

dp/dt

В

‘A-1

0

0 "

В

/

0

—r

9

dq/dt

=

0

4X

0

r

0

-P

dr/dt

0

0

4

-q

P

0

Подпись: /1 0 0" В p В mBl B X 0 h 0 9 + mBl 0 0 /3 r mB}_ / (10.15)

and it is easily expressed in three scalar equations:

~ = A-1 [(A ~ h)qr + mBl]

= 72_1[(7з – h)pr +mBl] (10.16)

^ = A_‘[(A – h)pq+mB3]

These first-order differential equations are nonlinear and coupled only through the aerodynamic moments mд, mj2, and mB, to the translational equations. Their integration yields the body rates p, q, and r. Let me point out again that the term body rates in flight mechanics refers to the angular velocity of the body wrt the inertial frame expressed in body axes, in short [шВЕ]в — [p q r].

Missile equations

Many missiles exhibit tetragonal symmetry, i. e., /3 = /2, and therefore possess rotational equations that are even simpler:

We are now in a position to summarize the equations that form the core of a missile’s six-DoF simulation. They are displayed in Fig. 10.1. The translational degrees of freedom, represented by the velocity components u, v, and w, are solved by Newton’s equation (10.3); and the rotational DoF, expressed in body rates p, q, and r, are governed by Euler’s equation (10.16). The kinematic equations block calculates body attitudes in the form of Euler angles ф, 6, and ф and the direction cosine matrix [T]BL from body rates using the quaternion methodology [Eqs. (10.11), (10.13), and (10.14)]. For aerodynamic table look-ups the incidence angles a1 and ф’, peculiar to missiles, are displayed [see Eqs. (3.22) and (3.23)]. Finally, the aerodynamic coefficients and propulsive forces are combined in the forces and moments block. Notice the special missile features of expressing the force coefficients as Сд, Су, and Сц in body axes. Refer to Sec. 10.2.1.3, in which I treat the modeling of missile aerodynamics in greater depth.

As I mentioned, these blocks form just the basic loop of a six-DoF simulation. In a full-up missile model they are joined by an autopilot and actuator to form the inner loop. An outer loop, supplying the navigation and guidance functions, completes the missile simulation. Modeling details of these additional components will be presented in Secs. 10.2.2, 10.2.4, and 10.2.5.