ORIENTATION OF THE AIRPLANE
We now need a set of differential equations from which the Euler angles can be calculated. These are obtained as follows: Let (i, j, k) be unit vectors, with subscripts 1, 2, 3 denoting directions (jc,, jq, z,), and so on of Fig. 4.2.
Let the airplane experience, in time At, an infinitesimal rotation from the position defined by ‘P, 0, Ф to that corresponding to (Ф + ДФ), (0 + Д0), (Ф + ДФ). The vector representing this rotation is approximately
Дп = i3 ДФ + j2 Д0 + ki ДФ
and the angular velocity is exactly
We begin with the force equation of (4.2,15):
fE = mVf (4.5,1)
Both vectors in (4.5,1) are now expressed in FB components; thus:
UJ„ = m 4 (LBBVf) = m{tEBWEB + Y,:J’n) (4.5,2)
at
The derivative of the transformation matrix is obtained from (A.4) as
= L eb&b (45,3)
With (4.5,3), (4.5,2) becomes
LBBfB = mil, EgtO/jV B + LEBVB)
Now premultiply by LB/: to get
fB = m(Vf + шв%) (4.5,4)
A similar procedure applied to the moment equation of (4.2,15) leads to
GB hB + <wBhB (4.5,5)
The force vector f is the sum of the aerodynamic force A and the gravitational force mg, that is,
f = mg + A (4.5,6)
where
AB = [X Y Z]T
and
mgB = mhBEgE = mLB/. 0 0 gf (4.5,7)
We denote VB = [u’ vE vva 17 and use (4.3,26) for wB. Equations (4.5,4 and 5) are then expanded using (4.3,4) to yield the desired equations. In doing so we note that

Leave a reply