Dynamics of Aircraft Flight

5 Movement of a Solid

5.1. Rotation of a solid around a fixed point

Consider a solid (S) mobile around a fixed point O: a point of the solid permanently corresponds with the point O.

Let (O, I₁, J, K₁) be a frame of reference (R₁) rigidly linked to (S) and (O, I, J, K) a frame of reference (R) with respect to which we propose to study the motion of (S).

We can transform (R) into (R1) with three successive rotations:

• – R) (O, I, J, K)→(R׳) (O, U, V, K)by rotation Ψ around
• – (R׳) (O, U, V, K)→(R׳׳) (O, U, V׳, K₁) by rotation θ around
• – (R״) (O, U, V׳, K₁)→(R₁) (O, I₁, J₁, K₁) by rotation φ around (O, K1).

The speed vector of a point of (S) is zero with respect to (R1), and with respect to (R”), it is:

In R’, we must add the drive speed of a point of (R”) with respect to (R’):

Figure 5.1. Rotation of a solid around a fixed point. For a color version of this figure, see www.iste.co.uk/louis/flight.zip

Likewise, we obtain the velocity vector of M with respect to (R):

with:

by letting: ...