Linear Aerodynamic Terms

In addition to the terms expressed previously, linear expressions are required for the aerodynamic angles and the total flightspeed.

Angle of Attack

Angle of attack is defined as

\[\alpha\triangleq\arctan\frac{W}{U}\]

which with the small perturbation theory is

\[\alpha=\arctan\frac{W_0+w}{U_0+u}\]

in stability axes, \(W_0=0\)

\[\alpha=\arctan\frac{w}{U_0+u}\]

and since \(w\) is small

\[\alpha\simeq\frac{w}{U_0+u}\]

the perturbational forward speed is much smaller than the trim forward speed and the linear angle of attack is:

(83)\[\alpha=\frac{w}{U_0}\]

Sideslip

Sideslip is defined as

\[\beta\triangleq\arcsin\frac{V}{V_f}\]

where \(V_f=\sqrt{U^2+V^2+W^2}\). Looking at a linear expression for the total flightspeed:

\[\begin{split}\begin{align}V_f&=\sqrt{\left(U_0+u\right)^2+\left(V_0+v\right)^2+\left(W_0+w\right)^2}\\ &= \sqrt{\left(U_0+u\right)^2+v^2+w^2}\end{align}\end{split}\]

the trim \(U_0\) is \(\gg\) all the perturbational terms so

\[V_f\simeq U_0\]

giving the linear sideslip, subject to small \(v\) as

\[\beta=\frac{v}{U_0}\]