# Linear Aerodynamic Terms

## Contents

# 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**:

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

## 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

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