Chapter 5.2 Coordinated turn

1. Used to find a simpler model between roll and heading/course rate
2. Can use aileron and roll to change the course over ground
3. No acceleration along the body y axis during a coordinated turn
4. Comfortable for passengers.
5. No sideslip is assumed during a coordinated turn
6. It is defined in Beard & McLain, but the assumptions are not described thoroughly Wish to find a connection between the yaw-rate $\dot{\psi }$ and roll- angle $\varphi$.

$${\delta }_a\to \varphi \to \psi$$

Wish to change to aileron to initiate a change in roll that in turn results in a change of heading.

Lift is decomposed into two equations when initiating a turn from level flight.

$$F_{lift}\sin \varphi \cos (\chi -\psi )=\frac{m{v}^2}{R}=mv\omega =m(V_g\cos \gamma )\dot{\chi }$$ $$F_{lift}\cos \varphi =mg\cos \gamma$$

Dividing the two expressions gives:

$$\dot{\chi }=\frac{g}{V_g}\tan \varphi \cos (\chi -\psi )$$

which is the coordinated turn condition in wind.

In the absence of wind we have $Va=Vg$ and $\psi =\chi$ which gives:

$$\dot{\psi }=\frac{g}{V_a}\tan \varphi$$

which is the expression commonly seen in the literature.

Turning radius

The turning radius is given by:

$$R=\frac{V_g\cos \gamma }{\dot{\chi }}=\frac{V_g^2\cos \gamma }{g\tan \varphi \cos (\chi -\psi )}$$

For level flight in the absence of wind we have the standard formula:

$$R=\frac{V_a^2}{g\tan \varphi }$$