# How I determine numerically the roll rate of an aircraft?

I'm a Spanish student of aerospace engineering and I'm doing my final bachelor work that consists in designing an unlimited aerobatic plane, and I wonder if it is possible to calculate an approximation of the roll rate depending on design parameters. ( I have aerodynamic analysis of the wing, CL, CD, MAC, W, ...)

And another question about it, how exactly the trailing edge sweep angle affects the stability? And some bibliography?

• Welcome to aviation.SE! – Pondlife Apr 27 '18 at 15:10
• Welcome to Aviation.SE. I'm not sure, but I feel that the second question is not completely related to the main one, so it might be better to ask it with a separate post. – Federico Apr 27 '18 at 15:39
• If you're in the ETSIA, you should go to the hangar building and ask whoever taught you Flight Mechanics I & II :P – AEhere Apr 28 '18 at 10:41

In order to determine the roll rate of an aircraft, all you need to do is to set the rolling moment from aileron deflection equal to the roll damping of the wing (actually, of the whole aircraft, but the wing dominates roll damping anyways). So you set $$c_{l\xi}\cdot\frac{\xi_{left}-\xi_{right}}{2} = -c_{lp}\cdot p = -c_{lp}\cdot\frac{\omega_x\cdot b}{2\cdot v_{\infty}}$$ $$\Rightarrow\omega_x = -\frac{2\cdot v_{\infty}}{b}\cdot\frac{c_{l\xi}}{c_{lp}}\cdot\frac{\xi_{left}-\xi_{right}}{2}$$

The symbols are:
$\kern{5mm} \xi\:\:\:\:\:$ aileron deflection angle
$\kern{5mm} v_{\infty}\:\:$ velocity
$\kern{5mm} b\:\:\:\;$ wing span
$\kern{5mm} c_{lp} \:\:$ coefficient of roll damping, using $\frac{b}{2}$ for its reference length
$\kern{5mm} c_{l\xi} \:\;$ coefficient of rolling moment due to aileron deflection
$\kern{5mm} p \:\:\:\;$ dimensionless roll rate
$\kern{5mm} \omega_x \:\:$ rolling speed in rad/sec

Damping decreases with altitude, and in this equation the increase of flight speed with altitude will make sure that roll damping decreases accordingly.

• Hi, I guess the question is also related to the geometric design parameters involved in roll rate, although the criteria you give definitely helps decide the required roll rate. Maybe, the answer can be expanded to include maneuver speed, max roll rate attainable, time to bank, and their relationship with the geometry and location of the ailerons, and maybe aileron tabs. – Gürkan Çetin Apr 27 '18 at 16:52
• @GürkanÇetin The geometric design parameters are reflected in the coefficients. By the way, the answer does give the max roll rate attainable. – Peter Kämpf Apr 28 '18 at 6:50