Measure angular velocity and acceleration from missing tooth wheel

[nick26]

Hi,
I need to come up with a math model for a digital ignition system. I've been thinking about it and I think that I need to measure 2 things to be able to calculate when I have to start charging the coil. They are the angular velocity and the acceleration but how can I do it? the idea is to use a 36-1 wheel and then measure each tooth periods. If I subtract the period of two teeth will that give me the acceleration?

Thanks!

 

[kuruman]

Subtracting the period of two teeth will give the difference of the periods of these two teeth.

You will need at least two tooth periods, that is measure the period of the same tooth two consecutive times. The average angular speeds for the two periods are $\bar \omega_1=\dfrac{2\pi}{T_1}$ and $\bar \omega_2=\dfrac{2\pi}{T_2}$.

Assuming constant angular acceleration (you need to convince yourself that this is the case in your system - see below how), the instantaneous angular speed is equal to the average at the half-time mark. In other words,

$\omega_1=\dfrac{2\pi}{T_1}$ at $t_1=\dfrac{T_1}{2}$
and
$\omega_2=\dfrac{2\pi}{T_2}$ at $t_2=T_1+\dfrac{T_2}{2}$.

Then the angular acceleration will be given by $\alpha=\dfrac{\omega_2-\omega_1}{t_2-t_1}.$

If you can collect many successive periods, say 10 or so, then you will improve your accuracy because you will be able to plot the $\omega_i$ points vs. the $t_i$ points. You should get a straight line with a slope equal to the angular acceleration. If the line is not straight, then the angular acceleration is not constant.