# Angular Velocity of a point on a deflated tire (gyroscope measurement)

Before I do the experiment, I am a bit confused about the expected signals provided by a 3-axis gyroscope (angular velocities vs. time) mounted on the edge of a (slightly) deflated car tire when the car is moving.

Consider a local coordinate system xyz

If the tire was rigid then you expect a constant measurement of

\begin{aligned} \omega_x & = 0 \\ \omega_y & =0 \\ \omega_z & =\Omega \end{aligned}

where $$\Omega$$ is the rotational speed of the tire. So the ideal orientation angle is $$\theta_z = \Omega t$$.

Now the fact that is is a pneumatic tire (I am not sure if underinflation makes a difference here) means that when the gyro goes under the contact orients itself with the ground for a short period of time.

If this process was instant, and the tire was perfectly flexible the orientation and rotational speed signals should look like this:

Rotationally speaking, the gyro is going to have a spike in speed going into the patch zone as well as leaving the patch zone. Within the spike zone the speed should near zero as it conforms to the road.

For an underinflated tire, you can expect the duration of such event to be longer.