## Tyred and motional: what a truck told me about frequency synthesis

Driving down to the seaside with my family this weekend, we passed an old Land Rover that had been 'pimped up' with wide tyres.  Though the wheel arches has been stretched a little, the outsides of the tyres still extended several inches further out than the vehicle body.  That's not permitted in the UK.  It's dangerous for pedestrians or cyclists who might be badly injured by the fast-moving tyre tread "because", as I said to my family, expectant as they were to hear my reasoning, "the top part of the tyre is moving with twice the forward velocity of the vehicle as a whole".

My family were skeptical about this claim, and demanded further explanation.  With calculus off-limits in a car full of artists, and the simultaneous requirement to concentrate on the driving, I think I pulled it off - but I decided not to push my luck by setting any homework questions...

Reflecting on this later, I realized that, with a periodic modulation of the tyre tread, the frequency at which tread blocks are passing a particular point in space is being frequency-modulated, and with a unit modulation index.  The tread blocks in contact with the asphalt aren't moving at all relative to it, while the ones at the topmost part of the tyre are passing at twice the rate that an unrolled piece of tyre tread strapped to the car would do.  The tread blocks travel an approximately cycloidal path; points on the tyre's sidewall that are closer to the axis travel on a smoother curve (a curtate cycloid) that tends towards being a sinewave as you get nearer to the axis.

Does this have a PSoC3 connection?  Well, there's some interest in creating quadrature clock oscillators using the Digital Filter Block.  This will use a well-known technique - see for instance Lyons - but I'm interested in adapting the technique so that the processor can actually run on the varying clock that it creates.  Looking back on my weekend tyre experience, I now see that the equations that fall out of that look very like the parametric equations for a cycloid.  This tells us how to modify the process to correct the modulation non-linearity.

So next time you see a beat-up old truck doing something unusual on the highway, ask yourself - what does this tell me about my project?  Happy treading!  best - Kendall

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