On 2/13/2018 9:27 AM, Henry Baker wrote:
There's a guy riding a bicycle with square wheels in a Repatha (drug) TV commercial which runs about every 2 minutes on some cable channels. I know this because the TV at my YMCA runs this channel while I'm on the treadmill. (I presume that the commercial runs even when I'm not on the treadmill, as well, although with modern "smart" technology, that is no longer a foregone conclusion.)
https://www.ispot.tv/ad/wuCV/repatha-on-the-right-path
Repatha 'On the Right Path' Commercial
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Thanks to square (Stan) Wagon wheels, we know that square bicycle wheels ride "smoothly" over a road filled with catenary-shaped speed bumps, whose curve length equals one side of the square wheel.
https://my.vanderbilt.edu/stacyfonstad/files/2011/10/squareWheels.pdf
http://www.exploratorium.edu/square_wheels/square_wheels.pdf
But while this ride is "smooth" in the sense that the center-of-gravity of the bike doesn't change height, it isn't "smooth" in the sense that the forward velocity isn't constant, nor is the rotational velocity of the wheels and pedals.
See Figure 4 in Stan's Feb. 1999 article "The Ultimate Flat Tire".
https://www.maa.org/sites/default/files/pdf/upload_library/22/Evans/february...
https://www.maa.org/sites/default/files/pdf/pubs/sampMMA.pdf
http://stanwagon.com/public/squarewheelposter2011.pdf
Q1: Is there a bicycle-cum-road design that provides a constant forward velocity for a square-wheeled bike?
Sure. That's what you would get on the catenary speed bumps. In practice the bicycle+rider are much more massive than the wheels, so as you rode over the bumps you would move smoothly at constant speed and the wheels would slow down and speed up as their "rolling radius" changed. Brent
Q2: Is there a bicycle-cum-road-cum-chainwheel design that provides a constant rotational pedal velocity for the square-wheeled bike? E.g., what if we also used a non-circular chain wheel -- perhaps a square chain wheel?
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BTW, the Antikythera Mechanism (c. 150 B.C. ?) used some pretty sophisticated mechanical linkages -- including gear wheels -- in order to provide the slowing and speeding up of the apparent planetary orbits.
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