Bike and Catenary

[ValeryOchkov]

Can you show it with Mathcad - the bikeway is parts of catenary!

[Werner_E]

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@ValeryOchkov wrote:

Can you show it with Mathcad - the bikeway is parts of catenary!

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Its more fun to create a wooden model

(see: https://www.macfunktion.ch/mathe/erstaunlich/quadrad/index.shtml)

or https://www.youtube.com/watch?v=QHLFJ2YoKGg

or https://www.youtube.com/watch?v=LgbWu8zJubo

But the ride would be very jerky in horizontal direction if the wheels are turned with constant angular velocity, even though some authors are talking about a "smooth" ride.

[Werner_E]

OK, making a Mathcad animation is fun, too, and as you can see, the horizontal movement is indeed quite jerky.

[ValeryOchkov]

Thanks Werner!

Today I woke up early and went out onto the veranda of my country house. The morning is beautiful, the sun is shining, the flowers are blooming in the garden and fragrant, birds are singing with might and main. I have ambitious plans for this day - to work even with the most interesting STEM-article on the Mathcad and the kinematics of a bicycle. On top of that, I open the Internet and see there an excellent solution to the problem by Werner. There are wonderful moments in life.

For complete happiness, I would like to see the Mathcad 15 sheet!

And What about a bicycle with three-cornered wheels, two-angled wheels (a straight line segment with a hole in the middle), and wheels with the shape of any convex polygon. After all, the wheel is a regular convex polygon with an infinite number of corners!

PS

Ten kilometers from my country house is the place where the great mechanical scientist Tchebyshev was born. Today I will sit on my bike and once again visit this place of pilgrimage for mathematicians.

[Werner_E]

Here's the file you asked for.

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And What about a bicycle with three-cornered wheels, two-angled wheels (a straight line segment with a hole in the middle), and wheels with the shape of any convex polygon.

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The problem with the two- and three-angled wheels is, they both would ruin the carriageway - in other words, it would not work from a mechanical point of view (but you sure could make a Mathcad animation nonetheless). Concerning the  equilateral triangle you can read about the problem here: https://www.walser-h-m.ch/hans/Miniaturen/Q/Quadratisches_Rad/Quadratisches_Rad.htm

But equilateral pentagons, hexagons, etc. are no problem and work quite similar as the square wheel. The carriageway would consist of a series of catenaries, too.

Concerning irregular polygons - I guess they should work, too, as long as the side lengths and angles meet certain conditions (to avoid the problem seen with equilateral triangle).

You could also go the opposite way - decide for a shape the road should have and create the suitable wheel shape. You may settle for a sine curve as the road and you will end up with a concave shaped wheel:

stolen from https://blog.matthen.com/post/4660429794/a-wheel-can-be-any-shape-you-want-it-to-be-if-you#_=_

The whole process reminds me a little bit on creating/constructing gearwheels using involute gearing or cycloid gearing.

So have fun and lets see your pentagon wheels or whatever you decide to create!