Thanks, David! Jim Propp On Thursday, July 10, 2014, David Wilson <davidwwilson@comcast.net> wrote:
Assume
- The rim of the wheel is a smooth closed curve without angles or cusps. - The road bed is a smooth curve. - The wheel rotates at constant angular velocity A about a fixed axle point O interior to the rim. - The axle point moves at a constant speed S in a straight line.
Let r be the distance from the axle point to the nearest rim point p. Let R be the distance from the axle point to the farthest rim point P.
Since the rim is smooth, a circle of radius r centered at O is tangent to the rim at p. As the axle point passes directly over p, there is no slippage, so its speed is S = Ad. Likewise, as the axle point passes over p, its speed is S = AD. Hence d = D and the curve is a circle.
This would be the simple case, I assume it can be generalized to other curves by limiting a smooth rimmed curve to the unsmooth rim in question. Hence, I suspect that if any gear actually rolls with constant linear and angular velocity at its axis point, it is due to slippage.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com <javascript:;>] On Behalf Of Dan Asimov Sent: Thursday, July 10, 2014 11:18 AM To: math-fun Subject: Re: [math-fun] Constant velocity implies round wheels?
P.S. Apparently Euler discovered that shaping each side of a gear tooth as the involute of a circle leads to exact constant rotary motion transmitted between circular gears -- and this is still the main design of circular gears today.
It *appears* that circular-to-linear motion, as transmitted by what's called rack-and-pinion gears (one circular, one straight), each using the Euler design, also maintains the same constant speed as a limit of the circular- circular case as one radius -> oo. But I haven't found anything authoritative that asserts this convincingly.
--Dan
On Jul 10, 2014, at 7:18 AM, Dan Asimov <dasimov@earthlink.net <javascript:;>> wrote:
Clearly a tricycle with gearlike wheels riding on a geared line would come pretty close. The theory of gears probably answers whether perfect linear motion can be obtained this way.
--Dan
On Jul 10, 2014, at 4:19 AM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
This question (in a square-wheeled-tricycle vein) is probably easy, but I haven't had any coffee yet so it's not easy for me!
If you want a tricycle with wheels of some shape, riding on a terrain of some shape, so that turning the pedals with constant angular velocity (relative to the pedal-axle) imparts constant linear velocity to the rider, must the wheels be round and the terrain be flat?
On a square-wheeled tricycle, the forward speed of the tricycle varies; this is one reason why the ride does not feel as smooth as one might like.
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