[math-fun] Constant velocity implies round wheels?
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. Jim Propp
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> 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.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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> 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> 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.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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] 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> 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> 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.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Based on what I read, the Euler design of using a circle-involute for the sides of circular gear teeth does not involve any slippage. Don't know if that generalizes to rack & pinion gears (one round, one straight). --Dan On Jul 10, 2014, at 4:56 PM, David Wilson <davidwwilson@comcast.net> wrote:
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.
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.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Arguably, the ride is bumpy, but how do Reuleaux triangular wheels (with central axle) on a flat surface do in regard to smooth velocity?
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp Sent: Thursday, July 10, 2014 7:19 AM To: math-fun Subject: [math-fun] Constant velocity implies round wheels?
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.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Dan Asimov -
David Wilson -
James Propp