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Square-Wheel Bike II

Rides on a flat road
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It is well known [links] that "A square wheel can roll smoothly, keeping the axle moving in a straight line and at a constant velocity, if it travels over evenly spaced bumps of just the right shape. This special shape is called an inverted catenary." The first link shows a Square-Wheel Bike.

My idea, which I am not sure will work, is to design a square-wheel bike that can operate on a flat road. Instead of making the road an inverted catenary, build in a vertically sliding rail that the wheel axle is mounted in. A mechanism powered by the rider (as are the wheels as usual), raises and lowers the axle height in a non-inverted catenary in synchrony with the wheel rotation so that the rider's seat is always a constant height off the ground.

What I am hoping is that the needed gearing and levering action automatically creates a leveraged force or torque on the wheel/road at just the right instant to overcome the difficult bit - getting off a flat side of the wheel and up onto the pointy corner. And I am fondly hoping that the rider will not feel any sudden increase of required pedal torque at this tricky moment, so all feels smooth?

Does it work?

sqeaketh the wheel, Sep 08 2012

Riding_on_Square_Wheels http://www.sciencen...ng_on_Square_Wheels
[sqeaketh the wheel, Sep 08 2012]

Square_wheel_communal_bike Square_20wheel_20communal_20bike
by pocmloc [sqeaketh the wheel, Sep 08 2012]

SquareWheelBikeFail http://www.zefrank....mage:SquareBike.jpg
[sqeaketh the wheel, Sep 08 2012]

Cam driven system https://picasaweb.g...vtA?feat=directlink
drawing with fatal error as pointed out by [spidermother] [pocmloc, Sep 09 2012, last modified Sep 10 2012]

Improved cam drive system https://picasaweb.g...vtA?feat=directlink
Rear wheel now has parallel motion. Both cams are now in the correct place. [pocmloc, Sep 10 2012]

[link]






       [+] Yes it will work but I want to see a drawing of the mechanism please.
pocmloc, Sep 08 2012
  

       Thinking about that. In the meantime, the third link shows an example that did not work out so well.
sqeaketh the wheel, Sep 08 2012
  

       If I had to bet a beer, I would bet this will not work.
sqeaketh the wheel, Sep 08 2012
  

       Are you seeking financial backing?
Lesser Spotted Kiwi, Sep 08 2012
  

       Completely ignoring the question of why you would do this, pneumatic tires would be difficult to design for this, and it would corner horribly.
MechE, Sep 08 2012
  

       I hadn't considered the necessity to turn the bicycle.
sqeaketh the wheel, Sep 08 2012
  

       //Are you seeking financial backing?//
I'm always seeking financial backing. You can send PayPal payments to sqeaketh- at- SqueakyEnterprises-dot-com
sqeaketh the wheel, Sep 08 2012
  

       Wasn't offering. Just curious.
Lesser Spotted Kiwi, Sep 08 2012
  

       I understand the Dead Poets Square Wheel Society are currently investigating the feasibility of building a traveling Square wheel wall of death, in an effort to stop the sport dying in it's tracks
Lesser Spotted Kiwi, Sep 09 2012
  

       //non-inverted catenary// I think the curve is sin(arccos(x)), i.e. a zig-zag of straight line segments (assuming constant forward velocity). It's definitely not a catenary. A square wheel rolling on a catenary makes progressive contact along the length of each side, while your wheel only makes contact at its corners (and briefly along the side). They are not symmetrical.   

       Anyone's geometry/algebra less rusty than mine? This would have been easy once :-/.   

       It should be entirely possible to power the vehicle using the vertical motion alone (without applying turning power to the wheels), by adjusting the phase. That would require a little sensing and (cerebellum or silicon) processing.   

       Aha! That may mean that a square wheel pogo- uni-cycle is possible.
spidermother, Sep 09 2012
  

       is there any reason why roulette wheels should not be fashioned in the same manner?
Lesser Spotted Kiwi, Sep 09 2012
  

       ^friction.   

       The bicycle would have to be preset to the rider's weight where weight = mass(rider) + the downward force of pedalling. You could get it smooth like that, not so much otherwise.
FlyingToaster, Sep 09 2012
  

       I have uploaded a drawing of a version that is actuated by cams. A rotating plate has a square internal track, which a bearing runs around the inside of. The bearing is on the end of a lever. The opposite end of the lever is the hub of the square wheel. The fulcrum is attached to the frame of the bicycle. The bearing has to run round the inside of the square cam otherwise it would follow a track with rounded corners. I think that the system can be scaled, and my drawing shows one wheel with a 4:1 reduction, i.e. the wheel arm of the lever is 4 times the length of the cam arm, and the wheel is correspondingly 4 times the diameter of the path taken by the centre of the bearing as it follows the inside track of the cam.   

       It goes without saying that the cam plate and the wheel have to be geared together 1:1. I have drawn the pedals on the cam plate but of course you would want to have some kind of reduction gearing between them. Imagine an internal hub gear connecting the pedals to the cam plate.   

       I think that a cam system is cheating and I would like to see it done entirely with linkages.   

       I think that the ride on this thing will be perfectly smooth and level with no humps or variations in the effort required to propel it. How could it be otherwise? The load is travelling in a dead straight line. What extra work is the "extra torque" doing at any point in the cycle? I am looking forward to my beer by the way.   

       However the steering is going to be hideous, since the point of contact of the wheel with the ground is going to oscillate forward and backwards relative to the steering axis. This bicycle will not be self-stabilising and will require a firm grip on the handlebars!
pocmloc, Sep 09 2012
  

       I love it! To collect your beer, go to your fridge, take out a beer and say, "This beer's on Sqeaketh the Wheel!"
sqeaketh the wheel, Sep 09 2012
  

       Excellent. Second one's on me.   

       My objection still holds. The bike in your diagram will still move up and down (though less than without the cam arrangement) because the cam and the wheel are not in symmetry. The hub of the road wheel (obviously) traces a circular path relative to the point of contact with the road. Your initial assumption that a catenary is involved is incorrect. Don't get me wrong, I like your idea, I'm just picking on the details because that' what I do best.
spidermother, Sep 10 2012
  

       //the cam and the wheel are not in symmetry// Ooops... correct. I have uploaded a second drawing which corrects the positioning of the cam.   

       Because each end of the lever traces a similar arc, the fulcrum of the lever should remain at a constant distance from the road surface. However, the machine rocks backwards and forwards by the distance between the chord and the tangent of the arc.   

       This could be solved by using a parallel linkage instead of a straight pivoted beam. This means that the cam and the wheel are no longer a fixed distance apart, so a chain cannot be used to gear them together. I have suggested a shaft drive with sliding link in the middle.
pocmloc, Sep 10 2012
  

       //traces a circular path relative to the point of contact with the road// Hmm, so it is a series of arcs, with the points pointing down, the wavelength w equal to the side of the square wheel, the arc radius =w/sqrt2, and the magnitude from top of arc to bottom of point =w/sqrt2-w/2.   

       Is it even possible to get that pointed shape using only linkages?
pocmloc, Sep 10 2012
  

       Excellent drawings [pocmloc]!!!
scad mientist, Sep 10 2012
  

       [pocmloc] Although the drawing is great, I'm not sure you've got all the mechanism just right yet. As [spidermother] pointed out, the hub traces a series of half circles. In order to maintain a constant horizontal velocity, the wheel hub would have to have an infinite vertical velocity as the flat spot in the wheel hits the ground, and then switch to an infinite velocity in the opposite direction. I can't tell for sure if your rear linkages might be doing that already, but your front wheel definitely doesn't account for this. You could use a second cam for each wheel to control the horizontal displacement of each wheel relative to the frame to allow the wheel to momentarily stop as it passes this discontinuity. I agree that linkages would be more elegant than cams, but I'm not sure if it could be done.   

       I was going to say that using a cam to control horizontal position would allow keeping the contact patch centered under the handle bar pivot, but to do that, the wheel would need to have 0 rotational velocity most of the time, and infinite rotational velocity as it flipped the square from one side to the next.   

       I think you'll need a good compromise to get a smooth ride. To avoid a bump as the flat part of he wheel hits the ground, you'll need the angular velocity of the wheel to reach 0 at least momentarily. I guess the way to analyze this is to look at the 3rd derivative: If you plot the x, and y position of every moving part of the bike vs. time, then plot the velocity, acceleration, and change in acceleration (3rd derivative) for each of these, as long as none of those lines is discontinuous, the ride ought to be smooth. Theoretically you could have a smooth ride if there was a sudden change in the acceleration of some object, but since nothing in a mechanical system is perfectly rigid, gradual changes in acceleration of all moving parts is probably needed to ensure a smooth ride and to reduce excess wear.   

       Once you've got a smooth ride, I'm not entirely sure that the moving contact patch would hurt steering that badly except in very tight slow situations. When traveling with reasonable speed, when turning with a constant radius, the force is always straight down relative to the bicycle. With the front wheel turned at a slight angle and the bike leaning, the line tangent to the front wheel and the road surface is sloping "uphill" in relation to the bicycle frame of reference, so as the contact patch moves forwards and backwards, the front of the bike will bump "up and down" (bicycle frame of reference). The force will also be moving side to side relative to the centerline of the bike, which could make it feel off balance, and cause the rider to naturally wobble the steering in an attempt to compensate, but I don’t think there will be a significant twisting force from the bicycle.
scad mientist, Sep 10 2012
  

       Sorry, I goofed in the statement above. I was thinking the path of the hub was half-circles for some reason. It would only be quarter circles, so there would be no velocities approaching infinity. I think we would still want the angular velocity of the wheels to hit 0 momentarily to avoid having a sudden 90 degree change in the velocity at that point.
scad mientist, Sep 10 2012
  

       For other shapes, a Reuleaux triangle might be fairly simple to implement. It shouldn't be too hard to rig it up so you have to pedal backwards to go forwards.
scad mientist, Sep 10 2012
  

       I think a friend had a toy like that, but I only played with it a couple times, so I don't remember it well.
scad mientist, Sep 10 2012
  

       This does sound a lot like re-inventing the wheel.   

       Perhaps the design could be initially analysed in terms of a regular polygon where the number of sides tends to infinity, and the length of each side tends towards zero ?
8th of 7, Sep 10 2012
  

       //that toy car with interchangeable cams// Yes, I had one, and I got some thick plastic and made a series of homemade cams to get strange snaking paths!
pocmloc, Sep 11 2012
  
      
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