Tire Overtreads--A Needed Invention
- Thread starter glennmac
- Start date
patrickg,
Nice challenge - will need some diagramming I do believe.
Of course, there can be no discontinuity as you state - the challenge is in clearing up some of the murky language so far used to describe the problem. I'll have a crack at a diagram tomorrow and see where it takes us.
It's always the small, unaccounted for variables that control these problems. They never simplify well - unless you want thrown tracks and snapped belts!! The difference between a bright idea in the bathtub and a 'I didn't expect that' in the field ...
Patrick
Nice challenge - will need some diagramming I do believe.
Of course, there can be no discontinuity as you state - the challenge is in clearing up some of the murky language so far used to describe the problem. I'll have a crack at a diagram tomorrow and see where it takes us.
It's always the small, unaccounted for variables that control these problems. They never simplify well - unless you want thrown tracks and snapped belts!! The difference between a bright idea in the bathtub and a 'I didn't expect that' in the field ...
Patrick
hayden
Veteran Member
Look at this in the time domain. Assume in our example that the drive wheel is rotating at 1 revolution per minute.
Every minute the following happens.
1) 3.14 ft of track travels over the contact area between the drive wheel and track inner surface. (rotating wheel in contact with a surface)
2) 3.14 ft of track travels along the two ft span between the wheels on the inner surface directly over the road surface contact area. (inner track is fixed length without contraction or compression hence movement at one point in contact with drive wheel has corresponding movement of another point anywhere else on same surface)
3) 3.14 ft of track travels in contact with the road (straight section of track is a rectangular solid without compression or expansion, so all points - inner surface and outer surface - are moving in the same straight line at the same speed.
4) 9.42 ft of track travels around the half circle forming the outer track surface on the same radius as the wheel-track contact area. (This is the distance around that path - basic geometry) The track achieves this by expanding on it's outer surfaces as it rounds the wheel.
5) The area where the track contacts the ground causes forward movement. 3.14 ft of track traverses this contact point so vehicle moves 3.14 ft.
6) A drive wheel in direct contact with the ground rotating at the same 1 revolution per minute will also move the vehicle 3.14 ft.
At this point hopefully we agree the tracked vehicle moves at the same speed as the untracked vehicle. If not, let me know which step above is off or missing.
7) If the ground contact point were on either of the two end half circles, 9.42 ft would move in contact with the ground. This is the only way to get speed and leverage the same as a 3 ft wheel.
Conclusion: The track propels the vehicle at the same speed as the wheels alone when the contact point is on the straight part of the track. If the contact point is on the half circle part of the track then propulsion will be equivalent to a wheel with diameter equal to the sum of the drive wheel plus 2x the track thickness.
As for the moment arm, it runs down vertically from the axle to the point on the wheel where the track departs the wheel. These are the end points of the moment arm so it's length is the wheel radius. Why doesn't it extend through the track to the ground? Because the line extending through the track is passing through a solid in flat contact with the ground and it's incapable of pivoting. Same reason the moment arm of a tire doesn't extend through the solid surface of a road. In contrast if you tip the track on end and place the half circle end in contact with the ground, the track is no longer a solid flat on the ground but rather a solid in full contact with the wheel such that it must be pivoting as part of the wheel radius, so the moment arm includes the track thickness in that case.
The discontinuity is in the transition between being on the circumference of a circle to being on it's tangent. When the ground contact is on the tangent, the world looks one way, and when it's on the circumference it looks another way. As the two drive wheels come closer together until they are concentric, the drive contact area makes an institaneous transition from being a tangent to being on the circle. The solid surface that the wheel moment arm is pushing against instantaneously disappears and the moment arm extends the full distance.
Every minute the following happens.
1) 3.14 ft of track travels over the contact area between the drive wheel and track inner surface. (rotating wheel in contact with a surface)
2) 3.14 ft of track travels along the two ft span between the wheels on the inner surface directly over the road surface contact area. (inner track is fixed length without contraction or compression hence movement at one point in contact with drive wheel has corresponding movement of another point anywhere else on same surface)
3) 3.14 ft of track travels in contact with the road (straight section of track is a rectangular solid without compression or expansion, so all points - inner surface and outer surface - are moving in the same straight line at the same speed.
4) 9.42 ft of track travels around the half circle forming the outer track surface on the same radius as the wheel-track contact area. (This is the distance around that path - basic geometry) The track achieves this by expanding on it's outer surfaces as it rounds the wheel.
5) The area where the track contacts the ground causes forward movement. 3.14 ft of track traverses this contact point so vehicle moves 3.14 ft.
6) A drive wheel in direct contact with the ground rotating at the same 1 revolution per minute will also move the vehicle 3.14 ft.
At this point hopefully we agree the tracked vehicle moves at the same speed as the untracked vehicle. If not, let me know which step above is off or missing.
7) If the ground contact point were on either of the two end half circles, 9.42 ft would move in contact with the ground. This is the only way to get speed and leverage the same as a 3 ft wheel.
Conclusion: The track propels the vehicle at the same speed as the wheels alone when the contact point is on the straight part of the track. If the contact point is on the half circle part of the track then propulsion will be equivalent to a wheel with diameter equal to the sum of the drive wheel plus 2x the track thickness.
As for the moment arm, it runs down vertically from the axle to the point on the wheel where the track departs the wheel. These are the end points of the moment arm so it's length is the wheel radius. Why doesn't it extend through the track to the ground? Because the line extending through the track is passing through a solid in flat contact with the ground and it's incapable of pivoting. Same reason the moment arm of a tire doesn't extend through the solid surface of a road. In contrast if you tip the track on end and place the half circle end in contact with the ground, the track is no longer a solid flat on the ground but rather a solid in full contact with the wheel such that it must be pivoting as part of the wheel radius, so the moment arm includes the track thickness in that case.
The discontinuity is in the transition between being on the circumference of a circle to being on it's tangent. When the ground contact is on the tangent, the world looks one way, and when it's on the circumference it looks another way. As the two drive wheels come closer together until they are concentric, the drive contact area makes an institaneous transition from being a tangent to being on the circle. The solid surface that the wheel moment arm is pushing against instantaneously disappears and the moment arm extends the full distance.
jdkid
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Hi ya's
hey i may have nutted something out Hayden ya right it's what i've been trying to say BUT in a patrickg is right too (ya both got to say how so !!!)hayden me and you have been looking at it like a dozer track (hinges in side out ie opens like this < ,while going around a driver.)that is right but if you take a "V" belt (one of them coged ones)the inside changes becouse it contracts on the pulley so the out side runs the same speed all the time but is faster than the inside .but in this case we are talking about tracks so inside speed remanes the same all the time
catch ya
JD Kid
hey i may have nutted something out Hayden ya right it's what i've been trying to say BUT in a patrickg is right too (ya both got to say how so !!!)hayden me and you have been looking at it like a dozer track (hinges in side out ie opens like this < ,while going around a driver.)that is right but if you take a "V" belt (one of them coged ones)the inside changes becouse it contracts on the pulley so the out side runs the same speed all the time but is faster than the inside .but in this case we are talking about tracks so inside speed remanes the same all the time
catch ya
JD Kid
jdkid
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Hi ya's
better way of putting it take 10 foot of flat rubber now cut V's every 2 inchs so there is just a strip holding it all together now make 2track shapes 1 with V's in ,one with V's out the one with V's out is a dozer track the inside stays the same ie10 foot..the one like a belt is 10 foot on the outside ie the in side is less .now if we made both tracks the same inside diamater the belt type would drive the tractor faster as the outside dosenot change but the track would drive the same speed as the driveing wheel .but back to the story, if all drivers were timed with each other it could be done, just that the belt type would in crease ya speed
catch ya
JD Kid
better way of putting it take 10 foot of flat rubber now cut V's every 2 inchs so there is just a strip holding it all together now make 2track shapes 1 with V's in ,one with V's out the one with V's out is a dozer track the inside stays the same ie10 foot..the one like a belt is 10 foot on the outside ie the in side is less .now if we made both tracks the same inside diamater the belt type would drive the tractor faster as the outside dosenot change but the track would drive the same speed as the driveing wheel .but back to the story, if all drivers were timed with each other it could be done, just that the belt type would in crease ya speed
catch ya
JD Kid
hayden
Veteran Member
Yes, you have a very good point. The track, or belt, or whatever can be designed such that the outter surface expands to go around the wheels, or the inner surface could contract to go around the wheel. In all my examples I've assumed the outer surface expands which is the case in drive tracks that I've seen, and I think is the case with a fan belt. In the case of a cogged fan belt it's not clear to me how much the coggs collaps when rounding the wheel verses simply provide an interlocking mechanism. Maybe it's a combination. Either way, at any point in time the distance around the inner path and outer path is the same regardless of HOW the track was formed into that shape. The "do you stretch it or do you compress it" question I think only affects the length of the track if you lay it out flat on the ground. An "expansion" track would lay out flat and measure the length of the inner path, where a "compression" track woudl lay out flat and measure the length of the outer track. With either track design I think the model still holds true and travel speed and leverage is the same.
Either way I think we've been able to show why you can't model a tracked vehicle as a wheeled vehicle with bigger wheels. My description I'm sure is far from a rigorous proof but I think it does answer the questions. Maybe if there's someone with more current and practiced acedemic skills they can help formalize it a bit. I haven't done calculus in close to 25 years and probably couldn't remember how to solve the simplest integral or differential. Also, the circle to tangent transition may be improperly called a discontinuity - I don't know. It clearly is an instantaneous transition point that appears to be fundamental in the mechanics of a track. Which side of that point you are one determines if you have a short moment arm pushing against a solid object, or a longer moment arm pushing against a single point of road contact and causing rolling action.
Anyway - got to get to work now.
Peter
Either way I think we've been able to show why you can't model a tracked vehicle as a wheeled vehicle with bigger wheels. My description I'm sure is far from a rigorous proof but I think it does answer the questions. Maybe if there's someone with more current and practiced acedemic skills they can help formalize it a bit. I haven't done calculus in close to 25 years and probably couldn't remember how to solve the simplest integral or differential. Also, the circle to tangent transition may be improperly called a discontinuity - I don't know. It clearly is an instantaneous transition point that appears to be fundamental in the mechanics of a track. Which side of that point you are one determines if you have a short moment arm pushing against a solid object, or a longer moment arm pushing against a single point of road contact and causing rolling action.
Anyway - got to get to work now.
Peter
jdkid
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Hi ya hayden
i think a fanbelt contracts as most i have seen have a stronger backing also fanbelt cracks on the inside from flexing ,most coged belts ya find on gear that has a pulley running off the back of the belt or a small pulley
i think a belt maybe faster tho and bigger would be faster still .
ok pic this 1 belt and 1 track both haveing 20 foot outside meserment (?)where they would touch the ground .both have 16 foot inside meserment (i don't know how thick etc etc ) now knowing the track expandes to go around wheels 1 full turn of the track = 16 foot (hell i hope i got this right) but cos the belt contracts 1 full turn =20 foot (of travel in both cases) meaning that patrickg's tractor would be faster on belts than ours on tracks
i know a heaps of maths would back this up but i'm a better driver than math's man
catch ya
JD Kid
i think a fanbelt contracts as most i have seen have a stronger backing also fanbelt cracks on the inside from flexing ,most coged belts ya find on gear that has a pulley running off the back of the belt or a small pulley
i think a belt maybe faster tho and bigger would be faster still .
ok pic this 1 belt and 1 track both haveing 20 foot outside meserment (?)where they would touch the ground .both have 16 foot inside meserment (i don't know how thick etc etc ) now knowing the track expandes to go around wheels 1 full turn of the track = 16 foot (hell i hope i got this right) but cos the belt contracts 1 full turn =20 foot (of travel in both cases) meaning that patrickg's tractor would be faster on belts than ours on tracks
i know a heaps of maths would back this up but i'm a better driver than math's man
catch ya
JD Kid
patrickg
Veteran Member
Patrick, I see we are converging. Usually do when everyone is patient and actually considers all points of view.
Now that you accept the NO_DISCONTINUTY concept, the implications of that is that you agree with my basic understanding of how it works pretty much in its entirety with the exception of some area(s)t you have called muddy language. Likely mine, yours, and everyone elses even peripherally involved. Or rephrased less egocentrically, our mutual ongoing understanding is approaching congruity.
Haydn, on the other hand is still out there holding onto the extreme and untenable idea that track thicknes doesn't equate to tire profile/thickness or effect the vehicle speed given the same RPM, assumining I recall his comments without reference to the written word.
I have difficulty with the delayed turnaround. I'm much more used to a fast paced give and take with a chalk or dryboard handy. Then usually diferences are either erased or solidified much more speedily and I come across much more better on a personal level when you can see me (no visible horns, tail, un-forked tongue).
If there are any residual REAL differences in our understanding of the physics, mechanics, or math beyond that masked with imprecise language, please point them out and I will try to agree with you, if able, or try to pose an example or question that I think will illuminate the issue.
If there are none, I'm ready to sign a separate peace with you and proceed to deal with Haydn. Or if we are in basic agreement perhaps you can have an exchange with Haydn, I may not have the requisite background experience to formulate communicative but definitive examples within the envelope of his personal experience base.
Patrick
Now that you accept the NO_DISCONTINUTY concept, the implications of that is that you agree with my basic understanding of how it works pretty much in its entirety with the exception of some area(s)t you have called muddy language. Likely mine, yours, and everyone elses even peripherally involved. Or rephrased less egocentrically, our mutual ongoing understanding is approaching congruity.
Haydn, on the other hand is still out there holding onto the extreme and untenable idea that track thicknes doesn't equate to tire profile/thickness or effect the vehicle speed given the same RPM, assumining I recall his comments without reference to the written word.
I have difficulty with the delayed turnaround. I'm much more used to a fast paced give and take with a chalk or dryboard handy. Then usually diferences are either erased or solidified much more speedily and I come across much more better on a personal level when you can see me (no visible horns, tail, un-forked tongue).
If there are any residual REAL differences in our understanding of the physics, mechanics, or math beyond that masked with imprecise language, please point them out and I will try to agree with you, if able, or try to pose an example or question that I think will illuminate the issue.
If there are none, I'm ready to sign a separate peace with you and proceed to deal with Haydn. Or if we are in basic agreement perhaps you can have an exchange with Haydn, I may not have the requisite background experience to formulate communicative but definitive examples within the envelope of his personal experience base.
Patrick
Assume the model with 2 equal size wheels with the tracks going around them. Now put a third wheel under the track. So the track goes under the front wheel, over the middle wheel, and under the back wheel. The track is now touching the ground only at the tangent points of the front and back wheels. What happens to the speed of the tractor? Is this like Peter's example of the tractor standing on its nose? Does this example affect anyone's viewpoint?
patrickg
Veteran Member
Hi Haydn, Finally got a chance to read your latest. Time domain? You got me all ready for fourier transforms or some sophisticated stuff and then nothing... back to rotating wheels with unstated error introducing assumptions.
OK a wheel rotates once a minute and goes 3.14 ft, assuming the wheel is 1 ft in diameter, which wasn't stated. Statement 1 is true. Statement 2 is conditionally true with the assumption that the track is infintesimally thin (at least much much thinner than the diameter of the wheel so that its thickness, which adds to the wheels radius, is not contributing significantly to the diameter of the wheel.
Not sure what this next part is trying to say...
3) 3.14 ft of track travels in contact with the road (straight section of track is a
rectangular solid without compression or expansion, so all points - inner surface and
outer surface - are moving in the same straight line at the same speed.
Unless this system is "laying rubber" i.e. spinning its wheels uh er ah "track", the part of the track in contact with the ground isn't moving. So, yes the top and bottom surfaces of this stationary track are going equal speeds, zero, the straight line part is mute.
You're really starting to lose me on this next part.
4) 9.42 ft of track travels around the half circle forming the outer track surface on the
same radius as the wheel-track contact area. (This is the distance around that path -
basic geometry) The track achieves this by expanding on it's outer surfaces as it rounds
the wheel.
I don't get the part just before, "The track achieves this by expanding on it's outer surfaces as it rounds
the wheel."
Give me some more detail or an example illustrating this last statement. I'm not sure why this track that was, a moment ago, rectangular solid and apparently infinitely rigid but bendable (strong thin steel band or something) as it could neither be compressed or stretched, suddenly expands on its outer surface as it rounds the wheel. This disagrees with and is inconsistent as regards the "thin" assumption that was required to get past an earlier statement.
OK I'm hanging in there to read, analyze, and comment on the conclusion B U T it is not logically consistent to disagree with the premises and accept the conclusion. If it were a logically constructed argument calling any premise into question, inescapably questions the conclusion. Still, I'll have a look.
OK, had a look, ought to say, "no comment", I'd get into less trouble but lets try this, I don't buy it for now and don't think we were close enough in the earlier stuff (numbered section) that I could have a valid surmise on the conclusion, see paragraph above.
e.g. you said "The discontinuity is in the transition between being on the circumference
of a circle to being on it's tangent."
I don't understand what you mean. By definition all tangents are on the circumference at the point of tangency. If you refer to the flat part of the track on the ground as a tangent to the circle at the bottom of the wheel then I still don't know what you mean by discontinuity or how it figures in.
A. Maybe you could get it through my thick skull faster if you explained to me the difference between your "two wheels running on a track laid out on the ground" thingy (with real world thickness to track N O T thin") and taking the tire off of a wheel cuting it into in one place laying it out on the ground and driving over it with the tireless wheel.
B. I would also like your "take" on the thought experiment where we shorten the wheel base of a thick track using vehicle until the axels merge and we have a wheel with a tire the thickness of the original track. Where is the discontinuity in this? At what point does the behavior "know" to change from the Haydn-Track tracked model to just about everybody in the world's wheel-with-tire model.
If you can explain these last two things, A and B above, in a clear, concise, and communicable manner then I'll probably throw in the towel, and have a rootbeer float in your honor. Probably to the undying (for several seconds) appreciation of the viewers at home.
Patrick
OK a wheel rotates once a minute and goes 3.14 ft, assuming the wheel is 1 ft in diameter, which wasn't stated. Statement 1 is true. Statement 2 is conditionally true with the assumption that the track is infintesimally thin (at least much much thinner than the diameter of the wheel so that its thickness, which adds to the wheels radius, is not contributing significantly to the diameter of the wheel.
Not sure what this next part is trying to say...
3) 3.14 ft of track travels in contact with the road (straight section of track is a
rectangular solid without compression or expansion, so all points - inner surface and
outer surface - are moving in the same straight line at the same speed.
Unless this system is "laying rubber" i.e. spinning its wheels uh er ah "track", the part of the track in contact with the ground isn't moving. So, yes the top and bottom surfaces of this stationary track are going equal speeds, zero, the straight line part is mute.
You're really starting to lose me on this next part.
4) 9.42 ft of track travels around the half circle forming the outer track surface on the
same radius as the wheel-track contact area. (This is the distance around that path -
basic geometry) The track achieves this by expanding on it's outer surfaces as it rounds
the wheel.
I don't get the part just before, "The track achieves this by expanding on it's outer surfaces as it rounds
the wheel."
Give me some more detail or an example illustrating this last statement. I'm not sure why this track that was, a moment ago, rectangular solid and apparently infinitely rigid but bendable (strong thin steel band or something) as it could neither be compressed or stretched, suddenly expands on its outer surface as it rounds the wheel. This disagrees with and is inconsistent as regards the "thin" assumption that was required to get past an earlier statement.
OK I'm hanging in there to read, analyze, and comment on the conclusion B U T it is not logically consistent to disagree with the premises and accept the conclusion. If it were a logically constructed argument calling any premise into question, inescapably questions the conclusion. Still, I'll have a look.
OK, had a look, ought to say, "no comment", I'd get into less trouble but lets try this, I don't buy it for now and don't think we were close enough in the earlier stuff (numbered section) that I could have a valid surmise on the conclusion, see paragraph above.
e.g. you said "The discontinuity is in the transition between being on the circumference
of a circle to being on it's tangent."
I don't understand what you mean. By definition all tangents are on the circumference at the point of tangency. If you refer to the flat part of the track on the ground as a tangent to the circle at the bottom of the wheel then I still don't know what you mean by discontinuity or how it figures in.
A. Maybe you could get it through my thick skull faster if you explained to me the difference between your "two wheels running on a track laid out on the ground" thingy (with real world thickness to track N O T thin") and taking the tire off of a wheel cuting it into in one place laying it out on the ground and driving over it with the tireless wheel.
B. I would also like your "take" on the thought experiment where we shorten the wheel base of a thick track using vehicle until the axels merge and we have a wheel with a tire the thickness of the original track. Where is the discontinuity in this? At what point does the behavior "know" to change from the Haydn-Track tracked model to just about everybody in the world's wheel-with-tire model.
If you can explain these last two things, A and B above, in a clear, concise, and communicable manner then I'll probably throw in the towel, and have a rootbeer float in your honor. Probably to the undying (for several seconds) appreciation of the viewers at home.
Patrick