A friend of mine recently raised the question why wheels roll easier than, say, cubes. I pointed out that in the non-round variety, the weight at least of the wheel itself would have to be lifted periodically, requiring additional energy. In response, he noted that a wheel with a deflated tire rolls much harder than with a fully inflated one, despite the center of gravity still describing a horizontal line. My explanation would be that the additional force is needed to “deform” the tire material. But is that really all about it, or am I missing something relevant? Wouldn’t that effect be counterbalanced by the tire’s backside reassuming its “natural” shape? And what happens when we abstract from vertical forces (like weight), as far as reasonable?
Could anyone help us put the mystery of wheel-rolling on solid theoretical grounds?
Thanks in advance,
Nicole
That’s about it. A non-round wheel would require force not just to lift the wheel but whatever the wheel is bearing. Of course, it would also come back down again, but the additional energy needed to constantly lift and lower something is huge. Cars have suspensions largely for passenger comfort but the suspension also increases efficiency by keeping the center of gravity relatively stable. Research on competitive runners shows that a gait that produces less up-and-down motion allows runners to conserve a lot of energy. Jockeys use their legs to keep their center of gravity at the same level relative to the ground so the horse doesn’t have to expend extra energy lifting and lowering the jockey.
The reason that properly inflated tires help fuel economy is they reduce the energy needed for deformation/reformation.
To expand on this part, all materials exhibit hysteresis (or, more properly, elastic hysteresis), which is a technical term meaning that the energy required to deform the material is always greater than the energy recovered after deformation–in other words, some energy is always lost in the deformation process. Deflated tires deform more than properly deflated tires. More deformation, more energy required to deform. More energy required to deform, more energy lost to hysteresis.
So even though a tire does “reassume its natural shape” on the rear side, it doesn’t do so with the same energy required to deform it in the first place.
Thanks to both of you. It’s kind of comforting (though also a bit disappointing) that sometimes, things really are as simple as they appear.
It’s just a bit surprising that hysteresis would account for such a great loss of energy. Are there differences between various materials, or would the ratio always be the same? What if, instead of an elastic material, I’d use some kind of magnetic construction?
I don’t know what you mean by that. You mean like a maglev train? Even with a conventional train, the metal wheels don’t deform like pneumatic rubber tires so the main loss there is heat from friction.
No; I was thinking of some quasi-circular arrangement of magnets. Now, my engineering skills are a bit underdeveloped, but if I’m lucky, this might work in theory: you have, say, four bar magnets, stuck on one common axis through their centers (ignore thickness, or bend them a bit), such that their poles create the corners of an octagon, with alternating poles. Now you arrange another eight bar magnets as the sides, with their poles each touching an opposite pole of the radial magnets. Each of the side magnets will now face the same pole of its neighbors on both sides; but supposed that the radial magnets are much stronger, they should stay in place. If I haven’t made a fundamental mistake so far, we let that “wheel” now “roll”, and exert some force downwards. It will certainly deform, like a tire. But would the forces involved in the deforming/reforming be basically the same as in elastic hysteresis?
You have to do extra work to raise the vehicle and get it over the “hump” (when the wheel is at its tallest), but you get all of that energy back when the vehicle settles back down again as the wheel reaches its shortest height. Note, for example, that a football on pavement will roll end-over-end quite nicely (provided it’s not going so fast that it hops). The only irretrievable loss involved is to hysteresis, which is a property of the material of the wheel (and ground); an elliptical steel wheel on a steel rail would be pretty much as efficient as a round wheel, although it will not give a very smooth ride.
A vehicle’s suspension is not intended to (and does not) increase efficiency; it’s intended to keep parts from breaking when the vehicle hits bumps in the road. This could be done without decreasing efficiency, IF the shock absorbers were removed, leaving behind just the springs. The shock absorbers themselves remove energy from the system, making it less efficient: anyone who has raced motocross will tell you that a dirt bike’s shock absorbers are SMOKIN’ hot after a race, which means that they have been dissipating a lot of mechanical energy from the suspension.
While I think I understand what you’re saying, this seems a bit counter-intuitive. Let’s take something other than steel, which is still rather soft. Maybe some super-hard, extremely smooth ceramics, same for the surface, a perfect plain. Remove all air (I like my thought experiments as simple as possible). Now we accelerate both wheels to a reasonable speed, and let them roll free. Intuition (which, of course, isn’t too reliable), would tell me that, while the round wheel could roll on pretty much forever, the elliptical one would come to a halt relatively soon. Is it just my intuition fooling me; am I underestimating the minimized hysteresis, or are there other factors?
OK, assume a zero-hysteresis material for wheel and road, zero aerodynamic drag. You think that round wheel would roll on forever, but the elliptal wheel would come to a stop at some point. This requires that the vehicle’s kinetic energy be pissed away somehow. Since it’s not lost as heat due to material hysteresis, and not lost to aero drag, where would the vehicle’s kinetic energy go?
When the elliptical wheel is at its tallest, the vehicle has high potential energy, and low kinetic energy. As it rolls forward, the wheel’s contact patch moves rearward of the axle, causing an accelerative moment. The vehicle accelerates forward because of this. When the wheel has rolled so that it’s at its lowest height, the vehicle’s potential energy is low, and its kinetic energy is high.
The reverse process happens during the next 90 degrees of wheel rotation, and we again end up with a tall wheel, high potential energy, and low kinetic energy. Kinetic and potential energy keep getting juggled back and forth indefinitely, two cycles per wheel revolution..
Thanks, Joe. Well, for a zero-whatsoever setting, it’s fairly clear. I had been thinking of a nearly-ideal situation, where losses are minimized, but not eliminated. And I think I’ve figured it out: it’s not that the elliptical wheel would lose more energy, but that (given a somewhat realistic weight) the lifting requires a lot more energy than to just keep rolling, i.e. at the point when the elliptical wheel is too slow for the next turn, the round one, while having the same level, still has enough left to really keep rolling a whole lot further. Which would mean that both you and my imagination were correct. How satisfying. Although, again, it’s embarrassingly simple.
I’m pretty sure that for any real material, the elliptical wheel would also lose more energy per revolution. Consider the extreme case, where the “wheel” is effectively just a stick: If you set a stick on end and let it fall over, it’s not going to bounce back up to anywhere near the same height.
I’m pretty sure that’s true to the extend as hysteresis plays in. In an ideal setting, it should keep… uhm, rolling; only that in your example, you have the turning center in the wrong place. It should be in the middle of the stick. Fix it on an axle, rotate it quickly, let loose (a bit above touch-down level). Now it should look a lot more realistic for the stick to perform some acceptable flip-flops. But I think that in practice, a great deal of energy would rapidly be absorbed by both the stick and the ground.