Why the Wheel Is Round

Muscles, Technology, and How We Make Things Move

The University of Chicago Press

Contents

Circling Bodies

First — don't be shy — try a few motions with your own body. Twist an extended arm as far as you can one way and then twist it the other way. Your wrist (mainly) can't even do a full 360-degree rotation. Twist your neck — your head won't rotate even as far as your hand did. Your lower back's mobility limits how far your torso can rotate just as severely, and feet (mainly ankles) feel still greater rotational constraint. All sorts of limbering and muscle-strengthening exercises depend on rotation — "curls" just put the matter more explicitly. After all, appendages hook on to us at pivot points around which they swing. But they swing through limited arcs, with varying degrees of constraint. Thus arms move around shoulders more freely; legs around hips less so, with flexibility evidently traded against stability and reliability. No picture need be provided; doing it yourself should be persuasive.

Continuous rotation, as with a proper wheel? For better or worse, no animal joint has ever managed that trick. Yes, we humans can rotate continuously — but only if we do it as a whole-body activity — as do somersaulting or rolling children. Almost all other creatures that rotate live within that general limitation as well. We're looking at tumbleweeds, a shrimp that rolls back to the water when washed up on a beach, a caterpillar that rotates head over heels, so to speak, and the helicopter-like seeds (really fruits, technically samaras) of trees such as maples. More about these systems in a few pages.

Then look around. Sure, we've created a host of devices that may turn but also face (by design) much the same limitation on rotation — most hinges, door handles, light switches, latches, staplers, scissors, pliers ... But playing a far more central role in our technology are things that rotate without limit as parts of otherwise non-turners, things that go around and around as long as they're driven and perhaps a little longer. I mean devices based on that marvelous invention, the wheel and axle. That includes almost all of our motors and their associated shafts, pulleys, gears, and so forth. It includes our diverse wheeled and propeller-driven vehicles. Plus all manner of hand tools, from eggbeaters to socket wrenches. Long ago that meant wagons and potter's wheels, and the diversity of our rotational contraptions has been on the increase throughout our history. No doubt at all — mechanisms that rotate as parts of otherwise non-rotating contrivances form the very core of our mechanical technology.

We thus glimpse a paradoxical problem. Through most of human history (and prehistory, if you prefer the distinction), muscle has been the main motor of our technology, whether we work our own personal meat or persuade that of our domesticated animals to do our jobs. Muscle can only pull, and it must remain attached at both its ends. How can a non-rotating engine drive truly rotational machinery? This book explores the diverse ways that humans have faced up to and managed to deal with that most basic of dilemmas. In essence, it explores one facet of the biomechanics behind history.

Your immediate rejoinder might be that the difficulty yields to a trivially simple fix. Specifically, just add a crank, a lever extending radially outward from the rotating shaft with a slip fitting on a sideways extension of that lever. No need for an illustration — we make such things all the time, from hand-operated household gear such as pencil sharpeners, eggbeaters, and meat grinders to the engines of our cars, in which pistons moving (for most cars) up and down crank and thereby turn drive shafts. That slip fitting might be nothing more than a greasy hand or a loosely fitting outer handle of wood or plastic. It seems reasonable that this obvious trick should have been particularly appropriate for ancient devices, with their slow rotation rates. Oddly enough, cranks remained unknown (or nearly so) until about a thousand years ago. Think of it — for all their sophistication, the classical Mediterranean civilizations made no significant use of this simple and now ubiquitous arrangement. Punning subtly, one might ask, where's the rub?

Muscle-powered rotational machinery obviously has a much longer history than cranks — think again about all those wagons, chariots, and potter's wheels. How, then, were they persuaded to rotate? And have these more ancient fixes persisted, even gained in importance, with the further proliferation of rotational devices? No surprise — one question leads to another.

First, then, what are the options for making shafts and wheels turn? If nothing else, its peculiar modernity tells us that a crank isn't the only thing that will work. Consider some other possibilities, put as a series that I don't assert is chronological, fully complete, or mutually exclusive — and at the expense of suspense ...

• Roll the top of a cylinder by pushing something across it while the bottom then rolls (at half the speed) along the ground — rolling a log or barrel, as in figure 1.1. Of course, sooner or later (more likely sooner), the propelling roller on top leaves the driven roller behind. So you can't cover much distance without fairly often moving the roller left behind from rear to front. Even with a series of driven rollers, creating a new front one with a rear reject remains required. The simpler French-style or rod-type rolling pin works this way; its task doesn't ask that it roll very far and allows easy lifting and repositioning

• Pull or push on the axis of a wheel while a part of its circumference contacts the ground with enough friction so it rotates rather than just sliding along — as a horse pulls a cart and as in figure 1.2; or as you use a conventional rolling pin, one with a rotating handle at each end, by pushing or pulling the handles. The rolling pin then rotates as it presses the pie crust, although the handles do not. Proper bearings aren't absolutely necessary — a person can pull along a bagel-shaped (toroidal) water tank, hauling on a rope that loops through its center hole.

• Make an animal (perhaps a person) walk while pushing or pulling in monotonous circles around a vertical shaft or drum from which a radial lever protrudes — for example, turning a large posthole digger (auger), as in figure 1.3. The motor itself then rotates at just the same speed as the shaft or drum, so no bearing need be supplied — at least between the two. (Of course, that shaft or drum will typically turn around its own bearing.) Years ago, playgrounds had small merry-go-rounds driven by one or more children as others sat on the deck and made encouraging noises.

• Grab the handle of a tool, turn it through an arc, then release it, grab it again after turning one's arm or body some ways opposite the direction of the tool's rotation, and turn it again, as in figure 1.4. The prose may imply complication, but the process could not be simpler or more familiar. It's what we do with the steering wheels of cars and with the knobs on such electronic gear as still has knobs. And we do the same with screwdrivers and screw-on jar lids.

• Design the tool so the activity you're performing with it includes a recovery phase in which the tool's shaft rotates back to its original orientation — as with the knobs of old wristwatches and in figure 1.5. Thus no net rotation occurs, and no problem arises. A yo-yo works that way, as did many ancient tools — drills for boring holes and starting fires, for instance. Turn and return, one might say.

• Roll something, perhaps a rope or bundle of fibers, up on a shaft — a shaft with one end free from any supporting bearing. Every so often pull the roll off that free end of the shaft without unrolling it, as in figure 1.6. Each of the original rolling turns then becomes a twist of that rope or bundle of fibers. That's the basic trick behind spinning thread or cordage of any kind, in effect making long, tension-resisting, flexible material from the short fibers we harvest from plants (cotton, for instance) or animals (wool and so forth). (Only the silk of silk moths comes in naturally long fibers, and we spin these mainly to bring them up to a convenient diameter for use as thread.)

We'll get back to each of these, exploring their advantages and disadvantages and how each has been used by technologies based on both muscle and other movers — in short, its functions, origins, and history. Of course, we have only spotty knowledge of the early history of devices as basic as these. Different cultures have taken different technological trajectories, and the extent to which the how-to-do-its of living have spread among them continues to generate controversy. Too often it's far from clear whether a technique was learned from another culture or whether it was independently invented — to say nothing of the matter of when either happened. Moreover, the history of technology has a problem of sourcing that's far worse than that of, say, the history of science. Craftsmen were not just secretive; until recently they were almost always illiterate. Science deliberately leaves a written record (even if it sometimes gets lost); technology rarely does so. Still, that may leave too bleak an impression — technology is the more likely of the two to leave behind some physical impression, a persistent archaeological record as artifacts. In addition, its history lives on in such things as common words and linguistic allusions. For instance, the expression "loose cannon" refers to the mayhem caused when a cannon of a sailing warship, weighing perhaps half a ton, came unhitched and careened around with the rolling of the ship, smashing almost anything in its erratic path.

Before going further, a distinction needs to be established, an absolutely critical matter here but one all too vague in everyday speech. Heading off in a constant direction will never be confused with rotation. But what if you move in a circle while never changing orientation, continuously facing the same direction? Admittedly this takes some unusual footwork inasmuch as, at times, you have to go sideways and backwards. If you trace your path on the floor, you certainly will find that you've made some kind of a closed loop, so you've undoubtedly gone around. At the same time, if you've faced the same way throughout, just as undoubtedly you haven't turned. So there are two ways to go around in circles. For practical reasons, mainly for describing motion with equations and for stating important conservation laws, physical scientists distinguish between these two kinds of circular motion. We need to do so as well.

Terminology. By definition, then, circular motion comes in two versions — not to exclude a mix of the two. In "rotation," orientation changes with time; in "translation," orientation doesn't change even if a body moves in a circle or part (an arc) of a circle. Figure 1.7 illustrates the difference. For present purposes, we'll rigorously restrict use of the term "rotation" to its proper physical kind. Yes, irrotational circular motion sounds oxymoronic, but clearly it's not. Moreover, it matters more than you might think. It takes on especial importance in fluid dynamics — as when a wing generates lift or a hurricane blows in a huge circle. We're really quite good at it ourselves, whether you exercise as a whole body or as you move a hand in a circle, signaling that someone might pass you. In American football, a ball carrier dodges and swerves and goes around while moving downfield, translating with the body ever facing the goal line. The carrier truly rotates only when shaking off a tackler with a whole body spin.

Our sensory equipment makes exactly this distinction, doing it without arousing your awareness. You translate in circles of any diameter and at any speed without getting dizzy, but when you rotate in circles, you have no such luck. Slow social dancing involves lots of circular translation, as do at least some maneuvers in square dancing. A Ferris wheel rotates, but its individual compartments, their orientation maintained gravitationally, translate in circles. By contrast, ballet and ice dancing go in for vertiginous levels of true rotation, no trivial matter for the performer. Still, for even these last, the motions consist entirely of whole body rotation; again, that's the best we can do with our lack of fully rotational joints.

I would have preferred more descriptive designations emphasizing the contrast between, say, "motion with change of heading" for rotation and "motion without change of heading" for circular translation, but we're stuck with the oddly specialized use of two ordinary (and thus easily misunderstood) words. Early in the twentieth century, the psychologist and philosopher William James offered an excellent illustration, even if coupled with a message that we have to reject quite explicitly. He imagined a hunter encountering a squirrel on the trunk of a tree. The squirrel runs around to the opposite side of the tree, so the hunter, at a much greater radius, moves around as well. The squirrel, no dodo, would like to survive the encounter, so it keeps moving in order to keep the trunk between itself and the hunter. Thus both squirrel and hunter make rotational motions. Does the hunter circle around the squirrel? He (male in the original) remains facing the squirrel, so he clearly does not. At the same time, he's north, then east, then south, and then west of the squirrel, so just as clearly he must circle the squirrel. James, illustrating the essence of pragmatism with the tale, said that the distinction is purely semantic and thus essentially meaningless. However, for our purposes, without a doubt both squirrel and hunter have engaged in true rotational motion, with the latter's motion describing a path around that of the former.

Not that translating around in a circle, without conversion into rotation by means of a crank, can't serve practical purposes. Think of what you do when stirring a pot or the batter for a cake. You make the stirring spoon translate around in circles, and it does its job at least as well as it would if it were truly rotating. A traditional mortar and pestle works the same way. These translational actions may even do better than their rotational equivalents — one translational turn will produce more movement of the pestle's periphery than would one rotational turn. Sometimes they can do very much better, since rotating a shaft in materials that retain odd traces of solidity often leads to undesirable effects — more on this business (strangulated flow) in chapter 10.

A complicated (and probably hypothetical) machine, a particularly ingenious contrivance, provides an especially neat and satisfying illustration of this distinction. Among much larger and more immediately important machines, Agostino Ramelli, a sixteenth-century military engineer, designed a vertical wheel that kept a set of books open for a single reader, as in figure 1.8. By turning the wheel, the end user (as we would now say) could select which volume to consult, and volumes stayed both in a fixed orientation and opened to preselected pages. So the wheel rotated but the individual book supports (and books) translated in a circle. Ramelli accomplished the trick with what are called "epicyclic" or "planetary" gears; in this particular case, the central (sun) gear doesn't either rotate or translate, and the outer planetary gears translate but do not rotate. To effect this marriage, the planetary gears need to have the same number of teeth as the sun gears. (Neither the number of teeth on the intermediate gears nor the number of intermediate gears matters — they just ensure the correct relative direction of turning of the planets or, put strictly, assuring their non-turning.) Ramelli's is a particular (and odd) application of this kind of gearing, which was known if not common at the time. It appears in Leonardo da Vinci's notebooks, for instance, and it had been occasionally used in clock movements. We've used it in many automobile transmissions, from that of the Ford Model T to modern overdrives and automatics. A lovely animation of such epicyclic gearing appears in the Wikipedia article on gears.

As well as introducing the underlying elements on which the story will turn, perhaps the author ought to expose his personal perspective. My main professional area has been biology, centering on biomechanics in the broadest sense — as might be suspected from an account that began with the range of motion of our appendages. As an experimentalist in an area without a stereotyped experimental armamentarium, I've repeatedly had to cobble together odd tools. I've long recognized that the more mechanical items one makes, the more adept one becomes at devising both quick fixes and generally useful pieces of apparatus. The various challenges, over more than fifty years, have often asked that I look into the state of one art or another — metalworking, devising simple electronic circuits, pipe-fitting, adapting motors, and so forth. Not only have I acquired some distinctly arcane abilities, but the problems, by yielding to solutions involving things no longer widely used, have often tickled my still older interest in history.