Old gears tell us a lot about the fundamentals of gearing
If you look into Hoover's translation of Georg Bauer's De Re Metallica ("On Mining Technology"), you will see woodcuts of machinery, or millwork, used to apply the power of water wheels, treadmills, and horse whims to pumping and other mechanical processes in the processing of ores, as it was done in 1556. This machinery, made of wood except for some iron fittings, is typical of millwork from Roman times. None of it would have seemed novel to Vitruvius. Because the wooden construction leaves few traces in the course of centuries, and it is industrial, not mentioned in literary works, very little is known of classical machinery, but the few archaeological discoveries that have been made show surprising ingenuity and skill. Bauer's machinery has been adapted to his environment, but otherwise is the same in principle as Vitruvius's of a millenium and a half earlier.
The problem of power transmission solved by the gearing in this machinery is to take a slow-moving shaft, rotated by some source of power and either horizontal or vertical, and to produce rapid rotary motion of other shafts, either horizontal or vertical. The engineer sketched out his needs to the millwright, a carpenter with special knowledge, who executed the work according to his craft skills. For this purpose, toothed wheels or gears were used. The large driving gear was mounted on the driving shaft, and was either a spur gear or a crown gear, depending on whether the driven shaft was parallel or perpendicular to the driving shaft. The driving gear engaged a lantern gear on the driven shaft. The names come from the distant resemblance of the gears to common objects. A gear used to drive a vertical shaft was called a wallower, and was usually a lantern gear or, later, a bevel gear.
If there are N1 teeth on the driving gear, and N2 teeth on the driven gear, or follower, then every revolution of the driving gear must cause N1/N2 revolutions of the follower. This is called the gear ratio. In the present case, it is always greater than unity and represents the factor by which the rotation of the driven shaft is speeded up relative to the input shaft. A gear ratio of, say, 3 to 8 would be typical in this case.
The teeth must be evenly spaced around both gears, and in order to mesh properly must be of the same pitch, or distance apart circumferentially. If we represent the two gears by circles that are tangent to one another, called the pitch circles, then the distances on each circumference must be equal to the width of a tooth and a space. Moreover, the width of a tooth must be less than the width of a space, to permit easy meshing. The necessary clearance results in backlash between the gears. This means that if the driver is reversed, it rotates a short ways until it again contacts a tooth. The pitch must allow more than one pair of teeth to be in mesh at a time, so that the force is transmitted continuously.
As the gears rotate, the teeth must not interfere with each other as they approach, come into mesh, and finally recede. This restricts the height and shape of the teeth, and the minimum number of teeth on a gear. The primitive shape of the teeth on a spur or crown gear is rectangular, with straight flanks, which are the sides of the teeth that come into contact on meshing gears. Lantern gears used cylindrical teeth, with circular flanks. Therefore, the straight flank of a driver tooth had line contact with the circular flank of a driven tooth.
The cylindrical teeth of a lantern gear may be considered equally spaced on the pitch circle of the gear, extending equally above and below it. The projections of a tooth above and below the pitch circle are called addendum and dedendum. The addendum of the lantern gear is half the diameter of the rods, while the dedendum is large. The rectangular teeth of a spur or crown gear may be considered to have equal addendum and dedendum. The dedendum of the spur gear must be greater than the addendum of the lantern gear to prevent interference. The addendum of the spur gear must also be limited to avoid interference. With wooden gears, the teeth were usually of hardwood fastened with pegs driven in on the back side. These teeth could easily be replaced.
The driving tooth presses on the driven tooth. The normal at the point of contact is the direction of this pressure, since the two surfaces can, and do, slide on each other in the tangent direction and cannot exert any force in this direction. The direction of the normal is called the line of pressure, which in this case passes through the center of the rod, and the point where the flank of the driver tooth contacts the rod. The behavior of this line as the gears rotate is of great importance in gearing, since it gives the instantaneous speed ratio of the gears. In the present case, when the gear flanks first come into contact on approach the driven gear is forced ahead faster than average. As the teeth mesh more deeply, the driven gear is moved more slowly than average. As the teeth recede, the speed of the driven gear is again urged more rapidly forward. Of course, the driven gear cannot respond without inertia, so the result is a periodic strain that causes torsional vibrations in the shafts, and extra stress on the gear teeth.
If more than one pair of teeth is in mesh simultaneously, which is naturally the case, they fight with one another as well. The result is a rotation at the speed determined by the gear ratio, accompanied by vibration and noise at a frequency at double the rate of tooth engagement. At low speeds, this was acceptable, although every millwright probably knew some tricks that would alleviate the problem. One modification seems to have been to round off the upper parts of the flanks of the spur or crown gear teeth. This makes them look more modern, but was done by cut-and-try, not theory, and could be only partially successful. The gears shown at the right had cast-iron frames, but wooden teeth, for easier running.
The smoothness and quietness of operation of wooden gears has been remarked, when the mill has been well built. This is mainly due to the low speed, less than about 200 ft. per minute peripheral speed, and the elasticity of the wood combined with its sound absorption properties. The gears would also "run in" to accommodate each other. Modern metal gears, operating at greater speeds and loads, must have accurate tooth profiles that ensure a constant velocity ratio. At first, these profiles were cycloidal, but now are almost exclusively involute. This development began in the 17th century with the rise of steam power.
Before the steam engine, mills were driven either by muscle power, water wheels, or sails, in order of decreasing reliability. Windmills proved their utility on the windy plains of Persia, or in pumping water from flooded Holland back into the ocean, where the irregularity of wind power was no drawback. The Romans introduced large-scale water-powered mills. Small, primitive water wheels, often driven by the impact of flowing water, had been used since remote antiquity. Mills were used for grinding, pounding, blowing, sawing, and lifting.
In the eighteenth century, the millwright's skills began to include work in iron, which was replacing more and more of the traditional wooden machinery. Scientific study of gearing showed how gear teeth could be shaped to ensure a uniform motion, which meant that gearing could be applied to higher speeds than before, greatly increasing the rate of power transfer, and allowing turbines to replace water wheels. Steam provided the added power reliably and where required, and its exploitation was added to the millwright's responsibilities. The millwright's finest hour probably came in the creation of textile machinery towards the end of the century. The millwright became the mechanical engineer when rational, rather than traditional, design became the rule, and when wood was forced into a minor role in millwork by iron and steel. This development took place in the first half of the nineteenth century, when machinery was assuming its modern appearance.
J. Reynolds, Windmills and Watermills (New York: Praeger, 1970). Excellent ilustrations of gears from medieval to modern times.
H. C. and L. H. Hoover, G. Agricola, De Re Metallica (New York: Dover, 1950).
Composed by J. B. Calvert
Created 22 August 1999
Last revised 21 February 2007