Nearly all practical gears now have involute tooth profiles. The involute tooth has great advantages in ease of manufacture, interchangeability, and variability of centre-to-centre distances. Originally, however, cycloidal profiles were used. Cycloidal profiles are as technically suitable as involute profiles, perhaps even slightly superior in some respects. They were satisfactory when gears were specially made in mating pairs, and were expensive. Accurate involute teeth became much easier and cheaper to produce; together with their other advantages, they have completely supplanted cycloidal gears. Many modern engineering texts and handbooks at most only mention cycloidal gears in passing. The inventor of cycloidal gears does not appear to be known; there are claims for Desargues, de la Hire, Roemer and Camus. They probably came into use early in the 17th century.

Cycloidal gears still have a few applications, however, and are of historic interest. More importantly, the understanding of cycloidal gears will help the understanding of gear action in general. The available references do not explain cycloidal gears very well, so it is hoped the present article will make things clearer. Incidentally, pin gearing is a third example of gear tooth form that may be of interest. There is a British Standard (BS978-2) for fine-pitch cycloidal gears. Although these may be used in mechanical clocks, the slow speeds and light loads in clocks do not require conjugate gear tooth profiles.

The *common cycloid* is the curve traced by a point on the circumference of a circle as the circle rolls without slipping on a straight line. Parametric equations are x = r(t - sin t), y = r(1 - cos t), where t is the angle of rotation. This graceful curve has several interesting properties. The one of most interest with respect to cycloidal gears is that the normal to the curve at any point passes through the point where the describing circle passing through the given point is in contact with the straight line. It is easy to see why this is the case, since the point of contact with the straight line is instantaneously at rest, and the circle is instantaneously rotating about this point. Therefore, the velocity of the point must be normal to a line from this instantaneous centre to the point. It is very important to grasp this notion, since it is the very basis of the use of cycloidal curves as gear tooth profiles.

This property is illustrated in the figure at the right. The circle, of radius r, has rolled through an angle θ from the position when the point P on the circle coincided with the origin. At this point on the curve, the normal passes through the instantaneous centre Q which is instantaneously fixed. It is exactly the same if the centre of the circle C is a fixed point, and the straight line (imagined as drawn on a surface) moves a distance rθ to the left. The instantaneous centre Q is now a fixed point, which is the pitch point for a set of gears. As the gears and describing circles rotate, the normal from the point of contact always passes through the pitch point.

The cycloid is an example of a *roulette*, a curve generated by a point on one curve that is rolled on another curve. If the describing circle rotates outside the circumference of another circle, the curve traced is an *epicycloid* ("epi"- upon). If it rotates on the inside, the curve is a *hypocycloid* ("hypo" - beneath). The curve traced by a point on a radius extended of the describing circle is *prolate*; if the point is on a radius inside the circle, it is *curtate*. Therefore, we have prolate and curtate cycloids, epicycloids and hypocycloids. These are also called *trochoids* ("trochos" - wheel), especially if they have loops, but a trochoid is a roulette where both curves are circles, in general. The normal to an epicycloid or a hypocycloid at any point passes through the point of tangency of the describing circle and the base circle, that is, through the instantaneous centre, as in the case of a common cycloid.

One gear drives another by pressure between the teeth in contact. The tooth profiles have a common tangent, and the pressure is normal to this tangent. The teeth slide on one another as they engage and disengage while the gears rotate. It is essential that if the driving gear rotates at a constant speed, so does the driven gear. This is achieved if the line along which the pressure acts passes through the point of tangency of the pitch circles that represent the idealization of the gears as cylinders in contact. In this case, the pressure acts tangent to two cylinders whose ratio of diameters is the same as the ratio of diameters of the pitch circles. In involute gearing, these are the *base circles* whose involutes are the tooth profiles. In cycloidal gearing, the corresponding circles vary in size, but always in a fixed ratio.

This fundamental condition must be satisifed in all gears that operate at high speeds or transmit considerable power. If it is not, the gears will vibrate, make loud noise and will wear rapidly.

Let us suppose we have two parallel shafts that we wish to connect with gearing. We idealize the gears as two cylinders rolling on one another without slipping. The diameters of these cylinders are the *pitch diameters* D of the gears. The sum of the diameters is twice the distance between the shafts, and the ratio of the diameters is the ratio of the desired speeds of the shafts. Now we choose the *diametral pitch* P of the gears that will be used. This is the number of teeth T per inch of pitch diameter D: T = PD. The distance on the pitch circle devoted to one tooth and its space is the *circular pitch* P_{c}. Clearly, P_{c} = πD/T = π/P. Gears that mesh together must have the same P.

We imagine the pitch circles tangent at the *pitch point* a on the line of centres. For each gear, we now select a *describing circle* with its centre on the line of centres and tangent to the pitch circles at point a. The size of the describing circles can be chosen arbitrarily, and need not be the same for the two gears. We shall see that describing circles from 1/2 to 1/4 the pitch diameters will be practical. Now imagine all these circles rolling without slipping at the point a. The upper describing circle will roll on the inside of the upper pitch circle and on the outside of the lower pitch circle. Imagine a stylus attached to the describing circle at the point initially at a, which can draw separately on surfaces fixed to the two gears. It will trace out a hypocycloid on the top gear, and an epicycloid on the bottom gear. These curves will be tangent at the instantaneous position of the stylus, and the normal to them will pass through the instantaneous centre of rotation, which is always a. If these curves are the tooth profiles, then the fundamental condition will be satisfied. In this way, the upper describing circle traces out the form of the flank (part inside the pitch circle) of the upper gear and the face (part outside the pitch circle) of the lower gear. Two surfaces in contact must be generated by the same describing circle.

In the same way, the lower describing circle traces out the flank of a tooth of the lower gear, and the face of a tooth of the upper gear. It is convenient to imagine rotation in the opposite sense in this case. Note that the profile of a tooth depends on both describing circles. This is the reason why cycloidal gears cannot be made interchangeable, since the shape of the teeth depends on both gears, not just on the gear considered.

The construction of the tooth profiles is shown at the left. The describing circles C and D have a diameter half that of the pitch circles, so that the hypocycloid is a straight radial line. All four circles rotate about their fixed centres as the tooth profiles are traced out by the point initially at the pitch point. Rotation to the left and right are indicated in the diagram. Each describing circle traces out the flank of one gear and the face of the other. Both circles are necessary to find the tooth profiles for both gears. Note that the face profiles are longer than the flank profiles; this reflects the sliding as the gears approach and recede. The path of contact will be from point a' through the pitch point a to point a". The pressure angle is not shown; it is the angle between a line from the point of contact to a and the tangent to the pitch circles at a. This figure should clearly show the conjugate action of cycloidal tooth profiles. It is not difficult to express the tooth profile curves in rectangular coordinates relative to each gear, using this diagram as a basis.

From the way the profiles are constructed, it is clear that the describing circles are also the circles of contact; that is, the point of contact is always somewhere on these circles. It begins at the point where the addendum circle (the outside of the gear) cuts the describing circle, at some considerable pressure angle, then moves down the circle to the pitch point where the pressure angle is zero. Then it moves along the other describing circle with increasing pressure angle until the addendum circle is reached. The path of contact is not a straight line at the pressure angle, as for involute gears, but parts of two circles.

If the describing circle is half the diameter of the pitch circle, the hypocycloid is a radial straight line. For this reason, such a describing circle is often chosen. If the describing circle is larger, the tooth exhibits a distinct neck, and is very weak. On the other hand, a smaller describing circle gives a better fillet and a strong tooth. However, the describing circle must not be too small, or the pressure angle becomes excessive. For normal spur gears, a describing circle half the pitch diameter or a bit smaller is usually chosen. A cycloidal gear may have a straight flank conjugate to a mating tooth, something that does not happen with involute gears.

A cycloidal rack is described on a straight line, and consists of cycloidal segments, quite different from the straight-sided involute rack. In general, a cycloidal gear cannot be distinguished from an involute gear by casual observation. A cycloidal gear will not run successfully with an involute gear of the same pitch, though they may apparently mesh. Size of the describing circles is analogous to pressure angle for involute gears. Two cycloidal gears will not run together unless the describing circles used to design them are the same. There has been limited success in designing interchangeable series of cycloidal gears, but this is not nearly as satisfactory as with involute gears.

An interesting application of cycloidal gears is to the design of Roots blower rotors. A pair of two-lobed rotors is shown at the right. The pitch and describing circles are shown. The describing circles are 1/4 the diameter of the pitch circles. The rotors consist of full arches of the epicycloid and hypocycloid. Since the pressure angle becomes 90°, the rotors will not drive each other, but must be driven by external gears. Still, they are cycloidal gears of two teeth. The rotors do not have to have this shape, but it is a possibility. The Roots blower handles large quantities of air at a small pressure difference. It was patented by the Roots brothers in 1860 as a blast furnace blower, but was invented in 1848 by Isaiah Davis. The air is carried from input to output in the space between a rotor and the casing, as in any gear pump. Normally, the rotors do not touch the casing and run without lubrication. Three-lobed rotors are also used, and can be designed on the same principles as the two-lobed rotors.

V. L. Doughtie, P. Schwamb, A. L. Merrill and W. H. James, *Elements of Mechanism*, 6th ed. (New York: John wiley & Sons, 1947). pp 254-259.

E. Buckingham, *Analytical Mechanics of Gears* (New York: Dover, 1988). pp 24-29. This book is out of print, but was still available from Amazon at about $45 in early 2007.

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Composed by J. B. Calvert

Created 19 February 2007

Last revised 21 February 2007