Trochoids, Etc.

Including the theory of gear teeth


  1. Introduction
  2. Gears
  3. References


When I wanted to refresh my knowledge of the meaning of trochoid, which I recalled was a plane curve, I was surprised to find that my usual references said nothing about it. The Cambridge Dictionary of Science and Technology, however, told me that trochoid = roulette. A roulette, it explained, is the curve traced out by a fixed point on one curve as it rolls without slipping on a second curve. As a slight generalization, the fixed point does not have to be on the first curve, but may be fixed in relation to it.

Surely the most familiar roulette is the cycloid, the path of a point on the circumference of a circle rolling on a straight line. It was given its name by Galileo in 1599, and was first studied by Nicholas of Cusa (1401-1464). This curve, with some of its properties, is illustrated in the diagram. If the circle rolls on the outside of another circle, we have an epicycloid, and if it rolls on the inside, a hypocycloid. If the point is not on the circumference, but on a radius from the centre to the circumference, a cycloid becomes curtate, and if the point is beyond the circumference, prolate. All these curves are probably familiar to the reader, and it would be a good idea to review them now, and perhaps to draw some, since it is inconvenient for me to draw them for HTML.

It seems that if the point tracing the curve is on the circle, then we have a cycloid, but if it is either inside or outside the circle, we have a trochoid. Hence, we have epitrochoids and hypotrochoids as well as epicycloids and hypocycloids. Trochoid, then, is not a term for a general roulette, but for these specific kinds.

The cycloid is the solution to the problem of the brachistochrone: to find the curve along which a mass point moving without friction under the action of gravity will travel between two points in the shortest time. This celebrated problem was proposed in 1696 by John Bernoulli, and was soon solved independently by him, Newton and Leibniz. The period of the motion of a particle moving without friction on a cycloid is independent of the amplitude of the motion. Huygens used this property to design a pendulum whose period was independent of its amplitude of oscillation, taking advantage of the fact that the cycloid is its own involute (see below).

These curves appear as the path of a charged particle in crossed electric and magnetic fields. If the particle starts from rest, we have the plain cycloid. The movement of charged particles is discussed in Motion of Charged Particles. In this application, it seems customary to refer to the trajectories as trochoids. Indeed, the most restricted meaning of trochoid seems to be that of a curtate or prolate cycloid, or a general cycloid. Another application, which will be discussed below, is to the form of gear teeth. The important curves here are epicycloids and hypocycloids, and the term trochoid does not appear to be used for them, though it would seem appropriate.

These terms are derived from Greek, and a little word study might be interesting. The Greek for circle is kuklos, and the word for wheel is trocos. In the Latin alphabet, these become cyclos and trochos, which are then pronounced as in English. The Greek suffix -eidhs, meaning "looks like," comes over as -eides, usually modified to -oid. Therefore, cycloid and trochoid should be things that look like circles or wheels, which indeed they are in Greek. When we apply these terms in English to plane curves, we do not mean curves that look like circles, but curves that are generated by circles or wheels, and that is the connection. Furthermore, 'epi means "on," and `upo means "under." These become epi- and hypo-, respectively, as used in epicycloid and hypocycloid.

A completely different sort of curves are the evolute and involute, which are associated in mutual pairs. The evolute of any curve is the locus of the centres of curvature of the curve, while the original curve is the involute. The simplest case is that of a circle, which has only one centre of curvature, its centre, which is a degenerate evolute. The circle itself is the involute of this point. Think of an inextensible string unwinding from this point. The end of the string describes a circle. We may freely choose the radius of the circle, and the string never gets any longer as it unwinds from the point of zero radius.

If we start with a circle and unwind an inextensible string around it, we get a sort of spiral. The centre of curvature of any point on the spiral is a point on the circle, so the circle is the evolute of this spiral. The spiral is, then, the involute of the circle. The involute is always the curve created by unwinding the taut, inextensible string from the evolute. The involute of a circle is the most common form of a gear tooth. We shall see below why it is used for this purpose.

An easy way to draw the involute of a circle is shown in the diagram. With dividers, lay off equal segments of the circumference. A setting of about an eighth of the diameter of the circle will do. Then draw radii to the points thus determined so that accurate tangents can be drawn at right angles; only one such line is shown. On each tangent, lay off the corresponding length with the dividers. Connect the points so determined with a smooth curve. Make a template if this curve must be used in several places, to avoid having to construct it anew. A similar procedure can be used in computer drafting. This method can also be used to draw the involutes of other curves, since it is equivalent to unwinding a taut string. The involutes of circles of different diameters have different shapes, and must be drawn separately.


Gear design is something that seems simple on the surface, but is remarkably complex in practice, and is by no means elementary. We shall treat here only the theory of spur gearing, not the application and manufacture of gears, which is a very involved subject. Basically, a pair of gears has interdigitating teeth that compel mutual rotation depending on the ratio of the numbers of teeth on the two gears. The force is transmitted by a sliding contact between the gear teeth. If the driving gear has T teeth, and the driven gear T' teeth, then the angular velocity of the driven gear is T/T' times the angular velocity of the driving gear. This can be imagined as the rolling of two cylinders on each other of diameters D and D', such that D/D' = T/T'. D and D' are called the pitch diameters, in contact at the pitch point P (really a line of contact across the face of the gear). From this relation, we have that the ratio T/D = T'/D' is the same for the two gears. This ratio is called Pd, the diametral pitch, of the gears. Given the distance between the shafts bearing the gears and the numbers of teeth, the pitch diameters can be exactly calculated.

If the ratio of the number of teeth on mating gears were a small integer, wear might be distributed periodically, not uniformly. This was pernicious with early hand-made gears, but is not a great problem with modern accurate gears. To eliminate this problem, an extra tooth was added to one gear, called a hunting tooth, which had only a small effect on the gear ratio. For example, a 4:1 reduction might be obtained with gears with 15 and 60 teeth. Each tooth on the gear would come into contact with the same tooth on the pinion. If 60 teeth were increased to 61, then the pinion would advance one tooth for each rotation of the gear, equalizing the wear. Of course, 59 teeth could be used as well as 61.

The gears are arranged to have sliding contact between pairs of surfaces formed by the teeth of the gears, such that more than one such pair is always in contact, the rotation bringing successive pairs into contact. The surfaces in sliding contact must have a common tangent. The pressure between the two surfaces is normal to this tangent. In order that the ratio of angular velocities of the two gears remain constant, the Law of Gearing must be satisfied. This is that the normal to the common tangent must pass through the pitch point of the gears. If the Law of Gearing is satisfied, the motion will be smooth, quiet and free of vibration. If it is not satisfied, the gears will shake and vibrate as they rotate, severely limiting their speed and the amount of power that can be transmitted. Good gearing requires very accurate tooth shapes that guarantee that the Law of Gearing is satisfied. Tooth profiles that satisfy this condition are called conjugate.

If the shape of a tooth on one gear is given, it is possible to construct the shape of the tooth on a mating gear that is conjugate. This is most clearly explained if one of the gears is a rack; that is, a gear of infinite radius. In the figure at the right, a tooth of the rack is represented by the profile OP, where P is an arbitrary point and O is the pitch point, where the pitch circle of the gear of radius R and the pitch line of the rack are tangent. O is a point on both the rack and gear teeth. By moving the rack to the left, and allowing the gear to rotate clockwise without slipping, the point P will eventually become the point of contact. The normal to the rack tooth at this point cuts the pitch line at S. When P becomes the point of contact P', the normal must pass through the pitch point O. The tooth has then been moved through a distance z, so that S moves to O. The dotted curve is the position of the tooth when this happens. At this position, P' is also a point on the gear tooth. The gear has rotated clockwise through an angle ε = z/R. If we now rotate the gear back to its original position through the anticlockwise angle ε, O and P" are two points on the gear tooth profile. P" has the polar coordinates r,θ on the gear, which can also be expressed as rectangular coordinates. It is not difficult to express these relations algebraically, so that the conjugate profile can be calculated numerically (See Buckingham, p.4). The difference between the arc lengths OP on the rack tooth and OP" on the gear tooth clearly shows the motion involves slipping. The same construction can be used for two gears of arbitrary radii.

The most familiar gears can be imagined as derived from surfaces rolling on each other without slipping. We have already mentioned cylinders that can connect parallel axes, giving spur gears. Each pitch surface is formed by rotating the line of contact around its shaft. The line becomes a tooth in the gear. A line that passes through a point on an axis generates a cone on rotation. Two axes that intersect can be connected by cones generated in this way, realized as bevel gears. An oblique line generates a hyperboloid of one sheet. Two nonintersecting axes generate mating hyperboloids. The contact in this case is not pure rolling, but there is some sliding along the oblique line, that is greater as the smallest distance between the axes increases. This gives hyperboloidal gears, which may resemble bevel gears but are quite different. An example is the hypoid gear used in automotive differentials.

A completely different action is seen in the case of helical or screw gears. The pitch surfaces are cylinders, but the teeth are based on helical curves. As a gear rotates, the helix advances, pressing on the helix on the other gear and causing it to rotate. The commonest example is the worm and gear. A simple worm may have only one tooth, wrapped in a helix, or may have multiple starts, or threads. The lead of a helix is the distance it advances for one full turn around the base cylinder. The pitch of a worm is the axial distance between teeth, equal to the lead divided by the number of starts.

In order to talk about gears, we must define a number of terms relating to them. We have already mentioned the diametral pitch Pd = T/D. The circular pitch is Pc = πD/T, the length of circumference on the pitch circle for each tooth. Of course, Pd Pc = π, so one can be found from the other. About half of the circular pitch is devoted to the tooth thickness, half to the tooth space. The tooth is made a little narrower to ensure free motion, and so the tooth space is a little wider. The difference between the space and the thickness is the backlash. The backlash can be small on accurately formed gears, but must be larger if the teeth are cast. Generally, then, to a good approximation, tooth thickness = tooth space = Pc/2. These distances are measured on the pitch circle.

The addendum is the height of the tooth outside the pitch circle, while the dedendum is the depth inside the pitch circle. The dedendum is a little greater, and the difference is called the clearance. A typical addendum is 1/Pd, and dedendum 1.157/Pd, so the clearance is 0.157/Pd. The fillet at the bottom of a tooth generally has the radius of the clearance. The face of a tooth is the part outside the pitch circle, while the flank is the part inside the pitch circle. We'll find that in general the face of one tooth presses against the flank of a mating tooth. The outside diameter of a gear is the pitch diameter plus twice the addendum, or OD = T/Pd + 2/Pd = (T + 2)/Pd. Therefore, to find the diametral pitch of a gear by measurement, count the number of teeth, add 2, and divide by the outside diameter as measured by calipers.

Because it is simpler, we shall discuss involute gearing first. Consider the two pitch circles, meeting at the pitch point P. Draw a line through P normal to the line of centres of the two shafts. Now draw a second line through P inclined at an angle θ. We will want this line to be the line of action of the forces between the gear teeth, so that θ is the pressure angle. Now draw normals from the pressure line to the centres of the two shafts, S and S'. These normals will be the radii of the base circles. From similar triangles, the ratio of the radii of the base circles is the same as the ratio of the radii of the pitch circles. Imagining the base circles as connected by a taut cord, they will rotate just as the gears should, while a point on the cord traces out involutes to the two base circles. Hence, if the tooth surfaces are involutes to the base circles, they will touch on the line of pressure constantly as the gears revolve, so that the Law of Gearing will be satisfied. This is not especially easy to see, but it is true. The face and flank of the tooth are parts of the same involute. Typical pressure angles for involute gearing are 14.5° and 20°. 14.5° was long standard in the U.S., but now 20° is the standard pressure angle. Some 20° teeth were called stub teeth, having a smaller addendum.

Consider a linear gear, or rack, that mates with a circular gear to produce linear motion. A rack can be considered as a gear of infinite radius, so the involute will go over to a straight line inclined at the pressure angle. Remember that the force on sliding surfaces is normal to the surfaces, so a constant pressure angle means straight-sided rack teeth. With a 14.5° pressure angle, the angle between the teeth will be 29°, the same as the angle of the Acme thread that is used for driving threads as an improvement on the square thread. To draw an involute rack, simply lay out the addendum and dedendum lines above and below the pitch circle, then mark intervals of the circular pitch along the pitch circle, and draw the tooth profiles. It is easy to see that the mating gear can be moved toward and away from the rack with only a change in the backlash.

Involute gears have the advantages that (1) it is possible to have a series of gears of different diameters that will mate with each other, and that (2) the distance between the shafts can be altered slightly with no ill effects. The gears must have the same pressure angle and diametral pitch if they are to mate. Except in certain cases, it is unnecessary to have a series of mating gears, since the same gears usually work in pairs. However, it is a convenience for manufacturing and the stocking of standard sizes.

Cycloidal gears now have their own article, which the reader should consult for their theory.

Since gears involve sliding contacts, friction reduces their efficiency. The forces between the teeth act to rotate the gears, and as well to force the shafts apart. With a given rate of power transmission, the forces can be calculated easily for involute gears, since the pressure angle is constant. The gear teeth must be lubricated. For small powers, a grease is satisfactory, but gears transmitting considerable power should run in oil, so that hydrodynamic lubrication is provided and heat is removed. High-pressure grease, typically including MoS2, is the usual lubricant. The frictional losses in well-designed gears are very small. The efficiency of spur gears is usually 0.995 or better. Gears with considerable sliding contact, such a worms and helical gears, have lower efficiencies. In small gears, it is sometimes helpful to make mating gears from two different materials to reduce friction. A typical coefficient of friction with hard, lapped teeth is 0.02.

Many spur gears, and most bevel gears, that transmit considerable power have helical teeth. These may also be called twisted or spiral teeth to avoid confusion with helical gears with sliding action. Spiral is a misnomer, since a spiral is a plane curve and there are no spirals involved. When the cylindrical pitch surface is unrolled flat, the teeth are parallel straight lines at the helix angle ψ. The teeth may be generated by the same cutters used for straight spur gears, but fed at an angle. This makes the normal section the same form, but the rotation plane section has a greater pressure angle and a different pitch, though it is still involute. For this reason, generated twisted gears should be run together. The benefit of helical teeth is that the action begins at one end of the tooth and progresses to the other, and that the load is divided between several teeth if the gear face is sufficiently wide. For this reason, the gears run quietly. A disadvantage is that there is now an axial force that must be resisted by thrust bearings. Alternatively, gears of opposite helical sense may be mounted on the same axis, as in herringbone gears. A right-handed helical gear meshes with a left-handed gear on a parallel axis. The action is the same as for straight spur gears, with line contact.

With bevel gears, the teeth may be slightly curved in addition to being twisted. This moves the load from the ends of the teeth to the centres, making the gears not as susceptible to the bad effects of misalignment. The ends of the teeth are, naturally, weaker than the more central parts because of the lack of support. The same cutter that makes twisted bevel gears of this type can also make gears of a zero angle of twist, the so-called Zerol gears which can be used like straight bevel gears. Automobile manufacturers have been able to nearly completely eliminate gear noise by using twisted gears in the transmissions and differentials.

Gears are made by either of two principal methods, forming or generation. In forming, the cutter directly shapes the tooth. A milling cutter may cut the spaces between the teeth, or the tool may be made to follow a template of the desired form. In generation, the gear blank is machined to be conjugate to the cutter. Most gears are made by generation. Since this requires special and expensive tools, a general machine shop is usually not well-equipped to make gears, and they are made by shops dedicated to the purpose. The gear shaper uses a tool shaped like a basic rack or gear, which is oscillated relative to a gear blank to shape the teeth. The gear blank can be oscillated rather than the tool, and then the machine becomes actually a planer, though quite different from the usual planer. The most popular method is hobbing, which uses a cutting tool, the hob, that is like a worm gear with the teeth segmented to form the cutters. The hob cuts as it is rotated and is fed into and along the gear blank. It is essentially a basic rack that generates the gear teeth while cutting due to its rotation. The same tool can be used to make a variety of gears using generation.

Pin gearing is an excellent example illustrating conjugate tooth profiles. In pin gearing, one of the gears has cylindrical "teeth." The pins generally connect two discs, so the gear is called a lantern gear, from its resemblance to a lantern. This gear is driven by a spur gear with epicycloidal teeth, which gives a constant velocity ratio.

The diagram shows the pitch circles of the lantern gear and the driver, where N = 16 and N' = 8, so that the lantern gear turns twice as fast as the driver. Points are marked at intervals of the circular pitch (c.p.) on each pitch circle. As the gears rotate, these points approach, coincide at the pitch point P, and separate. Imagine that the pins are of negligible diameter, which simplifies the analysis. These imaginary pins are the centres of the actual pins of a practical diameter. Suppose points a' on the lantern and a on the driver are initially at the pitch point P. When the gears rotate by one tooth, these points find themselves at a and a', and the point on the lantern has traced out an epicycloid. This curve is to be repeated at intervals of the c.p., and, reflected, forms the other half of the tooth profile. At P, then, two epicycloids c and c' meet. The pin at P has just run down epicycloid c, and is about to be pushed by epicycloid c' as rotation continues. This happens successively for each pin on the lantern gear as the gears rotate. The driver tooth is made high enough that there is always one pin in full contact with an epicycloid like c'; better, two pins should always be in contact, and a new pin engaged before an old one is released. It is clear that the driver can push a pin only when it is past the centre line, or in recession. To the left of the centre line, in approach, the pin is behind the gear tooth. It is worth studying the gears until these matters are clearly understood.

The reason why the epicycloidal profile is conjugate to the pins is that normals to the epicycloid, as we have seen, always pass through the pitch point P. The direction of the force exerted on a pin is always normal to the epicycloid, so the line of pressure always passes through the pitch point, making the velocity ratio constant and satisfying the Law of Gearing. This is especially easy to see in this example. When practical pins are used, some of the tooth is cut away parallel to the epicycloid to accommodate the radius of the tooth, and a semicircular recess is made between each tooth into the pitch circle of the driver for the same reason.

We noticed that when the gear drives the lantern, the action is completely in recession. If the lantern drives the gear, then the action is completely in approach. Action in approach essentially wedges the pins into the gear, and is much less favorable, with larger frictional forces than action in recession. Though the lantern will drive the gear, it will not do so as well as when the gear drives the lantern. This is the reason why the gear is always the driver, and the lantern the follower. The lantern can be larger than the gear, as well as smaller, so the velocity ratio can assume any practical value larger or smaller than unity.


P. M. B. Walker, ed., Cambridge Dictionary of Science and Technology (Cambridge: CUP, 1988).

P. Schwamb, A. L. Merrill and W. H. James, Elements of Mechanism, 6th ed. (New York: John Wiley & Sons, 1947). Chapters VII and X.

V. M. Faires, Design of Machine Elements, 4th ed. (New York: Macmillan, 1965). Chapters 13-16 of this excellent text treat gears, and even explain the Law of Gearing, but only involute gearing is mentioned.

E. Buckingham, Analytical Mechanics of Gears (New York: Dover, 1988).

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Composed by J. B. Calvert
Created 8 December 2003
Last revised 27 February 2007