Operating principle

A special case of cycloidal gearing with point tooth form is the so-called lantern pinion. The Animation below shows the operating principle of the lantern gear. A disc with pins (or trundles) engages with a (cycloidal) gear wheel and drives it.

Design of the lantern pinion and lantern gear wheel

The figure below shows the construction of a lantern pinion. As is usual with cycloidal toothing, a base circle is first required on which a rolling circle forms the cycloidal tooth flank. In the case of a lantern pinion, however, that the trajectory point (“pencil tip”) is no longer concentrated on a single point but is extended to form a circle (this reduces the wear on the tooth flank in later operation). The diameter of the circle corresponds to the diameter of the trundles.

If the rolling circle is now rolled on the base circle, the envelope of the trundle circles forms the shape of the tooth flank (equidistant to the cycloid). For the construction of the lantern pinion, the trundles are arranged on the circumfarance of the rolling circle (=pitch circle) .

Lantern pinions are a very old type of gearing, which used to be frequently found in mills. At that time, wooden bars were used. Nowadays these are low-friction trundles. Lantern pinions are still used when heavy loads have to be transmitted at low speeds, e.g. for driving large telescopes. Lantern pinions are not suitable for high-precision transmissions.

Lantern pinions are used for large gearboxes with high torques to be transmitted!

Due to the much easier manufacturing, the cycloid shape of the lantern pinion gear wheel is often approximated by an involute shape in practice.

Line of action

The animation below shows the meshing of a lantern pinion. The line of action no longer runs exactly on the rolling circle, because the trajectory point on the rolling circle during the construction of the tooth flank was extended to a circle.

The line of contact is determined by the intersection between the line \(\overline{MC}\) and the trundle (with \(M\) as the center of the trundle and \(C\) as the pitch point). This is a direct result of the law of gearing, which states that the normal (= radial of the trundle) must run through the pitch point.

Since the meshing of the trundles is essentially concentrated on that part of the line of contact which lies in front of the pitch point, lantern pinions generally have relatively low contact ratios.

Lantern pinion wheel with involute toothing

If a “rod” is used instead of a disc to which the trundles are attached (“driving rod”), the rolling circle is then a rolling straight line. If this straight line is used to construct the tooth flank of the gearwheel, one will obtain involute shape tooth flanks (involute gear wheel)!

Line of action

The figure below shows the meshing of two cycloidal gears. The dash dotted pitch circles are at the same time rolling circles and correspond to the base circles with which the cycloidal shape of the respective gear was created (see article Geometry of cycloidal gears). The rolling circles touch each other at the pitch point C. The line of contact highlighted in red, on which the tooth flanks touch, runs through this pitch point. The line of contact consists of arc sections of the rolling circles with which the cycloidal tooth form was constructed. The line of contact is limited by the point of intersection between the rolling circle (line of action) and the tip circle (addendum circle) of the opposite gear.

With cycloidal gears, the line of contact is formed by the rolling circles of the hypocycloids and limited by the tip circles of the gears!

A closer look at the meshing shows that first the concave dedendum flank of the driving gear (yellow) meets the convex addendum flank of the driven gear (blue). The situation is reversed at the pitch point (inflection point of the line of contact) and the convex addendum flank of the driving gear meets the concave dedendum flank of the driven gear. Thus, a cycloidal toothing always provides a smooth flank pairing and thus a significant reduction in the surface pressure in comparison to involute toothing. This also reduces wear on the teeth. Note that with cycloidal gears, a dedendum flank of one gear will never be in contact with a dedendum flank of the other gear (the same applies to the addendum flank)!

Cycloidal gears show a relatively low flank wear compared to involute gears!

Law of gearing

The reason that cycloidal gears also satisfy the law of gearing and thus provide a constant transmission ratio is due to the fact that the rolling circles for the construction of the cycloidal tooth shape are applied equally to both gears: the rolling circle for the construction of the hypocycloid of one gear is used for the construction of the epicycloid of the mating gear and vice versa (see article Geometry of cycloidal gears).

This fact finally results in the law of gearing, which states that for a constant transmission ratio the normal at the point of contact of two flanks in mesh must always run through the pitch point C.

The normal at the point of contact of two tooth flanks in contact must always run through the pitch point (general law of gearing)!

If this were not the case, then the torque would vary permanently. In contrast to involute gears, the (standard reference) pitch circles of two cycloidal gears must therefore touch each other exactly (inflection point in the line of contact), i.e. the centre distance as the sum of the pitch circles radii must be maintained exactly. Since this cannot always be guaranteed (e.g. by thermal expansion), the use of cycloidal gears in classical mechanical engineering plays hardly any role.

In order to comply with the law of gearing (constant transmission ratio), the centre distance for cycloidal gears must be maintained exactly!

Point tooth form

If the rolling circle for the dedendum flank is chosen larger and larger for the construction of the hypocycloid, then the dedendum flank decreases more and more. The animation below shows the construction of the hypocycloid for a ratio of base circle to rolling circle of 0.97. However, the later tooth flank is only a fraction of this drawn hypocycloid.

The animation below shows the engagement of two cycloidal gears designed with a rolling circle to base circle ratio of 0.97. Since the rolling circle lies almost identically on the pitch circle and the rolling circle ultimately determines the line of contact, the teeth are engaged almost completely on the pitch circles.

Thus the contact point of the tooth flanks on the first part of the line of contact up to the pitch point C remains almost exclusively on the pitch circle of the yellow (driving) gear (see point A). The wear of the dedendum flank on the pitch circle is therefore very high. Only with the second part of the line of contact (i.e. from the pitch point to the end of meshing), the entire addendum flank is successively used for meshing. However, the mesh on the blue (driven) gear is now almost completely on the pitch circle (see point B).

In the limiting case where the rolling circle corresponds to the base circle and therefore the diameter ratio becomes one, the dedendum flank concentrates only on a single point. This results in a so-called point tooth form, since the entire addendum flank of the mating gear slides only on this one point. The wear of such a point tooth form is correspondingly high.

A point tooth form is favourable only with regard to the line of contact, which increases to a maximum and thus ensures a large contact ratio. The figure below shows the lines of contact for different rolling to base circle ratios.

A point tooth form shows a relatively large contact ratio; however, the wear on the tooth flank is very high!

Note: The contact ratio is determined by the ratio of the length of the line of contact to the base pitch (distance between two adjacent flanks in contact).

Due to the unfavourable wear, point tooth forms are generally of no importance. An exception to this is the so-called lantern gear, which is dealt with in the following article.

Construction of a cycloid

The shape of the flank of a cycloidal gear is a so-called cycloid. A cycloid is constructed by rolling a rolling circle on a base circle. A fixed point on the rolling circle describes the cycloid as a trajectory curve. A distinction can also be made between an epicycloid and a hypocycloid. An epicycloid is obtained when the rolling circle is rolled on the outside of the base circle. If, on the other hand, the rolling circle is rolled on the inside of the base circle, this is referred to as a hypocycloid. Accordingly, rolling circles can be divided into inner rolling circles (-> hypocycloids) and outer rolling circles (-> epicycloids).

A cycloid is obtained by rolling a rolling circle on a base circle! Epicycloids result from rolling on the outside of the base circle and hypocycloids from rolling on the inside of the base circle!

Construction of cycloidal gears

With cycloidal toothing, the addendum flank of the tooth has the shape of an epicycloid and the dedendum flank the shape of a hypocycloid. The inner rolling circle with which the hypocycloid is constructed generally does not correspond to the outer rolling circle with which the epicycloid is constructed, i.e. different rolling circles are used.

So that the teeth of two cycloidal gears can mesh correctly without interference, the outer rolling circle which is used for the construction of the addendum flank (epicycloid) of one gear, is then used as the inner rolling circle for the construction the dedendum flank (hypocycloid) of the mating gear! Conversely, the inner rolling circle for the construction of the dedendum flank (hypocycloid) of one gear corresponds to the outer rolling circle for the construction of the addendum flank (epicycloid) of the mating gear. This connection of the identical rolling circles ensures the validity of the fundamental law of gearing, which is necessary for a constant transmission ratio.

The rolling circle for the construction of the addendum flank of one gear is used for the construction of the dedendum flank of the mating gear and vice versa!

Since a rolling circle is always used to construct the tooth shape of two gears to be mated, cycloidal gears are always specially matched to each other. A cycloidal gear cannot be replaced easily by a gear with a different number of teeth as is the case with involute gears. Such a interchangeability of gears with cycloidal toothing is only possible if the rolling circles are always chosen identically and are only related to the base circle of the “main gear”.

Cycloidal gears must always be specially matched to each other and can generally not be exchanged at will!

The rolling circles are generally matched to the base circles, i.e. they are in a certain ratio to each other, since the ratio of rolling circle diameter to basic circle diameter determines the shape of the cycloid and therefore the shape of the tooth flank. All cycloids with the same rolling circle to base circle ratio are geometrically similar to each other. A ratio of about 1:3 for rolling circle diameter to base circle diameter is often found (the rolling circle diameter refers to the inner rolling circle for the construction of the hypocycloid!).

The base circle of a cycloidal gear always corresponds to the standard reference pitch circle which is in the case of cycloidal gears identical to the operating pitch circle. The contact point of the pitch circles corresponds to the pitch point. The pitch circle diameter d is determined analogously to an involute gear by multiplying the module m and the number of teeth z:

\begin{align}
&\boxed{d = m \cdot z} \\[5px]
\end{align}

The base circle of a cycloidal gear corresponds to the pitch circle!

The addendum diameter d_a (tip diameter) of a cycloidal gear can also be determined analogously to an involute gear:

\begin{align}
&\boxed{d_a = d + 2 \cdot m} \\[5px]
\end{align}

Special types of cycloids

Special case of a hypocycloid: a straight line

A special case in the tooth form of cycloidal gears arises when the rolling circle corresponds to half the base circle. In this case, straight foot flanks are obtained which extend radially outwards. Such a toothing is also called “clock toothing“, since it was often found in clockworks in the past (nowadays, however, the circular arc toothing is mostly used).

Straight dedendum flanks are obtained if the rolling circle diameter for the construction of the hypocycloid corresponds to half the base circle diameter!

Note that a hypocycloid can only be constructed if the rolling circle diameter is smaller or, in extreme cases, equal to the base circle of the gear (in such an extreme case one also speaks of a point tooth form). Otherwise no rolling can take place on the inside of the base circle! However, this restriction does not apply to the construction of an epicycloid; in this case, rolling circles of any size can be used. An important special case will be discussed in more detail in the next section.

Special case of an epicycloid: an involute

An important special case of an epicycloid is obtained when the outer rolling circle is made larger and larger. In extreme cases, the result is a circle with an infinitely large diameter, which corresponds to a rolling straight line due to the infinitely small curvature. The resulting epicycloid is then called an involute and the toothing is correspondingly called an involute toothing.

Involute toothing is a special case of cycloidal toothing with a rolling circle of infinite diameter.

Pros and cons of cycloidal gears

The cycloidal shape of a tooth leads to less wear of the tooth flanks during meshing and thus to lower friction losses in comparison to the involute shape. The reason for this is the lower contact pressure (lower hertzian contact stress), since a convex and a concave flank always meet in mesh and “nestle” up against each other, so to speak.

Furthermore, cycloidal gears can be produced with a significantly lower number of teeth without undercutting compared to involute gears. In this way, gears with only three or even two teeth can theoretically be produced.

The lower friction and the low number of minimum teeth are the main reasons why cycloidal gears are/were often found in clocks.

Despite the mentioned advantages of cycloidal gears, involute gears are still the most commonly used type of gears in mechanical engineering! The reason is the relatively simple production of an involute shape (straight tool flanks) compared to a cycloidal shape (curved tool flanks).

Furthermore, cycloidal gears are very sensitive to an inaccurate adjustment of the centre distance, which then leads to a change in the transmission ratio. For these reasons, cycloidal gears are hardly found in mechanical engineering but are only used in special cases such as in the watch industry, for roots type blowers or for the drive of gear racks.

Operating principle

The animation below shows the operating principle of a Roots-type blower (Roots-type supercharger) with a pair of two-lobed rotary vanes (rotors), named after its inventor Francis Roots. The fluid is pumped from the suction side (inlet) to the pressure side (outlet) between the individual meshing lobes and the housing. Due to the counterpressure at the pressure side, the gaseous fluid is compressed.

The rotors are usually driven by a belt drive and a pair of gear wheels mounted on the rear. The rotors are therefore not driven by the contact of their meshing vanes! Rather, even a small clearance must remain between the rotary vanes, as otherwise the high speeds would result in inadmissible heat development and the rotors would wear out rapidly.

Instead of two-lobe rotors, however, three-lobe or even four-lobe rotors are usually used. In order to achieve a more continuous compression and thus avoid pressure surges, the rotors are also twisted along their axis of rotation. The cross-sectional profile of the rotors consists of cycloids. Due to the contact-free meshing, however, it is not a cycloid gearing in the true sense!

Roots-type blowers are used, among other things, as compressors for supercharging combustion engines. They are also used for pumping liquids or foodstuffs such as rice or cereals.

Roots-type supercharger

The animation below shows the structure and the functional principle of a Roots-type supercharger as it is used for charging the combustion air in automobiles.

The air is sucked in through a slot at the front. The air enters the housing through this air inlet, which is equipped with cooling fins.

Inside the Roots-type supercharger are the three-lobed rotary vanes (rotors), which pump the air between the rotors and the inner wall of the housing from the suction side (top) to the discharge side (bottom). Due to the “accumulated” air on the discharge side, the pressure there increases and the air is compressed accordingly.

The rotary vanes are driven by a belt drive, whereby the driving pulley drives only one of the two rotors directly. The second rotor is driven by two helical gears.

Once the air has been pumped to the pressure side, the compressed air can be used for the combustion process. The unwanted increase in temperature during compression is counteracted by the cooling fins on the housing.

The pressure ratio between the pressure side and suction side of a Roots-type supercharger is generally a maximum of 2. Depending on the rotational speed, the efficiency in these cases is around 50 %. With larger pressure ratios, the efficiency decreases rapidly! Basically, the smaller the pressure ratio, the more efficient Roots-type superchargers are.

Design of the rotors

The shape of the rotors in Roots-type blowers is made up of cycloids in the same way as cycloid gears. However, there is no power transmission between the rotary vanes as is the case with cycloid gears!

Two-lobed rotary vane

For the construction of two-lobed rotor, the rolling circles for the epicycloids and the hypocycloids are chosen to a quarter of the diameter of the base circle (see article “Geometry of cycloidal gears“).

Three-lobed rotary vane

For the construction of a three-lobed rotor, the rolling circles for creating the hypocycloids and epicycloids are to be chosen to a sixth of the diameter of the base circle.