Willis equation for planetary gears

In the article Willis equation for planetary gears, the following fundamental equation was derived describing the motion of sun gear (s), ring gear (r) and carrier (c) of a planetary gear:

\begin{align}
\label{pl}
&\boxed{n_r \cdot z_r = n_c \cdot \left(z_r + z_s \right) – z_s \cdot n_s} \\[5px]
\end{align}

In this equation, n denotes the rotational speed of the components and z the number of teeth of the respective gears. This equation can now be used to show the different transmission ratios of planetary gears.

Transmission ratios

With a single planetary gear set one will obtain three different modes of operation, depending on which component (sun gear, carrier or ring gear) is fixed. Input and output are then carried out by the other two components. Which transmission ratios result in each case, is shown in the next section.

Fixed sun gear

If the sun gear is fixed (n_s=0) and the gearbox input is carried out by the ring gear and the output by the carrier, the following transmission ratio i_s=n_r/n_c results according to equation (\ref{pl}):

\begin{align}
&n_r \cdot z_r = n_c \cdot \left(z_r + z_s \right) – z_s \cdot \underbrace{n_s}_{=0} \\[5px]
&n_r \cdot z_r = n_c \cdot \left(z_r + z_s \right)  \\[5px]
&\frac{n_r}{n_c} = i_s = \frac{z_r+z_s}{z_r}     \\[5px]
\label{i_s}
&\boxed{i_s = 1+\frac{z_s}{z_r}} ~~~1<i_s<2 \\[5px]
\end{align}

Equation (\ref{i_s}) shows that the transmission ratio is always greater than 1, i.e. the rotational speed is decreased by the planetary gearbox. But the transmission ratio is also limited to a maximum value, since the number of teeth of the sun gear must always be smaller than that of the ring gear (otherwise the sun gear would be larger than the surrounding ring gear). In the theoretical limiting case, if the sun gear is as large as the ring gear and therefore both have identical numbers of teeth, the teeth ratio becomes z_s/z_r=1 and the transmission ratio 2 at most.

If input and output are reversed, i.e. the gearbox input is carried out by the carrier and the output by the ring gear, then the transmission ratio range lies between 1 and 0.5.

Fixed ring gear

A further possibility for speed conversion is obtained, when the ring gear is fixed (n_r=0) and the gearbox input is carried out by the sun gear and the output by the carrier. This results in the following transmission ratio i_r=n_s/n_c:

\begin{align}
&\underbrace{n_r}_{=0} \cdot z_r = n_c \cdot \left(z_r + z_s \right) – z_s \cdot n_s \\[5px]
&0 = n_c \cdot \left(z_r + z_s \right) – z_s \cdot n_s \\[5px]
&\frac{n_s}{n_c} = i_r = \frac{z_r+z_s}{z_s} \\[5px]
\label{i_r}
&\boxed{i_r = 1+\frac{z_r}{z_s}} ~~~2<i_r<\infty \\[5px]
\end{align}

In the present case one also obtains a reduced rotational speed, because the transmission ratio will be greater than 2 in any case, since the number of teeth of the ring gear is always greater than that of the sun gear [the teeth ratio is thus greater than 1 (z_r/z_s>1)]. The transmission ratio is not limited to a maximum value, since the ring gear and thus its number of teeth can in principle be chosen as large as desired and the transmission ratio then strives towards infinity.

If, in the opposite case, the gearbox input is no longer carried out by the carrier but by the ring gear, then the reciprocal transmission ratios with a range between 0 and 0.5 are obtained.

Fixed carrier

A last possibility for the transmission ratio is obtained when the carrier ist fixed and the gearbox input is carried out by the sun gear and the output by the ring gear. In this case the following transmission ratio i₀=n_s/n_r results:

\begin{align}
&n_r \cdot z_r = \underbrace{n_c}_{=0} \cdot \left(z_r + z_s \right) – z_s \cdot n_s \\[5px]
&n_r \cdot z_r = – z_s \cdot n_s \\[5px]
&\frac{n_s}{n_r} = i_0 = -\frac{z_r}{z_s} \\[5px]
\label{i_0}
&\boxed{i_0 = -\frac{z_r}{z_s}} ~~~\text{“stationary transmission ratio”}~~~-\infty<i_0<-1 \\[5px]
\end{align}

First of all, the negative sign is noticeable in the transmission ratio of equation (\ref{i_0}). It indicates that the direction of rotation between input and output shaft changes (“reverse gear”). In the present case, the transmission ratio ranges between -∞ and -1 and in the opposite case (when input and output are reversed) between -1 and 0.

Note, that in this case the planetary gear works like a stationary gearbox without  moving rotational axes. For this reason, the transmission ratio in the case of a fixed carrier also called fixed carrier transmission ratio or stationary transmission ratio i₀!

Direct drive

A planetary gear can also be used as a so-called direct drive. The carrier and the sun gear are firmly fixed to the ring gear. In this case, the rotary motion is transmitted directly from the input shaft to the output shaft (transmission ratio 1:1).  Such a direct drive is used, for example, in three-speed gear hubs as the “2nd gear”.

Stationary transmission ratio (fixed carrier transmission ratio)

If one looks at the equations (\ref{i_s}), (\ref{i_r}) and (\ref{i_0}), then obviously all transmission ratios can also be expressed by the fixed carrier transmission ratio i₀=-z_r/z_s. For a fixed sun gear, the transmission ratio i_s then becomes:

\begin{align}
&\boxed{i_s = 1-\frac{1}{i_0}}  \\[5px]
\end{align}

For a fixed ring gear, the transmission ratio i_r can be calculated as follows using the fixed carrier transmission ratio i₀:

\begin{align}
&\boxed{i_r = 1-i_0}\\[5px]
\end{align}

Even the fundamental equation for planetary gears (\ref{pl}) can be expressed by the fixed carrier transmission ratio i₀:

\begin{align}
&n_r \cdot z_r = n_c \cdot \left(z_r + z_s \right) – z_s \cdot n_s \\[5px]
&n_r \cdot \frac{z_r}{z_s} = n_c \cdot \left( \frac{z_r}{z_s} + 1 \right) – n_s \\[5px]
& – n_r \cdot i_0 = n_c \cdot \left(1-i_0 \right) – n_s \\[5px]
&\boxed{ n_s = n_c \cdot \left(1-i_0 \right) + n_r \cdot i_0 }~~~\text{with}~~~\boxed{i_0=-\frac{z_r}{z_s}}~~~\text{fixed carrier transmission ratio} \\[5px]
\end{align}

In the article Derivation of the Willis equation, the fundamental equation for epicyclic gears was derived in the following form:

\begin{align}
\label{g}
&\boxed{n_p \cdot d_p = n_c \cdot \left(d_p + d_s \right) – n_s \cdot d_s} \\[5px]
\end{align}

In this equation, n_p denotes the rotational speed and d_p the diameter (pitch circle) of the planetary gear. For the sun gear, the speed is denoted by n_s and the diameter by d_s. The rotational speed of the carrier is denoted by n_c.

The Willis equation (\ref{g}) generally applies to all planetary gears. Although the planet gears of a classic planetary gearbox are enclosed by a ring gear, this does not change the derived relationships between sun gear, planet gear and carrier. The only question that arises is how the motion of the planet gears is transferred to the ring gear.

Since a mere rolling motion without sliding between the ring gear and the planet gear takes place (considered as pitch cylinders), the velocity at the contact point must be equal. If one knows the speed v_po with which the outermost point of the planet gear moves, then this corresponds to the velocity v_r of the ring gear. Otherwise, a relative motion would come up, which of course can not be the case with toothed wheels. The pitch circle radius r (or pitch circle diameter d) of the ring gear can then be used to determine its rotational speed n, since the following relationship applies between these parameters:

\begin{align}
\label{o}
&v = \omega \cdot r = \omega \cdot \tfrac{d}{2} ~~~ \text{with} ~~~ \omega = 2 \pi \cdot n ~~~\text{applies}: \\[5px]
\label{v}
&\underline{v = \pi \cdot n \cdot d} \\[5px]
\end{align}

The same situation applies to the rotational speeds at the contact point between the planet gear and the sun gear. At the innermost point, the speed of the planet gear v_pi must be equal to the speed of the sun gear v_s. The center of gravity of the planet gear moves with the velocity v_c of the carrier. There is a linear relationship between these velocities (see black dotted line in the figure above), so that the circumferential speed of the ring gear v_r can be determined for a given circumferential speed of the sun gear v_s and a given circumferential speed of the carrier v_c.

Why is there such a linear relationship?

Why there is such a relative simple, linear relationship of the speeds will be shown in the following. For the sake of simplicity, the gears are assumed to be pitch cylinders.

The motion of a point on the planet gear can be understood as the superposition of two motions. On the one hand, the planet gear first rotates around its own center of gravity. In this case the typical symmetrical and linear increase of the velocity according to the equation (\ref{o}) is obtained, starting from the axis of rotation of the planet gear. The maximum speeds v_p are obtained  at the pitch circle of the planet gear.

In the center of rotation, the speed is zero as long as the planet gear axis does not move. However, the axis of rotation now moves at the speed of the carrier v_c. Both motions can now be superposed to the total motion.

The velocity of the planet gear at the outmost contact point with the ring gear points in the same direction as the velocity of the carrier. At the innermost point of contact with the sun gear, however, in the opposite direction. Due to the symmetrical speed distribution, the resulting speed of the planet gear at the outermost point of contact with the ring gear is therefore higher (v_po=v_c+v_p) to the same extent as it is lower at the innermost point of contact with the sun gear (v_pi=v_c-v_p).

In other words, the velocity of a point on the planet gear increases linearly, starting from the point of contact with the sun gear. Since the speed of the carrier v_c is assumed to be given, only the speed of the planet gear at the point of contact to the sun gear v_s must be known in order to determine the circumferential speed at the opposite point of contact to the ring gear v_r.

As already explained, for a mere rolling process without relative motions, the speed of the planet gear at the contact point to the ring gear (v_po=v_c+v_p) must be equal to the circumferential speed of the ring gear v_r:

\begin{align}
&v_r\overset{!}{=}v_{po} \\[5px]
\label{v_r}
&\underline{v_r=v_c+v_p} \\[5px]
\end{align}

The same applies to the contact point between the planet gear and the sun gear. There, the speed of the planet gear (v_pi=v_c-v_p) must be equal to the circumferential speed of the sun gear v_s:

\begin{align}
&v_s\overset{!}{=}v_{pi} \\[5px]
\label{v_s}
&\underline{v_s=v_c-v_p} \\[5px]
\end{align}

If we subtract equation (\ref{v_s}) from equation (\ref{v_r}), we obtain the following relationship between the circumferential speeds of the sun gear v_s, the planet gear v_p and the ring gear v_r:

\begin{align}
&v_r – v_s = v_c+v_p-v_c+v_p \\[5px]
&v_r = 2 \cdot v_p + v_s \\[5px]
\label{vvv}
&\underline{ v_p = \frac{v_r}{2} – \frac{v_s}{2} } \\[5px]
\end{align}

If the relationship of equation (\ref{v}) ist used in equation (\ref{vvv}), then the relationship between the corresponding rotational speeds is obtained:

\begin{align}
&v_p = \frac{v_r}{2} – \frac{v_s}{2} \\[5px]
&\pi \cdot n_p \cdot d_p = \frac{\pi \cdot n_r \cdot d_r}{2} – \frac{\pi \cdot n_s \cdot d_s}{2} \\[5px]
\label{nn}
&\boxed{n_p \cdot d_p = n_r \cdot \frac{d_r}{2} – n_s \cdot \frac{d_s}{2}} \\[5px]
\end{align}

The relation resulting from equation (\ref{nn}) can now be equated directly with the fundamental equation (\ref{g}) and one finally gets the following relation between the rotational speeds of the sun gear (S), the carrier (T) and the ring gear (H):

\begin{align}
&n_r \cdot \frac{d_r}{2} – n_s \cdot \frac{d_s}{2} = n_c \cdot \left(d_p + d_s \right)  – n_s \cdot d_s \\[5px]
&n_r \cdot d_r – n_s \cdot d_s  = 2 \cdot n_c \cdot \left(d_p + d_s \right)  – 2 \cdot n_s \cdot d_s \\[5px]
\label{f}
&\underline{n_r \cdot d_r  = 2 \cdot n_c \cdot \left(d_p + d_s \right)  – d_s \cdot n_s} \\[5px]
\end{align}

Additionally, it can be used that the diameters of the ring gear, the planet gear and the sun gear are not independent from each other. The ring gear diameter d_r corresponds to the sum of the sun gear diameter d_s and twice the planet gear diameter d_p:

\begin{align}
&d_r = d_s + 2 \cdot d_p \\[5px]
&\underline{d_p = \frac{d_r-d_s}{2}} \\[5px]
\end{align}

This results in the planetary gear equation for classic single-stage planetary gears (independent of the properties of the planet gears!):

\begin{align}
&n_r \cdot d_r = 2 \cdot n_c \cdot \left(\frac{d_r-d_s}{2} + d_s \right) – d_s \cdot n_s \\[5px]
&n_r \cdot d_r =n_c \cdot \left(d_r – d_s + 2 \cdot d_s \right) – d_s \cdot n_s \\[5px]
&\underline{n_r \cdot d_r = n_c \cdot \left(d_r + d_s \right) – d_s \cdot n_s} \\[5px]
\end{align}

Since for toothed wheels the pitch circle diameters d are proportional to the number of teeth z, the equation above can also be expressed by the number of teeth of the ring gear (z_r) and the number of teeth of the sun gear (z_s):

\begin{align}
\label{pl}
&\boxed{n_r \cdot z_r = n_c \cdot \left(z_r + z_s \right) – z_s \cdot n_s} \\[5px]
\end{align}

This equation can be used to explain the different transmission ratios of planetary gears. This will be discussed in more detail in the article Transmission ratios of planetary gears.

Operating principle

The animation below shows the structure and operating principle of a cycloidal drive.

An eccentric shaft (drive shaft) first drives a cycloidal disk. Fixed ring pins are arranged in a circle around the eccentric shaft, in which the cycloidal disc engages. Due to the eccentric motion, the cycloidal disc is driven around these pins so that the cycloidal disc rotates around its axis of symmetry. There are holes in the cycloidal disc which, unlike the eccentric shaft, now rotate clockwise. Roller pins of a pin disc engage in these holes. In this way, the cycloidal disc drives the pin disc, to which the centrally mounted output shaft is attached and which is coaxial with the input shaft.

If one compares the rotational speed of the input shaft with the speed of the output shaft in the upper animation, a speed reduction obviously occurs in addition to the reversal of the direction of rotation. In this animation, the pin disc rotates by a total of 40° during a complete rotation of the eccentric shaft (360°). Only after 9 revolutions of the input shaft has the output shaft completed a complete revolution. The transmission ratio of the gearbox is thus 9:1.

The heart of a cycloidal drive is the cycloidal disc, whose geometry plays a central role in the kinematic of the gearbox. The profile of such a disc can be traced back to a cycloid. That’s why the gearbox is called a cycloidal drive. Since cycloidal drives are used to reduce the speed, they are also referred to as cycloidal speed reducers. The construction of the geometry of the cycloidal disc is described in more detail in the article “Construction of the cycloidal disc“.

Due to the symmetrical load distribution, two cycloid discs are often used in practice, which are then offset by 180°. This ensures that the unbalance forces compensate each other, resulting in smoother operation at high speeds. The double design of the cycloidal discs also allows very high torques to be transmitted. In addition, the cycloid discs are often designed with a so-called shortened cycloid, which results in a
less lobed profile with reduced eccentricity and smaller hole diameters in the cycloid discs (more on this in the article “Construction of the cycloidal disc“.

Demonstration of the operating principle

The kinematics of a cycloidal drive appear very complex at first. But the idea behind it, which leads to such a kinematics, is quite simple. First imagine a fixed ring, on the inside of which a disc rolls. The disc is driven around the inside of the fixed ring with the aid of an eccentric shaft. The rolling disc thus rotates around its axis of symmetry.

Such a gearbox does not yet provide a large power transmission, since the power transmission is only achieved by frictional forces between the ring and the disc. Therefore a positive fit is used. This is achieved by a toothing between the fixed ring and the rolling disc.

A cycloid is used as the tooth form for the rolling disc. The rolling disc serves as the base circle for the construction of the epicycloid. The fixed ring, in turn, serves as the reference circle on which the pins are arranged, in which the cycloid disc engages.

Transmission ratio

The transmission ratio of a cycloidal drive is determined by the number of the fixed ring pins N and the number of the lobes of the cycloidal disc n (which is identical to the number of the space between the lobes!). The number of lobes in the cycloidal disc must always be smaller than the number of surrounding pins, otherwise the disc would be larger than the reference circle of the pins and the disc would not even fit between the pins. In most cases the cycloidal disc has one lobe less than the number of pins.

Using the example with N=10 pins and n=9 lobes, the calculation of the transmission ratio will be derived in the following. After one revolution of the eccentric shaft, the cycloidal disc has obviously engaged with a total of N=10 pins. Since the disc has only n=9 lobes, it must have moved by one lobe during one revolution of the drive shaft. This corresponds to one ninth of a full revolution.

Thus, the input shaft (eccentric shaft) must rotate 9 times so that the cycloidal disc and with it the output shaft (pin disc) perform a full rotation. The transmission ratio in this case would therefore be 1:9.

If, for example, the cycloidal disc had only n=7 lobes on its circumference, then it would have moved on by 3 lobes during a revolution around the N=10 pins. With one revolution of the input shaft, the output shaft would then have moved on by 3 times 1/7 revolutions. The input shaft would then have to rotate 7 times for the output shaft to make 3 revolutions. In this case, the transmission ratio would be 7:3.

If this principle is generalized, the transmission ratio of a cycloidal drive can be determined as follows on the basis of the number of lobes of the cycloidal disc n and the difference to the number of pins N:

\begin{align}
&\boxed{i = \frac{n}{N-n} } \\[5px]
\end{align}

Note: Because the cycloidal disc rolls on the inside of the reference pitch circle of the pins, the rotational motion of the disc around its center of gravity is opposite to the circular motion of the center of gravity itself (moving around the center point of the pin ring). Therefore the disc does not rotate completely around its centre of gravity when rotating around the pins. This would only be the case if the disc would roll on the outside of the ring pins (the rotational motion of the disc around its center of gravity and the motion of the center of gravity itself would then be identical).

Cycloidal drive vs. planetary gear

When it comes to large transmission ratios in a compact design, two gear types are particularly suitable: The planetary gear and the cycloidal drive. The similarities between the two gear types become particularly clear when the ring gear of the planetary gearbox is fixed. The gearbox is driven by the sun gear and the output is done by the carrier.

In the case of the planetary gear, the rotary axes are the axes of the planet gears (shown in blue) and in the case of the cycloidal gear, the axes of the cycloidal discs. While the planet gears are driven by a sun gear, the cycloid discs are driven by an eccentric shaft. As they rotate, the planet gears move around the inside of the ring gear. In the same way, the cycloid discs rotate around the fixed pins arranged in a circle. The motion of the planet gears drives the carrier and transmits the power to the output shaft. In the same way, the motion of the cycloidal discs drives a pin disc, which in turn transmits the power to the output shaft.

Compared to planetary gears, cycloidal drives are very robust against shock loads. In addition, cycloidal drives achieve significantly better positioning accuracy due to their very low backlash and high torsional stiffness. Therefore cycloidal drives are ideally suited for all types of drive technologies (e.g. for servo motors in robotics) and especially for heavy loads.

Cycloidal drives generally have a lower mass and a more compact design than planetary gears, especially at high transmission ratios, and have a longer service life. At relatively low gear ratios of less than 20, however, planetary gears usually offer better advantages and higher efficiency, depending on the application.

Design of the cycloidal disc

Cycloidal disc with ordinary cycloid

As already explained in the article “Operating principle“, the basic shape of the cycloidal disc is a cycloid. Such a cycloidal shape is obtained by a rolling circle that rolls on a base circle (see also article “Geometry of cycloidal gears“). The cycloid then corresponds to the path described by a point at the circumference of the rolling circle. The resulting cycloidal shape (ordinary cycloid) is referred to as the reference profile.

However, it must be taken into account that the cycloidal disc must later roll around the fixed pins. For this reason, the drawing point (“pencil tip”) must be extended to a circle when constructing the cycloid, whereby the diameter of this drawing circle corresponds to the diameter of the fixed pins around which the cycloidal disc later rolls! The enveloping profile that is created when rolling the rolling circle with its drawing circle then corresponds to the actual profile of the dycloidal disc.

The construction of the actual disc profile from the reference profile can also be imagined as follows. If the center of a milling cutter (whose diameter corresponds to the later pin diameter) is placed on the reference profile and milled along that profile, then the actual (equidistant) disc shapeis obtained.

The shape of the cycloidal disc is obtained from an equidistant to the cycloid!

Cycloidal disc with contracted cycloid

The cycloidal disc designed in the previous section has a relatively high eccentricity, which leads to enormous unbalance forces at high speeds and results in an uneven run. The large eccentricity also leads to the fact that the holes in the cycloidal disc are relatively large and therefore close to each other. The low material thickness between the holes could lead to a deformation of the holes at high forces.

For these reasons, the cycloidal disc is often designed with a so-called contracted cycloid. The drawing point is no longer placed at the circumference of the rolling circle (distance R but is located within the rolling circle (distance r<R). In contrast to this, an extended cycloid is obtained if the drawing point is placed outside the rolling circle (distance r>R). However, the latter has no meaning in mechanical engineering, which is why only the contracted cycloid is used.

The figure below shows the effect of such a contracted cycloid on the shape of the cycloidal disc. The contour of the cycloidal disc is “softer”. Both the eccentricity and the later hole diameters in the cycloidal disc are significantly reduced.

The animation below shows the operation of the cycloidal disc which is constructed with a contracted cycloid.

Note, that the transmission ratio does not change due to the construction with contracted cycloids. The transmission ratio is only determined by the number of lobes n of the cycloidal disc and the number of pins N (for more information see article “Operating principle“):

\begin{align}
&\boxed{i = \frac{n}{N-n} } \\[5px]
\end{align}

The transmission ratio of a cycloidal drive is also reflected in the ratio between the base circle diameter and the rolling circle diameter, which is used to construct the cycloidal disc. From the point of view of the cycloidal disc, it is a matter of rolling the (“fixed”) pins on the cycloidal disc (even if the view as an outside observer is just the other way round: the cycloidal disc rolls on the fixed pins – just a question of perspective).

\begin{align}
\label{2}
&i=\frac{d}{\delta} \\[5px]
\end{align}

Construction of the cycloidal drive

The cycloidal disc shown above will in the following be used to show the determination of the parameters required for the construction of the cycloidal drive. The reference circle on which the fixed pins are arranged is chosen in this case with D = 160 mm. The pin diameter itself is d_p = 20 mm. A total of N=10 pins are used, which should lead to a transmission ratio of i = 9. The rollers of the roller disc have a diameter of d_r = 14 mm. The rollers themselves are arranged on a reference circle with the diameter d_d = 88 mm. The eccentricity of the rotating cycloidal disc is chosen with e = 4 mm.

In principle, the above-mentioned parameters are arbitrary, but should be chosen sensibly. The cycloidal disc can now be constructed using these parameters:

• reference circle diameter of the fixed ring pins D
• pin diameter d_p
• number of fixed pins N
• transmission ratio i
• diameter of the roller pins d_r
• reference circle diameter d_d of the roller pins on the roller disc
• eccentricity of the cycloidal disc e

Diameter of the rolling circle

The diameter of the rolling circle δ for the construction of the cycloidal disc must first be chosen so that the circumference of the rolling circle corresponds exactly to the pin pitch of the fixed pins. This is the only way to ensure that the spacing of the lobes on the cycloidal disc corresponds to the spacing of the fixed pins on the housing and that meshing is possible. Since the respective pitch circle diameter is proportional to the number of lobes or pins, the diameter δ of the rolling circle must therefore be smaller by the number of fixed pins N than the pitch circle diameter D of the fixed pins:

\begin{align}
\label{3}
&\boxed{\delta=\frac{D}{N}}~(= 16 \text{ mm})  \\[5px]
\end{align}

Thus, in combination with equation (\ref{2}), the base circle diameter d for the construction of the cycloidal disc can be calculated as follows:

\begin{align}
&i=\frac{d}{\delta}=\frac{d \cdot N}{D} \notag \\[5px]
\label{4}
&\boxed{d = \frac{i}{N} \cdot D} ~(= 144 \text{ mm})  \\[5px]
\end{align}

Eccentricity

The distance of the drawing point to the center of the rolling circle during the construction of the cycloidal disc corresponds directly to the later eccentricity e, because this distance ultimately determines the “amplitude” with which the drawing circle oscillates around the base circle during the construction of the cycloid. The eccentricity is always smaller or, in extreme cases, equal to half the diameter of the rolling circle.

\begin{align}
&\boxed{e \le \frac{\delta}{2}}  \\[5px]
\end{align}

In the present case, the eccentricity was chosen with e = 4 mm. However, the eccentricity should not be too small, as otherwise the shape of the cycloidal disc will become too “soft” and in extreme cases will change into a pure circular shape with an eccentricity of e=0. In this case, of course, a positive fit is no longer possible and even if the eccentricity is too low, there is a risk of the cycloidal disc slipping over the pins.

Hole diameter of the cycloidal disc

The eccentricity e in turn influences the hole diameter d_h of the cycloidal disc. On the one hand, the roller pins with the diameter d_r must fit through the holes and on the other hand, the holes must take into account the oscillation of the cycloidal disc with the eccentricity e. Therefore, the diameter of the holes d_h corresponds to the diameter of the roller pins d_r plus twice the eccentricity e:

\begin{align}
\label{exzenter}
&\boxed{d_h = d_r + 2 \cdot e} ~~~~(= 22 \text{ mm}) \\[5px]
\end{align}

The pitch circle diameter on which the holes are arranged around the centre of the cycloidal disc corresponds exactly to the pitch circle diameter on which the rollers are arranged around the centre of the roller disc (d_d = 88 mm)!

The geometry of the cycloidal disc or the cycloidal drive is thus completely determined!

Why does a car need a differential gear?

In automobiles, the wheels are usually driven by the engine using a bevel gear. This allows the rotational motion from the engine to the wheels to be deflected by 90°. If, however, the wheels were rigidly connected to each other by a common shaft, this would lead to problems when cornering. In such a case, the outer wheel must cover a greater distance than the inner wheel. However, since both wheels must travel around the turn at the same time, the outer wheel must rotate faster than the inner wheel.

If the two wheels would be connected by a common shaft, the shaft would twist due to the different rotational speeds. Sooner or later, such a twist is compensated by a slipping of one of the wheels. This slipping in the curve not only reduces driving safety but also leads to considerable tyre wear and, in the long run, to shaft breakage.

When cornering, the outer wheel must be able to rotate faster than the inner wheel!

For this reason, in the early days, only one of the wheels was driven. The other wheel was mounted freely on the shaft so that it could rotate with a different speed. Such a one-sided drive leads however to the fact that the vehicle tries to drive a slight curve. This reduces not only the driving fun but also the driving safety. It was therefore necessary to find a solution to drive both wheels at the same time while allowing different speeds: The differential gear was born.

The picture below shows the differential gear of a truck. One sees the pinion (shown yellow in the animation above) and the bevel gear (shown orange in the animation above). The other bevel gears are inside the housing and not visible from the outside.

Design of a differential gear

The design and operating principle of a differential gear are not easy to understand at first glance. The main question is how to come up with such an arrangement of gears. For the sake of simplicity, it makes sense to first understand the individual steps behind the idea of the differential gear.

1st step – drive of the separated shafts by pins and a freely rotatable bar

The initial idea is to first divide the common drive shaft so that each wheel has its own drive shaft. This ensures that the shaft does not twist if one of the two wheels rotates with a different speed. Two pins are now attached to each of the separate shafts. Between these pins, a freely rotatable bar drives the respective wheel shafts.

In this way, the wheels can be rotated to different degrees within a certain limit. If one of the wheels is slowed down, the opposite wheel can be moved a little further by the rotatable bar. However, the different rotation should not be too large, otherwise the rod will slide out of the pins and no more force can be transmitted.

2nd step – drive of the shafts by several pins and freely rotatable bars

In order to increase the yet very limited motion, one could just use several pins instead of only one as well as more rotatable bars. The pins and the bars can now slide into each other one after the other. The wheel drive is no longer limited. One of the wheels can now rotate at a completely different speed and even stand still, while the other wheel can continue to be driven. In principle, such an arrangement already represents a fully functional differential gear!

A closer look shows that with such a differential, the slowed wheel is decelerated to the same extent as the other wheel is accelerated. The speed loss on one side of the wheel is compensated by a speed gain of the same magnitude on the other side. This principle is based on the law of conservation of energy.

Such a kinematic behaviour of the wheels is exactly what is needed when cornering. When cornering, the inner wheel must rotate more slowly to the same extent as the outer wheel must rotate faster.

A differential gear ensures that the inner wheel rotates to the same extent more slowly as the outer wheel rotates faster when cornering!

3rd step – Replacing pins and bars with bevel gears

The power transmission by pins and bars is not very effective. Therefore they are replaced by gears, more precisely by bevel gears. The bevel gear shown in blue, which revolves around the shafts of the wheels, is also referred to as the spider gear. In principle, this spider gear is nothing else than a planet gear as it is known from planetary gears. And indeed, the differential gear can be seen as a special form of a planetary gearbox (more on this later).

4th step – drive of the shafts by further bevel gears

The drive of the spider gear is of course not done by hand but by the engine. The spider gear is in turn driven by a bevel gear unit (usually a hypoid gear), consisting of a pinion (shown in yellow) and a bevel gear (shown in orange). The spider gear ist mounted on this orange bevel gear. Since the orange bevel gear “carries” the revolving spider gear, the orange bevel gear is also referred to as carrier.

5th step – symmetrical arrangement of the bevel gears to avoid bending stresses

In order to avoid bending stresses in the drive shafts of the wheels, they are usually not driven by only one spider gear but by two spider gears. The second spider gear is offset by 180°.

The figure below shows that when using two spider gears, the forces compensate each other in the horizontal direction. The drive shafts of the wheels are then subjected purely to torsion, but not to bending!

Kinematics of a differential gear

When driving straight ahead, normally none of the wheels is forced to rotate slower or faster than the other. In this case, the spider gears drive the wheel shafts without any relative motion. The wheels then rotate at the same speed as the carrier.

If one now drives into a right turn, for example, the inner wheel is slowed down by the shorter distance to be travelled. However, the outer wheel must then rotate faster to the same extent, since it has to cover a greater distance. Due to its special design, a differential gear ultimately ensures exactly such a kinematic behavior! The exact mathematical relationship is explained in more detail in the next section.

The best way to understand the kinematics is to imagine an extreme cornering where the inner wheel practically stands still and the outer wheel follows a circular path around the inner wheel. In this case, the carrier drives the spider gears around the bevel gear (“side gear”) of the stationary wheel shaft. The spider gears then begin to rotate and now perform relative motions. The opposite bevel gear (“side gear”) of the left drive shaft is now driven by this rotation of the spider gears in addition to the already existing rotation of the carrier and thus rotates faster.

Compared to the carrier, the inner wheel rotates more slowly to the same extent as the outer wheel rotates faster when cornering.

Only when cornering is complete and the wheel speeds have been adjusted again, do the two wheel shafts no longer move relative to each other and the speed of the carrier corresponds to the wheel speeds.

Even if the speeds of the wheels differ when cornering, both wheels are always driven by the same torque! This is because in gearboxes the change in torque only results from the ratio of the number of teeth of the gears. However, the differential gear has a symmetrical design. It does not differ in the number of teeth between the left and right drive shaft. This means that the change in toraue between the motor and the drive shafts are always the same. Both gears therefore have the same torque.

Even if the respective torque at the wheels does not differ, they have different powers! This is because the power is determined by the product of torque M and rotational speed n:

\begin{align}
\boxed{P=2 \pi \cdot M \cdot n} \\[5px]
\end{align}

It should be noted, however, that when the differential is active when cornering, there are relative motions of the bevel gears which lead to an additional reduction in gear efficiency.

Although a differential gear provides different speeds and thus different power for the wheels, the torque on both wheels is identical!

Differential gear as a special case of a planetary gearbox

As already mentioned, a differential gear is a special type of a planetary gearbox. One of the bevel gears on the wheel shafts can be regarded as a sun gear while the other bevel gear then corresponds in a figurative sense to the ring gear.

Since a differential is a special type of a planetary gearbox, the relationship between the different rotational speeds can also be described by the fundamental equation for planetary gears (Willis equation):

\begin{align}
&\boxed{ n_s = n_c \cdot \left(1-i_0 \right) + n_r \cdot i_0} \\[5px]
\end{align}

For classic planetary gears, n_r refers to the rotational speed of the ring gear, n_s denotes the rotational speed of the sun gear and n_c refers to the rotational speed of the carrier. i₀ denotes the so-called fixed carrier transmission ratio.

In the case of a differential gear, the fixed carrier transmission ratio corresponds to the transmission ratio which is obtained when the carrier is fixed. If one of the wheels (the “ring gear”) is rotated in this state, then the other wheel (the “sun gear”) obviously rotates at the same speed, but in the opposite direction. The fixed carrier transmission ratio is therefore i₀=-1.

If the fixed carrier transmission ratio of i₀=-1 is used in the upper equation, then the following relationships apply:

\begin{align}
& n_s = n_c \cdot \left(1-i_0 \right) + n_r \cdot i_0 ~~~\text{with}~i_0=-1~~~~\text{:}  \\[5px]
&n_s = n_c \cdot \left(1-(-1) \right) + n_r \cdot (-1) \\[5px]
&n_s = n_c \cdot 2 – n_r \\[5px]
&n_r + n_s = 2 \cdot n_c \\[5px]
\end{align}

Since differential gears do not have a classic sun gear or ring gear, the corresponding rotational speeds of the gears are denoted by n₁ (=n_r) or n₂ (=n_s). Thus, the following relationship between the rotational speeds of the wheels n₁ or n₂ and the rotational speed of the carrier n_c applies:

\begin{align}
&\boxed{n_1 + n_2 = 2 \cdot n_c} \\[5px]
\end{align}

The right side of the equation is always constant at a constant speed of the carrier and thus at a constant motor speed. Now it can also be seen mathematically that at a constant motor speed, a reduction of the speed at one of the wheels results in an increased speed at the opposite wheel. By rearranging the equation, one can also see that the speed of the carrier corresponds to the mean speed of the two wheels.

\begin{align}
&\boxed{n_c = \frac{n_1 + n_2}{2}} \\[5px]
\end{align}

Differential lock

The big advantage of differential gears is that they can be used when cornering by dividing the rotational speed or power between the respective wheels according to their needs. In some situations, however, this can also be a disadvantage. For example, when starting on a smooth or slippery ground, one of the wheels may lose its grip and slip, while the other wheel remains on the ground. The differential gear now transmits the entire power to the rotating wheel, while no power is at the stationary wheel. The spinning wheel now turns at double speed, while the other wheel stands still. In this way one hardly obtains a forward driving force and if then only a one-sided force due to the sliding friction of the rotating wheel.

Such a case where one of the wheels has less grip than the other and is thus tempted to slip, occurs primarily during off-road driving, where the load on the wheels varies permanently. But even in fast cornering, where the inner wheel is greatly relieved by the centrifugal forces, the danger of slipping increases and the one-sided power distribution threatens. If, in the worst case, the vehicle tilts slightly and the inner wheel loses its grip, this wheel receives full power and rotates in the air at twice the speed. The opposite wheel, which still has the grip to the ground, does not get any power and therefore no drive of the car is possible anymore.

In the cases mentioned above, a differential gear is therefore more of an obstacle. For this reason, mainly off-road vehicles are equipped with so-called differential locks. Such a differential lock then rigidly connects the two drive shafts of the wheels with each other again and thus deactivates the differential. However, this leads to the twisting of the drive shaft when cornering, as already explained at the beginning. Differential locks should therefore only be activated in exceptional cases.

Assembly and function

The figure below schematically shows the assembled components of a three-speed gear hub as it can be found in bicycles.

The centerpiece of the gear hub is the planetary gear. The sun gear is fixed to the hub axle, which in turn is connected to the bicycle frame. The hub axle and the sun gear therefore do not perform any rotation during the ride. The planet gears run around the sun gear, which in turn are attached to the carrier and enclosed by the ring gear.

The shift rod is guided through the hollow hub axle and is provided with a shift pin at its end. The shift pin is guided in an elongated hole (slot). Depending on the gear, the shift pin can bring an axially displaceable sliding clutch into different positions by spring force. The position of the sliding clutch is controlled by a pull chain (usually with a wire rope connected) at the end of which the shift lever is located.

The sprocket is driven by the bicycle chain. It sits on the driver and is firmly connected to it. During rotation, the driver takes the axially displaceable sliding clutch with it in a positive fit.

For power transmission, the sliding clutch is provided with splines (“teeth”) on the circumference, which engage positively in the driver. Depending on the selected gear, the rotation of the sliding clutch is transferred – also by splines – either to the ring gear (1st and 2nd gear) or to the carrier (3rd gear). The different gears are described in more detail in the next section.

Switching of gears

In this section, the shifting of the three speed internal gear hub is explained in more detail. In all gears, the sprocket first drives the driver, which in turn transmits the torque to the sliding clutch. Depending on the position of the sliding clutch, this results in a total of three gears.

1st gear

In the first and second gear, the sliding clutch engages in the ring gear of the planetary gear unit. In the first gear, the ring gear drives the planetary gears around the fixed sun gear. This drives the carrier on which the planet gears are mounted. The ratchets attached to the carrier engage in the recesses of the hub sleeve and thus drive it. The hub rotates and so does the tire attached to it.

In the first gear, the carrier rotates at a lower speed compared to the ring gear, whereby the ring gear rotates at the frequency of the sprocket. This results in a transmission ratio greater than one. The torque is increased to the same extent as the speed is reduced. This provides a relatively large force on the rear wheel as is usual for a first gear.

2nd gear

To shift to second gear, the shift lever is “loosened” and the shift cable is extended. Both the sliding clutch and the overrunning clutch move forward due to the force of the spring. This also causes the ratchets of the overrunning clutch to now engage in the recesses of the clutch housing (in 1st gear the ratchets are retracted in the clutch housing and are therefore not engaged). The clutch housing is screwed directly into the hub sleeve and thus firmly connected to it.

The torque generated by the sprocket is thus transmitted directly to the hub via the driver, sliding clutch and clutch housing. It is basically a “rigid” connection of all power-transmitting components. The speed or torque of the sprocket thus corresponds exactly to the speed or torque of the rear wheel. The planetary gear is thus bypassed and a 1:1 ratio is obtained (direct drive).

Note: Although the carrier is also driven simultaneously by the ring gear, the overrunning clutch of the carrier sets the carrier into a so-called freewheeling mode and thus does not participate in power transmission (for further information see section overrunning clutch).

3rd gear

When shifting into third gear, the shift cable is “loosened” again by switching the gear lever. However, since the overrunning clutch of the ring gear is already at the stop, it cannot disengage any further. Only the slipping clutch pushes further forward and drives out of the ring gear and engages in the carrier.

In this way, the carrier is now driven by the sprocket and drives the planet gears around the fixed sun gear. Accordingly, the ring gear starts to move and rotates at a higher speed than the carrier. Finally one gets a transmission ratio smaller than one.

The higher rotational speed of the ring gear is transmitted to the clutch housing and then to the hub by the overrunning clutch attached with pins. As in second gear, in third gear the bridge, which rotates slower than the hollow gearwheel, is moved back into the freewheel via the overrunning clutch.

The third gear thus behaves exactly the opposite of the first gear in terms of transmission ratio. In the first gear the carrier is driven by the ring gear and in the third gear the carrier drives the ring gear. Thus the transmission ratios are reciprocal to each other.

Overrunning clutch

The ratchets mentioned in the previous section are part of the so-called overrunning clutch. This overrunning clutch has the task of enabling power transmission in one direction of rotation and locking in the opposite direction, i.e. not enabling power transmission.

The ratchets are simply inserted into the recesses of the overrunning clutch and secured with a wire ring. This wire ring also serves as a kind of “spring”, which is attempted to erect the ratchets. This ensures that the ratchets immediately engage in the provided “ramps” of the clutch housing.

Due to the “ramp shape”, the hub sleeve can also rotate when the sprocket and thus the entire drive train is at a standstill. This has the advantage that the pedals do not have to rotate permanently while driving, but can stand still, e.g. when driving downhill. In the case of such a freewheeling, the ratchets simply slide over the “ramps” of the sawtooth profile. This also explains the typical “ratcheting” noise of stationary pedals and rotating tires. If the rotational speed of the overrunning clutch is increased again, the ratches engage in the clutch housing again by the spring mechanism when the rotational speeds are adjusted.

When freewheeling, the clutch housing can “overrun” the ratchets, so to speak, but not vice versa, which gives this device the name “overrunning clutch” (also known as a locking device or freewheel).

A total of two overrunning clutches are installed in the internal-gear hub. Another one is located directly on the carrier. This is necessary because the carrier of a planetary gear always rotates more slowly than the ring gear. Especially in the second and third gear, the output takes place through the ring gear, which must be able to rotate faster than the carrier. In these cases the carrier must be set into a freewheel (for reasons of clarity the carrier is not shown in the animation below but only its ratchets and the wire ring).

Superposition of motions

The change in speed of planetary gearboxes are no longer as easy to understand as those of stationary transmissions. This is due to the fact that the motion of the rotating planet gears is ultimately a superposition of three different motions. The motion no longer consists of a simple rotation around its own axis, but the axis itself performs an additional circular motion around the axis of the sun gear, while the planet gear also performs an additional circular motion because of the rotation of the sun gear.

Thus, the motion of a rotating planet gear can be traced back to the superposition of three separately observable motions:

1. rotation of the carrier around the sun gear
2. rotation of the planet gear around its own center of gravity
3. rotation of the sun gear

1	2	3
Motion of the carrier	Motion of the planet gear	Total motion

However, the motions are not independent from each other, because the planet gear rotates on the sun gear. Thus the diameter ratio between the sun gear and the planet gear determines how often the planet gear rotates around its own axis while it moves once around the sun gear.

In order to derive the relationship of the rotational speeds between the sun gear, the planet gear and the carrier, the above-mentioned motions are first described separately and then superposed. For the sake of clarity, the gears are assumed to be (pitch) cylinders.

Rotation of the carrier around the sun gear

If the sun gear stands still and the planet gear is locked firm on the carrier, then the swept angle of the carrier φ_c corresponds to the angular position of the planet gear φ_p1.

\begin{align}
\label{P1}
&\underline{\varphi_{p1} = \varphi_c}  \\[5px]
\end{align}

Rotation of the planet gear around its own center of gravity

In fact, the planet gear will roll on the sun gear when mounted rotatably on the carrier and thus rotate around its own center of gravity. The planet gear will thus rotate by an additional angle φ_p2.

If one considers a mere rolling motion, then the arc length b_c, which the carrier has covered on the sun gear, corresponds exactly to the arc length b_p2, by which the planet gear has moved on its circumference. The additional angle φ_p2 can be determined by the radian measure as follows:

\begin{align}
&b_{p2} = b_c \\[5px]
&\tfrac{d_p}{2} \cdot \varphi_{p2} = \tfrac{d_s}{2} \cdot \varphi_c  \\[5px]
\label{P2}
&\underline{\varphi_{p2} = \frac{d_s}{d_p} \cdot \varphi_c}  \\[5px]
\end{align}

Rotation of the sun gear

The carrier is now held in position and the sun gear is rotated clockwise by an angle φ_s. In this case, the planet gear will turn counterclockwise by an angle φ_p3. Analogous to the case before, the following statement applies: The arc length b_s at the circumference of the sun gear corresponds to the arc length b_p3, by which the planet gear has moved on its circumference:

\begin{align}
&b_{p3} = – b_s \\[5px]
&\tfrac{d_p}{2} \cdot \varphi_{p3} = – \tfrac{d_s}{2} \cdot \varphi_s  \\[5px]
\label{P3}
&\underline{\varphi_{p3} = – \frac{d_s}{d_p} \cdot \varphi_s}  \\[5px]
\end{align}

The negative sign indicates that the motion of the planet gear is in the opposite direction to the motion of the sun gear.

Superposition of the different motions

The motions of the planet gear according to the equations (\ref{P1}), (\ref{P2}) and (\ref{P3}), which have been considered separately so far, can now be superposed to the total motion:

\begin{align}
&\varphi_p = \varphi_{p1} +\varphi_{p2} + \varphi_{p3}\\[5px]
\label{P}
&\underline{\varphi_{p} = \cdot \varphi_c + \frac{d_s}{d_p} \cdot \varphi_c     –    \frac{d_s}{d_p} \cdot \varphi_s   }  \\[5px]
\end{align}

The angular positions φ contained in this equation result from the respective angular velocity ω and the elapsed time t (φ=ω⋅t), whereby the angular velocity is directly related to the rotational speed n by ω=2π⋅n:

\begin{align}
&\varphi = \omega \cdot t  ~~~ \text{with} ~~~ \omega = 2 \pi \cdot n ~~~\text{applies:}  \\[5px]
\label{varp}
&\underline{\varphi = 2 \pi \cdot n  \cdot t}  \\[5px]
\end{align}

If equation (\ref{varp}) is used in equation (\ref{P}), the following relationship ultimately results between the rotational speed of the planet gear n_P and the rotational speeds of the sun gear n_s and the carrier n_c:

\begin{align}
&2 \pi \cdot n_p  \cdot t = 2 \pi \cdot n_c  \cdot t + \frac{d_s}{d_p} \cdot 2 \pi \cdot n_c  \cdot t – \frac{d_s}{d_p} \cdot 2 \pi \cdot n_s  \cdot t \\[5px]
&n_p =n_c + \frac{d_s}{d_p} \cdot n_c – \frac{d_s}{d_p} \cdot n_s ~~~~~~~~\text{|} \cdot d_p \\[5px]
&n_p \cdot d_p =n_c \cdot d_p + d_s \cdot n_c – d_s \cdot n_s  \\[5px]
\label{g}
&\boxed{n_p \cdot d_p = n_c \cdot \left(d_p + d_s \right) – n_s \cdot d_s} \\[5px]
\end{align}

Since the pitch circle diameter d of a gear is directly proportional to the number of teeth z, the equation above can also be expressed by the respective number of teeth:

\begin{align}
\label{pln}
&\boxed{n_p \cdot z_p = n_c \cdot \left(z_p + z_s \right) – n_s \cdot z_s} \\[5px]
\end{align}

This equation is called the fundamental formula of planetary gears  (also called Willis equation). The Willis equation is used to determine the different transmission ratios depending on the mode of operation, which will be explained in more detail in the article Willis equation for planetary gears.

Stationary gearbox

So-called stationary gearboxes are characterised by the fact that the gear wheels have stationary axes of rotation. The figure below shows a 2-stage stationary transmission with three gears. An input gear drives an output gear by an idler gear (intermediate gear).

The transmission ratio i₁ or i₂ of the respective gear stages is determined by the ratio of the corresponding number of teeth on the idler gear z_idler and input gear z_in or output gear z_out.

\begin{align}
&\text{1. Stage:} ~~~i_1 = \frac{z_{idler}}{z_{in}}  \\[5px]
&\text{2. Stage:} ~~~i_2 = \frac{z_{out}}{z_{idler}} \\[5px]
\end{align}

The total transmission ratio i_total of the 2-stage stationary gear unit (therefore also referred to as stationary transmission ratio or fixed carrier train ratio) is obtained by multiplying the transmission ratios i₁ and i₂:

\begin{align}
&i_{total} =i_1 \cdot i_2 =\frac{z_{idler}}{z_{in}} \cdot \frac{z_{out}}{z_{idler}}  \\[5px]
\label{i}
&\boxed{i_{total} = \frac{z_{out}}{z_{in}}} ~~~\text{stationary transmission ratio} \\[5px]
\end{align}

For the overall transmission ratio, obviously only the number of teeth of the output gear and the number of teeth of the input gear are of relevance! The number of teeth of the intermediate gear does not matter.

From stationary gearbox to planetary gearbox

Internal toothed gear as output gear

Instead of the externally toothed output gear, a ring gear with internal toothing can also be used in principle. As long as the number of teeth is not changed, this has no effect on the overall transmission ratio according to the equation (\ref{i}). Only the sense of rotation of the output gear will change as a result.

In general, the axes of rotation of the input shaft and the output shaft are not perfectly aligned on a common axis, but have an offset. However, by cleverly choosing the diameter and thus the number of teeth of the idler gear, it is possible to ensure that the input shaft and the output shaft are on a common axis.

Note that the number of teeth of the idler gear according to the equation (\ref{i}) has no effect on the overall transmission ratio anyway and can therefore be arbitrarily chosen in principle!

If the input and output shafts are to be coaxial (“coaxial” = “aligned on a common axis”), the pitch circle diameter of the intermediate gear must correspond exactly to the difference between the pitch circle radii of the output and input gear. Since the number of teeth is directly proportional to the pitch circle diameter, the respective number of teeth can be used instead of the pitch circle diameter. As a result, the number of teeth of the idler gear must be just half the difference between the number of teeth of the output and input gear.

\begin{align}
&d_{idler} = r_{out}-r_{in}=\frac{d_{out}}{2}-\frac{d_{in}}{2}=\frac{d_{out}-d_{in}}{2} \\[5px]
&\boxed{z_{idler} =\frac{z_{out}-z_{in}}{2}} \\[5px]
\end{align}

Inserting additional idler gears

A disadvantage of the present gearbox is that the input shaft (as well as the output shaft) is subjected to a bending load by the one-sided flank force. In the figure below, F denotes the reaction force of the idler gear acting on the tooth flank of the input gear.

However, bending stresses can be avoided if several intermediate gears are arranged symmetrically so that the flank forces compensate each other in their bending effect. In the case obtained with three idler gears, the input and output shafts are no longer subjected to bending load but only to torsion.

Mounting the idler gears on a carrier

In principle, the present gearbox already forms the pre-stage of a planetary gear. The last step consists only of mounting the idler gears on a so-called carrier. The carrier itself is connected to a shaft and is guided coaxially through the output shaft, which is designed as a hollow shaft.

With the carrier, the planetary gear is in principle complete. In this operating mode, it has a stationary transmission ratio (fixed carrier train ratio) according to the equation (\ref{i}). In terms of function, in this operating state the planetary gear does not yet differ from the stationary gearbox described above.

However, this changes if the planetary gear is used in a different way. This is because the output of the planetary gearbox does not always have to take place at the aforementioned “output shaft”. It is also possible to use the carrier as the output shaft, while the hollow shaft of the internally toothed gear is firmly locked. In this case, the idler gears now “orbit” the centrally located drive gear like planets a sun; hence the term planetary gear. The transmission ratio of the planetary gear in this mode of operation now differs from that of the stationary gearbox!

The gearwheels previously referred to as “idler gears” are generally referred to as planet gears and the centrally located external gear is referred to as the sun gear. The internally toothed gear wheel is generally called ring gear or annulus.

In contrast to stationary gears, planetary gears (also called epicyclic gears) are characterized by moving axis of rotation!

At this point, the necessity of a symmetrical arrangement of the planet gears becomes also apparent, as otherwise enormous unbalance forces would occur at high rotational speeds.

With a planetary gear it is not only possible to lock firm the ring gear and let the output take place through the carrier. Many other variants are possible, each with a different transmission ratio (see next section “Transmission ratios”). This makes the planetary gear particularly suitable for transmissions such as hub gears of bicycles or automatic transmissions of automobiles.

The advantage of planetary gears over conventional stationary transmissions is their compact design and the advantage that the shafts are all coaxial. For very large transmission ratios, it is also possible to connect several planetary gears in series.

Transmission ratios

With a planetary gear, different transmission ratios can be achieved, depending on the shaft at which the input or the output takes place and the gearwheel that is locked firm. The different transmission ratios will be explained using the example of the planetary gear shown below. The sun gear has z_s = 12 teeth, the planet gears z_p = 18 teeth and the ring gear z_r = 48 teeth.

The derivation of the transmission ratios shown in the following will be discussed in detail in a separate article due to their complexity.

Fixed ring gear

The highest transmission ratio of i=5 is obtained in this case when the gearbox is driven by the sun gear and the ring gear is fixed. The output then takes place by the carrier. Obviously, the lowest transmission ratio of i=0.2 (=1/5) is obtained when input and output are reversed while the ring gear remains fixed. The direction of rotation of the input and output shafts is maintained in both cases.

Fixed carrier

The second highest transmission ratio of i=4 is achieved in the present case if the planetary gearbox is still driven by the sun gear, but this time the carrier is fixed and the output is done by the ring gear. Consequently, the second lowest transmission ratio of i=0.25 (=1/4) is obtained by reversing the input and output. In both cases, however, the sense of rotation between input and output shaft is different! In such a case the transmission ratio has a negative sign (see table below). In this way a reverse gear can be generated.

Fixed sun gear

A further transmission ratio is obtained when the planetary gearbox is driven by the ring gear and the sun gear is fixed. In this case the carrier serves as the gearbox output. The transmission ratio now becomes i=1.25, while the sense of rotation is remained. If the output and the input of the gearbox are reversed, the transmission ratio is i=0.8 (=1/1.25).

Direct drive

With a planetary gear a so-called direct drive is also possible. All components of the planetary gear are then firmly connected to each other. The transmission ratio in this case is i=1. Such a direct drive is used, for example, in three-speed gear hubs as the “2nd gear”.

Summary

The various transmission ratios are summarised in the table below. In round brackets, the transmission ratios are given for reversed input and output (reciprocal transmission ratio). Negative signs indicate that the sense of rotation changes.

In manual transmissions, the shifting of various transmission ratios is performed by clutches that allow certain components (sun gear, carrier or ring gear) to be fixed depending on the desired gear. For functional reasons, however, not all of the transmission ratios shown in the table above can be achieved with a single planetary gear (planetary gear set). However, the transmission ratios can be increased enormously if several individual planetary gear sets are combined. In practice, up to three planetary gear sets are common in one gearbox.