Paradoxical as it may seem at first glance, a convex curvature of a pulley causes a flat belt to keep track and not jump off!

To understand this paradoxical behavior, let’s take a closer look at the belt on a crowned pulley. For this purpose, the figure below shows a flat belt attached to a cylindrical pulley (driving pulley) and to a crowned pulley (driven pulley). The curvature is greatly exaggerated for the sake of clarity. In practice, the curvature on pulleys is hardly visible and is often only a fraction of a millimeter.

Due to the curvature, the belt is tensioned differently on the two sides. The side of the belt closer to the center is tensioned more than the other side. This difference in tension means that the belt no longer wraps symmetrically around the pulley, but curves. This effect can be demonstrated relatively easily with a rubber band or even an elastic cloth. If you pull the rubber band apart on one side, the result is not a straight line, but the band curves outwards and forms an arc!

This effect of the bulge on the more tensioned side is also seen on the belt when it wraps around the crowned pulley. The belt is, so to speak, superelevated. This effect is all the greater the greater the differences in the tensions of the two sides of the belt and thus the more the pulley is curved.

Because the pulley and the belt itself are curved, the side of the belt with the bulge hits the pulley first.

This point of the belt, which runs onto the pulley first, is also under high tension and thus sticks very strongly to the pulley. This point, however, is tempted to run along a circular path around the pulley (line drawn in white with contact points marked in red). In this way, the belt thus pulls itself up towards the center of the pulley.

As long as the belt does not lie symmetrically in the center of the pulley, the more tensioned side will continue to bulge and thus always pull the belt further upwards. This effect only disappears when the belt lies symmetrically in the center of the pulley and both sides of the belt are equally tensioned. In this case, there is no bulge and the belt does not pull in any direction. This leads to the self-centering effect of the crowned pulleys and to the fact that the belt does not jump off the pulley.

Note that such a self-centering effect is only present with flat belts as long as the belt width is significantly greater than the belt height. With round belts or belts with a square cross-section, this self-centering effect no longer works with curved pulleys.

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.

Advantages of belt drives

Compared to a gear drive, a belt drive can be used to bridge greater distances between two shafts in a simpler way. Chain drives also offer this advantage and are therefore used for bicycles where a relatively large distance has to be covered between pedal and rear wheel.

Frictionally operating belts such as flat belts or V-belts also offer a natural overload function. In contrast to gear drives, overload simply causes the belt to slip through (sliding slip). This protects the transmission from major damage. In the worst case, only the belt needs to be replaced and not the entire gears and shafts as in the case of a damaged gear drive.

Another advantage of belt drives is the elasticity of the belts compared to rigid gears. This offers good damping characteristics (shock absorption), especially in the case of sudden torque changes. This is why belt drives are used, for example, in grinding plants or stone crushers. The starting and stopping behaviour is also damped accordingly and is not as jerky as with rigid gear drives. Note, however, that a high elasticity of the belt also results in increased elastic slip. Belts can therefore not be made too elastic, but neither can they be designed too inelastic, as otherwise the positive shock absorption properties would be missing.

An additional advantage of belt drives over gear drives is their insensitivity to angular misalignment as long as the axes continue to run in a parallel plane to each other. In many cases, such a misalignment is even deliberate. This makes it easy to redirect the direction of rotation. If the axis of the output shaft is turned by 180° and the belt is crossed, the original direction of rotation can easily be reversed. In contrast to an open belt drive, this is also referred to as a crossed belt drive.

Belt drives do not have to be lubricated in comparison to gear drives. This reduces maintenance costs accordingly. Belt drives also have lower noise emissions than gear drives, since no metallic teeth engage but only relatively soft, elastic belts drive the pulleys. This enables the transmission of high rotational speeds.

In addition, pulleys are usually not complete solid wheels, as is often the case with gears. Pulleys usually have recesses to reduce weight and manufacturing costs. As a result, belt drives are generally lighter than comparable gear drives.

Disadvantages of belt drives

However, the above-mentioned advantages of belt drives are also countered by disadvantages. Depending on the ambient conditions, belts are subject to more or less severe ageing phenomena, i.e. they lose their elastic properties over time and must be replaced. For this reason, belts can only be used within a certain temperature range. In addition, over time the belts become plastically stretched, so that they have to be re-tensioned at regular intervals.

Another disadvantage of some belt types such as flat belts or V-belts is the associated slip, which reduces the efficiency of the transmission accordingly. Slippage can only be prevented with timing belts due to the positive force transmission.

In some cases, the increased space requirement of a belt drive compared to a gear drive can also have a disadvantage. This is due to the fact that the belt pulleys cannot be placed directly against each other, while the toothed wheels of gear drives can even mesh with each other and thus be set up in a more space-saving manner. In addition, the wrap angle decreases with decreasing centre distance, so that wrapping can become unacceptably small. Although this can be compensated by idler pulley, it not only increases the design effort but may also increase the required space again.

Why mus belt tension be assured?

The importance of the belt tension for power transmission has already been explained in the article How does a belt drive work?. Despite the pre-tension (initial tension), however, the belt tension will change during operation due to plastic deformation or temperature. For this reason, belt drives must often be kept on tension by so-called tensioning systems.

It must also be borne in mind that a belt will have to be serviced over time and therefore have to be removed from the pulley and remounted. This is hardly possible when the belt is under tension, so that the tension must be removed when the belt is changed and must be tensioned again after installing by means of tensioning devices (belt tensioners).

Tensioning systems are used to generate and maintain the belt tension and thus ensure reliable power transmission!

Tensioner, idler and guiding pulleys

The belt tension can be maintained during operation, for example, by means of tensioner pulleys. Tensioning pulleys also serve to cushion heavy load changes. In addition, the wrap angle can be increased by tensioner pulleys. Tensioner pulleys are also often used for long belt lengths to prevent excessive belt vibration.

If such pulleys are merely used to deflect the belt, then these are generally referred to as idler pulleys. Idler pulleys are used, for example, in multiple drives in which one driving pulley drives several other pulleys. Idler pulleys can also be used for long belt lengths to reduce belt vibrations. Such pulleys can also take over the function of a guide at the same time, so that the belt does not jump off the pulley. Such pulleys are called guiding pulleys or guide rollers. Guiding pulleys often have protrusions (called flanged pulleys) on the left and right, between which the belt is held in track.

Tension pulleys do not yet have a tensioning effect; certain devices are required to achieve a tensioning effect (also known as tensioning systems). Tensioning devices are available in a wide variety of designs. The tensioning systems shown in the figures above were each designed with a simple spring mechanism. The springs offer the advantage that the belt tension can adapt dynamically to the operating state, e.g. strong load changes.

Eccentric tensioner pulley

Another way of generating a belt tension is by an eccentric mounting of the tensioner pulley. The desired belt tension can then be set by rotating the pulley to a certain position. In addition, torsion springs can be installed in the pulley, which then allow dynamic adjustment of the belt tension during load changes.

Hydraulic damping tensioners

Another way of tensioning a belt is by means of a lever arm to one end of which the tensioner pulley is attached and held in place by a spring.

However, if frequent load changes occur during operation, this tensioning system can be excited to strong vibrations. Hydraulic damping systems such as those found in door closers, for example, can therefore be used to absorb shocks. A piston is then placed in oil, which provides the necessary damping effect due to its viscosity.

Motor slide base

Besides the use of tension pulleys, the belt tension can also be applied by adjusting the driving pulley itself. When so-called motor slide bases are used, the entire motor is mounted on a movable slide which runs in a fixed guide.

The position of the carriage on the guide and thus the belt tension can be adjusted. However, if the belt tension decreases or if there are strong load changes, the motor slide base does not automatically adapt to the changed conditions but must be readjusted manually.

Pivoting motor base

A dynamic adaptation of the belt tension to the existing load conditions (or to plastic expansion processes) can be achieved by using a self-tightening motor base. The motor is screwed onto a pivoting motor base, whereby the centre of gravity of the entire system is designed in such a way that the motor tends to tilt backwards. At an inclination of about 15° to 20°, the motor base with its weight ensures a permanent and almost constant belt tension.

With the bevel gears considered so far, the axes of the ring gear and pinion intersect at a certain point. In these cases, an offset of the axes (skew axes) cannot be realized.

However, if rotational motions are also to be transmitted between axes that do not intersect, the pinion of a bevel gear unit in particular must be designed differently when the axes are offset. In particular, one no longer gets a rolling motion but a screwing motion. The pinion then forms a conical screw gear, which is then called a hypoid gear (hypoid gearbox).

With hypoid gears, rotary motions between non-intersecting axes can be realized!

As can be seen from the animation below, the positive offset increases the spiral angle of the screw gear. A positive offset means that the axis of the screw gear is shifted in the direction of the curved flanks of the ring gear (here: downwards). In the case of a negative offset, however, the screw gear is shifted against the curved flanks of the ring gear (here: upwards). If the offset is positive, the diameter of the pinion increases; or decreases if the axis offset is negative.

As the axial offset increases, the tooth line of the pinion must twist more strongly because the curved teeth of the ring gear are then inclined more strongly against the axis of the pinion.

Due to the stronger spiral shape of the teeth in the case of a positive offset, it is also achieved that several teeth are simultaneously involved in meshing (higher overlap ratio). This not only allows higher torques to be transmitted than with normal bevel gears, but also significantly reduces noise emissions.

Hypoid gears have higher load capacities and lower noise emissions than conventional bevel gears!

Hypoid gears are therefore used, for example, in differentials in the automotive industry. The figure below shows one of a total of two differential gears of a truck for the rear-wheel drive. The driving hypoid pinion and the driven ring gear with spiral toothing can be seen. The shown gearbox has a mass of about 150 kg.

The hypoid gear can ultimately be regarded as a mixture between a bevel gear and a worm gear and accordingly combines features of both variants. In particular, the bevel-shaped base bodies used for bevel gears and the screwing motion in worm drives.

Note that the basic bodies of the hypoid gears are no longer pitch bodies in the true sense. This is because the power transmission no longer takes place rolling in any contact point of the flanks but purely screwing, i.e. the tooth flanks slide permanently onto each other. Because of this gliding process, which is typical for screwing motions, hypoid gears require special lubrication with so-called hypoid gear oils.

Hypoid gears must be specially lubricated due to the screwing power transmission!

For cylindrical gears, a basic distinction can be made between external gears and internal gears. In the case of external toothing, the teeth are directed outwards on the circumference. In the case of internal gears, the teeth are directed inwards. An internal gear wheel is sometimes simply called a ring gear (although a ring gear can also have an external toothing!).

While the direction of rotation changes when two externally toothed gears are used, the sense of rotation remains the same when pairing with a internal gear. In addition, the centre distance can be shortened by using a ring gear with internal toothing instead of external toothing (with maintaining  the transmission ratio). This makes a space-saving gear design possible. Under certain circumstances, internally toothed gears can also offer better protection against dirt due to the internal teeth, if the gear unit has been designed accordingly.

The counterpart of the tooth flank profile of an external gear corresponds in principle to the tooth flank profile of an internally toothed gear. Thus the tooth profile of an external toothing is always convex, i.e. they have an outwardly curved shape (external curvature). With internal toothing, however, the tooth profile is concave, i.e. they are arched inwards (internal curvature).

When two external gears are meshing, a relatively narrow contact surface results due to the purely convex pairing of the tooth flanks. This in turn leads to a high tooth load (also called Hertzian contact stress). Therefore, the wear of the gears and tooth flanks is very high.

If, however, an externally toothed gear is paired with an internally toothed gear, the result is a convex/concave-flank pairing. The contact surfaces “nestle” up against each other, so to speak. This results in a larger contact area, which in turn results in lower tooth load. This reduces the wear of the gears. Conversely, this means that higher torques can be transmitted with the same wear with internal toothing than with the pairing of two externally toothed gears.

Even though internal gears offer many advantages compared to external gears, internal toothing is limited to a few special cases due to the relatively complex and thus expensive production. Internal gearing is used, for example, in planetary gears (epicyclic gears).

Herringbone gears

In order to combine the advantage of helical gears (higher load capacity and lower noise emission) with the advantage of spur gears (no axial forces and lower wear), so-called herringbone gears are used in special cases.

Due to the reciprocal arrangement of the helixes, each side generates an opposing axial force, which cancel each other out. This prevents axial thrusts that would have to be absorbed by bearings.

Due to the relatively long tooth length (due to the inclination), high torques can be transmitted with herringbone gearings. However, the complex and thus expensive production of such gear types is limited to special applications. (e.g. for large transmissions). Furthermore, subsequent fine machining of the teeth (e.g. by grinding) is almost impossible due to the difficult accessibility.

Herringbone gears allow high torques to be transmitted without generating axial forces. The bearing wear is correspondingly low. The production of such a gearing is very complex and therefore expensive!

Due to the complicated manufacturing process, the double helical gearing described below is often used in practice instead of herringbone gearing.

Double helical gears

The same effect as with herringbone gears is in principle achieved by the mirror-image arrangement of two helical gears, whose respective tooth flanks then also taper in the shape of an arrow. Such gears are then referred to as double helical gears.

The respective helix halves are produced on a common shaft, whereby a groove must exist in the middle for the manufacturing tool to exit. The production of a double helical gear is cheaper than the production of a herringbone gear.

Double helical gears consists of the mirror-image production of two helical gearing, with a groove in the middle between the helix halves!

In practice, it is almost not possible to assemble two separate helical gears in order to obtain a “double helical gear” due to the very precise arrangement with the mating gear.

When it comes down to reduce noises and transmit high torques, helical gears are often used. With such helical gears, the teeth no longer run as a straight line in axial direction as with spur gears, but at a certain angle (depending on the application between 20° and 45°). Since the gear wheel has a cylindrical basic shape, the tooth profile describes a segment of a helix (analogous to the spiral thread of a screw).

One can only get a straight tooth line if you imagine the teeth as a winding off (helical toothed rack), just as the unwinding of a thread also produces a straight thread line. The angle between the unwinded tooth line and the original axis of rotation is called the helix angle \(\beta\).

With helical toothing, the force for a pair of mating gears does not suddenly apply over the entire tooth width but is point-shaped (point contact!). At the end of the meshing, the force transmission does not drop abruptly, but the tooth gradually slips out, so to speak. This special meshing reduces the noise level of the gearbox significantly.

Since the circumferential forces at the beginning and end of the meshing only concentrate on a very small tooth area, these initially cause very high tooth loads. For this reason, several teeth should always be engaged at the same time in helical gears in order to distribute the load accordingly over several teeth (higher overlap ratio). If this is taken into account, helical gears can transmit higher torques than spur gears with the same dimensions.

Helical gears have a lower noise level and can transmit higher torques than spur gears!

The higher noise level of spur gears compared to helical gears can be heard very clearly, for example, in automobiles when reversing. In contrast to the gears for forward speed, the gears for the reverse speed are straight-cut toothed for cost reasons. This leads to the typical and significantly louder transmission noises while reversing!

While the tooth loads in a spur gear act purely in the circumferential direction, axial forces are generated by the pitch of the helix in helical gears. The larger the helix angle \(\beta\), the greater the axial forces will be. This must be taken into account when bearing the gear shafts. The direction of axial force depends on the sense of rotation of the helical gear.

Helical gears cause axial forces which must be absorbed by bearings!

This disadvantage due to the generation of axial forces can be eliminated by means of herringbone gears or double helical gears, as described in more detail in the next section.

Helical gears also have a negative effect on bearing wear, since the axial forces that occur lead to greater bearing forces.

The bearing wear is greater with helical gears than with spur gears!

When mating two helical gears, care must be taken to ensure that the helix angles are identical (and the module of course). Furthermore, the helix directions must be directed in the opposite direction. Analogous to screw threads, one speaks of left-hand helical gear or a right-hand helical gear (see figure above). This designation results from the direction in which the flank rises when the axis of rotation of the gear wheel is vertically aligned.

A spur gear can ultimately be regarded as a special case of helical gear with a helix angle of 0°. Accordingly, the properties of helical gears merge smoothly into those of spur gears with a decreasing helix angle. However, it should be noted that a helix angle of less than 10° offers hardly any advantage compared to spur gears!

With the gears considered so far, the axes of rotation are always parallel when meshing. With a special variant of helical gearing, gears can also be manufactured in such a way that the axes run skew, i.e. they cross each other without intersecting. In such a case one speaks of so-called screw gears or crossed helical gears (hyperboloid gears). Usually the axes of paired screw gears run at an angle of 90° to each other, but in principle any other angle is also possible.

Screw gears or crossed helical gears allow the skew mating of the gear shafts!

While the helix angles must be identical (but with different hand of helix) when pairing helical gears, paired screw gears have different helix angles (but with identical hand of helix)! The transmission ratio depends on the ratio of these helix angles by the way.

As the name suggests, the screw gears no longer show a pure rolling movement during engagement, but a screw motion. Typical for screw motions is the permanent sliding of the flanks. Thus, there are no points on the reference bodies of crossed helical gears to which a pure rolling process can be assigned (i.e. the circumferential speeds of the gears are not identical at any point). The reference bodies of screw gears are no longer “pitch bodies” but so-called rotational hyperboloids! A hyperbolioid is obtained by rotating a skew straight line around an axis of rotation.

The constant sliding of the flanks usually requires special lubrication of the screw gears (hypoid gear oil), otherwise increased wear is to be expected. Due to the screw course of the teeth, the flanks no longer touch each another line-shaped, but the contact is punctiform (exception: worm gears). In addition, the screw tooth path causes strong lateral forces, which must be absorbed constructively by an appropriate bearing.

Therefore, screw gears are designed for moderate torques and speeds, e.g. for drives for machine tools. The use of screw gears also has a disadvantageous effect on transmission efficiency, which is lower due to the sliding processes on the flanks.

The advantage of crossed helical gears, in addition to the already mentioned oblique arrangement of the gear axes, is their low-noise operation. In addition, screw gears can be shifted axially within relatively wide limits without having too much negative influence on power transmission.

Screw gears enable low noise emission in the medium load and speed range!

When pairing screw gears, which are designed as “cylindrical” helical gears, one also speaks of hyperboloid gears. However, the reference shape of screw gear can also be “conical” (see article bevel gears). Such screw bevel gears are also referred to as hypoid gears.

A special case of a screw gear is the so-called worm gear. Compared to the general case of a screw gear, the worm gear offers a line-shaped contact of the flanks and thus allows the transmission of higher torques.