• What are the pro and cons of internal gears compared to external gears?
• What are spur gears?
• Why can a toothed rack also be considered as a spur gear?
• Why are most gears straight-cut gears?
• What are pro and cons of helical gears compared to spur-toothed gears?
• In which cases are herringbone gears used and what is its advantage over helical gears?
• Why are double helical gears often used instead of herringbone gearing?
• What are screw gears and what are they used for?

## Introduction

The figure below shows a selection of different gear types as they are used in mechanical engineering. A rough classification can be made as follows:

• Cylindrical gears
• external toothed gears
• internal toothed gears
• toothed racks
• screw gears (hyperboloid gears)
• Bevel gears
• “normal” bevel gears
• screw bevel gears (hypoid gears)
• Worm and worm gears
• cylindrical worm and cylindrical worm gear
• globoid worm and globoid worm gear

The most common type of gears used in mechanical engineering are cylindrical gears; they can be produced very economically. In this type, the teeth are arranged on the circumference of a cylindrical disc (called pitch cylinder).

Cylindrical gears can only mesh with each other with their respective circumferences. For this reason, the rotary axes of the different gear shafts are always parallel to each other.

With cylindrical gears, the teeth are arranged on the circumference of a (pitch) cylinder! The gear axes always run parallel to each other.

## External and internal gears

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 $$a$$ 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).

## Rack (toothed bar)

While only rotary motions occur with conventional cylindrical gears, a rack can be used to generate a linear motion. In a rack, the teeth are no longer arranged on the circumference of a cylinder, but along a straight bar (toothed bar). The mating gear of a rack is always an cylindrical gear.

The rack can be considered as a unwinding of teeth from a cylindrical gear. In principle, the rack can also be regarded as a cylindrical gear with an infinitely large diameter. In this respect, the rack is only a limiting case of a cylindrical gear.

While the tooth profile for involute gears is convex with an external toothing (external curvature) and concave with an internal toothing (internal curvature), racks have straight tooth flanks (no curvature).

A rack corresponds in principle to a cylindrical gear with an infinitely large diameter. Racks for involute gears have straight tooth profiles.

A transmission which converts a rotary motion into a linear motion by means of a cylindrical gear (called pinion) and a rack is also called a rack gear. Such rack drives are used, for example, in machine tools for moving machine slides.

## Types of toothing

A further differentiation of cylindrical gears results in the actual form of the tooth line. The most important gearing forms as well as their advantages and disadvantages will be discussed in more detail in the next sections.

### Spur gears

If the teeth of a gearwheel run in a straight line, i.e. in the direction of the rotation axis, it is referred to as a spur gear or a straight-cut gear. Such a toothing can be produced very cost-efficiently by gear hobbing, gear planing or gear shaping.

With spur gears, up to three teeth mesh simultaneously with each other. However, at least one tooth must always engage the mating gear to ensure continuous power transmission. The more teeth are engaged at the same time, the lower the load for each tooth and the higher the power that can be transmitted.

Since with spur gears the entire width of a tooth engages at the beginning of meshing, the force transmission also suddenly starts and abruptly breaks off at the end of meshing. This leads to relatively high noise levels. Spur gears are therefore only suitable for low circumferential speeds.

Spur gears are the simplest and therefore most cost-effective type cylindrical gears! Spur gears do not allow the transmission of excessive torques and speeds.

Higher rotational speeds and torques can be achieved with the helical toothing described below.

### Helical gears

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 the axial direction, 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!

### 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.

## Screw gears (crossed helical 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.

## Preview

With the exception of screw gears, the other gears explained so far have the disadvantage that their axes can only be arranged parallel to each other. The bevel gears described in the next article offers the possibility of also realizing inclined shaft axes.