• For what purposes are transmissions used?
• What influence does a transmission have on the incoming and outgoing power?
• What is a positive-locked or a force-locked power transmission?
• How does the sense of rotation of mated gear wheels change?
• How can a reversal of the sense of rotation be achieved with belt drives?
• What is slippage in friction-locking power transmissions?
• What is meant by meshing of gears?
• How is the gear ratio defined and how can it be determined from the number of teeth of paired gears?
• What are the operating pitch circles?
• What is a gear stage?
• What is meant by speed ratio or power ratio?
• What statement does the spread of a shift gearbox provide?
• How can the torque change in a gearbox be explained?
• How is the change in torque related to the gear ratio?

## Introduction

In mechanical engineering there are many technical systems, which are driven either by muscle power or by motors. For example, the rear wheel of a bicycle is driven either by the muscles of the cyclist or by an electric motor. Electric motors are also used in drilling machines, while in cars internal combustion engines are used. These motors supply the energy needed to drive the respective components, e.g. to the drill chuck of a drilling machine or to the rear wheel of a bicycle.

However, all these different examples have one thing in common. The mechanical power of the motors is generally not directly used. Rather, the mechanical power must be provided in different ways, depending on the situation. When starting off with a car or a bicycle, the “force” behind the drive power should be as large as possible in order to be able to set the respective vehicle in motion.

Later on, the “speed” is more important in order to be able to cover a large in a short time. The mechanical power can therefore be used either for a high force or for a high velocity. Such a control between force and velocity, or more precisely between torque and rotational speed, is taken over by transmissions. Transmissions are sometimes simply referred to as gearings or gears, although not every transmission consists of gears. Transmissions are important elements in mechanical engineering.

Transmissions control the power supplied in favour of a high velocity (rotational speed) or in favour of a high force (torque)!

In addition, transmissions also have the task of influencing the direction of rotation. Think, for example, of the reverse gear of a car. Gear units therefore basically fulfil the following tasks:

• power transfer
• influencing the direction of rotation
• control of speed and torque

## Mechanical power

The mechanical power results with translatory motion by the multiplication of force $$F$$ and velocity $$v$$ and with rotary motion by the multiplication of torque $$M$$ and rotational speed $$n$$:

\begin{align}
&\boxed{P = F \cdot v} ~~~~~~~\text{translational power} \\[5px]
\label{p}
&\boxed{P = 2 \pi \cdot M \cdot n} ~~~~~~~\text{rotational power} \\[5px]
\end{align}

The article mechanical power deals with the derivation of these formulas in more detail.

Since a transmission can only ever change one of the two variables (velocity or force, or rather rotational speed or torque) in favour of or at the expense of the other variable, the mechanical power always remains constant. This fact is ultimately a direct consequence of the principle of energy conservation, because if both influencing variables could be reduced or increased simultaneously, then a transmission unit would destroy energy or generate it out of nothing.

So, a transmission does not change the mechanical power supplied but only the ratio of velocity and force or the ratio of revolution speed and torque, which is behind the power! Of course, this only applies as long as friction losses are not taken into account. Taking friction effects into account, the transmission output shaft will actually have slightly less power than the transmission input shaft. In no case, however, a transmission can increase the mechanical power. The term power is meant in the physical sense, as energy transmitted per unit time! This is actually the reason why it is called mechanical power transmission and not mechanical power transformation!

The mechanical power is not changed by a transmission (apart from friction effects which reduce the power)!

How torque and revolution speed (or force and velocity) are converted in a transmission is explained in more detail in the following sections.

## Transmission types

Transmission are available in many different types, depending on the application, e.g. as gear drive, traction drive, friction drive, worm drive, planetary drive, etc. However, the physical principle of transforming rotational speed (angular velocity) and torque or velocity and force are identical for all transmission types. However, before we go into more detail on how transmissions work, we will briefly explain the most important types. More detailed information can be found in the corresponding main articles.

### Gear drive

In gear drives, gear wheels, wich are also referred to as cogwheels or toothed wheels or simply gears, engage with each other and thus form-locking convert the revolution speed and torque of the drive shaft to the desired value on the output shaft. The gearbox input shaft is referred to as the drive shaft. This corresponds to the shaft which is connected to the motor and whose speed or torque is to be changed by the gear unit. The output shaft therefore corresponds to the gearbox output.

Form-locking (or positive locking) transmissions transfer the power through interlocking geometric forms!

The animation above schematically shows a three-stage gear transmission. If you look at it, you will notice that the direction of rotation of the gears changes with each gear pair! If, for example, the driving gear rotates counterclockwise, the driven gear will rotate clockwise. This reversal of the direction of rotation must be taken into account when designing gear drives.

The sense of rotation changes with each gear pair (gear stage)!

### Traction drive

In traction drives, the speed and torque are converted by wheels which drive each other via belts (belt drives) or chains (chain drives). The wheels of belt drives are also referred to as belt pulleys, and the those of a chain drives are referred to as chain wheels or sprocket wheels.

While the power transmission of chain drives is also form-locked, the power transmission of belt drives is not effected by interlocking forms but by frictional forces between belt and pulley. In such a case one speaks of a friction-locking transmission or somewhat imprecisely of a force-locking transmission.

Friction-locking (or force-locking) transmissions transfer the power through frictional forces!

The advantage of friction-locking power transmissions is the integrated overload protection. While with gear drives the teeth could break in case of overload or the chains could break in case of chain drives, with belt drives the belt is only pulled over the belt pulley in case of overload. Belt drives are therefore frequently used where many load peaks are to be expected, e.g. in cone crushers or jaw crushers for crushing stones.

The animation above schematically shows a three-stage traction drive. In contrast to gear drives or friction wheel drives (explained below), the direction of rotation of the individual pulleys does not change in the manner shown. However, this does not necessarily have to be the case with belt drives. To reverse the sense of rotation, the belts can also be crossed (crossed belt drive).

### Friction wheel drive

With special friction-locking drives, the toothless wheels can also roll directly onto each other. This is then referred to as a friction wheel drive, shown in the animation below.

The advantage of a friction drive compared to a gear drive is that in case of overload the friction wheels simply slip onto each other and thus protect the transmission from major damage. The disadvantage, however, is the lower efficiency, as relative movements occur due to non-optimal adhesion conditions between the friction wheels. Such a minimal slipping of the wheels will always be present with friction-locking power transmission. In technical terminology, this is also referred to as slippage and reduces the efficiency. Slippage also occurs between belt and pulley in belt drives.

Slippage is the relative movement between a driving element and a driven element in friction-locking power transmissions!

The elastic deformations of the friction wheels or belts at the contact points also lead to efficiency losses, since permanent “flexing” is associated with high forces. The flexing work becomes noticeable in a warming of the wheels or the belt.

## Operating principle

### Speed conversion

As explained in the introductory section, transmissions are used, among other things, to set the speed to a desired value. Such a speed conversion becomes obvious when looking at the animated gear transmission below. In this case, the speed is reduced from gear to gear.

The reduction in speed can be attributed to the different number of teeth between the respective pairs of gears. For example, the first driving gear (green) on the drive shaft has a total of 15 teeth. As a result, these 15 teeth rotate completely once when the gear wheel is turned. They push the following driven gear (orange) by 15 teeth further.

However, this driven gear has more teeth due to its larger diameter. As a result, it no longer moves by a full turn. In the present case, the driven gearwheel has a total of 30 teeth. Thus, during one rotation of the driving gearwheel, the driven gearwheel is pushed on by only half a rotation. This ultimately means a halving of the speed.

Note that the individual teeth of the larger gears also have the same dimensions as the teeth of the smaller gears, as the respective teeth must fit togehter. Such an interlocking of gears is also called meshing.

### Transmission ratio

The change in speed from a driving to a driven wheel is described by the so-called transmission ratio $$i$$. It is defined as follows:

\begin{align}
\label{def_uebersetzungsverhaeltnis}
&\boxed{i = \frac{n_1}{n_2}} \\[5px]
\end{align}

In this equation, $$n_1$$ denotes the rotational speed of the driving wheel and $$n_2$$ the rotational speed of the driven wheel. If the direction of rotation is reversed with a gearstage, this is usually indicated by a negative sign. For reasons of simplicity, however, this convention will not be applied in the following.

In the case described above, the transmission ratio between the two gears is $$i$$ = 2, which means that the driving wheel rotates twice as fast as the driven wheel or the driven wheel moves only half as fast as the driving wheel. Frequently, transmission ratios are also given in the form 2:1 (“two to one”).

The transmission ratio is defined as the ratio of the rotational speeds of the driving wheel to the driven wheel. It descriptively shows how often the driving wheel has to turn for one rotation of the driven wheel!

#### Gear drive

For two paired gears, the transmission ratio is determined by the (inverse) ratio of the number of teeth $$z$$ or the corresponding pitch circle diameter $$d$$:

\begin{align}
\label{zaehne_uebersetzungsverhaeltnis}
&\boxed{i = \frac{z_2}{z_1} = \frac{d_2}{d_1}} \\[5px]
\end{align}

The operating pitch circle diameter is the diameter of imaginary pitch cylinders that roll onto each other without sliding (somewhat a bit imprecise just referred to as pitch circle diameter). Consequently, the circumferential speeds on the operating pitch circle of both gears are identical. The pitch circle diameter of a toothed wheel is ultimately the equivalent of the pulley diameter of belt drives.

The operating pitch circle diameter is the diameter of imaginary cylinders that roll onto each other without slipping!

#### Traction drive

In the case of a friction wheel drive or traction drive, the transmission ratio can be determined by the (inverse) ratio of the respective wheel diameters $$d$$:

\begin{align}
&\boxed{i = \frac{d_2}{d_1}} \\[5px]
\end{align}

If, for example, the driven wheel is twice as large as the driving wheel, this also applies to the corresponding wheel circumferences. While the driving wheel rotates once, the double-sized wheel rotates only half a turn (either by rolling onto each other in the case of friction wheels or by chains or belts in the case of belt drives or chain drives). The speed is therefore halved and a transmission ratio of $$i$$ = 2 is again present.

### Gear stages

In principle, a certain transmission ratio can be assigned to each wheel pair within a transmission at which the speed changes. The animations above of the gear transmission and the belt transmission show that a transmission usually does not consist of one pair of wheels but of several, each mounted on a different shaft.

Each pair of wheels that mesh with one another represents a so-called gear stage and is characterized by a certain transmission ratio. In general, a gearbox has several gear stages, each with different transmission ratios.

A gear stage is a wheel pairing within a gearbox at which the speed or torque changes!

So when we talk about the transmission ratio of the entire gearbox, we mean the overall transmission ratio, i.e. the transmission ratio between input shaft and output shaft of the whole gear unit! The total transmission ratio $$i_{t}$$ can be calculated by multiplying the individual transmission ratios of the gear stages:

\begin{align}
&\boxed{i_{t} = i_1 \cdot i_2 \cdot i_3 \cdot \dots} \\[5px]
\end{align}

The overall transmission ratio of a gear unit results from the multiplication of the individual transmission ratios of the respective gear stages!

More detailed information on the function and structure of gear stages can be found in the corresponding main article.

### Forms of transmission ratios

Gearboxes do not always have to be designed to reduce the speed as is the case in the animations above. In many technical applications, an increase in speed is also desired. This is the case, for example, when driving on highways. In order to move forward as fast as possible, the wheels have to turn as fast as possible. Therefore, it is necessary to increase the speed of the motor shaft by means of a transmission. Then a large gear wheel must then drive a smaller wheel.

In such cases the transmission ratios are smaller than one and one also speaks of a speed ratio. At transmission ratios greater than one, the driven wheel rotates slower than the driving wheel and one speaks somewhat imprecisely of a power ratio. Note, that power in the physical sense ist not transformed but remains constant. Only the torque ist increased at a power ratio. Since the speed is reduced according to the increase of torque, the transmission is often called gear reducer or speed reducer.

A transmission ratio that leads to an increase in speed is called speed ratio. A transmission ratio that leads to an increase in torque is called power ratio.

For example, when starting off with a car in first gear, there is a power ratio with a maximum transmission ratio of about $$i_{max}$$ = 3.6. Accordingly, the speed is reduced by a factor of 3.6 compared to the motor speed. In top gear, on the other hand, the shiftable motor gearbox has a speed ratio with a minimum transmission ratio of approx. $$i_{min}$$ = 0.8. The speed is therefore increased by a factor of 1.25 (=$$\frac{1}{0.8}$$).

Gearboxes that can change their transmission ratio are also referred to as shiftable transmissions or manual transmissions or, for short, gearshifts. An important characteristic of shiftable transmissions is the increase in the transmission ratio from the minimum to the maximum. The greater this increase is, the larger the speed ranges can be shifted. This increase is also referred to as transmission spread $$S$$ and is calculated as follows:

\begin{align}
&\boxed{S = \frac{i_{max}}{i_{min}}} = \frac{3.6}{0.8}=4.5 \\[5px]
\end{align}

For the described gearbox, the spread is $$S$$ = 4.5, which means that the gear ratio can be increased by a factor of 4.5 starting from the minimum value.

The ratio of maximum to minimum gear ratio of a shiftable gearbox is called transmission spread!

### Torque conversion

In the previous section, the conversion of the speeds of two gears was described. Due to the energy conservation, a change in torque is always associated with this speed change! This is discussed in more detail in the following sections.

#### Gear drive

The change in torque within a pair of gears becomes clear when one looks more closely at the forces occurring. In the following it is assumed that the driving gear wheel has the torque $$M_1$$. The adjacent gear wheel is driven by this torque.

Depending on the diameter $$d_1$$ of the driving gear, a certain force $$F$$ is connected to the torque $$M_1$$. With this force, the tooth flanks on the pitch circle of the driving gear now press against the tooth flanks of the driven gear (also acting on the pitch circle).

The acting force $$F$$ can be determined from the definition of torque (“torque = force applied x lever arm”). Thus, at a given torque $$M_1$$, the corresponding force $$F$$ at the tooth flanks can be determined using the respective pitch circle diameter $$d_1$$:

\begin{align}
&M_1 = F \cdot r_1 = F \cdot \frac{d_1}{2} \\[5px]
\label{m_t}
&\underline{F = 2 \cdot \frac{M_1}{d_1}} \\[5px]
\end{align}

Note: To simplify matters, it was assumed that the force acts tangentially to the pitch circle, so that force and lever arm (= half pitch circle diameter) are perpendicular to each other. More detailed information on the actual direction of force of involute gears can be found in the corresponding article.

The calculated force $$F$$ of the driving gear from equation (\ref{m_t}) also acts on the driven gear. However, since the driven gear has a different pitch circle diameter, the force now acts on a changed lever arm ($$\frac{d_2}{2}$$). Consequently, this is also associated with a change in torque:

\begin{align}
&M_2 = F \cdot r_2 = F \cdot \frac{d_2}{2} ~~~\text{with equation (2)}~~~F = 2 \cdot \frac{M_1}{d_1} ~~~\text{:} \\[5px]
&M_2 = \underbrace{2 \cdot \frac{M_1}{d_1}}_{= F} \cdot \frac{d_2}{2} \\[5px]
\label{m_1}
&\underline{M_2 = M_1 \cdot \frac{d_2}{d_1}} \\[5px]
\end{align}

It is shown by equation (\ref{m_1}) that the torque $$M_2$$ on the driven gear is proportional to the ratio of the respective pitch circle diameters $$\frac{d_2}{d_1}$$. The larger the driven gear in relation to the driving gear, the greater the increase in torque will be.

For gears, the pitch circle diameter is directly proportional to the number of teeth. Because with a double (pitch circle) diameter, the gear wheel circumference is twice as large and thus also offers space for twice the number of teeth.

If the driven gear has twice as many teeth as the driving gear, the associated double lever arm ultimately doubles the torque. In this respect, the torque increase can also be expressed by the ratio of the number of teeth:

\begin{align}
\label{m_2}
&\underline{M_2 = M_1 \cdot \frac{z_2}{z_1}} \\[5px]
\end{align}

The ratio of pitch circle diameters in equation (\ref{m_1}) or ratio of the number of teeth in equation (\ref{m_2}) corresponds to the transmission ratio $$i$$ in equation (\ref{zaehne_uebersetzungsverhaeltnis}). This means that the change in torque can also be expressed directly by the transmission ratio:

\begin{align}
\label{1}
&\boxed{M_2 = M_1 \cdot i }~~~\text{with}~~~\underline{i = \frac{z_2}{z_1}= \frac{d_2}{d_1}=\frac{n_1}{n_2}} \\[5px]
\end{align}

Note that the transmission ratio is defined as the ratio of the rotational speeds of driving gear to driven gears. Thus, for the speed $$n_2$$ of the driven gear at a certain transmission ratio $$i$$ the following relationship to the original speed $$n_1$$ applies:

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

As the torque increases according to equation (\ref{1}) at a certain transmission ratio, the speed decreases to the same extent according to equation (\ref{2}) and vice versa. This is ultimately a direct consequence of the law of energy conservation. In the section “Energetic approach”, this relationship is explicitly derived using the law of energy conservation.

As the speed is increased by a gearbox, the torque is reduced to the same extent and vice versa!

Note that the equations above apply only to the ideal case of a non-dissipative gearbox. In general, friction cause a reduction in power and thus a reduction in the theoretically calculated torque for the driven shaft. These power losses are taken into account by a gear efficiency $$\eta_g$$:

\begin{align}
&\boxed{M_2 = M_1 \cdot i \cdot \eta_g } \\[5px]
\end{align}

For the calculation of the speed, however, the gear efficiency does not play a role, since the speed conversion results from the number of teeth (the teeth cannot penetrate each other and thus produce a lower speed than is preset by the ratio of the number of teeth).

#### Traction drive

Also with traction mechanism, the change of torque takes place in a similar manner as in gear transmissions. Depending on the diameter $$d_1$$, the driving wheel with the torque $$M_1$$ pulls on the belt or the chain with a certain force $$F$$ according to equation (\ref{m_t}).

The same force $$F$$ also acts on the driven wheel through the belt or the chain. Since the diameter $$d_2$$ of the driven wheel differs from the driving wheel, a change in torque $$M_2$$ results. The changed torque $$M_2$$ at the driven wheel again results from equation (\ref{m_1}). The exact forces acting on belt drives are described in more detail in separate articles.

## Energetic approach

The mechanical power $$P$$ of a rotating shaft results as a function of torque $$M$$ and rotational speed $$n$$ according to equation (\ref{p}) as follows:

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

If friction losses are not taken into account, the mechanical power supplied to the driving shaft must be the same as the power taken from the driven shaft due to the law of energy conservation. After all, the energy transferred within a certain time is ideally transmitted completely from the driving shaft to the driven shaft.

Thus, if the mechanical power of the driving shaft $$P_1$$ and the driven shaft $$P_2$$ are equated on the basis of the principle of energy conservation, this directly leads to the inverse relationship between speed and torque ratio:

\begin{align}
&P_1 = P_2 \\[5px]
&2 \pi \cdot M_1 \cdot n_1 = 2 \pi \cdot M_2 \cdot n_2 \\[5px]
&\boxed{\frac{M_1}{M_2} = \frac{n_2}{n_1}} \\[5px]
\end{align}