Gears are a vital component in machines, robotics, vehicles, aerospace products and so on. Probably everything that moves, every machine has at least one gear in it.

The most simple application is a **gear reducer**. This is a machine, equipment that transforms an input speed and torque to an output speed and torque. By transformation we mean **amplification** or **reduction**. The level of transformation is given by the **gear ratio**.

A gear reductor is basically a gearbox with only one gear. The gear ratio is obtained by meshing of two **spur gears**.

A gear reducer is usually used for as a **torque amplifier device**. For example you need to power a hydraulic pump which needs masiv amounts of torque. Your source of power is let say an electric motor. Between the motor and the pump we fit the gear reducer.

### How do we calculate the gear ratio ?

If you need to amplify the motor torque with a factor of 3 then you need a gear reducer with a 3.0 **gear ratio**. In order to calculate the gear ratio of the two meshed gears we need to know either:

- the
**number of teeth**of both input and output gears - the
**base diameter or radius**of both input and output gears

In the table below we have all the physical values we are going to use in our calculations. With question mark (?) we have all the variables we need to calculate.

Variable | Description | Value | Unit |

\[z_{IN}\] | number of teeth of the input gear | 16 | – |

\[z_{OUT}\] | number of teeth of the output gear | 24 | – |

\[r_{IN}\] | base radius of the input gear | 80 | mm |

\[r_{OUT}\] | base radius of the output gear | 120 | mm |

\[i\] | gear ratio | ? | – |

\[Tq_{IN}\] | input torque | 250 | Nm |

\[Tq_{OUT}\] | output torque | ? | Nm |

\[\omega_{IN}\] | input (rotational) speed | 1500 | rpm |

\[\omega_{OUT}\] | output (rotational) speed | ? | rpm |

\[F_{t}\] | contact (tangent) force | (no need) | N |

\[v_{t}\] | contact (tangent) speed | (no need) | m/s |

By input we understand the source of power, in our case could be an electric motor. By output we mean where the power is delivered (e.g. hydraulic pump).

The gear ratio *i* can be calculated in two ways:

- as a ratio between the number of teeth of the output gear and the number of teeth of the input gear

- as a ratio between the base radius of the output gear and the base radius of the input gear

The base radius is measured from the gear center of rotation up to the point of contact of the teeth. The same result is obtained by using the external radius, which is from the gear center up to the top of the teeth.

By replacing the mathematical expressions with the actual number of teeth and radius, we obtain the **gear ratio i**:

i &= \frac{z_{OUT}}{z_{IN}} &= \frac{24}{16} &= 1.5\\

i &= \frac{r_{OUT}}{r_{IN}} &= \frac{120}{80} &= 1.5

\end{split} \end{equation*} \]

The relationship between the output torque and the input torque is the following:

\[ \begin{equation*} \begin{split}Tq_{OUT} = i \cdot Tq_{IN}

\end{split} \end{equation*} \]

The gear reducer will amplify the input torque a number of times equal to the gear ratio:

\[ \begin{equation*} \begin{split}Tq_{OUT} = 1.5 \cdot 250 = 375 \text{ Nm}

\end{split} \end{equation*} \]

The relationship between the output speed and the input speed is the following:

\[ \begin{equation*} \begin{split}\omega_{OUT} = \frac{\omega_{IN}}{i}

\end{split} \end{equation*} \]

The gear reducer will reduce the input speed a number of times equal to the gear ratio:

\[ \begin{equation*} \begin{split}\omega_{OUT} = \frac{1500}{1.5} = 1000 \text{ rpm}

\end{split} \end{equation*} \]

Now we are going to demonstrate why the value of the output torque is the input torque multiplied with the gear ratio. Who came up with this formula?

At the contact point between the teeth of the gears there is a tangential force. This tangential force can be calculated function of the input gear and function of the output gear.

We know that the torque is the product between the force applied and the length of the lever arm. In our case the force is the tangential force between the teeth and the arm lever is the gear radius.

\[ \begin{equation*} \begin{split}Tq_{in} = r_{IN} \cdot F_{t}\\

\end{split} \end{equation*} \]

From this we extract the tangential force:

\[ \begin{equation*} \begin{split}F_{t} = \frac{Tq_{IN}}{r_{IN}}

\end{split} \end{equation*} \]

The same force is applied on the output gear:

\[ \begin{equation*} \begin{split}F_{t} = \frac{Tq_{OUT}}{r_{OUT}}

\end{split} \end{equation*} \]

By putting together the two mathematical expression of the tangential force, we get:

\[ \begin{equation*} \begin{split}\frac{Tq_{OUT}}{r_{OUT}} &= \frac{Tq_{IN}}{r_{IN}}\\

Tq_{OUT} &= \frac{r_{OUT}}{r_{IN}} \cdot Tq_{IN}\\

Tq_{OUT} &= i \cdot Tq_{IN}

\end{split} \end{equation*} \]

Now let’s demonstrate the effect of the gear ratio on the output speed. We use the same image as above but with speed notations instead of forces.

The tangential speed is the same in the contact point of the two gears. As the tangential force, we can write the tangential speed function of the input gear and output gear:

We know that the tangential speed is the product between the radius and the rotational speed.

\[ \begin{equation*} \begin{split}v_{t} = \omega_{IN} \cdot r_{IN}

\end{split} \end{equation*} \]

The same speed is applied on the output gear:

\[ \begin{equation*} \begin{split}v_{t} = \omega_{OUT} \cdot r_{OUT}

\end{split} \end{equation*} \]

By putting together the two mathematical expression of the tangential speed, we get:

\[ \begin{equation*} \begin{split}\omega_{OUT} \cdot r_{OUT} &=\omega_{IN} \cdot r_{IN}\\

\omega_{OUT} &= \frac{r_{IN}}{r_{OUT}} \cdot \omega_{IN}\\

\omega_{OUT} &= \frac{\omega_{IN}}{i}

\end{split} \end{equation*} \]

By the end of this tutorial you should know how to calculate a gear ratio function of the gears and also the effect of the gear ratio on torque and speed.

For any questions or observations regarding this tutorial please use the comment form below.

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