How Do You Calculate Gear Ratios and Rotational Speed at Different Points in a System?

Calculate the gear ratio by dividing the number of teeth on the driven gear by the number of teeth on the driving gear. Rotational speed changes inversely: a larger driven gear rotates more slowly, while a smaller driven gear rotates more quickly.

The Reasoning and Mechanism

For two meshing gears:

$$\text{Gear ratio} = \frac{T_{\text{driven}}}{T_{\text{driver}}} = \frac{n_{\text{driver}}}{n_{\text{driven}}}$$

Here, $T$ is the number of gear teeth and $n$ is rotational speed, normally measured in revolutions per minute (rpm). Rearranging gives:

$$n_{\text{driven}} = n_{\text{driver}} \times \frac{T_{\text{driver}}}{T_{\text{driven}}}$$

Consider a 20-tooth driving gear rotating at $1200\,\text{rpm}$, connected to a 60-tooth driven gear:

$$\text{Gear ratio} = \frac{60}{20} = 3$$ $$n_{\text{driven}} = \frac{1200}{3} = 400\,\text{rpm}$$

This is a speed reduction of $3:1$. Ideally, the output torque increases by the same factor, although friction makes real systems less efficient.

Speeds in a Compound Gear Train

Suppose the 60-tooth gear shares a shaft with a 15-tooth gear, which then drives a 45-tooth gear.

The overall ratio is the product of the stage ratios:

$$\frac{60}{20} \times \frac{45}{15} = 3 \times 3 = 9$$

Therefore, $1200 \div 9 = 133.3\,\text{rpm}$. Gears fixed to the same shaft always have equal rotational speed. Each pair of external meshing gears reverses the direction of rotation.

To determine rotation direction, mark the input direction and reverse it after every external gear mesh. In this example, there are two meshes, so the 45-tooth output rotates in the same direction as the 20-tooth input. A shared shaft does not reverse direction. This distinction matters because speed calculations use tooth ratios, while direction depends on the number of meshes rather than the number of gears.

Exam Technique

For B3.3 Mechanical systems application and selection, label the driver and driven gear at every stage, show the ratio formula, and include rpm units. A common misconception is that an idler gear changes the overall ratio; it changes direction and spacing, but not the ratio between input and output gears.