Gears, Gear Ratios, and Backlash

Small motors spin fast but weakly. A gearbox slows the output and multiplies the torque by the same ratio. Real gears have a little play between their teeth, called backlash, so the output does not move until the slack is taken up.

Gear ratio (output turns once per N input turns) Backlash Motor torque Gearbox efficiency

The gear ratio

With $N$ teeth on the output gear for every tooth on the input, the output turns $N$ times slower and, apart from losses with efficiency $\eta$, $N$ times stronger:

$$\omega_{\text{out}} = \frac{\omega_{\text{in}}}{N} \qquad \tau_{\text{out}} = \eta\,N\,\tau_{\text{in}}$$

Backlash

When the input reverses, its teeth must cross the gap before they push the other side of the output teeth. During that dead zone the output does not move at all.

What to notice

In the plot, the output follows a ramp in each direction, but the two ramps are offset, making a loop. The gap between them is the backlash. For a position-controlled joint this means the output can sit anywhere within the gap, which shows up as wobble or a small limit cycle in a controller. Precision robot joints use low-backlash gears such as harmonic drives or cycloidal reducers, or they measure the output angle directly.