Why a Gearbox Reduces Inertia
A motor does not directly “feel” the full inertia of the connected load when a gearbox is interposed between them. Just as the torque is amplified by the gear ratio i, the inertia calculated back to the motor shaft is reduced by i²:
J_red = J_Last / i²
J_red = moment of inertia reduced to the motor shaft (kg·m²) · J_Last = moment of inertia of the load (kg·m²) · i = gear ratio
This is the main reason why gearboxes in dynamic drives are used not only to adjust speed and torque, but also specifically to reduce inertia: Even a gearbox with i = 5 reduces the effective load inertia to one twenty-fifth. This effect works in both directions of the formula: If you have a critical inertia ratio for a given load, you can specifically improve it by using a higher gear ratio without having to replace the motor itself.
The inertia ratio J_red/J_Motor
What matters for a drive’s acceleration capability and control quality is not absolute inertia, but rather the ratio between reduced load inertia and the motor’s intrinsic inertia (J_Rotor, from the motor data sheet). The mass inertia calculator evaluates this ratio using a traffic light system:
- Marked point 1: Knowledge Check Example: i=8 → J_red=0.025 kg·m²
- Marked point 1: Knowledge Check Example: i=8 → J_red=0.025 kg·m²
Description and values of the figure
The reduced inertia decreases very rapidly with the square of the gear ratio (J_red=J_Last/i²): from i=1 to i=2, it already drops to one-quarter. Starting at about i=4, the curve flattens out because the absolute values become small; relatively speaking, however, each doubling of i continues to have a factor of 4— J_red drops again to one-quarter when i increases from 4 to 8. This decreasing effect is deliberately shown linearly so that the trend remains visible.
| Translation i | Reduzierte Trägheit J_red (kg·m²) |
|---|---|
| 1 | 1.600 |
| 2 | 0.400 |
| 3 | 0.178 |
| 4 | 0.100 |
| 5 | 0.064 |
| 6 | 0.044 |
| 7 | 0.033 |
| 8 | 0.025 |
| 10 | 0.016 |
Rule of thumb: Estimate J_red ( / J_Motor)
- Optimal—ratio up to 3:1, good inertia matching.
- Borderline—ratio 3:1 to 10:1; control performance may be compromised.
- Critical—ratio greater than 10:1; control loop is difficult to control.
If the ratio is critical, there are three options: use a gearbox with a higher gear ratio (which reduces J_red by i²), select a motor with a higher rotor moment of inertia, or review the mechanical design—reduce mass or move it closer to the axis of rotation. Otherwise, a ratio that is too high leads to control problems: The drive tends to overshoot and is more difficult to position precisely. The following also applies to acceleration: The greater the inertia reduced to the motor shaft, the more torque the motor must generate just to set the load in motion—regardless of the actual load torque of the application.
Mnemonic
Intermoment of inertia is also relevant for motor selection beyond servo axes: The accelerated moment of inertia of the load significantly determines how long it takes a drive to reach operating speed—an aspect that is incorporated into the complete motor selection process in the final unit of this module.
Knowledge check
Answer all three questions, then click "Check". From 2 of 3 correct answers, the unit counts as completed. You can retry at any time.
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