The Axis That Wouldn’t Tune Stiff
We tuned a servo axis (article 132). The motor was sized for the torque. But when we raised the position loop gain, the axis oscillated immediately (buzzed) at a low gain. We couldn’t get it stiff. The following error was high. The problem: the load inertia was too high compared to the motor’s rotor inertia. The load (a ball screw + a slide) had 20× the motor’s rotor inertia. The loop couldn’t handle the ratio. We either upsized the motor (bigger rotor inertia), added a gearbox (reduces the reflected inertia by the gear ratio squared), or softened the gains (accepting lower stiffness). We added a 5:1 gearbox. The reflected inertia dropped by 25×. The axis tuned stiff. The mistake was sizing the motor by torque only, not by inertia ratio.
Servo motor inertia ratio is the ratio of the load inertia (reflected to the motor) to the motor’s rotor inertia. This article covers the calculation.
What Is Inertia Ratio?
The motor’s rotor has an inertia (J_motor, kg·cm²). The load (the screw, the slide, the workpiece) has an inertia (J_load). Through the screw, the load reflects back to the motor shaft (reflected inertia). The ratio is:
J_ratio = J_reflected / J_motor
Where J_reflected is the load inertia as seen at the motor shaft.
For a ball screw drive, the reflected inertia is:
J_reflected = m × (p / (2π))²
Where m is the load mass (kg) and p is the screw lead (m/rev). For m = 50 kg, p = 0.01 m (10 mm lead): J_reflected = 50 × (0.01 / 6.28)² = 50 × (0.00159)² = 50 × 2.53e-6 = 1.26e-4 kg·m² = 126 kg·cm². If the motor rotor inertia is 5 kg·cm²: J_ratio = 126 / 5 = 25. That’s very high.
The inertia ratio rule: Aim for J_ratio under 5–10 (for general servo). Up to 30 is possible (with lower gains). Above 30, the loop can’t tune stiff. The axis that wouldn’t tune had J_ratio = 25. Add a gearbox (reduces by ratio²), or use a motor with more rotor inertia, or choose a finer lead (smaller p = lower reflected inertia).
Step 1: Calculate the Reflected Inertia
For a linear load (mass on a screw), the reflected inertia is m × (p/2π)². For a rotary load (a dial table), the inertia is the table’s inertia directly (no conversion). Calculate both, sum them (the screw’s own inertia plus the slide’s).
The screw’s own inertia (rotating) is about (π × d⁴ × L × ρ) / 32 (for a steel screw). Add it (for a long screw, it’s significant).
Step 2: Lower the Ratio
If J_ratio is too high (over 10), lower it:
- Use a finer lead (smaller p): A 5 mm lead (vs 10 mm) quarters the reflected inertia. But it reduces the top speed (article 57, 67). Trade-off.
- Add a gearbox (article 67): The reflected inertia drops by (1/gear_ratio)². A 5:1 gearbox divides the inertia by 25. This is the standard fix. But it adds backlash (article 68) and efficiency loss (article 102).
- Use a bigger motor (more rotor inertia): A larger motor has a bigger J_motor. The ratio drops. But it’s more expensive and heavier.
Step 3: What Happens at High Ratio
A high J_ratio means the load dominates (the motor’s rotor is small compared to the load). The loop can’t stiffen (the gain is limited by the load’s resonance). Symptoms:
- The axis oscillates (buzzes) at low gain.
- The following error is high (even at high gain).
- The axis is slow to settle (rings after the move).
These are the same symptoms as a mechanical resonance (article 132) — but the root cause is the inertia ratio. Check J_ratio first.
Step 4: The Gearbox Trade-off
A gearbox (article 67) reduces the inertia ratio, but it adds:
- Backlash (article 68): The gearbox has lost motion. For a precision axis, use a low-backlash gearbox (<1 arcmin).
- Efficiency loss (article 102): The gearbox loses 3–5% per stage. The motor must be bigger.
- Maintenance: The gearbox needs oil (article 102).
For a simple axis, size the motor so J_ratio is under 10 (no gearbox). For a high-inertia load (a big slide, a heavy table), use a gearbox.
| J_ratio | Result |
|---|---|
| <5 | Tunable to high stiffness (precision) |
| 5–10 | General (good tuning) |
| 10–30 | Possible (lower gain, slower response) |
| >30 | Problem (won’t tune stiff, resonance) |
An Inertia Checklist
- What is the load mass? (Slide + part?)
- What is the screw lead (p)?
- Calculate J_reflected = m × (p/2π)²?
- Add the screw’s own inertia?
- What is the motor’s rotor inertia (J_motor)?
- J_ratio = J_reflected / J_motor? (Under 10?)
- If too high: finer lead? (Speed trade-off?)
- If too high: gearbox? (Ratio² divides inertia?)
- If gearbox: low backlash? (Article 68?)
- Does the axis tune stiff? (Test, article 132?)
The Bottom Line
Servo motor inertia ratio sizes the motor for the load’s inertia, not just the torque. The axis that wouldn’t tune had J_ratio = 25. Calculate J_reflected = m × (p/2π)², divide by the motor’s rotor inertia. Aim for under 10. Lower a high ratio with a finer lead, a gearbox (ratio²), or a bigger motor. The axis that tuned stiff wasn’t the highest torque motor — it had the right inertia ratio.
