The Gearbox That Whined After a Month
We coupled a servo motor to a 100 RPM conveyor roller with a 10:1 gearbox. The conveyor needed 20 N·m at the roller. The gearbox was rated for 30 N·m output — enough. A month later, the gearbox whined and got hot. The problem: we’d sized for the steady-state torque, but the conveyor started under load (full of boxes). The startup torque spike was 40 N·m (2× the steady state). The gearbox’s peak rating was 45 N·m — barely. Over months of startup cycles, the gear teeth pitted. We stepped up to a 50 N·m gearbox. The whine stopped. The mistake was sizing for steady-state, not for the startup torque.
Gear reducer selection for servo drives is about more than the ratio. The output torque, the backlash, the duty cycle, and the overhung load all determine whether the gearbox lasts. This article walks through the sizing numbers.
The Gearbox Equations
A gearbox (reducer) takes the motor’s high speed, low torque output and converts it to low speed, high torque.
Output speed: nₒᵤₜ = nₘₒₜₒᵣ / i
Output torque: Tₒᵤₜ = Tₘₒₜₒᵣ × i × η
Reflected inertia: Jᵣₑբ = Jₗₒₐ / i²
Where i is the gear ratio, and η is the efficiency (0.95–0.98 for planetary, 0.85–0.9 for worm).
Step 1: Determine the Required Output Speed
What speed does the load need? For a conveyor, a rotary indexer, or a positioning axis, the output RPM is determined by the process.
Example: a conveyor roller needs to turn at 60 RPM to move the belt at 30 m/min. The motor’s rated speed is 3,000 RPM. So the ratio is i = 3000 / 60 = 50. But 50:1 is a big reduction. A more typical gearbox ratio is 10:1, 20:1, or 30:1. With a 20:1 gearbox, the output speed is 3000/20 = 150 RPM — too fast. We need either a 50:1 gearbox (two-stage) or a smaller motor speed.
For a positioning axis (ball screw), the motor speed maps to linear speed. With a 10 mm lead screw at 3,000 RPM, the linear speed is 30 m/min. If we need 15 m/min, a 2:1 gearbox halves it. But a gearbox adds backlash — for precision, direct drive (no gearbox) is preferred.
Step 2: Calculate the Required Output Torque
The output torque must drive the load. Break it down.
- Steady-state torque: Overcoming friction and moving the load at constant speed. For a conveyor: T = F × r, where F is the total friction force and r is the roller radius. For a 50 kg load on rollers with μ = 0.02, F = 50 × 9.81 × 0.02 = 9.8 N. Roller radius 0.05 m: T = 9.8 × 0.05 = 0.49 N·m. (Small.)
- Startup torque: Accelerating the load. This is 2–3× steady state for conveyors (overcoming static friction and inertia). Tₛₜₐᵣₜ = 3 × 0.49 = 1.5 N·m.
- Process torque: If the gearbox drives a press, mixer, or indexer, the process force adds torque. For a rotary indexer at 20 N·m output, the gearbox must deliver that continuously.
For our conveyor example, the steady torque is 0.5 N·m. But the conveyor starts loaded (boxes on the belt), so the startup torque is higher. We calculated 1.5 N·m. The gearbox output must handle this.
Step 3: Select the Gearbox by Rated Torque
The gearbox datasheet lists two torque ratings:
- Rated (continuous) torque: The torque the gearbox can deliver continuously. Must exceed the steady-state torque (with margin).
- Peak torque: The maximum torque the gearbox can handle for short periods (startup, shock). Must exceed the startup/peak torque.
For our example: steady 0.5 N·m, startup 1.5 N·m. A gearbox rated for 2 N·m continuous and 6 N·m peak works. But wait — we also need the motor torque reflected. The motor is 400 W (1.27 N·m rated, 3.8 N·m peak). With a 20:1 gearbox and 95% efficiency: Tₒᵤₜ = 1.27 × 20 × 0.95 = 24 N·m continuous. That’s way more than 0.5 N·m. We’re over-torqued.
Actually, for a conveyor we don’t need a servo at all — a gearmotor (AC induction with gearbox) is cheaper. But for positioning (where we need speed control), the servo + gearbox is overkill on torque. The point: the gearbox must be sized to the load, not to the motor. If the motor can deliver 24 N·m through the gearbox but the load needs 0.5 N·m, the gearbox is fine (it’s not over-stressed). The motor just doesn’t use its full torque.
| Gearbox Ratio | Output Speed (3000 RPM motor) | Output Torque (1.27 N·m motor) | Typical Use |
|---|---|---|---|
| 5:1 | 600 RPM | 6 N·m | High-speed, low-torque |
| 10:1 | 300 RPM | 12 N·m | Conveyors, mixers |
| 20:1 | 150 RPM | 24 N·m | Indexers, heavy conveyors |
| 50:1 | 60 RPM | 60 N·m | Low-speed, high-torque |
| 100:1 | 30 RPM | 120 N·m | Heavy duty, slow drives |
Step 4: Inertia Matching (Why the Gearbox Helps)
Recall from the servo sizing: the load inertia reflected to the motor is Jᵣₑբ = Jₗₒₐ / i². A gearbox divides the reflected inertia by the square of the ratio.
Example: a load of 0.01 kg·m² on the output. With no gearbox, the motor sees 0.01 kg·m². If the motor rotor is 0.0005 kg·m², the inertia ratio is 20:1 — too high. With a 10:1 gearbox: Jᵣₑբ = 0.01 / 100 = 0.0001 kg·m². Ratio to motor: 0.0001 / 0.0005 = 0.2:1 — excellent. The gearbox fixes the inertia mismatch.
This is the main reason to use a gearbox: not for torque, but to reduce the reflected inertia so the servo tunes well. For high-inertia loads (large belts, long screws), a gearbox makes the axis controllable.
Step 5: Backlash (For Precision Axes)
The gearbox has backlash (lost motion when direction reverses). This is critical for positioning axes.
- Worm gear: 1–3° backlash. Cheap, but not for precision. For conveyors, not for positioning.
- Planetary (standard): 3–8 arcmin (0.05–0.13°). Good for general positioning.
- Planetary (precision / preloaded): 1–3 arcmin. For high-precision axes.
- Zero-backlash (strain wave / harmonic): Under 1 arcmin. The most expensive, for direct positioning.
Backlash at the output shows up as positioning error. At 3 arcmin (0.05°) with a 10 mm lead screw, the linear error is 0.05/360 × 10 mm = 0.0014 mm — negligible. At 1° backlash, it’s 0.028 mm — measurable. For ±0.01 mm precision, use a preloaded planetary (1–3 arcmin).
Step 6: Duty Cycle and Thermal
A gearbox that runs continuously gets hot. The lubricant breaks down. The rated torque assumes a duty cycle.
- Continuous duty (24/7): The gearbox runs at rated torque continuously. Check the thermal rating — the gearbox must dissipate the heat. Larger gearboxes or cooling may be needed.
- Intermittent (cyclic): The gearbox runs part of the cycle. The peak torque can be higher (the gearbox cools between cycles). The RMS torque over the cycle must be under the continuous rating.
The gearbox whined because the startup torque (40 N·m) was near the peak rating (45 N·m). Every startup cycle loaded the gear teeth near the fatigue limit. Over months, they pitted. A gearbox rated for 50 N·m continuous / 150 N·m peak would have handled the startups with margin.
The gearbox sizing workflow: 1) Determine the required output speed (motor RPM / ratio). 2) Calculate steady and peak output torque. 3) Pick a gearbox with continuous rating > steady, peak rating > startup. 4) Check reflected inertia (J/i²) for servo tuning. 5) Check backlash for precision. 6) Check duty cycle / thermal. 7) Verify the mounting (overhung load, shaft size).
Gearbox Types: Pick by Application
| Type | Efficiency | Backlash | Cost | Best For |
|---|---|---|---|---|
| Planetary | 95–98% | 1–8 arcmin | Medium | Servo axes, general positioning |
| Worm gear | 70–90% | 1–3° | Low | Conveyors, non-precision, self-locking |
| Helical (inline) | 97–98% | 1–5 arcmin | Medium | High-power, continuous duty |
| Right-angle (bevel) | 95–97% | 1–5 arcmin | Medium-high | Space-constrained, motor perpendicular to load |
| Harmonic (strain wave) | 80–90% | <1 arcmin | High | High-precision, robots |
Overhung Load (The Output Shaft)
The gearbox output shaft carries the load (sprocket, pulley, coupling). The radial force on the shaft (overhung load) must be within the gearbox rating. A sprocket pulling sideways on the output shaft adds a bending load that wears the bearings.
If the overhung load is too high, use a larger gearbox (bigger shaft bearings) or support the load with an external bearing (not on the gearbox shaft).
A Gearbox Selection Checklist
- What is the required output speed? (Determines ratio.)
- What is the steady output torque? (Continuous duty.)
- What is the peak/startup torque? (Acceleration, shock.)
- Gearbox continuous rating > steady (with margin).
- Gearbox peak rating > startup (with margin).
- Reflected inertia J/i² (for servo tuning).
- Backlash acceptable for precision? (arcmin.)
- Duty cycle: continuous or intermittent? (Thermal.)
- Overhung load within rating?
- Mounting orientation (horizontal, vertical)?
- Efficiency (η) accounted for in torque calc?
- Lubrication (grease lifetime, oil change)?
The Bottom Line
Gear reducer selection isn’t picking a ratio and moving on. The gearbox must handle both steady and peak torque, the reflected inertia must tune well, and the backlash must suit the precision. The gearbox that whined wasn’t under-rated for steady state — it was under-rated for the startup torque spike. Size the continuous rating for steady, the peak rating for startup, and check the inertia ratio. The gearbox that runs cool and quiet for years wasn’t lucky — it was sized for the actual torque spikes, not just the nameplate number.
