The Two Functions of Gear Ratio — Torque Multiplication and Inertia Reduction
A सटीक ग्रहीय गियरबॉक्स placed between a servo motor and a load performs two simultaneous transformations. Both are governed by the gear ratio मैं — but they scale differently, and understanding this scaling difference is the core of correct ratio selection.
Standard torque sizing: T_required = T_load × SF, then i = T_required / (T_motor × η). Most engineers stop here. This gives the minimum ratio needed for torque — but not necessarily the ratio that gives the best servo dynamics.
The load inertia as seen by the motor shaft is divided by i². This means that a ratio change from 5:1 to 10:1 — a ×2 change — reduces the reflected inertia by a factor of 4. The inertia-matching effect of ratio is far more powerful than the torque-multiplication effect, yet it is the one most often absent from published selection guides.
In practice, i_optimal_inertia is often higher than i_min_torque — meaning inertia matching drives you toward a larger ratio than torque alone would require. The five-step decision framework later in this guide resolves conflicts between the two constraints.
The Inertia Ratio Target — Why 1:1 to 3:1 Is the Universal Standard
The inertia ratio (J_reflected / J_motor) determines how well the servo motor can control the load. A motor driving a perfectly matched load (1:1 ratio) can apply full Kv gain, achieve minimum settling time, and respond instantaneously to position error commands. As the inertia ratio increases beyond 3:1, the control loop must reduce its gain to avoid exciting the mechanical resonance of the system — and every unit of Kv reduction translates directly to slower settling time and reduced positioning accuracy.
| Inertia Ratio J_reflected / J_motor |
Max Kv Gain | Settling Time (relative) |
Dynamic Positioning | Gearbox Bearing Risk | Assessment |
|---|---|---|---|---|---|
| 1:1 | Full | 1.0× (fastest) | Best | Negligible | ✅ Ideal |
| 2:1 | Full | 1.0× | Excellent | None | ✅ Excellent |
| 3:1 | Full | 1.0× | Very good | None | ✅ Target maximum |
| 5:1 | ×0.77 | 1.3× | Reduced | Low | ⚠️ Acceptable |
| 8:1 | ×0.61 | 1.6× | Limited | Moderate | ❌ Avoid |
| 10:1 | ×0.55 | 1.8× | Poor | High | ❌ Requires low Kv |
| >10:1 | ×0.45 or less | >2.2× | Very poor | Very high | ❌ Redesign needed |
Kv reduction factors and settling time multiples are approximate, based on velocity-loop bandwidth limitation analysis for inertia-dominated servo systems. Actual values depend on motor type, servo drive tuning algorithm, and mechanical compliance. Gearbox bearing risk column reflects planet carrier pin fretting risk from cyclic resonance loading — see the failure causes guide for detail.
Why does high inertia ratio damage the gearbox? When the inertia ratio exceeds 5:1, servo engineers typically increase Kv to compensate for the sluggish response — pushing the gain toward mechanical resonance. The resulting drivetrain oscillation at 10–50 Hz imposes cyclic torque loading on the planet carrier bearings far beyond the smooth design load. Planet carrier pin bore fretting and bearing micro-pitting are the characteristic failure signatures of inertia-mismatch-driven oscillation in planetary gearboxes. Correct ratio selection eliminates this failure mode before commissioning.
The Formula — Calculating Optimal Gear Ratio from Inertia Data
The optimal gear ratio for inertia matching is the ratio that produces a reflected inertia equal to the motor rotor inertia (1:1 target). The formula derives directly from setting J_reflected = J_motor and solving for i:
i_max = √(J_load / J_motor)
≥ T_load · SF
- Calculate J_load — total load inertia including all rotating and linear masses reflected to the output shaft (see next section for component formulae)
- Read J_motor from the servo motor datasheet — this is the rotor inertia, specified in kg·m² or kg·cm²
- Calculate i_opt = √(J_load / J_motor) — this is the ideal ratio for 1:1 matching
- Identify EP series standard ratios within the acceptable band: i_min to i_opt
- For each candidate ratio, verify torque: T_available = T_motor × i × η ≥ T_load × SF
- Select the highest ratio that satisfies both inertia and torque constraints — higher ratio generally provides better inertia matching within the acceptable band
Calculating Load Inertia — Formulae for Common Machine Elements
J_load is the total inertia of all elements driven by the gearbox output shaft, expressed at the output shaft. For rotary loads this is direct; for linear loads the mass must be reflected through the mechanical transmission (rack-pinion, ballscrew, or belt-pulley) to obtain an equivalent rotary inertia at the gearbox output.
| Machine Element | Inertia Formula | Variables | Typical Applications |
|---|---|---|---|
| Solid cylinder (disk) | J = ½ m r² | m = mass (kg), r = radius (m) | Rotary tables, flywheels, pulleys, drive rollers |
| Hollow cylinder | J = ½ m (r_o² + r_i²) | r_o = outer, r_i = inner radius | Hollow shafts, pipe rollers, coil winders |
| Point mass at radius R | J = m R² | m = mass (kg), R = distance from axis | Workpiece on rotary table, cam follower, eccentric load |
| Linear mass via rack/pinion | J = m × r_pinion² | m = linear mass, r = pinion radius | Gantry axes, AGV drives, conveyor linear load |
| Linear mass via ballscrew | J = m × (pitch / 2π)² | pitch in metres (e.g. 0.01m = 10mm) | CNC feed axes, servo press, linear stages |
| Belt/pulley linear load | J = m × r_drive² | r_drive = drive pulley radius | Conveyor belts, vertical lift axes, timing belt drives |
The gearbox output shaft drives multiple elements simultaneously — the output shaft coupling, any mechanical transmission components (pinion, pulley, ballscrew), and the end load. All of these must be included in J_load before calculating the reflected inertia. Omitting the pinion or pulley inertia is common and produces an underestimate of J_load by 10–30% for typical drive configurations. For a ballscrew-driven axis, the ballscrew body inertia alone (J_screw = ½ × m_screw × r_screw²) can represent 40–60% of total reflected inertia when the linear load is light.
Three Fully Worked Examples — Indexer, AGV Drive, and CNC Rotary Axis
Index table: disc Φ500mm, 8kg steel
4 fixture blocks: 3kg each at R=200mm
Servo motor: 750W, J_motor = 0.00200 kg·m²
Required: index 90° in 0.5s, settle in 0.1s
J_table = ½ × 8 × 0.25² = 0.250 kg·m²
J_fixtures = 4 × 3 × 0.20² = 0.480 kg·m²
J_total = 0.730 kg·m²
i_opt = √(0.730 / 0.002) = 19.1
Nearest EP ratios: 16:1, 20:1
i=16: ratio=1.4:1 ✅ BEST CHOICE
i=20: ratio=0.9:1 ✅ (over-reduced)
Vehicle mass: 200kg, 2 drive wheels
Drive wheel: Φ150mm, 1.5kg
Motor: 400W, J_motor = 0.00080 kg·m²
Max speed: 1.2 m/s, max accel: 0.5 m/s²
J_wheel = ½ × 1.5 × 0.075² = 0.0042 kg·m²
J_vehicle = (200/2) × 0.075² = 0.5625 kg·m²
J_total = 0.5667 kg·m²
i_opt = √(0.5667/0.0008) = 26.6
i=16: ratio=2.8:1 ✅, n_motor=2,445rpm ✅
i=20: ratio=1.8:1 ✅ BEST BALANCE
i=20: n_motor=3,056rpm ⚠️ marginal
Table disc: Φ400mm, 25kg steel
Workpiece: 40kg, R=150mm (Φ300mm)
Motor: 1500W, J_motor = 0.00600 kg·m²
Peak cutting torque: 380 N·m, SF=1.5
J_table = ½ × 25 × 0.20² = 0.500 kg·m²
J_work = ½ × 40 × 0.15² = 0.450 kg·m²
J_total = 0.950 kg·m²
i_opt = √(0.950/0.006) = 12.6
i=12: ratio=1.1:1 ✅ (but check torque)
T_avail@12: T_m×12×0.94 ≥ 380×1.5?
→ Use EP-ZDS-142, 16:1 for torque+stiffness
The Speed-Inertia Trade-Off — When Both Constraints Cannot Be Met Simultaneously
In some applications, the ratio that gives optimal inertia matching produces a motor speed that exceeds the motor’s rated continuous speed at the required maximum output speed. This conflict — speed constraint versus inertia constraint — is the most common gear ratio dilemma in Korean servo automation design, particularly in AGV drives and high-speed conveyor systems.
| Ratio i | J_reflected / J_motor | Inertia OK? | n_motor at 60rpm output | Speed OK? | Overall |
|---|---|---|---|---|---|
| 3:1 | 27.8:1 ❌ | ❌ | 180 rpm | ✅ | Inertia fails |
| 8:1 | 3.9:1 ⚠️ | ⚠️ marginal | 480 rpm | ✅ | Acceptable with tuning care |
| 10:1 | 2.5:1 ✅ | ✅ | 600 rpm | ✅ | ✅ Best choice |
| 16:1 | 1.0:1 ✅ | ✅ ideal | 960 rpm | ✅ | ✅ Optimal inertia |
| 20:1 | 0.6:1 ✅ | ✅ over-matched | 1,200 rpm | ✅ | Motor under-utilised |
| 64:1 | 0.06:1 ✅ | ✅ but wasteful | 3,840 rpm ❌ | ❌ over speed | Speed fails |
Resolution rule: When the speed constraint limits how high the ratio can go, select the highest ratio that keeps motor speed within the recommended continuous range (3,000 rpm for EP series) at the required maximum output speed — then accept the inertia ratio that results. If this inertia ratio is above 5:1, compensate by specifying higher gearbox torsional stiffness (EP-ZDS series) to raise the resonant frequency and allow a higher servo Kv gain. Do not exceed motor speed limits for inertia matching — the motor thermal damage is irreversible.
EP Series Complete Gear Ratio Reference — All Available Ratios by Stage Count
The following table lists every standard gear ratio available across the EP series precision planetary gearboxes. Non-standard ratios can be manufactured to order — contact Korea Ever-Power application engineering with your i_optimal calculation for a custom ratio confirmation.
4:1
5:1
8:1
10:1
Highest efficiency (96%), lowest mass. Use for light loads with naturally good inertia matching (J_load/J_motor already 3–30).
12:1
15:1
16:1
20:1
25:1
32:1
40:1
64:1
94% efficiency. The primary range for inertia matching — covers the J_load/J_motor ratios of 80–4,000 with excellent inertia-optimal selection. Most industrial servo automation falls here.
80:1
100:1
120:1
160:1
200:1
256:1
320:1
516:1
90% efficiency. For very high J_load/J_motor ratios (10,000–270,000). Verify motor speed constraint carefully — at high ratios even modest output speeds require very low motor RPM, risking torque pulsation at low speed.
Five-Question Decision Framework for Gear Ratio Selection
Korea Ever-Power’s application engineering team performs complete inertia matching calculations — including J_load from your mechanical assembly data, i_optimal, standard EP ratio recommendation, and torque and speed verification. Provide your load mass, geometry, motor datasheet, and required speed/torque for a complete gear ratio recommendation in Korean or English, at no charge for qualified OEM enquiries.
संपादक: सीएक्सएम
