Simple, compound and epicyclic gear trains

Why it matters

Motors run fast with little torque; robot joints, conveyors, winches and wheels need slow speed with large torque. Gear trains do that conversion, and the choice between a simple, compound or epicyclic (planetary) train decides the size, weight, efficiency and backlash of the drive. Servo planetary gearheads, automatic transmissions, vehicle differentials and wind-turbine gearboxes are all epicyclic trains.

Key ideas

Train value and velocity ratio. For any train, define

• speed ratio (velocity ratio) = input speed / output speed;
• train value = output speed / input speed (its reciprocal). For a meshing pair, N₁·T₁ = N₂·T₂ (the pitch-line velocity is common), so speeds are inversely proportional to tooth numbers. External gears reverse direction; an internal (ring) gear keeps it.

Simple gear train. Each shaft carries one gear. The overall ratio depends only on the first and last gears: intermediate gears are idlers. Each idler adds one external mesh and so reverses the output direction; idlers also bridge a large centre distance, but they do not change the ratio. A single pair is limited to about 1:6–1:8 before the gear becomes unreasonably large.

Compound gear train. At least one intermediate shaft carries two gears rigidly fixed together. The ratio is the product of the individual pair ratios, so large reductions (1:50 or more) fit in a small space. A reverted compound train has input and output shafts coaxial; this requires equal centre distances, r₁ + r₂ = r₃ + r₄, or with equal modules T₁ + T₂ = T₃ + T₄ (as in lathe back gears and clocks).

Epicyclic (planetary) train. A sun gear S, planet gears P carried on an arm (carrier) C, and often an internal ring (annulus) R. The planets both spin on their own axes and revolve with the arm, so the train has 2 degrees of freedom: fixing one member (or tying two together) leaves 1 DOF and a definite ratio.

• Ring fixed, sun input, carrier output: the common reduction, ratio 1 + R/S (typically 3–10 per stage).
• Sun fixed: ring and carrier have ratio 1 + S/R (small ratio, overdrive or underdrive).
• Carrier fixed: an ordinary simple train with idler planets, ratio −R/S (reverse).
• Any two members locked together: the whole train turns as a block, ratio 1. Advantages: coaxial input and output, load shared between several planets (high torque density), several ratios from one train by braking different members, as in automatic transmissions. Geometry for equal modules: T_R = T_S + 2T_P.

Analysis methods. (1) Tabular method: first fix the arm and give the sun +x turns, writing the turns of each member; then add +y turns to every member (the whole train rotated with the arm); use the given conditions to find x and y. (2) Relative velocity (Willis) formula below. Both give the same answer; signs matter.

Torques in an epicyclic train. With steady speed and no losses: sum of external torques = 0, and power balance Σ T·ω = 0. The holding (fixing) torque on the fixed member is the reaction needed to keep it still. A differential is a bevel epicyclic train with the arm as the crown-wheel cage; it gives equal torques to the two wheels while letting them turn at different speeds.

Formulas

N₁ / N₂ = T₂ / T₁ (one pair; N in rpm, T teeth)

Simple train: N_in / N_out = T_out / T_in (idlers cancel). Output turns the same way as the input if the number of external meshes is even (an odd number of idlers between first and last gear), opposite if it is odd.

Compound train: N_in / N_out = (product of teeth on driven gears) / (product of teeth on driving gears)

Reverted train: T₁ + T₂ = T₃ + T₄ (equal modules)

Epicyclic (Willis): (N_B − N_C) / (N_A − N_C) = train value of A → B with the arm fixed

• e.g. sun to ring: (N_R − N_C) / (N_S − N_C) = −T_S / T_R
• sun to planet: (N_P − N_C) / (N_S − N_C) = −T_S / T_P

Ring fixed, sun in, carrier out: N_S / N_C = 1 + T_R / T_S

Torques: T_in + T_out + T_hold = 0 ; T_in·ω_in + T_out·ω_out = 0 (ideal)

Worked examples

Example 1 (standard): compound reduction. Given: motor 1440 rpm, 2 kW. Gear A (20 T, on the motor) drives B (60 T); C (18 T) on B's shaft drives D (54 T) on the output shaft.

1. N_in / N_out = (60 × 54) / (20 × 18) = 3240 / 360 = 9
2. N_out = 1440 / 9 = 160 rpm; two external meshes, so D turns the same way as A.
3. ω_out = 2π × 160 / 60 = 16.76 rad/s
4. Ideal output torque T = P / ω = 2000 / 16.76 = 119.4 N·m; with 95% overall efficiency, 113.4 N·m. Answer: 160 rpm, about 119 N·m ideal (113 N·m at 95% efficiency).

Example 2 (GATE level): planetary gearhead with holding torque. Given: sun S = 24 T, planets P = 30 T, ring R = 84 T (check: 24 + 2 × 30 = 84). Ring fixed; sun driven at 1200 rpm clockwise with 50 N·m. Find carrier speed, planet speed and the holding torque on the ring. Tabular method (clockwise positive):

1. Arm fixed, sun +x: planet −x·(24/30) = −0.8x, ring −x·(24/84) = −0.2857x.
2. Add +y to all: sun x + y, planet y − 0.8x, ring y − 0.2857x, arm y.
3. Ring fixed: y = 0.2857x. Sun: x + 0.2857x = 1200 → x = 933.3, y = 266.7.
4. Carrier: N_C = 266.7 rpm clockwise (check: 1200 / (1 + 84/24) = 1200 / 4.5 = 266.7).
5. Planet: 266.7 − 0.8 × 933.3 = −480 rpm, i.e. 480 rpm anticlockwise.
6. Output torque (ideal): T_C = T_S × N_S / N_C = 50 × 4.5 = 225 N·m, resisting.
7. Holding torque: T_hold = 225 − 50 = 175 N·m, applied to the ring in the same sense as the input torque. Answer: carrier 266.7 rpm CW, planets 480 rpm ACW, ring holding torque 175 N·m.

Common mistakes

• Including idler teeth in the ratio of a simple train.
• Dropping the minus sign for external meshes in the Willis formula or the table.
• Using N_S / N_C = R / S instead of 1 + R/S for a ring-fixed planetary.
• Forgetting the geometric condition T_R = T_S + 2T_P (or T₁ + T₂ = T₃ + T₄ in a reverted train) when finding unknown tooth numbers.
• Assuming the fixed member needs no torque; the holding torque is often the largest of the three.
• Mixing up "gear ratio" conventions; state whether you mean input/output speed or output/input.

For GATE ME

Very common: epicyclic train speed problems by the tabular method (often with a compound planet), holding torque on the fixed member, tooth numbers from the reverted or ring geometry, and compound train ratios. Practise setting up the table quickly with consistent signs and checking with the Willis formula.

Quick check

1. A 20 T gear drives a 50 T gear through a 35 T idler. Ratio and direction?
2. A reverted train has T₁ = 20, T₂ = 60, T₃ = 30. What is T₄ (equal modules)?
3. Sun 30 T, ring 90 T, ring fixed. Sun-to-carrier speed ratio?
4. In a planetary with the carrier fixed, sun 20 T and ring 80 T, sun at 400 rpm. Ring speed?
5. How many DOF does an epicyclic train with nothing fixed have?

Answers: 1. 2.5 : 1, output same direction as input. 2. 50. 3. 1 + 90/30 = 4. 4. −100 rpm (opposite direction). 5. Two.