Gear Trains
Simple, Compound & Epicyclic — Speed Ratios, Tabular Method & Solved Problems
Last Updated: March 2026
Key Takeaways 📌
- Gear train: A combination of gears used to transmit motion and change speed/torque between shafts.
- Simple train: Each shaft carries one gear. Speed ratio = product of driven teeth / product of driver teeth.
- Compound train: Some shafts carry two gears (one driven, one driver). Allows large speed ratios in compact space.
- Epicyclic (planetary): Gears rotate around a moving axis (arm). Use the tabular method to solve.
- Speed ratio: N1/N2 = T2/T1 (for meshing spur gears).
- Epicyclic gear trains are the most frequently tested TOM topic in GATE ME.
1. Gear Fundamentals
Speed Ratio — Two Meshing Gears
N₁T₁ = N₂T₂ (pitch line velocities must match)
N₁/N₂ = T₂/T₁
Where N = speed (rpm), T = number of teeth.
The gear with more teeth rotates slower. An external gear pair reverses rotation direction; an internal gear pair maintains direction.
Torque Relationship (Ideal)
Toutput/Tinput = Toutput gear/Tinput gear = Ninput/Noutput
Speed reduction → torque multiplication (and vice versa). Power is conserved (ideally).
Idler gears: Gears placed between the driver and driven gear that transmit motion without changing the overall speed ratio. They only affect the direction of rotation. An odd number of external idlers reverses the output direction; an even number maintains it.
2. Simple Gear Train
In a simple gear train, each shaft carries only one gear. Gears mesh in sequence: gear 1 → gear 2 → gear 3 → … → gear n.
Train Value — Simple Gear Train
Train value = Nlast/Nfirst = (−1)n × Tfirst/Tlast
Where n = number of external gear meshes. The (−1)n accounts for direction reversal.
Intermediate gears do NOT affect the speed ratio — only the first and last gears matter.
Limitation: simple gear trains cannot achieve large speed ratios in a compact space because each stage can only provide modest reduction.
3. Compound Gear Train
In a compound gear train, at least one shaft carries two gears — one driven by the previous stage and one driving the next stage. These two gears rotate at the same speed (locked on the same shaft), allowing the speed ratios to multiply.
Train Value — Compound Gear Train
Train value = Product of driven gear teeth / Product of driving gear teeth
= (T₂ × T₄ × T₆ × …) / (T₁ × T₃ × T₅ × …)
Where odd-numbered gears are drivers and even-numbered are driven (on each mesh).
Compound trains can achieve very large speed ratios. A clock mechanism, for example, uses compound gearing to convert the rapid oscillation of the escapement into the slow rotation of the hour hand.
Reverted gear train: A special compound train where the input and output shafts are coaxial (on the same axis). Used in clocks, lathe headstocks, and automotive transmissions. Requires: T₁ + T₂ = T₃ + T₄ (for equal centre distances).
4. Epicyclic (Planetary) Gear Train
In an epicyclic gear train, at least one gear revolves around another gear while also rotating on its own axis — like a planet orbiting the sun. The components are:
- Sun gear: Central gear, usually the input or fixed.
- Planet gear(s): Gears that rotate on their own axis while being carried around the sun by the arm.
- Ring gear (annulus): Internal gear that meshes with the planet gears from outside.
- Arm (carrier): Carries the planet gears around the sun gear.
Tooth relationship (for sun-planet-ring):
Tring = Tsun + 2Tplanet
This ensures the planet gears fit correctly between the sun and ring.
Epicyclic trains are used in automatic transmissions, bicycle hub gears, differential gears, and wind turbine gearboxes because they provide large speed ratios in a very compact package.
5. Tabular Method — Step by Step
The tabular (algebraic) method is the most reliable way to solve epicyclic gear train problems:
- Step 1: Create a table with columns for each component (Sun, Planet, Ring/other gear, Arm) and rows for operations.
- Step 2 — Fix the arm: Give the arm a rotation of 0 and rotate the gear train as if it were a simple train. If the sun rotates +x, the planet rotates −x(Tsun/Tplanet), etc.
- Step 3 — Rotate everything by +y: Add +y to every component (equivalent to rotating the whole assembly, including the arm).
- Step 4 — Total motion: Add rows. Each component’s net rotation = x-component + y.
- Step 5 — Apply boundary conditions: Use given information (e.g., sun is fixed → Nsun = 0, or arm speed = 100 rpm) to solve for unknowns x and y.
| Operation | Arm | Sun (TS) | Planet (TP) | Ring (TR) |
|---|---|---|---|---|
| Fix arm, rotate sun by +x | 0 | +x | −x(TS/TP) | −x(TS/TR) |
| Rotate everything by +y | +y | +y | +y | +y |
| Total | y | x + y | y − x(TS/TP) | y − x(TS/TR) |
6. Worked Numerical Examples
Example 1: Simple Compound Train
Problem: Gear A (20 teeth) drives gear B (50 teeth). Gear C (25 teeth) is on the same shaft as B and drives gear D (75 teeth). If A rotates at 1000 rpm, find the speed of D.
Solution
Train value = (TB × TD) / (TA × TC) = (50 × 75) / (20 × 25) = 3750/500 = 7.5
ND = NA / train value = 1000/7.5 = 133.3 rpm
Example 2: Epicyclic — Sun Fixed
Problem: An epicyclic train has sun gear (TS = 20), planet (TP = 30), and ring (TR = 80). The sun is fixed. The arm rotates at 100 rpm clockwise. Find the speed of the ring gear.
Solution
Verify: TR = TS + 2TP = 20 + 60 = 80 ✓
Using tabular method:
Arm speed = y = 100 rpm
Sun speed = x + y = 0 (sun is fixed) → x = −100
Ring speed = y − x(TS/TR) = 100 − (−100)(20/80) = 100 + 25 = 125 rpm (clockwise)
Example 3: Epicyclic — Ring Fixed
Problem: Same train as Example 2, but ring is fixed and sun rotates at 200 rpm clockwise. Find the arm speed.
Solution
Sun speed = x + y = 200
Ring speed = y − x(TS/TR) = y − x(20/80) = y − x/4 = 0 (ring fixed)
From ring: y = x/4 → x = 4y
Substituting into sun: 4y + y = 200 → 5y = 200 → y = 40 rpm (arm speed)
Speed ratio = Nsun/Narm = 200/40 = 5:1 — large reduction in compact space.
7. Common Mistakes Students Make
- Forgetting the sign convention in epicyclic trains: When a gear meshes externally, rotation reverses (negative ratio). When meshing internally (sun-ring through planet), the sign depends on the mesh type. Track signs carefully through the tabular method.
- Thinking idler gears change the speed ratio: In a simple gear train, idlers affect direction only — not the magnitude of the speed ratio. Only the first and last gears determine the ratio.
- Not verifying the tooth relationship TR = TS + 2TP: This must hold for the gears to physically fit. If the problem gives inconsistent values, check before proceeding.
- Mixing up compound and epicyclic analysis: Compound trains use direct ratio multiplication. Epicyclic trains require the tabular method because the arm (carrier) adds an extra degree of freedom.
8. Frequently Asked Questions
What is an epicyclic gear train?
An epicyclic (planetary) gear train has one or more gears that orbit around a central gear while rotating on their own axes. It consists of a sun gear, planet gears, a ring gear, and an arm (carrier). It can achieve large speed ratios in a compact package and is used in automatic transmissions, bicycle hubs, and wind turbine gearboxes.
How do you solve epicyclic gear problems?
Use the tabular (algebraic) method: (1) Fix the arm and express all gear speeds as ratios relative to one gear’s rotation (+x). (2) Add +y to all components (rotate the whole assembly). (3) Write total speed for each component. (4) Apply boundary conditions (which component is fixed, which has a known speed) to solve for x and y.