Gears and Gear Trains

Gears transmit motion between shafts at a fixed speed ratio. GATE asks about tooth geometry and about speed ratios in gear trains, especially epicyclic trains.

Terminology

• Pitch circle: the imaginary circle that rolls without slipping on the mating gear's pitch circle. Diameter d.
• Module m = d/T (mm per tooth). Mating gears must have the same module.
• Circular pitch p = πm. Diametral pitch = T/d = 1/m.
• Addendum (standard) = 1m; dedendum = 1.25m.
• Pressure angle φ: usually 20°.
• Centre distance = m(T₁ + T₂)/2.

Law of gearing

For a constant velocity ratio, the common normal at the point of contact must always pass through a fixed point, the pitch point. Involute and cycloidal profiles satisfy it. Involute gears also keep the velocity ratio constant if the centre distance changes slightly.

Velocity ratio: N₁/N₂ = T₂/T₁ = d₂/d₁.

Interference

If a gear has too few teeth, the tip of the mating tooth digs into the non-involute flank: interference. The minimum number of teeth on a pinion meshing with a rack to avoid it is 2a_w/sin²φ (a_w = addendum in modules), which is about 17 teeth for 20° full-depth teeth.

Gear trains

• Simple train: each shaft has one gear. Speed ratio = T_last/T_first; idlers change direction only.
• Compound train: two gears on an intermediate shaft. Train value = N_out/N_in = (product of driver teeth)/(product of driven teeth).
• Epicyclic train: the axis of a planet gear moves around the sun on an arm.

Epicyclic method (tabular): give the arm a fixed rotation, then lock the arm and rotate the sun. The basic relation, for a sun S, an annulus A (internal gear) and the arm a:

(N_S − N_a)/(N_A − N_a) = −T_A/T_S

Also, for meshing planets: T_A = T_S + 2T_P.

Worked example

An epicyclic train has a sun with 30 teeth and a fixed annulus with 90 teeth. The sun turns at 400 rpm. Find the arm speed.

• (400 − N_a)/(0 − N_a) = −90/30 = −3.
• 400 − N_a = 3N_a, so N_a = 100 rpm, same direction as the sun.
• Planet teeth: T_P = (90 − 30)/2 = 30.

Practice