Gear and Gear Trains: Theory and Design Concepts

Define key gear terminologies—pitch circle, pitch circle diameter, pitch point, pitch surface, base circle, pressure angle, addendum, addendum circle, root circle, circular pitch, module—and explain their role in gear design.

Explore the law of gearing by analyzing pinion–gear contact, the common normal and pitch point, and relate angular velocity to pitch circle diameters and tooth counts.

Explains velocity of sliding of teeth for meshing gears, showing sliding along the common tangent and that v equals (omega1 plus omega2) times JM from the pitch point.

Examine cycloidal and involute gear teeth, including cycloid, epicycloid, hypocycloid curves and the base-circle involute, plus how pitch point and pressure angle phi govern velocity ratio and torque.

Shows that altering the center distance in involute gears leaves the velocity ratio unchanged while shifting contact points and base circles, with the pressure angle increasing.

Involute gears offer a simple profile with constant pressure angle, enabling smooth, high-speed power transmission and misalignment tolerance. Cycloidal gears use a complex, variable pressure angle for high-load, low-speed durability.

Describe the four gear tooth systems used in industry: 14.5 degree composite, 14.5 degree full depth involute, 20 degree full depth involute, and 20 degree sub involute.

Compute length of the path of contact for a pinion and wheel pair, decomposed into the path of approach and recess, using base circles, addendum circles, and the pitch point.

Define the length of arc of contact as the path traced on the pitch circle from engagement start to end. It comprises arc of approach and arc of recess.

explains interference in involute gears, showing how extending pinion addendum or wheel addendum shifts contact points and undercuts teeth, defining the line of action and maximum path of contact.

Determine the minimum pinion teeth to avoid interference by analyzing addendum circles and the common tangent to base circles, and use the cosine law and table values.

Derives the minimum wheel teeth to avoid interference in involute gears, defining the wheel addendum W and fraction w, and linking teeth, gear ratio, addendum radii, and phi.

Derives the minimum number of teeth on a pinion for an involute rack to avoid interference, showing TPI equals 2 times the pitch circle radius over sin squared of the pressure angle.

Explore how gear trains transmit power between shafts using two or more gears. Identify four types—simple, compound, reverted, epicyclic—and note speed ratio and axis arrangement.

Explore the simple gear train, where a driver gear drives a driven gear on another shaft, with a single gear per shaft, and idle gears set speed ratio and direction.

Explore how compound gear trains transmit power through multiple gears on intermediate shafts, calculate speed ratios via teeth counts, and achieve greater reduction than simple gear trains.

Explore reverted gear trains where the first and last gears are coaxial and rotate in the same direction. Apply center-distance constraints and speed ratio in automotive transmissions and clocks.

Explains epicyclic gear trains, where an arm and gears move around a fixed axis, and uses tabular and algebraic methods to determine velocity ratios and applications.

Epicyclic gear train theory and design: an arm carries gears A (40 teeth) and B (50); B speeds 324 rpm with A fixed and 580 rpm when A turns clockwise.