Study Notes for Gear Trains | Theory of Machines (TOM) - Mechanical Engineering PDF Download

What is Gear Train?

A gear train is a combination of two or more gears arranged to transmit power and motion from one shaft to another. Gear trains change the speed, torque and direction of rotation between shafts. They are widely used in machines and mechanisms where controlled speed ratios or torque multiplication are required, for example in gearboxes, clocks, lathes and automotive transmissions.

• Basic function: transmit power and motion between shafts while obtaining required speed ratio and torque.
• Typical use: speed reduction (stepping down) or speed increase (stepping up) of the driven shaft relative to the driving shaft.

Types of Gear Trains

1. Simple Gear Train

   A simple gear train consists of a series of gears mounted on separate shafts so that each gear meshes with only one other gear. Intermediate gears that change the direction of rotation but do not change the overall speed ratio are called idlers. The presence of idlers does not affect the overall speed ratio, they only change the direction of rotation and increase centre distance.

   Train value: the train value is the product of speed ratios of successive engaged pairs taken appropriately (see formulae below). The speed ratio between input and output is related to the train value.

   Relation: Speed ratio = 1 / Train value (when train value is defined as gear ratio of output to input as product of tooth count ratios).

2. Compound Gear Train

   A compound gear train is one in which two or more gears are mounted on the same shaft so they rotate with identical angular velocity. This arrangement allows much larger overall speed ratios than are possible with simple trains of the same number of shafts. In a compound train the overall ratio is the product of ratios of successive pairs of meshing gears.

3. Reverted Gear Train

   A reverted gear train is a special arrangement of a compound train in which the first and last gears have parallel and coincident axes (they lie on the same two shafts) so that input and output shafts are in the same two bearing supports. This compact arrangement is useful where input and output must be coaxial or lie in fixed positions.

   For a reverted train with gears of pitch circle radii r1, r2, r3 and r4 arranged such that gear-1 meshes with gear-2 and gear-3 meshes with gear-4 and the axes of gear-1 and gear-4 coincide, the pitch circle radii satisfy:

   r1 + r2 = r3 + r4

4. Planetary (Epicyclic) Gear Train

   A planetary or epicyclic gear train is one in which one or more gears (planets) revolve about a central gear (sun) while also rotating on their own axes, usually held by a carrier or arm. In epicyclic trains at least one axis of a gear moves relative to the frame. These trains provide high reduction ratios in a compact space and are widely used in automatic transmissions, differential drives and reduction gears.

   If the arm is fixed, the system reduces to a simple train. If the central wheel is fixed and the arm is free to rotate, the planets move around the fixed wheel and the train is epicyclic.

5. Degrees of Freedom and Kinematic Coefficients

   In general, an epicyclic gear train has two degrees of freedom, because two independent inputs (for example a sun and a carrier) can produce a unique motion of the remaining element(s). The first-order kinematic coefficient (or velocity ratio) is used to describe the relation between angular speeds of the elements. For epicyclic systems these coefficients are obtained by relative motion methods or tabular methods described later.

6. Torque in Epicyclic Trains

   Consider the external torques on the elements of an epicyclic gear train. Let NS, Na, Np and NA be angular speeds and TS, Ta, Tp and TA be the external torques on gears S (sun), a (arm/carrier), P (planet) and A (annulus or another wheel) respectively. For equilibrium of external torques the algebraic sum of torques is zero:

   ∑T = 0

   Thus,

   TS + Ta + Tp + TA = 0

   For a planet P that is only carrying internal reaction and not connected externally,

   Tp = 0

   Therefore,

   TS + Ta + TA = 0

   Assuming no loss in power transmission, the sum of torque × angular speed for all external elements is zero:

   ∑(T ω) = 0

   So,

   TS · NS + Ta · Na + TA · NA = 0

   To include efficiency η when delivering power to or from an element, multiply the transmitted power terms by η accordingly (use sign conventions consistently).

7. Tabular Method for Kinematic Analysis

   The tabular method is a systematic procedure to determine final angular displacements or speeds of elements in epicyclic trains by assuming relative motions. The method proceeds as follows:

   • Assume one element (for example a gear) makes m revolutions and the arm makes n revolutions in a chosen reference direction.
   • Use the meshing constraints (relative motion between pairs of teeth) to derive algebraic relations between m and n. Consider clockwise rotations positive and anticlockwise negative (or use any consistent sign convention).
   • Apply given conditions (for example a particular element at rest or an element given a specific number of revolutions) to form simultaneous equations in m and n.
   • Solve for m and n; determine the number of revolutions and direction of all gears from these values.

   Example of applying constraints:

   If gear A is at rest, then its net revolutions are zero. If the assumed variables m and n correspond to revolutions of a particular gear and the arm respectively, then the condition gives

   N_A = 0

   which could lead to a relation such as

   m + n = 0

   If additionally the arm is given +k revolutions, then

   n = k

   By solving these simultaneous relations the unknown revolutions m and n are obtained and from them the revolutions and directions of other gears are deduced using the meshing relationships.

Practical Notes, Formulae and Keywords

• Gear ratio (i): ratio of angular speed of driving shaft to angular speed of driven shaft; can be expressed using number of teeth: i = N_driving / N_driven (sign indicates direction change).
• Train value: in compound trains the train value is the product of intermediate gear ratios; take care of sign (direction) changes due to idlers.
• Idler gear: changes direction of rotation but not the magnitude of speed ratio.
• Reverted train condition: equality of sums of pitch radii for corresponding meshes: r1 + r2 = r3 + r4 (useful for compact layouts).
• Epicyclic advantage: compactness and high reduction ratio capability; torques combine algebraically depending on which element is fixed, driven or taken as output.
• Tabular method: effective for kinematic problems in epicyclic trains where relative motions and constraints give linear relations between assumed revolutions.

Concluding Remarks

Understanding gear trains requires clear grasp of meshing relationships, sign conventions, and the role of compound and idler gears. Epicyclic systems demand special attention to relative motion and power balance. The tabular method and energy (torque × speed) balance are powerful tools for analysing kinematics and dynamics of gear trains encountered in mechanical design.