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Analysis of Gear Trains

In a Nutshell: The basic equations governing shear stresses and angles of twist are the

same for single shafts with single or multiple applied torques, for single shafts with

different diameters along the shaft, for single shafts with changes in material along the

shaft, for different combinations of shaft size, location of applied torque, etc, and for

multiple shafts connected by one or more gears leading to the term gear trains. The

table below lists the two basic equations governing shafts subjected to applied torques.

Shear stress developed in a shaft

τ = T c / J

Angle of twist of a shaft

φ = T L / J G

where τ = shearing stress - (MPa), (N/mm²), (lb/ft²), (lb/in²)

T = applied torque - (N m), (N mm), (lb ft), (lb in)

c = radius of the shaft - (m), (mm), (ft), (in)

J = polar moment of inertia of area of the shaft (circular shafts only)

Units for J - mm⁴, m⁴, in⁴, ft⁴

φ = angle of twist in radians

L = length of the shaft - m, mm, ft, in

G = shearing modulus of elasticity - (GPa), (Mpa), (lb/ft²), (lb/in²)

Strategy: Both shear stress developed in one or more shafts and angle of twist in one or

more shafts are directly related to torque. Thus, as a first step, one needs to determine the

torque developed in each shaft by passing sections in the shafts, constructing free body

diagrams, applying the equations of equilibrium, and recognizing that the force on one

gear tooth is the same on the mating gear tooth.

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