Gear teeth must be designed in fatigue to achieve both acceptable contact stresses on the flanks and acceptable bending stresses at the roots. In non-wind turbine applications, gearboxes typically operate at rated torque throughout their lives, so the gear strengths are traditionally modified by 'life factors' which are derived from the material S-N curves on the basis of the predicted number of tooth load cycles for the gear in question. The British code for gear design, BS436 (British Standards Institution, 1986), recognizes an endurance limit for both contact stress and bending stress, so that the life factors are unity when the number of tooth load cycles exceeds 109 and 3 X 106 respectively, but increase for lesser numbers of cycles.
The Hertzian compression stress between a pair of spur gear teeth in contact at the pitch point (i.e., at the point on the line joining the gear centres) is given by y bdi, oC =XI-TtE-T, --(7.46)
" (l — i2) u sin a cos a where Ft is the force between the gear teeth at right angles to the line joining the gear centres, b is the gear face width, d1 is the pinion pitch diameter, u is the gear ratio (greater than unity), and a is the pressure angle, i.e., the angle at which the force acts between the gears - usually 20°-25°.
Note that the contact stress increases only as the square root of the force between the teeth because the area in contact increases with the force as well.
The maximum bending stress at the tooth root is given by
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