Page 455 - Wind Energy Handbook

GEARBOX 429

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
9
6
10 and 3 3 10 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
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F t E u þ 1 1
ó C ¼ (7:46)
2
bd 1 ð(1 ı ) u sin Æ cos Æ
where F t is the force between the gear teeth at right angles to the line joining the
gear centres, b is the gear face width, d 1 is the pinion pitch diameter, u is the gear
ratio (greater than unity), and Æ is the pressure angle, i.e., the angle at which the
force acts between the gears – usually 208–258.
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

F t h
ó B ¼ K S (7:47)
1 bt 2
6
where h is the maximum height of single tooth contact above the critical root
section, t is the tooth thickness at the critical root section, K S is a factor to allow for
stress concentration at the root.
For gearing operating at rated torque only, the designer needs to show that the
resultant bending stress multiplied by an appropriate safety factor is less than the
endurance limit multiplied by the life factor, Y N , and a number of stress modifying
factors, as follows

ó B :ª < ó B lim :Y N :Y R Y X ....... .. (7:48)

A similar calculation is required in relation to the contact stress.
Given the predicted turbine load spectrum (Section 7.4.2), which should include
dynamic effects (see Section 7.4.3), it is then necessary to establish the required
design torque at the endurance limit. Normally this is done by invoking Miner’s
rule and determining the infinite life torque for which the design torque spectrum
yields unity fatigue damage in conjunction with the prescribed S–N curve. Y N in
Equation (7.48) can then be set to unity, as the life factor has been accounted for in
the derivation of the required infinite life torque.
Figure 7.27 shows specimen torque–endurance curves laid down by BS 436 for
case-hardened gears for tooth bending and tooth contact stress (with no pitting
allowed) plotted in terms of the torque at the endurance limit. Hence in each case
the design infinite life torque, T 1 , is calculated according to: