Optimized gearbox design - Wind Power Generation and Wind Turbine Design - page 513

pinion and wheel were of the same size. It follows that relative radius of curvature
is minimum at the lowest point of contact between the wheel tip diameter in the
root of the pinion. However, this is in the double tooth contact zone and thus the
chosen load point for calculating the highest surface stress is at the lowest point of
single tooth contact on the pinion fl ank.
The criterion calculated as above is known as the Sc factor whose value is
directly proportional to torque. It is therefore, valid for directly comparing load
capacity taking into account any linear application and service factors (factors of
ignorance!). While superfi cially, it has the dimensions of stress, in fact it is neces-
sary to take the square root of Sc (after it has been multiplied by the various fac-
tors) then further multiplying this by a constant (190 for N/mm 2 ) or (2290 for lb/
in 2 ) to get the “actual” compressive stress. The reason for the non-linear relation-
ship between load and stress is that the contact area increases as it fl attens so that
if load is increased by a factor of 4, stress is only doubled. Most international
design standards use this as their surface stress criterion. This leads to the anomaly
that an acceptable surface safety factor based on stress is the square root of the
associated Sc and bending safety factors directly related to load.
Historically, a simplifi ed surface criterion known as the “ K ” factor, has been
universally used for gear design. In effect, it is similar to Sc but as an approxima-
tion, it takes the pitch point as the chosen load point and further simplifi es calcula-
tion by treating the sine and cosine of pressure angle as constants. Arbitrary limits
for K may then be used as appropriate, for different applications, gear materials,
pressure angles, etc. Using this approach, it is much easier to relate the volume of
gears directly to the torque and ratio in a particular application viz.

1 T

⎛ ⎞

2

fd = 1 + nK (1 )

⎜ ⎟

⎝ ⎠

T

(2 )

2 w

fd = n ( + )

w

K

where K is the surface criterion, n the wheel/pinion ratio, f the face width, d the
pinion pitch diameter, d w the wheel pitch diameter, T the pinion torque and T w is
the wheel torque.
The chosen load point for calculating bending stress in both pinion and wheel,
is their respective highest point of single tooth contact, i.e. one base pitch from
either end of the contact path as appropriate. Figure 3 shows the angle at which the
normal tooth load at this point crosses the centre line of the tooth.
This load is then resolved into its tangential and radial components which
respectively, create bending and direct compressive stresses in the tooth root. The
resultant maximum tensile and compressive root fi llet stresses, in particular the
tensile, may be determined for the actual pitch, face width, etc. by an iteration
process which takes account of the precise geometric shape of the fi llet and the
associated stress concentration factor. The results are then compared with the per-
missible tensile fatigue limit suggested by the Goodman diagram as shown in Fig. 4.
This shows that when the load is unidirectional the mean stress is half the tensile