The importance of contact ratios

How to calculate the contact ratios for various styles of gearing.

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It is expected that when we flip the switch on a device that it will power up and some motion will occur. In the example of the desktop printer that we all have in our home office, we expect it to go through a warmup cycle that involves the device moving all of its internal parts in preparation of printing your document or image file. This moving occurs because the motor drives the gears, which, in turn, manipulate the working of the printer. If the motor is energized but the gears do not mesh, the printer cannot perform to your expectations. One of the driving factors in establishing the amount of load that a gear pair can transmit is the contact ratio.

The contact ratio is the numerical determination of the number of teeth on each gear when any gear set is in mesh. It accounts for the teeth that are sliding into the mesh, the teeth that are sliding out of the mesh, and those that are in full contact when the gears are engaged. 

The transverse contact ratio (εα) for spur gears is calculated using the following formula:

where: rk = tip diameter (mm)

rg = reference radius (mm)

a = center distance (mm)

α  b = working pressure angle (degrees)

α  0 = reference pressure angle (degrees)

m = module

When the working pressure angle is set to 20 degrees and module is set to 1, the values are calculated in Table 1.

Table 1

The transverse contact ratio (εα) for helical gears is calculated using the following formula:

where: rk = tip diameter (mm)

rg = reference radius (mm)

a = center distance (mm)

αα  bs = transverse working pressure angle (degrees)

αα  s = reference transverse pressure angle (degrees)

ms = transverse module

The transverse contact ratio (εα) for straight tooth bevel gears is calculated using the following formula:                

where: Rvk = tip diameter on the back cone for an equivalent spur gear (mm)

Rvg = reference radius on the back cone for an equivalent spur gear (mm)

Rv = back cone distance (mm)

r0 = pitch radius (mm)

δ 0 = reference cone angle (degrees)

hk = addendum at outer end (mm)

α  0 = reference pressure angle (degrees)

m = module

When the reference pressure angle is set to 20 degrees and module is set to 1, the values are calculated for straight tooth bevel gears produced in the Gleason® system in Table 2.

Table 2

When the reference pressure angle is set to 20 degrees and module is set to 1, the values are calculated for straight tooth bevel gears produced in the standard system in Table 3.

Table 3

The transverse contact ratio (εα) for spiral tooth bevel gears is calculated using the following formula:

where: Rvk = tip diameter on the back cone for an equivalent spur gear (mm)

Rvg = reference radius on the back cone for an equivalent spur gear (mm)

Rv = back cone distance (mm)

r0 = pitch radius (mm)

δ    0 = reference cone angle (degrees)

hk = addendum at outer end (mm)

α   s = mean transverse pressure angle (degrees)

m = module

α   n = reference normal pressure angle (degrees)

β   m = mean spiral angle (degrees)

When the reference pressure angle is set to 20 degrees, module is set to 1 and the spiral angle is set to 35 degrees, the values are calculated for spiral tooth bevel gears produced in the Gleason® system in Table 4.

Table 4

Using the tables and formulas, you will be able to determine the proper contact ratio for most styles of gearing. 

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Brian Dengel
is general manager of KHK USA Inc, a subsidiary of Kohara Gear Industry with a 24-year history of working in the industrial automation industry. He is skilled in assisting engineers with the selection of power-transmission components for use in industrial equipment and automation. Dengel is a member of PTDA and designated as an intern engineer by the state of New York. He is a graduate of Hofstra University with a Bachelor’s of Science in Structural Engineering.