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Internal Gear Technical Information
Internal Gear Calculations


The illustration above presents the mesh of an internal gear and external gear. Of vital importance is the operating (working) pitch diameters, dw, and operating (working) pressure angle, αw. They can be derived from center distance, ax, and the equation below.




The table above shows the calculation steps.
It will become a standard gear calculation if x1 = x2 = 0.
If the center distance, ax, is given, x1 and x2 would be obtained from the inverse calculation from item 4 to item 8 of the table above. These inverse formulas are in the table below.


Pinion cutters are often used in cutting internal gears and external gears. The actual value of tooth depth and root diameter, after cutting, will be slightly different from the calculation. That is because the cutter has a coefficient of shifted profile. In order to get a correct tooth profile, the coefficient of cutter should be taken into consideration.
Interference In Internal Gears
Three different types of interference can occur with internal gears:
(a) Involute Interference
(b) Trochoid Interference
(c) Trimming Interference
(a) Involute Interference
This occurs between the dedendum of the external gear and the addendum of the internal gear. It is prevalent when the number of teeth of the external gear is small. Involute interference can be avoided by the conditions cited below:


where αa2 is the pressure angle seen at a tip of the internal gear tooth.


The equation above is true only if the outside diameter of the internal gear is bigger than the base circle:


For a standard internal gear, where α = 20°, the equation above is valid only if the number of teeth is z2 > 34.
and αw is working pressure angle:


(b) Trochoid Interference
This refers to an interference occurring at the addendum of the external gear and the dedendum of the internal gear during recess tooth action. It tends to happen when the difference between the numbers of teeth of the two gears is small. The equation below presents the condition for avoiding trochoidal interference.


Below:


where αa1 is the pressure angle of the spur gear tooth tip:


In the meshing of an external gear and a standard internal gear α = 20°, trochoid interference is avoided if the difference of the number of teeth, z1 – z2, is larger than 9.
(c) Trimming Interference
This occurs in the radial direction in that it prevents pulling the gears apart. Thus, the mesh must be assembled by sliding the gears together with an axial motion. It tends to happen when the numbers of teeth of the two gears are very close. The equation below indicates how to prevent this type of interference.


Below:


This type of interference can occur in the process of cutting an internal gear with a pinion cutter. Should that happen, there is danger of breaking the tooling. The table below shows the limit for the pinion cutter to prevent trimming interference when cutting a standard internal gear, with pressure angle 20°, and no profile shift, i.e., xc = 0.


There will be an involute interference between the internal gear and the pinion cutter if the number of teeth of the pinion cutter ranges from 15 to 22 (zc = 15 to 22). The table below shows the limit for a profile shifted pinion cutter to prevent trimming interference while cutting a standard internal gear. The correction, xc, is the magnitude of shift which was assumed to be: xc = 0.0075 zc + 0.05.


There will be an involute interference between the internal gear and the pinion cutter if the number of teeth of the pinion cutter ranges from 15 to 19 (zc = 15 to 19 ).
Internal Gear With Small Differences In Numbers Of Teeth
In the meshing of an internal gear and an external gear, if the difference in numbers of teeth of two gears is quite small, a profile shifted gear could prevent the interference. The table below is an example of how to prevent interference under the conditions of z2 = 50 and the difference of numbers of teeth of two gears ranges from 1 to 8.


All combinations above will not cause involute interference or trochoid interference, but trimming interference is still there. In order to assemble successfully, the external gear should be assembled by inserting in the axial direction.
A profile shifted internal gear and external gear, in which the difference of numbers of teeth is small, belong to the field of hypocyclic mechanism, which can produce a large reduction ratio in one step, such as 1/100.


In the figure below the gear train has a difference of numbers of teeth of only 1; z1 = 30 and z2 = 31. This results in a reduction ratio of 1/30.

