Page 236 - Gear Technology Solutions

-h
Normal Distribution as First Level Function type: e
whereas:
-h… argument of Euler function

For the present 13 tooth example, the function should start at tooth number zi = 1
with DAmax = 0.01463 as the threshold value. The threshold value is controlled by
the chosen 0.1 multiplier in the exponent which delivers in the present example
1.463% of the Euler function magnitude. This results in a desirable cutoff before
the function travels to an infinitely far away zero value. It should also end at tooth
number zi = n+1 with DAmax = 0.01463 as the threshold value.

=

The exponent –h is developed as -h 0.1•(zi-n/2-1)² in order to fulfill the following
boundary conditions:

With zi = 1 and n = 13 or n/2 = 6.5 it becomes -h= -0.1▪(1-6.5-1)² = -4.225

also with zi = n+1 = 14 and n/2 = 6.5 it becomes -h= 0.1▪(14-6.5-1)² = -4.225
in case of zi = n/2 = 6.5 it becomes -h= 0

With these definitions, the normal distribution becomes:

e -0.1(zi-n/2-1)² (e.g. for zi = 1 → e -0.1(6.5)² = 0.01463) (4)

In order to achieve a positive maximum of 1.0 and a minimum of (2 • 0.01463-1.0)
= -0.971 the Euler function is multiplied by 2 and shifted in ordinate direction by
1.0:

2 • e -0.1(zi-n/2-1)² –1

which results in the first level function:

)
DAmax(zi) = Corr/( cosb • cosa • RM • sing • 2 • (e -0.1(zi-n/2-1)² –1) (5)

In order to receive an amplitude with the desired correction amount Corr normal to
the flank surface, in case of a modified A-axis rotation the cosine function has to
be multiplied with the following term from equation 2:

Corr/(cosb cosa • RM • sing
)
•

The maximum A-axis modification amount for a respective tooth becomes:

DAmax(zi) = Corr/(cosb • cosa • RM • sing) • [2 • e -0.1(zi-n/2-1)² -1] (6)

It is significant that both first level functions start at tooth number 1 and end at
tooth number n+1. If the function ended at the last tooth, number n, then the last

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