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tooth and the first tooth would receive the same modification which is not ideal with
respect to the scattering effect. The resulting function from equation 5 is graphical-
ly shown in Figure 13. The function will reach the -1.0 magnitude in the positive
and negative infinity. In order to design a useful normal distribution, the threshold
at the desired starting and ending point of the function has to be defined.

16.12 Second Level Modification Along the Path of Contact

The second level modifications use the first level tooth to tooth magnitudes DAmax
and apply them to a tooth bound modification function. Example functions are
shown in Figure 12. Three basic functions have been derived and successfully
tested.

The first order function boundary conditions are:
qs ≤ qj ≤ qe
if qj = qs => amplitude DA(zi,qj) = +1.0
if qj = qe => amplitude DA(zi,qj) = -1.0

with these boundary conditions the second level first order function (Figure 12)
becomes:
DA(zi,qj) = DAmax(zi) • 2 • (qj-q0)/(qs-qe) (7)
Whereas:

DA(zi,qj)… roll position depending ordinate value of the second level functions

The sinusoidal function boundary conditions are:
qs ≤ qj ≤ qe
if qj = qs => amplitude DA(zi,qj) = 0.0
if qj = qe => amplitude DA(zi,qj) = 0.0
if qj = q0 => amplitude DA(zi,qj) = 0.0

maximal amplitude between qs and q0 => +1.0
maximal amplitude between q0 and qe => -1.0

with these boundary conditions the second level sinusoidal function (Figure 12)
becomes:
DA(zi,qj) = DAmax(zi) • sin[2p• (qj-q0)/(qs-qe)] (8)

The third order function boundary conditions are:
qs ≤ qj ≤ qe
if qj = qs => amplitude DA(zi,qj) = +1.0
if qj = qe => amplitude DA(zi,qj) = -1.0
if qj = q0 => amplitude DA(zi,qj) = 0.0

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