Gear Cutting Tools

Calculation of the root
form circle diameter
The root form circle diameter can
be calculated using the software
developed by FETTE.
In theory, the root curve comprises
the region generated by the tooth
tip radius and that profiled by the
protuberance flank. The second
region is an involute profile, in
which the involute intersects the
root curve of the main involute.
The intersection is determined by
the root form circle diameter. In the
majority of cases examined, the involute
region of the undercut curve
is not present, however, and the
root rounding generated by the
tooth tip radius forms the intersection
with the main involute.
If the root form circle diameter or the effective root circle diameter are not
specified in the workpiece drawing, the effective root circle diameter
must be calculated from the gear pair data according to the following formulae:
(1) d Nf1 = (2 · a · sin α wt – d 2 Na2 – d 2 b2) 2 + d 2 b1
(2) d Nf2 = (2 · a · sin α wt – d 2 Na1 – d 2 b1) 2 + d 2 b2
(3) cos α wt = (z 1 + z 2 ) · m t · cos αt
2 · a
(4) m t =
m n
cos β
(5) tan α t = tan α n
cos β
(6) d b = z · m n · cos α t
cos β
Calculation of the effective root circle diameter
It has proved practical to plot the
computed root curve and to analyse
the result of the plot. The
intersection of the root curve with
the main involute following machining
is of decisive importance
for evaluation of the root form circle
diameter. On gears which have
been hardened and ground, it
must be considered that hardening
distortion and incorrect centring of
the grinding disk result in different
volumes being ground off the
roughed tooth flank. This may result
in the root form circle diameter
being displaced from the theoretical
dimension arrived at by calculation.
In such cases, it must be
ensured that an adequate reserve
remains between the calculated
root form circle diameter and the
requisite root form circle diameter.
Practical experience has shown
that gears with a small number of
teeth and only a small positive profile
displacement may lead to
problems if the root form circle diameter
is too large. The result can
be improved by a smaller protuberance
quantity, a larger addendum,
or a smaller tooth tip radius
on the basic hob profile.
In formulae (1) and (2), either the
tip circle diameter, or if a chamfer
is present, the tip form circle diameter
of the corresponding mating
gears, are employed as the effective
tip circle diameter.
Where:
d Nf1 , d Nf2 = effective root circle diameter
d Na1 , d Na2 = effective tip circle diameter
a = centre distance
α wt = operating pressure
angle
d b = base diameter
z 1 , z 2 = number of teeth
m t = real module
α t = real pressure angle
β = helix angle
machining
allowance
root form circle on
the gear following
machining
root form circle on
the finished gear
Tooth gap profile in the face plane
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