Page 39 - Gear Technology Solutions

Whereas:
a… Pressure Angle at Flank-Root Transition
M… Bending Moment
W… Section Modulus
b… Face Width
h… Tooth Root Thickness
l… Force Application arm = Module
F… Force
s… Root Bending Stress Calculated as Cantilever Deflection Beam

For example, the formal relationship in equations 1 through 8 applied to a
straight bevel gear with a straight root which has a module of 5mm, a face
width of 30 mm and a root tooth thickness at the center of 7.5mm shows that
for a force of 12500N the root bending stress is equal:

s = 12500 • 5 / (30 * 7.5²/6) = 222.21 N/mm² (6)

For a comparable Coniflex bevel gear with a root which is DR = 1mm deeper at
the center and 0mm deeper at the ends the following assumptions can be
made:

Average value of deeper root over entire face = 60% of value at center (1mm•
0.6=0.6mm). Pressure angle at the stress critical 30° tangent is 30°. The re-
sulting root bending stress is equal:

s Coniflex = 12500 • (5+0.6) / (30 * [7.5+2*0.6*sin30°] ²/6) = 213.38 N/mm² (7)

The comparison example in equations (6) and (7) show that the effect of taller
tooth at midface and larger root tooth thickness at midface cancel each other
out such that the Coniflex straight bevel gear shows even some reduction of
calculated root stress compared to the straight bevel gear with a straight root.

The limits of this principle are given by the fact that in case of a too large cur-
vature of the root line, the hourglass shaped root width will cause the cutter to
mutilate the opposite flank. The output of the Gleason Straight Bevel Gear
software gives a warning in cases where this is critical. The rule is that the re-
lation between face width and cutter radius should be below 40% in order to
achieve optimal root geometry.

Face Width / Cutter Radius < 0.26 (ideal) 0.4 (limit) (8)

40% of the radius of a cutter with 9” diameter is (4.5” or 114.3mm) • 0.4 =
45.7mm, which leads in case of a recommended face width of 33% of the out-
er cone distance to a maximal ring gear diameter of about 275mm. This di-
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mension is close to the limit of Phoenix 275 or Phoenix 280 machines, where
the maximal Coniflex cutter diameter is equal 9” “which closes the circle” [2].

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