In order to design gear pairs with high load carrying capacity and low noise emission, flank crowning is required to limit the
tooth contact area within the boundaries of the teeth. The art of selecting the optimal crown- ing is based upon satisfying two
basically opposite requirements:
A) the load-free or low-load operation should provide the maximal possible contact area
B) tooth contact must not reach the boundaries of the teeth in case of high-load operation.
Calculation of Blank Data
The blank data of pinion and ring gear will not change from the previous section. The required blank data — the basis for the
calculations in this section—are repeated here in Tables 1 and 2.
TheCreationofLengthCrowning The length curvature of a circular tooth flank is not consistent with the inverse of the cutter head radius, but is the curvature
of the cone section at the reference point perpendicular to the blade pressure angle. Figure 1 shows this principle with cylinder K1
which smooths onto the blade enveloping cone at the reference point.
Figure 1 Blade enveloping cone and cone curvature.
The tooth length curvature in Figure 1 is 1/ρ. A change in length crowning can now be realized if the axis of the
cutter head is rotated (tilted) around the reference point while the cutting edge remains in the same spatial position (Fig.
2). The rotation is conducted around an axis lying in the X4–Z4–plane and is perpendicular to the vector RWOB; i.e.,
In the current example, a length crowning by tilting the ring gear cutter head should be created. First, the outside
blade and the concave flank cut with it are observed (Fig. 2). If a length crowning of ψ = 50 μm (from center to heel and
from center to toe) is desired, then the required curvature change is calculated as follows:(1 )The function of the crowning parabola is: ψ = dξ2
(2) The second derivative of the crowning parabola is the curvature change: ψ" = 2d
With half the face width projected with the spiral angle β = 30° onto the flank line tangent direction:
(3) ξ = 15 mm / cosβ = 17.32 mm and
(4) ψZehe = ψFerse = 0.05 mm
substituted in Equation 1 and solving for coefficient d delivers:
(5) d = ψ / ξ2 = 0.000167 [1/mm]
Whereas the required curvature change is:
ΔK = ψ" = 0.000333 [1/mm]
(6) ρ = RN/cos (ALFW4-conjugate) = 81.17 / cos20° = 86.38 mm
Curvature changes are added with the correct sign to the inverted
Figure 2 Cutter head tilt for generating a length crowning.
Simulation of the Gear Cutting Process and Tooth Contact Analysis of the Example with Length and Profile Crowning
After inputting the blank data from Tables 1 and 2, the modified machine settings from Table 2 and the blade data
from Table 4 for creating length and profile crowning into the basic machine dataset of the flank generation and roll
The Ease-Off topographies for coast and drive side now have a circular curve in length as well as in profile direction. This
Ease-Off makes the gear design insensitive to manufacturing tolerances and load affected deformations. The motion
transmission error in the middle of the figure shows a parabola-shape graph for each of the three preceding pairs of teeth.
The tooth contact pattern in the lower part— typical for bevel gear pairs manufactured in the face hobbing process.
The mean points (stars at the contact center) are now centered at the middle of the profile. The results of this section not
only show a usable, but a well-designed, typical bevel gearset as it is developed for hard finishing by lapping. Small
deficiencies in the amount of active profile can be eliminated with a profile shift.
Figure 3 Extreme vector feed of a tilted cutter.
Figure 4 Pinion vector feed of a tilted cutter 180 – 24."
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