Gear Design Details
6.1 Gear Transmission Density Maximization
6.1.1 Introduction of Volume Function
Gear Design Details
This chapter addresses various specific tasks of gear design, such as trans-mission density maximization, achieving high gear ratio in the planetary drives, self-locking gear design, some plastic gear design issues, and tooth modeling technique.
6.1 Gear Transmission Density Maximization
Maximization of the gear transmission density allows the output torque to be increased within given gear drive dimensional constraints or to reduce gear drive size and weight with a given output torque to be reduced. Size and weight reduction often also accompanies cost reduction.
This chapter presents an approach that allows optimizing gearbox kinematic arrangement and gear tooth geometry to achieve high gear transmission den-sity. This approach uses dimensionless gearbox volume functions, which can be minimized by the gear drive internal gear ratio optimization [61].
6.1.1 Introduction of Volume Function
Load capacity or transmission density is defined by the gear tooth working flank surface durability that is limited, as a rule, by allowable contact stress level. For a pair of mating gears the gear transmission density coefficient Ko
(also known as the K-factor) [5, 38] is
K T
d b
u
o u
w w
= ×
× × ±
2 1 1
21 , (6.1)
where T1 is the driving pinion torque, dw1 is the pinion operating pitch diam-eter, bw is the effective gear face width in mesh, u = z2/z1 is the gear pair ratio, z1 is the driving pinion number of teeth, z2 is the driven gear number of teeth, + is for external gear mesh, and – is for internal gear mesh.
The gear pair transmission density coefficient Ko statistically varies about 0.5–4.0 MPa for commercial drives and about 4.0–12.0 MPa for more demand-ing applications such as aerospace, racdemand-ing, and automotive drives. So a wide range of Ko can be explained by the gear drive design (arrangement,
materials, heat treatment, lubrication, etc.), its application, operating condi-tions, and performance priorities, which may include size and weight, reli-ability, life, cost, noise and vibration, and many other characteristics. The gear pair volume definition is illustrated in Figure 6.1.
Weight of the pinion can be presented as
w1= ×ρ V1×Kv1, (6.2)
bw
dw2 dw1
(a)
dw2 dw1
bw
(b) FIGURE 6.1
Gear pair volume definition: (a) external gearing, (b) internal gearing. (From Kapelevich, A.L., and V.M. Ananiev, Gear Technology, November/December 2011, 46–52. With permission.)
where ρ is the material density, Kv1 is the volume utilization coefficient of the pinion (a ratio of the pinion volume to its operating pitch cylinder volume), and V1 is the operating pitch cylinder volume, which is equal to
V1 dw21 bw
= ×π4 × , (6.3)
which, considering Equation (6.1), also can be presented as
V T
The operating pitch cylinder volume of the mating gear is
V2 dw22 bw u2 V 4 1
= ×π × = × , (6.5)
where dw2 is the gear operating pitch diameter.
Assuming identical material density of both mating gears, the total weight of a gear pair is
w w= 1+w2 = ×ρ (V1×Kv1+V2×Kv2), (6.6) where Kv2 is the volume utilization coefficient of the mating gear.
The volume utilization coefficients Kv1 and Kv2 depend on the gear body shape (solid body or with central or lightening holes, rim, web, spokes, etc.). Their values for driving pinions (sun gears) statistically vary in a range of 0.8–1.0;
for driven (or planet) gears, 0.3–0.7, and for internal (or ring) gears, 0.05–0.1.
Then applying (6.5) the gear pair weight is w= ×ρ V1× Kv1+u2×Kv
where Fv is the dimensionless volume function.
For the cylindrical pair of gears the volume function is
F F F u
u K u K
v = v1+ v2= ± × v1+ 2× v
1 2
( ), (6.9)
where
F u
u K
v1 1 v1
= ± × (6.10)
is the pinion volume function, and
Fv2=(u± × ×1) u Kv2 (6.11) is the mating gear volume function.
The epicyclic gear stage volume definition is illustrated in Figure 6.2. In this case the subscript indexes 1, 2, and 3 are related to the sun gear, planet gear, and ring gear, respectively.
Operating pitch cylinder volume of the sun gear is defined by Equation (6.4) with the + sign, because the sun gear is in the external mesh with the planet gear. The planet gear operating pitch cylinder volume is defined by Equation (6.5). The operating pitch cylinder volume of the ring gear is
V3 dw23 bwi p2 V Kbw
4 1
= ×π × = × × , (6.12)
where dw3 is the ring gear operating pitch diameter, Kbw = bwi/bwe is the effec-tive gear face width ratio in the epicyclic gear stage, bwe is the effective gear face width in the sun/planet gear mesh, bwi is the effective gear face width in the planet/ring gear mesh, p = z3/z1 is the ring/sun gear ratio in the epicyclic stage, and z3 is the ring gear number of teeth.
dw1 dw3
bwe
dw2
bwi
FIGURE 6.2
Epicyclic gear stage volume definition. (From Kapelevich, A.L., and V.M. Ananiev, Gear Technology, November/December 2011, 46–52. With permission.)
Unlike the convex-convex sun/planet gear mesh tooth flank contact, the planet/ring gear mesh has the convex-concave tooth flank contact, resulting in significantly lower contact stress. This allows reducing the effective gear face width in the planet/ring gear mesh to achieve a similar level of contact stress as in the sun/planet gear mesh. This makes the effective gear face width ratio Kbw < 1.0. Typically it is 0.7–0.9.
Assuming the same density material for all gears, the total weight of gears in the epicyclic gear stage is
w w= 1+np×w2+w3= ×ρ (V1×Kv1+np×V2×Kv2+V3×Kv3), (6.13) where Kv3 is the volume utilization coefficient of the ring gear, and np is the number of planet gears.
Applying Equations (6.5) and (6.12) the total weight is w= ×ρ V1× Kv1+u2×np×Kv +p ×Kv ×Kbw
2 2
( 3 ). (6.14)
Then considering Equation (6.4) the epicyclic gear stage volume function is
F F n F F
is the sun gear volume function,
F u
n u K
ve
p v
2= + × ×1 2 (6.17)
is the planet gear volume function, and
F u
is the ring gear volume function.
The more planet gears in the epicyclic gear stage, the lower its volume function and more compact the gear stage.
When the input torque and gear ratio are given and the gear transmission density coefficient Ko is selected according to the application, volume func-tions allow estimating the size and weight of the gearbox at a very preliminary stage of design for different gear arrangement options.
6.1.2 Volume Functions for Two-Stage Gear Drives