Page 196 - Mechatronics with Experiments

182 MECHATRONICS
Of the four elements of a planetary gear, two of them can be specified as inputs. In
other words, planetary gears have two degrees of freedom that can be specified by the user,
that is, the speed (absolute angular velocity) of the any two components. Then, the absolute
speeds of the other two components can be found from the above equations. Once that is
known, the gear ratios can be expressed between any two absolute angular speeds, that is
w 3 w 4
N 31 = , N 41 = .
w 1 w 1
If we take one of the components as the output shaft of interest, and consider two
input shafts as input shafts, this configuration can be used as a “continuously variable gear
ratio” or a “continuously variable transmission” (CVT). The output shaft speed is a function
of two input shaft speeds. If we consider one of the input shafts as the primary input shaft
which has a gear ratio to the output shaft, then we can consider the second input shaft to
effectively change the gear ratio between the primary input shaft speed and output shaft
speed since we know how the second input shaft speed contributes to the output shaft speed.
For instance, let us consider that the input shafts are the ring and sun gears, and the
output shaft is the carrier. The carrier speed can be calculated as (Figure 3.25a)
w − w N
4 2 1
=− (3.280)
w − w 2 N 4
1
N 1 N 4
w = ⋅ w + ⋅ w 4 (3.281)
2
1
N + N 4 N + N 4
1
1
where the gear ratio between w and w is shown. But we can consider the contribution
2
1
of w as effectively changing that gear ratio. This method is used in continuously variable
4
transmissions (CVT). Similarly, for the same condition, we can derive the gear ratio rela-
tionship between the two input shafts and the planetary gear (component number 3) from
the above equations. It can be shown that the gear ratio relationship is
w − w N
3 2 1
=− (3.282)
w − w 2 N 3
1
( ) ( )
N 1 N + N 3 N 4 N + N 3
1
1
w = − 1 ⋅ w + ⋅ w 4 (3.283)
1
3
N 3 N + N 4 N 3 N + N 4
1
1
In most planetary gear applications, one of the three components other than the
carrier (typically the sun gear, planetary gear, or ring gear) is fixed, one of the remaining
two components acts as the input shaft, and the remaining two components, rotation is
kinematically determined by the appropriate gear ratio where one of the two remaining
components is the output shaft. If the carrier is fixed, the gear mechanism is in non-planetary
mode.
For the planetary gear type shown in Figure 3.26b, and for this special configuration
(ring gear is fixed) and the sun gear is input, the absolute gear ratios can be shown from
above equations as follows (set w = 0.0),
4
w 2 N 1
N = = (3.284)
21
w 1 N + N 4
1
( )
w 3 N 1 N + N 3
1
N = = ⋅ − 1 (3.285)
31
w N (N + N
1 3 1 4
It is trivial to show that N 32 = N ∕N , N 41 = N 42 = N 43 = 0.0.
31
21
This configuration (ring gear fixed) is used in final drive gear reducers inside the wheel
in large construction equipment applications. There is a planetary gear reducer between
the output shafts of the differential and the tire-wheel assembly. The sun gear is connected
to the differential output shafts (input to the planetary gear reducer of the final drive), the