Page 191 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 177
gear
w out
N = (3.260)
i
w in
N eng N i1
N = ⋅ ; i = 1, 2, 3, 4, 5 (3.261)
i
N 0 N i2
N eng N 61
N =− ⋅ ; reverse gear ratio (3.262)
r
N 0 N 62
where N ’s represent the number of teeth of each gear. Similarly, the torque output delivered
i
to the output shaft at any gear is
P = ⋅ P (3.263)
out eff in
T ⋅ w = ⋅ T ⋅ w (3.264)
out out eff eng eng
w = N ⋅ w (3.265)
out i eng
1
T out = eff ⋅ ⋅ T eng (3.266)
N i
where eff is the efficiency of the transmission, which is in the range of 85 to 97% (0.85
to 0.97) range, w eng = w input speed is same as the engine crank shaft speed. Assuming
in
100% efficiency for the transmission,
1
T = ⋅ T (3.267)
out eng
N
i
Notice that as the gear number in automotive applications increases (i.e., gear 1, gear 2,
gear 3, gear 4, gear 5), the gear ratio value N increases, that is N = 0.3, N = 0.5, N =
i 1 2 3
0.66, N = 0.75, N = 1.0. Hence, for a given engine speed, the vehicle speed gets larger.
4 5
However, in turn the torque delivered to the output shaft gets smaller as N increases. For
i
that reason, when a vehicle is climbing a hill, we need more torque, hence gear should be
shifted down to a lower gear.
The steady-state engine power is characterized by the so-called “lug curve,” which is
torque versus speed at full throttle for a given engine as discussed in Section 1.2. Each gear
of the transmission effectively scales this curve, assuming 100% transmission efficiency at
all gears for the sake of simplicity. Plotting the scaled lug curves (that is the output torque
and speed at the transmission output shaft) on the same figure for five different gears shows
us the speed–torque ranges each gear should be used in (Figure 3.24).
As an example, consider the following simple lug curve data for an engine under full
throttle conditions, and the gear ratios in gears 1 through 5,

w eng = [ 600 900 1200 1500 1800 2100 2400 ] (rev∕min) (3.268)
T eng = [ 25 50 100 95 80 75 70] (N ⋅ m) (3.269)
N = [1∕3.01∕2.01.∕1.51∕1.33 1∕0.9]; i = 1, 2, 3, 4, 5 (3.270)
i
Figure 3.24 shows the lug curve and the scaled versions of the lug curve to indicate the
torque at the transmission output shaft at different transmission output shaft speeds.
For instance, if the load torque is very large (i.e., climbing a very steep hill), the
transmission should operate in gear 1 and the maximum achievable speed is shown in the
figure. Likewise, if the load torque is very small and top speed is of importance, then
the transmission should be operated at a high gear (i.e., gear 5). Figure 3.24 shows the
maximum load torque and speed that can be supported at each gear.
Let us assume that the engine is operating at full throttle, and the transmission is in
gear 3. Then the operating curve of the vehicle is the gear 3 speed–torque curve shown. If