Page 152 - Mechatronics with Experiments

138 MECHATRONICS

r 2
r 1 Belt
Pulley
Pulley

FIGURE 3.2: Rotary to rotary motion conversion mechanisms: timing belt and toothed pulley.

The same concept of kinetic energy and work of the tool is used in all of the other
mechanisms to determine the reflected inertia and torque between output and input shafts.

3.2.2 Belt and Pulley
The gear ratio of a belt-pulley mechanism is the ratio between the input and output diameters.
Assuming no slip between the belt and pulleys on both shafts, the linear displacement along
the belt and both pulleys should be equal (Figure 3.2),

x =Δ ⋅ r =Δ ⋅ r 2 (3.35)
2
1
1
The effective gear ratio is by definition,
Δ 1
N = (3.36)
Δ 2
r 2
= (3.37)
r 1
d 2
= (3.38)
d 1

The inertia and torque reflection between the input and output shaft has the same relationship
as the gear mechanisms.

Example Consider a spur-gear mechanism with a gear ratio of N = 10. Assume that the
load inertia connected to the output shaft is a solid steel material with diameter d = 3.0in,
length l = 2.0 in. The friction related torque at the load is T = 200 lb ⋅ in. The desired
l
speed of the load is 300 rev∕min. Determine the necessary speed at the input shaft as well
as reflected inertia and torque.
The necessary speed at the input shaft is related to the output shaft speed by a
kinematic relationship defined by the gear ratio,

̇
= N ⋅ ̇ out = 10 ⋅ 300 rpm = 3000 rpm (3.39)
in
The inertia and torque experienced at the input shaft due to the load alone (which we call
the reflected inertia and the reflected torque) are

1
J in,eff = ⋅ J l (3.40)
N 2
1
T in,eff = ⋅ T l (3.41)
N