Page 156 - Mechatronics with Experiments

142 MECHATRONICS
Load

M

r Rack
Pinion

FIGURE 3.4: Rotary to translational motion conversion mechanism: rack and pinion
mechanism. The advantage of the rack and pinion mechanism over the lead-screw mechanism
is that the translational motion range can be very long. The lead-screw length is limited by the
torsional stiffness. In rack and pinion mechanisms, since the translational part does not rotate,
it does not have the reduced torsional stiffness problem due to the long length.

3.3.2 Rack and Pinion Mechanism
The rack and pinion mechanism is an alternative rotary to linear conversion mechanism
(Figure 3.4). The pinion is the small gear. The rack is the translational (linear) component.
It is similar to a gear mechanism where one of the gears is a linear gear. The effective gear
ratio is calculated or measured from the assumption that there is no slip between the gears,
Δx = r ⋅ Δ (3.73)
Δ ̇ x = r ⋅ Δ ̇ (3.74)
V = r ⋅ w (3.75)
where Δ is in radian units. Hence, the effective gear ratio is
1
N = ; if Δ is in rad units or w is in rad∕s units (3.76)
r
1
= ; if Δ is in rev units or w is in rev∕s units (3.77)
2 ⋅ r
The same mass and force reflection relations we developed for lead-screws apply for the
rack and pinion mechanisms. The only difference is the effective gear ratio.

3.3.3 Belt and Pulley
Belt and pulley mechanisms are used both as rotary to rotary and rotary to linear motion
conversion mechanisms depending on the output point of interest. If the load (tool) is
connected to the belt and used to obtain linear motion, then it acts as a rotary to linear
motion conversion device (Figure 3.5). The relations we developed for the rack and pinion

F, x

r r

T,
FIGURE 3.5: Rotary to translational motion conversion mechanism: Belt and pulley
mechanism where both pulleys have the same diameter. The output motion is taken from the
belt as the translational motion.