Page 106 - Robotics Designing the Mechanisms for Automated Machinery

3.5 Pneumodrive 95

There is an additional time component t 0, which is the time needed by the pressure
wave to travel from the valve to the orifice. This time can be estimated as follows:

where L is the length of the pipe from the valve to the input to the cylinder, and V s is
the sound speed V s = 340 m/sec.
In Figure 3.26 we show the pressure development versus time.
To calculate the movement of the piston, we must deduce the differential equation
for its displacement. This requires some intermediate steps. The thermodynamic equa-
tion for the air in the volume of the cylinder has the following form:

(The subscript c indicates values belonging to the cylinder volume.)
For the volume V c, we can substitute the obvious expression

where
5 = the displacement of the piston, and
F c = the cylinder's cross-sectional area
After differentiating Equation (3.129), we obtain

or

FIGURE 3.26 Pressure development versus time in a real
pneumatic drive. Here: t 0 is determined from (3.128); t a
from (3.120); fc from (3.127); p' and p" are intermediate
pressure values.