Page 107 - Robot Builders Source Book - Gordon McComb

96 Dynamic Analysis of Drives

For the supercritical airflow regime, the piston moves in such a way that G cr = constant
(see Equation (3.115)). Thus, after integration of Equation (3.130), we obtain

(This expression is written for the initial conditions: when t = 0, then s = S Q and p = p Q.)
For the layout shown in Figure 3.23 (where 3 = cylinder, 4 = piston, 5 = spring with
stiffness c), the differential equation of the movement of the piston 4 may be written as

where
V= speed of the piston,
Q = the load, which includes the useful and harmful forces.
Now, by expressing p c in terms of Equation (3.131) and substituting it into Equa-
tion (3.132), we obtain

This equation is essentially nonlinear even for the supercritical regime when G c r = con-
stant. It becomes more complicated for the subcritical regime of the air flow when we
have to substitute for the value of G from Equation (3.111) for this regime.
We give here a numerical example in MATHEMATICA language for Equation (3.133),
which also takes Expression (3.115) into consideration. The data for a device corre-
sponding to that shown in Figure 3.23 are:
m = 400 kg,
c = 100 N/m,
a = 0.5,
2
F p = 0.0002m ,
2
p r = 500000 N/m ,
2
p Q = 100000 N/m ,
S 0 = 0.05 m,
2
F c = 0.01m ,
# = 28.7Nm/kg°K.
The following three equations differ in the temperatures of the acting air. In Equa-
tions (fl), (£2), and (f3), the absolute temperatures are taken as T r = 293° K, T 2 = 340° K
and T 3 = 400° K, respectively. The higher the value of the temperature, the faster the
piston moves, and the longer the distance 5 it travels during equal time intervals. For
instance (see Figure 3.26a)), in the considered example during 0.8 seconds for given
temperatures, displacements of the piston correspondingly are

T! = 293°K ^ = 2.12 meters, [f 1 ]
T 2 = 340°K s 2 = 2.40 meters, [f2]
T 3 = 400°K s 3 = 2.76 meters. [f3]
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