The change of gas temperature in the working chamber V 0 of the cylinder is
T ¼ T 0
p 0
p
1Àk
ð
Þ=k
ð8:32Þ
where
p s Supply pressure;
T s Absolute gas temperature;
T 0 Initial temperature of gas in cylinder;
A 0 Effective cross-section area of channel;
c 0 Flow coefficient at intake passage.
With Eqs. (8.28)*(8.32), the relationship between piston starting delay time
and various factors can be investigated according to two flow states. Knowing the
factors affecting the starting delay time of piston is of great significance to the
dynamic characteristics analysis of pneumatic servo system, which can guide us to
consider the factors affecting the delay time from which aspects, so as to improve
the dynamic characteristics of servo system to meet the requirements of use. Next,
the starting delay time of piston will be investigated in the sonic (critical) and
subsonic (noncritical) flow states.
(1) Starting delay time of piston under critical flow condition. If the flow of air
through the intake port of the cylinder into the working chamber of the cylinder
is sonic adiabatic flow, and the process of gas state change in the cylinder is a
polytropic process (according to the previous analysis, the polytropic process is
closer to the actual situation), then according to Eqs. (8.29), (8.30), and (8.32),
it can be obtained,
c 0 A 0
p s
ffiffiffiffi ffi
T s
p
T s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
kg
R
2
k þ 1
k þ 1
ð
Þ = kÀ1
ð
Þ
s
¼
V 0
nRT 0
p
p a
1Àn
ð
Þ=n dp
dt
So,
c 0 A 0
p s
ffiffiffiffi ffi
T s
p
T s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
kg
R
2
k þ 1
k þ 1
ð
Þ= kÀ1
ð
Þ
s
dt ¼
V 0
nRT 0
p
p a
1Àn
ð
Þ=n
dp
ð8:33Þ
or
p
1Àn
ð
Þ=n dp ¼
nRT 0
V 0
p
1Àn
ð
Þ=n
0
G max dt
ð8:34Þ
30
8 Pneumatic Actuators, Driving Elements and Accessories
T ¼ T 0
p 0
p
1Àk
ð
Þ=k
ð8:32Þ
where
p s Supply pressure;
T s Absolute gas temperature;
T 0 Initial temperature of gas in cylinder;
A 0 Effective cross-section area of channel;
c 0 Flow coefficient at intake passage.
With Eqs. (8.28)*(8.32), the relationship between piston starting delay time
and various factors can be investigated according to two flow states. Knowing the
factors affecting the starting delay time of piston is of great significance to the
dynamic characteristics analysis of pneumatic servo system, which can guide us to
consider the factors affecting the delay time from which aspects, so as to improve
the dynamic characteristics of servo system to meet the requirements of use. Next,
the starting delay time of piston will be investigated in the sonic (critical) and
subsonic (noncritical) flow states.
(1) Starting delay time of piston under critical flow condition. If the flow of air
through the intake port of the cylinder into the working chamber of the cylinder
is sonic adiabatic flow, and the process of gas state change in the cylinder is a
polytropic process (according to the previous analysis, the polytropic process is
closer to the actual situation), then according to Eqs. (8.29), (8.30), and (8.32),
it can be obtained,
c 0 A 0
p s
ffiffiffiffi ffi
T s
p
T s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
kg
R
2
k þ 1
k þ 1
ð
Þ = kÀ1
ð
Þ
s
¼
V 0
nRT 0
p
p a
1Àn
ð
Þ=n dp
dt
So,
c 0 A 0
p s
ffiffiffiffi ffi
T s
p
T s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
kg
R
2
k þ 1
k þ 1
ð
Þ= kÀ1
ð
Þ
s
dt ¼
V 0
nRT 0
p
p a
1Àn
ð
Þ=n
dp
ð8:33Þ
or
p
1Àn
ð
Þ=n dp ¼
nRT 0
V 0
p
1Àn
ð
Þ=n
0
G max dt
ð8:34Þ
30
8 Pneumatic Actuators, Driving Elements and Accessories