starts at t ¼ 0 and gradually increases from p ¼ p a to the piston starting pressure
p ¼ p p . The time t ¼ t p À 0, i.e., the time interval t p between the piston’s stationary
motion and the piston’s starting motion, is the starting delay time of the piston.
If the airflow entering the cylinder through the intake passage of the cylinder is
sonic, and the flow process and the state change of the gas in the cylinder are
variable, the relationship between the starting delay time t p of the piston and the
starting pressure p p of the piston can be obtained by the integral of Eq. (8.36).
Z
p p
p 0 ¼p a
p
1Àn
ð
Þ=n dp ¼
Z t p
t¼0
nRT 0
V 0
p
1Àn
ð
Þ=n
0
G max
ð
Þ n dt
ð8:37Þ
Thus, it is obtained
p
1=n
p ¼ p
1=n
a
RT 0
V 0 p a
c 0 A 0 p s
T s R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ngRT s
2
n þ 1
n þ 1
ð
Þ = nÀ1
ð
Þ
s
t p þ 1
2
4
3
5
ð8:38Þ
After introducing the local sound speed a sn ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
ngRT s
p
and p 0 ¼ p a of the air
supply flow, the result is as follows:
p
1=n
p ¼ p
1=n
a
RT 0
V 0 p a
c 0 A 0 p s
T s R
a sn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2
n þ 1
n þ 1
ð
Þ= nÀ1
ð
Þ
s
t p þ p
1=n
0
2
4
3
5
ð8:38’Þ
After finishing, the expression of starting delay time t p of piston is obtained.
t p ¼
p a V 0 T s
c 0 A 0 p s T 0 a sn K
p x
p a
1=n
À1
"
#
ð8:39Þ
where
K ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2
n þ 1
n þ 1
ð
Þ= nÀ1
ð
Þ
s
Obviously, the starting delay time of piston is related to the state parameters of
working gas and the structural parameters of cylinder. For example, Eq. (8.39)
shows that the starting delay time t p of piston will increase with the increase of
initial volume V 0 . Physically speaking, when the cross-section area of the intake
passage of cylinder A 0 is constant, the increase of the initial volume of the working
chamber of cylinder V 0 will undoubtedly increase the charging time, so the time to
reach a certain starting pressure p p will increase, that is, the starting delay time of
piston t p will increase. If the flow into the cylinder is not a changeable process, but
an adiabatic flow state, and the change of the gas flow into the cylinder is not a
32
8 Pneumatic Actuators, Driving Elements and Accessories
p ¼ p p . The time t ¼ t p À 0, i.e., the time interval t p between the piston’s stationary
motion and the piston’s starting motion, is the starting delay time of the piston.
If the airflow entering the cylinder through the intake passage of the cylinder is
sonic, and the flow process and the state change of the gas in the cylinder are
variable, the relationship between the starting delay time t p of the piston and the
starting pressure p p of the piston can be obtained by the integral of Eq. (8.36).
Z
p p
p 0 ¼p a
p
1Àn
ð
Þ=n dp ¼
Z t p
t¼0
nRT 0
V 0
p
1Àn
ð
Þ=n
0
G max
ð
Þ n dt
ð8:37Þ
Thus, it is obtained
p
1=n
p ¼ p
1=n
a
RT 0
V 0 p a
c 0 A 0 p s
T s R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ngRT s
2
n þ 1
n þ 1
ð
Þ = nÀ1
ð
Þ
s
t p þ 1
2
4
3
5
ð8:38Þ
After introducing the local sound speed a sn ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
ngRT s
p
and p 0 ¼ p a of the air
supply flow, the result is as follows:
p
1=n
p ¼ p
1=n
a
RT 0
V 0 p a
c 0 A 0 p s
T s R
a sn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2
n þ 1
n þ 1
ð
Þ= nÀ1
ð
Þ
s
t p þ p
1=n
0
2
4
3
5
ð8:38’Þ
After finishing, the expression of starting delay time t p of piston is obtained.
t p ¼
p a V 0 T s
c 0 A 0 p s T 0 a sn K
p x
p a
1=n
À1
"
#
ð8:39Þ
where
K ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2
n þ 1
n þ 1
ð
Þ= nÀ1
ð
Þ
s
Obviously, the starting delay time of piston is related to the state parameters of
working gas and the structural parameters of cylinder. For example, Eq. (8.39)
shows that the starting delay time t p of piston will increase with the increase of
initial volume V 0 . Physically speaking, when the cross-section area of the intake
passage of cylinder A 0 is constant, the increase of the initial volume of the working
chamber of cylinder V 0 will undoubtedly increase the charging time, so the time to
reach a certain starting pressure p p will increase, that is, the starting delay time of
piston t p will increase. If the flow into the cylinder is not a changeable process, but
an adiabatic flow state, and the change of the gas flow into the cylinder is not a
32
8 Pneumatic Actuators, Driving Elements and Accessories