Z t p
0
c 0 A 0 T 0
V 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ngR
n À 1
ð
ÞT s
s
dt ¼
Z p p
p 0
dp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
p
p s
2=n À
p
p s
n þ 1
ð
Þ=n
r
t pT ¼
V 0
c 0 A 0 T 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n À 1
ð
ÞT s
2ngR
s
Z p p
p 0
dp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
p
p s
2=n À
p
p s
n þ 1
ð
Þ=n
r
ð8:45Þ
Similar to the sonic flow, the starting delay time t p of cylinder piston under
subsonic flow is also affected by the gas flow parameters, the state parameters, and
the motion parameters of the actuator (cylinder).
8.2.1.4 Cylinder Natural Frequency
The performance of the whole servo system will be affected by the performance of
the actuator in the pneumatic servo system. In addition to the piston, starting delay
time t p mentioned above is an important characteristic parameter of the cylinder, the
natural frequency of the cylinder is also an important characteristic parameter.
Referring to the method of calculating the undamped natural frequency of hydraulic
cylinder by sliding valve in hydraulic servo system, the cylinder is regarded as a
mass gas spring system. According to the definition of motion equation and volumetric elastic modulus
B ¼ ÀDp=
DV
V
Calculating undamped natural frequencies of cylinders
x nk ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
2kA 2 p
V 0 m
s
ð8:46Þ
The upper is the case in which the state change of gas in cylinder is the case of
adiabatic process. If the change of gas state in cylinder is an isothermal process, the
frequency formula given by Eq. (8.46) should be rewritten as follows:
x nT ¼
ffiffiffiffiffiffiffiffiffiffi
2A 2 p
V 0 m
s
ð8:46’Þ
8.2 Structure and Characteristics of Actuators
35