thermodynamic characteristics of gas need to be considered. As an ideal gas, the
equation of state of the gas is
pV
n
¼ const
ð8:2Þ
where
p Gas pressure;
V Gas volume;
n Variable index of gas state change.
Finding derivation on both sides of the formula, it is obtained
V
n dp þ nV
nÀ1 pdV ¼ 0; that is, Vdp þ npdV ¼ 0
The definition of gas volumetric elastic modulus b 0 in cylinder chamber is
b 0 ¼ ÀV
dp
dV
¼ np
ð8:3Þ
Here, gas is a compressible fluid, a function of pressure, which varies linearly
with the pressure. Volumetric elastic modulus A of gas is not a constant, its value is
related to gas pressure and variable index. For example, when the gas pressure is
5 Â 10
5 Pa and 10 Â 10
5 Pa, respectively, the volumetric modulus of elasticity of
chamber is 7 Â 10
5 Pa and 14 Â 10
5 Pa, which is two times different in value.
Assuming that the pneumatic cylinder system has a small disturbance near the
equilibrium position, the pneumatic cylinder system is equivalent to the load mass–
spring system. The equivalent spring stiffness acted by the pneumatic cylinder
chamber is
K eq ¼ À
dF
dx
¼ À
Adp
dV
A
¼
A
2
b 0
V
ð8:4Þ
V ¼ V 0 þ Ax
ð8:5Þ
where
V 0 Cavity gas volume including pipeline (m
3 );
V Volume of Gas in Gas Chamber (m
3 );
x Displacement of piston from initial position (m);
A Effective area of cylinder piston (m
2 ).
The natural frequency calculation method of pneumatic cylinder system is the
same as that of load mass–spring system K ¼ 0
ð
Þ:
4
8 Pneumatic Actuators, Driving Elements and Accessories