(1) Gas state equation
p i V i ¼ M i RT i
ð11:31Þ
where
p i Absolute pressure of gas in front and rear chamber;
V i Front and rear chamber volume;
M i Gas mass in front and rear chambers;
R Gas constant;
T i Thermodynamic temperatures of gases in front and rear chambers.
(2) Differential equation of piston acceleration
d
2 x
dt 2 ¼
1
M
p 1 A 1 À p 2 A 2
ð
ÞÀg
ð11:32Þ
(3) Differential equation of piston velocity
dx
dt
¼ V 0 þ
1
M
p 1 A 1 À p 2 A 2
ð
ÞÀg
!
dt
ð11:33Þ
(4) Differential equation of piston displacement
X ¼ S 0 þ V 0 Dt þ
1
2
1
M
p 1 A 1 À p 2 A 2
ð
ÞÀg
!
dt
2
ð11:34Þ
where
X
Piston displacement;
M
Piston mass;
p 1 ; p 2 Gas pressure in front and rear chamber;
A 1 ; A 2 Piston effective area of front and rear chamber;
V 0
Piston initial velocity;
S 0
Piston initial displacement;
g
Gravity acceleration.
11.4 Dynamic Process and Theoretical Model of Large Diameter …
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