changeable process, but an isothermal process, the expression of the starting delay
time t p of the cylinder piston can be simplified as follows:
t pk ¼
p 0 V 0 T s
c 0 A 0 p s T 0 a sk K k
p p
p 0
À 1
ð8:40Þ
or,
t pk ¼
V 0 T s
c 0 A 0 p s T 0 a sk K k
p x À p a
ð
Þ
ð8:40’Þ
where
a sk ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
kgRT s
p
;
K k =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
k þ 1
k þ 1
ð
Þ= kÀ1
ð
Þ
r
;
k
Adiabatic index (For atmosphere k ¼ 1:4).
Obviously, the piston starting pressure p p has a certain relationship with the
piston motion parameters (such as speed and acceleration), the sliding friction force
between piston and cylinder wall and piston rod at the sealing filler. In general,
these relationships can be expressed by the following differential equation of piston
motion:
m
d
2 x
dt 2 À p p A þ P þ f ¼ 0
ð8:41Þ
where
m Converted mass of all moving parts on piston;
x Piston displacement;
A Piston effective area;
P Total load converted to the piston (including the force of back pressure on the
piston);
f Total frictional force of piston and piston rod.
From the Eqs. (8.40)’ and (8.41), it can be seen that the delay time of piston
starting is related to the workload of cylinder, the position of piston before starting
or the initial volume of cylinder, friction force, and air source. Increasing friction
force and load on piston will increase the starting delay time, and increasing supply
pressure will help to shorten the delay time.
For pneumatic servo system, in order to obtain a relatively small starting delay
time t p , the supply pressure p s should be increased as much as possible. When the
supply pressure of the pneumatic system is limited, the flow coefficient c of the
intake passage can be increased. That is to say, t p should be shortened by increasing
the section area of intake passage A appropriately and reducing the friction between
8.2 Structure and Characteristics of Actuators
33
time t p of the cylinder piston can be simplified as follows:
t pk ¼
p 0 V 0 T s
c 0 A 0 p s T 0 a sk K k
p p
p 0
À 1
ð8:40Þ
or,
t pk ¼
V 0 T s
c 0 A 0 p s T 0 a sk K k
p x À p a
ð
Þ
ð8:40’Þ
where
a sk ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
kgRT s
p
;
K k =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
k þ 1
k þ 1
ð
Þ= kÀ1
ð
Þ
r
;
k
Adiabatic index (For atmosphere k ¼ 1:4).
Obviously, the piston starting pressure p p has a certain relationship with the
piston motion parameters (such as speed and acceleration), the sliding friction force
between piston and cylinder wall and piston rod at the sealing filler. In general,
these relationships can be expressed by the following differential equation of piston
motion:
m
d
2 x
dt 2 À p p A þ P þ f ¼ 0
ð8:41Þ
where
m Converted mass of all moving parts on piston;
x Piston displacement;
A Piston effective area;
P Total load converted to the piston (including the force of back pressure on the
piston);
f Total frictional force of piston and piston rod.
From the Eqs. (8.40)’ and (8.41), it can be seen that the delay time of piston
starting is related to the workload of cylinder, the position of piston before starting
or the initial volume of cylinder, friction force, and air source. Increasing friction
force and load on piston will increase the starting delay time, and increasing supply
pressure will help to shorten the delay time.
For pneumatic servo system, in order to obtain a relatively small starting delay
time t p , the supply pressure p s should be increased as much as possible. When the
supply pressure of the pneumatic system is limited, the flow coefficient c of the
intake passage can be increased. That is to say, t p should be shortened by increasing
the section area of intake passage A appropriately and reducing the friction between
8.2 Structure and Characteristics of Actuators
33