V 10 ; V 20 Fluid volume between pipes and control valves on both sides (m
3 );
p 1 ; p 2
Fluid Pressure in two cavities of actuator (Pa);
s
Maximum displacement of piston (m).
The effective spring stiffness of two parallel equivalent mass–spring systems is
K ¼ K eq1 þ K eq2 , that is,
K ¼
A
2
1 b 1
V 10 þ A 1 x
þ
A
2
2 b 2
V 20 þ A 2 s À x
ð
Þ
¼
A
2
1 np 1
V 10 þ A 1 x
þ
A
2
2 np 2
V 20 þ A 2 s À x
ð
Þ
ð8:10Þ
The natural frequency of the double acting cylinder is
x ¼
ffiffiffiffi ffi
K
M
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
A 2
1 np 1
ðV 10 þ A 1 xÞM
þ
A 2
2 np 2
½V 20 þ A 2 s À x
ð
ÞM
s
ð8:11Þ
When the piston is in two extreme positions of the cylinder, there is,
When x ¼ 0; K max1 ¼
A
2
1 np 1
V 10
þ
A
2
2 np 2
V 20 þ A 2 s
; x max1 ¼
ffiffiffiffiffiffiffiffiffiffiffi
K max1
M
r
ð8:12Þ
When x ¼ s; K max2 ¼
A
2
1 np 1
V 10 þ A 1 s
þ
A
2
2 np 2
V 20
; x max2 ¼
ffiffiffiffiffiffiffiffiffiffiffi
K max2
M
r
ð8:13Þ
The following assumptions are made for the position control system with
cylinder:
(1) There is p 1 A 1 ¼ p 2 A 2 in the initial position equilibrium state of the cylinder.
(2) The volume of gas in cylinder is much larger than that in pipeline, that is, the
volume of gas in pipeline can be neglected, that is V 10 ¼ V 20 ¼ 0.
By Eqs. (8.10) and (8.11), the equivalent gas spring stiffness and natural frequency of the cylinder system are, respectively,
K ¼
nA 1 p 1
x
þ
nA 2 p 2
s À x
¼ nA 1 p 1
1
x
þ
1
s À x
x 0 ¼
ffiffiffiffi ffi
K
M
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
nA 1 p 1
M
Á
1
x
þ
1
s À x
s
ð8:14Þ
From the above equation, it can be seen that the natural frequency of the cylinder
system varies with the initial position of the piston. The natural frequency characteristics of the double acting cylinder can be calculated by Eq. (8.14). The calculated results are shown in Fig. 8.5 and Table 8.2. According to Eq. (8.14), the
condition for the minimum natural frequency is dK=dx ¼ 0, that is when
x=s ¼ 0:5; K ¼ K min .
8.1 Pneumatic Cylinder and Hydraulic Cylinder
7
3 );
p 1 ; p 2
Fluid Pressure in two cavities of actuator (Pa);
s
Maximum displacement of piston (m).
The effective spring stiffness of two parallel equivalent mass–spring systems is
K ¼ K eq1 þ K eq2 , that is,
K ¼
A
2
1 b 1
V 10 þ A 1 x
þ
A
2
2 b 2
V 20 þ A 2 s À x
ð
Þ
¼
A
2
1 np 1
V 10 þ A 1 x
þ
A
2
2 np 2
V 20 þ A 2 s À x
ð
Þ
ð8:10Þ
The natural frequency of the double acting cylinder is
x ¼
ffiffiffiffi ffi
K
M
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
A 2
1 np 1
ðV 10 þ A 1 xÞM
þ
A 2
2 np 2
½V 20 þ A 2 s À x
ð
ÞM
s
ð8:11Þ
When the piston is in two extreme positions of the cylinder, there is,
When x ¼ 0; K max1 ¼
A
2
1 np 1
V 10
þ
A
2
2 np 2
V 20 þ A 2 s
; x max1 ¼
ffiffiffiffiffiffiffiffiffiffiffi
K max1
M
r
ð8:12Þ
When x ¼ s; K max2 ¼
A
2
1 np 1
V 10 þ A 1 s
þ
A
2
2 np 2
V 20
; x max2 ¼
ffiffiffiffiffiffiffiffiffiffiffi
K max2
M
r
ð8:13Þ
The following assumptions are made for the position control system with
cylinder:
(1) There is p 1 A 1 ¼ p 2 A 2 in the initial position equilibrium state of the cylinder.
(2) The volume of gas in cylinder is much larger than that in pipeline, that is, the
volume of gas in pipeline can be neglected, that is V 10 ¼ V 20 ¼ 0.
By Eqs. (8.10) and (8.11), the equivalent gas spring stiffness and natural frequency of the cylinder system are, respectively,
K ¼
nA 1 p 1
x
þ
nA 2 p 2
s À x
¼ nA 1 p 1
1
x
þ
1
s À x
x 0 ¼
ffiffiffiffi ffi
K
M
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
nA 1 p 1
M
Á
1
x
þ
1
s À x
s
ð8:14Þ
From the above equation, it can be seen that the natural frequency of the cylinder
system varies with the initial position of the piston. The natural frequency characteristics of the double acting cylinder can be calculated by Eq. (8.14). The calculated results are shown in Fig. 8.5 and Table 8.2. According to Eq. (8.14), the
condition for the minimum natural frequency is dK=dx ¼ 0, that is when
x=s ¼ 0:5; K ¼ K min .
8.1 Pneumatic Cylinder and Hydraulic Cylinder
7