When x ¼ 0; K max1 ¼
A
2
1 b
V 10
þ
A
2
2 b
V 20 þ A 2 s
When x ¼ s; K max2 ¼
A
2
1 b
V 10 þ A 1 s
þ
A
2
2 b
V 20
When the volume of oil in pipeline is neglected and only the volume of oil in the
cylinder is considered, the natural frequencies and their minimum values of the
hydraulic cylinder system are, respectively,
V 10 ¼ V 20 ¼ 0; x ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
b
M
Á
A 1
x
þ
A 2
s À x
s
ð8:17Þ
When,
x 0 ¼
ffiffiffiffi
A 1
p
ffiffiffi ffi
A 1
p þ
ffiffiffi ffi
A 2
p s,
the
minimum
natural
frequency
is
x 0 ¼ x min ¼
ffiffiffiffi ffi
b
Ms
q
ffiffiffiffiffi
A 1
p þ
ffiffiffiffiffi
A 2
p
À
Á :
By Eq. (8.17), the natural frequencies of double acting symmetrical hydraulic
cylinders are shown in Fig. 8.7 and Table 8.3. When the piston is in two extreme
positions of the hydraulic cylinder, the natural frequency reaches the maximum.
When the piston is in the middle position of the hydraulic cylinder, the natural
frequency reaches the minimum.
When x 0 ¼ 0:5s, the natural frequency reaches the minimum value x 0 ¼ 2
ffiffiffiffiffi ffi
A 2 b
sM
q
.
When the effective area of the left and right piston rods of the hydraulic cylinder
is not the same, the hydraulic cylinder is asymmetric. As shown in Eq. (8.17), for
example, in the case of A 1 ¼ 2A 2 , the natural frequency characteristics of the
cylinder are shown in Fig. 8.8 and Table 8.4. When the piston reaches the extreme
position of the end caps on both sides of the cylinder, the natural frequency reaches
Natural frequency
Initial position of piston
Fig. 8.7 Natural frequency
characteristics of double
acting hydraulic cylinder
(A 1 ¼ A 2 )
10
8 Pneumatic Actuators, Driving Elements and Accessories
A
2
1 b
V 10
þ
A
2
2 b
V 20 þ A 2 s
When x ¼ s; K max2 ¼
A
2
1 b
V 10 þ A 1 s
þ
A
2
2 b
V 20
When the volume of oil in pipeline is neglected and only the volume of oil in the
cylinder is considered, the natural frequencies and their minimum values of the
hydraulic cylinder system are, respectively,
V 10 ¼ V 20 ¼ 0; x ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
b
M
Á
A 1
x
þ
A 2
s À x
s
ð8:17Þ
When,
x 0 ¼
ffiffiffiffi
A 1
p
ffiffiffi ffi
A 1
p þ
ffiffiffi ffi
A 2
p s,
the
minimum
natural
frequency
is
x 0 ¼ x min ¼
ffiffiffiffi ffi
b
Ms
q
ffiffiffiffiffi
A 1
p þ
ffiffiffiffiffi
A 2
p
À
Á :
By Eq. (8.17), the natural frequencies of double acting symmetrical hydraulic
cylinders are shown in Fig. 8.7 and Table 8.3. When the piston is in two extreme
positions of the hydraulic cylinder, the natural frequency reaches the maximum.
When the piston is in the middle position of the hydraulic cylinder, the natural
frequency reaches the minimum.
When x 0 ¼ 0:5s, the natural frequency reaches the minimum value x 0 ¼ 2
ffiffiffiffiffi ffi
A 2 b
sM
q
.
When the effective area of the left and right piston rods of the hydraulic cylinder
is not the same, the hydraulic cylinder is asymmetric. As shown in Eq. (8.17), for
example, in the case of A 1 ¼ 2A 2 , the natural frequency characteristics of the
cylinder are shown in Fig. 8.8 and Table 8.4. When the piston reaches the extreme
position of the end caps on both sides of the cylinder, the natural frequency reaches
Natural frequency
Initial position of piston
Fig. 8.7 Natural frequency
characteristics of double
acting hydraulic cylinder
(A 1 ¼ A 2 )
10
8 Pneumatic Actuators, Driving Elements and Accessories