Page 534 - Mechatronics with Experiments

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Example Let us consider an electrohydraulic (EH) motion axis with the following
parameters for the cylinder and load.
A he = 2.0in 2 (7.298)
A = 1.0in 2 (7.299)
re
L = 20 in (7.300)
W = 1000 lb (7.301)
Let us consider the natural frequency at the mid-stroke point,

V = A ⋅ L∕2 = 20 in 3 (7.302)
he he
and
V = A ⋅ L∕2 = 10 in 3 (7.303)
re
re
5
Let us approximate the bulk modulus of hydraulic fluid to be 2.5 × 10 psi. The weight of
2
the rod and load W = 1000 lb, hence the mass, M = W∕g = 1000 lb∕386 in∕s . The open
loop natural frequency of the EH axis when the cylinder is at mid-stroke point is
√
w = K∕M (7.304)
n
1000 2 2
M = lb∕(in∕s ) = 2.59 lb∕(in∕s ) (7.305)
386
( 2 2 )
2 1 2 2 2 3
5
K = 2.5 ⋅ 10 ⋅ + lb∕in (in ) ∕in (7.306)
20 10
4
= 7.5 ⋅ 10 lb∕in (7.307)
Then,
√
w = K∕M (7.308)
n
√
2
4
= 7.5 ⋅ 10 ∕2.59 (lb∕in)∕(lb − s ∕in) (7.309)
= 170 rad∕s (7.310)
= 27 Hz (7.311)
Therefore, a closed loop controller should not attempt to reach a bandwidth higher than

w bm < (1∕3 ⋅ w ) = 9.0 Hz (7.312)
n
This particular EH motion axis cannot accurately follow cyclic small motion commands
with a frequency higher than 9.0Hz.

7.8 LINEAR DYNAMIC MODEL OF A ONE-AXIS HYDRAULIC
MOTION SYSTEM

Linear dynamic models assume that the hydraulic system operates about a nominal oper-
ating condition, that is the valve is operating about the null position with small movements
(Figure 7.64). Figure 7.80 shows a block diagram of a closed loop electrohydraulic (EH)
control system. Figures 7.81, 7.89, and 7.90 shows a closed loop position control (position
servo control) version of the system. Figure 7.82 shows a closed loop force control (force
servo loop) version of the system. Figure 7.91 shows a one-axis EH motion system where
the load pressure is regulated by the control element instead of the load position.