Page 569 - Mechatronics with Experiments

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JWST499-c07
JWST499-Cetinkunt
ELECTROHYDRAULIC MOTION CONTROL SYSTEMS 555
where Q (t) is the flow rate from the pump, Q (t) is the flow rate from pump to cylinder
p
v
side of the valve (P to A or P to B, that is Q (t) = Q (t) for up motion or Q (t) = Q (t)
v
PA
PB
v
for down motion, see below).
The two-stage servo valve (flapper nozzle type first stage, and spool type second stage
(Figure 7.60)) is modeled as a current command and actual current dynamic relationship,
torque generated from the current via a gain, and the spool displacement relationships of
the second stage. Once the spool position is known, the flow rate through the valve can
be determined as a function of the pressure differential across it and the spool position.
The current amplifier and the electrical dynamics of the servo valve are modeled as the
dynamic relationship between the commanded current and actual current. It is modeled as
a first-order filter. The current to torque relationship is modeled as a constant gain. The
torque to spool position is described as a second-order mass-force system, where the natural
frequency of the model (w ) varies in a range as a function of the magnitude of the spool
n
displacement as a percentage of the maximum value,
x (t)
v
(7.548)
x
v,max
(or as the ratio of the commanded current to the maximum commanded current).
i (t)
cmd
(7.549)
i cmd,max
In other words, the frequency response of the valve spool position to the current command
(Figure 7.70) is different for different values of the magnitude of the current command. For
instance, for 40% of maximum current command, corresponding to about 40% of maximum
spool displacement in steady-state, the bandwidth of spool position is 75 Hz, whereas for
100% of the commanded current, the bandwidth is 25 Hz. The dynamic model of the valve
and its current amplifier, relating the commanded current signal to the actual current, then
to the valve actuation torque, and then to the valve spool position, and finally to the flow rate
as function of spool position and pressure differential across the valve, are expressed below.
i (s) K a
v
= (7.550)
i cmd (s) s + 1
a
T (s) = K ⋅ i (s) (7.551)
v t v
( 2 2 ) x vmax 2
s + w s + w n ⋅ x (s) = ⋅ w ⋅ T (s); w (x ∕x vmax ) (7.552)
v
v
n
v
n
n
T v,max
For x ≥ 0.0 up motion (7.553)
v
√
(p (t) − p (t))
v
|x (t)| p A
Q (t) = Q ⋅ ⋅ (7.554)
PA r
x Δp
v,max r
√
(p (t) − p (t))
v
B
t
|x (t)|
Q BT (t) = Q ⋅ ⋅ (7.555)
r
x v,max Δp r
For x < 0.0 down motion (7.556)
v
√
(p (t) − p (t))
|x (t)| p B
v
Q (t) = Q ⋅ ⋅ (7.557)
PB r
x Δp
v,max r
√
(p (t) − p (t))
v
|x (t)| A t
Q (t) = Q ⋅ ⋅ (7.558)
AT
r
x v,max Δp r
where x v,max is the maximum spool displacement of the servo valve when maximum
actuation torque is applied T v,max = K ⋅ i v,max .
t