Page 544 - Mechatronics with Experiments

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Further detail can be added to the relief valve dynamics by modeling it as a second-
order filter, which means the inertial effects of the relief valve spool are taken into account.
if p (t) ≥ p relief (7.388)
P
2
d Q (t) dQ (t) 2
r
r
= −2 w n − w ⋅ Q (t) + K relief ⋅ (p (t) − p relief ) (7.389)
r
P
n
dt 2 dt2
else (7.390)
Q (t) = 0.0 (7.391)
r
end (7.392)
where and w represent the damping ratio and natural frequency of the second-order
n
dynamic model for the relief valve.
Directional Check Valves In many EH motion applications, it is not desirable to
allow flow from the cylinder back to the pump which can happen if P > P or P > P .
P
B
P
A
This is accomplished by a one directional load check valve on each line. The load check
valve closes to prevent back flow from cylinder to pump. During extension, if P > P ,
A P
then Q = 0.0; During retraction, if P > P , then Q = 0.0.
PA B P PB
Open-Center EH Systems An “open-center” EH system has a fixed displacement
pump and an open-center valve, where there is an orifice between pump and tank and the
pump displacement is constant,
A (x ) ≠ 0 (7.393)
PT
s
Q = w pump ⋅ D ( sw0 ) (7.394)
p
P
Closed-Center EH Systems A “closed-center” EH system has a variable displace-
ment pump and closed-center valve, where there is no orifice directly between pump and
tank, and the pump displacement is variable,
A (x ) = 0; (7.395)
PT
s
Q = w pump ⋅ D ( ) (7.396)
sw
P
p
Dynamic Model of an Accumulator The accumulator state at any given instant
of time is defined by two variables: accumulator pressure and fluid volume in the accumula-
tor, p (t), V (t). The operating parameters of the accumulator are precharge, minimum,
acc acc
and maximum pressures (p , p , p ) and the maximum discharge volume of the accu-
pre min max
mulator (the fluid volume difference in the accumulator between minimum and maximum
pressures, that is V ). If we assume that the flow between the line and accumulator
disch
follows the same relationship as the flow through an orifice or valve, and that the pressure
changes linearly as a function of the change in volume, the dynamic state of the accumulator
can be described by the following equations,
If V acc (t) ≤ 0.0 and p line (t) ≤ p acc (t) (7.397)
Q acc (t) = 0.0 (7.398)
else (7.399)
√
Q (t) = sign(p (t) − p (t)) ⋅ K ⋅ |p (t) − p (7.400)
acc line acc acc line acc (t)|
end (7.401)
( )
dp acc (t) p max − p min
= ⋅ Q acc (t) (7.402)
dt V disch
dV (t)
acc
= Q acc (t) (7.403)
dt