Page 612 - Mechatronics with Experiments

JWST499-Cetinkunt
JWST499-c07
598 MECHATRONICS Printer: Yet to Come October 9, 2014 8:41 254mm×178mm
x (t) √
(8) Q (t) = Q ⋅ s ⋅ (p (t) − p (t))∕Δp if x < 0 (7.802)
PB r P B r s
x s,max
x (t) √
(9) Q (t) = Q ⋅ s ⋅ (p (t) − p (t))∕Δp if x < 0 (7.803)
AT r P T r s
x s,max
Notice that the ideal relief valve model cancels the fluid compressibility whenever p (t) = p ,
P relief
by calculating the Q (t) such that the right hand side of the pump pressure dynamics is equal to zero.
r
The accumulator acts both as a hydraulic power storage device which increases the line
capacitance, hence absorb large pressure spikes, and as a hydraulic power source when line pressure
drops below the pressure setting of the accumulator to support the pump. Especially when fast
dynamic changes occur in demand for hydraulic power, the pump cannot react fast enough. But the
accumulator can provide the transient power needed for a short period of time. The flow into or out
of the accumulator is function of the line pressure, accumulator pressure and max-min pressure range
settings of the accumulator.
Accumulator state (volume and pressure) is modeled as,
If V (t) ≤ 0.0 and p (t) ≤ p (t) (7.804)
acc P acc
(10) Q (t) = 0.0 (7.805)
acc
else (7.806)
√
(10) Q (t) = sign(p (t) − p (t)) ⋅ K acc ⋅ |p (t) − p (t)| (7.807)
P
acc
acc
P
acc
end (7.808)
( )
dp (t) p max − p min
acc
(11) = ⋅ Q (t); p (t ) = p min (7.809)
acc
acc 0
dt V disch
dV (t)
(12) acc = Q (t); V (t ) = 0.0 (7.810)
dt acc acc 0
where initial conditions of the accumulator pressure and fluid volume in the accumulator are specified
(p (t )and V (t )). On power-up, as accumulator is charged by the line pressure, the accumulator
acc 0 acc 0
pressure increases. For a given accumulator, we know the discharge volume (V ), maximum,
disch
minimum, and precharge pressures (p max = p relief , p min , p ) and initial conditions on pressure and
pre
fluid volume in the accumulator, that is typical values p (t ) = p , V (t ) = 0.0.
0
acc
pre
acc 0
For simulation purposes, we can consider the initial pressure of the accumulator as p (t ) =
acc 0
p min (or a value between p min and p max ) and initial volume V (t ) = 0.0. Maximum discharge volume
acc 0
is reached at p acc = p max , and zero discharge volume is reached (no fluid volume left to discharge)
at p acc = p min . The net result of adding an accumulator between the pump and the valve is to reduce
the pressure variations by increasing the compliant fluid volume, while the accumulator acts both as
energy storage (during p > p ) and energy source (during p < p ) device.
acc
acc
s
s
An engine or electric motor provides the mechanical power to the pump. The engine/motor
speed is w pump rev∕min. The pump is of fixed displacement type, as indicated by its hydraulic symbol,
3
and the volumetric displacement of the pump is D m ∕rev. The leakage flow from the pump is
p
neglected. We assume that the pressure at the input port of the valve is the same as the pump output
pressure, p , and the pressure at the output port of the valve is the same as the pressure at the cylinder.
P
In other words, we neglect the pressure drop in the line between the pump and the main valve, and
the main valve and the cylinder.
Let us assume the following values for the parameters of the system components:
3
D = 0.0001 m ∕rev (7.811)
p
3
Q = 20.0 × 10 −4 m ∕s (7.812)
r
Δp = 1000 psi (7.813)
r
x db = 10.0% (7.814)
x s,max = 100.0% (7.815)
−6
2
A (x ) = (20 ⋅ 10 )∕(100 − x )) ⋅ (|x | − x )m ; x ≥ x db (7.816)
s
db
PA
s
s
db
−6
2
A (x ) = (10 ⋅ 10 )∕(100 − x )) ⋅ (|x | − x )m ; x ≤ −x db (7.817)
db
s
PB
db
s
s