Page 543 - Mechatronics with Experiments

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JWST499-Cetinkunt
JWST499-c07
ELECTROHYDRAULIC MOTION CONTROL SYSTEMS 529
adjustable. Let us assume that the maximum pressure is set to p max,p for the pump side,
and to p max,l for the line side. If P > p max,p , P = p max,p ; the pump side relief valve opens
P
P
and limits the maximum output pressure of the pump. If either of the cylinder side lines’
pressure exceeds the relief valve settings, P , P > p max,l , P , P = p max,l ; the cylinder
A
B
B
A
side relief valve opens.
If the relief valve is modeled to take all excess flow between the supply and flow that
goes through the main valve as soon as the pump pressure reaches the relief pressure (ideal
relief valve model, see Figure 7.41c),
Q (t) = Q (p relief , p (t)) (7.376)
r
P
r
= Q (t) − (Q (t) + Q (t)) ; for p = p relief and up motion (7.377)
PA
PT
P
P
= Q (t) − (Q (t) + Q (t)) ; for p = p relief and down motion (7.378)
PT
P
PB
P
= 0.0 ; for p (t) < p relief (7.379)
P
then, when p = p relief , the fluid compressibility effect on the pump-valve side is effectively
cancelled, since this type of relief valve model enforces that there is no net fluid volume
change in the hose volume between pump and valve. In other words, the right hand side
of Equation 7.297 (or Equation 7.305) would be zero when p = p , which effectively
relief
cancels the fluid compressibility. Fluid compressibility is still taken into account when
p < p .
relief
A more realistic relief valve model (non-ideal relief valve model) includes a small
increase in the pump pressure and the ability to dump the flow as a function of the difference
between its input pressure (pump pressure) and relief pressure with a high gain,
Q (t) = Q (p , p (t)) (7.380)
r r relief P
= K relief ⋅ (p (t) − p relief ); for p (t) ≥ p relief (7.381)
P
P
= 0.0 ; for p (t) < p relief (7.382)
P
where K relief is a high gain approximating the behavior of the relief valve (see Figure 7.41c,
where this equation is an approximation to the “actual” relief valve behavior). With this
model, flow through the relief valve is determined by the pressure relationships and it
does not cancel the right hand side of the pressure dynamics Equation 7.297 or 7.305
for p (t) ≥ p relief . Therefore, this type of relief valve model does not cancel the fluid
P
compressibility effect on the pump-valve side for p (t) ≥ p relief .
P
The non-compressible fluid condition can always be imposed in any section of the
hydraulic circuit by setting the left hand side of the pressure dynamics equations for
p (t), p (t), p (t) equal to zero.
B
A
P
Further accuracy can be added to the relief valve model by including its transient
dynamic response between pump pressure and and the flow rate through the relief valve,
that is using a first (or second) order filter model,
if p (t) ≥ p relief (7.383)
P
dQ (t)
r
relief = −Q (t) + K relief ⋅ (p (t) − p relief ) (7.384)
r
P
dt
else (7.385)
Q (t) = 0.0 (7.386)
r
end (7.387)
where relief is the time constant of the relief valve response when modeled as a first-order
filter, Q (t ) initial condition would be needed to integrate this equation over time.
r 0