Page 541 - Mechatronics with Experiments

JWST499-Cetinkunt
October 9, 2014 8:41 254mm×178mm
JWST499-c07
Printer: Yet to Come
ELECTROHYDRAULIC MOTION CONTROL SYSTEMS 527
The first form of the equation is the classic orifice equation relating the pressure differential
across the orifice, orifice area, and discharge coefficient to the flow rate. The second form
relates the flow rate for a given valve based on the valve ratings as percentage of the rated
flow. The second form of the flow rate equation is more useful in practice since for a
given valve, its performance ratings are measured and given by the manufacturer (or can
be determined experimentally):
Q , p , A max (7.357)
r
r
Then for a given orifice opening A and pressure differential across the valve Δp, we can
find the flow rate across the valve.

7.9 NONLINEAR DYNAMIC MODEL OF ONE-AXIS
HYDRAULIC MOTION SYSTEM

A nonlinear dynamic model for a single axis linear hydraulic motion system, where the
actuator is a cylinder, is discussed below. The same identical equations apply for a rotary
hydraulic motion system, where the actuator is a rotary hydraulic motor, with analogous
parameter replacements. The fluid compressibility is also taken into account in the model
(Figures 7.3 and 7.90). Let us consider the motion of the piston-rod-load assuming they
are rigidly connected to each other. Using Newton’s second law for the force–motion
relationships of the cylinder and the load,
(i) Extension motion:

m ⋅ ̈ y = P ⋅ A − P ⋅ A − F ;0 ≤ y ≤ l cyl (7.358)
A
ext
B
A
B
and the pressure transients in the control volumes on both sides of the cylinder,

̇ p (t) = (Q (t) − ̇ y(t) ⋅ A ) (7.359)
A PA A
(V + y(t) ⋅ A )
hose,VA A

̇ p (t) = (−Q BT (t) + ̇ y(t) ⋅ A ) (7.360)
B
B
(V hose,VB + (l cyl − y(t)) ⋅ A )
B

̇ p (t) = (Q (t) − Q (t) − Q (t) − Q (t)) (7.361)
P P PA PT r
V
hose,pv
where
Q = w ⋅ D ( ) (7.362)
P pump p sw
A (x ) √
PT
s
Q = Q ⋅ ⋅ (P − P )∕P (7.363)
PT r P T r
A
PT,max
A (x ) √
s
PA
Q PA = Q ⋅ ⋅ (P − P )∕P r (7.364)
A
r
P
A PA,max
A BT (x ) √
s
Q = Q ⋅ ⋅ (P − P )∕P (7.365)
BT r B T r
A
BT,max
Q = Q (p relief , p ) (7.366)
p
r
r
(ii) Retraction motion:
m ⋅ ̈ y = P ⋅ A − P ⋅ A − F ;0 ≤ y ≤ l cyl (7.367)
A
A
B
ext
B