Page 561 - Mechatronics with Experiments

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JWST499-c07
JWST499-Cetinkunt
ELECTROHYDRAULIC MOTION CONTROL SYSTEMS 547
F sol
Q 3t C P P 3
V 3 δ 3
F k
T shear p
Q leak d 3 A 3 l 3
A 4
Port 2
P
Solenoid m p X i
A 1 p
P 2 Q 4
V 2
Q 2
A
Poppet 2
valve

Port P = P
1 s 1
(a) (b)
FIGURE 7.99: Poppet valve cross-section and its dynamic model: (a) picture of a 2/2 way
poppet valve (by Rexroth), (b) cross-section of the 2/2 way poppet valve. Courtesy
Bosch-Rexroth.

√
Q (t) = C ⋅ A (x (t)) ⋅ p (t) − p (t) (7.506)
p
2
d
2
1
2
√
Q (t) = C ⋅ A ⋅ p (t) − p (t) (7.507)
l
4
d
2
4
√
Q = C ⋅ A ⋅ p (t) − p t (7.508)
t
3
3t
d
Q leak (t) = Q leak 3 3 3 2 3 leak ⋅ (p (t) − p (t)) (7.509)
(l , d , , p , p ) ≈ K
2
3
where
F shear (t) = F shear 3 3 3 p (7.510)
(l , d , , ̇ x (t)) ; shear force
which can be defined as a look-up table for the numerical simulation. The shear force is a
function of the length l , diameter d , clearance , and the speed of the spool ̇ x (t).
p
3
3
3
The force generated by the electric actuator of the valve is modeled as,
F (t) = K ⋅ i (t) (7.511)
sol fi sol
where i (t) is the current supplied to the solenoid (or to an electric actuator from the
sol
amplifier), K is the current to force gain.
fi
The parameters of the valve are;
m = mass of the poppet (7.512)
p
c = damping coeficient of the poppet motion (7.513)
p
k = spring of the poppet. The spring is preloaded to hold the valve closed (7.514)
p
when the valve is not actively actuated by the electric actuator (F (t)) (7.515)
sol
A = cross-sectional area of the valve where p (t) acts on (7.516)
1
1
A = orifice opening cross-sectional area of the valve (7.517)
2
A = area of the valve on the back side of the poppet (7.518)
3