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e
q = A x˙ + C
( p − p ) −C p
− V3 p˙ ,
(11)
L 3 ip2
B L ep2 L L
e
Fig. 5. Results of cavity filling flow simulation for WAIM
Fig. 6. CFD simulation results of load characteristic
where pA, pB, pL, qA, qL are respectively pressure and flow rate at port A, B and L of the cylinder, and x is the displacement of the piston. Besides m, Bp, f, Cip, Cep are respectively equivalent mass, visco-damping coefficient, static friction, internal and external leakage coefficient of the piston, and βe is the bulk modulus of water, the transmission medium of this system. The friction between the cylinder and piston is large, especially the static friction, for the reason of poor lubrication of water. Meanwhile, the internal leakage in the pressure cylinder is also large for the low viscosity of water and it can not be neglected. Non-external leakage consideration and Laplace transformation of Eqs. (9)–(11) yield:
2
4 Compensation for Pressure Control
pA A1 = pB A2 + pL A3 + ms X + Bp sX + F ,
(12)
q = A sX + C
( p − p ) + V1s p ,
(13)
An ordinary air accumulator which has been specially treated against corrosion and rust is used in the system. The
A 1 ip1 A B A
e
discharge process of accumulator is assumed as fast and
q = A sX + C
( p − p ) − V3 s p .
(14)
adiabatic, i.e., the polytropic exponent na=1.4. Fluid is incompressible in comparison to the gas. And no turbulent
L 3 ip2
B L L
e
flow appears in the accumulator[22]. Based on these assumptions and linearization of compressing process of gas, the linear model of accumulator with inlet
The flow resistance of the throttle valve at port L and the
water injector can be linearized as
characteristic can be written as
qL = ktL ( pL − p),
(15)
p m s2 + B s + k G (s) = A = − a a a ,
A q A 2 s
(8)
where ktL is the flow rate gain to pressure of the throttle valve and injector. By Eqs. (7) and (15), we obtain that
2 + ′ +
where
k = n p A 2 V
and
B = B
+ A 2 k
. ma, ka
G (s) = pL = mL s BL s kL ,
(16)
a a a 0 a a 0
a