Page 444 - Mechatronics with Experiments

JWST499-Cetinkunt
JWST499-c07
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3. Voltage is a relative quantity, there is no absolute zero voltage. However, there is
absolute zero pressure, which is the vacuum condition.
The capacitor (C) and accumulator (C ) analogy is as follows,
H
1 1
V = i(t) ⋅ dt p = Q(t) ⋅ dt (7.37)
C ∫ C ∫
H
where C is determined by the volume and stiffness of the accumulator, which has the
H
3
5
5
2
units of Length ∕Force (i.e., m ∕Nt as a result of volume∕pressure = [m ]∕[Nt∕m ]). The
capacitance of a hydraulic circuit is defined as the ratio of volumetric change to pressure
change.
dp(t) 1
= ⋅ Q(t) (7.38)
dt C H
Q(t)
C = (7.39)
H
dp(t)∕dt
(dV∕dt)
= (7.40)
(dp∕dt)
dV
= (7.41)
dp
For a given volume of fluid, V, and bulk modulus, , capacitance is
V
C = (7.42)
H

The inertia of moving fluid in a pipe acts like an inductor. Consider a pipe with cross-
sectional area A, length l, mass density of fluid , pressures at the two ends p , p , and flow
2
1
rate Q through it. Assume the following,
1. the friction in the pipe is neglected,
2. the compressibility of the fluid in the pipe is neglected.

The motion of the fluid mass in the pipe can be described as,

(p − p ) ⋅ A = m ⋅ ̈ x (7.43)
1 2
Q ̇
= ( ⋅ l ⋅ A) ⋅ (7.44)
A
where m = ⋅ V = ⋅ l ⋅ A, and Q = ̇ x ⋅ A. The pressure and flow relationship is
( )
⋅ l
p − p = ⋅ Q ̇ (7.45)
1 2
A
p − p = L ⋅ Q ̇ (7.46)
1 2
where the hydraulic inductance is defined as L = ⋅ l∕A. Notice the analogy between
pressure, flow rate, and hydraulic inductance versus the voltage, current, and self inductance,
dQ(t)
Δp(t) = L ⋅ (7.47)
dt
di(t)
ΔV(t) = L ⋅ (7.48)
dt