Page 609 - The Mechatronics Handbook

0066_Frame_C20 Page 79 Wednesday, January 9, 2002 5:49 PM

The continuity equation for chamber 2 is

V 0 − A c x + V sm dP 2
+
Q FI – Q C2 = A c x ˙ ------------------------------------- -------- (20.43)
b dt
where

Q C1 = flow entering chamber 1
Q C2 = flow leaving chamber 2
Q FI = leakage flow between piston and barrel
2
2
A c = thrust section = (D al – D st )π/4
D al = bore diameter
D st = rod diameter
V 0 = volume of the chambers with piston centered = A c L/2
L = stroke
x = piston displacement (x = 0 in centered position)
V sm = dead band volume
The dynamic equilibrium equation of the piston is

–
P 1 A c – P 2 A c – F e – Mx ˙˙ F A = 0 (20.44)
where

M = translating mass
F e = external force
F A = force of friction = gx ˙ F ATT sign(x ˙)+
γ = coefficient of viscous friction
F A = force of coulomb friction
Leaks can be modeled as resistances in laminar and steady-state conditions of the following type:

R = ∆P (20.45)
-------
Q
where
R = resistance
Q = flow rate
∆P = pressure difference
In the case of an annular pipe, we get

pDh 1 +( 1.5e )
3
2
Q = ---------------------------------------∆P (20.46)
12ml
where
D = seat diameter
h = meatus thickness = (D − d)/2
d = spool diameter
ε = eccentricity = 2e/(D − h)
e = distance between seat axis and spool axis
µ = dynamic viscosity
l = meatus length

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