Page 621 - The Mechatronics Handbook

0066_Frame_C20 Page 91 Wednesday, January 9, 2002 5:49 PM

The following magnitudes are differentiated (the subscripts 1 and 2 refer, respectively, to the rear 1
and front 2 chambers of the pistons):

A piston thrust section
disturbance of force acting on the piston rod
F e
G mass flow rate of air entering the chamber
M mass of the translating parts of the piston
n air polytropic coefficient
P cylinder chamber pressure
initial cylinder chamber pressure
P i
ambient pressure
P amb
R air constant
initial cylinder chamber air temperature
T i
x rod position measured starting from x 0
piston half stroke
x 0
dead band
x m
γ coefficient of viscous friction
The continuity and equilibrium equations are given by:

P 1 n dx
-------- = ------------------------------------------------------------------------- – -------------------------------- ------ (20.58)
G 1 nRT 1i
dP 1
(
(
dt A 1 x 0 + x m1 + x) P 1 /P 1i ) ( 1 – n)/n ( x 0 + x m1 + x) dt
P 2 n dx
-------- = ------------------------------------------------------------------------ + --------------------------------- ------ (20.59)
G 2 nRT 2i
dP 2
(
dt A 2 x 0 + x m2 – x) P 2 /P 2i ) ( 1 – n)/n ( x 0 + x m2 − x) dt
(
2
d x ( P 1 – P amb )A 1 – ( P 2 – P amb )A 2 – F e – g dx/dt
-------- = --------------------------------------------------------------------------------------------------------- (20.60)
dt 2 M
The flow proportional valve V 1 is modeled as a variable section pneumatic resistance. The equations
used for calculating mass flow rate G through a pneumatic resistance, characterized by a conductance C
and by a critical ration b, in accordance with ISO 6358, which connects two environments A and B, with
respective pressures of P A and P B , taken to be positive in the A → B direction, are

P B
sonic flow: G = r 0 P A C for 0 < ----- ≤ b (20.61)
P A

----- ≤
-----------------------
subsonic flow: G = r 0 P A C 1 –   P B /P A – b  2 for b < P B 1 (20.62)
1 b 
–
P A
P A
sonic flow: G = – r 0 P B C for 0 < ----- ≤ b (20.63)
P B

P A
-----------------------
subsonic flow: G = – r 0 P B C 1 –   P A /P B – b  2 for b < ----- ≤ 1 (20.64)
1 b 
–
P B
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