Geology Reference
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(used in aircrafts) with interacting components
from mechanical and hydraulic domains. The
actuator is composed of a single chamber and a
piston of cross-sectional area A. The amount of
fluid entering the chamber q in through a hydraulic
pump is controlled by the control valve. The pres-
sure difference between right and left sides of the
chamber ( p r - p l ) results in a force that controls
the movement of the piston a distance u ( t ). The
piston is attached by a rigid rod to a mass M that is
connected to a spring and a damper which model
the aerodynamic effects. The displacement is con-
trolled either manually or automatically based on
a feedback system. The description of the actuator
parameters are given in Table 2.
The bond graph model for the actuator of
Figure 5(a) is shown in Figure 5(b). Note that
Figure 5(c) represents the TCG which will be
discussed later. The reservoir is represented by
an effort source Se and the pump is represented
by a transformer TF 1 with efficiency parameter
PE. The valve is represented by a resistor R 1 = R V .
The difference in the chamber's pressure p r - p l
(i.e. e 3 ) is represented by a 0-Junction that ensures
equal pressures (e3 = e4 = e5). The fluid com-
pressibility is represented by a C-element
C
variables (two capacitors and one inductance)
namely e 4 , e 8 and f 7 . Thus the equation of motion
is of a third order. To verify the BG model, the
effort e 4 can be calculated as the force in the first
spring (bond 4) in Box 6.
Similarly, the equation governing the second
capacitor is given as:
(12)
e
=
( /
1
C f
)
=
kf
8
2 8
7
The equation for the inductance element is
given as:
Mf
= = − − =
e
   
e
e
e
Ae
  
− −
e
e
7
7
6
8
9
4
8
9
(13)
Substituting Eqs. (11) and (12) into Eq. (13)
and making use of
e
=
Bf
, we get:
9
7
Mf
=
( / )(
1
C Aq
A f
2
)
kf
Bf
(14)
7
in
7
7
7
  
Substituting f
u t
( ),
f
u t
( ),
f
u t
( )
=
=
=
7
7
7
into Eq. (14) and simplifying leads to:
3
2
u t
t
( )
u t
t
( )
u t
t
( )
2
MC
+
BC
+
(
kC
+
A
)
=
Aq in
1 = β . The pressure difference in bond 5 ( e 5
= p r - p l = e 3 = e 4 ) is multiplied by the cross-
sectional area of the piston A to provide the force
acting on the mass M. This is represented by the
transformer element TF 2 , connecting bonds 5 and
6, with the transformer parameter A. The inertial
effect of the mass M is modeled by an I-element.
The spring is represented by a C-element and the
damper is represented by an R-element R 2 = B .
The displacement u ( t ) is the time integral of the
flow (velocity) in bond 7. There are three state
V
1
1
1
3
2
(15)
Equation (15) represents the equation of motion
governing the dynamics of the hydraulic actuator
that can be obtained using traditional mechanical
system derivation. Therefore, the mathematical
basis of bond graphs is the same as the classical
dynamic theory. One should not, however, over-
look the advantages of bond graphs (e.g., topologi-
cal modeling tool, modeling sensors, qualitative
Box 6.
)
(11)
e
=
( /
1
C f
)
=
( /
1
C
)(
f
f
)
=
( /
1
C S f
)(
Af
)
=
( /
1
C q in
)(
Af
4
1 4
3
5
7
7
1
1
1
 
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