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2. Insert 0-J between 1-J to accommodate force-
generating elements (stiffness and damping
forces).
3. Simplify the resulting bond graph by remov-
ing the zero velocity at the support points.
4. Assign numbers, and directions to bonds
using the half arrow (defines the direction
of power).
5. Assign causality to bond graph elements and
bonds (details provided in next subsection).
energy storage elements. For the I-element, the
state variable is flow and for the C-element the
state variable is effort. The
n
-DOF system has
2
n
state variables (e.g.,
e
2
,
f
3
,
e
7
,
f
11
,
e
14
,
f
18
, …).
Each flow variable represents the velocity of the
associated mass and efforts represent the stiffness
forces in the springs. The inertial force of the first
mass is given as (bond 3) in Box 1.
Note the substitutions of equality of flows and
balance of efforts at 1-junctions and equality of
efforts and balance of flows at 0-junctions.
Similarly, the inertial force in bond 11, (see Box
2).
These procedures result in the lumped pa-
rameters of damping (
D
1
,
D
2
, …,
D
n
), stiffness
(
k
1
,
k
2
, …,
k
n
) and inertia (
m
1
,
m
2
, …,
m
n
) being
interconnected using energy conserving junctions
producing the topological bond graph model
shown in Figure 4(b). Each mass is represented
as a separate block connected to neighboring
blocks through bonds. Sensors are also modeled as
separate blocks for measuring displacements and
are discussed in Section 6. Note that Figure 4(c)
represents the TCG which will be discussed later.
Considering the inertial force of the (n-1)
th
mass, one gets: (see Box 3).
Finally, the equilibrium equation of the forces
acting on the
n th
mass can be shown to be shown
in Box 4.
Combining Eqs. (6- 9), the equations of motion
of the
n
-DOF structure is given in a matrix form
in Box 5.
Eq. (10) represents the well known equation
of motion of the
n
-DOF structure derivable using
structural dynamics principles (Clough & Penzien
2003, Chopra 2007). Note that, if the structure is
subjected to ground acceleration
y
( )
, the forces
at the floor levels are given as
p t
3.2.2 Causality Assignment and
Bond Graph Verification
To assign causality to BGs we assign fixed cau-
salities to sources. Integral causality is assigned
to storage elements (I- and C-elements) and is
propagated through junctions. I- and C-elements
have integral causality since cause is integrated
to provide effect. Causality is then assigned to
R-elements and bonds connecting junctions. The
procedure is continued until the entire BG is as-
signed causality.
The BG model of Figure 4(b) represents the
equations of motion of the
n
-DOF system in an
implicit form. The verification of the BG model
is performed by deriving the system equations
of motion from the BG and comparing them
with those from structural dynamics theory. The
derivation of the system equations from the bond
graph is systematic and can be easily coded. The
system state variables are associated with the
1 2
.
Thus the BG is simply a topological represen-
tation of the system equations of motion, using
elements that exchange efforts and flows. The BG
model is correct only if the correct equations of
motion of the system can be derived from it. The
above example derived the equations of motion for
an n-DOF system by hand for sake of explanation.
This derivation can be also automated (Manders
et al, 2006).
( )
= −
m y t
( );
i
=
,
, ...,
n
i
i
3.3 Hydraulic Actuator System
This section illustrates the modeling of multidis-
ciplinary dynamic systems using BG. Figure 5(a)
shows a simplified model of a hydraulic actuator
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