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that power flows from the source into the
system.
7.
Serial and parallel junctions (Figures
1(c1) & 1(c2)):
A serial or 1-junction has
equality of flows (velocities), and efforts
(forces) sum up to zero. The equality of
velocities or displacements represents the
continuity condition between substructures
that is known in FEM analysis. The summa-
tion represents the D' Alembert's principle
that implies dynamic equilibrium of forces
acting on the system or a substructure of it.
This junction represents a common veloc-
ity point in a structural system. Thus, a 1-J
can model the velocity at a floor level in a
multi-story building.
sum to zero. In a 0-J, the strong bond determines
the effort at the junction and is stroked nearer to the
junction. The constitutive relations for the 0-J are:
e
e
e
e
e
;
f
f
f
f
f
0
= = = =
− − − − =
1
2
3
4
5
1
2
3
4
5
(3)
Equations (1, 2 and 3 and the constitutive
equations for BG elements define the relations
among the system parameters and responses,
which represent the basis for the SI technique
developed in this chapter.
This section provided a brief description of
BG theory and terminology. The modeling of
discrete systems, including SDOF and MDOF
systems and an actuator composed of hydraulic
and mechanical components, using bond graphs
is developed in section 3. Continuous structures
are tackled in Sections 3 and 4, respectively. The
derivation of temporal causal graphs from bond
graphs is explained in Section 5. Section 6 devel-
ops the modeling of sensors using the bond graph
theory. The development of the health assessment
methodology is presented in Sections 7 and 8.
Numerical illustrations are provided in Section 9.
In a 1-junction only one bond, referred to as
strong bond, brings the information of flow (ve-
locity) and other bonds receive it (i.e., only one
bond is open ended and other bonds are stroked
away from the junction). According to Figure
1(c1), bond 3 is the strong bond and thus the flow
is governed by bond 3 and is imposed on other
bonds. Conversely, the effort is determined by
other bonds and is imposed on bond 3. Strong
bonds are used in constructing TCG from BG as
demonstrated in the next section.
The constitutive relation of power for the
1-junction of Figure 1(c1) is given by:
3. MODELING DISCRETE
DYNAMIC STRUCTURES
USING BOND GRAPHS
e f
−
e f
−
e f
−
e f
−
e f
=
(1)
0
This section demonstrates the construction of
bond graphs for discrete dynamic structures.
First, we consider SDOF and MDOF structures.
Subsequently, we demonstrate the modeling
procedure for an actuator composed of hydraulic
and mechanical components. Discrete structures
are those structures having lumped properties
(inertia, damping and stiffness). Verifications of
bond graphs are provided by deriving the equa-
tions of motion of the structure from the bond
graph model which are shown to be identical to
those obtainable from structural dynamics theory.
1 1
2 2
3 3
4 4
5 5
Since the 1-junction has equality of flows,
Eq. (1) leads to:
e e e e e
1
− − − − =
(2)
0
2
3
4
5
Eq. (2) represents the D' Alembert's dynamic
equilibrium principle. The 0-junction (0-J) is the
counterpart of the 1-junction and represents com-
mon effort (force) points in the system (Figure
1(c2)). Thus a 0-J has equal efforts, and the flows
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