Health Assessment of Engineering Structures Using Graphical Models - Structural Seismic Design Optimization and Earthquake Engineering - page 354

Table 2. Description of the actuator parameters

Figure 6(a) depicts the modeling of continuous
dynamic systems using the finite-mode bond
graphs. The transformers ensure that each mode
is excited by the corresponding modal force. The
number of modes to be retained in the dynamic
analysis varies based on the boundary conditions
and the dynamic characteristics of the structure
(Clough & Penzien 2003, Chopra 2007). The as-
sociated TCG is shown in Figure 6(b) and will be
discussed later.
The dynamic analysis of continuous struc-
tures using the finite-mode bond graph approach
requires performing free vibration analysis to
determine the natural frequencies and mode shapes
of the structure, see, e.g., Clough & Penzien, 2003
and Chopra, 2007. The free vibration analysis is
carried out based on the boundary conditions of
the structure and can be performed analytically or
numerically (e.g. using the finite element method).
The natural frequencies and mode shapes are
then used to quantify the parameters of the bond
graph elements (I, C, R and TF) for each mode of
vibration. In general, the first few modes usually
provide fairly acceptable results.

Parameter Description

A
V 0
R v
Se
p p p r - p l
M
k
B
u(t)

Cross sectional area of the piston
Volume of the chamber
Bulk density of the fluid
Resistance of the valve
Effort source
Chamber's pressure at right side
Chamber's pressure at left side
Pressure difference
Mass
Spring stiffness
Damping coefficient
Displacement of the mass

damage identification, etc.) which are not provided
by most existing system identifications methods.
The next section demonstrates the modeling of
continuous structures using bond graphs.

4. MODELING CONTINUOUS
STRUCTURES USING FINITE-
MODE BOND GRAPHS

This section demonstrates the modeling of con-
tinuous structures such as beams and frames using
bond graphs. The modeling methodology is based
on the normal mode approach (Clough & Penzien
2003, Chopra 2007) and is known as finite-mode
bond graph (Karnopp et al, 2006). To do this, the
external force (represented by an effort source S e )
is connected to a 0-junction that is connected to
n 1-junctions using transformers (Figure 6(a)).
The 0-J decomposes the external forces into their
modal components. The 1-junctions represent the
vibration modes retained and the transformers
represent the mode shape evaluated at the point
of application of the external force. The displace-
ment of the ith 1-J represents the contribution to
the structure response in the generalized coordi-
nates. The I-, C- and R-elements (inertia, stiffness
and damping forces) connected to the 1-junction
represent the modal forces.

5. CONSTRUCTING
TEMPORAL CAUSAL GRAPH
FROM BOND GRAPH

The step that follows the construction of the bond
graph model is to use this model to develop a
framework that facilitates damage detection and
isolation. This is achieved by deriving the tem-
poral causal graph (TCG) from the bond graph
(Mosterman & Biswas, 1999). The TCG repre-
sents the constitutive relations of BG elements
and junctions in graphical form. More precisely,
the TCG represents causal relations among the
system variables and parameters. The TCG is a
directed graph in which the vertices represent the
system variables and the directed edges express
the relation between the vertices. The labels on
the edges determine the type of causal relation