Geology Reference
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between the variables. These relations are similar
to the forms of Eqs. (1, 2 and 3 but in a qualita-
tive form (=, +1, -1). For instance, Eq. (3) reveals
that the efforts e1 and
e
2
are equal, the flows
f
1
and
f
2
have +1 relation (i.e., increase in
f
1
leads to
increase in
f
2
and vice-versa) and
f
2
and
f
3
have -1
relation (i.e., increase in
f
1
leads to decrease in
f
2
and vice-versa). The directions of the arrows in the
TCG are defined based on the strong bond. Thus,
a strong bond in a 1-junction governs the flow of
the strong bond with arrows directed towards the
flows of other bonds. The efforts of other bonds
will have arrows directed towards the effort of the
strong bond. A strong bond in a 0-J governs the
effort at the junction with arrows directed towards
efforts of other bonds.
To construct the TCG for the SDOF system
of Figure 3(a) from the BG model, it is first
noted that the 1-J has equal flows and efforts sum
to zero. We start with the external force
p
(
t
), be-
ing imposed on the system, directed towards
e
1
.
Since bond 3 is the strong bond, thus
f
3
determines
the flow and
e
1
,
e
2
,
e
4
determine the effort at the
junction. Therefore, the flow arrows are drawn
directed from
f
3
to
f
2
,
f
4
with equal signs (equal
flow joint) and the efforts arrows are directed
from
e
1
,
e
2
,
e
4
to
e
3
. The labels on the arrows con-
necting efforts are determined from the equation
e
3
=
e
1
-
e
2
-
e
4
. Thus, the arrow connecting
e
3
and
e
2
carries the label -1 indicating the - sign between
e
3
and
e
2
. To complete the TCG model we trans-
late the constitutive relations for I-, C- and R-
elements. For the I-element
e
mass and then the connecting bonds (e.g. bonds 5
and 12). Note that the TCG for the second, third,
…, nth masses are identical and that each bond
graph block is represented by an associated TCG
block. The procedures are systematic and can be
coded (Manders et al, 2006).
The temporal causal graph for the actuator of
Figure 5(a) is shown in Figure 5(c). The energy
source
S
e
defines the effort direction from
e
0
to
e
1
operated by the pump efficiency PE. The direction
of the flow takes the opposite direction (i.e. from
f
1
to
f
0
). Bond 2 is the strong bond (indicated by
the causal stroke or the perpendicular line away
from the junction) at the 1-junction (common flow
junction
f
1
=
f
2
=
f
3
) and thus
f
2
defines the flow
at the junction. Accordingly
f
2
is connected with
f
1
and
f
3
with arrows starting from
f
2
and pointing
towards
f
1
and
f
3
. Since
f
1
=
f
2
=
f
3
, therefore = sign
is shown in these arrows. Conversely, the effort
is determined by
e
1
and
e
3
and thus the arrows are
directed from
e
1
and
e
3
towards
e
2
due to the rela-
tion
e
1
-
e
2
-
e
3
= 0 (or
e
2
=
e
1
-
e
3
) at the 1-junc-
tion, +1 and -1 signs are shown above the arrows
that connect
e
2
with
e
1
and
e
3
, respectively. Bond
4 is the strong bond for the 0-junction (common
effort junction
e
3
=
e
4
=
e
5
). Therefore, the effort
is defined by bond 4. Accordingly,
e
4
is con-
nected with
e
3
and
e
5
by arrows pointing to
e
3
and
e
5
with equality sign. The flow is determined by
f
3
and
f
5
(
f
3
-
f
4
-
f
5
= 0 or
f
4
=
f
3
-
f
5
) and thus the
arrows connect
f
3
and
f
5
to
f
4
with +1 and -1 signs,
respectively. These signs are determined from the
relation
f
4
=
f
3
-
f
5
. The efforts
e
5
and
e
6
are re-
lated by the area of the piston with an arrow
pointing from
e
5
to
e
6
. The arrows for the flows
f
5
and
f
6
take the opposite direction. The strong
bond of the last 1-junction (
f
6
=
f
7
=
f
8
=
f
9
) is bond
7 and thus the flow at this junction is defined by
f
7
. Accordingly
f
7
is connected to
f
6
,
f
8
and
f
9
by
arrows pointing from
f
7
towards
f
6
,
f
8
and
f
9
with
equality sign. The effort at the junction is defined
by
e
6
,
e
8
and
e
9
and is shown by arrows connect-
ing these efforts to
e
7
. The effort relation at the
1-junction is
e
6
-
e
7
-
e
8
-
e
9
= 0 or
e
7
=
e
6
-
e
8
-
e
9
.
or
=
mf
3
3
1
=
∫
/
thus the label is dt/m (i.e. effort
is integrated to provide flow). For the R-element,
the equation is
e D
4
=
and thus the label is D
(being multiplied by f4 to give e4). For the C-
element,
e
f
m e dt
3
3
=
∫
, thus the arrow carries the
label k. This completes the TCG of the SDOF
system.
To build the TCG for the MDOF system of
Figure4, we start by constructing the TCG for each
k
f dt
2
2
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