<s>
A	O
bond	B-Application
graph	I-Application
is	O
a	O
graphical	O
representation	O
of	O
a	O
physical	O
dynamic	O
system	O
.	O
</s>
<s>
It	O
is	O
similar	O
to	O
a	O
block	B-Application
diagram	I-Application
or	O
signal-flow	O
graph	O
,	O
with	O
the	O
major	O
difference	O
that	O
the	O
arcs	O
in	O
bond	B-Application
graphs	I-Application
represent	O
bi-directional	O
exchange	O
of	O
physical	O
energy	O
,	O
while	O
those	O
in	O
block	B-Application
diagrams	I-Application
and	O
signal-flow	O
graphs	O
represent	O
uni-directional	O
flow	O
of	O
information	O
.	O
</s>
<s>
Bond	B-Application
graphs	I-Application
are	O
multi-energy	O
domain	O
(	O
e.g.	O
</s>
<s>
This	O
means	O
a	O
bond	B-Application
graph	I-Application
can	O
incorporate	O
multiple	O
domains	O
seamlessly	O
.	O
</s>
<s>
The	O
bond	B-Application
graph	I-Application
is	O
composed	O
of	O
the	O
"	O
bonds	O
"	O
which	O
link	O
together	O
"	O
single-port	O
"	O
,	O
"	O
double-port	O
"	O
and	O
"	O
multi-port	O
"	O
elements	O
(	O
see	O
below	O
for	O
details	O
)	O
.	O
</s>
<s>
If	O
the	O
dynamics	O
of	O
the	O
physical	O
system	O
to	O
be	O
modeled	O
operate	O
on	O
widely	O
varying	O
time	O
scales	O
,	O
fast	O
continuous-time	O
behaviors	O
can	O
be	O
modeled	O
as	O
instantaneous	O
phenomena	O
by	O
using	O
a	O
hybrid	O
bond	B-Application
graph	I-Application
.	O
</s>
<s>
Bond	B-Application
graphs	I-Application
were	O
invented	O
by	O
Henry	O
Paynter	O
.	O
</s>
<s>
Many	O
systems	O
can	O
be	O
expressed	O
in	O
terms	O
used	O
in	O
bond	B-Application
graph	I-Application
.	O
</s>
<s>
is	O
a	O
matrix	B-Architecture
object	O
;	O
</s>
<s>
is	O
the	O
Hermitian	B-Algorithm
of	O
;	O
it	O
is	O
the	O
complex	O
conjugate	O
of	O
the	O
transpose	O
of	O
.	O
</s>
<s>
If	O
is	O
a	O
scalar	O
,	O
then	O
the	O
Hermitian	B-Algorithm
is	O
the	O
same	O
as	O
the	O
complex	O
conjugate	O
;	O
</s>
<s>
Relationship	O
between	O
generalized	O
flow	O
and	O
generalized	O
momentum	B-Algorithm
.	O
</s>
<s>
Relationship	O
between	O
generalized	O
momentum	B-Algorithm
and	O
generalized	O
effort	O
.	O
</s>
<s>
The	O
negative	O
sign	O
is	O
dropped	O
in	O
this	O
equation	O
because	O
the	O
sign	O
is	O
factored	O
into	O
the	O
way	O
the	O
arrow	O
is	O
pointing	O
in	O
the	O
bond	B-Application
graph	I-Application
.	O
</s>
<s>
If	O
an	O
engine	O
is	O
connected	O
to	O
a	O
wheel	O
through	O
a	O
shaft	O
,	O
the	O
power	O
is	O
being	O
transmitted	O
in	O
the	O
rotational	O
mechanical	O
domain	O
,	O
meaning	O
the	O
effort	O
and	O
the	O
flow	O
are	O
torque	B-General_Concept
( τ	O
)	O
and	O
angular	O
velocity	O
( ω	O
)	O
respectively	O
.	O
</s>
<s>
A	O
word	O
bond	B-Application
graph	I-Application
is	O
a	O
first	O
step	O
towards	O
a	O
bond	B-Application
graph	I-Application
,	O
in	O
which	O
words	O
define	O
the	O
components	O
.	O
</s>
<s>
As	O
a	O
word	O
bond	B-Application
graph	I-Application
,	O
this	O
system	O
would	O
look	O
like	O
:	O
</s>
<s>
As	O
the	O
engine	O
is	O
applying	O
a	O
torque	B-General_Concept
to	O
the	O
wheel	O
,	O
it	O
will	O
be	O
represented	O
as	O
a	O
source	O
of	O
effort	O
for	O
the	O
system	O
.	O
</s>
<s>
Further	O
,	O
the	O
torque	B-General_Concept
and	O
angular	O
velocity	O
symbols	O
are	O
dropped	O
and	O
replaced	O
with	O
the	O
generalized	O
symbols	O
for	O
effort	O
and	O
flow	O
.	O
</s>
<s>
The	O
bond	O
number	O
will	O
be	O
important	O
later	O
when	O
converting	O
from	O
the	O
bond	B-Application
graph	I-Application
to	O
state-space	O
equations	O
.	O
</s>
<s>
Single-port	O
elements	O
are	O
elements	O
in	O
a	O
bond	B-Application
graph	I-Application
that	O
can	O
have	O
only	O
one	O
port	O
.	O
</s>
<s>
Examples	O
of	O
sources	O
include	O
:	O
motors	O
(	O
source	O
of	O
effort	O
,	O
torque	B-General_Concept
)	O
,	O
voltage	O
sources	O
(	O
source	O
of	O
effort	O
)	O
,	O
and	O
current	O
sources	O
(	O
source	O
of	O
flow	O
)	O
.	O
</s>
<s>
The	O
constant	O
is	O
called	O
a	O
transformer	B-Algorithm
constant	O
or	O
gyrator	O
constant	O
depending	O
on	O
which	O
element	O
is	O
being	O
used	O
.	O
</s>
<s>
A	O
transformer	B-Algorithm
applies	O
a	O
relationship	O
between	O
flow	O
in	O
flow	O
out	O
,	O
and	O
effort	O
in	O
effort	O
out	O
.	O
</s>
<s>
Examples	O
include	O
an	O
ideal	O
electrical	O
transformer	B-Algorithm
or	O
a	O
lever	O
.	O
</s>
<s>
where	O
the	O
r	O
denotes	O
the	O
modulus	O
of	O
the	O
transformer	B-Algorithm
.	O
</s>
<s>
In	O
an	O
electrical	O
circuit	O
,	O
the	O
1	B-Application
junction	I-Application
represents	O
a	O
series	O
connection	O
among	O
components	O
.	O
</s>
<s>
Bond	B-Application
graphs	I-Application
have	O
a	O
notion	O
of	O
causality	O
,	O
indicating	O
which	O
side	O
of	O
a	O
bond	O
determines	O
the	O
instantaneous	O
effort	O
and	O
which	O
determines	O
the	O
instantaneous	O
flow	O
.	O
</s>
<s>
Completing	O
causal	O
assignment	O
in	O
a	O
bond	B-Application
graph	I-Application
model	I-Application
will	O
allow	O
the	O
detection	O
of	O
modeling	O
situation	O
where	O
an	O
algebraic	O
loop	O
exists	O
;	O
that	O
is	O
the	O
situation	O
when	O
a	O
variable	O
is	O
defined	O
recursively	O
as	O
a	O
function	O
of	O
itself	O
.	O
</s>
<s>
In	O
bond	B-Application
graph	I-Application
notation	O
,	O
a	O
causal	O
stroke	O
may	O
be	O
added	O
to	O
one	O
end	O
of	O
the	O
power	O
bond	O
to	O
indicate	O
that	O
this	O
side	O
is	O
defining	O
the	O
flow	O
.	O
</s>
<s>
Consider	O
a	O
constant-torque	O
motor	O
driving	O
a	O
wheel	O
,	O
i.e.	O
</s>
<s>
It	O
is	O
possible	O
for	O
a	O
bond	B-Application
graph	I-Application
to	O
have	O
a	O
causal	O
bar	O
on	O
one	O
of	O
these	O
elements	O
in	O
the	O
non-preferred	O
manner	O
.	O
</s>
<s>
Transformers	B-Algorithm
are	O
passive	O
,	O
neither	O
dissipating	O
nor	O
storing	O
energy	O
,	O
so	O
causality	O
passes	O
through	O
them	O
:	O
</s>
<s>
In	O
order	O
to	O
determine	O
the	O
causality	O
of	O
a	O
bond	B-Application
graph	I-Application
certain	O
steps	O
must	O
be	O
followed	O
.	O
</s>
<s>
Next	O
apply	O
the	O
causality	O
for	O
the	O
0	O
and	O
1	B-Application
junctions	I-Application
,	O
transformers	B-Algorithm
,	O
and	O
gyrators	O
.	O
</s>
<s>
One	O
of	O
the	O
main	O
advantages	O
of	O
using	O
bond	B-Application
graphs	I-Application
is	O
that	O
once	O
you	O
have	O
a	O
bond	B-Application
graph	I-Application
it	O
does	O
n't	O
matter	O
the	O
original	O
energy	O
domain	O
.	O
</s>
<s>
Below	O
are	O
some	O
of	O
the	O
steps	O
to	O
apply	O
when	O
converting	O
from	O
the	O
energy	O
domain	O
to	O
a	O
bond	B-Application
graph	I-Application
.	O
</s>
<s>
The	O
steps	O
for	O
solving	O
an	O
Electromagnetic	O
problem	O
as	O
a	O
bond	B-Application
graph	I-Application
are	O
as	O
follows	O
:	O
</s>
<s>
The	O
steps	O
for	O
solving	O
a	O
Linear	O
Mechanical	O
problem	O
as	O
a	O
bond	B-Application
graph	I-Application
are	O
as	O
follows	O
:	O
</s>
<s>
Parallel	O
power	O
is	O
when	O
power	O
runs	O
in	O
parallel	O
in	O
a	O
bond	B-Application
graph	I-Application
.	O
</s>
<s>
The	O
next	O
step	O
is	O
to	O
draw	O
all	O
of	O
the	O
arrows	O
for	O
the	O
bond	B-Application
graph	I-Application
.	O
</s>
<s>
Now	O
that	O
we	O
have	O
the	O
Bond	B-Application
graph	I-Application
,	O
we	O
can	O
start	O
the	O
process	O
of	O
simplifying	O
it	O
.	O
</s>
<s>
The	O
final	O
step	O
is	O
to	O
apply	O
causality	O
to	O
the	O
bond	B-Application
graph	I-Application
.	O
</s>
<s>
The	O
final	O
bond	B-Application
graph	I-Application
is	O
shown	O
below	O
.	O
</s>
<s>
Following	O
the	O
steps	O
with	O
this	O
circuit	O
will	O
result	O
in	O
the	O
bond	B-Application
graph	I-Application
below	O
,	O
before	O
it	O
is	O
simplified	O
.	O
</s>
<s>
Simplifying	O
the	O
bond	B-Application
graph	I-Application
will	O
result	O
in	O
the	O
image	O
below	O
.	O
</s>
<s>
Lastly	O
,	O
applying	O
causality	O
will	O
result	O
in	O
the	O
bond	B-Application
graph	I-Application
below	O
.	O
</s>
<s>
The	O
resulting	O
bond	B-Application
graph	I-Application
is	O
below	O
.	O
</s>
<s>
Now	O
that	O
the	O
bond	B-Application
graph	I-Application
has	O
been	O
generated	O
,	O
it	O
can	O
be	O
simplified	O
.	O
</s>
<s>
The	O
simplified	O
bond	B-Application
graph	I-Application
can	O
be	O
seen	O
below	O
.	O
</s>
<s>
The	O
last	O
step	O
is	O
to	O
apply	O
causality	O
,	O
the	O
final	O
bond	B-Application
graph	I-Application
can	O
be	O
seen	O
below	O
.	O
</s>
<s>
Next	O
you	O
start	O
the	O
process	O
of	O
simplifying	O
the	O
bond	B-Application
graph	I-Application
,	O
by	O
removing	O
the	O
1-junction	O
of	O
the	O
wall	O
,	O
and	O
removing	O
junctions	O
with	O
less	O
than	O
three	O
bonds	O
.	O
</s>
<s>
The	O
bond	B-Application
graph	I-Application
can	O
be	O
seen	O
below	O
.	O
</s>
<s>
There	O
is	O
parallel	O
power	O
in	O
the	O
bond	B-Application
graph	I-Application
.	O
</s>
<s>
Lastly	O
,	O
apply	O
causality	O
,	O
the	O
final	O
bond	B-Application
graph	I-Application
can	O
be	O
seen	O
below	O
.	O
</s>
<s>
Once	O
a	O
bond	B-Application
graph	I-Application
is	O
complete	O
,	O
it	O
can	O
be	O
utilized	O
to	O
generate	O
the	O
state-space	O
representation	O
equations	O
of	O
the	O
system	O
.	O
</s>
<s>
where	O
is	O
a	O
column	O
matrix	B-Architecture
of	O
the	O
state	O
variables	O
,	O
or	O
the	O
unknowns	O
of	O
the	O
system	O
.	O
</s>
<s>
is	O
a	O
column	O
matrix	B-Architecture
of	O
the	O
inputs	O
of	O
the	O
system	O
.	O
</s>
<s>
The	O
first	O
step	O
to	O
solve	O
the	O
state	O
equations	O
is	O
to	O
list	O
all	O
of	O
the	O
governing	O
equations	O
for	O
the	O
bond	B-Application
graph	I-Application
.	O
</s>
<s>
Both	O
equations	O
can	O
further	O
be	O
rearranged	O
into	O
matrix	B-Architecture
form	O
.	O
</s>
<s>
A	O
bibliography	O
on	O
bond	B-Application
graph	I-Application
modeling	O
may	O
be	O
extracted	O
from	O
the	O
following	O
conferences	O
:	O
</s>
<s>
on	O
Bond	B-Application
Graph	I-Application
Modeling	O
and	O
Simulation	O
(	O
ICBGM’95	O
)	O
,	O
January	O
15	O
–	O
18	O
,	O
1995	O
,	O
Las	O
Vegas	O
,	O
Nevada	O
.	O
</s>
