18 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bond graph | 10/11 | https://en.wikipedia.org/wiki/Bond_graph | reference | science, encyclopedia | 2026-05-05T14:13:40.582713+00:00 | kb-cron |
These equations can be manipulated to yield the state equations. For this example, one is trying to find equations that relate
p
˙
3
(
t
)
{\textstyle {\dot {p}}_{3}(t)}
and
q
˙
6
(
t
)
{\textstyle {\dot {q}}_{6}(t)}
in terms of
p
3
(
t
)
{\textstyle p_{3}(t)}
,
q
6
(
t
)
{\textstyle q_{6}(t)}
, and
e
1
(
t
)
{\textstyle e_{1}(t)}
. To start, one should recall from the tetrahedron of state that
p
˙
3
(
t
)
=
e
3
(
t
)
{\textstyle {\dot {p}}_{3}(t)=e_{3}(t)}
starting with equation 2, one can rearrange it so that
e
3
=
e
1
−
e
2
−
e
4
{\displaystyle e_{3}=e_{1}-e_{2}-e_{4}}
.
e
2
{\displaystyle e_{2}}
can be substituted for equation 4, while in equation 4,
f
2
{\displaystyle f_{2}}
can be replaced by
f
3
{\displaystyle f_{3}}
due to equation 3, which can then be replaced by equation 5.
e
4
{\displaystyle e_{4}}
can likewise be replaced using equation 7, in which
e
5
{\displaystyle e_{5}}
can be replaced with
e
6
{\displaystyle e_{6}}
which can then be replaced with equation 10. Following these substituted yields the first state equation which is shown below.
p
˙
3
(
t
)
=
e
3
(
t
)
=
e
1
(
t
)
−
R
2
I
3
p
3
(
t
)
−
r
C
6
q
6
(
t
)
{\displaystyle {\dot {p}}_{3}(t)=e_{3}(t)=e_{1}(t)-{\frac {R_{2}}{I_{3}}}p_{3}(t)-{\frac {r}{C_{6}}}q_{6}(t)}
The second state equation can likewise be solved, by recalling that
q
˙
6
(
t
)
=
f
6
(
t
)
{\textstyle {\dot {q}}_{6}(t)=f_{6}(t)}
. The second state equation is shown below.
q
˙
6
(
t
)
=
f
6
(
t
)
=
r
I
3
p
3
(
t
)
−
1
R
7
⋅
C
6
q
6
(
t
)
{\displaystyle {\dot {q}}_{6}(t)=f_{6}(t)={\frac {r}{I_{3}}}p_{3}(t)-{\frac {1}{R_{7}\cdot C_{6}}}q_{6}(t)}
Both equations can further be rearranged into matrix form. The result of which is below.
[
p
˙
3
(
t
)
q
˙
6
(
t
)
]
=
[
−
R
2
I
3
−
r
C
6
r
I
3
−
1
R
7
⋅
C
6
]
[
p
3
(
t
)
q
6
(
t
)
]
+
[
1
0
]
[
e
1
(
t
)
]
{\displaystyle {\begin{bmatrix}{\dot {p}}_{3}(t)\\{\dot {q}}_{6}(t)\end{bmatrix}}={\begin{bmatrix}-{\frac {R_{2}}{I_{3}}}&-{\frac {r}{C_{6}}}\\{\frac {r}{I_{3}}}&-{\frac {1}{R_{7}\cdot C_{6}}}\end{bmatrix}}{\begin{bmatrix}p_{3}(t)\\q_{6}(t)\end{bmatrix}}+{\begin{bmatrix}1\\0\end{bmatrix}}{\begin{bmatrix}e_{1}(t)\end{bmatrix}}}
At this point the equations can be treated as any other state-space representation problem.
== International conferences on bond graph modeling (ECMS and ICBGM) == A bibliography on bond graph modeling may be extracted from the following conferences :
ECMS-2013 27th European Conference on Modelling and Simulation, May 27–30, 2013, Ålesund, Norway ECMS-2008 22nd European Conference on Modelling and Simulation, June 3–6, 2008 Nicosia, Cyprus ICBGM-2007: 8th International Conference on Bond Graph Modeling And Simulation, January 15–17, 2007, San Diego, California, U.S.A. ECMS-2006 20TH European Conference on Modelling and Simulation, May 28–31, 2006, Bonn, Germany IMAACA-2005 International Mediterranean Modeling Multiconference ICBGM-2005 International Conference on Bond Graph Modeling and Simulation, January 23–27, 2005, New Orleans, Louisiana, U.S.A. – Papers ICBGM-2003 International Conference on Bond Graph Modeling and Simulation (ICBGM'2003) January 19–23, 2003, Orlando, Florida, USA – Papers 14TH European Simulation symposium October 23–26, 2002 Dresden, Germany ESS'2001 13th European Simulation symposium, Marseilles, France October 18–20, 2001 ICBGM-2001 International Conference on Bond Graph Modeling and Simulation (ICBGM 2001), Phoenix, Arizona U.S.A. European Simulation Multi-conference 23-26 May, 2000, Gent, Belgium 11th European Simulation symposium, October 26–28, 1999 Castle, Friedrich-Alexander University, Erlangen-Nuremberg, Germany ICBGM-1999 International Conference on Bond Graph Modeling and Simulation January 17–20, 1999 San Francisco, California ESS-97 9TH European Simulation Symposium and Exhibition Simulation in Industry, Passau, Germany, October 19–22, 1997 ICBGM-1997 3rd International Conference on Bond Graph Modeling And Simulation, January 12–15, 1997, Sheraton-Crescent Hotel, Phoenix, Arizona 11th European Simulation Multiconference Istanbul, Turkey, June 1–4, 1997 ESM-1996 10th annual European Simulation Multiconference Budapest, Hungary, June 2–6, 1996 ICBGM-1995 Int. Conf. on Bond Graph Modeling and Simulation (ICBGM'95), January 15–18, 1995, Las Vegas, Nevada.
== See also == 20-sim simulation software based on the bond graph theory AMESim simulation software based on the bond graph theory Hybrid bond graph Coenergy
== Systems for bond graph == Many systems can be expressed in terms used in bond graph. These terms are expressed in the table below. Conventions for the table below:
P
{\displaystyle P}
is the active power;
X
^
{\displaystyle {\hat {X}}}
is a matrix object;
x
→
{\displaystyle {\vec {x}}}
is a vector object;
x
†
{\displaystyle x^{\dagger }}
is the Hermitian conjugate of x; it is the complex conjugate of the transpose of x. If x is a scalar, then the Hermitian conjugate is the same as the complex conjugate;
D
t
n
{\displaystyle D_{t}^{n}}
is the Euler notation for differentiation, where:
D
t
n
f
(
t
)
=
{
∫
−
∞
t
f
(
s
)
d
s
,
n
=
−
1
f
(
t
)
,
n
=
0
∂
n
f
(
t
)
∂
t
n
,
n
>
0
{\displaystyle D_{t}^{n}f(t)={\begin{cases}\displaystyle \int _{-\infty }^{t}f(s)\,ds,&n=-1\\[2pt]f(t),&n=0\\[2pt]{\dfrac {\partial ^{n}f(t)}{\partial t^{n}}},&n>0\end{cases}}}
{
⟨
x
⟩
α
:=
|
x
|
α
sgn
(
x
)
⟨
a
⟩
=
k
⟨
b
⟩
β
⟹
⟨
b
⟩
=
(
1
k
⟨
a
⟩
)
1
/
β
{\displaystyle {\begin{cases}\langle x\rangle ^{\alpha }:=|x|^{\alpha }\operatorname {sgn}(x)\\\langle {a}\rangle =k\langle b\rangle ^{\beta }\implies \langle b\rangle =\left({\frac {1}{k}}\langle a\rangle \right)^{1/\beta }\end{cases}}}