11 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bond graph | 2/11 | https://en.wikipedia.org/wiki/Bond_graph | reference | science, encyclopedia | 2026-05-05T14:13:40.582713+00:00 | kb-cron |
The definitions relating p to e and q to f are indicated diagrammatically. Physical properties are encapsulated in constitutive equations relating the energy and power variables. In the diagram, C represents the constitutive equation relating q and e, I represents the constitutive equation relating p and f and R represents the constitutive equation relating e and f. Instead of placing e, f, p and q on the four corners of a square as in the diagram, they can alternatively be placed at the four vertices of a tetrahedron; such a diagram is called the tetrahedron of state. These three constitutive equations
Φ
C
,
Φ
I
and
Φ
R
{\displaystyle \Phi _{C},\Phi _{I}~{\text{and}}~\Phi _{R}}
correspond to three components each relating e and f: two dynamic components, C and I, incorporating an integrator and the R-component. The C- and I- components store, but do not dissipate energy, R dissipates, but does not store energy. It is also possible to define a memristor component linking p and q, but this component is not commonly used.
Φ
C
,
Φ
I
and
Φ
R
{\displaystyle \Phi _{C},\Phi _{I}~{\text{and}}~\Phi _{R}}
may be linear or nonlinear functions relating the variables as:
q
=
Φ
C
(
e
)
,
p
=
Φ
I
(
f
)
and
e
=
Φ
R
(
f
)
{\displaystyle q=\Phi _{C}(e),~~p=\Phi _{I}(f)~~{\text{and}}~~e=\Phi _{R}(f)}
in the linear case:
q
=
C
e
,
p
=
I
f
,
e
=
R
f
{\displaystyle q=Ce,~~p=If,~~e=Rf}
where
C
,
I
,
and
R
{\displaystyle C,~I,~{\text{and}}~R}
are scalar constants representing generalised capacitance, inertia and resistance respectively. Because e and f are conjugate variables, they are carried on a single bond and therefore the three components C, I and R are connected to a single bond and thus have a single energy port though which energy flows. The components, and impinging bond are shown in the figure. By convention, bonds point into these three components.
==== Notation ==== The colon (:) notation is sometimes used to refer to the components; thus, for example I:M, C:K and R:D could be used in the bond graph of the mass-spring-damper system to emphasise the link between the bond graph components and their physical analogues.
=== Analogies between connections === Electrical circuit diagrams have two sorts of connection: parallel and series or common voltage and common current; they distribute, but do not store or dissipate energy. The bond graph analogy of the common voltage and common current connections are the 0-junction (common effort) and 1-junction (common flow) respectively; they both distribute, but do not store or dissipate energy. All bonds impinging on a 0-junctions have the same effort. As energy is distributed, not dissipated, it follows that the sum of energy inflows (indicated by bonds pointing in) must equal the sum of energy outflows (indicated by bonds pointing out). Hence, if the common effort is e, the power flow constraint implies that the sum of the
m
{\displaystyle m}
inflows
f
i
i
n
{\displaystyle f_{i}^{in}}
must equal the sum of the
n
{\displaystyle n}
outflows
f
j
o
u
t
{\displaystyle f_{j}^{out}}
:
∑
i
=
1
m
f
i
i
n
=
∑
j
=
1
n
f
j
o
u
t
{\displaystyle \sum _{i=1}^{m}f_{i}^{in}=\sum _{j=1}^{n}f_{j}^{out}}
Using the same argument, the efforts impinging on a 1-junction are constrained by:
∑
i
=
1
m
e
i
i
n
=
∑
j
=
1
n
e
j
o
u
t
{\displaystyle \sum _{i=1}^{m}e_{i}^{in}=\sum _{j=1}^{n}e_{j}^{out}}
=== Analogies between external connections ===
Systems such as that the simple electrical and mechanical systems in the figure have no connection to the environment. Such connections may include external voltages and external forces (that is efforts) and external currents and external velocities (that is flows). Similarly, external measurements of efforts and flows are also important. For the purposes of building hierarchical system, it is convenient to define energy ports though which energy can flow between systems. These five possibilities correspond to five bond graph components:
S
e
{\displaystyle {\text{S}}_{e}}
, an effort source analogous to applying external voltages and external forces
S
f
{\displaystyle {\text{S}}_{f}}
, a flow source analogous to applying external currents and external velocities
D
e
{\displaystyle {\text{D}}_{e}}
, an effort sensor (detector) analogous to measuring voltage and forces
D
f
{\displaystyle {\text{D}}_{f}}
, a flow sensor (detector) analogous to measuring currents and velocities
SS
{\displaystyle {\text{SS}}}
, a source/sensor component which acts both as
S
e
{\displaystyle {\text{S}}_{e}}
D
f
{\displaystyle {\text{D}}_{f}}
and as
S
f
{\displaystyle {\text{S}}_{f}}
D
e
{\displaystyle {\text{D}}_{e}}
pairs as well as an energy port for external connections. For example, the mass-spring-damper system can be augmented with an force
F
{\displaystyle F}
applied to the mass and a measurement of the mass velocity
v
{\displaystyle v}
using a single SS components. The colon notation has been included; the SS:io refers to the fact that the force and velocity pair could be considered as system input and output.
=== Analogies between energy transducers ===
The conjugate effort and flow variables have different units in each energy domain thus models in one domain cannot be directly be connected by bonds to a different energy domain. However, because power has the same units (J/s or W) in each domain, the two-port power transducing components TF and GY can be used to provide such connections. As the two components transmit, but do not store or dissipate power, it follows that the power associated with the conjugate variables of the left-hand and right-hand bonds must be the same:
e
2
f
2
=
e
1
f
1
{\displaystyle e_{2}f_{2}=e_{1}f_{1}}
Each component has a modulus
m
{\displaystyle m}
associated with it. In the case of the TF component:
e
2
=
m
e
1
and
f
1
=
m
f
2
{\displaystyle e_{2}=me_{1}~~{\text{and}}~~f_{1}=mf_{2}}
In the case of the GY component:
e
2
=
m
f
1
and
e
1
=
m
f
2
{\displaystyle e_{2}=mf_{1}~~{\text{and}}~~e_{1}=mf_{2}}