BOND GRAPH

A 'bond graph' is a graphical description of a physical dynamic system. It is an energy-based technique. A bond graph shows the energy flows around a system. Bond graphs have a number of advantages over conventional block diagram and computer simulation techniques (Gawthrop and Ballance, 1999):

★ Since they work on the principle of conservation of energy, it is difficult to accidentally introduce extra energy into a system.

★ Since each bond represents a bi-directional flow, systems which produce a "back force" (e.g., back emf in a motor) on the input are easily modeled without introducing extra feedback loops.
Bond graphs are based on the principle of continuity of power. If the dynamics of the physical system to be modeled operate on widely varying time scales, fast continuous-time behaviors can be modeled as instantaneous phenomena by using a hybrid bond graph.

Contents

Basics

Junctions

Causality

Example

Other components

Transformer

Gyrator

See also

External links

References

Further reading

Basics

The fundamental idea of a bond graph is that power is transmitted between connected components by a combination of "effort" and "flow". In the case of mechanical systems, effort is force and flow is rate; in the case of torsional systems, effort is torque and flow is angular velocity. So if an engine is connected to a wheel through a shaft, the power being transmitted is the product of the effort and the flow so power=ωτ. A 'word bond graph' is a first step towards a bond graph, in which words define the components. As a word bond graph, this system would look like
τ
engine ---------- wheel
ω
A half-arrow is used to provide a sign convention, so if the engine is doing work when τ and ω are positive, then the diagram would be drawn
τ
engine ---------- wheel
ω /

Junctions

Power bonds may join at one of two kinds of junctions: a '0 junction' and a '1 junction'.

★ In a 0 junction, the flow sums to zero and the efforts are equal. This corresponds to a node in an electrical circuit (where Kirchhoff's current law applies).

★ In a 1 junction, the efforts sum to zero. This corresponds to force balance at a mass in a mechanical system.
For an example of a 1 junction, consider a resistor in series:
v1 ---///---v2
i1 R i2=i1
As a bond graph, this becomes
R
|
v1 | v2
------ 1 ------
i1 / i2 /
From an electrical point of view, this diagram may seem counterintuitive in that flow is not preserved in the same way across the diagram. It may be helpful to consider the 1 junction as daisy chaining the bonds it connects to and power bond up to the ''R'' as a resistor with a lead returning back down, so the above is equivalent to
R
+//+
/
| |
v1 | | v2
------+ +------
i1 i2

Causality

Bond graphs have a notion of causality, indicating which side of a bond determines the instantaneous effort and which the instantaneous flow. By noting the causation within a bond graph, analysis becomes easier.
As an example of causality, consider a capacitor in series with a battery. It is not physically possible to charge a capacitor instantly, so anything connected in parallel with a capacitor will necessarily have the same voltage (effort variable) as the capacitor. Similarly, an inductor cannot change flux instantly and so any component in series with an inductor will necessarily have the same flow as the inductor. Because capacitors and inductors are passive devices, they cannot maintain their respective voltage and flow indefinitely—the components to which they are attached will affect their respective voltage and flow, but only indirectly by affecting their current and voltage respectively.
Note: Causality is a symmetric relationship. When one side "causes" effort, the other side "causes" flow.
Active components such as an ideal voltage or current source also are causal.
In bond graph notation, a 'causal stroke' may be added to one end of the power bond to indicate that the opposite end is defining the effort. Consider a constant-torque motor driving a wheel. That would be drawn as follows:
motor τ |
SE ----------| wheel
ω |
Symmetrically, the side with the causal stroke (in this case the wheel) defines the flow for the bond.
Causality results in compatibility constraints. Clearly only one end of a power bond can define the effort and so only one end of a bond can have a causal stroke. In addition, the two passive components with time-dependent behavior, I and C, can only have one sort of causation: an I component determines flow; a C component defines effort. So from a junction, J, the only legal configurations for I and C are
|
J -------| I
|
and
|
J |------- C
|
A resistor has no time-dependant behavior: you can apply a voltage and get a flow instantly, or apply a flow and get a voltage instantly, thus a resistor can be at either end of a causal bond:
| |
J -------| R J |-------- R
| |
Sources of flow define flow, sources of effort define effort. Transformers are passive, neither dissipating nor storing energy, so causality passes through them:
| .. | | .. |
------| TF -----| or |------ TF |------
| | | |
A gyrator transforms flow to effort and effort to flow, so if flow is caused on one side, effort is caused on the other side and vica versa:
| .. | | .. |
|------ GY -----| or ------| GY |------
| | | |
; Junctions
In a 0-junction, efforts are equal; in a 1-junction, flows are equal. Thus, with causal bonds, only one bond can cause the effort in a 0-junction and only one can cause the flow in a 1-junction. Thus, if the causality of one bond of a junction is known, the causality of the others is also known. That one bond is called the 'strong bond'
___
|
|
|
------| 0 ------|
strong bond ^
|
|
|
---
|
|
|
---
|------ 1 |------
strong bond ^ ___
|
|
|
One can continue, assigning causality using the above rules. Any model which results in inconsistent causality is not physically valid. For example, consider an inductor in series with an ideal current source—a physically impossible configuration. The bond graph would look like
SF ------ 1 ------ I
Assigning causality to the source bond we get:
SF |----- 1 ------ I
Propagating the causality through the junction gives
SF |----- 1 |----- I
But assigning causality to the inductor gives
SF |----- 1 |----| I
which is invalid, because the causality on the right bond is redundant. This ability to automatically identify impossible configurations is a major advantage of bond graphs.
In contrast, one can inadvertently draw an electrical diagram or mechanical schematic that, while possible to construct, would not behave as modeled. For example, one can connect a capacitor directly to a battery, but the assumption that the battery is an ideal voltage source would be violated corresponding with the fact that the theoretical flux would be infinite. The bond graph would tell you that a resistor needs to be put in series with the capacitor to keep the model realistic.

Example

Consider a simple RC circuit^[1]:
R
i1 --//-----+------ i2 →
v1 | v2
C = ↓ic
|
ground ----+------
As a bond graph, the same system looks like
R C
| |
v1 | | v2
in ------ 1 ----- 0 ------- out
i1 / / i2 /
If we assume $i\_2=0,mathrm\{A\}$, then we can find a transfer function relating $v\_1$ and $v\_2$,
:$\{\; v\_2(s)\; over\; v\_1(s)\; \}\; =\; \{\; 1\; over\; 1\; +\; RCs\; \}$
But because the state of a bond has an effort and a flow, the transfer function for the bond graph looks like
:.
Suppose we put two of these in series:
R → i2 R
i1 --//-----+------------//-----+------ i3 →
v1 | v2 | v3
C = ↓ic C = ↓ic
| |
ground ----+----------------------+---------
The corrsponding bond graph looks like
R C R C
| | | |
v1 | | v2 | | v3
in ------ 1 ----- 0 ------- 1 ----- 0 ------- out
i1 / / i2 / / i3 /
Now the transfer function for $v\_3(s)/v\_1(s)$ is not simply $1/(1+RCs)^2)$ because now $i\_2$
eq 0,mathrm{A}.
But the transfer function for the bond graph is simply
:.

Other components

Transformer

A transformer adds no power but transforms it, much as an electrical transformer or a lever.
Denoted
r
e1 .. e2
------- TF ------
f1 f2
where the ''r'' denotes the modulus of the transformer. This means
:$f\_2=rf\_1$
and
:.

Gyrator

A 'gyrator' relates flow to effort. It also adds no power and is written
e1 μ e2
------- GY ------
f1 f2
meaning that
:$e\_2=mu\; f\_1$
and
:$e\_1\; =\; mu\; f\_2$.

See also

★ Hybrid bond graph

★ 20-sim modeling and simulation software for bond graphs [1]

★ AMESim simulation software based on the bond graph theory

★ SYMBOLS Shakti SYstem Modeling by BOnd graph Language and Simulation [2]

External links

★ http://groups.csail.mit.edu/drl/journal_club/papers/Samantaray__2001__www.bondgraphs.com_about.pdf

References

★ Amalendu Mukherjee, Ranjit Karmakar (1999): "Modeling and Simulation of Engineering Systems Through Bondgraphs" CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. ISBN-13: 978-0849309823

★ Gawthrop, P. J. and Ballance, D. J., 1999: "Symbolic computation for manipulation of hierarchical bond graphs" in ''Symbolic Methods in Control System Analysis and Design'', N. Munro (ed), IEE, London, ISBN 0-85296-943-0.

Further reading

★ Karnopp, D. C., Rosenberg, R. C. and Margolis, D. L., 1990: ''System dynamics: a unified approach'', Wiley, ISBN 0-471-62171-4.

★ Thoma, J., 1975: '' Bond graphs: introduction and applications'', Elsevier Science, ISBN 0-08-018882-6.

★ Gawthrop, P. J. and Smith, L. P. S., 1996: ''Metamodelling: bond graphs and dynamic systems'', Prentice Hall, ISBN 0-13-489824-9.