A bond graph is a graphical representation of a physical dynamic system. It allows the conversion of the system into a state-space representation. It is similar to a block diagram or signal-flow graph, with the major difference that the arcs in bond graphs represent bi-directional exchange of physical energy, while those in block diagrams and signal-flow graphs represent uni-directional flow of information. Bond graphs are multi-energy domain (e.g. mechanical, electrical, hydraulic, etc.) and domain neutral. This means a bond graph can incorporate multiple domains seamlessly.
The bond graph is composed of the "bonds" which link together "single-port", "double-port" and "multi-port" elements (see below for details). Each bond represents the instantaneous flow of energy () or power. The flow in each bond is denoted by a pair of variables called power variables, akin to conjugate variables, whose product is the instantaneous power of the bond. The power variables are broken into two parts: flow and effort. For example, for the bond of an electrical system, the flow is the current, while the effort is the voltage. By multiplying current and voltage in this example you can get the instantaneous power of the bond.
A bond has two other features described briefly here, and discussed in more detail below. One is the "half-arrow" sign convention. This defines the assumed direction of positive energy flow. As with electrical circuit diagrams and free-body diagrams, the choice of positive direction is arbitrary, with the caveat that the analyst must be consistent throughout with the chosen definition. The other feature is the "causality". This is a vertical bar placed on only one end of the bond. It is not arbitrary. As described below, there are rules for assigning the proper causality to a given port, and rules for the precedence among ports. Causality explains the mathematical relationship between effort and flow. The positions of the causalities show which of the power variables are dependent and which are independent.
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. Bond graphs were invented by Henry Paynter.
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: is the active power; is a matrix object; is a vector object; is the Hermitian of ; it is the complex conjugate of the transpose of . If is a scalar, then the Hermitian is the same as the complex conjugate; is the Euler notation for differentiation, where:
Vergent-factor:
Other systems: Thermodynamic power system (flow is entropy-rate and effort is temperature) Electrochemical power system (flow is chemical activity and effort is chemical potential) Thermochemical power system (flow is mass-rate and effort is mass specific enthalpy) Macroeconomics currency-rate system (displacement is commodity and effort is price per commodity) Microeconomics currency-rate system (displacement is population and effort is GDP per capita)
Tetrahedron of state
The tetrahedron of state is a tetrahedron that graphically shows the conversion between effort and flow. The adjacent image shows the tetrahedron in its generalized form. The tetrahedron can be modified depending on the energy domain.
Using the tetrahedron of state, one can find a mathematical relationship between any variables on the tetrahedron. This is done by following the arrows around the diagram and multiplying any constants along the way. For example, if you wanted to find the relationship between generalized flow and generalized displacement, you would start at the and then integrate it to get . More examples of equations can be seen below.
Relationship between generalized displacement and generalized flow.
Relationship between generalized flow and generalized effort.
Relationship between generalized flow and generalized momentum.
Relationship between generalized momentum and generalized effort.
Relationship between generalized flow and generalized effort, involving the constant C.
All of the mathematical relationships remain the same when switching energy domains, only the symbols change. This can be seen with the following examples.
Relationship between displacement and velocity.
Relationship between current and voltage, this is also known as Ohm's law.