Bond graph
This means a bond graph can incorporate multiple domains seamlessly.
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.
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.
All of the mathematical relationships remain the same when switching energy domains, only the symbols change.
For power density systems, the formula for the velocity of the resonance wave is as follows:
If an engine is connected to a wheel through a shaft, the power is being transmitted in the rotational mechanical domain, meaning the effort and the flow are torque (τ) and angular velocity (ω) respectively.
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:
As the engine is applying a torque to the wheel, it will be represented as a source of effort for the system.
They are denoted by a capital "S" with either a lower case "e" or "f" for effort or flow respectively.
In an electrical circuit, the 1 junction represents a series connection among components.
In a mechanical circuit, the 1-junction represents a velocity shared by all components connected to it.
In formulating the dynamic equations that describe the system, causality defines, for each modeling element, which variable is dependent and which is independent.
Completing causal assignment in a bond graph model will allow the detection of modeling situation where an algebraic loop exists; that is the situation when a variable is defined recursively as a function of itself.
Symmetrically, the side with the causal stroke (in this case the wheel) defines the flow for the bond.
The reason that this is the preferred method for these elements can be further analyzed if you consider the equations they would give shown by the tetrahedron of state.
The resulting equations involve the integral of the independent power variable.
The results of a causal conflict are only seen when writing the state-space equations for the graph.
Transformers are passive, neither dissipating nor storing energy, so causality passes through them:
To solve parallel power you will first want to write down all of the equations for the junctions.
By manipulating these equations you can arrange them such that you can find an equivalent set of 0 and 1-junctions to describe the parallel power.
Once you have determined the relationships that you need you can redraw the parallel power section with the new junctions.
A simple electrical circuit consisting of a voltage source, resistor, and capacitor in series.
The arrows on junctions should point towards ground (following a similar path to current).
This is because flow and effort pass through these junctions without being modified, so they can be removed to allow us to draw less.
A more advanced electrical system with a current source, resistors, capacitors, and a transformer
A simple linear mechanical system, consisting of a mass on a spring that is attached to a wall.
Like the electrical examples, power should flow towards ground, in this case the 1-junction of the wall.
Once a bond graph is complete, it can be utilized to generate the state-space representation equations of the system.
