Bond graph
![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Mass-spring-damper-bond-graph.png/164px-Mass-spring-damper-bond-graph.png)
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 (dE/dt) 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.[1]
Systems for bond graph
[edit]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 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;
- is the Euler notation for differentiation, where:
- Vergent-factor:
Generalized flow | Generalized displacement | Generalized effort | Generalized momentum | Generalized power (in watts for power systems) | Generalized energy (in joules for power systems) | |
---|---|---|---|---|---|---|
Name | ||||||
Description | Time derivative of displacement | A quality related to static behaviour. | The energy per unit of displacement | Time integral of effort | Transformation of energy from one to another form | Conserved quantity in closed systems |
Elements | ||||||
Name | Hyperance , hyperrigitance | Compliance , rigitance | Resistance | Inertance (or ) | Abrahance | Magnance |
Properties | Power dissipative element | Charge storage element (State variable: displacement) (Costate variable: effort) | Power dissipative element | Momentum storage element (State variable: momentum) (Costate variable: flow) | Power dissipative element | Power dissipative element |
Quantitative behaviour | For 1-dimension systems (linear): For 1-dimension systems: Impedance: | Potential energy for N-dimension systems: Potential energy: Potential coenergy: For 1-dimension systems: Impedance: | For 1-dimension systems (linear): Power for 1-dimension non-linear resistances ( is the effort developed by the element): Rayleigh power: Rayleigh oower for non-linear resistances: Rayleigh effort: For N-dimension systems: For 1-dimension systems: Impedance: | Kinetic energy for N-dimension systems: Kinetic energy: Kinetic coenergy: For 1-dimension systems: Impedance: | For 1-dimension systems (linear): For 1-dimension systems: Impedance | For 1-dimension systems (linear) For 1-dimension systems Impedance |
Generalized behaviour | Energy from active effort sources: Hamiltonian effort: Lagrangian effort: Passive effort: Power equation: Effort equation: Lagrangian equation: Hamiltonian equation: If is the coenergy, is the energy, is the state variable and is the costate variable,
For linear elements: |
Flow-related variables | : pounce / pop | : flounce / crackle | : jounce / snap | : jerk |
---|---|---|---|---|
: acceleration | : velocity (flow) | : displacement (displacement) | : absement | |
: absity | : abseleration | : abserk | ||
Effort-related variables | : yank | : force (effort) | : linear momentum (momentum) | |
Passive elements | ||||
Compliance (C) | Resistance (R) | Inertance (I) | Abrahance (A) | Magnance (M) |
Spring where is the spring stiffness | Damper where is the damper parameter | Mass where is the mass | Abraham–Lorentz force where
| Magnetic radiation reaction force where
|
Cantilever
| Cyclotron radiation resistance where
| |||
Prismatic floater in a wide waterbody where
| Viscous friction where is the viscous friction parameter | |||
Elastic rod where
| Inverse of kinetic mobility where is the kinetic mobility | |||
Newton's law of gravitation where
| Geometry interacting with air (e.g. drag) where
| |||
Coulomb's law where
| Absquare[clarification needed] damper where is the Absquare damper parameter | |||
Casimir force where
| Dry friction where
| |||
Biot–Savart law where
| ||||
Piston pressing fluid inside an adiabatic chamber where
|
Flow-related variables | : angular jerk | : angular acceleration |
---|---|---|
: angular velocity (flow) | : angular displacement (displacement) | |
Effort-related variables | : rotatum | : rorque (effort) |
: angular momentum (momentum) | ||
Passive elements | ||
Compliance (C) | Resistance (R) | Inertance (I) |
Inverse of the angular spring constant where is the angular spring constant | Angular damping where is the damping constant | Mass moment of inertia Type where is the mass moment of inertia |
Rod torsion where
| Governor (e.g. used in music boxes) where is the governor constant | |
Bending moment (cantilever) where
| ||
Parallel force field where
|
Flow-related variables | : electric inertia | : electric current (flow) | : electric charge (displacement) | |
---|---|---|---|---|
Effort-related variables | : distension | : voltage (effort) | : flux linkage (momentum) | |
Elements | ||||
Hyperance (H) | Compliance (C) | Resistance (R) | Inertance (I) | Abrahance (A) |
Frequency-dependent negative resistor (FDNR)
| Linear capacitor where
| Linear resistor where
| Linear inductor (solenoid) where
| Frequency-dependent negative conductance (FDNC) Type |
Diode where
| Toroid where
| |||
Intra-gyrator | Inter-gyrator | |||
Compliant gyrator | Resistive gyrator | Inertant gyrator | ||
Hall effect device where
| Induction motor where
| DC motor where
| Faraday gyrator where
| |
Faraday disk where
| ||||
Intra-transformer | Inter-transformer | |||
Electrical transformer (only for AC signals)
|
Flow-related variables | : volumetric flow rate (flow) | : volume (displacement) |
---|---|---|
Effort-related variables | : pressure (effort) | : fluid momentum (momentum) |
Elements | ||
Compliance (C) | Resistance (R) | Inertance (I) |
Pipe elasticity where
| Darcy sponge where
| Fluid inertia in pipes where
|
Compressible fluid (approximation) where
| Valve
| |
Tank with area : where
| Poiseuille resistance for cylinders where
| |
Isothermal chamber where is chamber's constant | Turbulence resistance where is an empirical parameter | |
Compressible fluid where is the bulk modulus | Nozzle where
| |
Adiabatic bladder where
| Check valve where
|
Flow-related variables | : magnetic flux (displacement) | |
---|---|---|
Effort-related variables | : magnetomotive force (momentum) | |
Elements | ||
Compliance (C) | Resistance (R) | Inertance (I) |
Permeance () where
| Magnetic complex impedance ()
| Magnetic complex inductance ()
|
Flow-related variables | : gravitational current Loop (flow) | : gravitational charge (displacement) |
---|---|---|
Effort-related variables | : gravitational voltage (effort) | : gravitational momentum (momentum) |
Elements | ||
Compliance (C) | Resistance (R) | Inertance (I) |
Gravitational capacitance Type
| Gravitational orbital resistance Type
| Gravitational inductance Type
|
Flow-related variables | : current density (flow) | : electric displacement field (displacement) |
---|---|---|
Effort-related variables | : electric field (effort) | : magnetic potential vector (momentum) |
Elements | ||
Compliance (C) | Resistance (R) | Volumetric power density inertance () |
Electrical permittivity Type | Electrical resistivity Type | Magnetic permeability |
Flow-related variables | : magnetic flux density (displacement) | |
---|---|---|
Effort-related variables | : magnetic field strength (effort) | |
Elements | ||
Compliance (C) | Resistance (R) | Volumetric power density inertance () |
Magnetic permeability for magnetic circuits Type |
Flow-related variables | : flux of mass (flow) | : accumulated flux of mass (displacement) |
---|---|---|
Effort-related variables | : acceleration of gravity (effort) | |
Elements | ||
Compliance (C) | Resistance (R) | Volumetric power density inertance () |
Gravitational permittivity Type
| Gravitational permeability |
Flow-related variables | : gravitomagnetic field (displacement) | |
---|---|---|
Effort-related variables | : gravitomagnetic field strength (effort) | |
Elements | ||
Compliance (C) | Resistance (R) | Volumetric power density inertance () |
Gravitational permeability
|
Flow-related variables | : heat rate (flow) | : total heat (displacement) |
---|---|---|
Effort-related variables | : temperature (Effort) | |
Elements | ||
Compliance (C) | Resistance (R) | Inertance (I) |
Isobaric heat where
| Conduction resistance where
| |
Isocoric heat where is the constant-volume heat capacitance | Convection resistance where
| |
Isothermal heat where
| Stefan–Boltzmann law where
|
Flow-related variables | : strain rate |
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