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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bond graph | 1/11 | https://en.wikipedia.org/wiki/Bond_graph | reference | science, encyclopedia | 2026-05-05T14:13:40.582713+00:00 | kb-cron |
A bond graph is a graphical representation of the energy flows though and between physical dynamical systems including those in the electrical, mechanical, hydraulic, thermal and chemical domains. It is used to model and analyse systems relevant to engineering and to systems biology. Because the concept of energy is common to all physical domains, the bond graph provides a unified description of all of these energy domains and can be thought of as a systematic use of the physical analogies introduced by the 19th century scientists James Clerk-Maxwell and Lord Kelvin. The mechanical–electrical analogy is one example of a physical analogy. Bond graphs use the concept of analogous power conjugate variables whose product is energy flow, or power; these variable pairs are called effort and flow and, for example, correspond to voltage and current in the electrical domain and force and velocity in the mechanical domain. These power conjugate variables are transmitted by bonds which connect bond graph components. Bond graph components are also based on analogies and, using the electrical and mechanical domains as examples, include the C-component to represent both mechanical spring and electrical capacitor, the I-component to represent both a mechanical inertia and an electrical inductor and the R-component to represent both mechanical damper and electrical resistor. The electrical circuit notions of parallel and series connections are abstracted as 0-junctions and 1-junctions in bond graph terminology and again used as connection analogues for each physical domain. The bond graph transformer (TF) and gyrator (GY) components represent energy transformation within and between domains; thus an ideal gearbox in the rotational mechanical domain is represented by the TF component and an ideal DC motor transforming electrical into mechanical energy is represented by a GY component. Non-ideal transducers with flexibility, inertia and friction are modelled by including C, I and R components. The concept of causality in the context of bond graphs is used not only to generate system equations in a number of forms including ordinary differential equation (ode), state-space and differential-algebraic equations (dae) form suitable for simulation purposes but also to investigate dynamical system properties such as invertibility and zero dynamics. Causality can also be used to guide and correct modelling choices. The bond graph use of energy flows leads to the systematic construction of hierarchical models of large multi-domain systems; thus the bond graph method provides a basis for constructing large computer models, or digital twins, of multi domain physical systems including systems relevant not only to engineering but also to systems biology and the life sciences. The bond graph approach is related to the behavioral modelling approach of Jan C Willems, and the port-Hamiltonian approach of Arjan van der Schaft and B. M. Maschke. The bond graph method was originally proposed by Henry Paynter who applied the approach to engineering systems; the use of bond graphs to model biophysical systems was introduced by Aharon Katchalsky, George Oster, and Alan Perelson in the early 1970s.
== Analogies == The importance of analogies between physical domains was noted by Lord Kelvin and James Clerk-Maxwell. The bond graph can be thought of as a systematic approach to analogies. This section emphasises key features of bond graph analogies; more detail appears in a number of textbooks and tutorial papers. As an introduction to the key features of bond graphs, the figure shows the bond graph of two analogous systems: one electrical and one mechanical. A brief description is given here and expanded in the following sections.
The bond graph C-component represents either the electrical capacitor C or the mechanical spring K. The bond graph I-component represents either the electrical inductor L or the mechanical mass M. The bond graph R-component represents either the electrical resistor R or the mechanical damper D. The harpoon symbol is a bond transferring energy; the harpoon direction corresponds to positive energy flow and is a sign convention. The conjugate variables are displayed on each bond for the purposes of illustration. The 1-junction represents the series connection of the electrical circuit where the same current i flows in each component and the connection of the three components in the mechanical system which share a common velocity. Thus the bond graph components share the same flow f, but the efforts are different, corresponding to the voltages and forces of the electrical and mechanical components respectively. Energy is conserved at the junction by requiring that the three efforts add to zero. The bond graph uses three classes of analogy: analogies between variables, analogies between components and analogies between component connections; these are discussed in the following sections.
=== Analogies between variables ===
The bond graph use energy flow, or power, as the basis for abstracting analogies between different physical domains. Power conjugate variables are a pair of variables whose product is power and a list of these appears in the table for various physical domains. As indicated in the table, the bond graph uses the effort/flow analogy to categorise each of the two conjugate variables; the across/through analogy is also possible but not commonly used. The conventional bond graph symbol for effort is e, and that for flow is f. As well as the two conjugate power variables e and f, the bond graph uses two integrated variables p and q where:
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{\displaystyle p=\int ^{t}e(\tau )d\tau ~~~{\text{and}}~~~q=\int ^{t}f(\tau )d\tau }
equivalently:
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{\displaystyle {\dot {p}}={\frac {dp}{dt}}=e~~~{\text{and}}~~~{\dot {q}}={\frac {dq}{dt}}=f}
The harpoon symbol shown in the figure represents a power bond, or bond, transferring energy; the harpoon direction corresponds to positive energy flow and corresponds to a sign convention. The conjugate variables are displayed on each bond for the purposes of illustration.
=== Analogies between components ===