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domains that has been expounded in a number of successful textbooks (e.g., Karnopp et al., 1975,
1999), there are systematic methods for selecting sets of independent variables to describe a system,
ways to take advantage of the ease of identifying velocities and voltages, and matrix methods to facilitate
computer analysis. In fact, several computer-aided modeling support packages using the bond-graph
language are now available. Furthermore, bond graphs have been applied successfully to describe the
dynamics of spatial mechanisms (including gyroscopic effects) while, to the authors’ knowledge, linear
graphs have not.
Although the force-voltage analogy is most commonly used with bond graphs, the force-current
analogy can be used just as readily; the underlying mathematical formalism is indifferent to the choice
of which variables are chosen as analogous. In fact, pursuing this line of thought, the choice is unnecessary
and may be avoided; doing so affords a way to clarify the potential confusion over the role of intensive
variables and the dual types of connection available for some elements in some domains.
In the Generalized Bond Graph (GBG) approach (Breedveld, 1984) all energy storage becomes anal-
ogous and only one type of storage element, a (generalized) capacitor, is identified. Its displacement is
an extensive variable; the gradient of its energy storage function with respect to that displacement is an
intensive variable. In some (but not all) domains a particular kind of coupling known as a gyrator is
found that gives rise to the appearance of a dual type of energy storage, a (generalized) inertia as well as
the possibility of dual ways to connect elements. The GBG representation emphasizes the point that the
presence of dual types of energy storage and dual types of connection is a special property (albeit an
important one) of a limited number of domains. In principle, either a “mass-capacitor” analogy or a
“mass-inductor” analogy can be derived from a GBG representation by choosing to associate the gyrating
coupling with either the “equilibrium” or “steady-state” energy storage elements.
The important point to be taken here is that the basis of analogies between domains does not depend
on the use of a particular abstract graphical representation. The practical value of establishing analogies
between domains and the merits of a domain-independent approach based on intensive vs. extensive
variables remains regardless of which graph-theoretic tools (if any) are used for analysis.

15.7 Concluding Remarks

In the foregoing we articulated some important considerations in the choice of analogies between
variables in different physical domains. From a strictly mathematical viewpoint there is little to choose;
both analogies may be used as a basis for rigorous, self-consistent descriptions of physical systems. The
substantive and important factors emerge from a physical viewpoint—considering the structured way
physical behavior is described in the different domains. Summarizing:
• The “system-of-particles” model that is widely assumed in basic science and engineering naturally
leads to the intuitive analogy between force and voltage, velocity and current, a mass and an
inductor, and so on.
• The measurement procedures used to motivate the distinction between across and through vari-
ables at best yield an ambiguous classification.
• Nodicity (the property of “arbitrary connectability”) is not a general property of lumped-
parameter physical system models. Thus, electrical networks, which are nodic, can be quite
misleading when used as a basis for a general representation of physical system dynamics.
• The intuitive analogy between velocity and current is consistent with a thermodynamic classifi-
cation into extensive and intensive variables. As a result, the analogy can be generalized to dynamic
behavior in domains to which the “system-of-particles” image may be less applicable.
• The force-voltage or mass-inductor analogy reflects an important distinction between equilibrium
energy-storage phenomena and steady-state energy-storage phenomena: the constitutive equations
of steady-state energy storage phenomena require an inertial reference frame (or must be modified
in a non-inertial reference frame) while the constitutive equations of equilibrium energy storage
phenomena do not.

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