Page 329 - The Mechatronics Handbook

Electrical capacitors, for instance, may be combined in parallel or in series and the resulting equivalent
capacitor may readily be determined. In a parallel connection, equilibrium is determined by voltage (an
intensive variable) and the electric charges (extensive variables) are added as before. However, a series
connection is the “dual” in the sense that the roles of charge and voltage are exchanged: equality of charges
determines equilibrium and the voltages are added. Mechanical springs may also be combined in two ways.
However, that is not the case for translational masses and rotational inertias; they may only be combined
into a single equivalent rigid body if their velocities are equal and in that case their momenta are added.
The existence of two “dual” ways to combine some, but not all, of the energy storage elements based
on nonscalar quantities is somewhat confusing. It may have contributed to the lengthy debate (if we date
its beginning to Maxwell, lasting for over a century!) on the best analogy between mechanical and electrical
systems. Nevertheless, the important point is that series and parallel connections may not be generalized
in a straightforward way to all domains.

Nodicity

As insight is the foremost goal of modeling, analogies should be chosen to promote insight. Because there
may be fundamental differences between all of the physical domains, care should be exercised in drawing
analogies to ensure that special properties of one domain should not be applied inappropriately to other
domains. This brings us to what may well be the strongest argument against the across-through classification.
History suggests that it originated with the use of equivalent electrical network representations of nonelec-
trical systems. Unfortunately, electrical networks provide an inappropriate basis for developing a general
representation of physical system dynamics. This is because electrical networks enjoy a special property,
nodicity, which is quite unusual among the physical system domains (except as an approximation).
Nodicity refers to the fact that any sub-network (cut-set) of an electrical network behaves as a node
in the sense that a Kirchhoff current balance equation may be written for the entire sub-network. As a
result of nodicity, electrical network elements can be assembled in arbitrary topologies and yet still
describe a physically realizable electrical network. This property of “arbitrary connectability” is not a
general property of lumped-parameter physical system models. Most notably, mass elements cannot be
connected arbitrarily; they must always be referenced to an inertial frame. For that reason, electrical
networks can be quite misleading when used as a basis for a general representation of physical system
dynamics. This is not merely a mathematical nicety; some consequences of non-nodic behavior for control
system analysis have recently been explored (Won and Hogan, 1998).
By extension, because each of the physical domains has its unique characteristics, any attempt to
formulate analogies by taking one of the domains (electrical, mechanical, or otherwise) as a starting point
is likely to have limitations. A more productive approach is to begin with those characteristics of physical
variables common to all domains and that is the reason to turn to thermodynamics. In other words, the
best way to identify analogies between domains may be to “step outside” all of them. By design, general
characteristics of all domains such as the extensive nature of stored energy, the intensive nature of the
variables that define equilibrium, and so forth, are not subject to the limitations of any one (such as
nodicity). That is the main advantage of drawing analogies based on thermodynamic concepts such as
the distinction between extensive and intensive variables.

15.6 Graphical Representations

Analogies are often associated with abstract graphical representations of multi-domain physical system
models. The force-current analogy is usually associated with the linear graph representation of networks
introduced by Trent (1955); the force-voltage analogy is usually associated with the bond graph represen-
tation introduced by Paynter (1960). Bond graphs classify variables into efforts (commonly force, voltage,
pressure, and so forth) and flows (commonly velocity, current, fluid flow rate, and so forth). Bond graphs
extend all the practical benefits of the force-current (across-through) analogy to the force-voltage (effort-
flow) analogy: they provide a unified representation of lumped-parameter dynamic behavior in several

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