Page 323 - The Mechatronics Handbook

15.2 History

It is curious that one of the earliest applications of analogies between electrical and mechanical systems
was to enable the demonstration and study of transients in electrical networks that were otherwise too
fast to be observed by the instrumentation of the day by identifying mechanical systems with equivalent
dynamic behavior; that was the topic of a series of articles on “Models and analogies for demonstrating
electrical principles” (The Engineer, 1926). Improved methods capable of observing fast electrical tran-
sients directly (especially the cathode ray oscilloscope, still in use today) rendered this approach obsolete
but enabled quantitative study of nonelectrical systems via analogous electrical circuits (Nickle, 1925).
Although that method had considerably more practical importance at the time than it has today (we
now have the luxury of vastly more powerful tools for numerical computation of electromechanical system
responses), in the late ’20s and early ’30s a series of papers (Darrieus, 1929; Hähnle, 1932; Firestone, 1933)
formulated a rational method to use electrical networks as a framework for establishing analogies between
physical systems.

15.3 The Force-Current Analogy: Across
and Through Variables

Firestone identified two types of variable in each physical domain—“across” and “through” variables—
which could be distinguished based on how they were measured. An ‘‘across’’ variable may be measured
as a difference between values at two points in space (conceptually, across two points); a ‘‘through’’ variable
may be measured by a sensor in the path of power transmission between two points in space (conceptually,
it is transmitted through the sensor). By this classification, electrical voltage is analogous to mechanical
velocity and electrical current is analogous to mechanical force. Of course, this classification of variables
implies a classification of network elements: a mass is analogous to a capacitor, a spring is analogous to
an inductor and so forth.
The “force-is-like-current” or “mass-capacitor” analogy has a sound mathematical foundation. Kirchhoff’s
node law or current law, introduced in 1847 (the sum of currents into a circuit node is identically zero)
can be seen as formally analogous to D’Alembert’s principle, introduced in 1742 (the sum of forces on
a body is identically zero, provided the sum includes the so-called “inertia force,” the negative of the
body mass times its acceleration). It is the analogy used in linear-graph representations of lumped-
parameter systems, proposed by Trent in 1955. Linear graphs bring powerful results from mathematical
graph theory to bear on the analysis of lumped-parameter systems. For example, there is a systematic
procedure based on partitioning a graph into its tree and links for selecting sets of independent variables
to describe a system. Graph-theoretic approaches are closely related to matrix methods that in turn
facilitate computer-aided methods. Linear graphs provide a unified representation of lumped-parameter
dynamic behavior in several domains that has been expounded in a number of successful textbooks (e.g.,
Shearer et al., 1967; Rowell & Wormley, 1997).
The mass-capacitor analogy also appears to afford some practical convenience. It is generally easier to
identify points of common velocity in a mechanical system than to identify which elements experience
the same force; and it is correspondingly easier to identify the nodes in an electrical circuit than all of
its loops. Hence with this analogy it is straightforward to identify an electrical network equivalent to a
mechanical system, at least in the one-dimensional case.

Drawbacks of the Across-Through Classification

Despite the obvious appeal of establishing analogies based on practical measurement procedures, the
force-current analogy has some drawbacks that will be reviewed below: (i) on closer examination,
measurement-based classification is ambiguous; (ii) its extension to more than one-dimensional mechan-
ical systems is problematical; and (iii) perhaps most important, it leads to analogies (especially between
mechanical and fluid systems) that defy common physical insight.

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