Page 325 - The Mechatronics Handbook

Insight is the ultimate goal of modeling. It is a crucial factor in producing innovative and effective
designs and depends on developing and maintaining a “physical intuition” about the way devices behave.
It is important that analogies between physical effects in different domains can be reconciled with the
physical intuition and any method that requires a counter-intuitive analogy is questionable; at a minimum
it warrants careful consideration.

15.4 Maxwell’s Force-Voltage Analogy:
Effort and Flow Variables

An alternative analogy classifies variables in each physical domain that (loosely speaking) describe motion
or cause it. Thus fluid flow rate, electrical current, and velocity are considered analogous (sometimes
generically described as “flow” variables). Conversely, fluid pressure, electrical voltage, and force are con-
sidered analogous (sometimes generically described as “effort” variables).
The “force-is-like-voltage” analogy is the oldest drawn between mechanical and electrical systems. It
was first proposed by Maxwell (1873) in his treatise on electricity and magnetism, where he observed
the similarity between the Lagrangian equations of classical mechanics and electromechanics. That was
why Firestone (1933) presented his perspective that force is like current as “A new analogy between
mechanical and electrical systems” (emphasis added). Probably because of its age, the force-voltage
analogy is deeply embedded in our language. In fact, voltage is still referred to as “electromotive force”
in some contexts. Words like “resist” or “impede” also have this connotation: a large resistance or impedance
implies a large force for a given motion or a large voltage for a given current.
In fact, Maxwell’s classification of velocity as analogous to electrical current (with force analogous to
voltage) has a deeper justification than the similarity of one mathematical form of the equations of
mechanics and electromechanics; it can be traced to a similarity of the underlying physical processes.

Systems of Particles

Our models of the physical world are commonly introduced by describing systems of particles distributed
in space. The particles may have properties such as mass, charge, etc., though in a given context we will
deliberately choose to neglect most of those properties so that we may concentrate on a single physical
phenomenon of interest. Thus, to describe electrical capacitance, we consider only charge, while to describe
translational inertia, we consider only mass and so forth.
Given that this common conceptual model is used in different domains, it may be used to draw
analogies between the variables of different physical domains. From this perspective, quantities associated
with the motion of particles may be considered analogous to one another; thus mechanical velocity,
electrical current, and fluid flow rate are analogous. Accordingly, mechanical displacement, displaced
fluid volume, and displaced charge are analogous; and thus force, fluid pressure, and voltage are analo-
gous. This classification of variables obviously implies a classification of network elements: a spring relates
mechanical displacement and force; a capacitor relates displaced charge and voltage. Thus a spring is
analogous to a capacitor, a mass to an inductor, and for this reason, this analogy is sometimes termed
the “mass-inductor” analogy.

Physical Intuition

The “system-of-particles” models naturally lead to the “intuitive” analogy between pressure, force, and
voltage. But, is such a vague and ill-defined concept as “physical intuition” an appropriate consideration
in drawing analogies between physical systems? After all, physical intuition might largely be a matter of
usage and familiarity, rooted in early educational and cultural background.
We think not; instead we speculate that physical intuition may be related to conformity with a mental
model of the physical world. That mental model is important for thinking about physical systems and,
if shared, for communicating about them. Because the “system-of-particles” model is widely assumed

©2002 CRC Press LLC