Page 326 - The Mechatronics Handbook

(sometimes explicitly, sometimes implicitly) in the textbooks and handbooks of basic science and engi-
neering we speculate that it may account for the physical intuition shared by most engineers. If so, then
conforming with that common “system-of-particles” mental model is important to facilitate designing,
thinking, and communicating about mechatronic systems. The force-voltage analogy does so; the force-
current analogy does not.

Dependence on Reference Frames
The “system-of-particles” model also leads to another important physical consideration in the choice of
analogies between variables: the way they depend on reference frames. The mechanical displacement that
determines the elastic potential energy stored in a spring and the displaced charge that determines the
electrostatic potential energy stored in a capacitor may be defined with respect to any reference frame
(whether time-varying or stationary). In contrast, the motion required for kinetic energy storage in a
rigid body or a fluid must be defined with respect to an inertial frame. Though it may often be overlooked,
the motion of charges required for magnetic field storage must also be defined with respect to an inertial
frame (Feynman et al., 1963).
To be more precise, the constitutive equations of energy storage based on motion (e.g., in a mass or
an inductor) require an inertial reference frame (or must be modified in a non-inertial reference frame).
In contrast, the constitutive equations of energy storage based on displacement (e.g., in a spring or a
capacitor) do not. Therefore, the mass-inductor (force-voltage) analogy is more consistent with funda-
mental physics than the mass-capacitor (force-current) analogy.
The modification of the constitutive equations for magnetic energy storage in a non-inertial reference
frame is related to the transmission of electromagnetic radiation. However, Kirchhoff’s laws (more aptly
termed “Kirchhoff’s approximations”), which are the foundations of electric network theory, are equiv-
alent to assuming that electromagnetic radiation is absent or negligible. It might, therefore, be argued
that the dependence of magnetic energy storage on an inertial reference frame is negligible for electrical
circuits, and hence is irrelevant for any discussion of the physical basis of analogies between electrical
circuits and other lumped-parameter dynamic-system models. That is undeniably true and could be used
to justify the force-current analogy. Nevertheless, because of the confusion that can ensue, the value of
an analogy that is fundamentally inconsistent with the underlying physics of lumped-parameter models is
questionable.

15.5 A Thermodynamic Basis for Analogies

Often in the design and analysis of mechatronic systems it is necessary to consider a broader suite of
phenomena than those of mechanics and electromechanics. For instance, it may be important to consider
thermal conduction, convection, or even chemical reactions and more. To draw analogies between the
variables of these domains it is helpful to examine the underlying physics. The analogous dynamic
behavior observed in different physical domains (resonant oscillation, relaxation to equilibrium, etc.) is
not merely a similarity of mathematical forms, it has a common physical basis which lies in the storage,
transmission, and irreversible dissipation of energy. Consideration of energy leads us to thermodynamics;
we show next that thermodynamics provides a broader basis for drawing analogies and yields some
additional insight.
All of the displacements considered to be analogous above (i.e., mechanical displacement, displaced
fluid volume, and displaced charge) may be associated with an energy storage function that requires
equilibrium for its definition, the displacement being the argument of that energy function. Generically,
these may be termed potential energy functions. To elaborate, elastic energy storage requires sustained
but recoverable deformation of a material (e.g., as in a spring); the force required to sustain that
deformation is determined at equilibrium, defined when the time rate of change of relative displacement
of the material particles is uniformly zero (i.e., all the particles are at rest relative to each other).
Electrostatic energy storage requires sustained separation of mobile charges of opposite sign (e.g., as in

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