Page 324 - The Mechatronics Handbook

Measurement as a Basis for Analogies

Even a cursory review of state-of-the-art measurement technologies shows that the across-through
classification may be an anachronism or, at best, an over-simplification. Velocity (an “across” variable)
may be measured by an integrating accelerometer that is attached only to the point where velocity is
measured—that’s how the human inner ear measures head velocity. While the velocity is measured with
respect to an inertial reference frame (as it should be), there is no tangible connection to that frame. As
a further example, current in a conductor (a “through” variable) may be measured without inserting an
ammeter in the current path; sensors that measure current by responding to the magnetic field next to
the conductor are commercially available (and preferred in some applications). Moreover, in some cases
similar methods can be applied to measure both “across” and “through” variables. For example, fluid
flow rate is classified as a through variable, presumably by reference to its measurement by, for example,
a positive-displacement meter in the flow conduit; that’s the kind of fluid measurement commonly used in
a household water meter. However, optical methods that are used to measure the velocity of a rigid body
(classified as an across variable) are often adapted to measure the volumetric flow rate of a fluid (laser
doppler velocimetry is a notable example). Apparently the same fundamental measurement technology
can be associated with an across variable in one domain and a through variable in another. Thus, on
closer inspection, the definition of across and through variables based on measurement procedures is,
at best, ambiguous.

Beyond One-Dimensional Mechanical Systems
The apparent convenience of equating velocities in a mechanical system with voltages at circuit nodes
diminishes rapidly as we go beyond translation in one dimension or rotation about a fixed axis. A
translating body may have two or three independent velocities (in planar and spatial motion, respectively).
Each independent velocity would appear to require a separate independent circuit node, but the kinetic
energy associated with translation can be redistributed at will among these two or three degrees of freedom
(e.g., during motion in a circle at constant speed the total kinetic energy remains constant while that
associated with each degree of freedom varies). This requires some form of connection between the
corresponding circuit nodes in an equivalent electrical network, but what that connection should be is
not obvious.
The problem is further exacerbated when we consider rotation. Even the simple case of planar motion
(i.e., a body that may rotate while translating) requires three independent velocities, hence three inde-
pendent nodes in an equivalent electrical network. Reasoning as above we see that these three nodes
must be connected but in a different manner from the connection between three nodes equivalent to
spatial translation. Again, this connection is hardly obvious, yet translating while rotating is ubiquitous
in mechanical systems—that’s what a wheel usually does.
Full spatial rotation is still more daunting. In this case interaction between the independent degrees
of freedom is especially important as it gives rise to gyroscopic effects, including oscillatory precession
and nutation. These phenomena are important practical considerations in modern mechatronics, not
arcane subtleties of classical mechanics; for example, they are the fundamental physics underlying several
designs for a microelectromechanical (MEMS) vibratory rate gyroscope (Yazdi et al., 1998).

Physical Intuition

In our view the most important drawback of the across-through classification is that it identifies force
as analogous to fluid flow rate as well as electrical current (with velocity analogous to fluid pressure as
well as voltage). This is highly counter-intuitive and quite confusing. By this analogy, fluid pressure is
not analogous to force despite the fact that pressure is commonly defined as force per unit area. Fur-
thermore, stored kinetic energy due to fluid motion is not analogous to stored kinetic energy due to
motion of a rigid body. Given the remarkable similarity of the physical processes underlying these two
forms of energy storage, it is hard to understand why they should not be analogous.

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