Page 328 - The Mechatronics Handbook

All of the motion variables considered to be analogous (i.e., velocity, fluid flow rate, current) may be
associated with an energy storage function that is defined by steady state (rather than by equilibrium).
For a rigid body, steady motion requires zero net force, and hence constant momentum and kinetic
energy. For the magnetic field that stores energy in an inductor, steady current requires zero voltage, and
hence constant magnetic flux and magnetic energy.
It might reasonably be argued that any distinction between equilibrium and steady state is purely a
matter of perspective and common usage, rather than a fundamental feature of the physical world. For
example, with an alternative choice of reference frames, “sustained motion” could be redefined as “rest”
or “equilibrium.” From this perspective, a zero-relative-velocity “equilibrium” between two rigid bodies
(or between a rigid body and a reference frame) could be defined by zero force. Following this line of
reasoning any distinction between the mass-inductor and mass-capacitor analogies would appear to be
purely a matter of personal choice. However, while the apparent equivalence of “equilibrium” and “steady
state” may be justifiable in the formal mathematical sense of zero rate of change of a variable, in a
mechanical system, displacement (or position) and velocity (or momentum) are fundamentally different.
For example, whereas velocity, force, and momentum may be transformed between reference frames as
rank-one tensors, position (or displacement) may not be transformed as a tensor of any kind. Thus, a
distinction between equilibrium and steady state reflects an important aspect of the structure of physical
system models.

Analogies, Not Identities

It is important to remember that any classification to establish analogies is an abstraction. At most,
dynamic behavior in different domains may be similar; it is not identical. We have pointed out above
that if velocity or current is used as the argument of an energy storage function, care must be taken to
identify an appropriate inertial reference frame and/or to understand the consequences of using a non-
inertial frame. However, another important feature of these variables is that they are fundamentally
vectors (i.e., they have a definable spatial orientation). One consequence is that the thermodynamic
definition of extensive and intensive variables must be generalized before it may be used to classify these
variables (cf., Breedveld, 1984). In contrast, a quantity such as temperature or pressure is fundamentally
a scalar. Furthermore, both of these quantities are intrinsically “positive” scalars insofar as they have well-
defined, unique and physically meaningful zero values (absolute zero temperature, the pressure of a
perfect vacuum). Quite aside from any dependence on inertial reference frames, the across-through
analogy between velocity (a vector with no unique zero value) and pressure (a scalar with a physically
important zero) will cause error and confusion if used without due care.
This consideration becomes especially important when similar elements of a model are combined (for
example, a number of bodies moving with identical velocity may be treated as a single rigid body) to
simplify the expression of dynamic equations or improve their computability. The engineering variables
used to describe energy storage can be categorized into two groups: (i) positive-valued scalar variables
and (ii) nonscalar variables. Positive-valued scalar variables have a physically meaningful zero or absolute
reference; examples include the volume of stored fluid, the number of moles of a chemical species, entropy,
1
etc. Nonscalar variables have a definable spatial orientation. Even in the one-dimensional case they can
be positive or negative, the sign denoting direction with respect to some reference frame; examples include
displacement, momentum, etc. These variables generally do not have a physically meaningful zero or
absolute reference, though some of them must be defined with respect to an inertial frame.
Elements of a model that describe energy storage based on scalar variables can be combined in only one
way: they must be in mutual equilibrium; their extensive variables are added, while the corresponding
intensive variables are equal, independent of direction, and determine the equilibrium condition. For
model elements that describe energy storage based on nonscalar variables there are usually two options.

1
The term “vector variables” suggests itself but these variables may include three-dimensional spatial orientation,
which may not be described as a vector.

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