Page 316 - Mechanical Engineers' Handbook (Volume 2)

2 Ideal Elements 307

pure fluid inertance are examples of T-type storage elements. There is no corresponding
thermal element.
A-type or capacitive storage elements are defined by a single-valued constitutive rela-
tionship between the across-variable difference v (t) and the integrated through variable h(t).
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These elements store energy by virtue of the across variable. Differentiating the constitutive
relationship yields the element law. For a linear A-type element, the element law states that
the through variable is proportional to the derivative of the across-variable difference. Pure
translational and rotational inertia (masses) and pure electrical, fluid, and thermal capacitance
are examples.
It is important to note that when a nonelectrical capacitance is represented by an A-type
element, one terminal of the element must have a constant (reference) across variable, usually
assumed to be zero. In a mechanical system, for example, this requirement expresses the
fact that the velocity of a mass must be measured relative to a noninertial (nonaccelerating)
reference frame. The constant-velocity terminal of a pure mass may be thought of as being
attached in this sense to the reference frame.
D-type or resistive elements are defined by a single-valued constitutive relationship be-
tween the across and the through variables. These elements dissipate energy, generally by
converting energy into heat. For this reason, power always flows into a D-type element. The
element law for a D-type energy dissipator is the same as the constitutive relationship. For
a linear dissipator, the through variable is proportional to the across-variable difference. Pure
translational and rotational friction (dampers or dashpots) and pure electrical, fluid, and
thermal resistance are examples.
Energy storage and energy-dissipating elements are called passive elements, because
such elements do not supply outside energy to the system. The fourth set of one-port elements
are source elements, which are examples of active or power-supplying elements. Ideal sources
describe interactions between the system and its environment. A pure A-type source imposes
an across-variable difference between its terminals, which is a prescribed function of time,
regardless of the values assumed by the through variable. Similarly, a pure T-type source
imposes a through-variable flow through the source element, which is a prescribed function
of time, regardless of the corresponding across variable.
Pure system elements are used to represent physical devices. Such models are called
lumped-element models. The derivation of lumped-element models typically requires some
degree of approximation, since (1) there rarely is a one-to-one correspondence between a
physical device and a set of pure elements and (2) there always is a desire to express an
element law as simply as possible. For example, a coil spring has both mass and compliance.
Depending on the context, the physical spring might be represented by a pure translational
mass, or by a pure translational spring, or by some combination of pure springs and masses.
In addition, the physical spring undoubtedly will have a nonlinear constitutive relationship
over its full range of extension and compression. The compliance of the coil spring may
well be represented by an ideal translational spring, however, if the physical spring is ap-
proximately linear over the range of extension and compression of concern.

2.4 Multiport Elements
A physical device that exchanges energy with its environment through two or more pairs of
through and across variables is called a multiport element. The simplest of these, the idealized
four-terminal or two-port element, is shown in Fig. 3. Two-port elements provide for trans-
formations between the physical variables at different energy ports, while maintaining in-
stantaneous continuity of power. In other words, net power flow into a two-port element is
always identically zero: