Page 77 - Mechanical Engineers' Handbook (Volume 2)

66 Input and Output Characteristics

the proof mass to zero and measuring the force required to do it. The differential pressure-
cells used in the process control industry measure differential pressure across a diaphragm
or capsule while preventing the diaphragm or capsule (separating the two pressures) from
moving. They thus avoid having to worry about the nonlinear elastic characteristics of the
capsule and seals in the system.* Their output is either the current in a voice coil or the
pressure in a bellows necessary to oppose the motion. In all of these cases, however, there
is a transient displacement until the servo zeros it out, and some energy must be lost to
transfer the information required by the servo in the instrument. For this reason, null instru-
ments are no better at measuring stored energy, the voltage on a small capacitor for example,
than passive but high-impedance instruments, particularly if the energy consumed to reach
null is not extremely low.

7 DISTRIBUTED SYSTEMS IN BRIEF
While a detailed study of the input–output relationships for distributed systems is beyond
the scope of this chapter, a brief discussion can tie these in to the concepts already covered.
All of the systems discussed to this point have been lumped, a label that implies physical
dimensions are not of importance; the system parts can be considered as point objects.
Considering a tank, for example, to be a point does not mean that the tank has no dimensions;
it merely means that the internal pressure is considered to be the same everywhere within
it; conditions in its interior are absolutely uniform. When studying lumped circuits, we are
not concerned with the dimensions of the circuit elements or with their distances from each
other on the circuit board. In reality, the nodes of the circuit have lengths, but they are
ignored for the purposes of analyzing the lumped circuit.
In the mechanical domain, we consider that masses are rigid and behave as if the forces
acting were applied at a point, and if we are interested in the distributed properties of an
object, we consider only its moment of inertia. In each of these examples, changes in the
physical variables of our model are assumed to propagate instantaneously, even though it is
well known that all have finite propagation velocities. A system element must be considered
to be distributed to have properties that vary with a physical dimension if that assumption
is not true. This occurs whenever the dimensions of the object are large compared to the
characteristic size of the events occurring.
Mechanical disturbances of all kinds propagate at the speed of sound in the medium
involved, and electromagnetic disturbances propagate at speeds near the speed of light, de-
pending again on the medium. A hammer blow to the end of a long, slender bar, for example,
induces a strain pulse at the struck end which travels into the bar (informing the interior of
the event) at the speed of sound in compression in the bar material (c E/
, where E is
Young’s modulus of the material,
is its mass density, and c is the propagation speed of
compressive or tensile events). If the pulse duration is short, its physical length approaches
that of the bar or may be shorter than the bar. Our simple lumped models of the bar’s
behavior [F M¨y and Fdt Mv(t) Mv(0)], which treat it as a solid rigid object, are
incorrect. Similarly, if a small explosion, a spark, for example, is initiated in a tank, the
pressure in the tank will not remain uniform throughout. Instead, pressure waves will prop-
agate within the tank at the speed of sound until damping takes its toll; we can no longer

*Dry friction would nonetheless be fatal to the instrument because it would be fatal to its servo and
would lead to dynamic instabilities of the limit-cycle variety.