Page 48 - Mechanical Engineers' Handbook (Volume 2)

3 Energy, Power, Impedance 37

is changeable, but the answer is always a transformer or gyrator of some type (a gear ratio
in this case).

3 ENERGY, POWER, IMPEDANCE
3.1 Definitions and Analogies
Energy is the fundamental currency in the interactions between elements of a physical system
no matter how the elements are defined. In engineering systems, it is convenient to describe
these transactions in terms of a complementary pair of variables whose product is the power
or flow rate of the energy in the transaction. These product pairs are familiar to all engineers:
voltage current power, force displacement energy, torque angular velocity
power, pressure flow power, and pressure time rate of change of volume exchanged
power. Some are less familiar: flux linkage current energy, charge voltage
energy, and absolute temperature entropy flux thermal power. Henry M. Paynter’s 1
tetrahedron of state shows how these are related (Fig. 1). Typically, one of these factors is
extensive, a flux or flow, such as current, velocity, volume flow rate, or angular velocity. The
other is intensive, a potential or effort,* such as voltage, force, pressure, or torque. Thus P
extensive intensive for any of these domains of physical activity.
This factoring is quite independent of the analogies between the factors of power in
different domains, for which any arbitrary selection is acceptable. In essence, velocity is not
like voltage or force like current, just as velocity is not like current or force like voltage. It
is convenient, however, before defining impedance and working with it to choose an analogy
so that generalizations can be made across the domains of engineering activity. There are
2
two standard ways to do this: the Firestone analogy and the mobility analogy. Electrical
engineers are most familiar with the Firestone analogy, while mechanical engineers are prob-
ably more comfortable with the mobility analogy. The results derived in this chapter are
independent of the analogy chosen. To avoid confusion, both will be introduced, but only
the mobility analogy will be used in this chapter.
The Firestone analogy gives circuitlike properties to mechanical systems: All systems
consist of nodes like a circuit and only of lumped elements considered to be two-terminal
or four-terminal devices. For masses and tanks of liquid, one of the terminals must be
understood to be ground, the inertial reference frame, or atmosphere. Then one of the energy

Figure 1 H. M. Paynter’s tetrahedron of state.

*This is Paynter’s terminology, used with reference to his ‘‘Bond Graphs.’’