Page 312 - Mechanical Engineers' Handbook (Volume 2)

2 Ideal Elements 303

2.1 Physical Variables
An idealized two-terminal or one-port element is shown in Fig. 2. Two primary physical
variables are associated with the element: a through variable ƒ(t) and an across variable v(t).
Through variables represent quantities that are transmitted through the element, such as the
force transmitted through a spring, the current transmitted through a resistor, or the flow of
fluid through a pipe. Through variables have the same value at both ends or terminals of the
element. Across variables represent the difference in state between the terminals of the
element, such as the velocity difference across the ends of a spring, the voltage drop across
a resistor, or the pressure drop across the ends of a pipe. Secondary physical variables are
the integrated through variable h(t) and the integrated across variable x(t). These represent
the accumulation of quantities within an element as a result of the integration of the asso-
ciated through and across variables. For example, the momentum of a mass is an integrated
through variable, representing the effect of forces on the mass integrated or accumulated
over time. Table 1 defines the primary and secondary physical variables for various physical
systems.

2.2 Power and Energy
The flow of power P(t) into an element through the terminals 1 and 2 is the product of the
through variable ƒ(t) and the difference between the across variables v (t) and v (t). Sup-
1
2
pressing the notation for time dependence, this may be written as
P ƒ(v v ) ƒv 21
2
1
A negative value of power indicates that power flows out of the element. The energy
E(t , t ) transferred to the element during the time interval from t to t is the integral of
a
b
a
b
power, that is,

Figure 2 A two-terminal or one-port element, showing through and across variables. 1