Page 313 - Mechanical Engineers' Handbook (Volume 2)

304 Mathematical Models of Dynamic Physical Systems

Table 1 Primary and Secondary Physical Variables for Various Systems 1
Integrated
Through through Across Integrated across
System Variable f Variable h Variable v Variable x
Mechanical– Force F Translational Velocity Displacement
translational momentum p difference v 21 difference x 21
Mechanical– Torque T Angular Angular velocity Angular displacement
rotational momentum h difference 21 difference 21
Electrical Current i Charge q Voltage Flux linkage 21
difference v 21
Fluid Fluid flow Q Volume V Pressure Pressure–momentum
difference P 21 21
Thermal Heat flow q Heat energy H Temperature Not used in general
difference 21

t b t b
E Pdt ƒv dt
21
t a t a
A negative value of energy indicates a net transfer of energy out of the element during the
corresponding time interval.
Thermal systems are an exception to these generalized energy relationships. For a ther-
mal system, power is identically the through variable q(t), heat flow. Energy is the integrated
through variable H(t , t ), the amount of heat transferred.
a
b
By the first law of thermodynamics, the net energy stored within a system at any given
instant must equal the difference between all energy supplied to the system and all energy
dissipated by the system. The generalized classification of elements given in the following
sections is based on whether the element stores or dissipates energy within the system,
supplies energy to the system, or transforms energy between parts of the system.

2.3 One-Port Element Laws
Physical devices are represented by idealized system elements, or by combinations of these
elements. A physical device that exchanges energy with its environment through one pair of
across and through variables is called a one-port or two-terminal element. The behavior of
a one-port element expresses the relationship between the physical variables for that element.
This behavior is defined mathematically by a constitutive relationship. Constitutive relation-
ships are derived empirically, by experimentation, rather than from any more fundamental
principles. The element law, derived from the corresponding constitutive relationship, de-
scribes the behavior of an element in terms of across and through variables and is the form
most commonly used to derive mathematical models.
Table 2 summarizes the element laws and constitutive relationships for the one-port
elements. Passive elements are classified into three types. T-type or inductive storage ele-
ments are defined by a single-valued constitutive relationship between the through variable
ƒ(t) and the integrated across-variable difference x (t). Differentiating the constitutive rela-
21
tionship yields the element law. For a linear (or ideal) T-type element, the element law states
that the across-variable difference is proportional to the rate of change of the through vari-
able. Pure translational and rotational compliance (springs), pure electrical inductance, and