Page 49 - Mechanical Engineers' Handbook (Volume 2)

38 Input and Output Characteristics

variables is measured across the terminals of the element and the other passes through the
element. In a circuit, voltage is across and current passes through. For a spring, however,
velocity difference is across and the force passes through. Thus this analogy linked voltage
to velocity, angular velocity, and pressure as across variables and linked current to force,
torque, and flow rate as through variables. Clearly, across through power.
The mobility analogy, in contrast, considers the complementary power variables to con-
sist of a potential and a flux, an intrinsic and extrinsic variable. The potentials, or efforts,
are voltage, force, torque, and pressure, while the fluxes, or flows, are current, velocity,
angular velocity, and fluid flow rate.

3.2 Impedance and Admittance
Impedance, in the most general sense, is the relationship between the factors of power.
Because only the constitutive relationships for the dissipative elements are expressed directly
in terms of the power variables, V R i for example, while the equations for the energy
R R
storage elements are expressed in terms of the derivative of one of the power variables* with
respect to the other, i C(dV /dt) for example, these are most conveniently expressed in
C C
Laplace transform terms. Impedances are really self-transfer functions at a point in a system.
Since the concept was probably defined first for electrical systems, that definition is most
standardized: Electrical impedance Z is defined as the rate of change of voltage with
electrical
current:
d(voltage) d(effort)
Z electrical (7)
d(current) d(flow)
By analogy, impedance can be similarly defined for the other engineering domains:
d(force)
Z translation (8)
d(velocity)
d(torque)
Z rotation (9)
d(angular velocity)
d(pressure)
Z fluid (10)
d(flow rate)
Table 1 is an impedance table using these definitions of the fundamental lumped linear
elements. Note that these are derived from the Laplace transforms of the constitutive equa-

Table 1 Impedances of Lumped Linear Elements
Domain Kinetic Storage Dissipation Potential Storage
Translational Mass: Ms Damping: b Spring: k/s
Rotational Inertia: Js Damping: B Torsion spring: k ƒ /s
Electrical Inductance: Ls Resistance: R Capacitance: 1/Cs
Fluid Inertance: Is Fluid resistance: R Fluid capacitance: 1/Cs

*See Fig. 1 again. Capacitance is a relationship between the integral of the flow and the effort, which
is the same as saying that capacitance relates the flow to the derivative of the effort.