Page 101 - Modern Control Systems

Section 2.5 The Transfer Function of Linear Systems 75

Thus, substituting Equation (2.71) into Equation (2.72), we obtain
2
A , d y dy
- ( M -Q) = M- [Z + { . (2.73)
b
Furthermore, the volumetric fluid flow is related to the piston movement as
dy
Q = A-%. (2.74)

Then, substituting Equation (2.74) into Equation (2.73) and rearranging, we have

2 2 d
d y + + A \ y n i c ,
Ak x ( h
——x = M—r- + \ b + —• —. (2.75)
dt 2 \ kpjdt v '
k P
Therefore, using the Laplace transformation, we have the transfer function
Y(s) K
(2.76)
X(s) s(Ms + BY
where
2
A
Ak x
K = —^ and B = b + —.
k P k p
Note that the transfer function of the hydraulic actuator is similar to that of the elec-
tric motor. For an actuator operating at high pressure levels and requiring a rapid
response of the load, we must account for the effect of the compressibility of the
fluid [4,5].
Symbols, units, and conversion factors associated with many of the variables in
Table 2.5 are located at the MCS website. The symbols and units for each variable can be
found in tables with corresponding conversions between SI and English units. •

The transfer function concept and approach is very important because it pro-
vides the analyst and designer with a useful mathematical model of the system ele-
ments. We shall find the transfer function to be a continually valuable aid in the
attempt to model dynamic systems. The approach is particularly useful because the
5-plane poles and zeros of the transfer function represent the transient response of
the system. The transfer functions of several dynamic elements are given in Table 2.5.
In many situations in engineering, the transmission of rotary motion from one
shaft to another is a fundamental requirement. For example, the output power of an
automobile engine is transferred to the driving wheels by means of the gearbox and
differential. The gearbox allows the driver to select different gear ratios depending
on the traffic situation, whereas the differential has a fixed ratio. The speed of the
engine in this case is not constant, since it is under the control of the driver. Anoth-
er example is a set of gears that transfer the power at the shaft of an electric motor
to the shaft of a rotating antenna. Examples of mechanical converters are gears,
chain drives, and belt drives. A commonly used electric converter is the electric
transformer. An example of a device that converts rotational motion to linear mo-
tion is the rack-and-pinion gear shown in Table 2.5, item 17.