Page 132 - Modern Control Systems

106 Chapter 2 Mathematical Models of Systems

Table 2.8 Parameters of a Large DC Motor
= 10 / = 2
K m
= 1 b = 0.5
R a
= \ = 0.1
L a K b
When the system is in balance, Vi = 0, and when K ( = 0.1, we have

1 + *2/*l _ *2 - = i
l + i? 3/i?4 *1 ' '

This relation can be achieved when

R 2(Ri = 10 and R 3/R 4 = 10.
The parameters of the motor and load are given in Table 2.8. The overall system is
shown in Figure 2.44(b). Reducing the block diagram in Figure 2.44(c) or the signal-
flow graph in Figure 2.44(d) yields the transfer function

<o(s) 540Gi(s)G 2(s) 540GxG 2
co d(s) 1 + O . I G ^ + 540GxG 2 1 + 5 4 0 . ^ ^
5400 5400
(s + 1)(25 + 0.5) + 5401 Is 1 + 2.5s + 5401.5
2700
(2.129)
s 2 + 1.255 + 2700.75'
Since the characteristic equation is second order, we note that <o n = 52 and
£ = 0.012, and we expect the response of the system to be highly oscillatory (under-
damped). •

EXAMPLE 2.15 Mechanical accelerometer

A mechanical accelerometer is used to measure the acceleration of a rocket test
sled, as shown in Figure 2.45. The test sled maneuvers above a guide rail a small dis-
tance 5. The accelerometer provides a measurement of the acceleration a{t) of the
sled, since the position y of the mass M, with respect to the accelerometer case, is
proportional to the acceleration of the case (and the sled). The goal is to design an
accelerometer with an appropriate dynamic responsiveness. We wish to design an
accelerometer with an acceptable time for the desired measurement characteristic,
y{t) = qa{t), to be attained (q is a constant).
The sum of the forces acting on the mass is
dy_
-b± -ky J = M-t(y + x)
dt " '" df

or
2
l
d /2„ y dy t ^ r d Alx
J„
M-~r 2 + b-r + ky = -M—ir. (2.130)
dt dt * df