Page 168 - Modern Control Systems

142 Chapter 2 Mathematical Models of Systems

for the fluid-flow equation, (b) What happens to the
.J approximation obtained in part (a) if the operating
point is Pi - P 2 = 0?
Force < ; * ! L P2.6 Using the Laplace transformation, obtain the current
m •• > J " I 2(s) of Problem P2.1. Assume that all the initial cur-
AT, | v .<') rents are zero, the initial voltage across capacitor C\ is
zero, v{t) is zero, and the initial voltage across C 2 is 10
volts.
P2.7 Obtain the transfer function of the differentiating
circuit shown in Figure P2.7.
T>'2<'>
ftf 2
1
FIGURE P2.2 Vibration absorber.
P2.3 A coupled spring-mass system is shown in Figure
P2.3. The masses and springs are assumed to be equal.
Obtain the differential equations describing the system. +
VAs)
V 2[s)

-»• I'II O
Force v,(r) -*• A',(0
ru) FIGURE P2.7 A differentiating circuit.
I
M WWWH u Hi P2.8 A bridged-T network is often used in AC control
k systems as a filter network [8]. The circuit of one
k nzn bridged-T network is shown in Figure P2.8. Show that
the transfer function of the network is
FIGURE P2.3 Two-mass system.
V&) 1 + IR^Cs + RiRjpV
P2.4 A nonlinear amplifier can be described by the fol- Kn(i') 1 + (2«, + R 2)Cs + i?,i? 2 cV
lowing characteristic:
Sketch the pole-zero diagram when R x = 0.5,¾ = 1,
J 4 and C = 0.5.
"o(') =
«m < 0'
I 4
Tlie amplifier will be operated over a range of ±0.5
volts around the operating point for v in. Describe the wv
amplifier by a linear approximation (a) when the op-
erating point is sjj,, = 0 and (b) when the operating
point is w in = 1 volt. Obtain a sketch of the nonlinear
function and the approximation for each case.
P2.5 Fluid flowing through an orifice can be represented
by the nonlinear equation
2
Q = K(P, - A)" , FIGURE P2.8 Bridged-T network.
where the variables are shown in Figure P2.5 and K is
a constant [2]. (a) Determine a linear approximation P2.9 Determine the transfer function Xi(s)/F(s) for the
coupled spring-mass system of Problem P2.3. Sketch
the s-plane pole-zero diagram for low damping when
M = l,b/k = l,and

0.1.
4
2-
P2.10 Determine the transfer function Yi{s)jF(s) for the
FIGURE P2.5 Flow through an orifice. vibration absorber system of Problem P2.2. Determine