Block Diagram Drawing Problems
1. Draw a block diagram for the system modeled by the following equations:
m(t ) = r(t ) 10c(t )
dm(t )
f (t ) =
+ 25m(t )
dt
c (t ) = 4 ( f (t ) 60x (t )dt
1
2. Draw a block diagram for the system defined by the following
ME 446 - Homework Solutions - Fall 2008 - Chapter 2
E2.2
R = 10000e"0.1T , T # = 20 C
R $ R# +
%R
(T " T #)
%T T = T0
R# = 10000e"0.1T0 = 1353.4 &
%R
&
= "0.1(10000)e"0.1T
= "135.34
T = 20
%T T = 20
C
R $ 1353.4 " 135.34(T " 20) &
E2.2
!
f " 0.5 + k ( x #
ME 446 - Homework Solutions - Fall 2008 - Chapter 5
E5.1
a) For a step input, a Type 1 system is needed (1 integration)
b) For a ramp input, a Type 2 system is needed (2 integrations)
E5.7
The pole at -0.6 dominates the poles with real part at -2. Because
ME 446 - Homework Solutions - Fall 2008 - Chapter 7
E7.1
a)
G=tf([1 4 0],[1 2 2])
Transfer function:
s^2 + 4 s
-s^2 + 2 s + 2
rlocus(G)
b) K = 0.309
c) -1.24
d)
Ts =
!
4
4
=
= 3.2 s
"# n 1.24
CP7.1
a)
G=tf(10,[1 14 43 30])
Transfer function:
10
-s^3 + 14
ME 446 - Homework Solutions - Fall 2008 - Chapter 9
E9.8
a)
G=tf(4,[1 3 2 0])
Transfer function:
4
-s^3 + 3 s^2 + 2 s
margin(G)
b)
To obtain a 16 dB gain margin, the gain must be decreased by 16 - 3.52 = 12.48 dB.
"12.48
K c = 10
20
= 0.238
K # = K c K =
ME 446 - Homework Solutions - Fall 2008 - Chapter 12
E12.10
By trail and error using root locus, one possible PI control is
Gc =
!
2.08(s + 0.1)
s
Using Matlab, the closed-loop transfer functions and step response plot for 50%
change in K are as follows:
DRAFT
Solution of Basic Frequency Response Problems
1. Calculate the magnitude and phase of the frequency response of the following
transfer function at a frequency of 2 rad/sec.
G ( s) =
5 ( s + 2)
(s
2
+ 7s + 12 ) s
Solution
G ( s) =
5 ( s + 2)
(s
2
+ 7
DRAFT
Basic Frequency Response Problems
1. Calculate the magnitude and phase of the frequency response of the following
transfer function at a frequency of 2 rad/sec.
G ( s) =
5 ( s + 2)
(s
2
+ 7s + 12 ) s
1
DRAFT
2. Calculate the magnitude in dB and phas
Linearization Problems - Solutions
1. Friction force Ff is to be approximated by an equation of the form
dy dy
F f = F fo + k
dt dt
o
If the nominal velocity is 2.6 inch/s, only small variations away from the nominal
velocity are expected and the fri
Block Diagram Drawing Problems
1. Draw a block diagram for the system modeled by the following equations:
m(t ) = r(t ) 10c(t )
dm(t )
f (t ) =
+ 25m(t )
dt
c (t ) = 4 ( f (t ) 60x (t )dt
Solution:
M ( s ) = R( s ) 10C ( s )
F ( s ) = sM ( s ) + 25 M ( s
Final Value Problems
1. What is the final value of y(t) given the following Laplace transform:
Y ( s) =
25
s(0.1s + 1)(0.01s + 1)
2. What is the final value of y(t) given
dy(t )
+ Ky (t ) = Au(t )
dt
where u(t) is the unit step function? (Assume that the
Final Value Problems - Solutions
1. What is the final value of y(t) given the following Laplace transform:
Y ( s) =
25
s(0.1s + 1)(0.01s + 1)
Solution:
The roots of the denominator of sY(s) are:
(0.1s + 1)(0.01s + 1) = 0
( s + 10)( s + 100) = 0
s1 = 10
s2
Initial Value Problems
1. What is the initial value of the function y(t) represented by the following Laplace
transform:
Y ( s) =
25
s(0.1s + 1)(0.001s + 1)
1
2. What is the initial value of the function y(t) represented by the following Laplace
transform
Initial Value Problems - Solutions
1. What is the initial value of the function y(t) represented by the following Laplace
transform:
Y ( s) =
25
s(0.1s + 1)(0.001s + 1)
Solution
25
y(0) = lim s
s s(0.1s + 1)(0.001s + 1)
25
y(0) = lim
s (0.1s + 1)(0.001s
Laplace Transformation Problems
1. a) Write the differential equation that describes the system shown below where h(t) is
the system output, qi(t) is the system input, A = 200 cm2 and Gv = 15 cm2/s where
qo(t)=Gvh(t).
Note the basic principle qs(t) = qi(t
Laplace Transformation Problems - Solutions
1. a) Write the differential equation that describes the system shown below where h(t) is
the system output, qi(t) is the system input, A = 200 cm2 and Gv = 15 cm2/s where
qo(t)=Gvh(t).
Note the basic principle
Linearization Problems
1. Friction force Ff is to be approximated by an equation of the form
dy dy
F f = F fo + k
dt dt
o
If the nominal velocity is 2.6 inch/s, only small variations away from the nominal
velocity are expected and the friction force