Homework 8
Date: March 16th, 2026
ME 3017: System Dynamics
Spring Semester 2026
Total Points: 100
To be turned-in by Monday the 23rd March
1. (30 pts) A certain system is described by the following model. Time is measured in seconds.
Wn 2
2
=
=
X
=
52 + 22conStWn2
R
m
ẍ + bẋ + 4x = us (t)
(1)
-
a) Set the values of the damping ratio ω so that maximum overshoot is not greater than 20% of the
steady-state value.
Mp
= 20 +
= Xss
(h) .
2
.
6
-
2
.
62
!
=
9 8792
.
= 2 ET ( 2) 1-2456
.
b) The figure hereafter shows the plot of εn tr (in rad) versus ω for the previous system where tr is
the 100% rise time and εn the natural frequency. Using this figure, set the values of the damping
ratio ω so that the 100% rise time is not greater than 3 s.
n 2
Wn =
=
tr 3
=
21
-
·
tr 3 W
=
:
=
2
.
Untr
88
=
6
80
Contr 6
=
c) Can both specifications be satisfied at the same time (i.e. maximum overshoot not greater than
20% of the steady-state value and tr → 3 s)?
Yes
.
436
-21
.
87
d) Give priority to the specification that the overshoot should be as small as possible and deduce
the optimized value of the damping constant b.
42
=
b
52
b
4( 88) 3
=
.
.
1
=
7
2. (30 pts) Compute the damping ratio, the maximum overshoot, the peak time, the 100% rise time,
the delay time, and the 2% settling time for the following model. The initial conditions are zero.
Time is measured in seconds.
+
52 + 22cnS + Wn
ẍ + 4ẋ + 8x = 2us (t)
max overshoot :
tp =e
tr
=
Mp =
&
=
2 -d
=
-
Wniez
-In
=
=
08
Mc
-
Mp
=
q
2 22
2
=
arctan(e)
.
ts
1 67
.
==
ta =
2x + ux
(2)
4
Wn =
·
=w
=
3 93
=
2
.
747
3. (20 pts) Consider the following model where b is a constant.
>
I
ẍ + 2bẋ + (2b + 1)x = 2(1 ↑ e→7t ) + cos 0.3t
(3)
a) For what values of b is the system stable?
10
2b > 0
2b + 170
Sp
b 7
.
S
b) For what values of of b will the free response consist of decaying oscillations?
2
2bH
=
b 2 - 2b
-
0<2
<
/
10
&
It
<
2
<E
if combine /parta :
4. (20 pts) Below are given the poles of two second-order system’s transfer functions. When it is
appropriate, determine the damping ratio ω, the natural frequency εn , the time constant ϑ and the
damped frequency εd .
S
=
-I can
+
jon Mez
=
a) s1 = ↑5 + 5j and s2 = ↑5 ↑ 5j
+
2Wn
wn
=
+
writer: Beze
5
=
(
b) s1 = 3j and s2 = ↑3j
Brad/se
=
=0
soo
Zund/o
1
-
=
=
B
e2 =?
-
Bed
⑳Wn
=
2
2
=
/
==
e
Z
Hunts
cited
=