Chapter 4
Single Degree of Freedom Systems:
Forced Response
W
e consider the same single degree of freedom system we had earlier but subject it now to
a force,
F
(
t
). Frequently in practice, the force is a complicated function of time, and we
will look at some possibilities and the consequent response of the system.
4.1
Harmonic Excitation
The most common kind of excitation (in vibration, one often uses the word
excitation
to describe
a force) for vibrating systems is harmonic. This is a consequence of periodic motions that are very
common (by design) in most engineering systems. In addition, it is important to study harmonic
excitation since the solution obtained here forms the mathematical basis for getting the solution to
situations when the forces are
not
harmonic.
4.1.1
Equation of Motion
Consider the SDOF model again with a harmonic force as shown in Fig. 4.1. Then, the equation of
motion is
m
¨
x
+
c
˙
x
+
kx
=
F
0
sin
ωt
(4.1)
where,
F
0
is the amplitude of the force, and
ω
is the frequency of excitation. Note that a more
general harmonic force would be
F
0
sin(
+
φ
). We will consider that case later. Next, we write
the equation in standard form:
¨
x
+2
ζω
n
˙
x
+
ω
2
n
x
=
ω
2
n
F
0
k
sin
(4.2)
53
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CHAPTER 4. SINGLE DEGREE OF FREEDOM SYSTEMS: FORCED RESPONSE
F(t)
Figure 4.1: A Forced massspringdamper system
4.1.2
Solution
From the elementary theory of diﬀerential equations, the solution can be sought in the form,
x
(
t
)=
x
c
cos
ωt
+
x
s
sin
(4.3)
Then,
˙
x
(
t

ωx
c
sin
+
s
cos
(4.4)
¨
x
(
t

ω
2
x
c
cos

ω
2
x
s
sin
(4.5)
Substitution into the equation of motion and separation of the sine and cosine functions yields the
following set of linear algebraic equations in the unknown coeﬃcients.
•

ω
2
+
ω
2
n
2
ζωω
n

2
n

ω
2
+
ω
2
n
‚•
x
c
x
s
‚
=
•
0
ω
2
n
F
0
/k
‚
(4.6)
Solving the set of algebraic equations, we get the following.
•
x
c
x
s
‚
=
1
Δ
0
•

ω
2
+
ω
2
n

2
n
+2
n

ω
2
+
ω
2
n
0
ω
2
n
F
0
/k
‚
(4.7)
where,
Δ
0
=
(

ω
2
+
ω
2
n
)
2
+(2
n
)
2
(4.8)
Next, we deﬁne a convenient nondimensional parameter called the frequency ratio,
r
, that is a ratio
of the forcing frequency to the undamped natural frequency of the system.
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 '09
 harmonic excitation, single degree, frequency ratio

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