3.14 Response Due to Periodic Forces
105
Applying the principle of superposition, the total steady-state response is given
by
x =
∞
n =−∞
H (nω) C n e
in ω t
(3.114)
Example 3.16 Determine the steady-state response of an undamped SDF system for
an excitation in the nature of a square wave of Fig. 3.30, by using complex Fourier
series. Given p = 8ω. Also, draw the response spectra.
For a square wave excitation, the complex Fourier series is given by Eq. (3.108)
as
F (t) = −
2i A
π
∞
n=1, 3, ···
1
n
e
in ω t
The complex frequency response function is given by
H (nω) =
1/k
1 −
nω
p
2
+ i
2ζ
nω
p
(3.115)
Now, ζ = 0 and p = 8ω.
Therefore, H (nω ) =
1/k
1−(
n
8 )
2
Therefore, total response is given by [Eq. (3.114)]
x (t) =
∞
n=1, 3, ...
−
2i A
nπ
1/k
1 −
n
8
2
e
in ω t
(3.116)
or
x (t) =
∞
n=1, 3, ...
x n e
in ω t
(3.117)
where
| x n | = | H (nω ) F n (ω ) |
=
2 A
π kn
1 −
n
8
2
The sketch of | x n | is given in Fig. 3.32.