104
3 Forced Vibration of Single Degree of Freedom System
1 when n is even
e
− in π
= cos nπ − i sin nπ =
− 1 when n is odd
⎫
⎬
⎭
(3.107)
Therefore,
C n =
i A
2πn
2e
− in π
− 1 − e
−2in π
= −
2i A
nπ
when n is odd.
Substitution of the value of C n in Eq. (3.105) yields
F (t) =
∞
n=1, 3, ...
−
2i A
nπ
e
in ω t
= −
2i A
π
∞
n=1, 3, ...
1
n
e
in ω t
(3.108)
3.14.4 Response of SDF System to Periodic Forces
Represented by Complex Fourier Series
For a SDF system subjected to an exciting force F 0 e
iω t , the equation of motion is
m ¨
x + c ˙
x + kx = F 0 e
iω t
(3.109)
The steady-state solution of the SDF system is assumed in the complex form as
follows
x = H (ω) F 0 e
iω t
(3.110)
H (ω) is termed as complex frequency response function.
Substituting the value of x and the necessary derivatives from Eq. (3.110) into
Eq. (3.109), we get
H (ω) =
1
− mω 2 + icω + k
=
1/k
1 −
ω
p
2
+ i
2ζ
ω
p
(3.111)
or
H (ω) =
1/k
[1 − η 2 ] + i (2ζ η)
(3.112)
Therefore, the response of individual component of C n e
in ω t is given by
x n = H (nω) C n e
in ω t
(3.113)