3.14 Response Due to Periodic Forces
103
Fig. 3.31 Excitation and response spectra
F (t) =
∞
− ∞
C n e
in ω t
(3.105)
where ω =
2π
T
is the fundamental frequency.
The value of C n can be obtained as
C n =
1
T
T +τ
τ
F (t) e
− in ω t dt
(3.106)
Therefore,
C 0 =
1
T
T +τ
τ
F (t) dt = average value of F(t)
In this form of Fourier series, the coefficients are generally complex and for real
functions, C −n is the complex conjugate of C n .
Example 3.15 Determine the complex Fourier series expression for the square wave
of Fig. 3.30.
The coefficient of the complex Fourier series expression is determined as follows:
C n =
1
T
T /2
0
Ae
− in ω t dt +
1
T
T
T /2
− Ae
− in ω t dt
=
A
in ω T
− e
− in ω T /2
+ 1 + e
− in ω T
− e
− in ω T /2
=
A
in 2π
1 − 2e
− in π
+ e
− 2π in
Further,