2.5 Frequency Domain Analysis
33
by the steady-state solution, i.e., x(t) x ss (t) = x p (t), for sufficiently large times.
The focus of the frequency domain analysis is on the steady-state solution.
We first represent the forcing functions f (t) by sums of harmonics of different
frequencies and amplitudes and then construct similar representations for the
oscillator steady-state displacement. We show that the input f (t) and the output
x ss (t) have the same frequencies but different amplitudes. The output amplitudes
are controlled by DAFs.
2.5.1 Fourier Series Representation of Forcing Functions
This section provides a brief review of essentials on Fourier series which are needed
to find the response of SDOF systems in the frequency domain. Detailed technical
considerations can be found in [4] (Chaps. 1–3).
Suppose that the applied force f (t) is a periodic function with period τ , i.e.,
f (t) = f (t + τ ) for any t in the domain of definition of f (t), and consider the
representation
f (t) =
a 0
2
+
∞
i=1
a i cos(ν i t) + b i sin(ν i t)
,
(2.52)
where ν 1 = 2 π/τ and ν i = i ν 1 , i = 1, 2, . . ., are multiple of the fundamental
frequency ν 1 .
To determine the coefficients (a i , b i ) in Eq. 2.52, we assume that the series
representation of f (t) can be integrated term by term. Under this assumption, the
integrals over [0, τ ] of the both sides of Eq. 2.52 multiplied by 1, cos(ν j t) and
sin(ν j t), j = 1, 2, . . ., give
a 0 =
2
τ
τ
0
f (t) dt,
a j =
2
τ
τ
0
f (t) cos(ν j t) dt, j ≥ 1
b j =
2
τ
τ
0
f (t) sin(ν j t) dt j = 1, 2, . . . ,
(2.53)
by using the orthogonality of the trigonometric functions, i.e.,
τ
0
cos(ν i t) cos(ν j t) dt =
τ
2
δ ij ,
τ
0
sin(ν i t) sin(ν j t) dt =
τ
2
δ ij ,
τ
0
sin(ν i t) cos(ν j t) dt = 0,
(2.54)
33
by the steady-state solution, i.e., x(t) x ss (t) = x p (t), for sufficiently large times.
The focus of the frequency domain analysis is on the steady-state solution.
We first represent the forcing functions f (t) by sums of harmonics of different
frequencies and amplitudes and then construct similar representations for the
oscillator steady-state displacement. We show that the input f (t) and the output
x ss (t) have the same frequencies but different amplitudes. The output amplitudes
are controlled by DAFs.
2.5.1 Fourier Series Representation of Forcing Functions
This section provides a brief review of essentials on Fourier series which are needed
to find the response of SDOF systems in the frequency domain. Detailed technical
considerations can be found in [4] (Chaps. 1–3).
Suppose that the applied force f (t) is a periodic function with period τ , i.e.,
f (t) = f (t + τ ) for any t in the domain of definition of f (t), and consider the
representation
f (t) =
a 0
2
+
∞
i=1
a i cos(ν i t) + b i sin(ν i t)
,
(2.52)
where ν 1 = 2 π/τ and ν i = i ν 1 , i = 1, 2, . . ., are multiple of the fundamental
frequency ν 1 .
To determine the coefficients (a i , b i ) in Eq. 2.52, we assume that the series
representation of f (t) can be integrated term by term. Under this assumption, the
integrals over [0, τ ] of the both sides of Eq. 2.52 multiplied by 1, cos(ν j t) and
sin(ν j t), j = 1, 2, . . ., give
a 0 =
2
τ
τ
0
f (t) dt,
a j =
2
τ
τ
0
f (t) cos(ν j t) dt, j ≥ 1
b j =
2
τ
τ
0
f (t) sin(ν j t) dt j = 1, 2, . . . ,
(2.53)
by using the orthogonality of the trigonometric functions, i.e.,
τ
0
cos(ν i t) cos(ν j t) dt =
τ
2
δ ij ,
τ
0
sin(ν i t) sin(ν j t) dt =
τ
2
δ ij ,
τ
0
sin(ν i t) cos(ν j t) dt = 0,
(2.54)