34
2 Single Degree of Freedom (SDOF) Systems
where δ ij = 1 for i = j and zero otherwise. The coefficients of Eq. 2.53 are
called the Fourier coefficients of f (t) and the representation of Eq. 2.52 with these
coefficients is referred to as the Fourier series of f (t).
Consider the truncated version
f n (t) =
a 0
2
+
n
i=1
a i cos(ν i t) + b i sin(ν i t)
, n = 1, 2, . . . ,
(2.55)
of the Fourier series of f (t) which is used in numerical calculations. The accuracy of
this representation of f (t) depends on the truncation level. The following paragraph
provides some technical comments on the accuracy of f n (t) and the validity of the
calculations in Eq. 2.53 involving term by term integration. The reader may skip
these comments unless interested in technicalities.
It can be shown that f n converges uniformly to f on [0, τ ], i.e., max 0≤t≤τ |f n (t)−
f (t)| becomes small and remains small from an index n on, i.e., given ε > 0,
there is n ε such that max 0≤t≤τ |f n (t) − f (t)| ≤ ε for n ≥ n ε . If f (t) has
discontinuities, f n (t) converges to the arithmetic mean of the right and left limits
f (t+) − f (t−)
/2 of f (t) at its discontinues points [4] (Sect. 1.10). It can
also be shown that the term by term integration which allows us to calculate the
Fourier coefficient of the Fourier series of f (t) is valid for continuous, piecewise
differentiable functions [4] (Sect. 3.10).
Example 2.13 The solid lines in Fig. 2.12 are the graph of a target function f (t)
with the support [0, τ = 10]. The function is continuous but is not periodic in this
time interval as f (0) = f (τ ). The top left, top right, bottom left, and bottom right
panels show with dashed lines the truncated Fourier series representations f n (t) for
n = 0, n = 5, n = 10, and n = 30. Visual inspection suggests that the accuracy
of f n (t) improves with n. As expected, the representations f n (t) are in error at the
ends of the interval [0, τ ] since f (t) is not periodic.
Considerations as in Sect. 2.4.2.2 can be used to construct the following
amplitude-phase representation of f n (t)
f n (t) =
a 0
2
+
n
i=1
a 2
i + b 2
i sin(ν i t + ϕ i ), n = 1, 2, . . . ,
(2.56)
where tan(ϕ i ) = a i /b i . This follows from the equalities
a 2
i + b 2
i
⎛
⎝
a i
a 2
i + b 2
i
cos(ν i t) +
b i
a 2
i + b 2
i
sin(ν i t)
⎞
⎠
=
a 2
i + b 2
i
sin(ϕ i ) cos(ν i t) + cos(ϕ i ) sin(ν i t)
=
b i
a 2
i + b 2
i
sin(ν i t + ϕ i ),
2 Single Degree of Freedom (SDOF) Systems
where δ ij = 1 for i = j and zero otherwise. The coefficients of Eq. 2.53 are
called the Fourier coefficients of f (t) and the representation of Eq. 2.52 with these
coefficients is referred to as the Fourier series of f (t).
Consider the truncated version
f n (t) =
a 0
2
+
n
i=1
a i cos(ν i t) + b i sin(ν i t)
, n = 1, 2, . . . ,
(2.55)
of the Fourier series of f (t) which is used in numerical calculations. The accuracy of
this representation of f (t) depends on the truncation level. The following paragraph
provides some technical comments on the accuracy of f n (t) and the validity of the
calculations in Eq. 2.53 involving term by term integration. The reader may skip
these comments unless interested in technicalities.
It can be shown that f n converges uniformly to f on [0, τ ], i.e., max 0≤t≤τ |f n (t)−
f (t)| becomes small and remains small from an index n on, i.e., given ε > 0,
there is n ε such that max 0≤t≤τ |f n (t) − f (t)| ≤ ε for n ≥ n ε . If f (t) has
discontinuities, f n (t) converges to the arithmetic mean of the right and left limits
f (t+) − f (t−)
/2 of f (t) at its discontinues points [4] (Sect. 1.10). It can
also be shown that the term by term integration which allows us to calculate the
Fourier coefficient of the Fourier series of f (t) is valid for continuous, piecewise
differentiable functions [4] (Sect. 3.10).
Example 2.13 The solid lines in Fig. 2.12 are the graph of a target function f (t)
with the support [0, τ = 10]. The function is continuous but is not periodic in this
time interval as f (0) = f (τ ). The top left, top right, bottom left, and bottom right
panels show with dashed lines the truncated Fourier series representations f n (t) for
n = 0, n = 5, n = 10, and n = 30. Visual inspection suggests that the accuracy
of f n (t) improves with n. As expected, the representations f n (t) are in error at the
ends of the interval [0, τ ] since f (t) is not periodic.
Considerations as in Sect. 2.4.2.2 can be used to construct the following
amplitude-phase representation of f n (t)
f n (t) =
a 0
2
+
n
i=1
a 2
i + b 2
i sin(ν i t + ϕ i ), n = 1, 2, . . . ,
(2.56)
where tan(ϕ i ) = a i /b i . This follows from the equalities
a 2
i + b 2
i
⎛
⎝
a i
a 2
i + b 2
i
cos(ν i t) +
b i
a 2
i + b 2
i
sin(ν i t)
⎞
⎠
=
a 2
i + b 2
i
sin(ϕ i ) cos(ν i t) + cos(ϕ i ) sin(ν i t)
=
b i
a 2
i + b 2
i
sin(ν i t + ϕ i ),