4.6 Frequency Domain Analysis
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4.6 Frequency Domain Analysis
We extend developments for SDOF systems related to the Fourier series representations to MDOF systems with proportional and non-proportional damping. First, we
construct Fourier series for vector-valued forcing functions.
4.6.1 Fourier Series Representation of Vector-Valued Forcing
Functions
Suppose that the applied force f(t) is a periodic vector-valued function with period
τ , i.e., f(t) = f(t + τ ) for any t in the domain of definition of f(t). The components
{f (r) (t)} of this n-dimensional function admit the representation (see Eq. 2.52)
f
(r) (t) =
a
(r)
0
2
+
∞
j =1
a
(r)
j cos(ν j t) + b
(r)
j sin(ν j t)
, r = 1, . . . , n,
(4.94)
where ν 1 = 2 π/τ and ν j = j ν 1 , j = 0, 1, 2, . . ., are multiple of ν 1 .
The coefficients of the above representation of the real-valued functions {f (r) (t)}
result from Eq. 2.53 with f (r) (t) in place of f (t). For calculations, the Fourier series
representations of Eq. 4.94 are truncated by retaining the top m terms so that f (r) (t)
is approximated by
f
(r) (t) ˜
f
(r) (t) =
a
(r)
0
2
+
m
j =1
a
(r)
j cos(ν j t) + b
(r)
j sin(ν j t)
,
r = 1, . . . , n,
(4.95)
and
f(t) ˜ f(t) =
1
2
a 0 +
m
j =1
a j cos(ν j t) + b j sin(ν j t)
,
(4.96)
where a 0 , a j , and b j , j = 1, . . . , m, are n-dimensional column vectors with
components {a
(r)
0 }, {a
(r)
j }, and {b
(r)
j }. Note that, the components of the truncated
version ˜ f(t) of the periodic forcing function f(t) have energy at the same frequencies,
the discrete frequencies {ν j }, j = 0, 1, . . . , m. Generally, the energy associated with
these frequencies differs for different components of ˜ f(t). The Fourier transforms of
the components of ˜ f(t) have the form in Eq. 2.57.
The following subsections find the response of MDOF systems with proportional
and non-proportional damping subjected to vector-valued forcing functions represented by truncated Fourier series.