4.6 Frequency Domain Analysis
109
by using the notations
sin(θ j ) =
α i,j
α 2
i,j + β 2
i,j
and cos(θ j ) =
β i,j
α 2
i,j + β 2
i,j
(4.103)
so that θ j = tan −1 (α i,j /β i,j ). Note that the Fourier transform of the steady-state
solutions,
FT[ ˜
q ss,i ](ν) =
α i,0
2 ˜
m i ˜
k i
δ(ν) +
m
j =1
r d,i (ν j )
˜
m i ω 2
i
α 2
i,j + β 2
i,j δ(ν − ν j ),
i = 1, . . . , n,
(4.104)
and the input ˜
f (t) have energy at the same frequencies. However, the energies
associated with the input and output frequencies differ and so do the energies of
different modal coordinates.
4.6.3 Steady-State Solution: Non-proportional Damping
Consider the truncated Fourier series representation of the forcing function f(t)
given by Eqs. 4.95 and 4.96. The forcing functions of Eq. 4.88 are
v T b ˜ f(t)
i
v T
i u i
= α i,0 +
m
j =1
α i,j cos(ν j t) + β i,j sin(ν j t)
, i = 1, . . . , 2 n,
(4.105)
where
α i,0 =
v T b a 0
i
2 v T
i u i
, α i,j =
v T b a j
i
v T
i u i
, and β i,j =
v T b b j
i
v T
i u i
,
i = 1, . . . , 2 n.
(4.106)
If the real parts of the eigenvalues {λ i } are negative, the modal coordinates reach
the steady-state solution as time increases indefinitely so that for large times (see
Eq. 4.89)
q i (t) q ss,i (t) =
t
0
e
λ i (t−s)
v T b f(s)
i
v T
i u i
ds, i = 1, . . . , 2 n.
(4.107)
The steady-state system solution z(t) in Eq. 4.91 with f(t) approximated by ˜ f(t) has
the form