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7 Forced Vibration Analysis of Multiple Degrees of Freedom System
7.10 Complex Matrix Inversion Method for Forced
Vibration Analysis of MDF Systems
The external force acting on a system is of the following form:
{F(t)} =
F
e
iωt
(7.71)
The equations of motion for MDF system with damping then become
[M]{ ¨
x} + [C]{ ˙
x} + [K ]{x} =
F
e
iωt
(7.72)
The steady-state solution of Eq. (7.72) is
{x} = {X}e
iωt
(7.73)
Therefore
{ ˙
x} = iω{X}e
iωt
= iω{X}
(7.74)
and
{ ¨
x} = −ω
2
{x}
(7.75)
Combining Eqs. (7.72) to (7.75), we get
−ω
2 [M] + iω[C] + [K ]{x}
= {F}e
iωt
(7.76)
Equation (7.76) can be written as
[A]{x} = {F}e
iωt
(7.77)
where matrix [A] is a complex matrix of order n × n.
Let
[A] =
A
+ i
A
(7.78)
where [A
] and [A
] are real square matrices of same order as that of [A].
In the present case
[A
] = Real[A] = [K ] − ω
2
[M]
(7.79)
[A
] = Im[A] = ω[C]
(7.80)