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Analysis and results
Computational aspects
The problem is solved by the frequency response method using, as for the time domain,
- the full, coupled multi-degree-of-freedom (MDOF) system,
or
- superposition of a system of decoupled, single-degree-of-freedom (SDOF)
equations obtained by modal analysis; both eigenvectors and so-called load
dependent Ritz vectors may be used as modal coordinates.
Representation of stiffness, mass and damping is also the same as in the time domain.
The problem is to find the solution of
iΩt
Mr·· + Cr· + Kr = R ( Ω ) = R̃e
(12)
Ω is the frequency of the applied harmonic loading. We seek the particular solution of
eqn. (12), that is the so-called steady state solution
r = r̃e
iΩt
(13)
which has the same frequency as the loading. A tilde (~) on top of a symbol denotes a
complex quantity. Substituting eqn. (13) into eqn. (12) yields:
2
– Ω Mr̃ + iΩCr̃ + Kr̃ = R̃
(14)
K̃r̃ = R̃
(15)
or
where
2
K̃ = K – Ω M + iΩC
(16)
is the complex dynamic stiffness matrix. The solution of eqn. (15) gives the complex
response vector r̃ . An arbitrary response component (dof number j) may be expressed
as
r j = r j0 e
i ( Ωt + β j )
= r̃ j e
iΩt
(17)
where
r̃ j = r j0 e
iβ j
= r jR + ir jI
(18)
whose real and imaginary components are
r jR = r j0 cos β j
and
r jI = r j0 sin β j
(19)
Here
r j0 =
2
2
r jR + r jI
(20)
is the amplitude value of the response, and
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