Benchmark solution
=====================


Calculation method
------------------

The system of coupled second-order differential equations is of the form:

:math:`M\ddot{u}+Ku\mathrm{=}F`

with :math:`M\mathrm{=}\left[\begin{array}{ccc}0& 0& 0\\ 0& 10& 0\\ 0& 0& 5\end{array}\right]` and :math:`K\mathrm{=}28000\left[\begin{array}{ccc}1+0.1j& \mathrm{-}1\mathrm{-}0.1j& 0\\ \mathrm{-}1\mathrm{-}0.1j& 2+0.1j& \mathrm{-}1\\ 0& \mathrm{-}1& 1\end{array}\right]`

The :math:`\omega` solution to :math:`\mathrm{F}\mathrm{=}{\mathrm{F}}_{0}{e}^{j\omega t}({j}^{2}\mathrm{=}\mathrm{-}1)` harmonic excitation is of the form :math:`u\mathrm{=}{u}_{0}{e}^{j\omega t}`, which leads to: :math:`(\mathrm{K}\mathrm{-}\mathrm{M}{\omega }^{2}){u}_{0}\mathrm{=}{\mathrm{F}}_{0}`

This system is resolved for all :math:`\omega`.


Reference quantities and results
------------------------

Displacement along :math:`x` of point :math:`C` for some frequencies.

Reduced natural frequencies and damping.


Uncertainties about the solution
----------------------------

Semi-analytical solution.


Bibliographical references
---------------------------


1. J. PIRANDA: Instructions for using the modal analysis software MODAN - Version 0.2 (1990). Laboratory of Applied Mechanics - University of Franche Comté‑Besançon (France).