2. Benchmark solution#
2.1. Calculation method#
The system of coupled second-order differential equations is of the form:
\(M\ddot{u}+Ku\mathrm{=}F\)
with \(M\mathrm{=}\left[\begin{array}{ccc}0& 0& 0\\ 0& 10& 0\\ 0& 0& 5\end{array}\right]\) and \(K\mathrm{=}28000\left[\begin{array}{ccc}1+0.1j& \mathrm{-}1\mathrm{-}0.1j& 0\\ \mathrm{-}1\mathrm{-}0.1j& 2+0.1j& \mathrm{-}1\\ 0& \mathrm{-}1& 1\end{array}\right]\)
The \(\omega\) solution to \(\mathrm{F}\mathrm{=}{\mathrm{F}}_{0}{e}^{j\omega t}({j}^{2}\mathrm{=}\mathrm{-}1)\) harmonic excitation is of the form \(u\mathrm{=}{u}_{0}{e}^{j\omega t}\), which leads to: \((\mathrm{K}\mathrm{-}\mathrm{M}{\omega }^{2}){u}_{0}\mathrm{=}{\mathrm{F}}_{0}\)
This system is resolved for all \(\omega\).
2.2. Reference quantities and results#
Displacement along \(x\) of point \(C\) for some frequencies.
Reduced natural frequencies and damping.
2.3. Uncertainties about the solution#
Semi-analytical solution.
2.4. Bibliographical references#
PIRANDA: Instructions for using the modal analysis software MODAN - Version 0.2 (1990). Laboratory of Applied Mechanics - University of Franche Comté‑Besançon (France).