2. Reference solution#
2.1. Calculation method used for the reference solution#
The reference solution is the one given in sheet SDLD27 of guide VPCS.
The problem leads to the search for the eigenvalues and eigenvectors of the following dissipative system:
\(M\ddot{u}+C\dot{u}+Ku\mathrm{=}0\)
with \(M\) mass matrix, \(C\) damping matrix, \(K\) stiffness matrix.
This dissipative problem is associated with the conservative problem: \(Ku+M\ddot{u}\mathrm{=}0\). In harmonic form, it is written \(K\mathrm{-}{\omega }^{2}M\mathrm{=}0\).
Let \(\Lambda \mathrm{=}\mathrm{[}{\omega }_{v}^{2}\mathrm{]}\) be the spectral diagonal matrix of the eigenvalues of this conservative system and \(\phi \mathrm{=}\mathrm{[}{\varphi }_{v}\mathrm{]}\) the corresponding matrix of eigenvectors.
The \({\varphi }_{v}\) are standardized such as: \({\phi }^{T}M\phi \mathrm{=}\mathit{Id}\) \({\phi }^{T}K\phi \mathrm{=}\Lambda\).
The solutions of the dissipative system are of the form:
\(u\mathrm{=}{u}_{0}{e}^{\mathit{st}}\) from where \((M{s}^{2}+Cs+K){u}_{0}\mathrm{=}0\).
We break down \({u}_{0}\) in the base of \({\varphi }_{v}\). We then have \({u}_{0}\mathrm{=}\phi q\), from where:
\((I{s}^{2}+\gamma s+\Lambda )q\mathrm{=}0\) with \(\gamma \mathrm{=}{\phi }^{T}C\phi\) (full matrix)
This eigenvalue problem is solved by an inverse power method using \({s}_{v}\mathrm{=}j{\omega }_{v}\) as an initial estimate.
2.2. Benchmark results#
The 8 damping and frequencies specific to the system, as well as the 1st and 8th modes (complex).
2.3. Uncertainty about the solution#
Semi-analytical solution.
2.4. Bibliographical references#
PIRANDA - User manual for the modal analysis software MODAN - Version 0.2 (1990) Laboratory of Applied Mechanics - University of Franche‑Comté - Besançon (France)
Guide VPCS. Dynamic Group Supplement. September 94