Reference solution
=====================


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:

:math:`M\ddot{u}+C\dot{u}+Ku\mathrm{=}0`

with :math:`M` mass matrix, :math:`C` damping matrix, :math:`K` stiffness matrix.

This dissipative problem is associated with the conservative problem: :math:`Ku+M\ddot{u}\mathrm{=}0`. In harmonic form, it is written :math:`K\mathrm{-}{\omega }^{2}M\mathrm{=}0`.

Let :math:`\Lambda \mathrm{=}\mathrm{[}{\omega }_{v}^{2}\mathrm{]}` be the spectral diagonal matrix of the eigenvalues of this conservative system and :math:`\phi \mathrm{=}\mathrm{[}{\varphi }_{v}\mathrm{]}` the corresponding matrix of eigenvectors.

The :math:`{\varphi }_{v}` are standardized such as: :math:`{\phi }^{T}M\phi \mathrm{=}\mathit{Id}` :math:`{\phi }^{T}K\phi \mathrm{=}\Lambda`.

The solutions of the dissipative system are of the form:

:math:`u\mathrm{=}{u}_{0}{e}^{\mathit{st}}` from where :math:`(M{s}^{2}+Cs+K){u}_{0}\mathrm{=}0`.

We break down :math:`{u}_{0}` in the base of :math:`{\varphi }_{v}`. We then have :math:`{u}_{0}\mathrm{=}\phi q`, from where:

:math:`(I{s}^{2}+\gamma s+\Lambda )q\mathrm{=}0` with :math:`\gamma \mathrm{=}{\phi }^{T}C\phi` (full matrix)

This eigenvalue problem is solved by an inverse power method using :math:`{s}_{v}\mathrm{=}j{\omega }_{v}` as an initial estimate.


Benchmark results
----------------------

The 8 damping and frequencies specific to the system, as well as the 1st and 8th modes (complex).


Uncertainty about the solution
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Semi-analytical solution.


Bibliographical references
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* J. 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