Reminders: formulation of the problem
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We consider a mechanical structure represented, in the context of finite element modeling, by its matrices of stiffness :math:`K`, mass :math:`M` and possibly damping :math:`C`. The equation governing the evolution of the structure is written :math:`M\ddot{x}+C\dot{x}+Kx\mathrm{=}0`.

We want to characterize the free vibrations of the mechanical structure, defined by natural frequencies :math:`{f}_{i}\mathrm{=}\frac{{\omega }_{i}}{2\pi }` (:math:`{\omega }_{i}`: natural pulsation of mode no. :math:`i`) and the associated modal deformations :math:`{x}_{i}` (and the modal damping :math:`{\zeta }_{i}` if the model contains damping).

In the absence of depreciation (the simplest and most frequent case), the modal calculation consists in finding :math:`\left\{{\omega }_{i},{x}_{i}\right\}` couples such as :math:`(K\mathrm{-}{\omega }_{i}^{2}M){x}_{i}\mathrm{=}0`.


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