1. Reminders: formulation of the problem

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

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

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