8. Appendix 1. Interpreting complex eigenvalues#

In the case of symmetric damping and in the absence of internal dampening, orthogonality relationships and the fact that the eigenelements appear in conjugate pairs, we have the following relationships:

\(\begin{array}{c}\frac{{\Phi }_{i}^{{\text{*}}^{T}}\mathrm{C}{\Phi }_{i}}{{\Phi }_{i}^{{\text{*}}^{T}}\mathrm{M}{\Phi }_{i}}\mathrm{=}\frac{{c}_{i}}{{m}_{i}}\mathrm{=}2\text{Re}({\lambda }_{i})\\ \frac{{\Phi }_{i}^{{\text{*}}^{T}}\mathrm{K}{\Phi }_{i}}{{\Phi }_{i}^{{\text{*}}^{T}}\mathrm{M}{\Phi }_{i}}\mathrm{=}\frac{{k}_{i}}{{m}_{i}}\mathrm{=}{∣{\lambda }_{i}∣}^{2}\end{array}\)

If we write down \({\lambda }_{i}={\alpha }_{i}\pm i{\beta }_{i}\), we can then define

\(\begin{array}{}{\omega }_{i}=\mid {\lambda }_{i}\mid =\sqrt{{\alpha }_{i}^{2}+{\beta }_{i}^{2}}=\frac{{\beta }_{i}}{{\xi }_{i}}\\ {\xi }_{i}=\text{Re}\frac{({\lambda }_{i})}{{\omega }_{i}}=\frac{{\alpha }_{i}}{\sqrt{{\alpha }_{i}^{2}+{\beta }_{i}^{2}}}{\omega }_{i}\end{array}\)

We can write the complex eigenvalue in the following form

\({\lambda }_{i}=-{\xi }_{i}{\omega }_{i}\pm i{\omega }_{i}\sqrt{1-{\xi }_{i}^{2}}\)

This formulation induces the following physical interpretation:

  • The actual term represents the dissipative nature of the system.

  • The imaginary part represents the oscillatory part of the solution.

  • \({\omega }_{i}\) is the pulse of the \(i\) th mode.

  • \({\xi }_{i}\) is the depreciation of the \(i\) th mode,

  • \({\omega }_{{d}_{i}}={\omega }_{i}\sqrt{1-{\xi }_{i}^{2}}\) is the reduced damping of the \(i\) th mode.

As for the physical interpretation of eigenvectors:

  • The physical meaning of the existence of a complex eigenvector lies in the fact that while the structure vibrates in an eigenmode, its different degrees of freedom do not vibrate with the same phase in relation to each other.

  • The stomachs and the modal nodes do not correspond to stationary points, but move during the movement.

Notes:

  • We find the classic formulation of damped systems with 1 degree of freedom.

:math:`{k}_{i},{m}_{i}`*and:math:`{c}_{i}`* are real. They are indeed quantities intrinsic to a mode (modal quantities) and therefore dependent on its normalization. *

  • We remind you that the modes in GEP (just like the complex modes in GEP) do not diagonalize the matries \(\mathrm{M},\mathrm{K}\text{et}\mathrm{C}\mathrm{.}\)

  • If the real part of the eigenvalue is negative, then the eigenmode is a periodic damped pulsating movement \(\omega\) . If, on the contrary, it is positive, then the natural mode is a periodic movement of increasing amplitude and therefore unstable.