4. Formulation of standardization rules#

The various standards used as well as the definition of the various modal parameters are listed in the reference documentation [R5.01.03].

4.1. Real eigenmodes#

For modes_meca_r (real eigenmodes) the associated generalized eigenvalue problem is: \((K\mathrm{-}{\omega }^{2}M)x\mathrm{=}(K\mathrm{-}{(2\pi f)}^{2}M)x\mathrm{=}0\)

where \(K,M\) are respectively the mass matrix and the stiffness matrix of the mechanical system.

For the “MECANIQUE” models, we define the components of the eigenvector:

  • translation components \({u}^{T}\)

  • rotation components \({u}^{R}\)

  • components of the multipliers of LAGRANGE \(\lambda\)

  • other components (pressure and fluid potential) \({p}_{f}\)

We call:

  • \({u}^{\mathrm{TR}}\) translation and rotation components,

  • \(u\) components other than LAGRANGE multipliers.

which leads to:

\({u}^{\text{*}}=\left[\begin{array}{}u\\ \lambda \end{array}\right]=\left[\begin{array}{}{u}^{T}\\ {u}^{R}\\ {p}_{f}\\ \lambda \end{array}\right]\)

For models with translation and rotation components, the \({\phi }_{i}\) eigenmode provided by the modal analysis algorithms is by default:

\({\Phi }_{i}\mathrm{=}\frac{{u}^{\text{*}}}{\mathit{max}u}\mathrm{=}\frac{{u}^{\text{*}}}{\mathit{max}{u}^{\mathit{TR}}}\mathrm{=}{{\Phi }_{i}}^{\mathit{TR}}\)

which is equivalent to the normalization obtained by the “TRAN_ROTA” keyword.

With the keyword “TRAN” the mode obtained is defined by:

\({\Phi }_{i}\mathrm{=}\frac{{u}^{\text{*}}}{\mathit{max}{u}^{T}}\mathrm{=}{{\Phi }_{i}}^{T}\)

For models with translation components only, normalization is by default:

\({{\Phi }_{i}}^{T}\mathrm{=}\frac{{u}^{\text{*}}}{\mathit{max}u}\mathrm{=}\frac{{u}^{\text{*}}}{\mathit{max}{u}^{T}}\)

which is equivalent to the normalization obtained by the “TRAN” keyword.

Normalization by default leads to the following generalized parameters:

  • generalized stiffness \({}^{T}\Phi _{i}K{\Phi }_{i}\mathrm{=}{\gamma }_{i}\)

  • generalized mass \({}^{T}\Phi _{i}M{\Phi }_{i}\mathrm{=}{\mu }_{i}\)

  • Hence the natural pulsation \({{\omega }_{i}}^{2}\mathrm{=}\frac{{\gamma }_{i}}{{\mu }_{i}}\)

Normalization to the generalized unit mass is obtained by the keyword “MASS_GENE”:

\({{\Phi }_{i}}^{M}\mathrm{=}\frac{{\Phi }_{i}}{\sqrt{{\mu }_{i}}}\) from where \({}^{T}{\Phi }_{i}^{M}M{{\Phi }_{i}}^{M}\mathrm{=}1.\) and \({}^{T}{\Phi }_{i}^{M}K{{\Phi }_{i}}^{M}\mathrm{=}{{\omega }_{i}}^{2}\)

The one with generalized unitary rigidity is obtained by the keyword “RIGI_GENE”:

\({{\Phi }_{i}}^{K}\mathrm{=}\frac{{\Phi }_{i}}{\sqrt{{\gamma }_{i}}}\) from where \({}^{T}{\Phi }_{i}^{K}M{{\Phi }_{i}}^{K}\mathrm{=}\frac{1}{{{\omega }_{i}}^{2}}\) and \({}^{T}{\Phi }_{i}^{K}K{{\Phi }_{i}}^{K}\mathrm{=}1.\)

The normalization of the mode specific to the Euclidean norm “EUCL” is obtained naturally by:

\({{\Phi }_{i}}^{\mathrm{\parallel }u\mathrm{\parallel }}\mathrm{=}\frac{{u}^{\text{*}}}{\mathrm{\parallel }u\mathrm{\parallel }}\mathrm{=}\frac{{u}^{\text{*}}}{\sqrt{\mathrm{\sum }_{j}{({u}_{j})}^{2}}}\)

The normalization of the mode specific to the Euclidean norm “EUCL_TRAN” is:

\({{\Phi }_{i}}^{\mathrm{\parallel }{u}^{T}\mathrm{\parallel }}\mathrm{=}\frac{{u}^{\text{*}}}{\mathrm{\parallel }{u}^{T}\mathrm{\parallel }}\mathrm{=}\frac{{u}^{\text{*}}}{\sqrt{\mathrm{\sum }_{j}{({u}_{j}^{T})}^{2}}}\)

4.2. Complex eigenmodes#

For modes_meca_c modes (complex natural modes) resulting from a resolution of a quadratic problem with eigenvalues \({\lambda }^{2}M+\lambda C+K\mathrm{=}0\) where \(C\) is the damping matrix of the mechanical system, the \(\Phi\) modes are normalized in relation to the associated linearized problem:

\((\lambda \left[\begin{array}{cc}0& M\\ M& C\end{array}\right]+\left[\begin{array}{cc}\mathrm{-}M& 0\\ 0& K\end{array}\right])(\begin{array}{c}\lambda \Phi \\ \Phi \end{array})\mathrm{=}0\)

The eigenmode is normalized to the generalized unit mass (“MASS_GENE”), if \({\Phi }_{i}\) meets:

\((\lambda {}^{T}\Phi _{i}{}^{T}\Phi _{i})\left[\begin{array}{cc}0& M\\ M& C\end{array}\right](\begin{array}{}\lambda {\Phi }_{i}\\ {\Phi }_{i}\end{array})=1.\)

to the unitary generalized stiffness (“RIGI_GENE”), if \({\Phi }_{i}\) meets:

\((\lambda {}^{T}\Phi _{i}{}^{T}\Phi _{i})\left[\begin{array}{cc}-M& 0\\ 0& K\end{array}\right](\begin{array}{}\lambda {\Phi }_{i}\\ {\Phi }_{i}\end{array})=1.\)

For the other standards, the definitions are equivalent to those defined for real modes, simply replace the dot product by the Hermitian product.