6. Using the depreciation matrix#

6.1. Use of the viscous damping matrix#

6.1.1. Direct linear dynamic analysis#

The viscous damping matrix \(C\), regardless of how it is developed and whether it is proportional or not proportional, can be used for direct linear dynamic analysis (keyword MATR_AMOR) with the DYNA_VIBRA operator.

6.1.2. Dynamic analysis by modal recombination#

For modal recombination analyses, we must project this matrix into the subspace defined by a set \(\phi\) of real eigenmodes, obtained on the associated undamped problem \((K-{\omega }^{2}M)\phi =0\).

This operation is possible with the PROJ_BASE [U4.55.11] command or with the PROJ_MATR_BASE [U4.55.01] operator.

For the calculation of the dynamic response in force or in the movement imposed in the modal space, the following possibilities are available:

  • use of the generalized amortization matrix \({\phi }^{T}C\phi\):

    • in transitory analysis with the operator DYNA_VIBRA [R5.06.04] and [U4.54.03] and the keyword AMOR_GENE,

    • in seismic analysis by spectral method with the operator COMB_SISM_MODAL [R4.05.03] and [U4.54.04] and the keyword AMOR_GENE,

    • in harmonic analysis with the operator DYNA_VIBRA [R5.05.03] and [U4.54.03] and the keyword MATR_AMOR.

Recall that in the case of heterogeneous depreciation (use of localized depreciation options), the \({\phi }^{T}C\phi\) matrix is not a diagonal step.

  • use of viscous modal damping by providing constant reduced modal damping for all \(\xi\) modes or a list of \({\xi }_{i}\) values.

Several methods of identifying these coefficients are possible but there is no command for the systematic construction of the list of values. However, we can cite the use of the Basile hypothesis \((2{\xi }_{i}{\omega }_{i}=\text{diag}\frac{{\phi }^{T}C\phi }{{\phi }^{T}M\phi })\), the use of the RCC -G (or ETC -C) regulation for seismic analysis with ground damping, exploitation of experimental results,…

    • in transitory and harmonic analysis with the operator DYNA_VIBRA [R5.06.04] [U4.54.03] and the keyword AMOR_REDUIT.

    • in seismic analysis by spectral modal method with the operator COMB_SISM_MODAL [R4.05.03] [U4.54.04] with the [] [] operator and the key words AMOR_GENE or LIST_AMOR, or also with the keyword AMOR_REDUIT.

For analyses by dynamic substructuring, with the use of a modal base (Ritz base), refer to [R4.06.03] and [R4.06.04].

6.2. Use of the complex stiffness matrix#

The complex stiffness matrix \({K}^{\text{*}}\mathrm{=}K+{K}_{h}\), where \({K}_{h}\) is an imaginary matrix (in the sense of complexes!) , can be used for direct harmonic analysis with the operator DYNA_VIBRA [R5.05.03] and [U4.54.03] and the keyword MATR_RIGI.

For modal recombination analyses, no functionality is currently available for the use of the hysteretic damping model.

6.3. Complex modal analysis#

The viscous damping matrix \(C\) is indispensable for complex modal analysis with the operator dealing with the quadratic problem at eigenvalues [R5.01.02] and CALC_MODES [U4.52.02].

Recall that complex eigenmodes allow an approach better suited to the dynamic study of highly damped structures (reduced damping \(\xi\) > 20%). No dynamic response tool by modal recombination using a complex eigenmode base is available.