1. Viscous damping model#

The viscous damping model is the most commonly used. It corresponds to the modeling of a dissipated energy proportional to the vibratory speed:

\({E}_{d}=\frac{1}{2}{v}^{T}\text{Cv}=\frac{1}{2}{u}^{T}\text{Cu}\)

where \(C\) is the viscous damping matrix, with real coefficients.

It leads to the classical equations of structural dynamics:

\(\text{Mu}+\text{Cu}+\text{Ku}=f(t)\)

with \(K\) stiffness matrix and \(M\) mass matrix.

1.1. Proportional « global » viscous damping, or Rayleigh damping#

This modeling, which is easy to implement, corresponds to a linear combination of mass and stiffness matrices:

\(C=\alpha K+\beta M\)

It is currently available, using the operators DEFI_MATERIAU [U4.43.01 §3.1] and ASSEMBLAGE (OPTION =” AMOR_MECA “) [U4.61.21]. We can also use COMB_MATR_ASSE [U4.72.01], after assembling the stiffness and mass matrices with real coefficients. The option SANS_CMP =” LAGR “must be used when combining matrices in order to preserve the boundary conditions of the system (imposed by Lagrange relationships in the stiffness matrix).

This approach allows the validation of resolution algorithms,

It is not realistic for industrial studies, because it does not make it possible to represent the heterogeneity of the structure in relation to damping (dissipation at supports or assemblies). In addition, the global identification of the coefficients \(\alpha\) and \(\beta\) is only possible, in experimental modal analysis, for two distinct natural frequencies \(\left[{f}_{1},{f}_{2}\right]\); it gives, for the natural frequencies \(\omega \notin \left[{\omega}_{1},{\omega}_{2}\right]\) with \(\omega =2\pi f\), a law of evolution of the reduced damping of the form (see [R5.05.04]):

\(2\xi =\alpha \omega +\frac{\beta}{\omega}\)

For discrete elements, since the Rayleigh damping parameters cannot be defined in the operator AFFE_CARA_ELEM, if one wishes to take into account the contribution of these elements in the overall Rayleigh damping matrix, it is mandatory to go through the method of assembling this matrix with the operator COMB_MATR_ASSE. Without doing so, the overall damping matrix will only take into account the contribution of solid, surface, or beam-type elements. If the operator DYNA_NON_LINE is used, Rayleigh damping therefore does not take into account the contribution of discrete elements.

1.2. Viscous damping proportional to the elements of the model#

1.2.1. Depreciation characteristics#

It is possible to build a damping matrix from each element of the model, as for stiffness and mass.

Several functionalities can be used:

the assignment of discrete elements, on POI1 or SEG2 cells, by the AFFE_CARA_ELEM [U4.42.01] operator. This makes it possible to define, with several possible modes of description, a damping matrix for each degree of freedom.
the definition of a damping characteristic for any elastic material by the operator DEFI_MATERIAU [U4.43.01] by entering the keywords « AMOR_ALPHA »

This material is then assigned to the meshes in question.

1.2.2. Calculation of amortization matrices#

For all types of finite elements (continuous, structural or discrete media), it is possible to calculate the real elementary matrices corresponding to the calculation option « AMOR_MECA », after having calculated the elementary matrices corresponding to the calculation options « RIGI_MECA » and « » and « MASS_MECA » and « » or « MASS_MECA_DIAG », the elementary matrices are then of the form:

when material \(i\), with \(({\alpha}_{i},{\beta}_{i})\) proportional viscous damping characteristics, is assigned to an element \(C_{elem} = \alpha_{i} K_{elem} + \beta_{i} M_{elem}\)
for a discrete element \(C_{elem} = A_{elem}\). When the discrete is of the type « DIS_CHOC » (therefore defined with a penalization stiffness), it is also possible to assign a damping to it that will only be taken into account when there is contact [R5.03.17].

This operation is possible with the CALC_MATR_ELEM [U4.61.01] command with the « AMOR_MECA » option.

The assembly of all the elementary damping matrices is obtained with the usual ASSE_MATRICE operator [U4.61.22]. It should be noted that the same numberings and the same storage mode must be used as for the stiffness and mass matrices (operator NUME_DDL [U4.61.11]). It can also be recalled that the use of the macro-command ASSEMBLAGE [U4.61.21] makes it possible to combine these steps advantageously.

Note that the damping matrix obtained may not be proportional: \(C\ne \alpha K+\beta M\)

1.2.3. Use of the viscous damping matrix#

The matrix \(C\) is:

usable for direct linear dynamic analysis (keyword MATR_AMOR) with linear dynamic response operators.
essential for complex modal analysis with the eigenvalue search operator.

For modal-based analyses, we must project this matrix into the subspace defined by a set \(\Phi\) of real eigenmodes. This operation is possible with the operator PROJ_MATR_BASE [U4.63.12]. Note that in the general case (\(C\) not proportional), the projected matrix is not diagonal. However, it remains usable (keyword AMOR_GENE) for calculating the dynamic response in force or in imposed motion in the modal space, with the linear dynamic response operator:

Transient, Harmonic: DYNA_VIBRA [U4.53.03]
Complex modal analysis: CALC_MODES [U4.52.02]