2. Hysteretic damping model#

The hysteretic damping model can be used to treat the harmonic responses of structures with visco-elastic materials. The hysteretic damping coefficient \(\eta\) is determined from a test under harmonic cyclic loading at pulsation \(\omega\) for which a stress-strain relationship is obtained which makes it possible to define:

  • the energy dissipated per cycle in the form of:

    \({E}_{d}={\int }_{\text{cycle}}\sigma d\varepsilon\)

the complex Young’s modulus :math:`{E}^{text{}}` based on the stress-strain relationship:

\(\sigma \mathrm{=}{\sigma }_{0}{e}^{j\omega t}\) and \(\varepsilon \mathrm{=}{\varepsilon }_{0}{e}^{j(\omega t\mathrm{-}\varphi )}\)

with the amplitudes: \({\sigma }_{0}\) and \({\varepsilon }_{0}\), the phase: \(\varphi\)

\(\text{E*}\mathrm{=}\frac{\sigma }{\varepsilon }\mathrm{=}(\frac{{\sigma }_{0}}{{\varepsilon }_{0}}){e}^{j\varphi }\mathrm{=}(\frac{{\sigma }_{0}}{{\varepsilon }_{0}})(\text{cos}\varphi +j\text{sin}\varphi )\) where \(E\text{*}\mathrm{=}{E}_{1}+j{E}_{2}\mathrm{=}{E}_{1}(1+j\eta )\)

with:

  • real part: \({E}_{1}=(\frac{{\sigma }_{0}}{{\varepsilon }_{0}})(\mathrm{cos}\varphi )\)

  • imaginary part: \({E}_{2}=(\frac{{\sigma }_{0}}{{\varepsilon }_{0}})(\mathrm{sin}\varphi )\)

  • dissipation factor: \(\eta =\frac{{E}_{1}}{{E}_{2}}=\mathrm{tan}\varphi\)

This leads to the structural dynamics equations:

\(M\ddot{u}+ K (1+j.\eta) u=f(\Omega )\)

with \(K\) true elastic stiffness matrix, \(M\) mass matrix, and \(\eta\) the hysteretic damping coefficient. Note that we often talk about a complex stiffness matrix.

This model is a simplified version of the standard visco elastic model, and has several disadvantages:

  • The model obtained cannot be transposed into the time domain, since this model would not be causal,

  • This model does not have a frequency dependence of damping, like the standard viscoelastic model

However, it is a good approximation for calculating harmonic responses in a reasonably narrow frequency band, and also has the advantage of being simple to implement.

2.1. « Global » hysteretic damping#

This modeling, which is easy to implement, corresponds to:

\((-M{\omega }^{2}+j.\eta K+K)u=f(\Omega )\)

It is currently available, using the COMB_MATR_ASSE [U4.72.01] operator, after assembling the stiffness matrix with real coefficients, but it is of low use:

  • validation of resolution algorithms,

  • useless for industrial studies, as it does not make it possible to represent the heterogeneity of the structure in relation to damping (localized dissipation in particular areas of the structure treated with visco-elastic materials).

2.2. Hysteretic damping of model elements#

2.2.1. Depreciation characteristics#

It is possible to build a complex stiffness matrix from each element of the model, just like for real stiffness and mass.

Two functionalities can be used:

the assignment of discrete elements, on POI1 or SEG2 cells, by the AFFE_CARA_ELEM [:ref:`U4.42.01<u4.42.01>`] operator. This makes it possible to define, with several possible modes of description, a**real stiffness matrix**for each degree of freedom**and* a hysteretic damping coefficient to be applied to this « AMOR_HYST » matrix.

  • the definition of a damping characteristic for any elastic material by the operator DEFI_MATERIAU [U4.43.01] by the keyword « AMOR_HYST ».

This material is then assigned to the meshes in question.

In the event that certain materials are not considered to be affected by hysteretic damping, it is necessary to assign zero hysteretic damping to them.

2.2.2. Calculation of amortization matrices#

For all types of finite elements (continuous, structural or discrete media), it is possible to calculate the complex elementary matrices corresponding to the calculation option « RIGI_MECA_HYST », after having calculated the elementary matrices corresponding to the calculation options « RIGI_MECA ». Each elementary matrix is then of the form:

  • when material \(i\), with hysteretic damping characteristics \({\eta}_{i}\), is assigned to the element

    \({k}_{elem}^{*} = {k}_{elem} (1 + j.{\eta}_{i})\)

  • for a discrete element defined by a stiffness matrix K and a hysteretic damping coefficient \(\eta\)

    \({k}_{elem}^{*} = {k}_{discret} (1 + j.{\eta})\)

This operation is possible with the operator CALC_MATR_ELEM [U4.61.01] option « RIGI_MECA_HYST ». The assembly of the complex stiffness matrix \(K\ast\), from the elementary matrices, is obtained with the usual ASSE_MATRICE operator [U4.61.22]. Note that the same numbering and the same storage mode must be used as for the mass matrix (operator NUME_DDL [U4.61.11]).

The load used to calculate the real stiffness matrix (option « RIGI_MECA ») must be filled in with the keyword « CHARGE » for the calculation of the complex elementary stiffness matrix.

2.2.3. Using the complex stiffness matrix#

The complex stiffness matrix \(K\ast\) can be used for direct linear dynamic analysis (keyword MATR_RIGI) with the linear dynamic response operator DYNA_VIBRA [U4.53.03].

In the case where the model to be taken into account for the harmonic calculation is of a large size, it may be interesting to use model reduction methods. An effective approach for highly dissipative models is available in the documentation [U2.06.04].

Searching for eigenvalues can only be done with certain settings of the eigenvalue search operator CALC_MODES [U4.52.02].

The reference documentation [R5.01.02] specifies the types of problems that can be treated and the possible settings.

This research leads to complex eigenmodes, which cannot therefore be used for calculations in the time domain.

However, in cases where the amortization introduced remains low, it is possible to calculate the real modes associated with the associated non-dissipative model, and to assign to them the depreciation rates calculated by the calculation of complex modes in order to build a reduced model on a modal basis. This system can then be used to perform a transitory calculation in the modal space with DYNA_VIBRA [U4.53.03].