B3. Considering damping

The damping of a structure’s vibration is related to dissipation phenomena of various origins that can happen inside of a structure. It can be due to viscous phenomena specific to fluid dampers, which can be found in spring boxes or dampers (cable insulators, Jarret™ type dampers, viscous fluid dampers). It can also be caused by structural dissipation phenomena related to cracks and friction at the interfaces between materials. The energy is consumed in the behavioral hysteresis of the materials.

In most cases, the damping is considered in the dynamic calculations through a viscous damping matrix C. The components of this matrix can be exact at the local scale in the cases where the damping is completely controlled because the damping equipment used has well-known characteristics. However, most of the time, for civil engineering structures, the damping components are unknown because of the heterogeneity of the structures and the non-viscous characteristic of the materials.

It is preferable to build the damping matrix using critical damping ratios as defined in the regulation for various materials. Those ratios are defined using a viscous behavior hypothesis proportional to the solicitation velocity. It represents a fraction of the critical damping value that leads to, for a simple oscillator, a return to equilibrium without going through an oscillatory regime.

For the cases in which the damping components are known, the matrix C is constructed explicitly, and it is necessary to use a suitable temporal calculation method to respect its characteristics:

Either with a direct integration calculation: it requires discretizing the time and to solve the dynamic equilibrium at each time-step.
Either with a calculation after transformation to the modal bases, using complex vibration modes.

The analysis of rapid dynamic phenomena (explosion, impact) requires specific considerations.

It is not necessary to consider damping for local resistance verifications or vibrations over short periods, not long enough to “activate” harmonic responses of the structure. It is then not necessary for those cases to consider damping. It is interesting to note that this kind of calculations, realized with dynamic algorithms applying explicit time-dependent integration schemes (LS-Dyna, Radioss, PAM-Crash, Europlexus…), do not use the classical approach, which consists of inversing the stiffness matrix. On the contrary, for vibration analysis induced on a period following the impact duration, it is necessary to consider and characterize the damping.

Note: for rapid dynamic calculations (milliseconds), the damping component is often neglected because its order of magnitude is much smaller than the one of inertial local stiffness terms during the period of analysis

For the more “classic” cases, the damping matrix is generally constructed from the critical damping ratios defined for materials and a linear combination of the M and K matrices that were previously calculated by the FE computational model. This process enables us to considerably simplify the stages of calculation. This approach is related to the “Basile” hypothesis for modal analysis. This hypothesis was formulated to identify the structural damping (mainly in aeronautics), and is explained as follows:

Be careful! Considering the Basile hypothesis leads to an underestimation of the dynamic response elsewhere than at the edge of the materials or at the supports, which are subjected to a high damping ratio compared to a time-dependent calculation by means of direct integration, or using more complex methods such as the complex modes method. This justifies the need for direct integration calculations of the motion equations, or complex modes.

Most of the time, the Basile hypothesis is considered and the matrix is constructed starting from the Rayleigh method:

The terms α and β must be introduced. To be pragmatic, the following steps must be considered:

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f₁ is a lower bound of the first significant vibration mode of the structure, f_n, and the first mode encountered after the cut-off of the elastic response spectrum characterizing the applied temporal signal. The value of the reduced damping is calculated for all values of ω, applying the following formula:

Be careful! Since the damping is largely overestimated after the 2 frequencies of interest, it is necessary to master these two values. If fn is chosen adequately, the overestimation of the damping has no consequences because rigid body motions are found on these ranges of frequencies, which will be excited in the response, and the modes with frequencies greater than the critical frequency are insensitive to damping. However, one must be certain that there are no local vibration modes with frequencies worth considering.

The impact of an error in the choice of the first frequency can have the worst consequences if a significant mode was forgotten because its response will then be neglected.

It is important to note that in between the two frequencies, the damping is underestimated. The evaluation of the response of the structure is therefore conservative.