1. Notion of mechanical damping#
1.1. Depreciation models#
The movement of structures subjected to imposed forces or movements, which vary over time, depends, in particular, on damping properties, i.e. on the dissipation of energy in the materials constituting the structure and in the connections between the various structural elements and with the surrounding environment.
The physical phenomena involved in this energy dissipation are numerous frictions, shocks, viscosity and plasticity, and vibratory radiation at the supports.
Behavioral models representing these phenomena are often poorly understood and it is difficult to describe them explicitly at the elementary level. This is why the most used models are simple models that allow the main effects on structures to be reproduced on a macroscopic scale [bib1] [bib2]. The ones currently available are:
viscous damping: energy dissipated proportional to the speed of movement,
hysteretic damping (also called « structural damping »): energy dissipated proportional to the displacement such that the damping force is of the opposite sign to that of the speed.
Note that Coulomb damping, which corresponds to friction damping for which the energy dissipated is proportional to the reaction force normal to the direction of movement, requires modeling the contact, which goes beyond the strictly linear framework. Nonlinear operators can take it into account in all its generality [R5.03.50 & R5.03.52] while the modal transient resolution operator can model Coulomb friction in the context of point contacts [R5.06.03].
The values of the parameters of these models are deduced, when available, from experimental results. At the design stage, we are limited to the use of regulatory values.
1.2. General definitions to characterize depreciation [bib1]#
1.2.1. Loss coefficient#
The loss coefficient \(\eta\) is a dimensionless coefficient characteristic of the damping effect defined as the ratio of the energy dissipated \({E}_{d}^{\text{cycle}}\) during a cycle to the maximum potential energy \({E}_{p}^{\text{max}}\) multiplied by \(2\pi\):
1.2.2. Reduced amortization#
By definition, the reduced damping \(\xi\) is equal to half the loss coefficient observed for an excitation pulse equal to the natural pulsation of the non-damped system: