Asymptotic analysis of multiple mode structures equipped with multiple tuned mass dampers

The design of a mitigation dissipative system composed of several tuned mass dampers (TMDs) is not a simple task. As soon as several structural modes or several additional degrees-of-freedom are considered, the analytical response of the coupled system is not tractable anymore. The generic problem involving any number of structural modes and any number of attached tuned mass dampers is considered in this paper. The appropriate re-scaling of the problem is formulated in order to highlight the governing scales in the problem. An asymptotic analysis in the frequency domain, similar to the multiple timescale spectral analysis, is developed. The resulting expression for the frequency response function of the coupled system is expressed in a lower dimensional space, which offers the possibility to derive simple and accurate analytical solutions in other cases than the well-known single-mode single-TMD case. Although the major contribution of this paper consists in the derivation of this analytical formulation, two illustrations are also provided. The first one concerns the well-known single-mode single-TMD case, which illustrates the benefits of the proposed formulation and validates the resulting approximations against established literature. The second example concerns a singlemode double-TMD, where the proposed approximation is used to discuss the accuracy with which the two damping units need to be tuned in order to provide reasonable optimality with respect to mitigation performances.