General multi-DOF systems equipped with a tuned mass damper: Perturbation analysis and reduced models

A general multi- DOF system, passively controlled by a Mass Damper, tuned to one of its natural frequencies, is analyzed. The eigenvalues and eigenvectors of the controlled system are evaluated by a perturbation method based on an extrapolation from an ideal, non-physical, defective system, admitting two coincident eigenvalues, to which just one eigenvector is associated. All the small parameters of the augmented system are properly linked to a bookkeeping perturbation parameter ε, and series expansions of fractional powers of ε are used. The asymptotic series show that, similarly to what happens for the Den Hartog Oscillator, in which the structure possesses just one DOF , the multi- DOF system possesses two complex modes (revealing the non-proportional nature of damping) having close natural frequencies associated with two distinct eigenvectors, in which the main structure experiences displacements much smaller than the added mass. These two modes, suitably rendered real, are used to obtain a reduced-order model, whose two-mode basis is dictated by the perturbation analysis itself, rather than postulated, able to describe, in a simple analytical form, the Frequency Response Function of the controlled system, when excited by harmonic external forces of frequency close to the tuned frequency. The distinctive contribution of the work thus lies in reinterpreting the controlled system as a perturbation of an ideal defective one, and in grounding the modal reduction on the perturbation analysis rather than on heuristic assumptions. A low and a high shear-type multistory frame are taken as sample systems for validation purposes. Numerical results prove the accuracy of both the asymptotic expansions and the reduced-order model.