Vibration control of multi-degree-of-freedom structures using multiple tuned mass dampers and higher-order exceptional points

Abstract

Tuned mass dampers are among the most widely used devices for suppressing resonant vibration in buildings and other flexible structures, yet their behaviour on a multi-degree-of-freedom host with non-proportional damping is still only partly understood. This paper analyses a structure controlled by several tuned mass dampers as a perturbation of an ideal defective system. It shows that the double eigenvalue underlying the classical tuning is an exceptional point. We determine the Jordan structure produced by any number of absorbers and prove that exceptional behaviour arises only in the presence of damping. Algebraic conditions for an exceptional point of arbitrary order are derived, together with their coalescence variety and codimension, which fix the design freedom that a higher-order coalescence demands. A Newton and Puiseux perturbation theory then yields the fractional law by which the coalescing eigenvalues split, with a leading coefficient read from the biorthogonal Jordan chains of the quadratic pencil, and quantifies the resulting sensitivity. On this basis we construct a structure-preserving reduced-order model with a pseudospectral error bound, a branch-aware learning surrogate that represents the coalescing spectrum through its elementary symmetric functions, and a differentiable design principle whose zero set is the coalescence variety. We further quantify how amplitude-dependent mounts, manufacturing scatter, and long-term stiffness loss displace the coalescence, each of them within the same fractional law, and we obtain the spectral resolution that a monitoring system must reach before the enhanced sensitivity can be used. The theory is validated on three-storey and ten-storey shear frames and on a three-dimensional finite element model of a six-storey steel building with coupled sway and twist, and an even-handed comparison with established absorber designs is reported. The classical single-damper tuning is recovered as a limiting case.