Are tuned mass damper-controlled systems quasi-semi-simple or quasi-defective?

A two- DOF system, known as the Den Hartog Oscillator , made of a single- DOF, representative of a main structure, equipped with a Tuned Mass Damper , is analyzed as a prototype of passively controlled structures. The eigenvalues and eigenvectors of the system are asymptotically determined according to two different procedures. The first one, known in literature, takes as starting point of the integer series expansions a double semi-simple eigenvalue (i.e., admitting two distinct eigenvectors). The second, which is believed to be original, starts the expansion from a double defective eigenvalue (i.e., admitting just one eigenvector), and therefore is carried out in terms of fractional series expansions. The two approaches are found to lead to the same asymptotic results; however, the newer one is considered to be conceptually superior (and therefore more easily exported in other similar problems), since the fractional dependence on parameters, which is the main reason of the good performance of the Tuned Mass Damper, has not to be guessed, but naturally appears in the treatment. Moreover, the underlying quasi-defective nature of the Den Hartog Oscillator explains why the oscillation amplitude of the Tuned Mass Damper is larger than that of the main structure, since the modes are perturbations of a unique mode, in which just the added mass moves, while the structure is at rest.