On the stability of a tuned vibration absorber for time varying multiple frequencies

Abstract: A class of actively tuned vibration absorbers is presented. The absorber substructure is controlled using an unconventional position feedback with multiple, unrelated time delays. This control sensitizes the absorber at a number of time-varying frequencies concurrently. It converts the absorber into a multi-frequency resonator (the Multiple Frequency Delayed Resonator or MFDR). The resonator, in turn, acts as a perfect absorber of vibration at its resonance frequencies when it is attached to a harmonically excited primary system. This tuning scheme offers some practical benefits. The highlight of the paper is on the stability analysis of such systems with multiple, unrelated time delays. A new strategy to achieve this, the Directional Stability Chart Method (DSCM), is presented, which is based on a one-dimensional search. A complementary method is also explained, the Local Stability Radius Method (LSRM), which results in a regional stability understanding. Examples are given to demonstrate the capabilities of these methods. Tunability feature to the time varying frequencies is also shown through simulations. A single degree of freedom (SDOF) primary system and a two-degrees-of-freedom (TDOF) absorber are considered for dual frequency suppression as a case study