Vibration suppression in ultrasonic machining described by non-linear differential equations via passive controller

In the present work, the coupling of two non-linear oscillators of the main system and absorber representing ultrasonic cutting process is investigated. This leads to a two-degree-of-freedom system subjected to multi-external excitation force. The aim of this work is to control the main system behavior at simultaneous primary and internal resonance condition, where the system damage is probable. Multiple scale perturbation method is applied to obtain the solution up to the second order approximations. The different resonance cases are reported and studied numerically. The stability of the system is investigated applying both phase-plane and frequency response techniques. The effects of the different parameters of the absorber on the system behavior are studied numerically. Comparison with the available published work is reported.