Vibration suppression of non-linear system via non-linear absorber

In this paper, the coupled non-linear differential equations of the non-linear dynamical two-degree-of-freedom vibrating system including quadratic and cubic non-linearities are studied. The system consists of the main system and the absorber. The absorber is used to control the main system vibrations when subjected to multi-external excitation forces at simultaneous primary and internal resonance. This system represents many applications in machine tools, ultrasonic cutting process, etc. The method of multiple scales perturbation technique (MSPT) is applied throughout to determine the solution up to third order approximations. The different resonance cases are reported and studied numerically. Stability is studied applying frequency response functions. The effects of different parameters of the system are studied numerically. Optimum working conditions for the absorber where obtained at internal resonance ratio 1:3. This means smaller mass for the absorber which solves the problem of space limitation. A comparison is made with the available published work.