Steady state dynamics of a non-linear beam coupled to a non-linear energy sink

•A non-linear beam under harmonic excitation coupled to a NES is studied.•The NES has a non-linear stiffness of order three.•Complex averaging method combined with arc-length continuation is used.•Bifurcation analysis and Poincare׳ sections are used to study the effect of NES.•Robustness of NES against changes in the amplitude of excitation is investigated.•Also the performance of NES is compared with linear vibration absorber.

A non-linear, simply supported beam under harmonic excitation coupled to a non-linear energy sink (NES) is considered here. The NES has a non-linear stiffness of order three. Steady state dynamic of the beam is investigated by two different theories of Euler–Bernoulli and Timoshenko. Complex averaging method combined with arc-length continuation is used to achieve an approximate solution for the steady state vibrations of the system based on 1:1 resonance condition. In order to design an optimized NES for the purpose of reducing the vibration amplitude of the beam, the effect of NES parameters on the amplitude of the primary system is investigated by varying the parameters, individually. The results demonstrated a significant reduction in the vibration amplitude of the original system. By illustrating the frequency spectrum, other harmonic components are detected and the steady state dynamic of the non-linear primary system is computed including the higher harmonics. Non-linear dynamic studies such as bifurcation analysis and Poincare׳ sections are also applied in order to study the effect of NES on the vibration behavior of the beam, in a more accurate manner. Numerical simulations confirm the accuracy of the approximate solutions. Robustness of the NES against changes in the amplitude of excitation is also investigated. Also the performance of NES is compared with linear vibration absorber.