Effects of a Dynamic Vibration Absorber on Nonlinear Hinged-Free Beam

Abstract

This study examined the vibrations of a hinged-free nonlinear beam placed on a nonlinear elastic foundation. The authors found that specific combinations of an elastic modulus in the elastic foundation resulted in 1:3 internal resonances in the first and second modes of the beam. This prompted adding a dynamic vibration absorber (DVA) on the elastic beam in order to prevent internal resonance and suppress vibrations. When the DVA was placed at the free end of the beam, the boundaries were time dependent. Thus, a shifting polynomial function was used to convert the nonhomogeneous boundary conditions into homogeneous boundary conditions. The authors analyzed this nonlinear system using the method of multiple scales (MOMS). Fixed-point plots were also used to facilitate the observation of internal resonance. This made it possible to study the influence of nonlinear geometry and nonlinear inertia associated with the vibration of the elastic beam. The authors also examined the combination of optimal mass ratio and elastic modulus for the DVA in order to prevent internal resonance and achieve optimal damping effects. In the nonlinear beam investigated in this study, the placement of a DVA between 0.25l and 0.5l from the hinged end of the beam proved the most effective for damping; locations between 0.7l and 0.8l on the beam were ineffective. Finally, numerical methods and simple experiments were studied to compare the results from MOMS and demonstrate the effects of the DVA of this model.