Influence of tuned mass damper on fixed-free 3D nonlinear beam embedded in nonlinear elastic foundation

Abstract

This study investigated the damping effects of a tuned mass damper (TMD) on a fixed-free 3D nonlinear beam resting on a nonlinear elastic foundation with existing internal resonances. The nonlinear elastic foundation below (y-direction) and beside (z-direction) the nonlinear beam was simulated using cubic nonlinear springs via the method of multiple scales for the analysis of this problem. We first examined this system to ensure if the internal resonance exists. The 1:3 internal resonance was found in the 1st and 2nd modes of the system. This prompted us to add a TMD on the elastic beam in order to suppress internal resonance and vibrations. We also examined the influence of the mass and location of the TMD as well as damping and spring coefficients on the damping effects. Analysis data is presented in graphs, including 3D maximum amplitude plots of the modes and 3D maximum amplitude contour plots. Our results show that placing the TMD in the y-direction also suppresses vibrations in the z-direction. The locations of the maximum amplitudes between node points in the mode shapes and near the free end of the beam indicate where the TMD should be placed with regard to damping.