The objective of this study was to optimize the damping effects of a lumped-mass vibration absorber (LMVA) attached to a hinged–hinged nonlinear beam held on a nonlinear elastic foundation. The LMVA was located at various points along the beam and supported by a spring from beneath. Analysis was performed on the internal resonance conditions with the aim of eliminating internal resonance and reducing the amplitude of vibrations in the beam by altering the location and mass of the LMVA as well as the spring constant. We employed the method of multiple scales (MOMS) for the analysis of frequency response in various modes. Fixed points plots were constructed and 3D maximum amplitude contour plots (3D MACPs) were compiled to identify the LMVA combinations with the optimal damping effects. The best damping results in the 3rd mode were achieved when the LMVA was placed between 1/4l and 1/2l in combinations of View the MathML source (mass of LMVA/mass of beam) and View the MathML source (LMVA spring constant/elastic foundation spring constant) followed a linear relationship of View the MathML source=10View the MathML source.