This study investigated the damping effects of a dynamic vibration absorber (DVA) attached to a hinged-hinged nonlinear Euler-Bernoulli beam. The model constructed in this study was used to simulate suspended nonlinear elastic beam systems or vibrating elastic beam systems on a nonlinear Winkler-type foundation. This makes the modeling in this study applicable to suspension bridges, railway tracks, and even carbon nanotubes. The hinged-hinged beam in this study includes nonlinear stretching effects, which is why we adopted the method of multiple scales (MOMS) to analyze the frequency responses of fixed points in various modes. The use of amplitudes and vibration modes made it possible to examine the internal resonance. Our results indicate that particular elastic foundations or suspension systems can cause 1:3 internal resonance in a beam. The use of 3D maximum amplitude contour plots (3DMACPs) enabled us to obtain a comprehensive understanding of various DVA parameters, including mass, spring coefficient, and the location of DVA on the beam, and thereby combine them for optimal effect. Our results were verified using numerical calculations.