This study considered a slender hinged-free nonlinear beam embedded in a Winkler type elastic foundation simulated using nonlinear cubic springs. This also created multiple possibilities for mode coupling and internal resonance. The objective of this study was to avoid internal resonance within this system and achieve effective vibration damping. We added a time-dependent boundary dynamic vibration absorber (TDB DVA) that was suspended at the free end of the beam to reduce vibration and prevent internal resonance. The Mindlin-Goodman method was used to analyze this time dependent boundary condition problem. The internal resonance condition based on the ratio of the elastic foundation frequency to the beam frequency of the main structure was obtained. The vibration reduction effects of other positions of the DVA were also studied. The influence of shortening effect (nonlinear inertia) and nonlinear geometry of this beam were taken into account as well. We employed the method of multiple scales (MOMS) to analyze this nonlinear problem. The Fixed point plots (steady state frequency response) were obtained. DVAs with various locations and spring constants were considered and the optimal mass range for the DVA to reduce vibration in the main structure was also proposed. The results of this study were verified using numerical simulation, which, in addition to confirming the accuracy by through comparison, established the applicability in this study.