Forced vibrations of a simply supported beam having an attached non-linear dynamic vibration absorber and excited by sinusoidal motion of its supporting base are investigated analytically. The absorber produces a hardening spring force in the form of a cubic curve. The governing partial differential equation reduces to the well-known Duffing equation, which is solved by the harmonic balance method. Besides the simple harmonic motion solution, the third order superharmonic and one-third order subharmonic are obtained. The results of the present analysis are applied to a magnetic non-linear dynamic absorber, and the optimum tuning and optimum damping for the absorber are obtained by the simplex method.