In this paper we solve the synthesis problem of finding a completely passive electric circuit analog to a vibrating beam. The synthesis problem is of interest when one wants to suppress beam mechanical vibrations by using distributed piezoelectric transduction. Indeed, an effective electromechanical energy transduction is guaranteed when the electric circuit (interconnecting the transducers' terminals) is resonant at all mechanical resonance frequencies and is able to mimic all the mechanical modal shapes. The designed electric circuit behaves as an electric controller of mechanical vibrations (i.e. an electric vibration damper) once suitably endowed with a set of resistors. Because of its completely passive nature, it does not require external power units and stands as an economical means of controlling vibrations.