Structures are subject to extensive vibrations that can reduce their life and contribute to failure. Piezoelectric transducers in conjunction with appropriate electric networks, can be used as a mechanical energy dissipation device. Most of the previous work has focused on active control in which electric power is supplied to the actuators that exert actions on the host structure to suppress its vibrations. Alternatively, undesired mechanical energy of a host structure could be converted into electrical energy that can be dissipated through a set of resistor. This does not require an external power unit and is a more economical means of controlling vibrations of a structure, but an effective transduction of mechanical energy into electric energy has to be guaranteed. Therefore, in the present work a completely passive distributed electric controller is found to achieve simultaneous beam multimodal vibration suppression. The optimal electric network (interconnecting the piezoelectric transducers) may be synthesized as a finite differences approximation of the derived distributed (infinite dimensional) optimal controller.