This thesis analytically investigates the feasibility of passive vibration damping of a rotating beam-like structure, such as a helicopter rotor, through the use of piezoelectric materials. Piezoelectric materials are unique, in that, they produce an electrical charge under the presence of mechanical stresses. Conversely, they generate mechanical stresses under applied electrical loads. When mounted to a structure undergoing bending stresses, such as a cantilever beam, there is an exchange of mechanical and electrical energy between the beam and the piezoelectric material. This electrical energy can be used to power small electronics such as onboard data transmitters. This energy can also be dissipated through electrical shunt circuits rather than being harvested for external use. Electrical components in a shunt circuit (resistors, capacitors, and inductors) release energy from the system as Joule heat. Energy dissipation corresponds to a vibration damping effect in the electromechanical system. Numerous configurations of electrical components and mechanical structures are explored. First, the Rayleigh-Ritz method of assumed modes is adopted for a rotating uniform single degree-of-freedom cantilever beam. Both in-plane (lag) and out-of-plane (flap) bending directions are considered. The beam model is modified to include piezoelectric elements and electrical shunt circuits. Two types of shunt circuits are considered: one with a single resistive element and one with a resistor and inductor in series. Various resistances are used in finding the frequency and impulse responses of the rotating beam with a shunt circuit. The change in damping potential between resistors is evaluated for each electromechanical system. The effects of the number of modes assumed when modeling the beam are also highlighted. Single-mode approximations are found to be helpful in understanding the foundations of the physics in the beam/piezo systems. It is also determined that multiple-mode approximations account for important electromechanical behavior that is neglected by the single-mode formulations. The settling times of the impulse responses are used as the figure of merit to assess energy dissipation in the systems. Successful vibration damping of rotating cantilever beams is predicted through the piezoelectric shunt circuits.