Finite-element modeling is used for the study of structures with piezoelectric actuators and sensors augmented by viscoelastic passive dampers. The structure is excited by a transient force; the response of the structure as determined by the computed voltage response of the sensor is used to construct a feedback loop that excites the actuator. A constant gain P--D (proportional and derivative) controller is used. The controller changes the effective stiffness of the structure and the derivative feedback also introduces the desired damping. The increased stiffness also lowers the vibration amplitude. Active vibration damping is compared and contrasted with passive damping provided by a viscoelastic dampers. A BMG model is used for the viscoelastic phase. A hybrid approach using modal superposition in conjunction with time integration is used for solving the matrix equations in space and time in an efficient manner. A parametric set of results is presented showing the contribution of active and passive damping for the transient response of a clamped plate structure.