A vibration damping system includes a vibration damping element for a turbine nozzle or blade. A body opening extends through the turbine nozzle or blade between the tip end and the base end thereof, e.g., through the airfoil among potentially other parts of the nozzle or blade. A vibration damping element includes a plurality of stacked plate members within the body opening. Each plate member is in surface contact with at least one adjacent plate member to cause friction that dampens vibration of the nozzle or blade. The body opening has an inner dimension, and each plate member has an outer dimension sized to frictionally engage the inner dimension of the body opening to damp vibration. Plate members may each include a central opening therein, and a fixed elongated body or cable may extend through the central openings. The damping element may alternatively include a helical metal ribbon spring.