This paper deals with considerations concerning the reduction in amplitude of unwanted vibrations of machinery parts through the use of materials of high damping capacity. The problem is to reconcile the requirement for high damping with that for high mechanical quality (strength, fatigue resistance, etc.). Two ways are suggested: (1) the use of a material whose decrement generally is low but rises rapidly as the stress increases; (2) the use of a material whose damping capacity is low but rises to high peaks in certain frequency ranges. It is shown by numerical computation in two instances, namely, turbine blade vibrations and crankshaft oscillations in engines, how the suggested methods would work out in practice.