Vibrations in machines and structures should be analyzed and controlled if they have undesirable effects such as noise, unpleasant motions, or fatigue damage with potentially catastrophic consequences. Conversely, vibrations are sometimes employed to useful purposes, such as for compacting materials.

The simplest vibrating system has motion of one degree of freedom (DOF) described by the coordinate x in Figure 4. (An analogous approach is used for torsional vibrations, with similar results.)