In control theory, stability is investigated by established techniques such as the root-locus method. The same techniques may be applied to the control equivalent of a mechanical system and will show the transient state of the system. Vector methods such as are used to solve vibration problems may also be extended to give transients.

This note gives a vector representation of the transient vibration of a one degree of freedom system possessing inertia, elasticity and viscous damping. For the transient which is set up by a change in the steady vibration of the system, the initial amplitude and phase angle are readily found by vector methods. This information is sufficient to define the transient.