Mechanical systems are subject to various excitations and it is of interest to determine the response of the system. It is by no means a simple problem to determine the response of a complex real life engineering system to a general excitation. Consider for example, the response of an automobile when moving on a rough road. The excitation to the system comes from the road roughness which is random in nature and needs to be modeled using statistical techniques. Similarly the tire and the suspension of the vehicle itself could exhibit non-linear behavior and hence the mathematic model and its solution will turn out to be quite complex. As another example, consider the dynamic response of an aircraft (such as the indigenous Light Combat Aircraft) during take-off, flight maneuvers and landing. Accurate modeling and response estimation involves complex calculations often on a digital computer.

While we aim to perform complex vibration analysis on a computer on such practical systems, in order to get insight into the methods of analysis as well as the nature of the response, we begin our discussion (as we did for free vibration case) with a simple single degree of freedom spring – mass-damper system subjected to a sinusoidal excitation of single frequency. As mentioned earlier, it is not necessarily true that damping in every physical system is viscous in nature (i.e., damping force proportional to velocity) however we consider viscous damping as it is perhaps the easiest way of accounting for damping in the system as it leads to linear differential equations with constant coefficients.