There are two vibration situations in which statistical analysis is useful. The first situation occurs when the vibration amplitude of a system appears to be random in time. In this case, it is convenient to describe the response of the system statistically. This form of analysis is called random vibration analysis which is our main focus in this article. The second situation occurs when a system is complex enough that its resonant modes appear to be distributed randomly in frequency domain. This form of analysis is called statistical energy analysis.

A random vibration is vibration whose amplitude is unpredictable. This can be caused by random varying loads over time. Examples of random vibration is vibration generated by a car riding on a rough road, jet engine vibration due to load conditions during flight and vibration of grinding action of angle grinder. While the instantaneous vibration amplitude is unpredictable, it is possible to describe the vibration in statistical terms such as gaussian amplitude distribution, probability distribution of the vibration amplitude, and mean-square vibration level.

A random vibration may be categorized as transient (non-stationary) and steady-state (stationary). A stationary random vibration has vibration characteristics that do not change over time. For practical purpose, a random vibration is stationary if the mean-square amplitude and frequency spectrum remain constant over a specified time period.

Technically, the statistical measures of a random vibration must be averaged over a period of time of representative samples. For a stationary random vibration, it may be possible to obtain equivalent averages by sampling over time if each time record is representative of the entire random vibration. Such random phenomenon is called ergodic. However, not all stationary random vibrations are ergodic. For example, supposed it is desired to determine the statistical parameters of the vibration levels of a jet engine during representative in-flight conditions. On a particular flight the vibration levels may be sufficiently stationary to obtain useful time averages. However, one flight is unlikely to represent all of the expected variations in load and other conditions that affect vibration levels. In this case it is necessary to combine the time averages with an average over a number of different flight conditions which represent the entire range of possible conditions.