# Definitions for Random Vibration Analysis

## Random or Stochastic Process

A stochastic process is generally viewed as a family of random variables, or a collection of a large number of records that describe a physical phenomenon. The records can be a function of time {x_{k}(t)} or frequency {x_{k}(f)}. Each record is somewhat different from any other record. It is impossible therefore to include all possible records in the analysis. Instead, a random process is described in terms of statistical properties. Each load in a random vibration study is a random process. The response of a model to these loads is also a random process described in statistical terms.

## Autocorrelation Function

The autocorrelation function of a random process describes the correlation between the values in a record at different instants of time. It is defined as the expected value of the product of a random variable x(t) with a time-shifted version of itself.

(Eq.1)

## Root Mean Square (rms)

The mean square value provides a measure of the energy associated with the random process.

It is defined as the value of the autocorrelation function for τ = 0

(Eq.2)

where E is called the expectation operator. The positive square root of the mean value is known as the root mean square, or rms.

## Variance

The mean square value of a random process about its mean μ_{x}.

(Eq.3)

The positive square root of the variance is known as standard deviation.

## Power Spectral Density (psd)

The power spectral density is defined as the Fourier transform of the autocorrelation function of a random process.

(Eq.4)

Power spectral density describes how the energy of the random process is distributed in the frequency domain.

## White Noise

A white noise signal has a uniform power spectral density in all frequencies. In other words, the signal's energy is distributed equally in all frequencies.