# Random Vibration

Random vibration analysis Or Power spectral density (PSD) analysis, more commonly known as random response analysis, is used to determine the structure response under random loading and strains in a system that is subjected to random excitations. Power spectrum that can be displacement, velocity, acceleration or force power spectral density and other forms.

## How do you calculate random vibration in Grms?

The root mean square (rms) value of this signal can be calculated by squaring the magnitude of the signal at every point, finding the average (mean) value of the squared magnitude, then taking the square root of the average value. The resulting number is the Grms metric.

## PSD Curve

In vibration analysis, the PSD stands for the Power Spectral Density of a signal. The PSD represents the distribution of a signal over a spectrum of frequencies just like a rainbow represents the distribution of light over a spectrum of wavelengths.

## Fatigue analysis via Random Vibration

Determining the fatigue life of parts under periodic, sinusoidal vibration is a fairly straightforward process in which damage content is calculated by multiplying the stress amplitude of each cycle from harmonic analysis with the number of cycles that the parts experience in the field. The computation is relatively simple because the absolute value of the vibration is highly predictable at any point in time. Vibrations may be random in nature in a wide range of applications, however, such as vehicles traveling on rough roads or industrial equipment operating in the field where arbitrary loads may be encountered. In these cases, instantaneous vibration amplitudes are not highly predictable as the amplitude at any point in time is not related to that at any other point in time.

As shown in Figure 1, the lack of periodicity is apparent with random vibrations. The complex nature of random vibrations is demonstrated with a Fourier analysis of the random time–history shown in Figure 2, revealing that the random motion can be represented as a series of many overlapping sine waves, with each curve cycling at its own frequency and amplitude. With these multiple frequencies occurring at the same time, the structural resonances of different components can be excited simultaneously, thus increasing the potential damage of random vibrations.

## Statistical Measures of Random Vibration

Because of the mathematical complexity of working with these overlapping sine curves to find instantaneous amplitude as an exact function of time, a more efficient way of dealing with random vibrations is to use a statistical process to determine the probability of the occurrence of particular amplitudes. In this type of approach, the random vibration can be characterized using a mean, the standard deviation and a probability distribution. Individual vibration amplitudes are not determined. Rather, the amplitudes are averaged over a large number of cycles and the cumulative effect determined for this time period. This provides a more practical process for characterizing random vibrations than analyzing an unimaginably large set of time–history data for many different vibration profiles.