How are fast Fourier transforms used in vibration analysis?

Fast Fourier transforms are mathematical calculations that transform, or convert, a time domain waveform (amplitude versus time) into a series of discrete sine waves in the frequency domain.

Machine vibration is typically analyzed with measurements of the vibration frequency, displacement, velocity, and acceleration. The latter three — displacement, velocity, and acceleration — are time domain measurements, meaning their amplitudes are plotted versus time. But these vibration signals contain useful information, such as noise and harmonic content, that are difficult or impossible to detect when their amplitudes are plotted in the time domain.

However, when displacement, velocity, and acceleration amplitudes are expressed in the frequency domain —that is, amplitude versus frequency — abnormalities, in the form of high amplitudes at certain frequencies, become visible. And because many vibration-related issues occur at specific frequencies, the cause and location of the vibration can be narrowed down or identified based on variations in amplitude at certain frequencies.

A complex vibration signal expressed in both the time and frequency domains.
Image credit: Power-MI

Note: A time domain plot is referred to as waveform, and a frequency domain plot is referred to as a spectrum.

Every waveform can be expressed as the sum of simple sine waves with varying amplitudes, phases, and frequencies. A Fourier transform is a mathematical process that converts a time domain waveform into these individual sine wave components in the frequency domain — a process often referred to as “spectrum analysis” or “Fourier analysis.”

To understand fast Fourier transforms, it’s helpful to first understand the underlying process, known as discrete Fourier transform (DFT). A discrete Fourier transform tests the time domain waveform for discrete, or individual, frequencies based on the length of the signal (N). The number of frequencies, or samples, required is equal the signal length squared (N²). Even for small signals, this can take significant time and computing power. To make the Fourier transform faster and more efficient, a method known as the fast Fourier transform is used.

The complex signal (red) is made up of simple sine waves (pink and green). A fast Fourier transform expresses these sine waves in the frequency domain, where high amplitudes at certain frequencies become apparent.
Image credit: National Instruments

Fast Fourier transforms (FFT) significantly reduce the number of complex calculations that must be undertaken by assuming that N (the length of the signal) is a multiple of 2. The underlying mathematics of this assumption eliminates redundant calculations and those that have no value (multiplying by “1” for example), which provides significant computational efficiencies and reduces the number of required samples to N*log₂(N) — an amount significantly less than N². This allows fast Fourier transforms to provide close approximations of the more time-consuming discrete Fourier transforms, but with significantly faster computing time.

It’s important to note that the sampling rate must be greater than the highest frequency component of the signal to ensure the sampled data accurately represents the input signal, according to the Nyquist sampling theorem.

The instrument for analyzing signals via fast Fourier transforms is the digital signal analyzer (also referred to as a spectrum analyzer). This device captures the vibration signal, samples it, digitizes it, and performs the FFT analysis. The resulting FFT spectrum helps pinpoint the location, cause, and severity of the vibration, based on the amplitude of the displacement, velocity, and frequency spectra.