Frequency Analysis

Excerpted From Introduction to Machine Vibration by Glenn D. White
(ISBN: 978-0-9820517-4-0)

To get around the limitations in the analysis of the wave form itself, the common practice is to perform frequency analysis, also called spectrum analysis, on the vibration signal. The time domain graph is called the waveform, and the frequency domain graph is called the spectrum. Spectrum analysis is equivalent to transforming the information in the signal from the time domain into the frequency domain. The following relationships hold between time and frequency:

A train schedule shows the equivalence of information in the time and frequency domains:

The frequency representation in this case is much shorter than the time representation. This is a "data reduction".

Note that the information is the same in both domains, but that it is much more compact in the frequency domain. A very long schedule in time has been compressed to two lines in the frequency domain. It is a general rule of the transformation characteristic that events that take place over a long time interval are compressed to specific locations in the frequency domain.

Why perform Frequency Analysis?

In the figure below, note that the individual frequency components are separate and distinct in the spectrum, and that their levels are easily identified. It would be difficult to extract this information from the time domain waveform.

It has been argued that the primary reason for the widespread use of frequency analysis is the wide availability of the inexpensive FFT analyzer!

In the next figure, we see that events that are overlapped and confused in the time domain are separated into individual components in the frequency domain. The vibration waveform contains a great deal of information that is not apparent to the eye. Some of the information is in very low-level components whose magnitude may be less than the width of the line of the waveform plot. Nevertheless, such very low-level components may be important if they indicate a developing problem such as a bearing fault. The essence of predictive maintenance is the early detection of incipient faults, so we must be sensitive to very small values of vibration signals, as we will see shortly.

In the next figure, a very low-level component represents a small developing fault in a bearing, and it would have been unnoticed in the time domain or in the overall vibration level. Remember that the overall level is simply the RMS level of the vibration waveform over a broad frequency range, and that a small disturbance such as the bearing tone shown here could double or quadruple in level before the overall RMS would be affected.

On the other hand, there are circumstances where the waveform provides more information to the analyst than does the spectrum.

How to perform Frequency Analysis

Before we investigate the procedure of performing spectrum analysis, we will look at the various types of signals we will be working with.

From a theoretical and practical standpoint, it is possible to divide all time domain signals into several groups. These different signal types produce different types of spectra, and to avoid errors in performing frequency analysis, it is instructive to know their characteristics.

Stationary Signals

The first natural division of all signals is into either stationary or non-stationary categories. Stationary signals are constant in their statistical parameters over time. If you look at a stationary signal for a few moments and then wait an hour and look at it again, it would look essentially the same, i.e. its overall level would be about the same and its amplitude distribution and standard deviation would be about the same. Rotating machinery generally produces stationary vibration signals.

Stationary signals are further divided into deterministic and random signals. Random signals are unpredictable in their frequency content and their amplitude level, but they still have relatively uniform statistical characteristics over time. Examples of random signals are rain falling on a roof, jet engine noise, turbulence in pump flow patterns and cavitation.

Deterministic Signals

Deterministic signals are a special class of stationary signals, and they have a relatively constant frequency and level content over a long time period. Deterministic signals are generated by rotating machines, musical instruments, and electronic function generators. They are further divisible into periodic and quasi-periodic signals. Periodic signals have waveforms whose pattern repeats at equal increments of time, whereas quasi-periodic signals have waveforms whose repetition rate varies over time, but still appears to the eye to be periodic. Sometimes, rotating machines will produce quasi-periodic signals, especially belt-driven equipment.

Deterministic signals are probably the most important in vibration analysis and their spectra resemble the following:

Most quasi-periodic signals are actually a combination of several harmonic series.

Periodic signals always produce spectra with discrete frequency components that are a harmonic series. The term "harmonic" comes from music, where harmonics are multiples of the fundamental frequency.

Non-Stationary Signals

Non-stationary signals are divided into continuous and transient types. Examples of non-stationary continuous signals are the vibration produced by a jackhammer and the sound of a fireworks display. Transient signals are defined as signals which start and end at zero level and last a finite amount of time. They may be very short, or quite long. Examples of transient signals are a hammer blow, an airplane flyover noise, or a vibration signature of a machine run up or run down.

Examples of some wave forms and their spectra

Following are some waveforms and spectra that illustrate some important characteristics of frequency analysis. While these are idealized in the sense that they were made from an electronic function generator and analyzed with an FFT analyzer, they do show certain attributes that are commonly seen in machine vibration spectra.

A sine wave consists of a single frequency only, and its spectrum is a single point.

Theoretically, a sine wave exists over infinite time and never changes. The mathematical transform that converts the time domain waveform into the frequency domain is called the Fourier transform, and it compresses all the information in the sine wave over infinite time into one point. The fact that the peak in the spectrum shown above has a finite width is an artifact of the FFT analysis, which will be discussed later.

A machine with imbalance has an excitation force that is a sine wave at 1X, or once per revolution. If the machine were perfectly linear in response, the resulting vibration would be a pure sine wave like the one shown above. In many poorly balanced machines, the waveform does resemble a sine wave, and there is a large vibration peak in the spectrum at 1X, or one order.

Here we see that a harmonic spectrum results from a periodic waveform, in this case a "clipped" sine wave. The spectrum contains equally spaced components, and their spacing is equal to 1 divided by the period of the waveform. The lowest of the components above zero frequency is called the fundamental, and the others are called harmonics. This waveform came from a signal generator, and it can be seen that it is not symmetrical about the zero line. This means it has a "DC." component, and this is seen as the first line at the left in the spectrum. This is to illustrate that a spectrum analysis can go all the way to zero frequency, or in common terminology, to DC.

In vibration analysis of machinery, it is not usually desirable to include such low frequencies in the spectrum analysis for several reasons. Most vibration transducers do not have response to DC, although there are accelerometers that are used in inertial navigation systems that do have DC response. For machine vibration, the lowest frequency that is generally considered of interest is about 0.3 orders. In some machines this will be below 1Hz. Special techniques are required to measure and interpret signals below this frequency.

Note that because this spectrum consists of discrete points, the signal is by definition deterministic!

It is not uncommon in machine vibration signatures to see a waveform which is clipped something like the one shown above. What this usually means is there is looseness in the machine, and something is restricting its motion in one direction.

The signal shown above is similar to the previous one, but it is clipped on both positive and negative sides, resulting in a symmetrical waveform. This type of signal can occur in machine vibration if there is looseness in the machine and motion is restricted in both directions. The spectrum seems to have harmonics, but they are actually only the odd-numbered harmonics. All the even-numbered harmonics are missing. Any periodic waveform that is symmetrical will have a spectrum with only odd harmonics! The spectrum of a square wave would also look like this.

Sometimes the vibration spectrum of a machine will resemble this if there is extreme looseness and the motion of the vibrating part is restricted at each extreme of displacement. An unbalanced machine with a loose hold-down bolt is an example of this.

Shown above is a short impulse produced by a signal generator. Note that its spectrum is continuous rather than discrete. In other words, the energy in the spectrum is spread out continuously over a range of frequencies rather than being concentrated only at specific frequencies. This is characteristic of non-deterministic signals such as random noise and transients. Note that the level of the spectrum goes to zero at a particular frequency. This frequency is the reciprocal of the length of the impulse, therefore the shorter the impulse, the greater its high frequency content. If the impulse were infinitely short (the so-called delta function, in mathematics), then its spectrum would extend from 0 to infinity in frequency.

By examining a continuous spectrum, it is usually impossible to tell whether it is the result of a random signal or a transient. This is an inherent limitation of Fourier-type frequency analysis, and for this reason it is a good idea to look at the wave form when a continuous spectrum is encountered. As far as machine vibration is concerned, it is of interest to the analyst whether impacting is occurring (causing impulses in the wave form) or random noise (for example, from cavitation) exists in the signal.

A rotating machine seldom produces a single impulse like this, but in the "bump test", this type of excitation is applied to the machine. Its vibration response will not be a classic smooth curve like this one, but it will be continuous with peaks corresponding to the natural frequencies of the machine structure. This spectrum shows that the impulse is a good input force to use in this type of test, for it contains energy over a continuous frequency range.

If the same impulse that produced the previous spectrum is repeated at a constant rate, the resulting spectrum will have an overall envelope with the same shape as the spectrum of the single impulse, but it will consist of harmonics of the pulse repetition frequency rather than a continuous spectrum.

A bearing produces this type of signal with a definite defect in one of the races. The impulses can be very narrow, and they will always produce an extensive series of harmonics.

Modulation Effects

Modulation is a non-linear effect in which several signals interact with one another to produce new signals with frequencies not present in the original signals. Modulation effects are the bane of the audio engineer, for they produce "intermodulation distortion", which is annoying to the music listener. There are many forms of modulation, including frequency and amplitude modulation, and the subject is quite complex. We will now look at the two primary types of modulation individually.

It is rare to see frequency modulation by itself; most machines will produce amplitude modulation at the same time as frequency modulation!

Frequency modulation (FM) is the varying in frequency of one signal by the influence of another signal, usually of lower frequency. The frequency being modulated is called the "carrier". In the spectrum shown above, the largest component is the carrier, and the other components which look like harmonics, are called "sidebands". These sidebands are symmetrically located on either side of the carrier, and their spacing is equal to the modulating frequency.

Frequency modulation occurs in machine vibration spectra, especially in gearboxes where the gear mesh frequency is modulated by the rpm of the gear. It also occurs in some sound system loudspeakers, where it is called FM distortion, although it is generally at a very low level.

This example shows amplitude modulation at about 50% of full modulation

Notice that the frequency of the waveform seems to be constant and that it is fluctuating up and down in level at a constant rate. This test signal was produced by rapidly varying the gain control on a function generator while recording the signal.

This type of signal is often produced by defective bearings and gears, and can be easily identified by the sidebands in the spectrum.

The spectrum has a peak at the frequency of the carrier, and two more components on each side. These extra components are the sidebands. Note that there are only two sidebands here compared to the great number produced by frequency modulation. The sidebands are spaced away from the carrier at the frequency of the modulating signal, in this case at the frequency at which the control knob was wiggled. In this example, the modulating frequency is much lower than the modulated or carrier frequency, but the two frequencies are often close together in practical situations. Also these frequencies are sine waves, but in practice, both the modulated and modulating signals are often complex. For instance, the transmitted signal from an AM radio station contains a high-frequency carrier, and many sidebands resulting from the carrier modulation by the voice or music signal being broadcast.

A vibration and acoustic signature similar to this is frequently produced by electric motors with rotor bar problems.

It is almost impossible to tell beating from amplitude modulation by looking at the waveform, but they are fundamentally different processes, caused by different phenomena in machines. The spectrum tells the story.

This waveform looks like amplitude modulation, but is actually just two sine wave signals added together to form beats. Because the signals are slightly different in frequency, their relative phase varies from zero to 360 degrees, and this means the combined amplitude varies due to reinforcement and partial cancellation. The spectrum shows the frequency and amplitude of each component, and there are no sidebands present. In this example, the amplitudes of the two beating signals are different, causing incomplete cancellation at the null points between the maxima. Beating is a linear process -- no additional frequency components are created.

Electric motors often produce sound and vibration signatures that resemble beating, where the beat rate is at twice the slip frequency. This is not actually beating, but is in fact amplitude modulation of the vibration signature at twice the slip frequency. Probably it has been called beating because it sounds somewhat like the beats present in the sound of an out of tune musical instrument.

The following example of beats shows the combined waveform when the two beating signals are the same amplitude. At first glance, this looks like 100% amplitude modulation, but close inspection of the minimum amplitude area shows that the phase is reversed at that point.

This looks like 100% amplitude modulation!

This example of beats is like the previous one, but the levels of the two signals are the same, and they cancel completely at the nulls. This complete cancellation is quite rare in actual signals encountered in rotating equipment.

Earlier we learned that beats and amplitude modulation produce similar waveforms. This is true, but there is a subtle difference. These waveforms are enlarged for clarity. Note that in the case of beats, there is a phase change at the point where cancellation is complete.