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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.

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