Logarithmic Frequency Scaling
So far, the only type of frequency analysis discussed has been on a linear frequency scale, i.e., the frequency axis is set out in a linear fashion. This is suitable for frequency analysis with a frequency resolution that is constant throughout the frequency range, commonly called "narrow band" analysis. The FFT analyzer performs this type of analysis.
There are several situations where frequency analysis is desired, but narrow band analysis does not present the data in its most useful form. An example of this is acoustic noise analysis where the annoyance value of the noise to a human observer is being studied. The human hearing mechanism is responsive to frequency ratios rather than actual frequencies. The frequency of a sound determines its pitch as perceived by a listener, and a frequency ratio of two is a perceived pitch change of one octave, no matter what the actual frequencies are. For instance if a sound of 100 Hz frequency is raised to 200 Hz, its pitch will rise one octave, and a sound of 1000 Hz, when raised to 2000 Hz, will also rise one octave in pitch. This fact is so precisely true over a wide frequency range that it is convenient to define the octave as a frequency ratio of two, even though the octave itself is really a subjective measure of a sound pitch change.
This phenomenon can be summarized by saying that the pitch perception of the ear is proportional to the logarithm of frequency rather than to frequency itself. Therefore, it makes sense to express the frequency axis of acoustic spectra on a log frequency axis, and this is almost universally done. For instance, the frequency response curves that sound equipment manufacturers publish are always plotted in log frequency. Likewise, when frequency analysis of sound is performed, it is very common to use log frequency plots.
The octave is such an important frequency interval to the ear that so-called octave band analysis has been defined as a standard for acoustic analysis. The figure below shows a typical octave band spectrum where the ISO standard center frequencies of the octave bands are used. Each octave band has a bandwidth equal to about 70% of it center frequency. This type of spectrum is called constant percentage band because each frequency band has a width that is a constant percentage of its center frequency. In other words, the analysis bands become wider in proportion to their center frequencies.
It can be argued that the frequency resolution in octave band analysis is too poor to be of much use, especially in analyzing machine vibration signatures, but it is possible to define constant percentage band analysis with frequency bands of narrower width. A common example of this is the one-third-octave spectrum, whose filter bandwidths are about 27 % of their center frequencies. Three one-third octave bands span one octave, so the resolution of such a spectrum is three times better than the octave band spectrum. One-third octave spectra are frequently used in acoustical measurements.
A major advantage of constant percentage band analysis is that a very wide frequency range can be displayed on a single graph and the frequency resolution at the lower frequencies can still be fairly narrow. Of course, the frequency resolution at the highest frequencies suffers, but this is not a problem for some applications such as fault detection in machines.
In the chapter on machine fault diagnosis, it will be seen the narrow band spectra are very useful in resolving higher-frequency harmonics and sidebands, but for the detection of a machine fault, no such high resolution is required. The vibration velocity spectra of most machines will be found to slope downwards at the highest frequencies, and a constant percentage band (CPB) spectrum of the same data will usually be more uniform in level over a broad frequency range. This means that a CPB spectrum takes better advantage of the dynamic range of the instrumentation. One-third octave spectra are sufficiently narrow at low frequencies to show the first few harmonics of run speed, and can be used effectively for the detection of faults if trended over time.
The use of constant CPB spectra for machine monitoring is not very well recognized in industry with a few notable exceptions such as the US Navy submarine fleet.
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