Vibration Frequencies

Real ground vibration wave traces from a seismograph, like the one at left, don't look very much like the simple, regularlyvarying sine and cosine waves discussed in the CVDG's chapter, Vibration 101. Vibration waves are usually much more complex. However, ground vibration seismograph traces can be analyzed mathematically to reveal a complex sum of waves of different frequencies, intensities, and phase relationships (i.e. relative positions of wave peaks in the various components). Vibration standards all over the world recognize that the potential for damage from ground vibration is strongly dependent on the frequencies present in the vibration. They set different "allowable" vibration intensities for different frequencies of vibration. Thus, finding out what wave frequencies and intensities contribute to the overall vibration is extremely important in gaining an understanding of its damage potential.
Waves and Frequencies A graph of "amplitude" ("intensity" or "volume") on the vertical axis vs. time on the horizontal axis, for a single piano note sound (a vibration wave in air) lasting about two seconds, is shown at right. The sound wave starts when the note is struck, oscillating rapidly and fading away over time, as shown in the graphic. A welltuned piano or a tuning fork will have most of its volume at the main ("fundamental") frequency for the note. We could simply count the number of the largest "peaks" occurring in one second in this graphic to get a good approximation of the fundamental frequency of the note, in cycles per second (Hertz, Hz). However, there are some other little "squiggles" on the large peaks of the graphic which show that there are other, higher, frequencies present besides the fundamental frequency. These are called "overtones" and several of them occur at even multiples of the fundamental frequency. It quickly gets difficult to find out much more than the fundamental frequency by simple inspection of such a graphic. Ground vibration amplitude vs. time plots are much more complicated than this single piano note. So, if we are going to be able to understand the frequency makeup of even relatively simple ground vibrations, we need some help. One way of getting at some frequency information from a vibration is to use what is basically a measuring technique, much like that described above for the piano note. One simply looks at the largest peak in the vibration intensity vs. time plot like the one at the top of the page, measures its width in fractions of a second at the baseline of the waveform (i.e. between the adjacent points where the intensity is zero), then calculates 1 divided by this width (the "inverse" of the width) to calculate the number of such waves which one could "fit" in one second. This number approximates the frequency of the largest peak. Seismograph software does this kind of calculation to give what is called a "zero crossing frequency" ("ZC frequency").^{[1] }Indeed, the frequency numbers indicated on seismograph data reports are usually ZC frequencies, since they are available even when waveform data do not exist for a more accurate and complete analysis. The ZC frequency can be reasonably accurate for simple waveforms with sinelike shapes (like those shown at left for sine (blue) and cosine (green) waves) having few frequency components. It can be quite inaccurate for complex waveforms like those from ground vibration, especially when the largest peak is distorted in shape from that of a pure sine wave, due to the overlap of multiple frequency components. Fast Fourier Transform Frequencies A far more accurate and informative method for determining all the frequency components of a complex wave is the use of Fast Fourier Transform (FFT). This is a mathematical technique, usually done with a minicomputer, in which the entire waveform of the ground vibration is "fitted" to a whole set of wave component frequencies, so as to obtain a full analysis of both the frequencies and their intensities which contribute to the vibration. FFT is extremely powerful and widely used. To learn more about how FFT is done and its many uses (e.g. medical MRI depends on FFT), but especially in ground vibration studies, see the CVDG Pro page, Fourier Transform. FFT can only be done when the full waveform is available. It cannot be applied to a vibration whose only record is a set of PPV's in histogram data. Comparing ZC and FFT Frequencies Some FFTderived ground vibration dominant frequencies are compared with the corresponding zero crossing frequencies for the same events in the table at right. In each of these instances, the seismograph was located well within 50 feet of the vibration source.
There are always differences between the ZC and FFT frequencies; some of the differences are quite large. This is especially true for those vibrations which have broad frequency distributions, e.g. those involving ground impacts. Such broad frequency vibrations cannot be adequately characterized with a single vibration frequency, no matter how derived. Most FFTderived dominant frequencies are lower than the corresponding ZC frequencies for the same vibration. This is a particularly important difference, since it is the lower frequencies which have the greatest damage potential. For these reasons, it is generally advised that one use FFTderived frequencies in analyzing vibration data, not ZC frequencies.^{[2] }ZC frequencies should only be used where no waveform data exist to apply FFT, usually due to seismograph Memory Full Exits or "loss" of data. In that case, ZC frequencies should be considered as rough approximations to the real frequencies and given less weight in any analysis of damage potential. FFT Vibration Spectra Unlike ZC crossing calculations, Fast Fourier Transform (FFT) analysis (CVDG Pro) of the vibration waveform provides the full vibration frequency distribution, not just an approximation of the dominant frequency. A full plot of the vibration frequency distribution vs. intensity, like the one at right, is called a "vibration spectrum" and is one of the most useful tools in determining the damage potential of a given vibration. Typically, vibrations generated by mechanical equipment (e.g. engine and other equipment with rotating or oscillating parts) have relatively narrow frequency distributions. Those generated by impact (e.g. pile driving or pounding on pavement with an excavator bucket) or impactlike activities (e.g. driving tracked equipment, pavement milling) have broad frequency ("broadband", "broad spectrum") distributions, like the impact vibration frequency (in Hz) plot at right. These "vibration signatures" are discussed in much more detail on the CVDG Pro page, Vibration Signatures. Vibration monitoring reports will often show only the Zero Crossing frequency of the major vibration component and its velocity (PPV), often without even identifying it as a ZC frequency. This will then be used as the basis for a claim that the vibration "met the standard" or was "nondamaging". A comparison of the ZC and FFT frequencies for this vibration is shown in the "milling cleanup, concrete drop" entry in the table above. Setting aside the inaccuracies of Zero Crossing frequencies, such single frequencybased conclusions have little meaning and virtually no import with respect to the potential for damage when the vibration has a broad range of frequencies, as in impactcaused vibrations like that of the vibration spectrum just above. The dominant frequency there is about 42 Hz, with a low peak particle velocity of 0.185 in/sec. Thus, most vibration reports would indicate that this vibration would be "nondamaging". However, when we look at the whole spectrum, we find that there are multiple vibration frequencies below the 40 Hz criterion indicated in USBM RI 8507. Moreover, their intensities are very nearly the same as the 42 Hz peak. Many of them overlap home resonance frequencies. Even though the naive and incomplete single frequency analysis would claim this vibration to have been nondamaging, the fact of the matter is that this particular vibration is known from videotape records to have caused additional damage in a home already damaged by the road construction operations, which caused both this vibration and the prior damage. Thus, conclusions in vibration monitoring reports based on single zero crossing frequency analyses, especially when there are no available FFTderived vibration spectra and no detailed information on the construction operations occurring at the time of the vibrations, should be viewed as poorly supported, at best. They may well be literally invalid, as they were for this vibration. This topic is discussed in much greater detail on the CVDG Pro page, Reporting Vibration Data. Frequency and Damage Potential As discussed in some detail on the page, Vibration Standards, most vibration standards recognize, and try to account for, the fact that vibration damage potential is frequencydependent. Generally, frequencies below 40 Hz have the potential for most damage, because they can directly interact with the home's natural vibration frequencies in resonances to build higher and higher levels of home vibration (a process called "amplification") as vibrations continue. Unfortunately, the studies on which the vibration standards are based are almost entirely ones of blasting vibrations, where the short durations (typically, 23 seconds) of the blasting vibrations inherently limit the amount of resonant amplification which can occur to between factors of 2 and 4.^{[3]} Blasting vibrations have shorter durations than the vibrations in the homes which they produce.^{[3],[4]} On the other hand, construction vibrations last much longer than either blasting ground vibrations or groundvibration caused home vibrations, so there is a greater opportunity for amplification in the house vibration.^{[6]} Unfortunately, there are few, if any, scientific studies which examine home resonance effects from longlived construction vibrations. While the dominant ground vibration frequencies vary in mine blasting, depending on the setting of the blasting,^{[5]} construction vibrations generally have more intensity at lower frequencies than those from blasting (see road construction example diagram at right). What all this means from the standpoint of damage potential is that resonance effects, particularly the amplifications achieved, on house vibrations caused by construction are virtually unknown and not well accounted for in existing standards.


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