# Understanding and Measuring Noise Sources in Vibration Isolation Systems

TMC is a manufacturer of precision vibration isolation systems which are used to support a wide range of vibration sensitive equipment in both commercial and academic research environments. Though many people know that vibrational noise is a problem, few understand how it is measured and quantified. This is particularly true with random sources of vibration. Because of this, noise often is measured improperly. This can lead to incorrect or ambiguous floor vibration requirements for commercial tools, misleading or inaccurate site surveys, and ultimately incorrect conclusions about system susceptibility and performance. This paper briefly introduces the key concepts in noise measurement, and how to interpret the results.

##### Introduction

To measure vibrational noise (we will use simply noise here), three tools are typically used. These are a vibration sensor (sensor) which is usually a *seismic grade* accelerometer, a storage oscilloscope (scope), and a two-channel spectrum analyzer with sub-Hertz capability (analyzer). The scope can be used to look at signal from the sensor in the time domain, and the analyzer can perform a Fourier Analysis on the sensor's signal to give a frequency domain picture of the same data.

Sounds simple, but is it? A scope can tell you the peak level of noise, but it can't give you the frequency of the noise or the RMS (Root-Mean-Squared) value of the noise. What is the meaning of a "peak" value? Is it a sensible question to ask what the "frequency" of noise is? What is the meaning of an RMS value? It turns out that it is not proper in most cases to ask what the frequency of noise is, and an RMS value is meaningless without also specifying a frequency range over which it is calculated.

Likewise, an analyzer can calculate either an amplitude spectrum or an amplitude spectral density. One has the units of Volts and the other units of V/√Hz (Volts per root Hertz). Which should you use? Most analyzers default to the amplitude spectrum, which is the incorrect thing to use to characterize most noise sources. Not knowing better, most users end up taking meaningless data because the analyzer settings are incorrect.

To understand how to use these tools, it is important first to understand what you are trying to measure: noise. Noise can be characterized as either random or coherent. As the next section shows, these can be further sub-classified as either stationary or nonstationary. These are fundamentally different sources of noise, and must be analyzed using different methods. It is up to you (the person doing the measurement or survey) to determine what type of noise you have, and to measure it appropriately.

##### Types of Noise - Random vs. Coherent

*A: Purely periodic (coherent) noise*

*B: Purely random noise*

*C: Periodic and random noise together*

*D: Non-stationary periodic noise*

*E: Random and non-stationary periodic noise*

*F: Non-stationary random noise*

Here are several examples of what different types of noise might look like on a scope. These examples were chosen so that the different types are easily identified.

Curves A and B represent what most people would recognize as noise – however they are also the least likely to be observed in the real world. Curve C represents the most common type of noise: a combination of A and B. In this case, the coherent noise source (periodic noise) dominates over the random component (for ease of illustration); however it is also very common to have random noise dominating. Most often an analyzer is required to "see" the coherent noise components in a signal, as they are not generally evident from a scope trace.

In the real world, noise is almost never constant in amplitude, but changes with time. Such noise is called non-stationary. Curve D shows a periodic source being randomly turned on and off. Curve E shows what you might see in a building with an air compressor or HVAC system cycling on and off: there is a background of random noise, with a periodic component that comes and goes. Nonstationary does not always mean "on" and "off", but can mean that noise gradually builds or wanes. Curve F shows non-stationary random noise. You might see noise like this holding a microphone near a highway, or in seismic noise data as a storm comes and goes.

##### Measuring Noise

Scopes are useful to get a quick indication of which type of noise dominates your measurement, and to determine the peak level of noise (something analyzers do not provide). These, however, are of limited use, and by far the most important tool for quantifying noise is the spectrum analyzer. Unfortunately, this is where things get tricky.

**Stationary coherent noise** sources are the simplest to measure and understand. In the case of Figure A, the signal is made up of a pure sine wave, plus some higher harmonics (2f_{0}, 3f_{0}, 4f_{0} and so on). Each component of the signal has a definite frequency and amplitude. A spectrum analyzer can digitize this signal, perform a Fast Fourier Transform (FFT) on the digitized data, and give you an amplitude spectrum which plots the amplitude of the components (in volts or microns) vs frequency. This can be plotted, or the height and frequency of each peak simply tabulated. Even in the presence of random noise, this is the correct method for measuring the strength of coherent noise sources.

**Random noise** is fundamentally different, and requires an alternate approach. This is because, for random noise, there is no longer a certain level or amplitude of signal at a given frequency. In fact, the is no noise at any single frequency at all! Instead, random noise must be thought of in terms of density.

Consider a volume of air (say the room you are sitting in). There are clearly a lot of air molecules in the room, but there are no molecules of air at any single point. To define a number of air molecules, you have to define two things: the density of the air, and a finite volume of air. If you have no volume, you also have no air. It is perfectly legitimate to talk about the average density of air at a single point, and that density can even differ from point to point (for example, the air is slightly denser at your feet than at your head). The same is true with random noise.

For random noise, the equivalent density is called the amplitude spectral density (ASD) at a given frequency, and the equivalent volume is the frequency bandwidth. The RMS level of the signal is analogous to the number of air molecules, which in the context of this paper is the RMS ground motion. The RMS amplitude is given by:

where the amplitude spectral density ASD(ƒ) is a function of frequency, and has units of µm/√Hz. The form of the integral makes it clear why the ASD is expressed "per-root-Hertz": this cancels the units of dƒ, and gives the RMS result in the proper units (microns). An analyzer displays the ASD(ƒ) when 1/√Hz or "per-root-Hz" units are selected.

Just as the number of air molecules is defined only for a fixed volume, the RMS motion is defined only for a fixed frequency range, in this case, from ƒ1 to ƒ2. It is very common to use the 1/3 octave plot to express RMS motions. In these, the frequency spectrum is broken up into 1/3 octave bins (from 1 to 1.23 Hz, 1.23 to 1.59 Hz, 1.59 to 2.0 Hz and so on). This integral is performed over each of these frequency ranges, and the resulting RMS values (now in µm) are plotted as a bar graph vs. frequency. Many analyzers will do this as a built-in function. To calculate the RMS motion over a larger span in frequency, just take the quadrature sum of all the bins in that range.

In a spectrum analyzer, you must select amplitude spectral density units of [Volts or units]/√Hz for the vertical scale. Otherwise, you will not measure random noise properly. Likewise, if the analyzer is set up to measure a spectral density, it will not measure the correct amplitude for coherent noise sources (in either ASD or 1/3 octave formats). The analyzer must be told whether it is to measure an amplitude or a density.

##### What the Analyzer Does

To further illustrate how random and coherent noise sources behave differently, consider the following figure:

A simple band-pass filter centered on ƒ_{0} with a width δƒ is connected to a speaker. The filter has a bandpass (peak) gain of 1. In scenario A, a sine wave of frequency ƒ_{0} is applied to the input. The output of the speaker is the full amplitude of the input, independent of the width of the filter δƒ. This is because the filter only eliminates components of the input away from ƒ_{0} (and there are none). By contrast, scenario B shows that the effect on random noise is dramatic. The filter removes energy above the frequency ƒ_{0} + 1/2 δƒ and below ƒ_{0} – 1/2 δƒ, leaving a signal which sounds closer and closer to a pure tone at ƒ_{0} as the width of the filter δƒ gets narrower. More importantly, as the filter becomes narrower, and it blocks more energy, the amplitude of the output drops proportional to √δƒ.

A spectrum analyzer is basically identical to a series of band-pass filters like that shown, except instead of speakers, it measures the amplitude of the filter output(s), and displays those numerical results in graphical form. It can be shown that the width δƒ of the filters in Hz is approximately 1/T, where T is the duration of the analyzer's measurement in seconds. FFT analyzers space the "filters" δƒ apart, which gives complete coverage of the frequency spectrum. For example, a 10 second measurement would give a 0.1 Hz frequency resolution (filter widths), resulting in 100 data points between 1 and 10 Hz (for example).

The picture above illustrates what an analyzer setup to measure an amplitude spectrum would see: each filter faithfully passes the amplitude of any coherent noise at the filter's center frequency. The output amplitudes are measured, and plotted by the analyzer. The measurement time T changes the filter widths δƒ, but this does not affect the result (as in scenario A).

It is immediately evident from this discussion that the filters in the picture above produce a nonsensical result for random noise: the amplitude of the noise spectrum would change as you change the measurement time T! The longer the measurement, the less noise you would appear to see. Clearly, the level of random noise is not a function of how long you choose to look at it. To correct for this effect, when the analyzer is set to measure an amplitude spectral density, it multiplies the filter outputs by √T. Since T has units of 1/Hz, the resulting plot has units of Volts (or microns) per-root-Hertz. As you change the measurement time T, this factor corrects for the filter bandwidths, and presents a consistent spectral density level.

##### Conclusion

Obviously, a factor of √T can be significant, but more importantly, amplitudes and amplitude densities are as fundamentally different from each other as a number of air molecules is from a gas density. They are in no way comparable. A comprehensive site survey should consist of the following steps:

- Configure the analyzer for a 1-100 Hertz measurement of the amplitude spectrum. Locate all sharp "spikes" in the spectrum, and tabulate their frequencies and heights. These will correspond to single-frequency noise sources (building fans, compressors, transformer hum, etc.). Associate the peaks with specific noise sources whenever possible (by turning off the possible sources, and seeing which spikes change or disappear).
- Change the vertical units to [units]/√Hz. Many analyzers will convert the amplitude spectrum to an ASD spectrum without recollecting data. Plot the results. Note that the height of the spikes from step 1 have changed their value. It is convenient to label the spikes on the ASD plot with their (Volt or micron) values you measured in 1. The "background" noise floor around the spikes accurately reflects the noise density for random noise sources (people walking, seismic and wind noise). Now you have a single (annotated) plot which accurately represents the noise in your environment.
- If your analyzer will calculate 1/3 octave plots from the ASD, this is a nice additional plot to have. Notice that the 1/3 octave "bins" at the spike frequencies may look like "spike bins". You should note the values measured in step 1 on these bins, if you are confident the noise at those frequencies is dominated by coherent noise sources. Otherwise, the RMS values for those bins will be incorrect by a factor of √T.
- Make note of any noise sources which might be non-stationary. Most of these sources will be coherent noise sources: compressors, fans, HVAC systems. Since the "worst case" condition is usually of greatest interest, care should be taken to make the noise measurement when all noise sources are present. Random noise can change with the weather (literally), local road traffic, etc. It typically varies over longer time periods (hours to days) so thought should be given to conditions which might influence ASD measurements.

It is the responsibility of the person making the measurements to determine what types of noise sources are present, and to collect the appropriate data. Even following the simple steps above can be complicated by non-stationary sources. With the proper awareness, and by taking careful notes about possible noise sources, there will be much less confusion about what vibrational noise issues exist in your environment.

*Author: Dr. Peter G. Nelson, VP/CTO, TMC*