Basic Signal Processing For Vibration

Abstract

Often when setting up measurement points for vibration data collection many choices selected such as the maximum frequency range, the lines of resolution, the window type and the integration mode are made based on rules of thumb instead of an understanding of how each of these are related to the frequency spectrum and the time waveform. A digital signal analyzer is a powerful tool that can present some problems for the uneducated user. With an understanding of signal processing basics these problems can be addressed, understood and avoided.

Fast Fourier Transform

The method used to convert time domain information to frequency domain information is the Fast Fourier Transform (FFT).

Often a frequency spectrum is referred to as an FFT. However, the FFT is the mathematical conversion from the time domain to the frequency domain. Since the signal that comes into the analyzer is an analog signal as discussed in the previous section, it must be digitally sampled by the analyzer. Therefore, the process used by digital analyzers is actually a variation of the FFT, called the Discrete Fourier Transform (DFT).

The DFT is similar as the analog time waveform is recreated in the analyzer by digitally sampling, and then the transformation into the frequency domain is done. Part of the reason the FFT process works is the assumption that the signal measured and digitally sampled is one period of a periodic signal that extends to minus infinity and to plus infinity. Normally, this is true for most vibrating pieces of equipment.

It is the digital sampling process that makes the signal processing more complicated. The information here unlocks the mysteries of digital signal processing without getting bogged down in too much theory.

In order to understand the FFT digital sampling process, you must understand the relationship between lines of resolution (LOR), Maximum frequency (Fmax), length of time waveform (Tmax), the digital sample size, aliasing, windowing, filters, and unit conversion.

Resolution

Once data has been converted to the frequency domain from the time domain, view all of the frequencies of interest in as much detail as possible. Resolution is the number of parts that the spectrum is broken into, usually called lines of resolution (LOR). The number of lines of resolution selected are divided into the maximum analysis frequency (Fmax) to arrive at the bandwidth (BW).
BW = Fmax /LOR

The lines are actually the center frequencies of what are often called “bins of energy”. Each bin actually contains an infinite number of frequencies, and all of the energy in the bin is summed and represented by a single amplitude at the center frequency identified at each line of resolution.

First, identify your frequencies of interest so that enough resolution is chosen to separate closely spaced frequencies. Also, be aware that more lines of resolution affects the length of the time waveform. Increased resolution can also decrease the actual amplitude of the vibration amplitudes due to the separation of energy into more energy bins.

Time Record Length

Calculate the time record length of the time waveform, Tmax, from the following basic relationship.
Tmax = 1/BW

At face value this is a simple and often used equation. However, to understand the limitations of some analyzers, it is important to more fully investigate the relationship between the Fmax the LOR, and the Tmax.
Tmax = sample size/sample rate
Sample size = 2.56 * Lines of Resolution
Sample rate = 2.56 * Fmax

These terms have already been defined, but be aware that some analyzers have an upper limit on the sample size. Usually this number will be 1024 or 2048. Therefore, a 400 line spectrum would require 2.56*400 = 1024 samples and an 800 line spectrum would require 2.56* 800 = 2048 samples. However, if the analyzer is limited to 1024 samples, then the 800 line spectrum will be created from 1024 samples since it is the upper limit of the analyzer. This is important when discussing the Tmax in the time waveform, because, in general, raising the Fmax decreases Tmax and raising LOR increases Tmax until the point that the multiple of 2.56 * LOR reaches the sample limit in the analyzer. In this case, the sample size for anything greater than 400 lines is forced to be 1024. The table below demonstrates how this limitation affects the Tmax for a maximum 1024 sample size analyzer.
Tmax = sample size/sample rate

Fmax *2.56 =sample rate LOR*2.56 = sample size Tmax
400 1024 100 256 0.25
400 1024 200 512 0.50
400 1024 400 1024 1.00
400 1024 800 2048 1.00

The last entry in the table may seem incorrect, but remember 1024 is the maximum sample size, and anything that would be greater than 1024 is forced to be 1024 for the calculation of Tmax. This is the reason why your time waveform is not affected when following the “raise-the-LOR-to-lengthen-the-time-waveform” rule. You must be aware of the upper limit of the sample size and the number of lines of resolution to which this number corresponds.

Aliasing

The term aliasing brings up the idea of someone hiding as someone else. Criminals often do this to elude law enforcement agencies. In the world of vibration analysis, an aliased frequency is one that appears as something it is not. Fortunately, it is not trying to hide from us but was forced into existence by an incorrect digital sampling process.

Now, in a digital analyzer, the time domain data is digitally sampled at the sampling rate. We already know that the sampling rate is controlled by the maximum analysis frequency, Fmax.

In order to get good data, the sampling rate must be set higher than two times the maximum frequency of interest, Fmax. Two times Fmax is known as the Nyquist frequency. A very common sampling rate is set at 2.56 times the Fmax.

Now, an aliased frequency occurs when frequencies higher than the Fmax are present in the signal. The sampling rate under samples this high frequency and creates a low (aliased) frequency equal to the high frequency minus sampling frequency.

For example, if the Fmax is 1000 Hz, then the sampling frequency will be 1000 * 2.56 = 2560. If a frequency is in the data at 2768 Hz, then an aliased frequency can be expected in the data at 2768-2560 = 208 Hz.

To avoid aliasing, digital analyzers use anti-aliasing filters which remove all frequencies above 40% of the sampling rate before the time data is converted to frequency data.

Therefore, with most digital analyzers, aliasing is no longer a concern.

Leakage

Leakage is a problem that can occur when complete cycles of the vibration signal in the time waveform are not captured in the time record. Instead of a discrete frequency in the spectrum, the result is energy spilling into adjacent bands or lines of resolution. A rectangular window, which is really not a window at all, can sometimes allow time data to be captured that can not be transformed well into the frequency domain. If a non-integer multiple of the periods of vibration appear in the time waveform for all the frequencies present, then the frequencies will smear over many lines of resolution.

Digital analyzers use windowing functions to avoid this problem. Typical window choices are Uniform, which is really no window at all, Hanning (often called a cosine taper), Flat top, Kaiser Bessel, Force, and Exponential as well as many others not listed here. The purpose of windowing data is to eliminate leakage problems. For most vibration analysis applications, the window of choice is the Hanning window. The Hanning window does a great job of forcing the beginning and the end of the time record to a zero amplitude. This allows for the reconstructed time waveform to be continuous with no amplitude variations.

A sinewave acquired with a uniform window that has an integer number of periods in the waveform will transform into the frequency spectrum with all of the energy contained in one spectral line of resolution. However, this condition is difficult to achieve without a lot of effort.

If the time waveform contains a non-integer multiple of periods, then the frequency data will smear over many lines of resolution.

The most typical case involves using a Hanning window on a non-integer multiple of periods in the time waveform. In this case, the data transformed to the frequency domain is contained primarily in one line of resolution with some frequency data smeared into the adjacent bands on either side of the primary frequency. This, however, tends to be the most acceptable method of data collection.

The Hanning window will still spread a pure discrete tone over three lines of resolution. To separate very closely spaced frequencies, increase the lines of resolution a little more. However, this can lead to another problem called the Picket Fence Effect.

Picket Fence Effect

The Picket Fence Effect occurs when the frequencies transformed to the frequency domain fall between the lines of resolution so that the amplitudes observed smear between adjacent lines of resolution. It’s like looking at the data though a picket fence with all of the peak amplitudes not necessarily showing up.

Some analyzers and some analysis software has the ability to look between the lines of resolution and pick the peak value. This is often called a peak detection routine or a locate feature.

Overlap Averaging

This is the process of re-using data stored in the input time buffer of the analyzer.

When using a Hanning window on our vibration data, most of the data has been multiplied by a factor of one at the center of the window to zero at the beginning and the end of the time window. In order to fully use all of the data being acquired in the analyzer, use overlap averaging. Fifty percent overlap averaging reuses 50% of the time data saving acquisition time and generally improving amplitude accuracy.

Integration and Differentiation

The vibration data that is the input signal into the analyzer is a time varying voltage proportional to the vibration measured by the transducer. In other words, an accelerometer produces a voltage that varies over time relative to the acceleration measured by the transducer. The voltage amplitude in the time waveform is converted to the desired amplitude units based on the sensitivity and conversion factor of the transducer.

Most analyzers have the ability to convert from the measurement units of the transducer to either of the other two units in the time domain or the frequency domain. At CSI integration of the time signal is called analog integration and integration of the frequency domain is called digital integration.

The process of integration is that of converting from acceleration to velocity or displacement, or converting from velocity to displacement.

The process of differentiation is that of converting from displacement to velocity or acceleration, or converting from velocity to acceleration.

Mathematically, how are these unit types related?

D (Displacement) = distance traveled by vibrating object
V (Velocity) = change in Displacement/change in Time
A (Acceleration) = change in Velocity/change in Time

Mathematically these terms are often represented with the following equations:

D=X
V=X/T
A = X/T/T = V/T

Therefore, if any one of these terms has been measured, integration and differentiation allows any of the other terms to be calculated provided the analyzer or software used is capable of this conversion process.

One drawback to integration is a flare-up of the lower frequency data due to the integration process. This effect is often called “integration noise” or a “ski-slope effect.” This is typically only noticeable when integrating from acceleration to displacement and tends to affect only the lower 1% of the frequencies. This may cause the overall vibration level to be higher than usual, and it may be excluded from the calculation of the overall vibration level.

Normally, this is not of any real concern unless the analyst is concerned with frequency analysis below 2-4 Hz. If this is the case, an investment in a low frequency accelerometer may be beneficial.

Filters

The vibration industry generally uses three types of filters:
Low Pass
Band Pass
High Pass

Each one filters data out of a signal, which you may find useful when analyzing signals with large dynamic ranges. For example, some spectra have both large and small amplitudes relative to each other. Because of the dynamic range of the analyzer, however, you cannot analyze the low amplitude vibration in the same plot as the high amplitude vibration. You would then use a filter to resolve the problem.

As shown above, the low pass filter removes data above the selected frequency. Use this filter during low speed (<600 RPM) frequency analysis. Data collection times can exceed 2-10 minutes per average. Therefore, you can increase data collection times with a low pass filter data by retaining only those events that take place in the selected frequency range.

You have two types of bandwidth filters from which to choose: Constant Percentage Bandwidth and Constant Bandwidth. These filters primarily serve the same function. The Constant Percentage Bandwidth filter changes width depending on the selected frequency.

Note the difference between the two types of filters in the example above. As shown, the filter to choose is the Constant Bandwidth Filter, because it provides the best resolution between both high and low frequency components.

The high pass filter displayed above gives you the ability to filter out low frequency components for detailed analysis. This proves useful when low frequency, high amplitude data swamps the high frequency, low amplitude data you want to see. This situation often occurs when high frequency events appear in the same plot as Run Speed and its relative harmonics.

Frequency Demodulation

Several publications have described the application of demodulation to vibration measurements for machinery defect analysis. As an application tool, demodulation proves helpful in a wide variety of applications.

The demodulation primarily increase the effective dynamic range of the analyzer for certain types of low level measurements. This increased range enhances defect indicators for fault analysis.

There are three primary functions for a demodulator.
A demodulator lets you use a low noise pre-amplifier to improve performance with very small input signals.
It provides the ability to filter input signals for specific analysis requirements.
It gives you envelope demodulation as an input signal processor for amplitude modulation analysis.

Summary

The key to collecting worthwhile data is understanding how the analyzer collecting the data performs its job. When setting up for data collection, remember that the spectral Fmax, the lines of resolution, the waveform Tmax, and the waveform size are all related. Don’t forget to choose the correct analysis window for each measurement application.

The conversion from analog data to digital data can alter the way that the data appears. Remember to recognize the limitations of the measurement process. Using the information in this section should help unlock the mysteries of digital signal processing.

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