Acquiring vibration data is only part of the challenge of vibration measurement; the other part is the analysis of the data acquired. It’s important to understand the types of wave forms associated with vibration analysis, the important differences between them and when it is appropriate to use each type of vibration analysis tool. Here’s a quick overview of some of the basics.
Time Domain Vibration Analysis
Vibration analysis starts with a time-varying, real-world signal from a transducer or sensor. Analyzing vibration data in the time domain (amplitude plotted against time) is limited to a few parameters in quantifying the strength of a vibration profile: amplitude, peak-to-peak value, and RMS, which are identified in this simple sine wave.
Fast Fourier Transform (FFT)
The fast Fourier transform (FFT) is an efficient algorithm used to compute a discrete Fourier transform (DFT). This Fourier transform outputs vibration amplitude as a function of frequency so that the analyzer can understand what is causing the vibration. The frequency resolution in an FFT is directly proportional to the signal length and sample rate. To improve the resolution, the time of the recording must be extended, but be careful of a changing vibration environment.
Spectrogram
A spectrogram takes a series of FFTs and overlaps them to illustrate how the spectrum (frequency domain) changes with time. If vibration analysis is being done on a changing environment, a spectrogram can be a powerful tool to illustrate exactly how that spectrum of the vibration changes.
Power Spectral Density
A power spectral density (PSD) takes the amplitude of the FFT, multiplies it by its complex conjugate and normalizes it to the frequency bin width. This allows for accurate comparison of random vibration signals that have different signal lengths. For this reason, PSDs are typically used to describe random vibration environments like those specified in military and commercial test standards.
Octave band
Analyzing a source on a frequency by frequency basis is possible but time consuming . The whole frequency range is divided into sets of frequencies called bands. Each band covers a specific range of frequencies. For this reason, a scale of octave bands and one-third octave bands has been developed. A band is said to be an octave in width when the upper band frequency is twice the lower band frequency. A one-third octave band is defined as a frequency band whose upper band-edge frequency (f2) is the lower band frequency (f1) times the cube root of two.
When two frequencies are present in a machine and a cause and effect
relationship is not present, the high frequency will be riding the low
frequency and the Fast Fourier Transform (F’FT) will yield spectral
lines at frequency one and frequency two. If there is a cause and effect
relationship and the two frequencies can mix together, the result is
amplitude modulation. Without getting mathematical, amplitude modulation
is a time varying amplitude. Amplitude modulation is caused when the
equipment has some form of non linearity. This non linearity permits the
amplitude of the two signals to add together when the signals are in
phase, or subtract when the signals are out of phase. With amplitude
modulation, the carrier frequency will be the frequency with the highest
amplitude. The envelope of the varying amplitude will be the difference
between the two frequencies. An FFT of these signals can yield spectral
lines at frequency one, and frequency one plus and/or minus frequency
two.
For example, suppose gear mesh frequency is modulated by gear speed,
gear mesh frequency is 1200 Hz, and gear speed is 20 Hz. An FFT of this
signal would then yield spectral lines at 1200 Hz, 1200 + 20 = 1220 Hz,
and/or 1200 – 20 = 1180 Hz.
Descriptions of these frequencies are:
1. 1200 Hz is gear mesh frequency.
2. 1220 Hz is gear mesh frequency plus gear speed. This is a sum frequency.
3. 1180Hz is gear mesh frequency minus gear speed. This is a difference frequency.
4. The difference between 1200and 1220Hz, or 1200and 1180Hz is 20 Hz, and this is also a difference frequency.
5. The source of excitation, or the problem shaft or gear is usually expressed as a difference frequency.
Frequency modulation is a time-varying frequency,as opposed to amplitude modulation, which is a time-varying amplitude. The lower frequency is the carrier, and the higher frequency is the modulator. The modulator is normally an excited frequency, and the source of excitation is normally the speed of the rotating unit. Pulse Excited Natural Frequency
Frequency Modulation
Frequency modulation can be a series of high frequency bursts similar to a pulse, or the high frequency can occur periodically with a low frequency. Since the frequency response of an accelerometer is best at high frequencies, such problems may be best measured in acceleration. Frequency modulation occurs most often in impacts, such as defects on the inner race of cylindrical roller bearings, or when two shafts are rotating very close to each other. Frequency modulation can occur in screw compressors, vacuum pumps, and blowers when one shaft is bent enough to permit an impact once each revolution.
One last comparison should be noted to clarify the differences between a high frequency riding a low frequency, amplitude modulation, and frequency modulation.
1. High frequency riding a low frequency - No looseness is present. High and low frequencies may be exact multiples of each other. No mixing of signals occurs. Changes in the phase have little or no effect.
2. Amplitude modulation - High frequency is the carrier; low frequency is the modulator. Signals go into and out of phase.
3. Frequency modulation - Low frequency is the carrier; high frequency is I the modulator.
A pulse is caused by a hit or an impact. Pulses fall into one of three categories: empty pulses,
frequency modulation, and amplitude modulation.This occurs when excited
or generated frequencies are not present. It is called empty because it
contains no generated or excited frequencies. A pulse is identified by a
series of spectral lines. The repetition rate of the pulse is equal to
the difference frequency between the spectral lines. The empty pulse has
a low level spectral line at shaft speed, and the amplitude increases
with each succeeding harmonic. Empty Pulse
Frequency modulation is a time-varying frequency. This frequency
modulation can appear as a series of bursts or beats. Generated or
excited pulses are usually caused by a once-per-revolution impact or
excitation. Generated or Excited Pulse
Another type of amplitude modulation occurs when one component is eccentric. One example of sum and difference frequencies is gear eccentricity. When one gear is eccentric or out of round, the amplitude of gear mesh frequency increases when the high place or places go into mesh. If the gear has only one high place, the signal amplitude will be higher once each revolution. In either case, amplitude modulation is caused by the eccentric gear. The associated spectra contain a spectral line at gear mesh frequency with side bands of gear speed. If the gear has more than one high place, then the difference frequency between the gear mesh frequency and the side bands is equal to the number of high places times the speed of the problem gear. If two high places are present, the difference frequency is two times gear speed. Three high places would generate a difference frequency of three times gear speed, four high places would generate four times gear speed, etc.
If the eccentric gear has not caused looseness, side bands will occur at gear mesh frequency plus gear speed or multiples of gear speed. In other words, the side bands will be on the high side of gear mesh frequency. The frequencies add, in this case, because the phase relationship between the carrier and the modulator is constant. As stated earlier, the machine is behaving in a linear manner. One Revolution of Gear with Four Eccentricities.
Sum and Difference Frequency with No Phase Shift.
Sum and Difference Frequency with Phase Shift.
When a gear or geared shaft system is loose, the looseness causes the modulator to subtract from the carrier because the two frequencies are out of phase. When the two frequencies are in phase, they add. Looseness causes an out-of-phase condition. Eccentricity is an in-phase condition.
If gear eccentricity has caused looseness (non linearity) associated with the problem gear, side band scan occur on both sides of gear mesh frequency. If looseness is the more severe problem, the amplitudes of the side bands will be higher on the low side of gear mesh frequency. If eccentricity is the more severe problem, the amplitudes of the side bands will be higher on the high side of gear mesh frequency. If looseness is the only problem, then the side bands occur only on the low side of gear mesh frequency.
The frequencies subtract when looseness is present because the phase relationship between the signals is not constant, which means the machine is acting in a nonlinear manner.
The principles described for gear mesh frequency apply to other generated frequencies such as blade or vane pass frequencies, bearing frequencies, frequencies from multiple defects, and frequencies from bars or corrugations on press rolls.
Two or more independent frequencies can be mixed together if a machine has some form of non linearity or other problem. There are many forms and degrees of frequency mixing. Examples of amplitude modulation, sum and difference frequencies, pulses, and frequency modulation are discussed in the following sections.
Amplitude Modulation :
Amplitude modulation occurs when two frequencies are added together algebraically. Frequencies will not add in a machine that behaves in a linear manner. Therefore, a problem must exist before amplitude modulation can occur. There are several forms of amplitude modulation; one form is a beat. A beat occurs when the amplitudes of two frequencies are added together.
When the amplitudes of the two frequencies go into phase, they add together. Then, as the two frequencies go out of phase, the amplitudes subtract until they are 180 degrees out of phase. The two frequencies continue to go into and out of phase, forming a time varying amplitude signal called a beat.
Beat-Amplitude Modulation of Two Frequencies.
Amplitude modulation also occurs when two frequencies are not exact multiples
High Frequency Riding a Low Frequency :
When two independent frequencies are present in a linear system, they cannot add together in amplitude or frequency. When this occurs, the two frequencies mix, and the high frequency will ride the low frequency, At first glance, amplitude modulation appears to be present.
High frequency riding on low frequency
Since harmonics are not present and the high frequency is riding the low frequency, there is not
a
cause and effect relationship. The two signals are generated
independently. It is important to note that a high frequency which is an
exact multiple of a low frequency will cause the amplitude of the high
frequency peaks to be the same in each period of the low frequency. A
high frequency that is not a multiple will cause the amplitude of the
high frequency peaks to vary during each period of the low frequency.
A special case to note is a single frequency with only odd harmonics
present. The harmonics tend to cancel each other out, except for one
positive and one negative peak per time period of the fundamental. The
peaks have an amplitude equal to the sum of all the amplitudes added
together.
A special case of odd harmonics is has only odd harmonics, and every other odd harmonic is 180
degrees out of phase. The resultant signal is a square wave. The
amplitudes correspond to the amplitude of the fundamental divided by the
harmonic number.
Single Frequency with Only Odd Harmonics.
Note that the signal is not exactly square, but has ripples. This is
due to the fact that a limited number of odd harmonics are contained in
the signal. A true square wave contains all odd harmonics, which cannot
be truly simulated on a computer as a sum of the cosine functions.
Single Frequency with Only Odd Harmonics.
It may appear impossible for a piece of equipment to generate a
square wave, but it is possible to generate a signal with square wave
features. This can occur in a motor if the motor has a loose mount. If
the mounting bolts are loose, the motor will tend to move up and down.
If the motor moves up and is stopped by the mounting bolt, and then
moves down and is stopped by the motor support, a square wave can be
generated. If clipping occurs on both the top and bottom of a signal and
the clipping is significant,the result will resemble a square wave.
CLIPPING :
A signal is said to be clipped when a slight amount of the positive
or negative signal is flattened. The upper signal is an undistorted time
signal. The lower signal is clipped at the bottom.
Clipped Signal
a clipped time signal and a spectrum from a motor 1800 RPM Belt Driving a Fan.
Such a signal can be generated when a machine goes against a stop in
one direction and cannot move further in that direction for a small
period of time. As the cycle continues, the machine moves away from the
stop in a relatively linear manner. The signal is distorted because
the time period for the negative and positive portion is not the same.
Clipping is also a “form of distortion. The frequency spectrum contains
very little harmonic content, because in order for harmonic content to
be generated, the signal distortion must be repeatable.
A harmonic is some exact multiple of a discrete frequency. The
discrete frequency,called the fundamental, is the first harmonic. The
second frequency, which is two times the fundamental frequency, is the
second harmonic. The second, third, fourth, etc., harmonics can be
either in phase or out of phase with the fundamental.
The phase
relationships between the fundamental and the harmonics are valuable in
diagnosing problems in rotating machines. Failure to understand and
use the time signal and harmonic phase can result in diagnostic errors .
A single frequency without harmonics will have one positive-going
peak per time period. The number of positive-going peaks in one time
period of the fundamental frequency identifies the highest number of
true harmonics. This is true for a single frequency with harmonics
only, and is true regardless of the phase relationships between the
fundamental and the harmonics. The amplitudes of the fundamental and
the harmonics determine the amplitudes of the positive-going peaks.
However, the phase relationships of the harmonics to the fundamental
determine the locations of the positive going peaks in the signal.
Single Frequency with an In-Phase Harmonic.
Single Freq. with a 180 Degree Phase Shift and Harmonic.
Single Freq. with a 180 Degree out-of-Phase Harmonic.
However, both positive-going peaks are at the top of the signal. This
can only occur if the second harmonic is 180 degrees out of phase with
the fundamental.
These phase and amplitude relationships hold true for linear systems.
However, most real applications contain nonlinearities,called
distortion. The distortion can appear in the signal as a phase shift in
one or more of the harmonics. Distortion of the signal can also generate
additional harmonics in the frequency domain which are not true
harmonics of the signal. Therefore, the number of peaks in the time
signal must be checked for true harmonic content.
Continuing with phase relationships, the next step is to observe a phase shift of 90 degrees.
Single Freq. with 90 Degree Phase-Shifted Harmonic.
A good rule to remember is that if the source of the harmonic is tied
to the source of the fundamental, such as a fixed, geared, or bolted
coupling, the harmonic should be in phase. If the source of the harmonic
is not tied to the source of the fundamental, the harmonic should be
out of phase.
Single Frequency with a Lower Amplitude Harmonic.
After seeing the effect of changing
amplitude , one can identify the effect of changing the amplitudes in
other ways. Changing amplitude only affects the amplitude of the
composite peak. It does not affect the number of peaks or the phase
relationship of the composite.
Single Frequency with Two Harmonics.
The addition of a third harmonic will now be
examined, along with the effects of changing the phase and amplitude.
Changing the amplitude changes the amplitude of the individual peaks, as
with two harmonics. Three positive peaks per cycle are present,
indicating the three harmonics.
Single Frequency with Only Third Harmonic.
Single Frequency with Two Harmonics.
Single Frequency with Two Phase-Shifted Harmonics.
The time domain signal is necessary to verify which harmonics are true and which harmonics are caused by distortion.
