25.9 Display Formats and Analysis Tools

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Vibration signals can be displayed in a variety of different formats. Each format has advantages and

disadvantages, but generally the more processing that is done on the dynamic signal, the more specific

information is highlighted and the more extraneous information is discarded. The broad display formats

that will be discussed here are the time domain, the frequency domain, the modal domain, and the

quefrency domain. Within each of these display formats, several different analysis tools (some specific to

that display format) will be described.

25.9.1 Time Domain

The time domain refers to a display or analysis of the vibration data as a function of time. The

principal advantage of this format is that little or no data are lost prior to inspection. This allows for a

great deal of detailed analysis. However, the disadvantage is that there is often too much data for easy

and clear fault diagnosis. Time-domain analysis of vibration signals can be subdivided into the

following sections.

25.9.1.1 Time-Waveform Analysis

Time-waveform analysis involves the visual inspection of the time-history of the vibration signal. The

general nature of the vibration signal can be clearly seen and distinctions made between sinusoidal,

random, repetitive, and transient events. Nonsteady-state conditions, such as run-up and coast-down,

are most easily captured and analyzed using time waveforms. High-speed sampling can reveal such

defects as broken gear teeth and cracked bearing races, but can also result in extremely large amounts of

data being collected — much of which is likely to be redundant and of little use.

25.9.1.2 Time-Waveform Indices

A time-waveform index is a single number calculated in some way based on the raw vibration signal and

used for trending and comparisons. These indices significantly reduce the amount of data that is

presented for inspection, but highlight differences between samples. Examples of time-waveform-based

indices include the peak level (maximum vibration amplitude within a given time signal), mean level

(average vibration amplitude), root-mean-square (RMS) level (peak level=

ffiffi

2 p ; reduces the effect of

spurious peaks caused by noise or transient events), and peak-to-peak amplitude (maximum positive to

maximum negative signal amplitudes). All of these measures are affected adversely when more than one

machinery component contributes to the measured signal. The crest factor is the ratio of the peak level to

the RMS level ðpeak level=RMS levelÞ; and indicates the early stages of rolling-element-bearing failure.

However, the crest factor decreases with progressive failure because the RMS level generally increases with

progressive failure.

25.9.1.3 Time-Synchronous Averaging

Averaging of the vibration signal synchronous with the running speed of the machinery being monitored

is called time-synchronous averaging. When taken over many machine cycles, this technique removes

background noise and nonsynchronous events (random transients) from the vibration signal. This

technique is extremely useful where multiple shafts that are operating at only slightly different speeds and

in close proximity to one another are being monitored. A reference signal (usually from a tachometer) is

always needed.

25.9.1.4 Negative Averaging

Negative averaging works in the opposite way to time-synchronous averaging. Rather than averaging all

the collected data, a baseline signal is recorded and then subtracted from all subsequent signals to reveal

changes and transients only. This type of signal processing is useful on equipment or components that are

isolated from other sources of vibrations.

25-16 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

25.9.1.5 Orbits

As described above, orbits are plots of the X direction displacement vs. the Y direction displacement

(phase shifted by 908). This display format shows journal bearing relative motion (bearing wear, shaft

misalignment, shaft unbalance, lubrication instabilities [whirl, whip], and seal rubs) extremely well, and

hence is a powerful monitoring and diagnostic tool, especially on relatively low-speed machinery.

25.9.1.6 Probability Density Functions

The probability of finding the instantaneous

amplitude value from a vibration signal within a

certain amplitude range can be represented as a

probability density function. Typically, the shape

of the probability density function in these cases

will be similar to a Gaussian (or normal)

probability distribution. Fault conditions will

have different characteristic shapes. Figure 25.10

shows two probability density functions. One is

characteristic of normal machine operating conditions,

and the other represents a fault condition.

A high probability at the mean value with a wide

spread of low probabilities is characteristic of the

impulsive time-domain waveforms that are typical

for rolling-element-bearing faults. This type of display format can be used for condition trending and

fault diagnostics.

25.9.1.7 Probability Density Moments

Probability density moments are single-number indices (descriptors), similar to the time-waveform

indices except they are based on the probability density function. Odd moments (first and third, mean

and skewness) reflect the probability density function peak position relative to the mean. Even moments

(second and fourth, standard deviation and kurtosis) are proportional to the spread of the distribution.

Perhaps the most useful of these indices is the kurtosis, which is sensitive to the impulsiveness in the

vibration signal and therefore sensitive to the type of vibration signal generated in the early stages of a

rolling-element-bearing fault. Because of this characteristic sensitivity, the kurtosis index is a useful fault

detection tool. However, it is not good for trending. As a rolling-element-bearing fault worsens, the

vibration signal becomes more random, the impulsiveness disappears, and the noise floor increases in

amplitude. The kurtosis then increases in value during the early stages of a fault, and decreases in value as

the fault worsens.

25.9.2 Frequency Domain

The frequency domain refers to a display or analysis of the vibration data as a function of frequency. The

time-domain vibration signal is typically processed into the frequency domain by applying a Fourier

transform, usually in the form of a fast Fourier transform (FFT) algorithm. The principal advantage of

this format is that the repetitive nature of the vibration signal is clearly displayed as peaks in the

frequency spectrum at the frequencies where the repetition takes place. This allows for faults, which

usually generate specific characteristic frequency responses, to be detected early, diagnosed accurately,

and trended over time as the condition deteriorates. However, the disadvantage of frequency-domain

analysis is that a significant amount of information (transients, nonrepetitive signal components) may be

lost during the transformation process. This information is nonretrievable unless a permanent record of

the raw vibration signal has been made.

Probability Density(dB)

Normalized Vibration Amplitude

Normal Bearing

Faulty Bearing

FIGURE 25.10 Normalized vibration amplitude vs.

probability density (normal and faulty bearings).

Machine Condition Monitoring and Fault Diagnostics 25-17

© 2005 by Taylor & Francis Group, LLC

25.9.2.1 Band-Pass Analysis

Band-pass analysis is perhaps the most basic of all frequency-domain analysis techniques, and involves

filtering the vibration signal above and/or below specific frequencies in order to reduce the amount of

information presented in the spectrum to a set band of frequencies. These frequencies are typically where

fault characteristic responses are anticipated. Changes in the vibration signal outside the frequency band

of interest are not displayed.

25.9.2.2 Shock Pulse (Spike Energy)

The shock-pulse index (also known as spike energy; Boto, 1979) is derived when an accelerometer is

tuned such that the resonant frequency of the device is close to the characteristic responses frequency

caused by a specific type of machine fault. Typically, accelerometers are designed so that their natural

frequency is significantly above the expected response signals that will be measured. If higher

frequencies are expected, they are filtered out of the vibration signal. High-speed rollingelement

bearings that are experiencing the earlier stages of failure (pitting on interacting surfaces)

emit vibration energy in a relatively high, but closely defined, frequency band. An accelerometer

that is tuned to 32 kHz will be a sensitive detection device. This type of device is simple, effective,

and inexpensive tool for fault detection in high-speed rolling-element bearings. The response from

this type of device is load-dependent and may be prone to false alarms if measurement conditions are

not constant.

25.9.2.3 Enveloped Spectrum

Another powerful analysis tool that is available in the frequency domain and can be effectively applied to

detecting and diagnosing rolling-element-bearing faults is the enveloped spectrum (Courrech, 1985).

When the vibration signal time waveform is demodulated (high-pass filtered, rectified, then low-pass

filtered) the frequency spectrum that results is said to be enveloped. This process effectively filters out

the impulsive components in signals that have high noise levels and other strong transient signal

components, leaving only the components that are related to the bearing characteristic defect

frequencies. This method of analysis is useful for detecting bearing damage in complex machinery where

the vibration signal may be contaminated by signals from other sources. However, the filtering bands

must be chosen with good judgment. Recall also, the impulsive nature of the fault signal at the

characteristic defect frequency leaves as the fault deteriorates.

25.9.2.4 Signature Spectrum

The signature spectrum (Braun, 1986) is a baseline frequency spectrum taken from new or recently

overhauled machinery. It is then later compared with spectra taken from the same machinery that

represent current conditions. The unique nature of each machine and installation is automatically taken

into account. Characteristic component and fault frequencies can be clearly seen and comparisons made

manually (by eye), using indices, or using automated pattern recognition techniques.

25.9.2.5 Cascades (Waterfall Plots)

Cascade plots (also known as waterfall plots) are successive spectra plotted with respect to time and

displayed in a three-dimensional manner. Changing trends can be seen easily, which makes this type of

display a useful fault detection and trending tool. This type of display is also used when a transient event,

such as a coast-down, is known to be about to occur. Cascade plots can also be linked to the speed of a

machine. In this case, the horizontal axis is labeled in multiples of the rotational speed of the machine.

Each multiple of the rotational speed is referred to as an “order.”

“Order tracking” is the name commonly used to refer to cascade plots that are synchronously linked to

the machine rotational speed via a tachometer. As the speed of the machine changes, the responses at

specific frequencies change relative to the speed, but are still tracked in each time-stamped spectra by the

changing horizontal axis scale.

25-18 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

25.9.2.6 Masks

Like negative averaging in the time domain, masks are baseline spectra that are used with an allowable

tolerance limit to “filter out,” or block, specific frequencies. This technique is similar to band-pass

analysis and requires a good knowledge of the full range of each machine’s operating limits (varying load

or speed).

25.9.2.7 Frequency-Domain Indices

It has been noted that frequency spectra are more sensitive to changes related to machine condition

(Mathew, 1987). Because of this sensitivity, several single number indices based on the frequency spectra

have been proposed. Like the time-waveform indices, frequency-domain indices reduce the amount of

information in frequency spectra to a single number. Because they are based on the frequency spectra,

they are generally more sensitive to changes in machine condition than time domain indices. They are

used as a means of comparing original spectra or previous spectra to the current spectra. Several

frequency domain indices are listed below:

* Arithmetic mean (Grove, 1979):

20 log

1

N

XN

i¼1

Ai

􀁻 !􀀝

1025

( )

Ai ¼ amplitude of ith frequency spectrum component

N ¼ total number of frequency spectrum components

* Geometric mean (Grove, 1979):

1

N

XN

i¼1

20 log

Affiiffi

2 p

􀀏 􀀐􀀝

1025

( 􀀏 􀀐)

* Matched filter RMS (Mathew and Alfredson, 1984):

10 log

1

N

XN

i¼1

Ai

Aiðref Þ

􀀏 􀀐2

( )

Aiðref Þ ¼ amplitude of ith component in the reference spectrum

* RMS of spectral difference (Alfredson, 1982):

1

N

XN

i¼1 ðLci 2 LoiÞ2

( )1=2

Lci ¼ amplitude (dB) of ith component

Loi ¼ amplitude (dB) of ith reference component

* Sum of squares of difference (Mathew and Alfredson, 1984):

1

N

XN

i¼1

􀀑

ðLci þ LoiÞ £ lLci 2 Loil

􀀜1=2

( )

25.9.3 Modal Domain

Modal analysis is not traditionally listed as a machine condition monitoring and fault diagnostics tool,

but is included here because of the ever-increasing complexity of modern machinery. Often, unless the

Machine Condition Monitoring and Fault Diagnostics 25-19

© 2005 by Taylor & Francis Group, LLC

natural (free and forced response) frequencies of machinery, their support structure, and the

surrounding buildings are fully understood, a complete and accurate assessment of existing

machinery condition is not possible. A complete overview of modal analysis will not be provided

here, but a specific approach to modal analysis (operational deflection shape [ODS] analysis) will

be described.

ODS analysis is like other types of modal analysis in that a force input is provided to a structure or

machine and then the response is measured. The response at different frequencies defines the natural

frequencies of the structure or machine. Typically, an impact or constant frequency force is used to

excite the structure. In the case of ODSs, the regular operation of the machinery provides the

excitation input. With vibration sensors placed at critical locations and a reference signal linking

together all the recorded signals, a simple animation showing how the machine or structure deflects

under normal operation can be generated. These animations, along with the frequency information

contained in each individual signal, can provide significant insights into how a machine or structure

deforms under a dynamic load. This information, in turn, can be a useful addition to other data when

attempting to diagnose problems.

25.9.4 Quefrency Domain

A quefrency-domain (Randall, 1981, 1987) plot results when a Fourier transform of a frequency spectra

(log scale) is generated. As the frequency spectra highlight periodicities in the time waveform, so the

quefrency “cepstra” highlights periodicities in a frequency spectra. This analysis procedure is particularly

useful when analyzing gearbox vibration signals where modulation components in spectrum (sidebands)

are easily detected and diagnosed in the cepstrum.

* Generally, the more processing that is done on the dynamic signal, the more specific useful

information is highlighted and the more extraneous information is discarded.

* The primary display formats used in machine condition monitoring are the time domain,

the frequency domain, the modal domain, and the quefrency domain.

* The time domain refers to a display or analysis of the vibration data as a function of time,

allowing for little or no data to be lost prior to inspection.

* Time domain analysis includes: waveform analysis, time waveform indices, time

synchronous averaging, negative averaging, orbit analysis, probability density functions,

and probability density moments.

* The frequency domain refers to a display or analysis of the vibration data as a function of

frequency, where the time domain vibration signal is typically processed into the

frequency domain by applying a Fourier transform, usually in the form of a FFT algorithm.

* The principal advantage of frequency-domain analysis is that the repetitive nature of the

vibration signal is clearly displayed as peaks in the frequency spectrum at the frequencies

where the repetition takes place. This allows for faults, which usually generate specific

characteristic frequency responses, to be detected early, diagnosed accurately, and trended

over time as the condition deteriorates.

* Frequency-domain analysis includes the use of band pass analysis, shock pulse (spike energy),

envelope spectrum, signature spectrum, cascades (waterfall plots), masks, and frequencydomain

indices.

* Quefrency-domain analysis involves a Fourier transform of a frequency spectra (log scale). As

the frequency spectra highlight periodicities in the time waveform, so the quefrency “cepstra”

highlights periodicities in a frequency spectra.

25-20 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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