HOS Analysis of Measured Vibration Data on
Rotating Machines with Different Simulated
Faults
Akilu Yunusa-Kaltungo*, Jyoti K. Sinha, Keri Elbhbah
School of Mechanical, Aerospace and Civil Engineering. The University of Manchester,
Oxford Road, Manchester, M13 9PL, UK.
Email: Akilu.kaltungo@postgrad.manchester.ac.uk,
jyoti.sinha@manchester.ac.uk, keri.elbhbah@postgrad.manchester.ac.uk.
Abstract Vibration-based condition monitoring (VCM) has gained tremendous
successes in the detection and differentiation of faults associated with rotating machines, through the installation of various numbers of vibration transducers at individual bearing pedestals of the monitored machine. This paper however exposes
the future potentials of the use of the higher order spectra (HOS) i.e., the bispectrum and the trispectrum for rotating machines faults diagnosis (FD). The aim of
this is to achieve a significant reduction in the number of vibration transducers required at each bearing pedestal, without necessarily compromising valuable information required for the diagnosis. Four cases (healthy, shaft misalignment,
cracked shaft and shaft rub) were simulated on an experimental rig with two rigidly coupled shafts supported by four ball bearings. Only four accelerometers (one
at each bearing pedestal) were used for this study. The HOS results were compared for the different conditions of the rig. The observations and findings are presented in the paper.
Key words Rotating machines, Condition monitoring, Spectrum, Higher Order
Spectrum, Bispectrum, Trispectrum.
1.0 Introduction
The impacts of machine failures with respect to safety, environment, profit and
market share losses are increasingly becoming enormous by the minute [1]. A
huge proportion of the operations in most manufacturing as well as service providing industries are dependent on rotating machines, which forms the basis for a
continuous search for tools and techniques that will effectively enhance the early
detection of incipient failures in these machines [2].
Vibration-based condition monitoring (VCM) over the years has been effectively
used for the diagnosis of faults in rotating machines, with one of its most estab-
2
lished and successful diagnostic techniques based on spectrum and other signal
analysis. The use of spectrum analysis for fault diagnosis in rotating machines has
been principally done through the examination of the presence of different harmonics and sub-harmonics of the machine’s rotational speed in the vibration spectrum, and this was clearly elaborated in the studies by Goldman and Muszynka
[3]. Similarly, studies by Sinha [4] provided tangible details about the concepts
and applications of VCM. Despite the tremendous successes that have been
achieved by these conventional techniques, their requirements for numerous vibration transducers (accelerometers and/or proximity probes) at each bearing location
could be overwhelming. If a large rotating machine with an appreciable number of
bearings (such as some turbo-generators) is to be monitored, the number of vibration transducers needed, the data to be analyzed and the time will be enormous,
which may in turn complicate the entire process of fault diagnosis. Hence, capitalizing on the recent advances achieved in the area of computational sciences may
enhance the feasibility of tremendously reducing the transducer requirements at
each bearing location without necessarily compromising any of the information
needed for fault diagnosis.
Emerging studies have revealed the capabilities of higher order spectra (HOS),
namely the bispectrum and trispectrum [5] for the diagnosis of various faults related to different rotating machines [6-11]. The greatest strength of HOS is in the
fact that it achieves a combination of the various frequency components present in
a signal; thereby providing the relationships between the harmonics and subharmonics responses of the running speed of the rotating machine, through the aid
of one-point measurement per bearing [6, 12]. Through this, there exists a great
possibility of reducing the amount of transducers required per bearing, during machinery vibration measurements. Therefore, this study simulates four different
cases (healthy, misalignment, crack and rub) on a medium scale experimental rig;
where two rigidly coupled shafts were supported on a relatively stiff support
through the aid of 4 ball bearings. The vibration experiments have been conducted
using 4 accelerometers for measurement (one installed at each bearing pedestal) in
the horizontal direction. Hence, the spectrum, bispectrum and trispectrum have
been computed. The paper provides the results as well as the potentials of using
trispectrum in vibration-based fault detection and analysis in rotating machines.
2.0 HOS
Bispectrum [6, 13-15] and trispectrum [5, 8] are the two types of HOS [10-11]
used in the present study. The computational approaches used for the spectrum
and the HOS,-bispectrum and trispectrum, are briefly discussed here.
3
2.1 Spectrum
The power spectrum of a time domain signal 𝑥(𝑡), is calculated by the discrete
Fourier transform (DFT) of the signal as follows;
𝑆𝑥𝑥 ( 𝑓𝑘 ) = 𝐸[𝑋(𝑓𝑘 )𝑋 ∗ (𝑓𝑘 )] ,
𝑘 = 1,2,3, … … . . , 𝑁
(1)
Where 𝑆𝑥𝑥 ( 𝑓𝑘 ) is the power density, 𝑋(𝑓𝑘 ) and 𝑋 ∗ (𝑓𝑘 ) are respectively the DFT
and its complex conjugate at frequency ( 𝑓𝑘 ) for the considered time domain signal𝑥(𝑡). N is the number of frequency points while the mathematical operator E [.]
denotes the mean.
2.2 Bispectrum
The bispectrum on the other hand is the double Fourier transform of the thirdorder moment of a time signal 𝑥(𝑡), which is computed by the DFT as [14-15];
𝐵𝑥𝑥𝑥 ( 𝑓𝑙 , 𝑓𝑚 ) = 𝐸[𝑋(𝑓𝑙 )𝑋(𝑓𝑚 )𝑋 ∗ (𝑓𝑙 + 𝑓𝑚 )] ,
l+ m ≤ N
(2)
The bispectrum gives the coupling between the frequencies at 𝑓𝑙 , 𝑓𝑚 and 𝑓𝑙 + 𝑓𝑚
for the considered time domain signal 𝑥(𝑡). Assuming that the frequencies 𝑓𝑙 and
𝑓𝑚 denote the pth and qth harmonics of the running speed of a rotating machine respectively, then the bispectrum (𝐵𝑥𝑥𝑥 ) component could also be written as 𝐵𝑝𝑞 [6].
2.3 Trispectrum
Similarly, the trispectrum represents the triple Fourier transform of the fourthorder moment of a time signal, which is computed thus [5, 8];
𝑇𝑥𝑥𝑥𝑥 (𝑓𝑙 , 𝑓𝑚 , 𝑓𝑛 ) = 𝐸[𝑋(𝑓𝑙 )𝑋(𝑓𝑚 )𝑋(𝑓𝑛 )𝑋 ∗ (𝑓𝑙 + 𝑓𝑚 + 𝑓𝑛 )], l+ m + n ≤ N
(3)
Just as in the case of the bispectrum, if 𝑓𝑙 , 𝑓𝑚 and 𝑓𝑛 respectively denote the pth,
qth and rth harmonics of the running speed of a rotating machine, then the trispectrum could also be represented by 𝑇𝑝𝑞𝑟 [5].
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3.0 Experimental Set-up
The photographic representation of the experimental rig is shown on Figure 1,
which is situated in the Dynamics Laboratory of the University of Manchester.
The rig principally consists of two rigidly coupled steel shafts of uniform diameters (20mm) but varying lengths (1m and 0.5m respectively), which were supported by four ball bearings mounted on relatively stiff pedestals (just as indicated by
Figure 1). The 1m shaft is connected to the electric motor via a flexible coupling.
There are three balance steel discs of dimensions 125mm (OD) x 15mm (thickness), with two of the discs fitted on the long shaft (first disc is 30mm from the
drive motor and the second is 19mm from the second bearing) and the third on the
shorter shaft (21mm from both bearings 3 and 4) as shown on Figure 1 [6].
Fig. 1 Photographic representation of the experimental rig.
4.0 Simulation of Faults
The following four cases (healthy, misalignment, cracked shaft and shaft rub)
have been simulated in the experimental rig and vibration data have been collected
at a constant rotational speed of 2040 RPM (34Hz), which corresponds to half of
the first natural frequency of the rig. For all four cases, only four accelerometers
(one at each bearing pedestal in the horizontal direction, due to reduced stiffness
in this direction) were used for the collection of the vibration responses. All vibration data were recorded on to a PC through the aid of a 16-channel, 16-bit Data
Acquisition Card (NI 6229), for subsequent signal processing using the MatLab
code. Further details about the simulated faults are also available in Elbhbah and
Sinha [6].
5
Fig. 2 Typical amplitude spectra for bearings 1 and 3; a-b (healthy), c-d (misalignment), e-f
(crack) and g-h (shaft rub).
5.0 Data Analysis and Results
The measured vibration data at 4 bearings have been analyzed using spectrum,
bispectrum and trispectrum signal processing techniques.
5.1 Spectrum Analysis
The averaged spectra for all four cases (healthy, misalignment, crack and rub)
were calculated using a 80% overlap with a Hanning window, frequency resolu-
6
tion of 0.6104 at sampling frequency (𝑓𝑠 ) of 5000 and number of data points, N =
8192. The study by Elbhbah and Sinha [6] has provided a detailed spectrum analysis (Figure 2) of the considered cases (using Equation 1), and has also identified
some of the difficulties associated with machinery vibration analysis using spectrum analysis technique.
5.2 HOS Analysis
It was observed in the earlier study that the bispectrum analysis shows much better
diagnosis features [6]. Now, the trispectrum analysis is also introduced and then
both bispectrum and trispectrum are combined for the observation of the best possible diagnostics feature for different faults. The merit of HOS is that they merge
different frequency components of a signal, so that the coupling between harmonics and sub-harmonics responses may generate peculiar characteristics at several
running speeds that could enhance fault diagnosis. The measured vibration data
from this experiment were also used to compute both the bispectrum and the
trispectrum. Some of the HOS plots from two bearings (1 and 3) at a running
speed of 2040 RPM (34 Hz) are shown on Figures 3 & 4. In the bispectrum, 𝐵11
signifies the coupling between 1x (twice) and 2x components of the spectrum; 𝐵12
signifies coupling between 1x, 2x and 3x components in the spectrum; 𝐵13 signifies coupling between 1x, 3x and 4x components in the spectrum; 𝐵22 signifies the
coupling between 2x (twice) and 4x components in the spectrum; 𝐵𝑠𝑠 signifies
coupling between 0.5x (sub-harmonic components twice) and 1x; and 𝐵𝑠1 signifies coupling between 0.5x, 1x and 1.5x components in the spectrum. Similarly, in
the trispectrum, 𝑇111 signifies coupling between 1x (thrice) and 3x components in
the spectrum; 𝑇112 = 𝑇211 = 𝑇121 signifies coupling between 1x (twice), 2x and 4x
components in the spectrum; 𝑇𝑠𝑠𝑠 signifies coupling between sub-harmonic components 0.5x (thrice) and 1.5x components in the spectrum; 𝑇𝑠𝑠1 = 𝑇1𝑠𝑠 = 𝑇𝑠1𝑠 signifies coupling between sub-harmonic components 0.5x (twice), 1x and 2x components in the spectrum. It was crystal clear that both the bispectrum and
trispectrum plots shown in the figures provided distinctions between the four simulated cases (healthy, misalignment, crack and rub).
The research study by Elbhbah and Sinha [6] only made a comparison between the
spectrum and the bispectrum, and concluded that the appearance of components
𝐵11 and 𝐵12 (=𝐵21 ) in the healthy case could be as a result of some residual rotor
unbalance and little misalignment between bearings 2 and 3. Furthermore, the
healthy case peaks were of significantly lower magnitudes when compared to either the misalignment or crack cases. In addition to 𝐵11 and 𝐵12 (=𝐵21 ), the misalignment case also possessed a 𝐵22 component. Although the crack case contained
similar components as the misalignment case, it possessed an additional 𝐵13 =𝐵31
components, which were of higher magnitudes than those of the misalignment
case. The rub case displayed an entirely different feature from the other three cas-
7
es, owing to the fact that most of the rotor’s unbalance energy have been converted to sub-harmonic responses, which was responsible for the cluster of peaks
around 𝐵𝑠𝑠 and 𝐵𝑠1 (=𝐵1𝑠 ) [6].
In the current study however, the trispectrum has been calculated (using Equation
3) and included. For the trispectrum plots, the healthy case at both bearings (1 and
3) only possessed the 𝑇111 component, which was also as a result of the residual
misalignment between bearings 2 and 3. It must be mentioned that the trispectrum
responses were quite consistent at both bearings. The misalignment case at bearing
1possessed components 𝑇112 = 𝑇121 = 𝑇211 which were small and equal in size,
and a large 𝑇111 component. Misalignment at bearing 3 showed small
𝑇212 =𝑇122 =𝑇221 and a large 𝑇222 . The crack case possessed a response that was
somewhat a reverse of the misalignment case in size, but similar in components
(i.e. 𝑇112 = 𝑇121 = 𝑇211 were large and the 𝑇111 was small). Just as in the case of
the bispectrum, the rub case displayed
several sub-harmonics
(𝑇𝑠𝑠𝑠 , 𝑇𝑠𝑠1 , 𝑇𝑠1𝑠 , 𝑇1𝑠𝑠 , 𝑒𝑡𝑐.). These observations conformed to the findings of Sinha
[13], except for the misalignment response at bearing 3, which possessed
𝑇212 =𝑇122 =𝑇221 and 𝑇222 components. This variation is due to the fact that the initial experiment involved the use of just 2 bearings and a single coupling, as opposed to the current work that had four bearings and 2 couplings (one flexible and
one rigid). However, the appearance of the 𝑇212 =𝑇122 =𝑇221 and 𝑇222 components
is due to the fact that bearing 3 is located next to the rigid coupling, while bearing
1 is located near the flexible coupling and therefore some of the energy generated
by the misalignment at bearing 1 are absorbed by the flexible coupling. The current study showed a strong consistency in the responses at all 4 bearings for all 4
cases, except for the misalignment case which had responses at bearings 2, 3 & 4
being similar, but different from the response at bearing 1 (which is due to the fact
that the misalignment is at bearing 1and the flexible coupling location).
From the HOS analysis, the features of bispectrum and trispectrum that aid their
differentiation of healthy and faulty conditions have been presented. For the four
cases, both bispectrum and trispectrum showed some consistent trends across all
bearing pedestals. In the healthy case at all bearings, the bispectrum showed dominant 𝐵11 as well as small peaks of 𝐵12 (=𝐵21 ), while the trispectrum displayed the
𝑇111 component, which may be due to residual unbalance and small misalignment.
The components present in the misalignment case for both bispectrum and trispectrum are quite similar to the healthy case, with 𝐵11 being dominant but with higher
amplitude than in the healthy case, while the trispectrum also displayed a dominant𝑇111 . In addition to the dominant 𝐵11 and 𝑇111 components in the misalignment case, both HOS displayed additional components (𝐵22 for bispectrum and
𝑇112= 𝑇121 =𝑇211 for the trispectrum). The crack case for both bispectrum and
trispectrum contained multiple harmonics of the running speed, while the shaft rub
case was characterized by several sub-harmonics and cluster of peaks.
8
Fig. 3 Typical amplitude bispectra (a, c, e & g) and trispectra (b, d, f & h) for bearing 1; a-b
(healthy), c-d (misalignment), e-f (crack) and g-h (shaft rub).
6.0 Conclusion
The potentials of applying HOS for faults identification and differentiation in rotating machines have been explored with the experimental simulation of four cases
(healthy, misalignment, crack and shaft rub) on an experimental rig. It was noticed
that the HOS provided a clear distinction between healthy and faulty conditions,
and also indicates the possibilities of identifying different faults using a certain
combinations of bispectrum and trispectrum components, for example B 11, B22 &
B21 (for bispectrum) and T 111 & T112 (for trispectrum) significantly high at all bearings for the crack. Hence, this present study highlights the possibility of eliminating the use of multiple sensors in orthogonal directions and phase analysis. However, the consistency of the fault classification and identification needs to be
further established by simulating more faults of varying sizes and locations on different rotating rigs.
Fig. 4 Typical amplitude bispectra (a, c, e & g) and trispectra (b, d, f & h) for bearing 3; a-b
(healthy), c-d (misalignment), e-f (crack) and g-h (shaft rub).
7.0 Acknowledgement
Akilu Yunusa-Kaltungo wishes to thank the Petroleum Technology Development
Fund (Federal Government of Nigeria) for sponsoring his PhD study.
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