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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: [email protected], [email protected], [email protected].
Abstract Vibration-based condition monitoring (VCM) has gained tremendous successes in the detection and differentiation of faults associated with rotating ma-chines, through the installation of various numbers of vibration transducers at indi-vidual 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 bispec-trum 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 re-quired at each bearing pedestal, without necessarily compromising valuable infor-mation 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 provid-ing 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-
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 har-monics and sub-harmonics of the machine’s rotational speed in the vibration spec-trum, 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 vibra-tion 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 vibra-tion 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, capital-izing 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 re-lated 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 sub-har-monics 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 pos-sibility of reducing the amount of transducers required per bearing, during ma-chinery 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.
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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(f ¿¿k )=E ¿¿ , k=1,2,3 ,…….. , N (1)
Where Sxx(f ¿¿k )¿ is the power density, ¿ and X ¿ ( f k) are respectively the DFT and its complex conjugate at frequency ( f ¿¿k )¿ 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 third-or-der moment of a time signal x (t), which is computed by the DFT as [14-15];
Bxxx (f ¿¿l , f m)=E ¿¿ , l+ m ≤ N(2)
The bispectrum gives the coupling between the frequencies at f l , f mand f l+ f m for the considered time domain signal x (t). Assuming that the frequencies f l and f mdenote the pth and qth harmonics of the running speed of a rotating machine re-spectively, 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 fourth-or-der moment of a time signal, which is computed thus [5, 8];
T xxxx ( f l , f m , f n) = E ¿, l+ m + n ≤ N (3)
Just as in the case of the bispectrum, if f l, f m and f n respectively denote the pth, qth and rth harmonics of the running speed of a rotating machine, then the trispec-trum could also be represented by T pqr [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 diame-ters (20mm) but varying lengths (1m and 0.5m respectively), which were sup-ported 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 cou-pling. 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 vibra-tion 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].
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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-
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tion of 0.6104 at sampling frequency (f s) of 5000 and number of data points, N = 8192. The study by Elbhbah and Sinha [6] has provided a detailed spectrum analy-sis (Figure 2) of the considered cases (using Equation 1), and has also identified some of the difficulties associated with machinery vibration analysis using spec-trum 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 pos-sible diagnostics feature for different faults. The merit of HOS is that they merge different frequency components of a signal, so that the coupling between harmon-ics 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 signi-fies 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 signi-fies coupling between 0.5x, 1x and 1.5x components in the spectrum. Similarly, in the trispectrum, T 111 signifies coupling between 1x (thrice) and 3x components in the spectrum; T 112 = T 211 = T 121 signifies coupling between 1x (twice), 2x and 4x components in the spectrum; T sss signifies coupling between sub-harmonic com-ponents 0.5x (thrice) and 1.5x components in the spectrum; T ss 1 = T 1 ss = T s1 s 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 sim-ulated 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 ei-ther the misalignment or crack cases. In addition to B11 and B12 (=B21¿, the mis-
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alignment case also possessed a B22 component. Although the crack case con-tained similar components as the misalignment case, it possessed an additional B13 =B31 components, which were of higher magnitudes than those of the mis-alignment case. The rub case displayed an entirely different feature from the other three cases, 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 (=B1 s) [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 T 111 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 T 112 = T 121 = T 211 which were small and equal in size, and a large T 111 component. Misalignment at bearing 3 showed small T 212=T 122=T 221 and a large T 222. The crack case possessed a response that was somewhat a reverse of the misalignment case in size, but similar in components (i.e. T 112 = T 121=T 211 were large and the T 111 was small). Just as in the case of the bispec-trum, the rub case displayed several sub-harmonics (T sss , T ss1 , T s1 s , T 1ss , etc .). These observations conformed to the findings of Sinha [13], except for the mis-alignment response at bearing 3, which possessed T 212=T 122=T 221 and T 222 com-ponents. 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 ap-pearance of the T 212=T 122=T 221 and T 222 components is due to the fact that bear-ing 3 is located next to the rigid coupling, while bearing 1 is located near the flexi-ble 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 mis-alignment case which had responses at bearings 2, 3 & 4 being similar, but differ-ent 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 domi-nant B11 as well as small peaks of B12 (=B21¿, while the trispectrum displayed the T 111 component, which may be due to residual unbalance and small misalign-ment. 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
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higher amplitude than in the healthy case, while the trispectrum also displayed a dominantT 111. In addition to the dominant B11 and T 111 components in the mis-alignment case, both HOS displayed additional components (B22 for bispectrum and T 112=¿T121 ¿=T 211 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.
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 ro-tating 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 B11, B22 & B21 (for bispectrum) and T111 & T112 (for trispectrum) significantly high at all bear-ings for the crack. Hence, this present study highlights the possibility of eliminat-ing the use of multiple sensors in orthogonal directions and phase analysis. How-ever, the consistency of the fault classification and identification needs to be fur-ther established by simulating more faults of varying sizes and locations on differ-ent 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).
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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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