F. E. Hernández1, O. Caveda1, V. Atxa2, J. Altuna2
1University of Pinar del Río, Marti 270, Pinar del Río, Cuba
2University of Mondragón, Loramendi 4, Mondragón, Spain
Abstract-The aim of this work is to evaluate, in a theoretical sense, the current application of bispectrum on rolling element bearings diagnosis. A mathematical model of the vibration generated by defective rolling element bearings is used and substituted into bispectrum formulas. This work demonstrated that using this statistical tool in order to detect a local fault on rolling element bearings is not effective, contrasting with practical results achieved in other papers. In that sense, some arguments concerning such a contradiction are exposed.
I. INTRODUCTION
Many signal processing techniques have been applied on machine diagnosis via vibration analysis. Among them, spectral analysis is highlighted due to the low cost-to-benefit rate obtained from its implementation. However, in many applications, the characteristics of the vibration to analyze (e.g., nonstationary, low signal to noise rate, etc.) cause the worsening of the effectiveness of this technique, and, in such cases, it is justified the application of advanced signal processing techniques [1].
The vibration emitted by defective rolling element bearings is an example of signal with features that could make the spectral analysis perform an inappropriate fault detection task.
That is why other signal processing techniques are being applied on the vibration analysis for bearings diagnosis.
Higher-order statistical signal processing, in particular, the bispectrum, is one of the actual signal processing techniques, through which better practical results have being obtained [2-4].
The linear modulation process that appears in the vibration produced by defective rolling element bearings makes possible to infer that bispectrum is suitable to be applied. It is well known that higher-order statistical signal processing allows to detect phase-related spectral components, a feature of the modulation signals.
However, as it will be mathematically demonstrated in next sections, the employment of bispectrum for detecting rolling element bearings faults is irrelevant in the sense that no parameters related to the failure characteristic frequency of the bearings are obtained. Obviously, this theoretical result hardly contrasts with practical results actually achieved by different
authors, thus some criteria about this contradiction are presented.
II. USEFUL STATISTICAL FOUNDATIONS
All the features described are statistical because they are based on statistical distributions of the vibration samples. Such features are moments and cumulants.
The signal moments can be expressed as mn =E
{ }
xn , where E{}
⋅ is the expectation operator which can be estimated (assuming that the signal is ergodic and stationary) using∑
== N
i n i
n x
m N
1
1 [5].
The signal moments are related to the probabilistic density function, p(x), by the moment generating function,
dx e x p s
∫
∞ sx∞
−
= ( ) )
φ( . The nth signal moment is calculated by evaluating the nth derivative of φ(s) at s=0,
0 n ( )
n n
s
d s
m ds
φ
=
= .
The second characteristic function is the logarithm of the moment generating function. The cumulants are calculated evaluating the derivatives of this function at s=0,
0
ln( ( ))
n
n n
s
d s
c ds
φ
=
= .
Cumulants have several useful properties that make their use more convenient than the use of moments. Firstly the higher-order cumulants of a Gaussian random variable are all zero.
Secondly the cumulant of the sum of two random variables is the sum of the cumulant of the random variables. Therefore if a Gaussian random variable is added to a non-Gaussian random variable, the resulting signal’s higher-order cumulants are the cumulants of the non-Gaussian signal.
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© 2007 Springer.
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A. Moment and cumulant functions and spectra
If a zero-mean real stationary random process, x(t), is considered [6], the moment and cumulant functions, are expressed as follows:
1 1
m = =c mean value, (1a)
[ ]
2( ) 2( ) ( ) ( )
m τ =c τ =E x t x t+τ , (1b)
3( , )j k 3( , )j k ( ) ( j) ( k)
m τ τ =c τ τ =E x t x t⎡⎣ +τ x t+τ ⎤⎦, (1c)
and so on.
The term “higher-order” is used when the order is higher than two.
On the other hand, the Wiener-Khintchine relation can be generalized by transforming cumulant functions, which results in the spectral cumulant functions as follows:
1
2( ) 2( )
c τ ←⎯→C f , (2a)
2
3( , )j k 3( , )j k
c τ τ ←⎯→C f f , (2b)
and so on.
The elements ←⎯→1 and ←⎯→2 denote the one and two dimensions Fourier transform. The second-order cumulant spectrum is the traditional power spectral density and the third-order cumulant spectrum is known as bispectrum.
III. VIBRATION EMITTED BY DEFECTIVE ROLLING
ELEMENT BEARINGS
The main function of rolling element bearings is to provide low friction conditions for supporting and guiding a rotating shaft.
The parts of the rolling element bearings are: rolling elements, inner race, outer race and cage. They remain in contact and their failures can be caused by manufacturing problems, inadequate usage or wearing.
One of the most important defects to detect on rolling element bearings is the local failure. This type of fault makes the bearings produce a vibration that corresponds to a linear modulation signal (see Fig. 1) which usually superimposes on other vibration sources in the rotating machine.
Fig. 1. Vibration produced by defective rolling element bearings.
IV. MODEL OF THE DEFECTIVE ROLLING ELEMENT
BEARINGS VIBRATION
One of the models that better characterizes the vibration produced by defective rolling element bearings is the one provided by Randall et al. in [7]:
( ) i ( 0 i) ( )
i
x t =
∑
a s t iT− −τ +n t , (3) where T0 is the average time between impacts, s(t) is the oscillating waveform generated by a single impact, n(t) is a zero-mean stationary noise, τi is a zero-mean delta-correlated point process with probability density function ρτ( )t , and ai is a periodically delta-correlated point process.V. PROCEDURE FOR DETECTING THE VIBRATION
GENERATED BY DEFECTIVE ROLLING ELEMENT BEARINGS
Most of the techniques involved on bearings diagnosis by vibration analysis are based on the identification of some pattern related to the failure characteristic frequency. This frequency equals
1/T
0; it depends upon mechanical characteristics of the rolling element bearings and can be calculated by well stated expressions [8, 9].Practical results achieved by the application of the bispectrum on the detection of local faults in rolling element bearings, suggest that the procedure in this case consists of identifying bispectral lines separated at the failure characteristic frequency,
1/T
0.In this work, the bispectrum of the vibration generated by defective rolling element bearings is theoretically calculated, using the model described in section IV and substituted in the bispectrum expressions. It will be shown that in theory no bispectral lines can appear in the result, leading to a contradiction with those practical outcomes obtained by other researchers.
VI. THEORETICAL BISPECTRUM OF THE VIBRATION
GENERATED BY DEFECTIVE ROLLING ELEMENT BEARINGS
The theoretical calculation of the bispectrum of the vibration produced by defective rolling element bearings is performed by substituting the vibration model in the third-order cumulant function, and then, by transforming the result (as expressed in (2b)).
In other words, the third-order cumulant function is written as follows:
HERNÁNDEZ ET AL.
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3 1 2
The calculation of (4) results in:
3 1 2 Substituting the third-order cumulant function in (2b) results in the bispectrum. In this case, the starting expression is:
1 1 2 2
which leads to the final expression of the bispectrum of x(t), as follows:
An inspection of the final bispectrum expression shows that no bispectral information in respect to the failure characteristic frequency, 1/T0, is achieved. In other words, the theoretical bispectrum does not result in discrete bispectral components equally spaced at the failure characteristic frequency.
VII. CONSIDERATIONS ABOUT THE PRACTICAL
“EFFECTIVENESS”OF THE BISPECTRUM APPLICATION IN
CONTRAST WITH THEORETICAL RESULTS
Despite the mathematical result reached in Section VI, several authors have previously presented the “benefits” of calculating the bispectrum of the vibration generated by a rotating machine in order to detect bearings failures [2-4]. In such works, when the bearings fault exists, bispectrum exhibits clear bispectral components separated at 1/T0, which is used as indication of failure existence. However, these components do not arise in the theoretical bispectrum, as shown in (7).
Attending to the practical difficulties in estimating the first-order moment of the vibration generated by defective rolling element bearings, it can be ensured that this contradiction is due to the fact that practical applications of the bispectrum are performed assuming that the mean value of x(t) is constant and equals / 2 shown in Fig. 2. Then it’s clear to realize that the result of calculating the bispectrum in both a theoretical and a practical sense is not the same.
APPLICATION OF HIGHER-ORDER STATISTICS 147
Fig. 2. Bispectrum of the vibration produced by a defective rolling element bearings (characteristic failure frequency equals to 1/T0).
VIII. CONCLUSIONS
This work demonstrated the theoretical inefficiency of calculating the vibration bispectrum in order to detect local faults in rolling element bearings. This conclusion constitutes a novel result since there are not previous references about it.
The contradiction in the results obtained by practical and theoretical applications of the bispectrum is evaluated in this paper. In fact, problems in the practical estimation of first-order moment of the vibration do not lead to the same results when applying both theoretical and practical bispectrum.
This study contributes to clarify the possibilities of applying advanced signal processing techniques on vibration analysis, specifically, the higher-order statistical processing.
REFERENCES
[1] F.E. Hernández, “Aplicación del procesamiento cicloestacionario de vibraciones, avanzado y de segundo orden, a la detección de fallos locales en cojinetes de rodamientos,” PhD dissertation presented at the Mechanic Department, Instituto Superior Politécnico José Antonio Echeverría, La Habana, Cuba, january 2006.
[2] A.C McCormick, “Cyclostationary and higher-order statistical signal processing algorithms for machine condition monitoring,” dissertation presented at Department of EEE, University of Strathclyde, United Kingdom, September 1998.
[3] J. Piñeyro, A. Klempnow, and J. Lescano,
“Effectiveness of new spectral tools in the anormaly detection of rolling element bearings,” Journal of Alloys and Components, vol. 310, pp. 276-279, 2000).
[4] F.E. Hernández, and V. Atxa, “Diagnóstico de maquinarias a partir del análisis de vibraciones,” VI Seminario Anual de Automática, Electrónica Industrial e Instrumentación, Vigo, Spain, September 2003.
[5] A. Papoulis, Probability, random variables, and stochastic processes, 3rd ed., McGraw-Hill, Inc., 1991, pp.
109-119.
[6] B. Boashash, E.J. Powers, and A.M. Zoubir, Higher-order statistical signal processing, Longman House, Melbourne, Australia, 1995.
[7] R.B. Randall, J. Antoni, and S. Chobsaard, “The relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other cyclostationary machine signals,” Mechanical Systems and Signal Processing, vol. 15, pp. 945-962, 2001.
[7] N. Tandon, and B.C. Nakra, “Vibration and acoustic monitoring techniques for the detection of defects in rolling element bearings. A review,” The Shock and Vibration Digest, vol. 3, pp. 3-11, 1992.
[8] R.B. Randall, “State of the art in monitoring rotating machinery,” International Conference on Noise and Vibration Engineering, Belgium, September 2002.
1/T0
1/T0
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Abstract-Independent Quality of Service (QoS) models need to be set up in IP and ATM integration and they are difficult to coordinate. This gap is bridged when MultiProtocol Label Switching (MPLS) is used for IP-ATM integration purposes.
Guarantee of Service (GoS) allows MPLS to improve performance of privileged data flows in congested domains. We first discuss the GoS requirements for the utilization in conjunction with MPLS. Then we propose a minimum set of extensions to RSVP-TE that allow signaling of GoS information across the MPLS domain.
I. INTRODUCTION
Multiprotocol Label Switching (MPLS) is currently mainly used to provide Virtual Private Networks (VPNs) services [1]
or IP-ATM with QoS integration purposes [2], combining ATM traffic engineering capabilities with flexibility of IP and class-of-service differentiation [3]. In this way MPLS bridges the gap between IP and ATM avoiding the need of setting up independent QoS models for IP and for ATM, which are difficult to match. ATM switches can dynamically assign Virtual Path Identifier/Virtual Channel Identifier (VPI/VCI) values which can be used as labels for cells. This solution resolves the problem without the need for centralized ATM-IP integration servers. This is called Label-Controlled ATM (LC-ATM) or IP+ATM.
Like ATM Virtual Circuits (VCs), MPLS Label Switched Paths (LSPs) let the headend Label Edge Router (LER) control the path its traffic takes to a particular destination [4]. This method is more flexible than forwarding traffic based on destination address only. LSP tunnels also allow the implementation of a variety of policies related to network performance optimization [5][6]. For example, LSP tunnels can be automatically or manually routed away from network failures or congestion points. Resource ReSerVation Protocol (RSVP) is a signaling mechanism used to reserve resources for these LSP tunnels throughout a network. So MPLS reserves bandwidth on the network when it uses RSVP to build LSPs.
Using of RSVP to reserve bandwidth for a particular LSP introduces the concept of consumable resource in the network, that allows to RSVP nodes to find paths across the domain which have bandwidth available to be reserved. Unlike ATM, there is no forwarding-plane enforcement of a reservation. A reservation is made in the control plane only, which means that if a Label Switch Router (LSR) makes an RSVP reservation
for 10 Mb and later it needs 100 Mb, it will congest that LSP.
The network attempts to deliver that 100 Mb, damaging performance of other flows that can have even more priority, unless we attempt to police the flows using QoS techniques.
Although RSVP with Traffic Engineering (TE), (performance optimization of operational RSVP networks), is expected to be an important application in this problematic [7] an extended RSVP-TE protocol can be used in a much wider context for performance improvement. In this way, MPLS-TE is providing fast networks but with no local flow control, so assuming that devices are not going to fail or there will be no data loss.
However, resource failures and unexpected congestion cause a great part of lost traffic. In these cases, upper layers protocols will request lost data retransmissions at end points, but the time interval to obtain retransmitted data can be significant. For some types of services with special requirements of delay and reliability, as stock-exchange data or medical information, MPLS is not able to ensure that performance will not be worse due to lost traffic end-to-end retransmissions.
In this work we describe a set of extensions to MPLS RSVP-TE signaling required to support GoS over MPLS. This technique will allow to offer Guarantee of Service to privileged data flows [8], allowing discarded packets due to congestion to be locally recovered, avoiding in this way, as far as possible, end to end retransmissions requested by upper layers.
Following section shows what is GoS and how it can be applied to privileged flows in an MPLS domain. In the third section we study the RSVP-TE extensions to transport GoS information through a MPLS domain. In fourth section an analysis of the proposal is shown and finally this article concludes indicating the contributions of the research.
II. GOS OVER MPLS
The GoS capacities for a MPLS privileged data flow is the capacity of a particular node to local recovering of discarded packets belonging to the data flow. This work proposes up to four GoS levels (see Table I), codified with two bits; so each packet can be marked with this information throughout all the route. A greater GoS level implies a greater probability that a packet can be found in the GoS buffer of any node it has been passing through. Thus the need of end to end retransmissions is avoided, recovering lost data in a much rather local environment.