Proceedings of the DETC99: 1999 ASME Design Engineering Technical Conferences September 12–15, 1999, Las Vegas, Nevada
DETC99/VIB-8332 USE OF HIGHER ORDER SPECTRA IN CONDITION MONITORING: SIMULATION AND EXPERIMENTS Alessandro Rivola DIEM, University of Bologna I-40136 Bologna, Italy.
[email protected]
Paul R. White Institute of Sound and Vibration Research University of Southampton Southampton SO 17 1BJ, U.K.
KEYWORDS: Higher Order Spectra; Non-linearity; Condition Monitoring; Machine faults.
ABSTRACT In the field of machine condition monitoring one can observe that a link exists between machine vibrations and its health condition, that is, there is a change in the machine vibration signature when machine faults occur. Damages occurring in machine elements are often related to non-linear effects, which may lead to non-linearities in the machine vibration. This paper concerns the study of some systems by means of techniques based on Higher Order Spectra (HOS). These techniques are particularly useful in the situation where only a single measurement sensor is available. If a process is Gaussian then HOS provide no information that cannot be obtained from the second order statistics. On the contrary, HOS give information about a signal’s non-Gaussianity. Since a Gaussian input passing through a linear system leads to a Gaussian output, assuming the signal as an output of a system with a Gaussian input, then HOS make it possible to analyse the structure of the output signal and to provide information related to the non-linearity within the system. A simple model is presented with the aim of showing the effectiveness of the normalised version of polyspectra in detecting different kinds of system non-linearities. HOS are used to interpret the signal structure and the system’s physical characteristics. Moreover, two experimental cases are presented. The HOS are applied to detect the presence of a fatigue crack in a straight beam and to analyse the vibration signal measured on a test bench for rolling element bearings. Both third and fourth order spectra seem to provide a possibility of using HOS as a condition monitoring tool. 1 INTRODUCTION In the field of manufacturing technology, the machine
condition monitoring significantly contributes to improve methods and techniques. Within condition monitoring techniques, the most established is vibration analysis which often produces satisfactory results. As a matter of fact, a link exists between machine vibrations and the health condition; moreover, transducers can be easily attached to machines and measurements are quick and efficient (Lyon, 1987). In addition, different mechanical phenomena (e.g. unbalance, gear mesh and faults, bearing faults, impacts) produce energy at different frequencies. Therefore, by signal processing techniques, diagnostic information can be obtained and the stage of the fault development can be evaluated. In particular, it can be observed that machine faults are often related to nonlinearities occurring in the machine itself (e.g. fatigue crack arising in a machine components, impacts in kinematic pairs, friction). This kind of phenomena leads to non-linearities in the machine vibration signature too. This paper concerns the analysis of some mechanical systems by means of signal processing techniques based on Higher Order Spectra (HOS) - a new area of signal processing - which has a wide variety of practical applications. HOS have been applied in fields as diverse as vibration analysis, underwater acoustics, speech processing, chaos and condition monitoring (Fackrell and Mc Laughlin, 1994; Nandi and Tutschku, 1994; Fackrell et al., 1995a; Fackrell et al., 1995b; Rivola and White, 1998a). These techniques are particularly useful when a single signal is available (i.e. vibration measurement using a single transducer) and are strongly sensitive to signal non-linearities. Since only Gaussian process can be completely described by their second order statistics, HOS give information about the signal’s non-Gaussianity. Considering the signal as an output of a system which has
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Gaussian input, the system may either be linear or non-linear. The case of a Gaussian input to a linear system leads to a Gaussian output, then HOS yield no information. Conversely, in the case of a Gaussian input passing through a non-linear system, the output signal will be non-Gaussian and so HOS allow the structure of the output signal to be analysed in order to provide information related to the non-linearity within the system (Collis et al., 1998; Rivola and White, 1998b). In this paper the bispectrum, the trispectrum and their normalised versions are used. They are briefly introduced in Section 2 from the frequency domain point of view. The bispectrum (i.e. third order spectrum) is a decomposition of the skewness of a signal over frequency and as such can detect asymmetric non-linearities. The bicoherence is a normalisation method for the bispectrum, which is often used in the detection of quadratic phase coupling, that is to say the interaction that can occur when a random signal is passed through systems containing quadratic non-linearities. Analogously, the trispectrum is the fourth order spectrum and is sensitive to the signal kurtosis; therefore it can detect symmetric nonlinearities. The normalisation of the trispectrum leads to the tricoherence which is sensitive to the cubic phase coupling. A useful feature of the bicoherence and tricoherence functions is that they are bounded between 0 and 1 (Collis et al., 1998). Some difficulties arising in HOS estimation in the case of narrowband signals are discussed. In Section 3 a simple single-degree of freedom (SDOF) model is used with the aim of showing the effectiveness of the bicoherence and the tricoherence in detecting different kinds of non-linearities (i.e. non-symmetric and symmetric). These non-linearities can be related to various mechanical phenomena such as the non-linear effects due to faults occurring in machine components. Since a system excited by random signal tends to distribute the energy between frequencies in a way that reflects the type of non-linearity contained in the system itself, particular effort is spent in order to relate information obtained from HOS to the signal frequency content and to the system’s physical characteristics. Section 4 presents two experimental cases. Firstly, the HOS are employed to analyse the dynamic behaviour of a straight beam with a fatigue crack. The beam is excited with Gaussian noise by means of a vibration shaker. The HOS show high sensitivity to the fatigue crack in the structure and the fourth order spectral analysis proves that the cracked beam could be considered as a system having a non-symmetric nonlinearity. Secondly, the HOS are applied to vibration signals measured on a test bench for rolling element bearings. Both sound and damaged condition of the bearing are considered; the fault is artificially produced. In the case of a bearing in good condition the normalised versions of HOS have low magnitude and no remarkable structure, whilst both bicoherence and tricoherence exhibit significant conformation and high magnitude in the case of a damaged bearing. In this case, particular effort is made in order to give an interpretation
of the bicoherence and tricoherence structure. The goodness of the experimental results seems to provide a possibility of using HOS as a condition monitoring tool. 2 THEORETICAL BACKGROUND 2.1 The higher order spectra. As is well known, one can define the rth statistical moments of real valued signal x(t) as µr = E[xr(t)], where E[·] denotes the expectation operator (Bendat and Piersol, 1980). The first moment is the mean of the signal, while the second is defined as the signal’s mean squared value. The moments are commonly defined about the mean so as to obtain the statistical central moment. The second central moment is called variance and is usually denoted as σ2. The third and fourth central moments are usually normalised; as a result of the normalisation the skewness and kurtosis are, respectively, obtained (Bendat and Piersol, 1980). The skewness, denoted as γ1, measures the asymmetry of the process’s PDF, and the kurtosis, γ2, is a measure of the degree of flatness of the PDF distribution near its centre. In the case of a Gaussian process - i.e. a stationary signal with a Gaussian probability density function (PDF) - the first and second order statistics completely describe the properties of the signal. In particular, a Gaussian process has null skewness while its kurtosis value is 3. One may conclude that, if a signal is non-Gaussian, then higher order moments are needed to completely describe its properties. Therefore, higher order measures may provide details about the signal which the conventional second order statistics cannot. One of the most used tools in signal analysis is the power spectrum. For a discrete time series x(n), the power spectrum can be defined in terms of the signal's Discrete Fourier Transform (DFT) X(k) as (Fackrell et al., 1995b): * S ( k ) = E[ X ( k ) X ( k )] ,
(1)
where the symbol * denotes the complex conjugate. The power spectrum can be considered as the decomposition of the signal power, i.e. the signal’s second moment, over frequency. HOS are the extension to higher orders of the concept of the power spectrum. The HOS are also called polyspectra. The third-order spectrum is termed the bispectrum and, in the same way that the power spectrum decomposes the power of a signal, the bispectrum decomposes the signal’s third moment over frequency. In other words, the bispectrum is related to the skewness of a signal and, as such, it may be used to detect asymmetric non-linearities (Hinich, 1982). For a discrete time series, the discrete bispectrum B(k, l) can be defined as (Fackrell et al., 1995b):
B( k , l ) = E[ X ( k ) X ( l ) X * ( k + l )] .
(2)
The bispectrum is complex-valued, contains phase
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information, and is a function of two independent frequencies, k and l. Since several symmetries exist in the bifrequency plane (k, l), it can be shown that it is necessary to compute the bispectrum only over a limited bifrequency region which is called the principal domain or the non-redundant region (Fackrell et al., 1995a; Nikias and Petropulu, 1993). The power spectrum is of limited value in analysing vibrations where non-linearities are involved; whereas the bispectrum provides supplementary information. The bispectrum is closely related to the third-order moment of a signal; if a signal is not skewed it will have zero bispectrum. Moreover, the amplitude of the bispectrum at the bifrequency (k, l) measures the amount of coupling between the spectral components at the frequencies k, l, and k+l, as shown by Eq. (2). When the frequency components couple in such a way it means that a quadratic non-linearity exists in the signal (Kim and Powers, 1979). In other words one can say that the bispectrum is sensitive to a particular type of non-linearity, that is non-symmetric non-linearity. Large numbers of systems are symmetrical and result in unskewed output signals. In these cases the third order spectrum contains no information. Consequently one has to extend the analysis to the fourth order spectrum which is called the trispectrum. As the bispectrum decomposes the skewness of a signal over frequency, in a similar way the trispectrum can be viewed as the decomposition of the signal's kurtosis over frequency. The discrete trispectrum T(k, l, m) can be defined as (Collis et al., 1998):
T ( k , l , m) = E[ X ( k ) X ( l ) X ( m) X * ( k + l + m)] . (3) The trispectrum has three independent frequencies and is sensitive to the coupling between the spectral components at the frequencies k, l, m and k+l+m. Since the trispectrum is connected to the fourth order measures of a signal, it can be employed to detect symmetric non-linearities. The trispectrum has 96 regions of symmetry (Chandran and Elgar, 1994). The principal domain partially fills the positive octant (k≥0, l≥0, m≥0) and the octant (k≥0, l≥0, m≤0) of the trifrequency domain. 2.2 The bicoherence and tricoherence functions. There are two methods of estimating HOS: the indirect and the direct method (Collis et al., 1998). The last one, which will henceforth be employed, is based on a segment averaging approach. The data is divided into N segments; an appropriate window is applied to each segment to reduce leakage; the quantities in Eq. (2) and (3) are computed for each segment by using the DFT; finally, the HOS are averaged across segments in order to reduce the variance of the estimator. It has been shown by Kim and Powers (1979), that the bispectral estimate obtained from this method has a variance which is proportional to the triple product of the power spectra. Consequently, the second-order properties of the signal
could dominate the estimate, which can result in a misleading interpretation of the bispectrum as discussed in (Collis et al., 1998). For this reason the bispectral analysis often deals with normalised versions of the bispectrum. One method for normalising the bispectrum employs the bicoherence, b2(k, l), which is defined by (Kim and Powers, 1979):
b2 ( k , l ) =
[
B( k , l )
E X ( k ) X (l )
2
2
] E[ X ( k + l ) ] 2
.
(4)
The bicoherence estimate has a variance which is approximately flat across all bifrequencies, as shown in (Kim and Powers, 1979). Other methods of normalising the bispectrum are discussed in (Collis et al., 1998; Hinich, 1982). The concept of normalisation can be extended to the fourth order, so the tricoherence can be defined as (Collis et al., 1998):
t ( k , l , m) = 2
[
T ( k , l , m)
E X ( k ) X ( l ) X ( m)
2
2
] E [ X ( k + l + m) ] 2
.
(5) A useful feature of both the bicoherence and tricoherence is that they are bounded between 0 and 1 (Fackrell et al., 1995a; Collis et al., 1998). The main disadvantage in computing the HOS is the data length needed in order to ensure statistical confidence. Dalle Molle and Hinich (1989) stated that the segment size should be the (r-1)th root of the total length of the data when handling the rth order spectrum. This means that for the computation of the trispectrum by using a DFT size of 64, the data length should be 643=262144, causing problems due to the fact that the data should be stationary over the measurement period. In the case of the computation of the bicoherence and tricoherence for a bandlimited signal, there are some problems due to small values occurring in the denominator of the estimator, which produce spurious effects. These effects can be reduced by adding a small constant to the denominator prior to calculating the bicoherence and the tricoherence (Collis et al., 1998). 2.3 Pre-whitening technique. Some problems due to spectral bias errors can occur in estimating the trispectrum of a narrowband signal (Collis et al., 1998). Bias errors can be reduced by means of large segments during the averaging process. However, the segment size is limited in the trispectral analysis owing to the large computational demand (the largest segment length that can reasonably be used is 64), therefore the estimated trispectrum may be significantly biased. The mentioned errors are considerable in case of narrowband
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signal. In particular, significant bias errors occur if the bandwidth of the signal is less than the width of the trispectrum cell, which depends on the sampling frequency and the number of samples contained in each segment. Collis et al. (1998) suggest applying a pre-whitening technique in order to cope with large bias errors. The methodology is applied to the signal in the time domain before the trispectrum is calculated: the spectrum of the signal is estimated (at high resolution); a linear phase finite impulse response filter - that has the same frequency response as the square root of the inverse of the spectrum of the signal - is designed; finally, the filter is applied to the original signal in order to produce a signal having an approximately flat spectrum. After the pre-whitening, the trispectral estimation errors should be reduced due to flat shape of the spectrum. The trispectrum of such a signal has spectral effects which have been removed and so it is not necessary to normalise it to form the tricoherence. However, in order to obtain the advantage of the tricoherence being a bounded measure, it is still possible to compute the tricoherence of a pre-whitened signal. Theoretically, the trispectrum is unaffected by pre-whitening technique because the operation is linear. However, small imperfections of the filter always exist but they are of little consequence.
linearities which may be related to various mechanical phenomena such as the non-linear effects due to faults occurring in machine components. For example Rivola and White (1998a) employed a similar model to simulate a fatigue crack in a beam. They proved that the cracked beam has a behaviour which is related to asymmetric non-linearities and illustrated how the crack can be detected by means of bispectral analysis. The present model can simulate a similar kind of non-linearity when δ1=δ2=0 and µ1≠µ2.
3.
Another significant mechanical phenomenon which is associated with the non-linearity arising in the machine itself is the impact in the kinematic pairs. This kind of fault is due to the increment of the backlash in kinematic pairs - as a result of the wear - and it may deeply affect the machine performances (Dalpiaz and Rivola, 1997). Contrary to the previous case, the effect of the backlash increment might lead to symmetric nonlinearities of the system. The SDOF model simulates a system with symmetric backlash when δ1=δ2≠0 and µ1=µ2. In the next sections some results of HOS analysis of the response of the SDOF system are presented and discussed in the case of symmetric and asymmetric non-linearity. The linear case is assumed to be characterised by the following parameters: m = 1 kg, c = 2.75 Ns/m, k= 5⋅103 N/m, µ1=µ2=9, δ1=δ2=0. In this case the linear radian frequency of the system is 223.6 rad/s (35.6 Hz). The excitation f(t) is random with Gaussian PDF. If such an excitation operates on the linear system the resulting output, that is the motion of the mass, will be Gaussian. On the contrary, the non-linear system leads to an output which is not Gaussian and as such cannot be described by second order statistics. In the latter case HOS can be employed to obtain information related to the system nonlinearities and to study the structure of the system’s output signal.
NON-LINEARITY ANALYSIS BY MEANS OF HOS: SIMULATION
3.1 The sdof model. In this Section, an analytical SDOF system is considered in order to show the effectiveness of the HOS to detect the presence of different kinds of system nonlinearities. The model is a forced oscillator with a pair of elastic stops with clearances δ1 and δ2 as shown in Fig. 1. The governing equation of motion of the system is:
m x + c x + k x + g( x ) = f ( t )
(6)
where m denotes the mass, c the damping coefficient, k the linear stiffness, and g(x) is a non-linear function of the displacement x defined as:
− δ 2 ≤ x ≤ δ1 0, g( x ) = µ 1 k ( x − δ 1 ), x > δ1 . µ k ( x + δ ), x < −δ 2 2 î 2
(7)
The constants µ1 and µ2 are the ratio of the stiffness of the elastic stops to the linear stiffness k. Obviously, when µ1=µ2=µ (i.e. the stiffness ratios are the same) and both the clearances δ1 and δ2 have zero value, the system is linear, has a linear stiffness k(1+µ) and its radian frequency is ω = k (1 + µ ) m . By modifying the values of the parameters µi and δi (i=1, 2) the SDOF model can simulate several kinds of non-
δ µ
δ
µ
Fig. 1 - The SDOF model.
3.2 Asymmetric non-linearity. The parameters of the SDOF model are adjusted in order to introduce an asymmetric non-linearity into the system. In particular, by assuming both the clearances δi to be zero, the stiffness ratio µ2=9, and
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µ1=βµ2 (with 0≤β≤1), the relationship between the restoring force, FR=kx+g(x), and the displacement, x, is consequently asymmetric with respect to the null position of the system mass.
β = 1.0
Amplitude [dB]
(a) 0 -10
0.2 0.1 0 0 150
50 [Hz] 100 0
50 100 Frequency [Hz]
10
100 50 [Hz] 150
150 0.4
β = 0.9 Bicoherence
(b) Amplitude [dB]
β = 1.0
(a)
0.3
-20 -30
0 -10
0
β = 0.9
(b)
0.3 0.2 0.1
-20 -30
0 0 0
50 100 Frequency [Hz]
10
150
-10
0.3
Bicoherence
0.4
-20
0
50 100 Frequency [Hz]
100 50 [Hz] 150
0
-30
150
50 [Hz] 100
β = 0.8
(c) Amplitude [dB]
Bicoherence
10
0.4
150
β = 0.8
(c)
0.2 0.1 0 0
Fig. 2 -Asymmetric non-linearity; power spectrum of the mass acceleration: a) linear system; b) and c): non-linear system.
Figure 2 presents the result of the power spectrum estimation of the mass acceleration; the estimate is obtained by using a DFT size of 512 and 64 averages. The shape of the power spectrum hardly changes with the decrement of the coefficient β: the resonant frequency slightly decreases and for the lowest value of β a higher harmonic component seems to arise at the frequency of ≈70 Hz.
0
150
50 [Hz] 100
100 50 [Hz] 150
0
Fig. 3 - Asymmetric non-linearity; bicoherence magnitude of the mass acceleration: a) linear system; b) and c): non-linear system.
The normalised higher order measures γ1 and γ2 of the mass acceleration are also computed and are listed in Table 1. They show that the PDF of the system response is
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predominantly asymmetric. In fact, the skewness increases with the decrement of β, whilst the kurtosis oscillates around 3. These higher order parameters give an overall measure of system non-linearity. More detailed information can be obtained by examining the HOS. Table 1 - Asymmetric non-linearity: features of the system response.
Parameter β Skewness γ1 Kurtosis γ2 max[b2(k,l)]
1.0 -0.00437 3.314 0.0254
0.9 0.0588 3.417 0.122
0.8 0.130 2.970 0.311
The third order spectrum, that is the bispectrum, should be able to detect the non-linearity because of its asymmetry. In order to demonstrate this, the bicoherence of the mass acceleration was estimated. The results are displayed in Fig. 3; the bicoherence is computed from a total data length of 32768 with a DFT size of 128. The bicoherence appears to be highly influenced by the system non-linearity, as shown in Table 1 where the value of the bicoherence maximum is evaluated over the non-redundant region. In particular, in Fig. 3(b-c) a significant bicoherence peak appears at the frequency pair (≈35, ≈35) Hz. This feature indicates that a coupling exists between frequency components at the triplet (35, 35, 70) Hz. In fact, as stated in (Rivola and White, 1998a), the system has a bilinear nature and its response contains the even harmonic 10
(b)
-10 -20
0
50 100 Frequency [Hz]
10 (c)
50 100 Frequency [Hz]
0
50 100 Frequency [Hz] (d)
-20
0
-20
10
-10
-30
-10
δ1 = δ2 = 0.05 mm
0
150
δ1 = δ2 = 0.025 mm
0
-30
150
Amplitude [dB]
Amplitude [dB]
10
δ1 = δ2 = 0.0 mm
0
-30
3.3 Symmetric non-linearity. In the following examples, the stiffness ratios µi are assumed to have the same value that they have for the linear case (i.e. 9), whilst the clearances δi are different from zero and have the same value which is increased from 0.0 to 0.8 mm step by step (see the first row of Table 2). In such a system there is then a symmetric backlash between the mass and the elastic stops. The restoring force FR can be considered as symmetric about the mass null position. The computation of the skewness and the kurtosis of the model response shows that the system has a non-linearity which is predominantly symmetric. In fact, as reported in Table 2, the skewness does not change significantly, while the kurtosis increases with the increment of the backlash. For the highest value of the clearance (i.e. 0.8 mm) the kurtosis falls to a value of about 3, which is characteristic of the response of a linear system with Gaussian input.
Amplitude [dB]
Amplitude [dB]
(a)
of the resonant frequency. The interaction between frequency components is evident in the bicoherence; on the contrary it scarcely appears in the power spectrum (see Fig. 2). One can observe that, even with the smallest degree of non-linearity (β=0.9), the bicoherence results in a clear indication of the presence of non-linearity and gives an idea of its nature. The only difference between the power spectra of Fig. 2(a) and 2(b) is that the resonance frequency has decreased: such an effect cannot be directly related to the form of the non-linearity.
150
δ1 = δ2 = 0.8 mm
0 -10 -20 -30
0
50 100 Frequency [Hz]
150
Fig. 4 - Symmetric non-linearity; power spectrum of the mass acceleration: a) linear system; b), c) and d): non-linear system.
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0.05 100
(a)
0.1 100
(b)
0.008
0.02
0
0
−100
−100
−100
100
0 100
−100
100
0
0
100
−100
0 −100
0.2 100
(c)
0.05 100
(d)
0.04
0.008
0
0
−100
−100
−100
100
0 100
−100
100
0
0
100
−100
0 −100
Fig. 5 - Symmetric non-linearity; tricoherence of the mass acceleration: a) linear system; b), c) and d): non-linear system. Table 2 - Symmetric non-linearity: features of the system response.
δ1=δ2 [mm] Skewness γ1 Kurtosis γ2 max[b2(k,l)] max[t2(k,l,m)]
0.0 -0.00437 3.314 0.0254 0.00937
0.025 -0.0115 3.522 0.0315 0.0397
0.05 -0.00344 4.202 0.0314 0.0580
0.8 0.00408 2.973 0.0244 0.0107
As a matter of fact, when the backlash reaches high levels the mass does not impact the stops and the system can be considered as linear. Actually, in this case the system is different from the original one due to the fact that the value of the stiffness is not affected by the stiffness of the elastic stops. The system changes are clearly visible in the power spectrum of the model response which is shown in Fig. 4. The resonant frequency decreases with backlash increment due to the decrement of the average stiffness. In addition, since the position of the resonant peak depends on the displacement x(t), when the backlash increases the resonant peak enlarges due to
the frequency modulation of the process. Moreover, wideband component arises at frequencies which are approximately three times the resonant peak. Such a phenomenon is related to the intrinsic nature of the system non-linearity. In the case of the highest backlash [Fig. 4(d)], the superharmonic band disappears and the power spectrum has only one peak which is shifted with respect to the resonant frequency of the original system. By observing the power spectrum trend and knowing the system, it is possible to obtain information related to the nonlinearity of the system itself. However, in general this is not the case, therefore HOS can be used in order to extract more details about the non-linearity within the system. In this case the third order spectrum is not expected to be a good tool due to the symmetry of the non-linearity. As a matter of fact the maximum of the bicoherence is hardly affected by the backlash increment (see Table 2). Therefore, one has to look to the fourth order spectrum. After pre-whitening performed with a spectrum based on 512 points, the tricoherence of the mass
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acceleration is estimated employing 32-point segments. The results are reported in Fig. 5 with reference to the same cases as Fig. 4. The tricoherence is displayed by drawing spheres in the trifrequency space: the size of the spheres represents the magnitude of the tricoherence (Collis et al., 1998). For the linear system [Fig. 5(a)], the tricoherence has no remarkable structure, whilst as the backlash increases, the conformation of the tricoherence acquires significance. The interaction between spectral components begins to appear even for the smallest value of the backlash (δ1=δ2=0.025 mm) as one can see in Fig. 5(b). In particular, denoting the resonant frequency of the system as fr, peaks arise at the triplet (fr, fr, fr) and the symmetrical reflections of this point. This means that the non-linearity is concentrated around the resonant frequency. In addition, the magnitude of the tricoherence is related to the degree of non-linearity as shown in Table 2 where the maximum of the tricoherence evaluated over the principal domain is listed. By taking into account more values of the backlash, the biggest spheres can be seen getting closer to the origin of the trifrequency space, because of the decrement of resonant frequency (Rivola and White, 1998b). The tricoherence of Fig. 5(d), which has no structure, confirms that in the case of high backlash the system develops into a linear system. 4.
APPLICATION OF HOS TO EXPERIMENTAL CASES
4.1 Test on a cracked beam. It is known that fatigue cracks in a structure are associated with non-linearities of the system. As a matter of fact, the crack alternately opens and closes: when the crack is closed the beam acts, approximately, as a homogeneous beam with no crack, while, when the crack is opened, a local reduction of flexural rigidity occurs (Ismail et al., 1990; Friswell and Penny, 1992). Traditional linear spectral analysis is obviously of limited value in studying vibrations where non-linearities are involved. Furthermore, linear spectral analysis requires two sensors. On the other hand the HOS analysis can be performed from a single sensor measurement and, as shown above, provides useful information about non-linearities in a system. In this Section, some of the results of the experimental tests conducted by Rivola and White (1998a) are summarised and additional HOS analysis is performed. These tests were carried out on a bronze straight beam of uniform cross section which was excited by means of a vibration shaker fed with Gaussian noise. Two conditions were examined: the integral beam and the beam with a fatigue crack. More details on the tests can be found in Rivola and White (1998a). The severity of the structural damage is expressed in terms of the ratio between its depth, a, and the height of the beam, h (see Fig. 6). In this paper only the cases a/h = 0.0, that is no crack is present, and a/h = 0.6 are discussed. The analysed signal is the vibration response of the beam
due to the Gaussian excitation. It is measured by means of an accelerometer located at the free end of the beam. The sampling frequency is 1500 Hz and a signal length of 32768 points is taken into account. Firstly, the higher order measures (i.e. the skewness and the kurtosis) of the beam acceleration response in the time domain are considered. The values of these statistical parameters are reported in Table 3: they do not give clear information about the beam condition because of their small changes. The linear spectral analysis is then carried out. Figure 7 shows the power spectra of the vibration signals; they are computed by averaging 64 segments which have 512 points each (hanning window is used). The presence of the crack causes small changes in the shape of power spectrum. In particular the resonant frequencies decrease in the case of a damaged beam, but this phenomenon might not be related to the presence of the crack. As a matter of fact the modal properties of the system may change due not only to structural damage, but also to other factors, such as environmental factors (e.g. the temperature). Due to the non-linear behaviour of the damaged beam, the bispectral analysis can be used as a crack detection tool (Rivola and White, 1998a). Figure 8 reports the result of the bicoherence estimate in case of undamaged structure and when the fatigue crack is present. The bicoherence is computed from the total data length with a DFT size of 128 points. It clearly appears that the bicoherence is very sensitive to the presence of the damage: the bicoherence rises in the case of the cracked beam, thus showing a significant coupling between frequencies due to the system non-linearities. It is noteworthy that the nonlinear behaviour of the system is linked to the presence of the crack; thus, the arising of bicoherence peaks is undoubtedly related to the structural damage. In order to better interpret the results achieved by means of the bispectral analysis it may be useful to inspect the power spectrum of Fig. 7(b): this shows a weak peak at about 570-575 Hz which is not visible in the case of the integral beam. Such a peak is due to the non-linear behaviour of the cracked beam. As a matter of fact, in the bicoherence plot of Fig. 8(b), the highest peak is located at the frequency pair (167, 406) Hz, which indicates a coupling between frequency components at the triplet (167, 406, 573) Hz, that is, between the second (167 Hz) and the third (406 Hz) beam’s resonance, and the small peak of the power spectrum at about 573 Hz. In order to analyse in detail the non-linear behaviour of a beam with a crack, the tricoherence of the system response is also estimated. The tricoherence is estimated by employing 32point segments. The pre-whitening technique is previously applied to the data by using a spectral estimate based on 512 points. The maximum of the tricoherence evaluated on the principal domain is reported in the last row of Table 3. As one can see, the maximum does not change with the presence of the crack, proving that the system has a non-linearity which is
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predominantly asymmetric. Table 3 - Features of the beam acceleration response.
a/h Skewness γ1 Kurtosis γ2 max[b2(k,l)] max[t2(k,l,m)]
0.0 -0.0131 3.229 0.0920 0.0293
0.6 0.0487 3.0678 0.389 0.0298
a
machinery. A large number of vibration signal processing techniques for bearing condition monitoring have been published in the literature across the full range of rotating machinery. The more common techniques are time and frequency domain techniques. These comprise techniques such as RMS, crest factor, probability density function, correlation function, power spectral density functions, cepstrum analysis, narrow band envelope analysis, shock pulse method (Braun and Datner, 1979; Mc Fadden and Smith, 1984; Wang and Harrap, 1996).
h (a)
Beam Bicoherence
Crack Accelerometer Shaker
0.2
0 0
Fig. 6 - Set-up of the experimental apparatus.
500 (a)
a/h = 0.0
[Hz]
500
40
250 [Hz] 750
0
30 (b) 20 10 0
0
200 400 Frequency [Hz]
600
(b)
0.2
0 0
50 a/h = 0.6
40
750
250 500
30 [Hz] 20
500
250 [Hz] 750
0
Fig. 8 - Bicoherence of the beam acceleration: a) integral beam; b) cracked beam.
10 0
a/h = 0.6
0.4 Bicoherence
Amplitude [dB]
750
250
50
Amplitude [dB]
a/h = 0.0
0.4
0
200 400 Frequency [Hz]
600
Fig. 7 - Power spectrum of the beam acceleration: a) integral beam; b) cracked beam.
4.2 Vibration signals from a test bench for rolling bearings. Rolling element bearings are among the most common components to be found in industrial rotating
It is well known that a bearing subjected to normal loading will fail due to material fatigue after a certain running time. The surface damage severely disturbs the rolling motion of the rolling elements. When a defect in one surface of a rolling element strikes another surface, it produces an impulse which may excite resonances in the bearing and in the machine structure. The impulses occur periodically with a
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frequency which is determined by the location of the defect. These impulses represent the forcing function, and as such form the input of the mechanical system comprising the bearing and the machine structure. The vibration which follows is the response of the system to this excitation and will contain the system resonances modulated at the characteristic defect frequency (Braun and Datner, 1979; Mc Fadden and Smith, 1984). The system resonances could be the action of the rolling element vibrating between the inner and outer races, the vibration of the races themselves in a ring mode, the vibration of the bearing housing or machine structure, or maybe a combination of these. In the following, HOS are applied to vibration signals measured from a test bench for rolling bearings. The vibration was sensed by a piezoelectric accelerometer mounted on the test bench casing close to the bearing to be monitored, which was a FAG 1024 double row self-aligning ball bearing. The measurements were carried out on the bearing both in sound condition and with a single artificial fault that was created by means of an electric pen on the outer race of the bearing to simulate a spalling damage. The bearing was placed in pure radial load of 500 N. Throughout the tests, the shaft was driven at a constant speed of 1600 rpm; the bearing had a fixed outer race and, from the knowledge of the bearing geometry, the nominal outer race element passing frequency at this shaft speed was calculated to be 130 Hz. More details on the test apparatus can be found in Rubini and Meneghetti (1998). The aim of the analysis is not to perform a diagnosis of the bearing fault (i.e. to determine the characteristic frequency of the damage), but to interpret and explain the results of the HOS analysis applied to experimental signals measured on a machine casing which contains a damaged mechanical component. The vibration signals are sampled at 12800 Hz and the data size used is 262144 for both bispectral and trispectral analysis. The results of the HOS are reported in Fig. 9 and Fig. 10; in particular Fig. 9 shows the plots of the bicoherence function, estimated by using a DFT length of 256 points, over the non-redundant region, and Fig. 10 displays the tricoherence of the measured signals which is computed with a DFT size of 64. When the bearing is in normal condition the machine casing vibration can be considered as the output of a system with a random input. In this case, assuming the system comprising the bearing and the machine structure to be linear and the input to be almost Gaussian, the system response will also be approximately Gaussian. These assumptions are confirmed by the results of HOS analysis. As a matter of fact the bicoherence is almost flat over the bifrequency plane [see Fig. 9(a)] and the tricoherence function has very low value and no significant structure [Fig. 10(a)]. Actually, the bicoherence plot of Fig. 9(a) is not exactly flat within some regions of its domain; this could be due either to the imperfect linearity of the mechanical system or to the
input non-Gaussianity. In the faulty state (i.e. there is a local bearing defect) both the bicoherence and tricoherence show high increment and conformation changes, as can be seen in Fig. 9 and Fig. 10. The maximum values of the normalised HOS are listed in Table 4 for both the bearing conditions; the tricoherence maximum is particularly sensitive to the presence of the damage. As a matter of fact, the ratio of the tricoherence maximum in the case of damaged bearing to the value in sound conditions is about 25. Table 4 - Test on rolling bearing: results of the HOS analysis performed on the casing acceleration.
Condition max[b2(k,l)] max[t2(k,l,m)]
normal 0.245 0.0251
fault 0.639 0.642
The explanation of the HOS changes might be the following. When the fault occurs, the input of the system can be considered as an impulse train which is due to the contact between the defect and the mating surfaces in the bearing. The vibration response which follows will have strong periodic content; in particular it will consist of periodic bursts of exponentially decaying sinusoidal vibration and, obviously, noise. The frequency of vibration is due to the natural frequencies of the system which are modulated by the characteristic defect frequency (Braun and Datner, 1979; Mc Fadden and Smith, 1984). As stated in (Fackrell et al., 1995a; Fackrell e al., 1995b) the bicoherence of such a signal should theoretically take the “Bed of Nails” form but, in practice, this happens only partially because of the narrow-band nature of the system’s impulse response function which causes the output signal to be dominated by measurement noise in the frequency region where the impulse response function is very small. Since the noise is likely to have a symmetric PDF, it will have a nearzero bicoherence. A diagnosis of the defect could be performed by observing that the location of the bicoherence peaks is related to the coupling between the system resonances and the characteristic bearing defect frequencies (Braun and Datner, 1979; Mc Fadden and Smith, 1984; Li et al., 1991), but this is outside the scope of this paper and will be investigated in a future study. A similar explanation can be given for the tricoherence function in the case of the damaged bearing. If the impulse response function of the system is wideband, the tricoherence of the response to an impulse train will take the form of a “Box of Balls” (Collis, 1996) but, as mentioned above, the impulse response of the considered system has narrow-band resonances. Nevertheless, one should remember that, in order to overcome problems due to large bias errors, it is opportune to apply a pre-whitening technique, as mentioned in Section 2.3. Therefore the system response will have an approximately flat spectrum which would allow the “Box of Balls” structure.
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However, in the present trispectral analysis, the 64-point DFT does not actually make it possible to have an impulse train contained in each averaging segment. In other words the tricoherence resolution is not high enough to obtain the form of the “Box of Balls”. On the other hand, to employ a longer DFT segment would require too large a computational effort and too much data.
signature when machine faults occur. Machine faults are related to non-linear effects, which may lead to non-linearities in the machine vibration signature.
(a)
0.1
5000
(a) Bicoherence
0.6 0
0.4 0.2
−5000
0 0
−5000
6000
0 5000
4000
[Hz] 4000
5000
0
6000
2000 2000 [Hz]
−5000
(b)
0
0.65
5000
(b) Bicoherence
0.6 0
0.4 0.2 −5000
0 0
−5000
6000
2000 4000
[Hz] 4000 6000
2000 [Hz] 0
Fig. 9 - Test on rolling bearing; bicoherence plot: a) normal condition; b) fault condition.
In spite of these limitations and effects, by comparing Fig. 10(a) and Fig. 10(b) the tricoherence function is clearly able to distinguish the presence of the damage even if there are still some difficulties in interpreting its meaning and the diagnosis of the defect is not an easy task. Therefore further investigations are required, but it seems that the tricoherence might be a useful tool in monitoring the machine health condition. 5 CONCLUSIONS In the field of machine condition monitoring it is worth noting that there is a change in the machine vibration
0
5000 0
5000 −5000 Fig. 10 - Test on rolling bearing; tricoherence: a) normal condition; b) fault condition.
In this paper a SDOF model has been used in order to simulate several non-linearities which can occur in mechanical systems. In particular, asymmetric and symmetric nonlinearities have been examined. Considering the input of the system as Gaussian, the model response non-Gaussianity is related to the non-linearity existing in the system. In particular, a system excited by Gaussian input distributes the energy between frequencies in a way that reflects the type of non-linearity contained in the system itself. The HOS have been employed in order to relate the frequency content of the model response to the structure of any non-linearity present. It has been shown that HOS can provide information which second order statistics cannot. The application of HOS to the simulated signals proves that the bicoherence (i.e. the normalised version of the third order spectrum) is sensitive to the asymmetric non-linearities.
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Conversely, it does not help to investigate symmetric nonlinearities, therefore one has the need of the fourth order spectrum, that is the trispectrum. This latter polyspectrum has been adopted to study the behaviour of a system with backlash and to interpret the system physical characteristics. The analysed signals have narrowband characteristics, therefore one has to perform the pre-whitening before the trispectrum estimation. Both the examined polyspectra give a quantitative measure of the non-linearity degree of the system. Two experimental cases are also presented. The HOS are employed to analyse the dynamic behaviour of a straight beam with a fatigue crack. The beam is excited with Gaussian noise by means of a vibration shaker. The bicoherence shows high sensitivity to the fatigue crack and gives information about the frequency content of the system response which the power spectrum analysis cannot. Therefore, the bicoherence is an useful crack detection tool. In addition the fourth order spectral analysis, namely the tricoherence function, proves that the cracked beam could be considered as a system having a nonsymmetric non-linearity. Further, the HOS are applied to vibration signals measured on a test bench for rolling element bearings. Both sound and damaged condition of the bearings are considered. In the case of a bearing in good condition both bicoherence and tricoherence have low magnitude and no remarkable structure, whilst they exhibit significant conformation and high magnitude in the case of a damaged bearing. Particular effort is dedicated to interpreting the structure of HOS. The experimental results presented in this paper show that there is scope for using HOS as a condition monitoring tool.
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Fackrell, J. W. A., and Mc Laughlin S., 1994, “The Higher Order Statistics of Speech Signals,” Proceedings of the IEE Colloquium on Techniques in Speech Signal Processing, London, UK, Vol. 138(7), pp. 1-6. Fackrell, J. W. A., White, P. R. , Hammond, J. K., Pinnington, R. J., and Parsons A. T., 1995a, “The Interpretation of the Bispectra of Vibration Signals: Part 1: Theory,” Mechanical Systems and Signal Processing, Vol. 9(3), pp. 257-266. Fackrell, J. W. A., White, P. R. , Hammond, J. K., Pinnington, R. J., and Parsons A. T., 1995b, “The Interpretation of the Bispectra of Vibration Signals: Part 2: Experimental Results and Applications,” Mechanical Systems and Signal Processing, Vol. 9(3), pp. 267-274. Friswell, M. I., and Penny, J. E. T., 1992, “A Simple Nonlinear Model of a Cracked Beam,” Proceedings of X International Modal Analysis Conference, San Diego, CA, Vol. 1, pp. 516-521. Hinich, M. J., 1982, “Testing for Gaussianity and linearity of a stationary time series,” Journal of Time Series Analysis, Vol. 3(3), pp. 169-176. Ismail, F., Ibrahim, A., and Martin, H. R., 1990, “Identification of Fatigue Cracks from Vibrating Testing,” Journal of Sound and Vibration, Vol. 140(2), pp. 305-317. Kim, Y. C., and Powers, E. J., 1979, “Digital Bispectral Analysis and its Applications to Nonlinear Wave Interactions,” IEEE Transactions on Plasma Science, PS-7, pp. 120-131. Li, C. J., Ma, J., Hwang, B., and Nickerson, G. W., 1991, “Bispectral analysis of vibrations for bearing condition monitoring,” Proceedings of the 3rd International Machinery Monitoring and Diagnostics Conference, Las Vegas, Nevada, pp. 225-231. Lyon, R. H., 1987, “Machinery Noise and Diagnostics,” Boston: Butterworths. McFadden, P. D., and Smith, J. D., 1984, “Vibration monitoring of rolling element bearing by the high frequency resonance technique - a review,” Tribology International, Vol. 17(1), pp. 3-10. Nandi, A., and Tutschku, K., 1994, “Machine Condition Monitoring Based on Higher Order Spectra and Statistics,” Proceedings of ATHOS 94, Edinburgh, UK. Nikias, C. L., and Petropulu, A. P., 1993, “Higher - Order Spectra Analysis. A non-linear signal processing framework,” New Jersey: Prentice Hall. Rivola, A., and White, P. R., 1998a, “Bispectral Analysis of the Bilinear Oscillator with Application to the Detection of Fatigue Cracks,” Journal of Sound and Vibration, Vol. 216(5), pp. 889-910. Rivola, A., and White, P.R., 1998b, “Detecting system nonlinearities by means of higher order statistics,” Proceedings of the 3rd International Conference on Acoustical and Vibratory Surveillance Methods and Diagnostic Techniques, Senlis, France, Vol. 1, pp. 263272. Rubini, R., and Meneghetti, U., 1998, “Use of the Wavelet Transform for the diagnosis of incipient faults in ball bearings,” Proceedings of the 3rd International Conference on Acoustical and Vibratory Surveillance Methods and Diagnostic Techniques, Senlis, France, Vol. 1, pp. 371-378. Wang, W. Y., and Harrap, M. J., 1996, “Condition monitoring of ball bearings using envelope autocorrelation technique,” Machine Vibration, Vol. 5, pp. 34-44.
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