DETECTING SYSTEM NON-LINEARITIES BY MEANS OF HIGHER ORDER STATISTICS †A. RIVOLA, ‡P.R. WHITE † DIEM - University of Bologna - Viale Risorgimento, 2 - I-40136 Bologna - Italy
[email protected] ‡ Institute of Sound and Vibration Research - University of Southampton - Southampton SO17 1BJ, U.K.
ABSTRACT In the field of machine condition monitoring one can observe that there exists a link between machine vibrations and its health condition. Machine faults are often related to non-linearities occurring in the machine itself. This paper concerns the detection of non-linearities in mechanical systems by means of diagnostic techniques based on Higher Order Spectra (HOS). Since HOS give information about a signal’s non-Gaussianity, assuming the signal as an output of a system with a Gaussian input, then HOS allow one to provide information related to the non-linearity within the system. A simple model is used with the aim of showing the effectiveness of the normalised version of polyspectra in detecting different kinds of system non-linearities. In addition, HOS are used to interpret the signal structure and the system’s physical characteristics.
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, there exists a link between machine vibrations and the health condition; moreover, transducers can be easily attached to machines and measurements are quick and efficient [1]. 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, one can observe that machine faults are often related to non-linearities 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 detection of non-linearities in mechanical systems by means of diagnostic 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 field as diverse as vibration analysis, underwater acoustics, speech processing, chaos and condition monitoring [2-7]. 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. Unfortunately, there are still some difficulties in interpreting information obtained by these techniques and further investigations are required. Since only Gaussian process can be completely described by their second order statistics, HOS give information about a signal’s non-Gaussianity. A Gaussian input passing through a linear system leads to a Gaussian output. Therefore, assuming the signal as an output of a system with a Gaussian input, then HOS allow one to analyse the structure of the output signal and to provide information related to the non-linearity within the system. In this paper the bispectrum, the trispectrum and their normalised versions are used. They are briefly introduced in Section II 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 non-linearities. 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 [8]. Some difficulties arising in HOS estimation in the case of narrowband signals are discussed. In Section III 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). 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.
2 - THEORETICAL BACKGROUND 2.1 - The higher order measures Given a real valued signal x(t), its rth moment, denoted µr, is defined as:
µ r = E[xr (t)] ,
(1)
where E[] denotes the expectation operator [9]. The first moment (r=1) is the mean of the signal. For second and higher moments, it is often convenient to calculate moments about the mean, referred to as central moments; the rth central moment is given by:
µ c r = E[(x(t) - µ 1 ) r ] .
(2)
The second central moment (r=2), which measures the dispersion of the data about the mean, is called variance and is usually denoted as σ2. 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 [9]. The third moment about the mean, µc3, measures the asymmetry of the process’s PDF; it is often normalised as γ1=µc3/σ3 and the normalised version is called skewness. One can observe that a Gaussian process has null skewness. The fourth central moment, µc4, measures the PDF distribution near its centre. The normalised version of µc4 is termed kurtosis and is defined as γ2=µc4/σ4. The kurtosis value for a Gaussian process is three. 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, such as skewness (3rd-order) and kurtosis (4th-order), may provide details about the signal which the conventional second order statistics cannot. 2.2 - The bispectrum One of the most used tools in signal analysis is the power spectrum. For a discrete time series, the power spectrum can be defined in terms of the signal's Discrete Fourier Transform (DFT) X(k) as [4]:
S ( k ) = E [ X ( k ) X * ( k )] .
(3)
The power spectrum can be considered as the decomposition over frequency of the signal power, i.e. the signal’s second moment. Since it is a real quantity, it contains no phase information. 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 [10]. The power spectrum is of limited value in analysing vibrations where non-linearities are involved; whereas the bispectrum provides supplementary information. For a discrete time series, the discrete bispectrum B(k, l) can be defined as [4]:
B ( k , l ) = E [ X ( k ) X ( l ) X * ( k + l )] .
(4)
The bispectrum is complex-valued, contains phase information, and is a function of two independent frequencies, k and l. Since several symmetries existing in the bifrequency plane (k, l), it can be shown that it is only necessary to compute the bispectrum in the following region:
f {k , l}: 0 ≤ k ≤ s , l ≤ k , 2 k + l ≤ f s , 2
(5)
where fs is the sampling frequency [5, 11]. Such a non-redundant region is called the principal domain. As above mentioned, the bispectrum is closely related to the third-order moment of a signal; if a signal is not skewed, that is the skewness is zero, 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 is shown by equation (4). When the frequency components couple in such a way it means that there exists a quadratic non-linearity in the signal [12]. In other words one can say that the bispectrum is sensitive to a particular type of non-linearity, that is non-symmetric non-linearity.
2.3 - The trispectrum 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. For a discrete time series, the discrete trispectrum T(k, l, m) can be defined as [8]:
T ( k , l , m) = E [ X ( k ) X ( l ) X ( m) X * ( k + l + m)] .
(6)
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, as is shown by equation (6). Since the trispectrum is connected to the fourth order measures of a signal, that is the kurtosis, it can be employed to detect symmetric non-linearities. The trispectrum has 96 regions of symmetry [13]. 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. Above the plane m=0, all the frequencies are positive and so their sum is equal to the fourth frequency k+l+m. Below the plane, only k and l are positive, therefore this octant contains information about interactions between the sum k+l and the sum of the other two frequencies. 2.4 - Normalisation: the bicoherence and tricoherence There are two methods to estimate HOS: the indirect and the direct method [8]. 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 equations (4) and (6) 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 [12], 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 misleading interpretation of the bispectrum as discussed in [8]. 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 [12]:
b2 ( k , l) =
B( k , l)
[
E X ( k ) X (l)
2
2
] E[ X ( k + l ) ]
2 .
(7)
The bicoherence estimate has a variance which is approximately flat across all bifrequencies, as shown in [12]. Moreover, a useful feature of the bicoherence is that it is bounded between 0 and 1 [4-5]. Other methods to normalise the bispectrum are discussed in [8, 10]. The concept of normalisation can be extended to the fourth order case to produce the tricoherence which is defined as:
t 2 ( k , l , m) =
T ( k , l , m)
[
E X ( k ) X ( l ) X ( m)
2
2
] E[ X (k + l + m) ]
2 .
(8)
According to the direct approach described above, the bicoherence and the tricoherence can be calculated using the following estimators: 2
N
∑ X j ( k ) X j ( l ) X j * ( k + l)
∧2
b ( k , l) =
j =1
N
N
∑ X j ( k ) X j ( l) ∑ X j ( k + l ) j =1
2
j =1
2
,
(9)
2
N
∑ X j ( k ) X j (l) X j (m) X j * ( k + l + m)
∧2
t ( k , l , m) =
j =1
N
N
∑ X j ( k ) X j (l) X j (m) ∑ X j ( k + l + m) j =1
2
2
.
(10)
j =1
The main disadvantage in computing the HOS is the data length needed in order to ensure statistical confidence. In [14] the authors state 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 [8]. 2.5 - Pre-Whitening technique Some problems can occur in estimating the trispectrum of a narrowband signal. These problems are due to spectral bias errors as discussed in [8]. 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 around the peaks in the true trispectrum; in other words in the cases where the signal is narrowband. 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. In [8] the authors suggest to apply 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 is not necessary to normalise it to form the tricoherence. However, in order to get the advantage that the tricoherence is a bounded measure, it is still possible to compute the tricoherence of a pre-whitened signal. Theoretically, the trispectrum is unaffected by prewhitening technique because the operation is linear. However, small imperfections of the filter always exist but they are of little consequence.
3 - NON-LINEARITY ANALYSIS BY MEANS OF HOS 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 non-linearities. As mentioned in the introduction, considering the system with a Gaussian input, if the system is linear the output will have Gaussian PDF. 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 non-linearities and to study the structure of the system’s output signal. The considered 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 )
(11)
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 ) = µ1 k ( x − δ 1 ), x > δ1 µ k ( x + δ ), x < −δ 2 2 2
(12)
δ1
δ2 µ2 k
µ1 k
x m
k
f(t)
c
Fig. 1 - The SDOF model.
FR
FR
δ1
k x ( 1+ µ 1 )
x
δ2
x
k x ( 1+ µ 2 )
Fig. 2 - Restoring force: a) asymmetric; b) symmetric. The constants µ1 and µ2 are the ratio of the stiffness of an elastic stop 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 nonlinearities which may be related to various mechanical phenomena such as faults occurring in machine components. For example in [6-7] the authors employ a similar model to simulate a crack in a beam. They prove that the cracked beam is a system having asymmetric non-linearity and illustrate how the crack can be detected by means of bispectral analysis. With reference to the model used in this paper, it can simulate a similar kind of non-linearity when δ1=δ2=0 and µ1≠µ2. In this case the relationship between the restoring force FR and the displacement x of the system is depicted in Fig. 2(a). The impact in the kinematic pairs is another significant mechanical phenomenon which is associated to the non-linearity arising in the machine itself. This kind of fault is linked to the increment of the backlash in kinematic pairs - which is due to the wear - and it may deeply affect the machine performances [15]. Contrary to the previous case, the effect of the backlash increment may lead to symmetric non-linearities of the system. Therefore, the phenomenon cannot be described by the third order statistics and one has to extend the analysis to the higher order. The SDOF model can be used with the aim of demonstrating the capability of the fourth order spectrum to detect symmetric non-linearities. When δ1=δ2≠0 and µ1=µ2 the model simulates the effect of a symmetric backlash in the system: the relationship between FR and x is shown in Fig. 2(b). 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. Since all the spectra of order higher than two are equal to zero for a Gaussian process, HOS analysis can extract information regarding deviations from Gaussianity, that is deviations from linearity of the system. 3.2 - Asymmetric non-linearity The parameters of the SDOF model are adjusted in order to generate a system non-linearity as that displayed in Fig. 2(a). In particular both the clearances δi are zero and the stiffness ratios are µ2=9 and µ1=βµ2. The acceleration of the system is computed by means of numerical integration for different values of the coefficient β (β=1, 0.9, 0.8 and 0.7).
The power spectrum of the mass acceleration was obtained by using a DFT size of 512 and 64 averages. Fig. 3 presents the result of the power spectrum estimation. The shape of the power spectrum hardly changes with the decrement of the coefficient β; a significant higher harmonic component at the frequency of ≈70 Hz only arises for low values of β. The normalised higher order measures γ1 and γ2 (i.e. the skewness and the kurtosis) of the mass acceleration are also computed and are listed in Table 1. They show that the PDF of the system response is 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 third order spectrum, that is the bispectrum. As a matter of fact, the system has asymmetric non-linearity which should be detected by the third order spectrum. TABLE 1 Asymmetric non-linearity: features of the system response. Parameter β
Skewness γ1
Kurtosis γ2
Bicoherence maximum
Tricoherence maximum
1.0
-0.004369
3.3139
0.02539
0.009371
0.9
0.05881
3.4166
0.1216
0.01064
0.8
0.1302
2.9700
0.3109
0.01087
0.7
0.2042
2.9395
0.5366
0.008306
In order to demonstrate the sensitivity of the bispectral analysis to asymmetric non-linearity, the bicoherence of the mass acceleration was estimated. The results are displayed in Fig. 4; the bicoherence was computed from a total data length of 32768 with a DFT size of 128. The bicoherence appears to be very influenced by the system non-linearity as shown in Table 1 which reports the relationship between the parameter β and the value of the bicoherence maximum evaluated over the non-redundant region which is defined by equation (5). In particular, in Fig. 4(b-d) a significant bicoherence peak appears at the frequency pair (≈35, ≈35) Hz. This feature indicates that there exists a coupling between frequency components at the triplet (35, 35, 70) Hz. In fact, as stated in [7], the system has bilinear nature and its response exhibits higher harmonic components 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. 3). 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. 3(a) and 3(b) is that the resonance frequency has decreased: such an effect cannot be directly related to the form of the non-linearity.
PSD [dB]
(b)
β = 1.0
60 PSD [dB]
(a) 60 40 20 0
40 20 0
-20
-20 50 100 Frequency [Hz] (c)
60
150
0
50 100 Frequency [Hz] (d)
β = 0.8
60 PSD [dB]
0
PSD [dB]
β = 0.9
40 20 0
150
β = 0.7
40 20 0
-20
-20 0
50 100 Frequency [Hz]
150
0
50 100 Frequency [Hz]
Fig. 3 - Asymmetric non-linearity; power spectrum of the mass acceleration: a) linear system; b), c) and d): non-linear system.
150
β = 1.0
0.3 0.2 0.1 0 0
50 100 150 0
50
100
0.3 0.2 0.1 0 0
50 100 150 0
50
100
150
β = 0.9
0.3 0.2 0.1 50 100 150 0 (d)
Bicoherence
0.4
0.4
0 0
150
β = 0.8
(c) Bicoherence
(b) Bicoherence
Bicoherence
(a) 0.4
0.4
50
100
150
β = 0.7
0.3 0.2 0.1 0 0
50 100 150 0
50
100
150
Fig. 4 - Asymmetric non-linearity; bicoherence magnitude of the mass acceleration: a) linear system; b), c) and d): non-linear system. In order to more deeply inspect the nature of the system non-linearity, the tricoherence of the system response was also estimated. The pre-whitening technique was performed using a spectral estimate based on 512 points and the trispectral estimate employed 32 point segments. The last column of Table 1 reports the maximum value of the tricoherence: as one can see the maximum does not change significantly with the parameter β, confirming that the system has a non-linearity which is asymmetric. 3.3 - Symmetric non-linearity In order to simulate the mechanical phenomenon of the backlash, the stiffness ratio µi are assumed to have the same value that they have for the linear case (i.e. 9), whilst the value of the clearances δi is different from zero [Fig. 2(b)]. In particular, both the clearances have the same value which is increased from 0.0 to 0.8 mm: the set of δ1=δ2 values is listed in the first column of Table 2. TABLE 2 Symmetric non-linearity: features of the system response. Clearances δ1=δ2 [mm]
Skewness γ1
Kurtosis γ2
Bicoherence maximum
Tricoherence maximum
0.0
-0.004369
3.3139
0.02539
0.009371
0.025
-0.01149
3.5219
0.03147
0.03967
0.05
-0.003442
4.2024
0.03139
0.05799
0.1
-0.006482
5.5193
0.06954
0.1131
0.2
-0.03463
6.3674
0.07631
0.1538
0.8
0.004081
2.9725
0.02437
0.01074
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. As a matter of
(a)
60
δ 1 = δ 2 = 0.0 mm
PSD [dB]
PSD [dB]
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. 5. 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 nonlinearity. As mentioned above, in the case of the highest backlash [Fig. 5(f)], 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 [Fig. 5(a)].
40 20 0 -20 50 100 Frequency [Hz]
40 20 0
(c)
60
150
δ 1 = δ 2 = 0.05 mm
40 20 0
0
PSD [dB]
PSD [dB]
δ 1 = δ 2 = 0.025 mm
-20 0
-20
50 100 Frequency [Hz] (d)
60
150
δ 1 = δ 2 = 0.1 mm
40 20 0 -20
50 100 Frequency [Hz] (e)
60
150
δ 1 = δ 2 = 0.2 mm
0
PSD [dB]
0
PSD [dB]
(b)
60
40 20 0 -20
50 100 Frequency [Hz] (f)
60
150
δ 1 = δ 2 = 0.8 mm
40 20 0 -20
0
50 100 Frequency [Hz]
150
0
50 100 Frequency [Hz]
150
Fig. 5 - Symmetric non-linearity; power spectrum of the mass acceleration: a) linear system; b), c), d), e) and f): non-linear 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 to be expected as a good tool due to the symmetry of the non-linearity. As a matter of fact the maximum of the bicoherence is not affected by the backlash increment (see Table 2). Therefore, one has to look to the fourth order spectrum, that is the trispectrum. After pre-whitening, the tricoherence of the mass acceleration was estimated; the results are reported in Fig. 6 with reference to the same cases of Fig. 4. The tricoherence is displayed by drawing spheres in the trifrequency space: the size of the spheres represents the magnitude of the tricoherence [8]. For the linear system [Fig. 6(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. 6(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. As the resonant frequency decreases due to the increment of the backlash, the biggest spheres get closer to the origin of the trifrequency space. This means that the non-linearity is concentrated around the resonant frequency. Moreover, 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. The tricoherence of Fig. 6(f), which has no structure, confirms that in the case of high backlash the system develops into a linear system.
0.06 100
(a)
0.15 100
−100
(b)
−100
−100
100 100
−100
100 100
−100
−100
0.3 100
(c)
0.4 100
−100
(d)
−100
−100
100 100
−100
100 100
−100
−100 0.06
0.4 100
(e)
100
(f)
−100
−100
−100
100 100
−100
−100
100 100
−100
Fig. 6 - Symmetric non-linearity; tricoherence of the mass acceleration: a) linear system; b), c), d), e) and f): non-linear system.
4 - CONCLUSIONS 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 non-linearities 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.
Some theoretical background of HOS are given and it has been shown that they can provide information which second order statistics cannot. Since 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, HOS have been employed in order to relate the signal frequency content to the structure of any non-linearity present. In particular, the bicoherence (which is the normalised version of the third order spectrum) is sensitive to the asymmetric non-linearities. Conversely, it does not help to investigate symmetric non-linearities, therefore one has the need of the fourth order spectrum, that is the trispectrum. The 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 prewhitening before the trispectrum estimation. Both the examined polyspectra give a quantitative measure of the non-linearity degree of the system. Since machine faults are often related to non-linearities occurring within the machine itself and, consequently, to the machine vibration signature, this paper shows that there is scope for using the HOS as a condition monitoring tool.
ACKNOWLEDGEMENTS This work was partially supported by a grant from the CNR - Italian National Research Council.
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