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Vibration Analysis with Generalised Norms in Condition Monitoring 583

Institut für Bergwerks- und Hüttenmaschinenkunde, RWTH Aachen, 2008

Vibration Analysis with Generalised Norms in Condition Monitoring

Prof. D.Sc.Tech. S. Lahdelma,

M.Sc. Tech. E. Juuso

Universität Oulu, Finnland

Prof. Dr.-Ing. habil. J. Strackeljan

Otto-von-Guericke Universität Magdeburg Institut für Mechanik, Lehrstuhl für Technische Dynamik

Abstract Advanced signal processing methods combined with automatic fault detection enable reliable condition monitoring for long periods of continuous operation. Root-mean-square and peak values obtained from vibration signals are useful features for detecting various faults. Unbalance, misalignment, bent shaft, me-chanical looseness and some electrical faults, for example, can be detected using features of displacement and velocity. Higher order derivatives provide additional possibilities for detecting faults that introduce high-frequency vibra-tions or impacts. Real order derivatives increase the number of signal alterna-tives. New generalised moments and norms related to lp space have been used for diagnosing faults in a roller contact on a rough surface. Kurtosis provides a strong indication if the order of derivation, , is at least 4. For peak values, the change is smaller but already starts at .3 The generalised moment and norm can be defined by the order of derivation, the order of the moment, p, and sample time, . Reliable results can be obtained by relative norms if and p are in the range between 4 and 6. For the impact of a small scratch, sensitivity is further improved with short sample times, but several sequential samples are required to guarantee the detection of impacts. Then the order p can also be reduced.

1 Introduction Any attempt to detect different types of machine faults reliably at an early stage requires the development of improved signal processing methods. Vibration measurements provide a good basis for condition monitoring. Early vibration measurements by means of mechanical or optical instruments used displace-ment x = x(t). The next step was the adoption of velocity, i.e. x(1) signals, which were obtained either by differentiating displacement or using sensors whose

584 Vibration Analysis with Generalised Norms in Condition Monitoring


output was directly x(1). The drawback with the signals x and x(1) is that they do not usually allow the detection of impact-like faults at a sufficiently early stage. Examples of these faults are defective bearings and gears. However, e.g. un-balance and misalignment can be detected successfully with x and x(1). Acceleration measurements have been performed more frequently upon the introduction of accelerometers. The signals x and x(1) can be obtained from the x(2) signal through analogue or numerical integration. The jerk, i.e. x(3) signal, has been used for examining slowly rotating bearings [1]. The first time deriva-tive of acceleration had been used earlier for assessing the comfort of travelling e.g. in designing lifts. Higher, real and complex order derivatives bring addi-tional methods to signal processing [2,3,4,5]. Different approaches have been reviewed in [6]. Vibration indices based on several higher derivatives in different frequency ranges were already introduced in 1992 [2]. Fractional integrals and derivatives are discussed in [7]. Higher and real order derivatives in processing vibration measurements and feature extraction by generalised moments and lp norms have been discussed in [8,9,10,11]. This paper deals with real order derivatives and different features in diagnosing faults in a roller contact with a rough surface of an inclined plane.

2 Signal Processing Some faults, such as unbalance, misalignment, bent shaft and mechanical looseness, can be detected by means of displacement and velocity, i.e. signals

)0(xx and )1(xx . On the other hand, bearing faults or faults in gears, as well as cavitation can be detected more efficiently with the acceleration signal. Often higher order derivatives provide more sensitive solutions, i.e. the ratios of fea-tures between the faulty and non-faulty cases become higher [6,12,13]. The presence or absence of bearing faults cannot always be determined from the raw acceleration signal. In general, the signal contains a multitude of differ-ent vibration components. To get a better understanding of the random vibration generated by rough surfaces, we tried to isolate all periodic, load- dependent and external vibration sources. This led to a very simple test rig describing the pure vibration generated by a rolling ball on an inclined plane [14]. Different sur-face structures are achieved by means of different types of files and a fault was simulated by scratching the file surface. This model could describe the realistic situation in many contact problems. The signal contains components of random excitation during the contact of the roller with a more or less rough surface and possibly a contact with an imperfect surface area. High frequent vibrations are excited as the rolling element passes the damaged area. Obviously the rolling ball on the file will not generate a periodic excitation like a roller bearing, be-cause there is only one contact situation during the measuring time. A set of features to classify surfaces with different roughness is described in [14]. For the investigations presented here only one file type was used.



The raw signals x(2) were measured with a sampling rate of 131072 Hz from a rough file by means of an accelerometer. The signals with a length of 0.5 s shown in Figure 1 are obtained from a file without a scratch (Figure 1a) and from a file with a small scratch on the surface (Figure 1b). From the original and non-filtered time signal, it is not possible to detect the impact of the small scratch. But after zooming into the range from 0.03 to 0.05 s, one can see how the signal structure of the faulty file changes at about 0.041 s (Figure 1c). A spike can be seen at about 0.041 s if a band pass filter (10-40 kHz) is used (Figure 1d). Another way of making this change noticeable is the calculation of the amplitude spectra of the signal with data sets before and after the excite-ment (Figure 2).

Figure 1: Time domain analysis of the signals caused by a roller contact on a rough surface with and without a scratch.

Two sets (each 1024 data points) were taken out of x(2). The first data set ends at 0.04 s and the second one starts at 0.04 s. The comparison of amplitude spectra in Figure 2 shows that a high frequent vibration generated by the rolling contact between the ball and the scratch is not present before the excitement. The spectrum on the right shows the high frequent vibration in 40 – 60 kHz range shortly after the excitement.



Figure 2: Amplitude spectra of the signal shown in Figure 1c: before and after the impact of the scratch. The severity of faults can be assessed by comparing signals and features in different orders of derivation [3,4,15]. For sinusoidal signals tXtxx sin)( we obtain

),sin()2

sin()(

tXtXxdt

xd.................... (1)

where is a real number, the amplitude ,XX and the change of the

phase angle . Figure 3 shows the two signals from the scratched surface

problem in two different orders of derivative.

Figure 3: Higher derivatives of order 3 and 4.5 calculated from the raw signal which is generated by a roller contact on a rough surface with a scratch (fault) and without a scratch (no fault).



3 Feature extraction 3.1 Statistical features Real order derivatives may improve the sensitivity of the features. The file signal has been analysed with kurtosis and peak value (Figure 4). The maximum sen-sitivity for kurtosis was 14.2, which was achieved with 75.3 , as compared with 1.3 obtained for the acceleration ( 2 ). The maximum sensitivities of the peak value were 4.00 and 1.6 for α values 5.5 and 2, respectively. In the range

3,2 , the effect of the impact is hidden by the random variation of the sig-nals. Kurtosis provides no indication in this range, and peak value is already slightly higher for 3 . However, it is hardly enough for automatic fault diag-nosis. The sensitivities are problem-specific, e.g. for an inner race fault in a roller bear-ing, the sensitivity for kurtosis (Figure 5 a) and peak value (Figure 5 b) [4] have a clear maximum. The measurements were performed in the frequency range 3-2000 Hz, and the rotation frequency was 2 Hz. For signals with a constant amplitude X, the derivation results increase with srad /1 if .0 A detailed discussion about the selection of the order of derivation is available in [3,4,5,15].

Figure 4. Sensitivity curves for kurtosis and peak value obtained from signals )(x by using a step 0.25 for .

(a) Kurtosis [4]. (b) Peak values [4].

Figure 5. Sensitivity curves for an inner race fault in a roller bearing.



3.2 Moments and norms New features are calculated by means of a generalised moment about the ori-gin:

,11

)(

N

i

p

ip x

NM

............................................. (2)

where the real number is the order of derivation, the real number p is the or-der of the moment, is the sample time, i.e. the moment is obtained from the absolute values of signals )(x . The number of signal values sNN where Ns

is the number of samples per second. Alternatively, the signals values )(ix can

be compared to the mean )(x :

,1 )(

1

)( pN

ii

p xxN

M

................................... (3)

The generalised central moment can be normalised by means of the standard deviation of the signal )(x :

.1 )(

1

)( pN

iip

p xxN

M

........................ (4)

which was presented in [8]. The order of derivation ranges from 1 correspond-ing to velocity to 4, which corresponds to the signal x(4). The moment 12

M ,

and the moment 4

M corresponds to the kurtosis of the signal. The standard

deviation can be obtained from (3) by taking the square root [9]:

.)1()( 2/12)(

1

)(2/12

xx

NM

N

ii

............ (5)

There are many alternative ways of normalisation, e.g.

)(

1

)(1 1

xxN

MN

ii

....................................... (6)

can be used. The norm can be defined for example by

,)1()( /1

1

)(/1 pN

i

p

ippp x

NMM

................ (7)

which is the lp norm

.)(

p

p xM

.................................................... (8)



This norm has same dimensions as the corresponding signals )(x . The lp norms are defined in such away that .1 p In this study, the order p is al-lowed to be less than one. The absolute mean,

,11

)()(

1

)(

N

iiav x

Nxx

...................................(9)

and the rms value,

,)1( 2/1

1

2)()(

2

)(

N

iirms x

Nxx

.......................(10)

are special cases of (8). Faults can also be detected with other types of norm, e.g. maximum norm

,max )(

,...,1

)( i

nixx

...........................................(11)

or norm

,0

)(

n

iC

i

nxx

..............................................(12)

which is the sum of the norms obtained for different orders of derivatives. In a special case, integer orders can be used: .,...,1,0 10 nn

The rms values of displacement and velocity can be obtained from (10) by using the values zero and one for the order of derivation, respectively:

,)1( 2/1

1

22

N

iirms x

Nxx .................................(13)

and

.)1( 2/1

1

22

N

iirms v

Nvv .................................(14)

These norms can be used for detecting unbalance, misalignment, bent shaft and mechanical looseness. For bearing faults, displacement and velocity should be replaced by acceleration or higher derivatives. For a very low rotation speed, the rms values are not sensitive for bearing faults, because the effect of few weak impacts is small in the sum (10) where N is a large number. For high rota-tion speeds, frequent strong impacts affect rms values significantly. The rms values can then be used to detect bearing faults, especially if 2 . Bearing faults in slowly rotating machinery can be detected with the maximum norm presented by (11) which in diagnostics is called peak value. To avoid the domi-nation of a distinct peak, the peak value can be calculated e.g. as an average of the highest three peaks.



3.3 Order of moment and sampling time The peaks of the signal have a strong effect on the moments (2), (3) and (4). The norm (7) combines two trends: a strong increase caused by the power p and a decrease with the power 1/p. For the order p = 1, there is no amplifica-tion. The significance of the highest peaks will decrease if p < 1. The moment (4) can be used in the same way as kurtosis in the previous studies [8]. The moments calculated for higher order derivatives are more sensitive to impacts than the ones calculated for the acceleration. The sensitivity of the moment im-proves when the order p of the moment increases. In Figure 6, the norm (7) was used in a relative form, i.e. the norm of the signal in faulty case (Figure 1b) is divided by the norm calculated from the signal in the non-faulty case (Figure 1a). The sample time 5.0 is equal to the complete measurement time.

Figure 6. Relative norm, pMrelative 5.0 , for the file signals.

In this application, a high value for α and p gives the highest sensitivity for the detection of the difference between the two states. It is interesting to see that α works like a filter. For 3 , the sensitivity is too low and does not improve with increasing order p. When 3 , the increase of p will raise the sensitivity. The signal can be analysed in short samples and features calculated for these samples. The norm values of several samples are then combined in fault detec-tion, e.g. by calculating an average from the highest three values of the sample



features or by taking the maximum of all samples. The sample time is an es-sential parameter in the calculation of moments and norms (Figure 7).

Figure 7. Comparison of pMrelative 05.0 and pMrelative

15.0 for two different

sample times s05.0 and s15.0 . The sample time s05.0 provided the most sensitive results. This is quite reasonable because the fault causes an impact which occurs in a time instant within the first sample of 0.05 s. A longer time interval will include parts of the



signal without a significant difference between the fault and non-fault states in the norm calculation resulting in a decrease of sensitivity. If is sufficiently short, the order of p could be smaller than 3. Since the time instants of the im-pacts are normally unknown, a larger or several samples should be consid-ered. The use of an unspecific sample time has to be compensated by an in-crease of p. For other monitoring problems, you need some other combinations of sample time and order of p. For the detection of cavitation in a Kaplan water turbine [9], the best parameter setting for long sample times was p = 4 (Figure 8). The best overall performance in the full power range was achieved with relative

)max( 75.24

3 M .

Figure 8. Coefficient of determination R2 calculated as the square of the correla-

tion coefficient between the relative )max( 4pM and the knowledge-based

cavitation index in a Kaplan water turbine [9].

4 Conclusion Advanced signal processing and feature extraction methods for vibration meas-urements are chosen in a specific way in order to detect different faults. Gener-alised moments and norms obtained from higher or real order derivatives pro-vide informative features for diagnosing faults in a roller contact on a rough sur-face. Kurtosis, peak values and relative norms provide an indication with higher order derivatives. For the impact of a small scratch, sensitivity is further im-proved with short sample times, but several sequential samples are required to guarantee the detection of impacts. Then the order of moment can also be re-duced.



5 References 1. J D Smith, ‘Vibration monitoring of bearings at low speeds’, Tribology Inter-

national Vol 15, No 3, pp 139-144, 1982. 2. S Lahdelma, ‘New vibration severity evaluation criteria for condition moni-

toring’ (in Finnish), Research report No. 85, University of Oulu, 1992. 3. S Lahdelma, ‘On the higher order derivatives in the laws of motion and their

application to an active force generator and to condition monitoring’, D.Sc.Tech. thesis. Research report No. 101, University of Oulu, 1995.

4. S Lahdelma and V Kotila, ‘Real Order Derivatives – New Signal Processing Method’ (in Finnish), Kunnossapito, Vol 17, No 8, pp 39-42, 2003.

5. S Lahdelma and V Kotila, ‘Complex Derivative – A New Signal Processing Method’, Kunnossapito, Vol 19, No 4, pp 39-46, 2005.

6. S Lahdelma and E Juuso, ‘Advanced signal processing and fault diagnosis in condition monitoring’, Insight, Vol 49, No 12, pp 719–725, 2007.

7. S G Samko, A A Kilbas, and O I Marichev, ‘Fractional Integrals and Deriva-tives. Theory and Applications’, 976 pp, Gordon and Breach Science Pub-lishers, 1993.

8. S Lahdelma and E Juuso, ‘Vibration Analysis of Cavitation in Kaplan Water Turbines’, Proceedings of the 17 World Congress, The International Fed-eration of Automatic Control, Seoul, Korea, July 6-11, 2008, pp 13420-13425.

9. S Lahdelma and E Juuso, ’Signal Processing in Vibration Analysis’, Pro-ceedings of Fifth International Conference on Condition Monitoring & Ma-chinery Failure Prevention Technologies CM 2008 – MFPT 2008, Edin-burgh, 15-18 July, 2008, pp 867-878.

10. S Lahdelma and E Juuso, ’Advanced Signal Processing in Mechanical Fault Diagnosis’, Proceedings of CM 2008 – MFPT 2008, Edinburgh, pp 879-889.

11. E Juuso and S Lahdelma, ’Intelligent Condition Indices in Fault Diagnosis’, Proceedings of CM 2008 – MFPT 2008, Edinburgh, pp 698-708.

12. S Lahdelma, E Juuso and J Strackeljan, Neue Entwicklungen auf dem Ge-biet der Wälzlagerüberwahung, in A Seeliger and P Burgwinkel (Ed.) Ta-gungsband zum AKIDA 2006, Aachen, Germany, pp 447-460, ASRE, Band 63, 2006, RWTH Aachen, November 2006.

13. S Lahdelma and E K Juuso, ’Intelligent Condition Monitoring for Lime Kilns’, in A Seeliger and P Burgwinkel (Ed.) Tagungsband zum AKIDA 2006, Aa-chen, Germany, pp 399-408, ASRE, Band 63, RWTH Aachen, November 2006.

14. T Doguer and J Strackeljan, ‘New Time Domain Method for the Detection of Roller Bearing Defects’, Proceedings of CM 2008 – MFPT 2008, Edin-burgh, pp 338-348.

15. S Lahdelma, ‘On the Derivative of Real Number Order and its Application to Condition Monitoring’, Kunnossapito, Vol 11, No 4, pp 25-28, 1997.

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