**1. Introduction**

352 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

Junsheng, C., Dejie, Y. and Yu, Y. (2007): "Application of an impulse response wavelet to

Kahaei, M. H., Torbatian, M. and Poshtan, J. (2006): "Detection of Bearing Faults Using Haar Wavelets", *IEICE Transaction fundamentals*, vol.E89a (3), pp.757-763. Khalid F. Al-Raheem, Roy, A., Ramachandran, K. P., Harrison, D.K. and Grainger, S.

Khemili, I. and Chouchane, M. (2005): "Detection of rolling element bearing defects by adaptive filtering", *European Journal of Mechanics and Solids*, vol.24, pp. 293–303. Lind, R. and Brenner, M. J. (1998): "Correlation filtering of modal dynamics using the

Orhan, S., Akturk, N. and Celik, V. (2006): "Vibration monitoring for defect diagnosis of

Peng, Z.K. and Chu, F.L. (2004): "Application of the wavelet transform in machine

Reeves T. (1994): "Failure modes of rolling element bearings", *Proceedings of 8th annual* 

Tandon N. (1994), "A comparison of some vibration parameters for the condition monitoring of rolling element bearings", *Measurements*, vol.12, pp.285-289. Thanagasundram, S. and Schlindwein, F. S. (2006): "Auto-regression based diagnostics scheme for detection of bearing faults", *Proceeding of ISMA*, pp. 3531-3546. Wang, C. and Gao, R. X. (2003): "Wavelet transform with spectral post-processing for

Weller, N. (2004): "Acceleration enveloping- higher sensitivity- earlier detection", *Machinery* 

Yang, W. and Ren, X. (2004): "Detecting Impulses in Mechanical Signals by Wavelets",

Yanyang, Z., Xuefeng, C., Zhengjia, H. and Peng, C. (2005): "Vibration based Modal

*EURASIP Journal on Applied Signal Processing*, vol.8, pp.1156–1162.

Engineering Materials, vol. 293-294, pp.183-190.

Mallat, S. (1999): *a wavelet tour of signal processing*, 2nd edition, Academic Press.

*Mechanical systems and signal processing, vol.* 18, pp. 199–221.

studies", *NDT & E International*, vol.39, pp. 293-298.

fault diagnosis of rolling bearings", *Mechanical systems and signal processing*, vol. 21,

(2008):"Application of Laplace Wavelet Combined with Artificial Neural Networks for Rolling Element Bearing Fault Diagnosis", ASME J. of vibration and Acoustics,

Laplace wavelet". NASA Dryden Flight Research center Edwards CA 93523-0273,

rolling element bearings as a predictive maintenance tool: comprehensive case

condition monitoring and fault diagnostics: a review with bibliography",

enhanced feature extraction", *IEEE transactions on instrumentation and measurement*,

Parameters Identification and wear fault diagnosis using Laplace wavelet", Key

**7. References** 

pp. 920–929.

pp.1-10.

Vol.130 (5), pp. 051007(1)-051007(9).

*meeting vibration inst., pp. 209-217*.

vol.52 (4), pp. 1296-1301.

*message*.

Condition monitoring is used for extracting information from the vibro-acoustic signature of a machine to detect faults or to define its state of health. A change in the vibration signature not only indicates a change in machine conditions but also points directly to the source of the signal alteration.

Fault diagnosis, condition monitoring and fault detection are different terms which are sometimes used improperly. Condition monitoring and fault detection refer to the evaluation of the state of a machine and the detection of an anomaly. Fault diagnosis could be set apart from other diagnoses since it is more rigorous and requires the type, size, location and time of the detected faults to be determined.

Due to their non-intrusive behaviour and use in diagnosing a wide range of mechanical faults, vibration monitoring techniques are commonly employed by machine manufacturers. Moreover, increases in computing power have helped the development and application of signal processing techniques.

Firstly, the monitoring procedure involves vibration signals to be acquired by means of accelerometers. Due to the selection of acquisition parameters being critical, the data acquisition step is not of minor importance. Sometimes, several steps, such as the correct separation of time histories, averaging and digital filtering is required in order to split the useful part of the signal from noise (electrical and mechanical), which is often present in industrial environments.

Secondly, signal processing techniques have to be implemented by taking into account the characteristics of the signal and the type of machine from which the signal is being measured (i.e. rotating or alternative machine with simple or complex mechanisms). In the final analysis, several features have to be extracted in order to assess the physical state of the machine or to detect any incipient defects and determine their causes.

When the nature of the signal varies over time, repeating the Fourier analysis for consequent time segments could describe the temporal variation of the signal spectrum. This well known technique is called Short Time Fourier Transform (STFT). The principal limitations of this approach are:


A more rigorous explanation of the latter is the Uncertainty Principle or Bandwidth-Time product that can be easily proved in [1] using the Parseval theorem and Schwartz inequality. This Principle states that:

$$
\Delta f \cdot \Delta t \geq \frac{1}{4\pi} \tag{1}
$$

On the Use of Wavelet Transform for Practical Condition Monitoring Issues 355

On the one hand, Short-Time Fourier Transforms (STFT), Wigner-Ville Distributions (WVD) and Continuous Wavelet Transforms (CWT) are usually used in order to distinguish faulty conditions for practical fault diagnosis and not to obtain reliable parameters for an

On the other hand, Discrete Wavelet Transforms (DWT) could be applied in order to extract informative features for an automatic pass/fail decision procedure [12]. Moreover, due to their power in identifying de-noising signals, the latter can be used in order to select

The aim of this study is to assess the effectiveness of both CWTs and DWTs for machine condition monitoring purposes. In this chapter, WTs are set up specifically for vibration signals captured from real life complex case studies which are poorly dealt with in literature: marine couplings and i.c. engines tested in cold conditions. Both Continuous (CWT) and Discrete Wavelet Transforms (DWT) are applied. The former was used for faulty event identification and impulse event characterization by analysing a three-dimensional representation of the CWT coefficients. The latter was applied for filtering and feature extraction purposes and for detecting impulsive events which were strongly masked by

This paragraph introduces the theory of fundamental background in order to understand

When referring to the definition of Fourier Transforms [1], it can be observed that this

basis for signal expansions. These functions are continuous and of infinite duration. The spectrum in question corresponds to the expansion coefficients. An alternative approach consists of decomposing the data in time-localised waveforms. Such waveforms are usually referred to as wavelets. In recent decades, the theoretical background of wavelet transforms

<sup>1</sup> \* ( , ) () *t b CWT a b x t dt*

This is a linear transformation which decomposes the original signal into its elementary

*t b <sup>t</sup> a a*

 

*a a* 

(3)

(4)

which form the

frequency bands which are mostly characterised by impulsive components.

achievements concerning the application of CWT and DWTs on real signals.

formulation describes the signal *x t*( ) by means of a set of functions *<sup>j</sup> <sup>t</sup> e*

The Continuous Wavelet Transform (CWT) of the time signal *x t*( ) is defined as:

, <sup>1</sup> ( ) *a b*

automatic procedure led by a data acquisition system [9].

noise.

**2. Background theory** 

with *aR bR* 0 , .

*a b*, :

functions

**2.1 Continuous Wavelet Transforms** 

has been extensively reported ([14]-[19]).

where *f* is the frequency resolution expressed in Hertz and *t* is the time resolution expressed in seconds. It can be easily understood that Eq. 1 points to a limitation in STFT analysis methods: fine resolution in both time and frequency domains cannot be obtained at the same time.

Several techniques have been developed [2][3] to overcome this problem and to analyse different types of non-stationary signals.

As is reported in [2], one can distinguish between three important classes of non-stationary signals:


Another important class of non-stationary signals is represented by Cyclostationary Signals which are not described here. Since this study deals with Transient signals, Wavelet Transforms (WT) have been proposed as an appropriate analysis tool.

In general, each type of fault produces a different vibration signature which might be detected by means of suitable signal processing techniques. Concerning i.c. engines, fault detection and diagnosis can be carried out using different strategies. One strategy can consist in modelling the whole mechanical system using lumped or finite element methods in order to simulate several faults and compare the results with the experimental data [4][5]. Another strategy is to adopt signal processing techniques in order to obtain features or maps that can be used to detect the presence of the defect [6][7]. Regarding the latter, a decision algorithm is require for a visual or automatic detection procedure. Moreover, maps can also be analysed for diagnostic purposes [8]. This method is used most commonly and is well suited to judgements involving expert technicians.

The latter strategy involves the application of time-frequency distribution techniques which are well suited for the analysis of non-stationary signals and have been widely applied to engine monitoring [9]-[11].

On the one hand, Short-Time Fourier Transforms (STFT), Wigner-Ville Distributions (WVD) and Continuous Wavelet Transforms (CWT) are usually used in order to distinguish faulty conditions for practical fault diagnosis and not to obtain reliable parameters for an automatic procedure led by a data acquisition system [9].

On the other hand, Discrete Wavelet Transforms (DWT) could be applied in order to extract informative features for an automatic pass/fail decision procedure [12]. Moreover, due to their power in identifying de-noising signals, the latter can be used in order to select frequency bands which are mostly characterised by impulsive components.

The aim of this study is to assess the effectiveness of both CWTs and DWTs for machine condition monitoring purposes. In this chapter, WTs are set up specifically for vibration signals captured from real life complex case studies which are poorly dealt with in literature: marine couplings and i.c. engines tested in cold conditions. Both Continuous (CWT) and Discrete Wavelet Transforms (DWT) are applied. The former was used for faulty event identification and impulse event characterization by analysing a three-dimensional representation of the CWT coefficients. The latter was applied for filtering and feature extraction purposes and for detecting impulsive events which were strongly masked by noise.
