**2.1 Continuous Wavelet Transforms**

When referring to the definition of Fourier Transforms [1], it can be observed that this formulation describes the signal *x t*( ) by means of a set of functions *<sup>j</sup> <sup>t</sup> e* which form the 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 has been extensively reported ([14]-[19]).

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

$$\text{CWT}(a,b) = \frac{1}{\sqrt{a}} \int\_{-\infty}^{+\infty} x(t) \wp^\* \left(\frac{t-b}{a}\right) dt\tag{3}$$

with *aR bR* 0 , .

This is a linear transformation which decomposes the original signal into its elementary functions *a b*, :

$$
\psi\_{a,b}(t) = \frac{1}{\sqrt{a}} \nu \left( \frac{t-b}{a} \right) \tag{4}
$$

On the Use of Wavelet Transform for Practical Condition Monitoring Issues 357

Firstly, as an initial improvement, the Impulse mother wavelet was taken into account in this work due to its capability in analysing impulses in vibration signals. It is defined as

> <sup>0</sup> () 2 cos 2 *if t t impulse t e f t*

Its capabilities and the comparison between Morlet and Impulse mother wavelets in analysing transient signals are well reported in [27] and [28]. In this study, <sup>0</sup>*f* assumes the most common values found in literature: 0.8125 Hz for the Morlet mother wavelet and 20

Secondly, a purification method inspired by the work of Yang [29] was considered in order to improve the accuracy of CWT representations and to try and solve the problem of frequency overlapping which has already reported in [10]. In [25] Yang applied the purification method using the Morlet wavelet, while in this paper the Impulse wavelet was

By means of purification methods, new CWT coefficients (*CWT* ) were calculated using the

ˆ *CWT a b t a t CWT a b t* (, ,) (,) (, ,) 

cov ( ), ( , ) (,)

deviation, *H* is the sinusoidal function and a indicates the wavelet scale. The expression

 12 1 1 2 2 *Cov x x E x x* (,) 

It can be noted that the correlation between the signal and the sinusoid *H* is evaluated over

window moves for the whole duration of the signal. After several tests, the choice of time

obtaining a higher correlation coefficient and the computational time needed for the

 

sinusoidal function with the frequency of the present wavelet scale given by

'cov' means covariance and is defined for the two data histories 1 *x* and 2 *x* as:

*a t*

a short time period defined by a time window with duration

(,) *a t* is the coefficient of correlation between the original signal and the

 ( ) (, )

is the time duration of the signal *x(t)*,

<sup>1</sup> *E x <sup>i</sup>* .

 

is based on a reliable compromise between the requirement of

(8)

(9)

*f T H aT xT H aT*

where 0*f* is the central frequency of the mother wavelets.

<sup>0</sup> 2 2

 

(6)

(7)

<sup>0</sup> /*a* with

is the standard

. In addition, the time

0


Hz for the Impulse wavelet.

also taken into account.

where *Tt t* 

window duration

correlation calculation.

as the central frequency of the mother wavelet. The correlation coefficient can be written as:

> /2; /2,

where *E* is the mathematical expectation and

following equation:

The term

follows:

which are determined by the translation (parameter *b* ) and the dilation (parameter *a* ) of a so called "mother (analyzing) wavelet" ( )*t* .

The *b* translation parameter describes the time localization of the wavelet, while the *a* dilation determines the width or scale of the wavelet. It is worth noting that, by decreasing the *a* scale parameter, the oscillation frequency of the wavelet increases, but the duration of the oscillation also decreases, so it can be noted that exactly the same number of cycles is contained within each wavelet.

Therefore, an important difference when compared to the classical Fourier Analysis, in which the time window remains constant, is that the time and frequency resolution now becomes dependent on the *a* scale factor. For CWTs, in fact, the width of the window in the time domain is proportional to *a*, while the bandwidth in the frequency domain is proportional to 1 / *a* . Thus, in the frequency domain, WTs have good resolution for low frequencies and, in the time domain, good resolution for high frequencies; the latter property makes CWTs suitable for the detection of transient signals. More details and applications for CWTs can be found in literature ([20]-[24]).

Two kinds of mother wavelet are known in literature:


Concerning the former, one of the most interesting is the Morlet wavelet which is defined as:

$$\left(\varphi\left(t\right)\_{morlet} = \frac{1}{\pi^{-1/4}} e^{-t^2/2} e^{i2\pi f\_0 t} \tag{5}$$

where 0*f* is the central frequency of the mother wavelet. The term 1/4 1 / is a normalization factor which ensures that the wavelet has unit energy; the Gaussian envelope 2 *<sup>t</sup>* /2 *e* modulates the complex sinusoidal waveform. Since the Morlet wavelet is a complex mother wavelet, one can separate the phase and amplitude components within any signal when using it. The CWT result is graphically represented in the time-scale plane, while in this chapter the maps are displayed in the time-frequency domain, using the relationship *f f* <sup>0</sup> /*a* between the central frequency of the analyzing wavelet and the scale. Moreover, when complex analyzing wavelets are used, only the amplitude is considered and represented using a linear scale.

Concerning CWT implementation, the algorithm proposed by Wang and Mc Fadden was applied taking advantage of the FFT algorithm [26].

#### **2.1.1 CWT improvements**

Several improvements have been taken into account in this chapter in order to improve CWT power in detecting and localizing transients within a signal. These enhancements concern:



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

which are determined by the translation (parameter *b* ) and the dilation (parameter *a* ) of a

The *b* translation parameter describes the time localization of the wavelet, while the *a* dilation determines the width or scale of the wavelet. It is worth noting that, by decreasing the *a* scale parameter, the oscillation frequency of the wavelet increases, but the duration of the oscillation also decreases, so it can be noted that exactly the same number of cycles is

Therefore, an important difference when compared to the classical Fourier Analysis, in which the time window remains constant, is that the time and frequency resolution now becomes dependent on the *a* scale factor. For CWTs, in fact, the width of the window in the time domain is proportional to *a*, while the bandwidth in the frequency domain is proportional to 1 / *a* . Thus, in the frequency domain, WTs have good resolution for low frequencies and, in the time domain, good resolution for high frequencies; the latter property makes CWTs suitable for the detection of transient signals. More details and


Concerning the former, one of the most interesting is the Morlet wavelet which is defined as:

1/4 <sup>1</sup> ( ) *<sup>t</sup> i ft morlet t ee*

normalization factor which ensures that the wavelet has unit energy; the Gaussian envelope 2

Concerning CWT implementation, the algorithm proposed by Wang and Mc Fadden was

Several improvements have been taken into account in this chapter in order to improve CWT power in detecting and localizing transients within a signal. These enhancements

 modulates the complex sinusoidal waveform. Since the Morlet wavelet is a complex mother wavelet, one can separate the phase and amplitude components within any signal when using it. The CWT result is graphically represented in the time-scale plane, while in this chapter the maps are displayed in the time-frequency domain, using the relationship *f f* <sup>0</sup> /*a* between the central frequency of the analyzing wavelet and the scale. Moreover, when complex analyzing wavelets are used, only the amplitude is considered and

where 0*f* is the central frequency of the mother wavelet. The term 1/4 1 /

<sup>2</sup> <sup>0</sup> /2 2

(5)

is a

( )*t* .

applications for CWTs can be found in literature ([20]-[24]).

Two kinds of mother wavelet are known in literature:

so called "mother (analyzing) wavelet"

contained within each wavelet.

represented using a linear scale.

**2.1.1 CWT improvements** 



applied taking advantage of the FFT algorithm [26].

*<sup>t</sup>* /2 *e*

concern:

Firstly, as an initial improvement, the Impulse mother wavelet was taken into account in this work due to its capability in analysing impulses in vibration signals. It is defined as follows:

$$\left(\psi\left(t\right)\_{impulse} = \sqrt{2\pi}e^{2\pi i f\_0 t - \left|2\pi t\right|}\cos\left(2\pi f\_0 t\right)\tag{6}$$

where 0*f* is the central frequency of the mother wavelets.

Its capabilities and the comparison between Morlet and Impulse mother wavelets in analysing transient signals are well reported in [27] and [28]. In this study, <sup>0</sup>*f* assumes the most common values found in literature: 0.8125 Hz for the Morlet mother wavelet and 20 Hz for the Impulse wavelet.

Secondly, a purification method inspired by the work of Yang [29] was considered in order to improve the accuracy of CWT representations and to try and solve the problem of frequency overlapping which has already reported in [10]. In [25] Yang applied the purification method using the Morlet wavelet, while in this paper the Impulse wavelet was also taken into account.

By means of purification methods, new CWT coefficients (*CWT* ) were calculated using the following equation:

$$\text{CV}\hat{\text{V}}\text{T}(a,b,t) = \gamma(a,t) \cdot \text{CV}\text{T}(a,b,t) \tag{7}$$

The term (,) *a t* is the coefficient of correlation between the original signal and the sinusoidal function with the frequency of the present wavelet scale given by <sup>0</sup> /*a* with 0 as the central frequency of the mother wavelet.

The correlation coefficient can be written as:

$$\gamma(a,t) = \left| \frac{\text{cov}\left(\mathbf{x}(T), H(a,T)\right)}{\sigma\_{f(T)}\sigma\_{H(a,T)}} \right| \tag{8}$$

where *Tt t* /2; /2, is the time duration of the signal *x(t)*, is the standard deviation, *H* is the sinusoidal function and a indicates the wavelet scale. The expression 'cov' means covariance and is defined for the two data histories 1 *x* and 2 *x* as:

$$\text{Cov}(\mathbf{x}\_1, \mathbf{x}\_2) = E\left[ (\mathbf{x}\_1 - \mu\_1)(\mathbf{x}\_2 - \mu\_2) \right] \tag{9}$$

where *E* is the mathematical expectation and <sup>1</sup> *E x <sup>i</sup>* .

It can be noted that the correlation between the signal and the sinusoid *H* is evaluated over a short time period defined by a time window with duration . In addition, the time window moves for the whole duration of the signal. After several tests, the choice of time window duration is based on a reliable compromise between the requirement of obtaining a higher correlation coefficient and the computational time needed for the correlation calculation.

On the Use of Wavelet Transform for Practical Condition Monitoring Issues 359

*j k i i <sup>j</sup> <sup>i</sup> c x t t k dt X Y* 

, () 2 2 *j j*

 

and where *j k*, *c* are the wavelet coefficients or detail coefficients representing the timefrequency map of the original signal *x t*( ) . This logarithmic scaling of both the dilation and

The dyadic grid can be considered as the most efficient in discretization terms and leads to the construction of an orthonormal wavelet basis. In fact, discrete dyadic grid wavelets are commonly chosen to be orthonormal, i.e. orthogonal to each other and normalized to have unit energy. This means that the information stored in a *j k*, *c* wavelet coefficient is not repeated elsewhere and allows for the complete regeneration of the original signal without redundancy. Orthonormal dyadic discrete wavelets are associated with scaling functions

/2

\*

*j*

, () 2 2 *j j*

The scaling function is orthogonal to the translation of itself, but not to dilations of itself.

By means of the scaling function, it is possible to obtain the approximation coefficients *j k*, *d* with the same procedure as the wavelet function (i.e. convolving the scaling function with

<sup>1</sup> () 2

*d x t t k dt* 

It is well known that diesel engines run very roughly at low speed ranges between 500-1000 rpm and that in marine applications they create vibrations in the body of the boat; moreover overall customer satisfaction with marine engines is based on performance in quietness

Since speed limits (4 knots) are usually required to be respected on leaving ports, the duration of the departure is quite high; thus, it is necessary to maintain engine speeds at a minimum. Smooth running with very low vibration levels result from a dynamically balanced design with counterweights. Good torsional vibration analysis is required to enable low speeds without noise and vibration effects. Another typical vibration source at low speeds is diesel engine combustion pressure. If fuel is injected after a small delay, the rapid combustion causes a quick rise in the pressure with high-frequency excitation force

 

<sup>1</sup> () 2

,

/2

translation steps is known as the dyadic grid arrangement.

where the elementary function is

, ( ) *j k* 

the signal):

terms.

2

*j k* 

*t* . The scaling function has the same form as the wavelet, given by

*j k* 

,

**3. Condition monitoring of marine couplings** 

2

*j k <sup>j</sup>*

\*

 

*<sup>n</sup> <sup>j</sup>*

1

(11)

*<sup>t</sup> t k* (12)

*t tk* (13)

(14)

In terms of the last CWT improvement, a new method which was recently proposed by Halim [30] was applied in order to compute the angular domain which averages across all the scales (TDAS) after CWT calculation. TDAS combines both wavelet analysis and the angular domain average in order to improve the time-frequency representation of the TSA of a signal. While the traditional method consists in taking wavelet transforms of the Time Synchronous Average, this new method performs the wavelet transformation first and then takes the time synchronous averages, obtaining the so-called TDAS distribution.

Assuming that the period of a time series is *P* and the time series has exactly *M* periods, the number of the total time samples is *N PM* . If the number of wavelet scales *s* is *S* the wavelet transformation of the time series generates the complex matrix CWT (since both complex Morlet and Impulse mother wavelets have been applied) of *S N* dimensions. It can be noted that each row of the absolute value of the CWT matrix is a time series corresponding to one *s* scale with a *P* period. If each of these time series is synchronously averaged (based on the period of the time series), the average of all the time series across all the scales can be computed obtaining the final TDAS matrix. Each row of the TDAS matrix represents the time synchronous average of the time series located at each scale. This method has the following advantages:


On the basis of these considerations, this method appears to be helpful when a lower number of averages is available.

It is worth noting that Halim obtained the TDAS matrix using a geometric average and the Morlet wavelet as its basis. In this work, the effectiveness of the method using the Impulse mother wavelet is verified and the linear average is also taken into account in order to be consistent with the traditional method.

#### **2.2 Discrete Wavelet Transforms**

A Discrete Wavelet Transform (DWT) is a technique which enables discrete coefficients to be calculated by replacing the continuous coefficients obtained through CWT calculation [31]. Due to this fact, the *a* and *b* parameters in Eq. 2 become to the power-of-two:

$$a = \mathsf{2}^{\dagger}, b = k\mathsf{2}^{\dagger}, \mathsf{j}, \mathsf{k}, \mathsf{e} \in \mathsf{Z} \tag{10}$$

where *j* is called level, 2 *<sup>j</sup>* denoted the scale and 2*<sup>j</sup> k* denotes the shift in the time direction. The DWT is defined as:

$$\mathbf{c}\_{j,k} = \frac{1}{\sqrt{2^j}} \int\_{-\infty}^{+\infty} \mathbf{x}(t) \boldsymbol{\nu}^\* \left(\mathbf{2}^{-j}t - k\right) dt \sum\_{i=1}^n \mathbf{X}\_i \mathbf{Y}\_i \tag{11}$$

where the elementary function is

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

In terms of the last CWT improvement, a new method which was recently proposed by Halim [30] was applied in order to compute the angular domain which averages across all the scales (TDAS) after CWT calculation. TDAS combines both wavelet analysis and the angular domain average in order to improve the time-frequency representation of the TSA of a signal. While the traditional method consists in taking wavelet transforms of the Time Synchronous Average, this new method performs the wavelet transformation first and then

Assuming that the period of a time series is *P* and the time series has exactly *M* periods, the number of the total time samples is *N PM* . If the number of wavelet scales *s* is *S* the wavelet transformation of the time series generates the complex matrix CWT (since both complex Morlet and Impulse mother wavelets have been applied) of *S N* dimensions. It can be noted that each row of the absolute value of the CWT matrix is a time series corresponding to one *s* scale with a *P* period. If each of these time series is synchronously averaged (based on the period of the time series), the average of all the time series across all the scales can be computed obtaining the final TDAS matrix. Each row of the TDAS matrix represents the time synchronous average of the time series located at each scale. This




On the basis of these considerations, this method appears to be helpful when a lower

It is worth noting that Halim obtained the TDAS matrix using a geometric average and the Morlet wavelet as its basis. In this work, the effectiveness of the method using the Impulse mother wavelet is verified and the linear average is also taken into account in order to be

A Discrete Wavelet Transform (DWT) is a technique which enables discrete coefficients to be calculated by replacing the continuous coefficients obtained through CWT calculation [31].

where *j* is called level, 2 *<sup>j</sup>* denoted the scale and 2*<sup>j</sup> k* denotes the shift in the time direction.

2, 2,, , *j j a bk <sup>j</sup> k Z* (10)

Due to this fact, the *a* and *b* parameters in Eq. 2 become to the power-of-two:

processed since the transformation is computed over the entire time series.

transformation is computed after the averaging process;

takes the time synchronous averages, obtaining the so-called TDAS distribution.

method has the following advantages:

the wavelet transform operator;

number of averages is available.

consistent with the traditional method.

**2.2 Discrete Wavelet Transforms** 

The DWT is defined as:

$$\boldsymbol{\nu}\_{j,k}(t) = 2^{-j/2} \boldsymbol{\nu}\left(2^{-j}t - k\right) \tag{12}$$

and where *j k*, *c* are the wavelet coefficients or detail coefficients representing the timefrequency map of the original signal *x t*( ) . This logarithmic scaling of both the dilation and translation steps is known as the dyadic grid arrangement.

The dyadic grid can be considered as the most efficient in discretization terms and leads to the construction of an orthonormal wavelet basis. In fact, discrete dyadic grid wavelets are commonly chosen to be orthonormal, i.e. orthogonal to each other and normalized to have unit energy. This means that the information stored in a *j k*, *c* wavelet coefficient is not repeated elsewhere and allows for the complete regeneration of the original signal without redundancy. Orthonormal dyadic discrete wavelets are associated with scaling functions , ( ) *j k t* . The scaling function has the same form as the wavelet, given by

$$
\phi\_{j,k}(t) = 2^{-j/2} \phi \left( 2^{-j} t - k \right) \tag{13}
$$

The scaling function is orthogonal to the translation of itself, but not to dilations of itself.

By means of the scaling function, it is possible to obtain the approximation coefficients *j k*, *d* with the same procedure as the wavelet function (i.e. convolving the scaling function with the signal):

$$d\_{j,k} = \frac{1}{\sqrt{2^j}} \prod\_{-\phi}^{+\phi} \varkappa(t) \phi^\* \left( 2^{-j}t - k \right) dt \tag{14}$$

#### **3. Condition monitoring of marine couplings**

It is well known that diesel engines run very roughly at low speed ranges between 500-1000 rpm and that in marine applications they create vibrations in the body of the boat; moreover overall customer satisfaction with marine engines is based on performance in quietness terms.

Since speed limits (4 knots) are usually required to be respected on leaving ports, the duration of the departure is quite high; thus, it is necessary to maintain engine speeds at a minimum. Smooth running with very low vibration levels result from a dynamically balanced design with counterweights. Good torsional vibration analysis is required to enable low speeds without noise and vibration effects. Another typical vibration source at low speeds is diesel engine combustion pressure. If fuel is injected after a small delay, the rapid combustion causes a quick rise in the pressure with high-frequency excitation force

On the Use of Wavelet Transform for Practical Condition Monitoring Issues 361

One experimental investigation was carried out on a marine propulsion package (Fig. 1). The 4-cylinder 4-stroke diesel engine with eight valves was located inboard just forward of the transom. The engine was turbocharged with an exhaust-driven turbo-compressor: the

Marine propulsion was assured by a stern-drive unit that contains the transmission and carries the propeller. The boat was steered by pivoting this unit with good characteristics in terms of speed, acceleration, steering and manoeuvring. In fact, the main advantage of sterndrives versus straight inboards is the possibility of changing the drive angle in order to obtain an optimum angle for speed or acceleration. The flexible coupling was mounted between the flywheel of the marine diesel engine and the propeller shaft. The primary side of the coupling was bolted to the flywheel, the secondary side was mounted onto the output shaft; between the two sides there were rubber elements which compensate for all types of

The vibration signals were measured from two points (see Fig. 2), which were close to the coupling, in order to analyse the vibration induced by the couplings at different angular positions of the stern drive when gears were repeatedly changed. The two accelerometers were mounted on both sides of the transom, that is, the left and right side. In order to compare the vibration behaviour of the two couplings, all compared vibration signals were

Vibration signals were measured by means of piezoelectric tri-axial accelerometers (frequency range: 1-12000 Hz). All signal records were acquired starting from a crankshaft reference position: a tachometer signal was taken using an inductive proximity probe close

In this context, DWTs were used to analyse the transom right-side signal in order to extract the scaling coefficients *j k*, *d* . In fact, the signals in time domain obtained during tests, when the gears were repeatedly changed, revealed a train of impulsive components. Fig. 3 shows that the acceleration peaks are unclear in the signal measured from the transom right-side where the noise level was too heavy. In order to indicate which type of coupling provides better vibrational behaviour, the mean value of the acceleration peaks was obtained directly from the original time history from the transom left-side. Concerning the signal measured at the right side, the mean value was obtained for low frequency components at the first level

Both types of couplings are very sensitive to transient dynamic phenomena due to gear changes. Table 1 shows that the mean value of the acceleration peaks for the Type 1 coupling is higher than the value for Type 2. Thus Type 2 gives better vibrational behaviour than the first type. It can be concluded that the time domain analysis of the coupling acceleration gives good condition monitoring information, if the DWT technique is used for signal denoising purposes. In order to precisely localise the impulsive phenomena in the time-frequency domain and to validate the previous thesis about Type 2 vibrational behaviour, the Continuous Wavelet Transform for a frame of the transom left-side signal is applied. The impulse with the highest amplitude is isolated for this signal in the time domain, (Fig. 5) and the CWT of this part of the signal is calculated for the two different coupling types. In this work, a Morlet

turbo was controlled by a waste-gate valve.

misalignment, particularly angular, and dampen vibrations.

picked up under exactly the same operating conditions.

to a gear wheel mounted onto the engine crankshaft.

1,*<sup>k</sup> d* , after DWT application (Fig. 4) with the Symlet analysing wavelet.

analysing wavelet was used, since its shape is similar to an impulse component.

components. An optimized injection system may eliminate fuel injection delay and the improved design of rigid cylinder blocks can reduce combustion pressure sources.

Finally, with regard to boat quietness, a general analysis is normally insufficient in evaluating the possibility of avoiding most noise and vibrations passing into the body of the boat through the crankshaft and the rigid coupling between the flywheel and the propeller shaft. It is necessary to highlight which parts are mainly related to vibration absorption. Coupling transmits torque and absorbs vibrations from the engine crankshaft. Placing a highly flexible coupling between the crankshaft and the propeller shaft will bring about further noise and vibration reduction. The vibration levels measured by the accelerometers mounted on different parts of the engine are a means of indicating which coupling works well.

Fig. 1. Propulsion package with the flexible coupling under study.

Fig. 2. Transom: position of accelerometers.

In the present study, Continuous (CWT) and Discrete Wavelet Transforms (DWT) are used to process the signals taken from a marine diesel engine in several operating conditions. The experimental results are presented and the capability of the above-mentioned analysis techniques are discussed.

components. An optimized injection system may eliminate fuel injection delay and the

Finally, with regard to boat quietness, a general analysis is normally insufficient in evaluating the possibility of avoiding most noise and vibrations passing into the body of the boat through the crankshaft and the rigid coupling between the flywheel and the propeller shaft. It is necessary to highlight which parts are mainly related to vibration absorption. Coupling transmits torque and absorbs vibrations from the engine crankshaft. Placing a highly flexible coupling between the crankshaft and the propeller shaft will bring about further noise and vibration reduction. The vibration levels measured by the accelerometers mounted on different parts of the engine are a means of indicating which coupling works

In the present study, Continuous (CWT) and Discrete Wavelet Transforms (DWT) are used to process the signals taken from a marine diesel engine in several operating conditions. The experimental results are presented and the capability of the above-mentioned analysis

improved design of rigid cylinder blocks can reduce combustion pressure sources.

Fig. 1. Propulsion package with the flexible coupling under study.

Fig. 2. Transom: position of accelerometers.

techniques are discussed.

well.

One experimental investigation was carried out on a marine propulsion package (Fig. 1). The 4-cylinder 4-stroke diesel engine with eight valves was located inboard just forward of the transom. The engine was turbocharged with an exhaust-driven turbo-compressor: the turbo was controlled by a waste-gate valve.

Marine propulsion was assured by a stern-drive unit that contains the transmission and carries the propeller. The boat was steered by pivoting this unit with good characteristics in terms of speed, acceleration, steering and manoeuvring. In fact, the main advantage of sterndrives versus straight inboards is the possibility of changing the drive angle in order to obtain an optimum angle for speed or acceleration. The flexible coupling was mounted between the flywheel of the marine diesel engine and the propeller shaft. The primary side of the coupling was bolted to the flywheel, the secondary side was mounted onto the output shaft; between the two sides there were rubber elements which compensate for all types of misalignment, particularly angular, and dampen vibrations.

The vibration signals were measured from two points (see Fig. 2), which were close to the coupling, in order to analyse the vibration induced by the couplings at different angular positions of the stern drive when gears were repeatedly changed. The two accelerometers were mounted on both sides of the transom, that is, the left and right side. In order to compare the vibration behaviour of the two couplings, all compared vibration signals were picked up under exactly the same operating conditions.

Vibration signals were measured by means of piezoelectric tri-axial accelerometers (frequency range: 1-12000 Hz). All signal records were acquired starting from a crankshaft reference position: a tachometer signal was taken using an inductive proximity probe close to a gear wheel mounted onto the engine crankshaft.

In this context, DWTs were used to analyse the transom right-side signal in order to extract the scaling coefficients *j k*, *d* . In fact, the signals in time domain obtained during tests, when the gears were repeatedly changed, revealed a train of impulsive components. Fig. 3 shows that the acceleration peaks are unclear in the signal measured from the transom right-side where the noise level was too heavy. In order to indicate which type of coupling provides better vibrational behaviour, the mean value of the acceleration peaks was obtained directly from the original time history from the transom left-side. Concerning the signal measured at the right side, the mean value was obtained for low frequency components at the first level 1,*<sup>k</sup> d* , after DWT application (Fig. 4) with the Symlet analysing wavelet.

Both types of couplings are very sensitive to transient dynamic phenomena due to gear changes. Table 1 shows that the mean value of the acceleration peaks for the Type 1 coupling is higher than the value for Type 2. Thus Type 2 gives better vibrational behaviour than the first type. It can be concluded that the time domain analysis of the coupling acceleration gives good condition monitoring information, if the DWT technique is used for signal denoising purposes.

In order to precisely localise the impulsive phenomena in the time-frequency domain and to validate the previous thesis about Type 2 vibrational behaviour, the Continuous Wavelet Transform for a frame of the transom left-side signal is applied. The impulse with the highest amplitude is isolated for this signal in the time domain, (Fig. 5) and the CWT of this part of the signal is calculated for the two different coupling types. In this work, a Morlet analysing wavelet was used, since its shape is similar to an impulse component.

On the Use of Wavelet Transform for Practical Condition Monitoring Issues 363

a

b

Fig. 5. Highest amplitude impulse for transom left-side acceleration. Coupling: Type 1 (a)

a b

Fig. 6. Continuous wavelet transform (Morlet wavelet) for transom left-side acceleration,

700-1500 Hz frequency range; Coupling: Type 1 (a) and Type 2 (b).

and Type 2 (b).

Fig. 6 reports the wavelet analysis results revealing that the highest amplitude wavelet coefficients for Type 1 (Fig. 6(a)) are in the frequency range of around 1100 Hz. Regarding Type 2 (Fig. 6(b)), the wavelet transform amplitude during the transient phenomena assumes lower values and reveals an appreciatively constant amplitude in the 700-1500 Hz frequency range. The time-frequency plot is able to clearly show the frequency content during the impulse and gives a clearer interpretation of the difference vibrational behaviour of two coupling types.

Fig. 3. The vibration signal (TYPE 2) from the transom left-side (a) and the transom rightside (b).

Fig. 4. DWT of the transom right-side signal (TYPE 2).

Fig. 6 reports the wavelet analysis results revealing that the highest amplitude wavelet coefficients for Type 1 (Fig. 6(a)) are in the frequency range of around 1100 Hz. Regarding Type 2 (Fig. 6(b)), the wavelet transform amplitude during the transient phenomena assumes lower values and reveals an appreciatively constant amplitude in the 700-1500 Hz frequency range. The time-frequency plot is able to clearly show the frequency content during the impulse and gives a clearer interpretation of the difference vibrational behaviour

Fig. 3. The vibration signal (TYPE 2) from the transom left-side (a) and the transom right-

Fig. 4. DWT of the transom right-side signal (TYPE 2).

of two coupling types.

side (b).

Fig. 5. Highest amplitude impulse for transom left-side acceleration. Coupling: Type 1 (a) and Type 2 (b).

Fig. 6. Continuous wavelet transform (Morlet wavelet) for transom left-side acceleration, 700-1500 Hz frequency range; Coupling: Type 1 (a) and Type 2 (b).

On the Use of Wavelet Transform for Practical Condition Monitoring Issues 365

The first faulty condition concerned an engine with a connecting rod with incorrectly tightened screws, that is, screws which were only tightened with a preload of 3 kgm, instead of the correct torque of 9. The second faulty condition concerned an engine with an inverted piston, with incorrectly positioned valve sites. This incorrect assembly hindered the correct correspondence between the valve plates and the valve sites. Since the exhaust valve site area is larger than the intake valve site, the exhaust valves knocked against the non-

Fig. 8(a) shows that the CWT map (Impulse wavelet) of the Time Synchronous Average (TSA) detected four cylinder pressurizations and two events related to the faulty condition. Even if a remarkable vertical line at 100 degrees was present in the CWT map of the TSA (Fig. 8 (a)), it is not sufficient to assure the presence of a mechanical fault since its amplitude is comparable to the pressurization peak amplitudes. Therefore, the CWT of the residual signal (i.e. the signal obtained by subtracting the time synchronous average from the raw signal) is an expected step in mechanical fault localization within engine kinematics (Fig. 8(b)). As depicted in Fig. 8(b), the presence of the pre-loaded rod is highlighted by a marked vertical line at about 100º.

As explained in [32] the peak is caused by the absence of controlled bush deformation when the correct tightening torque is not applied. This clearance is abruptly traversed whenever a change in the direction of the resultant force occurs on the rod. In particular, it was demonstrated that the acceleration peak took place at the beginning of the cylinder 3 intake stroke, corresponding to cylinder 2 pressurization (i.e. 'Press 2' in Fig. 8(a)). Hence, fault location can be only achieved by the analysis of the residual signal. It is worth noting that better angular fault localization can be achieved using the Morlet mother wavelet (Fig. 8(c)) which gives lower frequency resolution but higher angular localization of the anglefrequency map. Since the purpose of the proposed approach is to obtain reliable fault diagnostics through accurate angular transient event localization, the Morlet wavelet can be

In order to improve the CWT of the TSA, the purification method was firstly carried out

select which coefficient gives the best match between the frequency of the signal and the

Fig. 9(a) shows that this method provides a clearer representation in terms of sensitivity to background noise. However, the use of the coefficient correlation method does not improve the angular localization of the main engine events. As noted earlier, this enhancement can be obtained using the Morlet mother wavelet. The Morlet mother wavelet was used to compute the wavelet transform by means of both traditional and TDAS methods. No significant improvements in angular faulty localization can be obtained by using the TDAS method (Fig. 9(b)). Therefore, it can be concluded that a traditional CWT map with a Morlet mother

It should be noted that CWT is used in order to distinguish faulty conditions from normal ones for practical fault diagnosis and not to obtain reliable parameters for an automatic

In order to overcome this issue, the DWT technique for the extraction of faulty components from the signal, proposed by Shibata, was evaluated for the second fault which was

(,) *a t* used in this method is able to

using correlation weighted CWT coefficients, i.e. ˆ *CWT* , as described in Section 2.1.1.

considered the most desirable if compared with the Impulse wavelet.

As previously mentioned, the correlation coefficient

frequency corresponding to the Impulse wavelet scale.

wavelet is sufficient for faulty localization purposes.

procedure led by a data acquisition system.

condition tested, i.e. the inverted piston.

correspondent intake valve sites.

Fig. 7. Continuous wavelet transform (Mexican hat wavelet) for transom left-side acceleration, 700-1500 Hz frequency range; Coupling: Type 1 (a) and Type 2 (b).

Moreover, a comparison between two different Morlet and Mexican hat wavelet functions was evaluated. Undeniably, the Mexican hat wavelet function has a shape which is totally inadequate for analysing the signal impulse components. This is shown by the results in Fig. 7 which indicate that the Mexican hat highlighted the different frequency contempt of two coupling types but was not able to precisely localise the higher frequency components of the impulse signal.
