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Chapter Five

Testing and materials-damage detection methods

M. J. Roan The Applied Research Laboratory,The Pennsylvania State University, USA

Abstract

Damage to a system can be defined as changes that are introduced into a system or structure that adversely affect the state and/or the future performance of the system. This chapter introduces some of the most common types of damage that occur in materials of interest and the methods and science used to detect or even predict these failures. Primarily, this chapter will focus on vibration-based damage detection. The reason for this is that recent focus in the literature has been on approaches to damage detection that are more comprehensive or global in nature. These approaches differ significantly from past non-destructive testing methods that are local in nature such as visual or ultrasonic technologies. These techniques require a priori knowledge of the location of damage. This requires that the area to be examined is readily accessible, which may not always be the case. New approaches in vibration-based structural-health monitoring work under the assumption that the damage will significantly alter the dynamical response of that system. The problem then is to understand the physics of common failures, the affect they have on systems of interest, and to devise methodologies for predicting a system failure based on vibrational measurements of the system dynamics. This chapter will provide a review of the physics of the most common types of failures. Considering the large number of types of materials, the scope will be narrowed to failures in rotating machinery, namely gear systems. Next, state-of-the-art techniques for detection of damage in these systems are given. This is followed by a discussion of the direction of future research on damage detection and failure analysis.

1 Introduction

The detection of failures in mechanical systems and structures has been of great interest throughout history. The more difficult problem of the prediction of

WIT Transactions on State of the Art in Science and Engineering, Vol 1, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/1-85312-836-8/05

116 Advances in Fatigue, Fracture and Damage Assessment

failures has been of more recent interest. This problem, the prediction of failures in structures and mechanical systems, requires a fundamental understanding of the physics of the types of failures in materials and structures. In this chapter, basic types of failures that are of widespread interest will be discussed: failures in rotating machinery. Also, the methods used to detect and/or predict their occurrence will be presented. The interest in the ability to monitor, detect, or predict incipient failures in mechanical systems or structures is widespread across many disciplines. In each of these disciplines, the primary interest is in non-invasive or non-destructive damage detection. In civil engineering, the ability to test the health of a steel- structured, high-rise building, or a suspension bridge is of interest. In mechanical engineering, the ability to monitor the health of a gear-box using non-invasive techniques, track changes in the vibration signals arising from the gear-box, and predict that a gear will fail in a specified time may be the primary focus. In aerospace engineering, the interest may be to monitor the health of a passenger-jet airframe or predict the fatigue-induced delamination failure of a composite helicopter rotor blade. In each of these scenarios, a fundamental understanding of the physics of the types of failures is necessary. If it is understood, for example, how a gear tooth becomes weak at the base, and then flexes, causing a sudden change in the meshing frequency, then a failure- detection algorithm designed to identify this type of change can be implemented. The basic assumption in vibration-based damage-detection methodologies is that the damage, either to the mechanical system or structure will fundamentally alter the physical properties (i.e. stiffness, impulse response, damping) of the system. Alteration of these physical properties in turn alters the response of the system to vibrations moving through the system. This also, in turn, assumes that the original response of the undamaged system is known. The goal of monitoring the system then is to detect the changes that occur due to the damage. The organization of the chapter is as follows. First, a review of the underlying mechanisms that cause the common types of failures in rotating machinery is given. Secondly, vibration-based methods currently used to detect these types of failures are reviewed. The detection methods used, starting from the most simple to the more recent are then discussed with examples based on experimental failures. This is followed by a discussion of the future directions of damage-detection and failure-prediction research.

2 The physics of failure

In order to predict or even detect that a structure or mechanical system has failed or will fail, an understanding of the underlying physics of common types of failure is necessary. Due to the wide variety of materials and structures, the discussion in this chapter will be limited to failures in rotating machinery. The interest here is to provide background on failures that occur in systems that are life-critical as the human and economic cost of catastrophic failures are extremely high, as was the case in the failure of the composite tail section of Flight 93.

WIT Transactions on State of the Art in Science and Engineering, Vol 1, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line)

Advances in Fatigue, Fracture and Damage Assessment 117

Almost everything that is used in daily life has been fabricated. Structures and mechanical systems are made up of materials. In many cases, the structure is made up of several types of materials in combination. Automobiles are a prime example of the types of complicated mechanical systems and structures that are used every day by millions of people around the globe. Upon closer inspection, the modern automobile is an engineering marvel. The engine, for example, is subjected to extremes of temperature, loading, shock, and vibration (both self- induced and externally induced). A sizeable body of work has been done on the condition monitoring of internal combustion engines [1]. The loading conditions in a modern gearbox or transmission can also be extreme. One end of the loading spectrum occurs at acceleration from standstill. In this phase of operation, the loading (in this case a bending moment) on the gear and pinion teeth is extremely high. In highway cruising, the load on the overdrive is significantly lower. The automobile body is itself a structure, sometimes a body on a frame, or a unibody construction. This structure is subjected to many stresses, strains, bending moments, wide temperature shifts, and corrosive elements that all affect the safe lifetime of the structure. Inspection of a common mechanical system like the automobile provides insight into many of the difficulties in detecting a certain type of failure using vibrational analysis. Detecting changes “on the bench” differs vastly from fielding an automated condition-based monitoring system. For instance, a change in the type of road surface could cause changes in the vibrational signal measured by sensors on the transmission. The process of identifying the failure from all of the other external influences that change the system dynamics is referred to as feature extraction and identification. A great body of work has been done in identifying useful features that correspond to different types of failures. It is these features that tie the physics of the failure in the material to the algorithms that are designed to detect or predict the failures. This section will include a survey the important features used in detection or prediction of failures in rotating machinery and composite structures.

2.1 Rotating machinery – gear-tooth failures

In the case of rotating machinery, the focus in this chapter will be on the failure of gear teeth in a gear–pinion arrangement. Figure 1 shows a typical gear–pinion setup. In this section, the common failure of stress-fatigue cracking of the gear teeth will be reviewed. By gaining an understanding of the way the tooth fails, features specific to the failure can be developed. These will be extracted from the raw data and used to identify the failure from all of the other normal changes that occur in the gearbox.

2.1.1 Definitions and gear nomenclature

One of the most fundamental components in rotating machinery is the gear–pinion pair (Fig. 1). Gears transmit motion from one rotating shaft smoothly to



another rotating shaft. Design of gears has been the subject of extensive publication [2,3]. Spur gears, as seen in Fig. 2, have a specially shaped, involute tooth profile that is designed to allow the gear-mating action to produce a constant angular-velocity ratio. If we define the input shaft as the driven shaft and the output shaft at the loaded shaft, then the output shaft rate of rotation is a constant multiple of the input shaft rate of rotation. As an example, consider the input shaft as having 31 teeth. This smaller gear, also known as the pinion, drives a larger gear having 80 teeth. Having the gear ratio, in this case 31/80, gives an output shaft rotation rate of 31/80ths times the input shaft rate.

Figure 1: Gear–pinion arrangement.

In order to understand the affect that fatigue induced damage will have on a vibrational signal, it is first necessary to understand the nature of a healthy gear–pinion meshing signal. Within a gearbox there are two essential components: a smaller driving gear, the pinion, and the larger driven gear, the gear. The period of time between the initial contact and final contact of the teeth is called the tooth-meshing period [4]. Even though the gear and the pinion will have different rotational speeds and periods, the tooth-meshing period will be the same for each. In a perfect rotational setting, the vibration from the gears,

g_ideal ( )s t , will be the addition of sinusoids of the fundamental tooth-mesh

frequency, m , and its harmonics, 2 , 1,2,...,mft m P .



Figure 2: Spur-gear nomenclature.

Faults in the rotation of each gear will produce a vibration function, g_mod ( )s tthat is both amplitude and phase modulated,

, (1) where ( )ma t is the amplitude modulation and ( )mb t is the phase modulation. If

_ ( )g mods t is considered to be a projection of a complex-valued analytical function it can be represented by

, (2)

where ( )H is a Hilbert transform. This allows zg(t) to be expressed in terms of the amplitude-envelope function, ( )mA t and the instantaneous phase function,

( )m t ,

. (3) The vibration signals measured by the accelerometers are harmonically related due to the gear-meshing frequency (see Fig. 3).

g_mod0

( ) (1 ( )) cos(2 ( ))P

m m m mm

s t X a t mTft b t

( )( ) ( ) mj tg mz t A t e

g_mod g_mod( ) ( ) ( ( ))gz t s t j s tH

Center Line

Pitch Point

Pressure Angle

BaseCircle

Pitch Circle

Pitch Circle

BaseCircle



(2 ) (2 )(2 ( )

0 0, 0,( ) ,r n r mr j nf t j mf tj n Nf t

n mn n N m m M

s t c e c e c e (4)

where N is the number of pinion teeth, M is the number of gear teeth, rf is the

pinion rational frequency and 0,1,2,...,c are load-coefficient multipliers for each of the components (for simplicity, the amplitude and phase-modulation terms have been left out). The first summation is at the gear-mesh frequency and the other two components are due to the pinion and gear separately. The accelerometer measurements are dominated by tonal components and the time series will be the addition of multiple sinusoids given by

(5)

Figure 3: The vibration spectrum arising from pinion and gear meshing.

The vibration signals from the gears propagate through the gearbox. The resulting vibration measurements taken by accelerometers will have the following form:

(6)

where there are N gears with Q transmission paths, ( )knh t is the impulse

response of the kth input signal via the nth path, ( )ks t is the kth gear motion signal and w(t) is external noise from other sources. Because the accelerometers are mounted on rigid gearboxes, the vibration propagation can be assumed to be time invariant. When a gearbox is stressed from overloading, the most common failure will be tooth breakage. This fatigue failure can be caused by many different factors like misalignment, torsional vibration or loose bearings. As a tooth is failing or

_0

( ) cos(2 ).P

g ideal m mm

s t X mTft

1 1

( ) ( )* ( ) ( ),QN

kn kk n

s t h t s t w t



after it has broken, the characteristic vibration will change and the accelerometer-measured signal will be altered. The changes will appear as amplitude and phase modulations in frequency and as impulses in the time domain. These impulsive variations in the data will be the basis for locating gear-tooth failures in the adaptive algorithm. The most common failure in overloaded gearboxes is the fatigue failure of gear teeth. This type of failure is induced by excessive bending stresses [5]. In the gear–pinion arrangement shown in Fig. 1, it is most common for the gear teeth to fail before the pinion teeth. This is due to the fact that the gear teeth have a wider inter-tooth spacing. It has been shown [5] that a bending moment stress- induced crack forms at the base of the failing gear tooth. This is due to the fact that the tensile strength is greatest at the tooth root [2]. The forces and crack propagation are illustrated in Fig. 4.

Figure 4: Forces on a loaded gear tooth with crack at tooth root.

As the crack propagates in the direction of the forces that the pinion exerts on the gear, a cracked tooth will flex. This flex causes a short duration pause in the meshing. This happens as the tooth moves but the load remains stationary as the tooth bends. Eventually, the bending stops and the loaded gear turns at which point there is a short snap-back effect as the tooth returns to its initial unloaded position. The effect seen in the measured vibrational meshing signal is a modulation of the meshing frequency. Changes in the healthy gear-meshing

ForcesFrom Pinion

ReactiveForce From Loading

Crack at ToothBase

CrackPropagationDirection

DirectionOfRotation



vibration signal will therefore manifest themselves as phase modulations in the frequency domain and as short impulses in the time domain [6–8]. Algorithms designed to detect this abrupt change in the meshing frequency will be discussed in Sect. 3. The goal of these algorithms is to reliably predict the total failure of the tooth based on the vibrational measurements. Should a tooth break completely, the lifetime of the entire gearbox is drastically shortened. The reason for this is that the debris from the failed tooth can become caught in the gear–pinion mesh causing instant failure. Also, the teeth adjacent to the missing tooth will experience even higher bending moments as they must now carry the load for the missing tooth. This drastically increases the chance of crack formation at the base of adjacent teeth.

3 Damage-detection methods

In this section, the methods used to detect the types of failures mentioned above are presented. Having gained an understanding of how damage occurs in gear systems, methodologies to extract features of the damage from vibrational measurements have been devised. This chapter will review these methodologies and tie their operation to the physics of the failures discussed in Sect. 2.

3.2 Gear-tooth fatigue-failure analysis

For rotating mechanical designs, gears are the fundamental components. Many areas of signal processing have been applied to the problem of gear-health monitoring with varying success. This section only outlines a few of the general areas of ongoing work. Failure detection using time-series averaging [8], amplitude demodulation [9], phase demodulation [9], time–frequency distribution analysis [10], wavelet filtering [11], bispectral domain analysis [12,13], multivariate statistics from principal-component analysis [14] and others have been proposed for condition monitoring. Each of these techniques relies on the measurement of vibrational signals propagating through the gearbox and measured by possibly many accelerometers. These accelerometers are typically mounted on the gearbox exterior at a variety of locations, see Fig. 5. The data in the form of time series from these accelerometers is collected by a data-analysis system and stored for the purpose of feature extraction. The next several sections describe the types of features extracted, their relationship to the physics of the failure and the methods used to detect the damage and the pros and cons of each method.



Figure 5: Mounting of accelerometers on a gearbox test bed.

3.1.1 Time-synchronous averaging Time-domain averaging is used to examine the vibration produced by a single gear in a system that contains possibly a large number of mixed vibrational sources. This is a theme that is consistent among the various techniques, that is, separation of the desired signal from a large mixture of signals and contamination by noise. Time-domain averaging is a first-order technique. Later is this chapter, various higher-order techniques will be discussed. Time-domain averaging is a process whereby the measured vibration signal is ensemble averaged over the period of vibration of the gear that is to be analyzed. Taking a large number of averages produces a nearly periodic output. If the meshing signal g(t) is treated as a continuous signal then the averaging operation is given by:

(7)

In the above equation, T is the period of rotation of the gear of interest, N is the number of ensemble averages. If the signal g(t) is sampled as is the case in real systems, the discrete signal is given by

(8)

0

1( ) ( ).m

g t g t mTN

0

1( ) ( ).m

g n g n mTN



Taking the z-transform of (8) leads to the transfer function H(z)

(9)

where G(z) is the z-transform of g(n) and G is the z-transform of ( )g n .Evaluating (9) on the unit circle produces the frequency-response magnitude and phase responses that are given by

(10)

(11)

This analysis provides the insight that the operation of time-domain averaging is the equivalent of filtering the signal in the frequency domain with an ideal impulse train [15,16]. Observing (10) shows that the frequency response magnitude is unity for integer multiples of the averaging frequency but falls off quickly for other frequencies. Therefore, the noise reduction goes as (1/N). This greatly enhances the signal of interest, i.e. vibration that arises from the gear that has a period of rotation of interest. Several important drawbacks, however, exist in using this approach. The first is that the method requires many averages and long signal lengths. If the shaft rate is not constant, which is the case in most real-world applications and in most types of rotating machinery, then the comb filter becomes misaligned. This in turn causes a broadening of the spectral peaks allowing more out-of-band noise to enter the averaging process. Observing (10), one can see one of the most important drawbacks in this method. The equation dictates that the number of samples per sample period (to have unity frequency magnitude response) be an integer number. Therefore, should the shaft rate vary even by a small amount, the sample rate would have to be varied to correspond correctly. This is impractical in a real-world sense as most analog-to-digital conversion devices operate at a fixed sample rate. The last drawback of this approach is that a different vibration measurement and sample rate is necessary for each gear. This restriction makes any real-world system both complicated and expensive. The main positive characteristic of this type of monitoring system is the simplicity of the processing. Averaging requires little processing power for a simple system (i.e. a single gear–pinion pair system).

3.1.2 Amplitude and phase demodulation As mentioned in Sect. 2, the common physical model of gear-tooth failure is a modulation of the meshing vibration signal caused by the bending moment

( ) 1 1[ ] ,( ) 1

NT

T

G z zH zG z N z

sin( ) 1 2[ ]( ) sin

2

NTG zH z

TG z N

arg( ) ( 1) .2TH N



induced flexing of the failing gear tooth. Should a modulation occur as modeled, then demodulating the meshing vibration should reveal the presence of a cracked tooth. This section describes the mathematical analysis that relates the physical properties of the modulation to the failure-detection algorithm. The analysis for amplitude and phase demodulation of the vibration signal typically begins with a Fourier cosine representation of the healthy gear–pinion meshing signal. This is given by

(12)

Here, the fundamental period of the vibration signal has a period of Nt fr, where Nt is the number of teeth on the gear to be analyzed, fr is the rotation frequency of the gear, Xm is the amplitude of the mth term of the cosine series, and m is the phase of the mth term in the cosine series. Should a tooth on the healthy gear whose meshing signal is represented by (12) begin to break, the healthy signal should contain a local amplitude change (t) and a phase change (t). Equation (12) can then be changed to represent the signal arising from a non-healthy gear and becomes

(13)

The task then remains to estimate the modulation as a function of the dynamics of the gear so that a detection algorithm can be designed to detect an incipient fault. The amplitude and phase modulations apparent in (13) can be written in a generic form again through the use of the cosine series.

The amplitude modulation is given by

(14)

and the phase modulation is given by

(15)

To estimate the amplitude and phase modulations, the signal is bandpass filtered about the dominant mth harmonic. Assuming that the harmonic signal has enough power, the entire series is then approximated by the mth term.

(16)

0( ) cos(2 ).m t r m

mg t X mN f t

0

( ) (1 ( )) cos(2 ( )).f m t r mm

g t X t mN f t t

0

( ) cos(2 ),m mn rn

t A f t mn

0

( ) cos(2 ).m mn rn

t B f t mn

( ) (1 ( )) cos(2 ( )).m m m t r m mz t X a t mN f t b t



The Hilbert transform is then taken to form an analytical signal representation.

(17)

Now that an analytical representation of the modulations exists as a function of the system dynamics (number of teeth and period of rotation) a detection algorithm can be designed to look for the particular modulation. There are many difficulties that arise in actually implementing the amplitude and phase-demodulation technique. The fact, again, that the shaft rotation may not be absolutely constant makes this method unreliable. Also, the question of which mesh harmonics to examine may be difficult to answer. This technique also is different from gearbox to gearbox and is difficult to automate across all types of gearboxes.

3.1.3 Stationary metrics for damage detection Based on the analysis of vibration signals arising from healthy gearboxes, several metrics have been developed for fault detection under the assumption of stationarity of the mesh-frequency signature and its harmonics. Equation (5) gives the ideal transmission vibration signal (in the absence of noise). The presence of a fault superimposes additional dynamics on the vibrations signal [17]. The occurrence of a single fault abruptly causes the peak-to-peak value of the vibrations signal to increase. If the damage is distributed around the gear (i.e. from spalling or pitting) the peak-to-peak value can remain constant. However, it has been observed [18] that the amplitude of the meshing signal and harmonics Xm decrease for distributed damage. This section discusses some of the stationary measures to exploit these observations for detection of faults in rotating machinery.

3.1.3.1 FM0 Parameter The FM0 parameter was developed [19] based on the observation of the changes that occur in the healthy gear vibrations signal. FM0 is a metric that is used to detect major changes in the meshing vibration signal. FM0 is defined as

(18)

where PPA is the peak-to-peak amplitude of the time synchronous averaged waveform. A(fi) is the amplitude of the gear-mesh vibration fundamental and harmonics in the frequency domain. The algorithm used to calculate FM0 is given in Fig. 6.

1

0 ,( )

Ni

i

P P AF MA f

( ( )).manalytical signal H z t



Figure 6: Algorithm for the calculation of the FM0 parameter.

Observing (18), it is noted that where the PPA increases, FM0 obviously increases (corresponding to a point of major damage). Also, where A(fi)decreases while the PPA remains constant (corresponding to distributed damage) the FM0 parameter again increases. This measure, therefore, provides a metric on which detection algorithms can be executed.

3.1.3.2 FM4 Parameter In order to augment FM0 for the cases where faults occur at a limited number of teeth, the FM4 parameter was developed [19]. The FM4 parameter is a metric that is based on the method of removing the mesh frequencies, harmonics, and 1st-order sidebands from the measured vibration signal. The signal that is left over after removing these components is called the difference signal [19]. The normalized kurtosis of the difference signal is then computed as the kurtosis of the difference signal divided by the variance of the difference signal squared. The FM4 parameter is given by

(19)

Here di is the difference signal, d is the mean of the difference signal and N is the number of samples used in the calculation. Equation (19) shows that, when the energy in the higher-order sidebands increases due to breakage, the FM4 metric deviates from its nominal value of 3, the normalized kurtosis for Gaussian noise. The algorithm for calculating the FM4 parameter is given in Fig. 7.

VibrationMeasurements

SignalConditioning

DC Offset Removal

ShaftTachometer

Time-Syncronous Averaging

FM0Parameter

4

12

1

4 .

Ni

iN

ii

N d dFM

d d

TSACalculation



Figure 7: Algorithm for calculation of the FM4 parameter.

3.1.3.3 NA4 Parameter The NA4 parameter was developed in order to detect the onset of damage and then to continue to increase in magnitude as the damage progresses from mild to severe [18] . The NA4 parameter is given by

(20)

In (20), ri is the residual signal. This is the time-synchronous averaged difference signal without having removed the sidebands. NA4 is the quasi-normalized kurtosis, divided by the time-averaged variance. N is the number of samples in the current record in the run ensemble, i is the sample number in the time signal and j is the time sample in the run ensemble. If there is no damage, the NA 4 parameter should have a nominal value of 3, the case where only

Figure 8: Algorithm for calculating NA4 metric.

Vibration Measurements

Signal Conditioning

DC Offset Removal

Shaft Tachometer Time-Synchronous Averaging

CalculateFM4

Parameter

Remove mesh frequency, fundamental, and 1st-order sidebands

TSACalculation

TSACalculation

Remove mesh-frequency fundamental

CalculateNA4

4

1

2

1

4 .1

Ni

iM N

ij i i

r rN A N

r rm



Gaussian background and measurement noise is present. The algorithm for calculating the NA4 parameter is given is Fig. 8 Note that is this figure the several steps to calculate the TSA are combined in one block.

3.1.3.4 NA4* Parameter In order to provide enhanced trending capabilities, the NA4 parameter in the above section was developed. This was modified to develop the NA4* parameter by changing the denominator so that it is “locked” should the variance of the residual signal exceed a statistically determined value [20]. The NA4* parameter is given by

(21)

Here, M2 is the variance of the residual signal for a healthy vibration signal.

3.1.3.5 M6A and M8A These metrics were originally developed to detect surface damage on machinery components (such a spalling, and pitting). M6A and M8A work on the same principals as FM4, but are more sensitive to peaks in the difference signal. M6A is given by

(22)

And the M8A parameter is given by

(23)

It is simple to see the historical trend in the various measures that have been developed in that in order to provide more sensitivity, the measures are calculated using higher-order moments of the difference signal. In Sect. 3.1.5, a measure that tracks all of the higher-order moments is introduced. All of the above measures, FM0, FM4, NA4, NA4*, M6A and M8A are measures that operate on the time-synchronous averaged data. These assume stationarity of the signal that may not always be the case. The following section introduces the methods that have been developed for the cases of non-stationary vibration signals.

6

2 13

2

1

6 .

Ni

iN

ii

d dM A N

d d

8

2 14

2

1

8 .

Ni

iN

ii

d dM A N

d d

4

* 122

2

4 .

Ni

ir r

N A NM



3.1.4 Time–frequency methods for fault detection Several metrics based on time–frequency analysis have been developed [21] in order to detect fault signatures in measured vibration signals arising from gearboxes. These methods including spectrum analysis, spectrum analysis, and matched filtering, are designed to work when the time series arising from the accelerometers placed on the gearbox, are stationary. In real systems, the vibration frequency being measured can change rapidly with time. Also, the stationarity of the signal, on the time scale of the tooth meshing, is an issue especially in the presence of a fault. Purely frequency-domain methods of analysis have the drawback that upon decomposition of the signal into its composite frequencies, there is no way to determine when each of the frequencies has occurred [22]. In order to localize the damage in time and frequency, new methods of time–frequency analysis must be employed. Many time–frequency methods have been studied and applied to the vibration-based analysis of fault detection. This section will provide an outline of some of the more recent of these.

3.1.4.1 Short-time Fourier transform (STFT) The short-time Fourier transform (STFT) is also known as the spectrogram or windowed Fourier transform. In order to produce the spectrogram, short segments of the measured time series are windowed and Fourier transformed. Usually, overlapping windows are employed in a fashion where the windows move along the time series. Figure 9 shows the methodology for a linear chirp. The STFT was developed in order to handle the cases where the frequencies of the vibration signal are changing rapidly with time. If the time duration of the window being employed in the STFT is shorter than the time duration of the changes in either the frequency or changes in the statistical nature of the signal (changes in the probability density function of the amplitude distribution) then the algorithm is useful for fault detection. If the window is too long, then the effect of the faults cannot be accurately localized in time. The STFT is a transform that maps 1D time-series data to a 2D map. This complicates the portion of a damage-detection algorithm, as the tell-tale features that imply damage are now in a two-dimensional domain.



Figure 9: Short-time Fourier transform methodology for a linear FM sweep.

The STFT is given by:

(24)

where g is the window function, f(u) is the data to be transformed, and the sare the frequencies to be analyzed. The main drawback of applying the STFT is that the time durations and frequency resolutions are fixed by the window size and the FFT size, respectively. In physical terms, this means that the signal components that change rapidly within a time window, or lie outside the FFT range will be obscured.

2( , ) ( ) ( ) ,i uf t g u t f u e du

FFT

Overlapping windows moving in time



Figure 10: STFT for breaking tooth.

3.1.4.2 Wigner–Ville and Choi–Williams distributions applied to fault analysis The Wigner–Ville (WV) distribution [22] is a transfer function that, like the STFT, transforms 1D time-series data to a 2D time–frequency space. The WV distribution of a signal s(t) is given by

(25)

This can be discretized for use on sampled data as

(26)

Here, hn(k) is the data window that performs the function of frequency smoothing and x is the data to be analyzed.

Although the WV distribution provides high time–frequency resolution, it suffers from aliasing and, more importantly, significant cross-terms between the time and frequency components that make analysis of the 2D map produced by the WV transform difficult.

* 2( , ) .2 2

j fWD t f s t s t e d

2 * 4( , ) ( ) .j fkn

kWD n f h k x n k x n k e

Breakage



In order to suppress the cross-terms that arise in the WV distribution, modifications [22] were proposed. This modification includes the multiplication inside the integral of the WV distribution by a kernel given by

(27)

where controls the amount of attenuation of the cross-terms. By choosing the proper value of , the cross-terms in the VW distribution can be suppressed without losing the resolution in the auto terms. The WV method has been used to localize failures in time and frequency for spur gears. Several other papers [23] have been published using methods based on the WV-CW distribution.

Figure 11: The Wigner–Ville distribution of spur-gear breakage.

2 2

2( , ) ;e

Cross-Terms

MeshingFrequency

Breakage

,



Figure 12: The Choi–Williams distribution of spur-gear breakage.

3.1.4.3 Continuous-wavelet transform The continuous-wavelet transform (CWT) is also a time–frequency analysis tool that has recently received attention in the literature [24,25] in a wide array of applications. The CWT has also been applied to the analysis of signals arising from gearboxes with the goal of detecting or predicting failures. The CWT remedies the fixed window-length problem encountered in the use of the STFT through the use of variable window sizes. The CWT used short window lengths for the higher frequencies and longer window lengths for the low frequencies. The CWT, as a generalization of the STFT can therefore extract local features in both time and frequency. This makes the CWT a useful tool for detection of faults that are of an impulsive nature and that alter the frequency content of the non-stationary vibration signal being analyzed.

3.1.4.4 Continuous-wavelet analysis In continuous-wavelet analysis the analysis begins with a quantity referred to as the mother wavelet [24]. The mother wavelet is translated (delayed as in the overlapping windows in the STFT) and time scaled. Time scaling is the time dilation or the time compression of the mother wavelet. The analysis begins with a complex-valued window function or the mother wavelet is given by

(28) ( ) .ps

uu ss

MeshingFrequencies

Breakage



Translation is accomplished via a time shift thus

(29)

where 0p and 0s and is real. The CWT is given by the integral equation

(30) Here, f(u) is the vibration measurements to be analyzed.

Also, a condition that must be met exists on the choice of the mother wavelet. That is, the mother wavelet must be what is referred to as an admissible function. It must have finite energy satisfying

(31)

The natural question to ask then is which mother wavelet should be used for fault and failure analysis? This question has been recently researched [26,27] for various types of faults. Mother wavelets such as the Morlet wavelet, Shannon wavelet and the Hybrid wavelet [28] that combines the exponential decay of the Morlet with the flat passband of the harmonic wavelet, and a frequency-domain Morlet comblet [28] have all been used with varying success. The next challenge in using wavelet analysis for fault detection is that one cannot choose a continuous set of time scales. The methodology of choosing a set of time scales is referred to as multi-resolution analysis. Several books have been written on the subject [29]. In some applications, octave band-scale analysis is used where s=2k. However, typically in gear-fault analysis, a linear multi-resolution analysis is used. The linear variation of time scales over a range of interest forms a highly redundant set of basis functions. In this way, gear-meshing signals can be wavelet transformed and when sidebands arise that are localized in time, this causes a distortion in the delay-time scale plane. These features are then used for fault detection.

3.1.4 Independent-component analysis approach to health monitoring Independent-component analysis (ICA) is a generalization of principal- component analysis (PCA). ICA techniques depend on all of the higher-order moments of the signal being analyzed, whereas PCA is a second-order technique. ICA, based on information theory provides a powerful tool for signals analysis. In this section, an algorithm based on blind source separation (BSS) is used to provide a measure for fault detection in an operating gearbox by analyzing a quantity referred to as the learning curve W. For gear-fault detection, a trace of the changes in W, is the measurement used to provide evidence of failures. The BSS approach provides a measure of any change in the higher-order statistics of

, ( ) ,ps t

u tu ss

,( , ) ( ) ( ) .s tf s t u f u du

2( ).d



the gear signature. The BSS algorithm uses a non-linear function g(x), through which the input data is passed. The algorithm converges when the high-density part of the probability density function (PDF) of the input signal, x, is aligned with the highly sloping parts of non-linear, sigmoidal function g, and the slope of g is matched to the variance of x. Any changes in the higher order statistics of x(i.e. the PDF of x), the gear signature, will cause the trace of the weights to diverge. It is the change in the learning curve, caused by abrupt changes in the statistics of the gear signature that are used for fault detection. A detailed discussion of BSS convergence is provided by Bell and Sejnowski [29]. This section provides a method by which a simple measure of the changes in the statistics of a gear signature can be obtained. This avoids the difficulties involved with learning all parameters of the signature model, (5). As discussed in Sects. 3.1.3.1–3.1.3.5, the historical trend has been to use metrics for fault detection that rely on higher-order moments of the difference signal. Measures based in ICA inherently contain information on changes occurring in all of the higher-order moments. The method for BSS presented by Bell and Sejnowski [29] uses mutual information as the updated separating criterion for sources. The core idea behind the process is to minimize the mutual information, ( , )I , that the output components, Y, of a neural-network processor has about the input, X. The mutual information I(X,Y) is related to the entropy H(Y) and the conditional entropy H(Y|X) by

(32)

Entropy of the output is given by,

(33)

where [ ]E is the expectation function, ( )yf y is the probability density function (PDF) of Y. Maximization of the mutual information is at the point in the weight space where the gradient of I(X,Y) with respect to the weight matrix W is zero. Due to the complexities of the mathematical expressions a non-linear gradient descent algorithm is used. The gradient is found by differentiating. (32) with respect to the learned parameter W. The result is

(34)

(the second term is zero because it does not depend on W).

For an N N network, the BSS algorithm passes the input X through a non-linear function and the result is

(35)

( , ) ( ) ( )I Y X H Y H Y XW W

( , ) ( ) ( | ).I Y X H Y H Y X

( ) [ln( ( ))],yH Y E f Y

( ),Y g XW



where g is a nonlinear function that is twice differentiable, monotonically increasing and has highly sloping sections that match the PDF of the original inputs. The information-maximization approach learns a matrix W, by minimizing the mutual-information between the components of ( )Y g U ,with U XW . The learned un-mixing matrix W will be the inverse of the environmental mixing matrix A, 1W A (see Fig. 13). Bell and Sejnowski have shown that minimizing the mutual information between components of Y is the same as maximizing the entropy of Y. It has been proved that the PDF of Ycan also be expressed as,

(36) where |J| will be the absolute value of the transformation between X and Y and is a determinant of the matrix of partial derivatives,

(37)

By substituting (36) and (37) into (33), the entropy can be expressed as

(38)

The entropy of X will not be changed by updating W and can be ignored, therefore to maximize the entropy of Y, |J| is the parameter that is maximized.

If a natural gradient approach is taken, the following is the updating equation for BSS,

(39)

where ( )f X X , ( ) tanh( )g X X and ( )X is a pre-defined learning rate [30]. The un-mixing matrix weights in W are tracked to form a learning curve during the execution of the BSS algorithm. Changes in the learning curve W reflect changes in the statistics of the meshing signal due to tooth weakening and breakage.

( )( ) ,| |X

Yf Xf Y

J

1 1

1

1

.K

K K

K

y yx x

y yx x

J

( ) [ln ] [ln ( )].XH Y E J E f X

( 1) ( ) ( ) ( ( )) ( ( )) ( ),Tk k k I g y k f y k kW W W



Figure 13: A block diagram of the information maximization approach to blind source separation (BSS).

The BSS algorithm will be attempting to maximize the entropy of the output (produce a uniform flat distribution). This is achieved by changing the coefficients of W. So, if there is a change in the statistics of the input signal, the weights of W will be changed also. Since the algorithm is optimizing the weights of W for a static situation with little change in the PDF of the input, when impulsive changes occur there will be dramatic changes in W.

The BSS algorithm learns an un-mixing matrix in the form,

(40)

The weights w11 and w22 are set equal to 1 and the weights w12 and w21 are changed by the weight-updating equations. The un-mixing matrix is constructed in this form because it is a simple matrix to filter the signals and it is still able to detect changes in the statistics.

3.1.5. Illustrative experimental results for ICA-based processingTypically, to develop and test new metrics for fault detection, a test bed is designed to produce the failures of interest under controlled conditions. The data used in this section was collected using the Mechanical Diagnostics Test Bed (MDTB) at the Pennsylvania State University Applied Research Laboratory (ARL). This system records transitional sensor data with a PC data-acquisition system from gearboxes during excess loading (see Fig. 14). For this test bed, the

11 12

21 22

.w ww w

W



gearbox is driven by a 22.4-kW electric motor and the load is provided by a 56-kW electric motor/generator. There are several single-axis accelerometers and the data is recorded at ten second periods in half-hour intervals. The times given for each plot from Figs. 16–43 are measurements taken at half-hour increments, therefore the time between Fig. 23, time 200 and Fig. 24, time 204 represents 2 h of the gearbox running at 300% of the rated output torque. Each run begins with a healthy gear–pinion pair and is run until failure. For example, in Fig. 26 one can observe at gear tooth 40 a failure beginning to form. As the run progresses over the next 2 h, the breakage worsens and is reflected in the divergence of the learning curve. The same is also true in Fig. 32 at tooth 20, the learning curve begins to diverge at time 206 and worsens until time 265, 29.5 h later. A test system of this type is critical to the understanding of fundamental mechanics of failures in any structure or system and for the development of methods for detecting and predicting them.

Figure 14: The mechanical Diagnostics Test Bed (MTDB).

The runs chosen for this section show failures of a fatigue type caused by exceeding the rated-load capacity of the gear. For each case, the load is given as a percentage over the rated-load capacity. The learning-curve graphs that are shown are averages for 5 revolutions. The w12 weight is shown, but weight, w12or w21, can be chosen for the detection, because both weights will change as the statistics of the measured sources change.

3.1.5.1 Single gear-tooth failure In this example, the gear was driven until there was a single tooth broken (see Fig. 5). Figures 16 and 17 show the learning curves for time 134 and 136 when the gear was healthy. The rippling variations



in the curve are due to individual gear-tooth-meshing periods. The ripples are about 32 time samples wide, which is consistent with an input speed of 1750 rpm on the pinion and a 20-kHz sampling period. These ripples are an affect of minor changes in the statistics of the accelerometer measurements. The rising and falling effect of the curve is a result of gear-shaft imbalance causing amplitude modulation in the measurement.

Figure 15: Single gear-tooth failure (MDTB run #020).

Figure 16: Learning curve run #20 time 134.

Once a weakening in a tooth occurs, the PDF of the measured signal will change, as shown in Baydar et al. [31]. If there is a change in the PDF, then the weights of W will also change because the updating parameters are trying to achieve maximum entropy at the output. The learning curve for the un-mixing matrix will begin to diverge and become erratic due to the statistical uncertainty in the measured signals (see Fig. 18, time 137).

Ripple



This appears near teeth 45–55. Then, once tooth #48 actually breaks in times 138–140, (see Figs. 19–21) there will be an easily detectable change in the learning curve. This divergence in the learning curve is isolated for a small contact time (around 35 data samples). This is a direct detection of the impulsive nature of a breakage concentrated to a single tooth.



Area of Interest



Figure 19: Learning curve run #20 time 138


Fig. 21: Learning curve run #20 time 140.

Tooth breakage

Tooth breakage

Tooth breakage



3.1.5.2 Adjacent gear-tooth failureThe gearbox in run #12 had a 70:21 tooth ratio and was driven at 300% the maximum rated output torque. The gear was driven until there was a break in two teeth (see Fig. 22). Times 200 and 204 show the learning curve of the gear when there was still a healthy rotation (see Figs. 23 and 24). Since the gear was held at a 300% torque rate, the vibration on the gearbox increased over run #20. Time 205, Fig. 25, shows the onset of damage to the gear. The deviation in the curve around tooth #40 is almost unnoticeable unless it is carefully compared to the previous measurement, time 204. Observing changes in times 206–209 show how the breakage developed. The divergence in the learning curve is much greater in this instance than that of the single-tooth breakage in the previous section. The longer time of divergence indicates that there is a breakage in more than one gear tooth and that the breakage was concentrated to adjacent teeth. The learning-curve divergence lasts for almost 80 time samples in these detections, which is consistent with a breakage in 2 adjacent teeth. This shows the advantage over a time-averaging scheme where the length of the window used for averaging may be too long and may obscure the appearance of damage to multiple teeth.

Figure 22: Adjacent gear-tooth failure (MDTB run #012).

AdjacentToothBreakage






Onset of damage





Fig. 28 Learning curve run #12 time 209.

Onset of damage

Tooth broken

Tooth broken



3.1.5.3 Two non-adjacent gear-tooth failures In this example, the gearbox had a 46:30 tooth ratio and was driven at 200% the maximum rated output torque. The gear was driven until there was a break in two non-adjacent teeth (see Fig 29). Time 202 shows the graph of the healthy learning curve (see Fig. 30). The developing break around tooth #20 can be seen in times 204-207 (Figs. 31-33). In this case, the divergence is consistent with a single tooth breaking, but the large change in the rotational consistency is causing false deviation in later learning curves (see time 209 and 215, teeth 20–30, Figs. 34 and 35). As the gears are allowed to rotate longer though, these effects are reduced (time 262, Fig. 36).

Figure 29: Non-adjacent gear-tooth failure (MDTB run #008).




Figure 31: Learning curve run #008 time 204



Tooth broken

Onset of damage



Figure: 37 Learning curve run #008 time 265.





Times 262–268 and 285 show a detection of another tooth failure that is not adjacent to the initial break (Figs. 37–39). The failure appears as a new deviation at tooth #22. When the first breakage is noticed in time 207, there are actually two deviations that are detected. The first predicts the weakening failure of tooth #18. The second weakening predicts the latter break in tooth #22, almost 30 h before the failure occurs.

3.1.5.4 Multiple adjacent-gear-teeth failures The gearbox in run #6 had a 46:30 tooth ratio and was driven at 300% the maximum rated output torque. The gear was driven until there was a break in 10 teeth (see Fig 40 for a photograph of the broken gear). Figure 41, time 195, shows the healthy gear learning curve and then Fig. 34, time 198, shows the onset of two teeth weakening at numbers 23 and 25. Figure 43, time 199, shows the time period when there were multiple breaks in the teeth. This shows damage for 10 teeth, #22–#32, which is consistent with the photo and also shows a weakening at tooth #37. The erratic behavior of the signal vibration is clearly visible in the learning curve.

Figure 40: Multiple adjacent-gear-tooth failures (MDTB run #006).






4 Summary

In this chapter, approaches to damage detection in rotating machinery, namely gearbox systems, have been reviewed. Although the focus has been on gear-tooth failure, the methods presented in this chapter can be and have been adapted for use in other vibration-based failure-detection applications. For systems, especially rotating machinery systems where the period of rotation is constant (like in the case of power-generation equipment), time-syncronous averaging provides good results. In the case of mechanical systems where the

Onset of damage

Multiple broken teeth



vibration signals are stationary, the methods in Sect. 3.1.3 can be used successfully. For non-stationary signals, which make up the bulk of real-world systems, time–frequency methods have proven successful for systems ranging from delamination failure of composites [32] to bearing failure [33]. Methods for blind source separation and blind deconvolution based on independent- component analysis have also been recently applied to health monitoring [1,14]. These new methods provide convenient measures that inherently monitor all of the higher-order statistics of the vibration signal arising for the structure of interest. New work in the area of data fusion is also being applied to structural and mechanical health monitoring [34]. Lastly, research into how to handle the effect of contamination via external vibration is of vital importance. Limited work has been done in this area [35].

5 Acknowledgements

The data used in this chapter was developed under grant N00014-95-1-0461, sponsored by the Department of the Navy, Office of the Chief of Naval Research. It was collected, stored and distributed by the Condition-Based Maintenance (CBM) Department at the Applied Research Laboratory.

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chapter five - wit press...gives an output shaft rotation rate of 31/80ths times the input shaft...

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