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SIRM 2009 - 8th International Conference on Vibrations in Rotating Machines, Vienna, Austria, 23 - 25 February 2009
Vibration Analysis using Time Domain Methods for the Detection of small Roller Bearing Defects
Tahsin Doguer Jens Strackeljan Institut für Mechanik
Otto-von-Guericke-Universität Magdeburg,Fakultät für Maschinenbau
30106, Magdeburg, Germany [email protected]
Institut für Mechanik Otto-von-Guericke-Universität Magdeburg,
Fakultät für Maschinenbau 30106, Magdeburg, Germany
ABSTRACT The analysis of vibration signals is a major technique for monitoring the condition of machine components.
A focus of this paper is given to the early detection of very small bearing damages like false brinelling faults, which occur in the presence of a small relative motion between the rollers and raceways during non-rotation times. This leads to a small damage which is characterized by elliptical wear marks in the axial direction at each roller position. The paper shows that the vibration structure generated by a small surface defect differs from a normal state even if the signal energy is eliminated by normalisation of the data. Suitable time domain features are a mathematical description of the shape of selected time domain peaks, which could easily be calculated by the higher derivatives of the time acceleration signal and some parameters characterizing the randomness of the peak positions. After the step of extracting 32 features from the time signal a feature selection process is executed automatically. This enables the selection of a feature subset which is best suited to the present fault situation. Test rig results indicate the high potential of the new time domain features for both fault types. The last chapter gives a short introduction in an algorithm for bearing fault simulation.
1 INTRODUCTION Condition monitoring techniques have the objective of achieving the most effective, safe and efficient
operation of mechanical plant, machines or engines. In recent years there has been an increasing interest due to the requirement of reduced maintenance costs, improved productivity and safety. Roller bearings are used in a wide range in industrial rotating machinery and the robustness and reliability of roller bearings are essential for the machine health. Damages can put human safety at risk, cause long term machine down times, interruption of production and result in high costs. Main steps in condition monitoring are applying measurement techniques, signal processing and signal categorisation combined with classification algorithms, see [13], [6], [21], [12] and [1]. The healthy signature can be measured on operating machines but data with seeded faults are more difficult to obtain.
This paper presents an approach for detection of roller bearing defects using time domain methods, without the necessity of special information about bearing type and other operating parameters. The vibration signals of a roller bearing deliver a large content of information about its structural dynamics and operating conditions. Typical representatives as measurement parameters are displacement, velocity and acceleration.
Depending on bearing condition, signal can have various forms: • Structured signals due to rolling of a ball on a pitting, • Noise signals with stochastic excitation due to rolling on a large-scale damage or on smooth surface of
an intact bearing. An automated condition monitoring system works according to the main steps of measuring a signal, which represents the vibrations of the bearing, analysing the signal according to features and assigning the signal to a predefined damage category.
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2 THEORY
2.1 Derivative of time signal The presence or absence of bearing faults can be determined form the raw acceleration signal only in few
cases. In general the signal contains a multitude of different vibration components. To get a better understanding of the random vibration generated by rough surfaces we tried to isolate all periodic, load dependent and external vibration sources. This lead to a very simple test rig describing the pure vibration generated by a rolling ball on an inclined level, where different surface structures are achieved by using two types of files, see [2].
It is a highly demanding problem of deciding which features are to be formed from the time signal in order to
achieve the most error-free classification between different surface structures. The main idea is to perform a statistical analysis of the time signal or higher derivatives of this. Previous work in higher derivatives has been reported by J.D. Smith, Lahdelma and others, see [10], [11] and [14]. Specially Lahdelma has published a couple of papers describing the theoretical background of higher derivatives in combination with condition monitoring tasks, see [9]. The general suitability of this technique even in practical real world applications is without any controversy. In general the displacement, velocity or acceleration as a basic signal could be taken into account. This data set (time response), containing the acceleration values for discrete time steps can be seen as a function of time and hence derivable with respect to time. One possibility is to calculate the higher derivative for the complete time signal and extraction of characteristic features from the new time signal. RMS, Peak Value and statistical parameters like Kurtosis and Crest Factor are typical features for the detection of faults in roller bearings. Applying the Fast Fourier Transformation (FFT) on a time signal, performing the derivation in frequency domain and reconstructing the time signal via inverse FFT is an efficient method for calculating derivatives of arbitrary order. An advantage of this method is the ability to obtain integer, real or complex order derivatives, which all can be used for machine diagnosis, see [10].
High frequent vibrations are excited in roller bearing as the rolling element passes a damaged area. Figure 1a
shows the raw acceleration time signal )2(x gained by an accelerometer with a length of 0.5 s, which is in accordance to a number of 65536 digital data and a sampling rate of 131072 Hz. In original and non-filtered time signal, it is not possible to detect a high frequent vibration in signal. But after zooming in at 0.04 s, one can see how the signal structure changes at about 0.041 s.
Another way to make this change noticeable is calculation of the amplitude spectra of the signal with data
sets before and after the excitement. Two sets (each 1024 data points) were taken out from )2(x . Both have the same length. The first data set ends at 0.04 s and the second one starts at 0.04 s. The comparison of amplitude spectra in Figure 1e shows that the high frequent vibration generated by the rolling contact between ball and the damaged surface is not present before the excitement. The spectrum on the right shows the high frequent vibration in 40 – 60 kHz range shortly after the excitement.
High frequent vibration can be shown also via filtering. In Figure 1c and 1d filtered signal is shown by two different band pass settings (1-40 kHz and 10-40kHz). The first band pass filtered signal in Figure 1c (1-40 kHz) does not significantly differ from the raw signal in Figure 1a. A significant change can be seen in Figure 1d, in which the second band pass filtered signal (10-40kHz) is shown, at about 0.041 s. The similar effect can be obtained via second derivative )4(x of the raw time signal. The advantage is that it is not necessary to filter the raw signal (Figure 1f). This method can be very helpful in searching small faults in roller bearings, hence high frequent vibrations are excited via small damages.
2.2 Feature extraction The idea suggested in this paper is to use only parts of the time signal which stand in close relation to
possible bearing faults. For that purpose a peak in the time signal is defined as a local extremum in the measured acceleration signal )2(x , where x is the displacement. All peaks in the time signal which fulfil the requirements of the peak definition are detected and for each all significant information like peak position, amplitude and adjacent data points are stored. This data are used as an input for the calculation of features. The calculation of the fourth derivative x(4) was carried out by using a fourth-order centered difference formula on uniform grid. Therefore a set of adjacent data points on the left and right side of each peak have to be considered. Now the calculation of different features on the basis of this set of peak information containing amplitudes, position and adjacent data points is possible.
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-100
0
100x(2)(t) raw signal
x(2) (m
/s2 )
0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048 0.05-40-20
02040
zoom in x(2)(t), from 0.03 to 0.05 s
x(2) (m
/s2 )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-100
0
100filtered signal, bandpass 1 - 40 kHz
x(2) (m
/s2 )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-10
0
10filtered signal, bandpass 10 - 40 kHz
Time (s)
x(2) (m
/s2 )
0 20 40 600
2
4
61024 points, left side of t= 0.04 s
Frequency (kHz)
x(2) (m
/s2 )
0 20 40 600
2
4
61024 points, right side of t= 0.04 s
Frequency (kHz)
x(2) (m
/s2 )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
-2000
0
2000
2. derivative of raw signal
Time (s)
x(4) (G
m/s
4 )
Figure 1: Example for demonstrating the usefulness of higher derivatives for the detection of bearing faults
(b)
(a)
(c)
(d)
(e)
(f)
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A feature vector is defined as a set of parameters extracted from the considered signal, which gives indications about the current state of the operating system. In condition monitoring statistical methods have been widely used for investigation, where measured data are time series. Extensive literature is available on diagnostic techniques using RMS, Kurtosis, Crest Factor and histograms, see [13], [6], [11] and [9].
The method suggested here uses peaks as a source of information corresponding to bearing faults. Possible features may be calculated from the ratio of local maxima of measured )2(x and )4(x . Also the distance variation between local maxima on adjacent or non-adjacent locations over a predefined offset value can be considered. Further features can be obtained by the number of local maxima with absolute values, which are over some predefined threshold values. The histogram of peak amplitude and peak distance distribution are additional features which could be considered. Taking the norm of signal values - for instance with root mean square - may be useful to eliminate the influence of signal energy of the measured impact sound. A selection of features from a total of 32 is listed in Table 1. The feature extraction and analysis are performed in MATLAB 7.0. For a fast, on-line condition monitoring, measurement equipment and the software routines can be combined.
3 APPLICATIONS
3.1 Detection of small bearing fault in simple demonstrator For the investigation of the method described in Section 2 a simple bearing test rig was used. In Figure 2, on
the left the very simple test assembly is shown. The components are the outer race of a roller bearing and a cage, in which the outer ring is mounted. Only one ball driven by compressed air is rotating and the inner ring is replaced by a whole shaft with eight nozzles around it. All nozzles are placed with an offset angle of 45° to assure a continuous load to the ball. The vibration signal is measured by an accelerometer mounted at the outer ring. This assembly allows the measurement of the isolated vibration generated by the rotating ball and all other sources of excitation are omitted. The pathway from the source of vibration and the sensor is well defined and the number of join patches is minimized. Figure 2, right side, shows the fault in the outer race, which was induced by creating a small groove using electric spark erosion.
Figure 2: (left) Test rig is air driven and consists of outer race of a deep groove ball bearing type 6310, cage and
accelerometer. (right) Outer race of the ball bearing. A point fault (diameter 510 µm) was introduced using electric spark erosion.
The comparison of the signals in Figures 3 (bearing without a fault) and 4 (faulty bearing) indicates that a separation of the signal from a bearing without a fault and the outer race groove is possible in the time and frequency domain without any problems. The overall vibration level differs significantly. To eliminate the influence of the vibration level signals are normalised. The impulse of the ball in contact with the groove excites natural frequencies of the bearing and the surrounding elements (assembly parts). The amplitude spectra in Figures 3 and 4 have amplitudes in a frequency range up to 30 kHz, which is typical for a small damage size but also for a random excitation of an intact bearing. One has to consider that the duration of contact between the damage and the ball is very short in comparison to the complete measurement time. In consequence the normalised spectra in Figure 3 (lower right) and 4 (lower right) do not show significant differences between the two states.
510 µm
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0 0.1 0.2 0.3 0.4
-20
0
20x(2
) (m/s
2 )
Time (s)0 10 20 30 40
0
0.5
1
x(2) (m
/s2 )
Frequency (kHz)
good bearing
0 0.1 0.2 0.3 0.4-10
-5
0
5
10
x(2) rm
s-no
rmed
Time (s)0 10 20 30 40
0
0.05
0.1
0.15
0.2
x(2) rm
s-no
rmed
Frequency (kHz)
good bearing
Figure 3: Time signals and the amplitude spectra of intact outer race. Above measured signal, below signal was
normalised by its rms value.
0 0.1 0.2 0.3 0.4-500
0
500
x(2) (m
/s2 )
Time (s)0 10 20 30 40
0
2
4
6
8
x(2) (m
/s2 )
Frequency (kHz)
faulty bearing
0 0.1 0.2 0.3 0.4-10
-5
0
5
10
x(2) rm
s-no
rmed
Time (s)0 10 20 30 40
0
0.05
0.1
0.15
0.2
x(2) rm
s-no
rmed
Frequency (kHz)
faulty bearing
Figure 4: Time signals and the spectra of faulty outer race. Above measured signal, below signal was
normalised by its rms value.
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All feature combinations in Figure 5 are calculated after the normalisation of )2(x , and are well suited to separate the both classes. The distances between the class centres are much higher then the variation of the feature within a single class. In Table 1 a description of the selected features used in the figures and the corresponding dimensions are listed. The numbering indicates that these features are only a subset of the complete feature pool. All feature combinations could be used to design an automatic classification algorithm, but it is not the objective of this investigation to test such a classifier.
The two combinations of features in Figure 5 are selected to demonstrate the potential of the peak features.
Comparing the scatter-plots of various feature combinations from different surfaces, one can observe the separation of data points which belong to different surfaces. This property can be used to monitor bearing health state and to perform damage detection. When using the combination 5 | 8 and 9 | 10 an 100% classification is possible. In general the selection of feature combinations could be executed by a software algorithm, see [3] and [4]. Strackeljan has developed different tools for the task of automatic feature selection considering vibration signals, see [16], [18] and [19]. In addition a couple of algorithms are available for a broad range of classification tasks, see [17].
Table 1: Description of features, which are used for roller bearing tests. No. Description 5 Mean value of the peak amplitudes from )2(x at local maxima (peak) after normalisation.
)2(x is the normalised form of )2(x . Dim: - 7 RMS of )4(x . )4(x was built, using local maxima (peak) in )2(x . Derivation method was
named in Section 2.2. )2(x is the normalised form of )2(x . )4(x and )4(x respectively. Dim: (1/s2)
8 Mean amplitude value of )4(x . )4(x was built as described in Feature 7. Dim: (1/s2) 9 Standard deviation of )4(x . )4(x was built as described in Feature 7. Dim: (1/s2) 10 Mean value of the ratios, which are obtained using the amplitude of the local maxima
(peak) in )(x 2 and )(x 4 at the corresponding position. Dim: (1/s2)
∑=
=pn
i )(i
)(i
p x
xn
F1 2
4
101 np = Number of peaks
1 1.5 2 2.50
1
2
3x 10
10
Feature 5
Feat
ure
8
0.5 1 1.5 2
x 1010
0
5
10x 10
10
Feature 9
Feat
ure
10
Figure 5: Scatter-plots of good and faulty outer races in air driven test rig. Each measured time signal was
normalised by its RMS value before feature extraction. Dim: -.
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3.2 Detection of small faults in the bearings of vehicle wheels
To investigate the potential of the features calculated from higher derivatives we use a test rig for the diagnosis of a complete vehicle wheel bearing assembly. The test rig allows a radial of axial loading of the bearing with realistic forces, which were obtained from measurements in a car during different drive manoeuvres. Figure 6 shows the damaged section of the outer ring as a 3D surface measurement and a photograph of the damage in the outer ring of the demounted bearing. Objective of the investigation is the determination of detection limits of small faults by a measured acceleration signal. The focus of the car manufacturers is oriented to the problem whether the vibration will generate an acoustic emission which could be noticed inside the vehicle interior.
Figure 6: Small fault at the outer ring of a roller bearing.
In Figure 7 good and damaged bearings are compared. The acceleration )2(x was measured directly on the
housing of the outer race of the bearing. The data set consists of 65536 equidistant data points by a sampling frequency of 131072 Hz. In a, b the raw signals and c, d filtered signals are compared. Noticeable difference in amplitude levels between good and damaged bearings has occurred. Comparing the amplitude spectra in 7e with 7f, one can see the increased amplitude level in the range 12 – 40 kHz, which is caused by the small fault on the outer ring, shown in Figure 6. On the other hand, the comparison of )4(x in c and d shows also an increasing of the amplitude level but no significant change in the signal structure.
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-20
0
20x(2)(t) raw signal, good bearing
x(2) (m
/s2 )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-20
0
20x(2)(t) raw signal, damaged bearing
Time (s)
x(2) (m
/s2 )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-500
0
5002nd derivative of raw signal, good bearing
x(4) (G
m/s
4 )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-500
0
5002nd derivative of raw signal, damaged bearing
Time (s)
x(4) (G
m/s
4 )
0 5 10 15 20 25 30 35 400
0.2
0.4FFT with raw signal, good bearing
x(2) (m
/s2 )
0 5 10 15 20 25 30 35 400
0.2
0.4FFT with raw signal, damaged bearing
Frequency (kHz)
x(2) (m
/s2 )
Figure 7: Comparison of good and damaged bearings in the time and frequency domain
(b)
(a)
(c)
(d)
(e)
(f)
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Nevertheless the features calculated from the )2(x and )4(x , which are described in table 1 allows a 100% classification of the complete data set for the states: bearing in good condition and bearing with the small fault (Figure 8).
1 1.05 1.1 1.152
3
4
5x 10
10
Feature 5
Feat
ure
8
2 3 4 5
x 1010
1
1.5
2
2.5x 10
10
Feature 7
Feat
ure
9
Figure 8: Scatter-plots of good and damaged bearings described in section 3.2.
4 SIMULATION OF ROLLER BEARING FAULTS
4.1 Contact model Considerable attention is being carried out for the bearing faults and condition monitoring. In this area the
main interest lays on fault detection and bearing life expectation. However the gaining of data from the real machinery to study these subjects may be cost-intensive and time-consuming. Simulations may supply a considerable help by giving a better understanding in occurrence, shape and effects of faults in roller bearings.
Previous work has been reported for fault simulation, in which roller bearings and different kinds of faults are modelled as spring-mass-damping systems, see [13]. The simulation model presented in this paper considers the roller bearing as a multi body system, which consists of inner race, outer race, cage and balls, and takes into consideration the non-linear forces between elements of the roller bearing, which are calculated via Hertzian contact theory. Based on the idea of contact between two circular elements, an approach for fault simulation is presented, in which the faulty region is described as a collection of adjacent circles with variable size, location and number (Figure 9). Then as the rolling element moves on the outer race, the contact forces are calculated between two circular elements (roller element and fault element). Each element is considered as a rigid body. Three degrees of freedom are allowed, x , y translational and ϕ rotational about z -axis, which describe the motion of a body on a plane. Each body is connected to origin of inertial system.
In the next step the contact condition between different elements of the multi body system is defined in a way, that the algorithm defines contact always between two bodies. In a system consisting of n elements, there are 2/)1( −⋅ nn possible contacts between two bodies, which must be determined for each time step, see [5]. The algorithm takes the position of each body as input, calculates the distance to other bodies and determines whether, and between which elements contact has occurred. In case of a penetration between two bodies a contact force is calculated. The Contact force CF
r contains the normal contact force ( NF ) and tangential contact
force ( TF ). NF is the sum of the load in direction of surface normal ( Q ) and the contact damping force ( DF ). As shown in Eq. (1), Q is calculated according to the theory of Hertz for contact between elastic bodies, see [15]. Ns∆ is the depth of penetration. µ is the Hertzian coefficient, see [20]. For roller element and fault element, fb EE , are modulus of elasticity, fb νν , Poisson’s ratios, fb RR , radii and ∑ ρ total radius of curvature, see [8]. Figure 9 shows the basic components of the bearing model. DF is the product of contact damping coefficient ( d ) and normal component of relative velocity ( Nv∆ ) between contact partners at the contact point.
NDfbf
f
b
bN vdFRREEE
sEQ ∆⋅=+∑ =−
+−
=′⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅∆⋅
⋅∑
′⋅= ,11,
111,3
23 2232ρ
ννµρ
iNNiNNDN nFFnFFFQFji
rrrr⋅−=⋅=+= ,, (1)
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After multiplication of NF by the unit vector of surface normal ( inr
) the vector NFr
for each contact partner is obtained as shown in Eq. (1). TF is calculated via Coulomb friction coefficient ( Rµ ), and Rµ is a function of relative tangential velocity ( Tv∆ ) between contact partners. The vector TF
r is obtained via multiplication by the
unit vector of tangential component of relative velocity ( itr
) as shown in Eq. (2).
iji TTiNTRT FFtFvF
rrrr−=⋅⋅∆−= ,)(µ (2)
The contact force ( CFr
) on each body is given by addition of NFr
and TFr
. The momentum at the centre of mass due to CF
r is calculated as shown in Eq. (3).
ii CiSZ FrM
rrr×= (3)
To simulate the air drive mechanism (Figure 2) eight so called drive regions are defined in our program, where external load is introduced to rolling element each time it passes by. In case of a frictionless rolling the rotating frequency increases continuously. If the friction is considered and the initial rotating frequency is given zero, the value increases until reaching a steady level.
4.2 Fault Simulation The fault on a roller bearing race is simply thought to be consisting of a gap on the bearing race. At both ends
of this gap two circles are attached tangentially to the outer race. As the bearing element moves, it enters the gap by rolling on the so called fault balls, as it leaves the faulty region it continues rolling on outer race. Figure 9 shows the idea in an overstated way. In our simulation the fault width ( bf ) is set to 520 µm, radius of the fault balls ( bR ) is 2 mm, fault depth is 40 µm. Program requires only the angular position of the fault centre ( fϕ ) and
bf . The fault balls are then placed via simple trigonometric constrains.
Figure 9: Model for fault simulation in a roller bearing.
The depth of fault can be controlled by radius of circular fault elements. It also gives an indication about the steepness of fault region. Raw surfaces can be simulated by increasing the number of balls in the fault region. Hence the fault is body fixed, this model can be applied not only to simulate outer race faults, but also to simulate faults on inner race and on rolling elements.
4.3 Results As simulation results displacement, velocity, acceleration and contact force can be plotted over time or an
FFT analysis can be performed. In Figure 10 the )2(x (t) and )2(y (t) are shown. A small fault was introduced to the bearing model, with a size described as in 4.2 at position 0°. The result is noticeably similar to the acceleration signal, which was measured on the real damaged outer ring in Figure 4.
After validation of the program with the real test rig, the results may be used as a learning set for further
investigations like statistical calculations.
T
ω
Ra
fb
φf
Rf + +
Outer race
Rf
T
Rb +
Rolling element
Fault element
y
x 0°
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2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3-40
-20
0
20
40x(2) (t)
x(2) (m
/s2 )
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3-40
-20
0
20
40y(2) (t)
y(2) (m
/s2 )
Time (s)
Figure 10: )2(x (t) and )2(y (t) of the simulated roller bearing with a small fault on the outer ring.
5 CONLUSION The results obtained from two different test rigs demonstrate that the proposed feature generation is a
promising method for the detection of faults in roller bearings. The first test rig represents the rolling contact between files with different grades of cut and a roller ball. These test conditions are a simple model of an extended fault in roller bearings, which exceed the spacing between the balls and lead to a permanent contact between damaged surface and the ball. The second one considers a single small size pitting fault on a bearing race. In both cases the separation of different surfaces and also faulty and non-faulty bearing conditions is possible. The method enables a new introduction of higher derivatives because the algorithm is oriented on single peaks and a complete time signal. In the next steps the algorithm will be tested in combination with real world application.
First investigations concerning vehicle wheel bearings have been done and show that the method in general is
a suitable tool in condition monitoring. The results from the simulation program indicate clearly, that the simulation of bearing faults in combination with multi-body-system could help to improve the understanding of fault induced vibrations. Further work in progress is directed towards gaining the necessary improvement of the simulation programme and the implementation in standard MBS-Software like ADAMS or SIMPACK.
REFERENCES
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