Mechanical Systems and Signal Processing 166 (2022) 108349
0888-3270/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Identification of different manifestations of nonlinear stick – slip phenomena during creep groan braking noise by using the unsupervised learning algorithms k-means and
self-organizing map
Jurij Prezelj
a, Jure Murovec
a, Severin Huemer-Kals
b,*, Karl H ¨ asler
c, Peter Fischer
baUniversity of Ljubljana, Faculty of Mechanical Engineering, Aˇskerˇceva cesta 6, 1000 Ljubljana, Slovenia
bGraz University of Technology, Institute of Automotive Engineering, Inffeldgasse 11/II, 8010 Graz, Austria
cMercedes-Benz AG, 71059 Sindelfingen, Germany
A R T I C L E I N F O Keywords:
Brake NVH Signal processing Acoustic emission Signal features
Unsupervised classification Real-time AE envelope
A B S T R A C T
Creep groan is a friction-induced, low-frequency vibration and noise phenomenon of a vehicle’s brake system which is excited by a repeating stick–slip effect. Together with high influences of design and operational parameters, the non-linear stick–slip leads to an interesting bifurcation behaviour of creep groan. For objective rating procedures, detection and classification methods considering this bifurcation behaviour are necessary. Within this study, an approach based on acoustic emission is presented. The approach harnesses high-frequency acceleration contents that accompany creep groan’s characteristic stick–slip transitions. Whereas low-frequency vibration contents below 500 Hz are mainly defined by the characteristics of the brake system and the suspension of the vehicle, vibrations in the high-frequency range above 10 kHz exhibit patterns of waveforms similar to the patterns of acoustic emission bursts. By applying non-overlapping high- and low-pass filters, a novel signal, enveloping these bursts, was created. This envelope bursts signal enables a precise detection and quantification of stick–slip transitions directly in time domain, and led to the development of a whole new set of vibration signal features. These nine signal features were used to feed the unsupervised classification algorithms k-means and Koho- nen’s self-organizing map, which delivered robust and meaningful results. Four different creep groan classes were detected, where each has shown to be linked to a specific creep groan manifestation: Low-frequency groan, high-frequency groan and two transition phenomena with two/three stick–slip events per cycle were found. Classification results and their linked me- chanical behaviour suggest an interaction between two significant vibration patterns during creep groan, probably a longitudinal and a torsional displacement of the axle. Aside of deeper insights in creep groan’s bifurcation behaviour, the presented study enables not only the identification of creep groan, but also the automatic classification of its manifestations in real-time, and therefore provides further possibilities for creep groan control methods.
* Corresponding author.
E-mail address: severin.huemer-kals@tugraz.at (S. Huemer-Kals).
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing
journal homepage: www.elsevier.com/locate/ymssp
https://doi.org/10.1016/j.ymssp.2021.108349
Received 11 September 2020; Received in revised form 30 April 2021; Accepted 14 August 2021
1. Introduction 1.1. Motivation
According to an estimation of the Society of Automotive Engineers, the annual costs associated with brake noise and vibration problems are estimated to be up to 100 million €, [1]. Well-researched brake squeal remains one of the main contributors to this amount. However, low-frequency problems such as the creep groan noise gained attention due to current trends such as drive train electrification or automated driving functions – even though the problem is also well-known for conventional drive trains, [2]. The creep groan phenomenon is undesirable because it annoys passengers and gives drivers a feeling of a damaged braking system, probably reducing quality and safety impressions. Creep groan vibrations are generated by abrupt, non-linear force fluctuations be- tween the brake pads and the brake disc, which repeat with frequencies between approx. 15 and 500 Hz and result in unpleasant noise as well, [3–7,2,8–11]. Hence, sophisticated creep groan noise rating procedures are necessary for simulative prediction as well as experimental validation. These rating procedures can consist of subjective or objective assessments, see also the VDA recommendation 314 within [12].
Subjective assessments of creep groan phenomena remain an important part of practical studies due to the extreme complexity of the problem. The automotive industry spends substantial time and effort to evaluate and quantify the occurrence, intensity and harshness of creep groan noise, see again [12]. These tests are time-consuming and many resources are needed to obtain useful results – also with the background of different manifestations of creep groan phenomena. In addition, subjective results tend to vary due to the different perception of noise by each individual. Consequently, many tests have to be performed in order to achieve a representative rating. Results of subjective tests can be used only partially for the development of control measures, aiming to control and prevent creep groan vibrations.
Objective assessments are able to conquer many of the mentioned drawbacks such as result variation or high effort. Certainly, objective approaches need to be sophisticated enough to deal with the high complexity of creep groan phenomena. Therefore, an attempt was made to minimize the need for subjective testing. This approach allows the real-time identification and evaluation of creep groan vibrations on the basis of measurable quantities, which could also be used for possible control algorithms such as presented within [13].
1.2. Creep groan phenomena
Creep groan phenomena occur during braking at low speeds and during slow acceleration from a standstill, especially on a slope.
This means the action of releasing high brake pressures while simultaneously applying torque on the wheel, which especially happens when starting a vehicle with automatic transmission or electric drivetrain, [14]. Creep groan phenomena can then lead to a high level of vibrations and noise. Main attribute of friction-induced creep groan noise and vibration phenomena is their rather low dominant
a) Braking to standstill on a 15% hill at a max.
speed of vveh = 2.8 km/h.
b) Braking from vveh = 93 to 43 km/h at a max. brake pressure of pB = 25 bar.
c) Creep groan action from standstill on a 15% hill: 2 short events, 3 long events.
0 5 10 15
time in s 0
0.5 1 1.5 2 2.5
frequency Hzin
104
10-310-210-1100
acceleration m/sin 2
0 0.5 1 1.5 2 2.5 3 3.5
time in s 0
0.5 1 1.5 2 2.5
frequency Hzin
104
0 2 4 6 8 10 12
time in s 0
0.5 1 1.5 2 2.5
frequency Hzin
104
Fig. 1.Frequency contents of a brake calliper’s tangential accelerations during standstill, braking and creep groan action.
frequency.
Creep groan is inherently correlated with stick–slip transitions in the contact between disc and pads as stated within many pub- lications, e.g. [2,8,15]. These stick–slip transitions separate stick-phases (suspension wind-up) and slip-phases (release, damped oscillation), [16]. Creep groan is therefore composed from a series of stick–slip occurrences; in order to understand creep groan vi- brations, many authors focused on the analysis of stick–slip, see e.g. [17]. These transitions can be seen as an excitation source to the mechanical system. Simultaneously, the mechanical system can be regarded as a feedback loop, acting back on the stick–slip effect.
Therefore, stick–slip is a non-linear physical phenomenon comprising of abrupt phase transitions of the friction partners’ relative velocity, from zero to a finite relative velocity. Forces acting on a mechanical system have the shape of impulses due to virtually instantaneous changes of relative velocity and accelerations. Consequently, the energy of stick–slip transitions is distributed across a wide frequency range (Fig. 1c), similar to the broadened frequency range of brake actions without creep groan (Fig. 1a-b).
Vibrations depend on many suspension components, see e.g. [16], which ultimately introduce the time-variant forces into the vehicle body and generate structure-borne noise. This structure-borne noise is further transferred to the interior of the vehicle and the driver’s ear via the chassis components, as demonstrated by Bettella et al. [15]. They found that both airborne and structure-borne paths are present, with the structure path being typically more important. An experimental investigation on a test vehicle using ac- celerometers and microphones showed that the front brake callipers were the main source of creep groan noise.
The interaction between an axle system’s excitation and its response during creep groan is not unidirectional. Generally, me- chanical systems respond to excitations with movement - in the case of creep groan with dominant low-frequency contents. Vice-versa, this response movement affects the occurrence of the stick–slip excitation as well. The repetition frequency of stick–slip impulses is therefore related to the mechanical system’s eigenfrequencies, which can be seen in the typically measured acceleration spectra of the brake calliper’s top.
Simulations have been performed in order to increase the understanding of basic mechanisms and the influences of components and parameters such as friction laws or bushing elasticity behaviour. Coming from (multi-body) minimal models with two, three, four or five degrees of freedom (DOFs), even half-axle finite element models have been investigated for their creep groan behaviour as e.g.
within [2,8,11,18,19].
It turns out that in typical brake systems of passenger cars, creep groan appears basically in two different manifestations: As low- frequency groan from 15 to 25 Hz and as high-frequency groan from 60 to 100 Hz, [14,19,20]. The mechanism of the low-frequency manifestation is dominated by longitudinal displacements of the whole axle and contains highly impulsive contents, consequently it could be described as hard. It has distinct stick phases, superposed by high-frequency vibrations. This type of creep groan was also found to be favoured by decreasing instead of increasing speeds, [21]. High-frequency creep groan has been identified as “soft”
stick–slip and it may be expected in the range from 60 to 100 Hz with characteristics of harmonic-like vibrations. This type of creep groan occurs at rather low brake pressures which are usually more relevant for practical vehicle operating conditions.
In addition to these two basic manifestations, transition phenomena were found both by Meng et al. as well as by the authors of this work [21,22,23]. Here, basic low-frequency cycles were found to be superposed by multiple, high-frequency stick–slip transitions.
Similar behaviour was also found by Stender et al. during stick–slip experiments of strongly reduced models within [17]: Here, the term “driven stick–slip motion” was used.
1.3. Detection and quantification of creep groan
During vehicle tests, the most obvious approaches for detecting creep groan use sound pressure data or accelerometer data. Both approaches show advantages and disadvantages.
A detection based on measured sound pressure (microphone signal) may have problems with low-frequency contents and noisy signals due to other noise sources such as engine or drivetrain noise. Furthermore, a distinction between different brake NVH phe- nomena can be difficult; problems with transition groan as defined within [21] could confuse a threshold-based detection algorithm as well. Dobrynin et al. used audible sound generated by friction for the evaluation of friction. They concluded that sound is saturated with information and that it is hard to extract useable information about the stable friction process. However, transient phenomena can be easily observed from sound pressure results, [24]. Abdelhamid and Bray, [8], tried to identify creep groan from the audio signal by means of psychoacoustic features first. Tonality has been found to correlate well with the investigated creep groan noise. Loudness and dB(max) were found to be even better metrics for the creep groan quantification.
A detection based on accelerometer sensor signals, e.g. based on the calliper vibration, can be used for creep groan quantification in a variety of ways: the length of the creep groan event, the peak-to-peak range, the kurtosis or the root mean square (RMS) value could be used, [18]. Nevertheless, disturbances by engine and/or drive train noise and vibration can be present within accelerometers as well and need to be eliminated within such procedures, [9]. When using the kurtosis of the vibration signal as suggested in [25], one can utilize creep groan’s characteristic non-linear behaviour for its detection and quantification. This non-linear characteristic is utilized as well in frequency-based detection methods as explained within [14,18]. Multiple manifestations can be found by peak detection in frequency domain exhibiting bifurcations [26,27,28]. A map of different manifestations of the creep-groan phenomena was proposed by Fuadi et al., [29].
The majority of the experimental results of creep groan manifestations using vibrations are based on signal processing in frequency domain, focusing on the main creep groan frequency. Dobrynin et al. showed that that the time–frequency analysis of audible acoustic emission in a tribosystem can be applied, along with traditional approaches, to extract features of the wear and friction process, [24].
Rastegaev et al. presented a study on the classification of different types of tribological events using a R2 clustering algorithm in combination with 5 signal features derived from time domain (RMS, energy, variance, kurtosis and skewness) and 7 features derived
from frequency domain (median frequency, signal power, spectrum RMS, PSD entropy, PSD variance, PSD kurtosis and PSD skewness), [30].
Various research groups have already tried to harness learning methods for creep groan detection and classification. Stender et al.
[31] implemented deep learning for the detection of high-frequency brake squeal and related problems such as wirebrush sounds or click sounds. Region based convolutional neural networks (R-CNN) and region based fully convolutional neural networks (R-FCN) were considered for the detection and characterization. Supervised learning was performed based on spectrogram figures. Here, the best model (R-CNN) reached an average precision per class of 59% at an Intersection over Union (IoU) level of 90%. Furthermore, a digital twin for the investigation of a brake system’s NVH-behaviour was presented.
Pürscher et al. introduced supervised learning based on artificial neural networks (ANN) within [32]. Here, synthetically produced creep groan spectra were used to train a framework of two ANNs: One for creep groan detection, one for evaluation of its basic repetition frequency. White noise of different level was used to test the system’s performance on synthetic creep groan under dis- turbing signals. Furthermore, creep groan spectra measured during matrix tests on a half-axle test bench were classified by the trained network as well. Depending on the test matrix, 59% or 71% of test matrix values were correctly detected here.
Naturally, frequency analyses need to deal with a trade-off between frequency resolution and time resolution. If one of these parameters is increased, the other one will suffer. For a high-quality analysis and (learning-based) classification of creep groan signals, both high frequency resolution and high time resolution are desired: High frequency resolution classically for distinguishing different manifestations, high time resolution due to the fact that these manifestations can change fast. To analyse in time domain and identify the specific creep groan manifestation in time domain as well is therefore the central idea of the presented approach.
1.4. Scientific approach
The aspired goal of this study was to develop a novel method to identify and classify creep groan phenomena in real-time. A time- domain approach combined with unsupervised classification algorithms was taken. The study was conducted in five steps:
1) In the first step, measurements were performed according to the experimental design in Section 2.1. These measurements were optimized to ensure the occurrence of different manifestations of creep groan on the vehicle. Accelerometer data was favoured over sound pressure data due to the higher signal-to-noise ratio.
2) In the second step, different features were tested on sound and accelerometer signals: Crest factor, kurtosis, signal level, weighted signal levels dB(A), dB(B), dB(C) as well as the psychoacoustic features sharpness, roughness, fluctuation strength, tonality and many others. In total, more than 100 different features from 8 different sensors were analysed. As none of the investigated features could provide satisfactory results of creep groan manifestation classification, further steps were necessary.
3) In the third step, a new set of features for vibration signals was developed based on acoustic emission (AE), see [33] and Section 2.2.
This analysis of AE was related to the detection of stick–slip transitions in the friction contact. Therefore, an envelope signal for the stick–slip transitions’ bursts was developed: Enveloped Bursts (EB) signal.
4) The fourth step consisted of feeding unsupervised classification algorithms with relevant signal features derived from the EB signal, see Section 2.3. Two unsupervised classification algorithms were used: k-means algorithm and Kohonen’s Self Organizing Map (SOM). First findings within these steps can already be found within [34].
5) In the last step, the results of the unsupervised classification were analysed, see Section 3. Both unsupervised classification algo- rithms classified all acceleration data in six different classes, four of them related to creep groan. The robustness of the procedure was analysed. Different creep groan classes were found to be linked to the specific mechanic behaviour of different bifurcations.
2. Methodology and application 2.1. Creep groan experiments
Creep groan experiments were performed on a compact executive car with an automatic transmission. The test car was equipped with a double wishbone front axle setup with separate lower links and a floating calliper brake system with a ventilated grey iron brake disc. Two different sets of friction materials were used; one set of European (ECE) pads and one set of Non-Asbestos Organics (NAO)
Fig. 2. Placement of accelerometers on the front calliper carrier brackets.
pads. In this case, the ECE pads were known to be more prone to creep groan.
Accelerometers were mounted on the top of the brake calliper anchor brackets of the front wheels, as shown in Fig. 2. Furthermore, a microphone was placed inside the vehicle as well as within the wheel housing. Different driving parameters were selected with the aim to achieve different manifestations of creep groan. Three parameters have been included into the experimental plan: Direction of the driving (forward or backward), acceleration (from or to standstill), inclination of the road (15% or flat track). Each different setting was tested a minimum of five times with trained test drivers subjectively rating the creep groan behaviour from 1 (very poor) to 10 (excellent).
Signals from the 3D-accelerometers were recorded with a sampling frequency of 51200 Hz and 16-bit amplitude resolution. An analogous anti-aliasing filter with a passband frequency of 23.2 kHz, stopband frequency 32 kHz, passband ripple ±0.01 dB and a stopband attenuation of min. 80 dB was used. Although 3D-accelerometers have been used and signals from all orientations have been recorded, it was found out that the most prominent direction was perpendicular to the mounting plane as indicated in the Fig. 2. This relates mostly to the vertical calliper carrier acceleration. The sensor in use showed a sensitivity of 10 mV/g (±10%) and a nominal frequency range of 2 Hz–10 kHz (±5%). With its resonant frequency above 55 kHz, this piezoelectric acceleration sensor was not specifically designed for AE. However, as the rather stiff adhesive mount was chosen, one can assume to still receive plausible results near the upper frequency bound of 25.6 kHz.
A total of 558 measurements were performed. The duration of recordings differs from measurement to measurement due to different occurrences of creep groan at different times and its duration within each measurement.
2.2. Feature extraction: detection of stick-slip transitions from acoustic emission signals
This section explains how new signal features based on acoustic emission (AE) have been derived. All signal processing was performed using National Instruments LabVIEW 16.
2.2.1. Acoustic emission in friction processes
Acoustic emission (AE) is classically known as the emission of elastic, mechanical waves resulting from deformation and fracture of materials. AE is especially used in non-destructive testing of a mechanical part’s quality. As summarized by Baranov et al. in [35], AE- related phenomena occur during virtually all phenomena of solid and surface mechanics, including tribological processes. During such a tribological process, e.g. a brake pad sliding on a brake disc, high-frequency vibration bursts (AE) are caused by different effects.
These effects comprise elastic impacts, plastic deformation and the appearance of wear debris, changes in the local stress–strain states amongst other things. These effects are superposed from a typically high number of rather small, randomly occurring contact patches, which poses serious challenges for researchers and practitioners when extracting useful information from the upcoming acoustic emission, [30,36].
The correlation between acoustic emission and vibrations has been investigated in different studies, preliminary intended for machine health monitoring, [37,38,39]. Various friction and wear experiments have been performed to clarify their relationships. It was found that acoustic emission generated by sliding friction (slip event) can be detected in the frequency range from 2 kHz up to 100 kHz, [36]. The characteristics of the acoustic emission signals at the onset of a slip are insufficiently clarified. Taura and Nakayama [40] conducted friction tests and simultaneously measured the AE signal. The AE signal clearly increased during the slip and the effective value of the AE signal directly related to the sliding speed. From the obtained data, they modelled the correlation between AE energy and the friction work during the slip. Both values shared a strong proportional relationship, [40].
Ferrer et al. published a research work where the acoustic emission technique was used to record and study the elastic waves that appear during the transition from static to kinetic friction in a stick–slip experiment, carried out by using a sheet of soft steel and a clamp of quenched steel. They observed that the onset of the slip event is accompanied by an increase in the continuous AE activity and, immediately afterwards, a long train of AE waves. They identified the main characteristics of these trains to have a typical sequence: This sequence is defined by a first wave with relatively low amplitude and three or four following waves of very high amplitude. After these, the train of waves shows decaying-amplitude behaviour, [41]. Results provided by Ferrer et al. [41] and Stoica et al. [42] indicate that a single stick–slip effect is accompanied with a train of AE bursts, each shorter than 30 μs. However, the full burst train contains many AE bursts and lasts much longer than 30 μs. The burst train always begins with a low-amplitude wave.
According to results provided by Ferrer et al. and Stoica et al., the overall duration of the stick–slip effect reverberates in the form of AE bursts up to 10 ms. This corresponds to a frequency range around 100 Hz, similar to creep groan’s 1st order frequencies.
2.2.2. Signal processing approach: enveloped bursts signal
The aim of the signal processing was to harness AE-related acceleration contents for detection of stick–slip transitions and the
Fig. 3. Processing of an accelerometer’s vibration signal to an enveloped bursts (EB) signal.
extraction of features, which is similarly explained in [33]. Therefore, we created an enveloped burst (EB) signal: By transforming the high-frequency contents accompanying each stick–slip event, which can be considered as a train of AE bursts, we create a signal of individual, easily detectable pulses. Two non-overlapping high-pass and low-pass filters with cut-off frequencies fHP and fLP were implemented as depicted in Fig. 3. Coefficient of amplification a was set in relation to fHP and fLP. Eq.1 shows the applied approxi- mation of a, which was determined numerically based on the assumption that fHP >fLP. The amplification factor a is not significant (after the selection of fHP and fLP) because all features, that are extracted from the AE, are normalised before being fed into the un- supervised learning algorithms. The amplification factor is only important during robustness analysis to keep the AE signal approx- imately at the same level, which is required for robust peak detection of different combinations of fHP and fLP values.
a= (fHP
fLP
) 10
(
35000fHP+1000fLP
)
(1) The high-pass filter cut-off frequency fHP is chosen in order to filter out low-frequency vibrations not correlated to the stick slip occurrence. However, fHP should not be set too high to preserve sufficient signal to noise ratio. The low-pass filter is used to reshape the squared signal into clear peaks, representing an envelope of the squared signal. Effects from choosing different combinations of cut-off frequencies can be seen in Fig. 4 (left).
As one can conclude from Fig. 4, the proposed transformation of the original signal into a baseline signal with superimposed peaks is useful, because it enables easier detection and counting of pulses. Notice that the pulses’ maxima are still located approximately where stick–slip phenomena occur. This EB signal also provides relative information about the released energy of individual stick–slip transitions by measuring the signal’s height and surface area below the peak, see Fig. 4 (right). This way, one can accurately determine the time between two successive stick–slip occurrences and their amplitude, providing the exact frequency of creep groan based on the peak to peak duration within the observation interval. The observation interval was chosen with 250 ms. The features in Table 1 can be extracted from the EB signal for each observation interval.
The extraction of EB pulses from vibration signals depends on signal filtering, amplification and integration time. For high-pass and low-pass filters, Butterworth filters were selected: 4th order for the high-pass filter, 1st order for the low-pass filter (to prevent the creation of ghost peaks). Three parameters of data processing have been left to optimize: high-pass filter cut-off frequency, amplifi- cation and low-pass filter cut-off frequency. High-pass filter cut-off frequency has a strong influence on the peak values. The low-pass filter provides smoothing of the peaks’ shape, which is convenient for the peak detection algorithm. However, depending on the chosen low-pass cut-off frequency it might blur the boundary between two close neighbouring peaks if its value is set to low. Higher values of low-pass filter cut-off frequency provide sharp peaks which closely follow the stick–slip event, as shown in Fig. 4 (left). This kind of digital signal processing shows little need of processing power and is rather simple to implement.
2.2.3. Quality and robustness of feature extraction
The selection of high-pass and low-pass filter cut-off frequencies affects the shape of the resulting EB signal and therefore also the number of detected EB peaks. In order to estimate the robustness of the proposed signal processing procedure, effects of the chosen high-pass and low-pass frequencies were analysed. For a randomly selected experimental run, two relevant indicators of robustness are shown in Fig. 5: The averaged indicated width of peaks and the number of detected peaks.
The averaged indicated width of each peak in Fig. 5 (left) is calculated by dividing each peak’s surface area by the peak’s amplitude.
It describes the shape of each peak: Higher indicated width relates to a less pronounced, less pointy peak. As one can observe, the averaged indicated width of the peaks depends primarily on the choice of the low-pass filter frequency. As the low-pass filter frequency is increased, the indicated width of the peak decreases. Although the indicated width of the peaks changes significantly from 26 samples to 195 samples, it does not significantly affect the number of peaks detected.
Fig. 4.Transformation of a creep groan related vibration signal into a stick–slip related, enveloped bursts (EB) signal for further feature extraction.
Fig. 5 (right) shows the combined effect of high-pass and low-pass frequencies on the number of detected peaks. In the whole parameter range, the number of detected peaks varies approx. ±1%. Also, the numbers vary less for high-pass filter frequencies above 5 kHz. This result is consistent with the spectrum of the vibration signal in Fig. 1 (c), where most of the energy during creep groan is in the frequency range below 5 kHz, relating more to the macroscopic displacements of the calliper and less to AE. These frequency contents below 5 kHz have proven to show less value for the detection of AE related bursts.
To confirm the robustness of feature extraction for all the experimental data, 9 filter parameter sets of high-pass and low-pass frequencies were selected and labelled with numbers from j =1, 2, … M =9 in Fig. 5. To stress the robustness of the method, the lower value of the high-pass frequency was chosen to be 3 kHz. The next value chosen is 10 kHz, which corresponds to the upper frequency limit of the accelerometer’s typical working range. The highest value chosen was 16.5 kHz, which is still well below the accelerometer’s eigenfrequency, but outside of its intended use for reliable measurements. The first low-pass frequency of 250 Hz was chosen to be just above the limit that would provide peaks of sufficient prominence. A further two frequencies of 550 Hz and 850 Hz were selected for the low-pass frequency to stress the robustness.
Robustness of feature extraction was calculated for five features i =1, 2, … N =5 which do not depend on the amplitude of the measured signal. According to Table 1 these were: 1) Number of detected EB peaks, 4) Relative standard deviation of detected peak amplitudes 5) Maximum detected frequency of EB peaks, 6) Average frequency of EB peaks, 7) Relative standard deviation of EB peak frequency. All events were taken into calculation, n =1,2, … P =25030.
For each event, five feature vectors were extracted and formed a 5 by 9 matrix Rij of extracted feature values. Deviation Dn,i was calculated for each event and each feature according to Eq.2.
Table 1
Extracted features from EB signals, for each observation interval of 250 ms.
# EB signal feature
1 Number of detected EB peaks
2 Maximum amplitude of the highest detected EB peak
3 Average amplitude of all detected EB peaks
4 Relative standard deviation of detected peak amplitudes
5 Maximum frequency of EB peaks
(defined by the shortest duration between two detected EB peaks)
6 Average frequency of EB peaks
(defined by the average duration between detected EB peaks) 7 Relative standard deviation of EB peak frequency
(defined by the durations between detected EB peaks)
8 Level of the original accelerometer signal
9 Level of the HP-filtered signal (3 kHz, 10 kHz and 16.5 kHz)
Fig. 5.Influence of the high-pass and low-pass filter cut-off frequencies on average indicated width and number of detected peaks of the accel- erometer’s EB signal.
Table 2
Average deviations from the mean value over all 9 filter parameter sets of the extracted features from EB signals (Table 1), calculated according to Eq.
(2). Low deviations indicate a high robustness toward the chosen high-pass and low-pass cut-off frequency.
# EB signal feature Average deviation from the mean feature value over all measurements
1 Number of detected EB peaks 0.184
4 Relative standard deviation of detected peak amplitudes 0.365
5 Maximum detected frequency of EB peaks 0.279
6 Average frequency of EB peaks 0.197
7 Relative standard deviation of EB peak frequency 0.524
Dn,i=MargmaxRij− argminRij
∑M
j=1Rij
(2) Finally, the deviation of each feature was averaged over all events. Values of averaged deviations for each feature are given in Table 2. Lower values indicate less deviation and hence higher robustness of feature extraction toward the selection of high-pass and low-pass cut-off frequencies. The number of detected EB peaks varies from 81,731 to 92,866, with the highest number of peaks detected when the high-pass frequency is set to 3 kHz and the low-pass frequency is set to 250 Hz. The highest average frequency of peaks is 18.75 Hz and the lowest average frequency of the peaks is 16.43 Hz. As one can see, the selection of high-pass and low-pass frequencies affects the feature extraction substantially. However, its effect on classification will later be found minor.
2.3. Unsupervised classification
Unsupervised learning, in contrast to supervised learning, is not exposed to subjective, human, perception, because no data classified by humans is used. This advantage enables an independent, unbiased classification of data, [43]. Therefore, instead of su- pervised learning methods, such as k-NN and feed forward neural networks, two algorithms for unsupervised classification were selected for this study: k-means and Kohonen’s Self Organizing Maps (SOM). For both algorithms, classification is based on clustering.
Corresponding subjective attributes can only be defined after this process.
2.3.1. K-means algorithm
K-means is one of the basic unsupervised algorithms and can be thought of as a self-learning k-NN algorithm. The classification is random and depends on the initial conditions of the centroids. Vectors of measurement data, which are “near” to each other in terms of a low Euclidean distance in the input space, are classified into the same class.
In more detail, the k-means algorithm is an iterative algorithm that attempts to classify the data set into K different, non- overlapping clusters, where each data point can only belong to one group. Data points are D-dimensional vectors, with each component representing a feature extracted from observations. The k-means algorithm attempts to arrange N observations in K clusters so that the data points within the clusters are adjacent, while keeping the clusters as far away from each other as possible. The evaluation is based on the Euclidean distance between the points. The distance between a centroid with the index cm and the observed point xn is defined by an equation:
|cmxn|2=
∑D
i=1
c2m,i− x2n,i (3)
K-means algorithm assigns each data point to the nearest cluster defined with its centroid. The algorithm is based on minimizing the arithmetic mean of all data points that belong to the same cluster. The smaller the variation within clusters, the more homogeneous the data points are within the same cluster. K-means clustering therefore aims to classify the N observations into K (≤N) clusters defined with the centroids C ={c1, c2, …, ck} to minimize the sum of the squares within the cluster. Formally, the objective is to find:
argmin
⏟̅̅̅⏞⏞̅̅̅⏟
C
∑K
i=1
∑
x∈ci
‖xi− εi‖2=argmin⏟̅̅̅⏞⏞̅̅̅⏟
C
∑k
i=1
|ci|Var(ci) (4)
where εi is the average of points attributed to ci. This is equivalent to minimizing the pairwise squared deviations of points in the same cluster:
argmin
⏟̅̅̅⏞⏞̅̅̅⏟
C
∑K
i=1
1
|2ci|
∑k
x,y∈ci
‖x− y‖2 (5)
The equivalence can be deduced from identity:
∑k
x∈ci
‖x− ε‖2= ∑
x∕=y∕∈ci
(x− εi)(εi− y) (6)
Because the total variance is constant, this corresponds to maximizing the sum of the squared deviations between points in different clusters, which follows from the law of total variance. The k-means algorithm is usually initiated with a randomly filled matrix of centroids C. Euclidean distances are then calculated for each pair of data points to each centroid representing a class. Each data point is then assigned to a class, whose distance to the centroid is minimal. The new centroid is calculated from the new distribution of data points for each class. The new centroid of a given class is an averaged value of all data points assigned to this class. The Euclidean distances are calculated again and processed continuously until there are no more transitions of data points between the classes.
2.3.2. Kohonen’s self organizing map – SOM
A self-organizing map (SOM) or Kohonen map [44] is a type of artificial neural network (ANN) that is trained using unsupervised learning to produce a low-dimensional, discretized representation of the input space. It differs from other artificial neural networks in a way that it applies competitive learning as opposed to error-correcting learning (such as backpropagation with gradient descent). SOM
algorithm uses a neighbourhood function to preserve the topological properties of the input space. The SOM normally consists of a two- dimensional grid containing a certain number of neurons. These neurons are usually arranged rectangular or hexagonally. The position of the units in the grid, especially the distances between them and the neighbourhood relations, are very important for the learning algorithm. Self-organizing map has been implemented in numerous fault diagnosis studies, [45,46,47,48], and has proven to be a useful tool for classification when using vibration and sound signals. Kane and Andhare have shown that extracted psychoacoustic and statistical features are able to classify different gear faults, [49].
On an intuitive level, both k-means and SOM move the neurons to denser areas of the input space. With k-means, the nodes move freely and without direct relationship to each other. In SOM, as one neuron moves towards the data, it drags neighbouring neurons on the 2D grid with it. This of course maintains a topology embedded in the data space, and the clusters are formed geometrically. Unlike k-means, a neuron in SOM can easily be responsible for zero data points. This indicates that such neurons are located in empty space and are pulled in all directions by their neighbours. It has been shown that self-organizing maps with a small number of neurons behave similar to k-means, while larger self-organizing maps rearrange data in a way that is of a fundamentally topological character. The SOM algorithm shows better results when using small data sets. It is also less susceptible to noisy input space, [43,50]. The con- ventional SOM algorithm consists from the following steps:
Step 1: Initialize the weight vectors ci′s on the grid of a £b neurons with arbitrary values.
Step 2: Randomly select an input vector xn and feed all the neurons at the same time (parallel).
Step 3: Find the winning neuron, i.e. BMU (best matching unit), using the same equation as with k-means algorithm, i.e. the Euclidean distance measure.
Step 4: The weight vector of the neurons is updated using the following equation:
cBMU(n+1) =ci(n) +hc,i(n)[xn− ci(n)] (7)
where hc,i(n) is a Gaussian neighbourhood function:
hc,i(n) =α(n)⋅exp (
‖rBMU− ri‖2 2σ2(n)
)
(8)
where r is the coordinate position of the neuron on the map, α(n) is the learning rate and σ(n) is the width of neighbourhood radius.
Both α(n) and σ(n) decrease monotonically using the following equations:
α(n) =α(0)⋅ (α(N)
α(0) )n/N
(9)
σ(n) =σ(0)⋅ (σ(N)
σ(0) )n/N
(10) For all the input data N, steps 1–4 are repeated.
2.3.3. Chosen number of classes and application of unsupervised classification
Both algorithms were fed with the extracted features from the EB signal. The central question for application is finding a suitable number of classes.
k-means application. All of the classification was performed using 6 classes. This value was developed empirically, also based on a priori information: Starting point was the subjective rating by the trained test drivers. Although values from 1 (very poor) to 10 (excellent) could have been assigned, most tests were rated from 5 to 10 (meaning 6 integer values).
Fig. 6.Reducing data dimensionality with SOM: The neuron lattice and the neighbourhood function aim to preserve the topology embedded in the data (input) space.
In the following, a statistical approach was taken to verify this assumption of 6 classes. The probability of the k-means optimi- zation’s convergence, in a sense that no empty classes result, was evaluated while varying the number of classes. Randomized initial centroids were used. It was found that, by increasing the number of classes, the probability of the algorithm to converge is reduced, as it was already shown in [33]. The number of classes is almost perfectly correlated to the logarithm of the convergence probability for a higher number of classes: The logarithm represents the statistical properties of averaging while performing k-means classification.
When the number of initial classes is reduced, the probability of convergence stops following the interpolated curve. In our case, this occurred at 6 classes, leading towards the assumption that the presented data set can be classified well into 6 classes. As the results will show, one of the classes is reserved for standstill, one for braking without creep groan whereas the other 4 classes are reserved for different manifestations of creep groan.
SOM application. Again, 6 classes were chosen for the application of SOM, even though SOM could handle empty classes. This was done to allow a direct comparison with k-means algorithm. Certainly, performance could be reduced by choosing a low of number classes, which is discussed further in the results section.
Effects from introducing SOM’s neighbourhood function were to be investigated: In the presented case, a 1-dimensional grid was used, Fig. 6. This was chosen because the output layer (subjective quantification) is quite straightforward and ranges linearly from 1 (very poor) to 10 (excellent). The topological layer (input layer) of SOM correlates to the order of the neurons, which must be selected manually when using the k-means algorithm. The neurons on each side of the 1-dimensional grid automatically land on the opposite sides of the 9-dimensional input space due to the neighbourhood function. This topology preservation directly correlates to the linear evaluation of subjective quantification.
K-means algorithm has been prepared in LabVIEW, whereas Kohonen’s Self Organizing Map was performed in Mathworks Matlab.
Both algorithms were fed with the selected EB signal features given in Table 1, calculated from all the 558 measurements. Mea- surement data was split in windows of 250 ms, leading to a total of 25,030 feature vectors (9x1 each). For an analysis of robustness, these 25,030 feature vectors were calculated for nine different filter parameter sets according to Fig. 5. Hence, 225,270 different feature vectors were analysed both with k-means and SOM.
3. Results
Two kinds of results are presented here. First, the robustness of the proposed method is shown by analysing influences from 9 different filter parameter sets on the classification results. Second, classification results for one of these filter parameter sets are studied in detail, enabling a detailed comparison between the two algorithms k-means and SOM.
3.1. Overall classification results and robustness
Fig. 7 shows the mean values and standard deviations of each individual class’ share over all 9 different filter parameter sets, with blue dots for k-means algorithm and red squares for SOM algorithm.
Generally, both algorithms show comparable shares of the different classes in Fig. 7. The majority of inputs, around 53%, were classified to Class 0. The occurrence of Class 3 was the lowest, while also being the one where the difference between classified inputs of both algorithms was the largest. Class 5 was represented equally, deviating only by a few inputs. Shares of classes 4 and 5 were comparable to classes 1 and 2. Overall it can be observed that k-means algorithm reached smaller standard deviations of the different classes’ shares across all 9 filter combinations, indicating a higher robustness and consistency. Class 4 was the only one where SOM classification reached a similarly low standard deviation. For both algorithms, Class 2 tends to be the most sensitive regarding the filter parameter set, as indicated by the high standard deviation.
A more detailed insight in the characteristics of the different classes is given in Fig. 8. Here, influences from the 9 different filter parameter sets on the resulting k-means centroids and SOM neuron weights for each class are shown. In Fig. 8a-c, k-means centroids
Fig 7.Mean values and standard deviations of share of samples classified into individual classes for k-means and SOM classification.
and their standard deviations are drawn. Fig. 8d-f displays the same evaluation for SOM neuron weights. All 9 EB signal features are shown.
A quick view reveals that these 9 EB signal features were grouped for similar behaviour of the mean values: The left subfigures 8a and 8d show features relating to the level of the signal, which increase in a monotonous, convex curve with the class. The middle subfigures 8b and 8e show frequency-related features, which increase monotonously as well but in a concave curve. The right sub- figures 8c and 8f show features relating to irregularities in amplitudes and frequency as well as features relating to the amplitude of the EB peaks. The irregularities measured by the RSD of frequencies and amplitudes increase up to Class 4 until they drop again with Class 5. EB peak amplitude related features show increased values for Class 3, 4 and 5. These characteristic groups could be important for further practical application, where detecting and classifying creep groan in real-time shall be based on a minimal number of features.
Furthermore, Fig. 8 explains the different classes’ characteristics. Whereas Class 0 and Class 1 represent no creep groan due to the lack of EB peaks in both algorithms’ results, one can see that the other 4 relate to different manifestations of creep groan. Two irregular and two regular phenomena can be distinguished: Class 3 and especially Class 4 show increased standard deviations (RSTD) of EB peak frequency, indicating (rather irregular) transition creep groan. With Class 2 (low-frequency of EB peaks) and Class 5 (high-frequency of EB peaks), two classes representing the well-known regular creep groan phenomena were found. Here, the relative standard deviation (RSTD) of EB was found to be rather low or almost zero.
To assess the robustness towards different filter parameter sets in signal processing, the standard deviations of the centroids/neuron weights in Fig. 8 can be analysed. As one can see, strong differences in standard deviations depending on the class and the feature chosen can be seen. Fig. 8a and 8d show, that both centroids/neuron weights, which describe the level of the vibration signal, deviate significantly from group to group. However, “no creep groan” Classes 0 and 1 can be distinguished from the classes related to creep groan in almost any case.
Fig. 8b and 8e show that the centroids/neuron weights for the number of detected EB peaks is exceptionally robust. This centroid/
neuron weight is highly important because it defines the classification according to the occurrence of peaks per repeating cycle, which consequently describes the modes of creep groan as shown later in Fig. 10. Centroids/neuron weights for the average frequency and for the maximum detected frequency are less robust, especially for distinguishing between Class 4 and Class 5.
Fig. 8c and 8f show rather low standard deviations of the centroids/neuron weights and therefore indicate high robustness. Only exception is the relative standard deviation of EB peak frequency for the SOM algorithm: In contrast to the upper k-means results, a very high range of these values was found for different filter parameter sets.
A tabular overview of the robustness analysis for the classification is given in Table 3. On the left side, deviations from the median Fig. 8.Classification robustness for different 9 filter parameter sets: Means and standard deviations of k-means centroids and SOM neuron weights drawn over the resulting class.
Table 3
Robustness of classification and detection of creep groan: deviations from the median of all classifications and deviations in the classification of “creep groan” vs. “no creep groan”.
Deviation of classification from class median [%] Deviations in the classification of “creep groan” versus “no creep groan”
[%]
Algorithm: k-means SOM k-means SOM k-means SOM k-means SOM k-means SOM k-means SOM
LP filter↓, HP filter → 3000 Hz 10,000 Hz 16,500 Hz 3000 Hz 10,000 Hz 16,500 Hz
850 Hz 11.10 13.27 1.33 3.05 1.59 3.39 2.04 7.53 0.98 0.12 1.25 0.37
550 Hz 10.64 11.48 1.05 2.73 2.96 4.30 1.87 2.00 0.88 1.84 1.00 6.42
250 Hz 9.23 13.91 3.66 4.41 6.77 8.34 1.93 4.24 0.50 3.18 0.56 0.99
of each resulting class are given for all 9 filter parameter sets. On the right side, deviations in the detection of creep groan (Class 2–5) vs. no creep groan (Class 0–1) are shown, based on the class median as well.
As a result, the selection of the high-pass and low-pass filter cut-off frequencies affects not only the EB signal feature extraction but also the classification of creep groan. Especially the selection of the high-pass filter cut-off frequency is important, as indicated by the high deviations for a high-pass filter cut-off frequency of 3 kHz. Here, elevated vibration amplitudes up to 5 kHz, such as during slow braking without creep groan in Fig. 1a, are not cut-off by the filter. These vibration contents can cause the detection of additional, spurious peaks, see also Fig. 5 (right) on this issue. Certainly, Table 3 shows only deviations from the median of all the classifications:
No comparisons to “true” values were taken.
For the detection of “creep groan” vs. “no creep groan” only, the procedure is very robust for k-means algorithm, as all of the filter combinations resulted in less than 2.04 % misclassification. SOM also provided very good results, except for two outliers, where over 5
% of creep groan occurrences were falsely classified. Overall, k-means algorithm proved to be more robust to the selection of high-pass and low-pass filter frequencies.
The results above indicate acceptable robustness of the presented approach for classification of different creep groan manifesta- tions. For further in-depth analyses of the performance of both algorithms in Section 3.2, filter parameter set 5 (Fig. 5) with a low-pass filter cut-off frequency of 550 Hz and a high-pass filter cut-off frequency of 10 kHz was chosen. Reason was its relative proximity to the median over all classifications results in Table 3. Also, the parameters of filter parameter set 5 are in the middle of both ranges and roughly represent the mean value of both cut-off frequencies.
3.2. Exemplary results for filter parameter set 5
Table 4 provides a more detailed comparison between the two classification algorithms. All in all, k-means and SOM have classified 96.14% of inputs into the same class for this filter parameter set. The biggest deviation can be seen for those that were classified into Class 3 by k-means, but were put in Class 2 by SOM. These inputs account for more than half of all misclassifications. For Class 0 (no creep groan), the cumulative error was only 0.19% or 49 out of 25,030 input vectors. We can also observe that the further the classes are apart, the smaller the error was. Almost no misclassifications spanned over more than one class, with only two exceptions. Largest number of misclassifications was 2.02% and happened between k-means Class 3 and SOM Class 2. There were no inputs simultaneously classified as no creep groan and severe (Class 5) creep groan by either k-means or SOM. We can conclude that both algorithms showed exceptional results and almost maximum classification correlation.
To get a better perception of the distribution of input vectors and the centroids/neurons in the 9-dimensional input space, we have calculated the Euclidean distances between the input vectors and all the centroids (k-means) and neurons (SOM). Fig. 9 depicts the mean values and corresponding standard deviations of Euclidean distance of input vectors for each class (coloured entries). Also, the Euclidean distances between individual centroids/neurons is drawn: Circles on the left of each class refer to k-means results, squares on the right of each class refer to SOM results.
First, the mean values and standard deviations of the input vectors in Fig. 9 shall be discussed (coloured entries). If we observe Class 0, we can see that the majority of k-means input vectors (blue circle) is in the vicinity of the centroid (light grey circle) with minimum spread. The same can be said for SOM (orange square vs. light grey square), although the mean Euclidean distance of classified inputs is larger. The more we move towards Class 4, the larger the distances of the input vectors (coloured values) to centroids/neurons be- comes, together with an increasing spread of the classified inputs. For both algorithms in Class 5, the Euclidean distances between the input vectors and the Class 5 centroid/neuron as well as the standard deviations drop.
Differences between k-means and SOM regarding this Euclidean distance from input space to centroid/neuron exist especially for Class 0 and Class 1. The reason for these differences lies in the essence of both classification algorithms: In its learning phase, k-means moves its centroids independently from each other while SOM’s lattice in the output space affects all the neurons simultaneously. The topology preservation is in full effect for the neurons on the edge of the lattice, in our example: 1 and 6, corresponding to Class 0 and 5.
Most of the feature values in Class 0 are zero, which means the input vectors are put near the absolute edge of the 9-dimensional input space. While centroids can reach this regions, SOM’s neurons have difficulties as they are being held back and being pulled to the centre due to the neighbourhood function of the lattice.
Second, the Euclidean distances of centroids/neurons between different classes in Fig. 9 shall be discussed (grey entries). High Euclidean distances are preferable, as these would indicate a clear differentiation between different classes in the 9-dimensional parameter space. Although k-means achieved greater overall Euclidean distances between individual centroids and smaller spread Table 4
Classification results and correlation between k-means and SOM for filter parameter set 5.
Classification correlation Kohonen’s SOM classification percentage per class, [%]
Class 0 Class 1 Class 2 Class 3 Class 4 Class 5
K-means classification percentage per class, [%] Class 0 52.57 0.01 0 0 0 0
Class 1 0.10 12.81 0 0 0 0
Class 2 0.08 1.27 6.28 0.07 0 0
Class 3 0 0.02 2.02 2.63 0.22 0
Class 4 0 0 0 0 7.91 0.04
Class 5 0 0 0 0 0.03 13.94
Fig. 9. Euclidean distances of inputs, centroids and neurons in the 9-dimensional space, derived from features based on filter parameter set 5.
0.00 0.04 0.08 0.12 0.16 0.20
time in s -150
-100 -50 0 50 100
acceleration in m/s2 Class 0
0.00 0.04 0.08 0.12 0.16 0.20
time in s -150
-100 -50 0 50 100
acceleration in m/s2 Class 1
0.00 0.04 0.08 0.12 0.16 0.20
time in s -150
-100 -50 0 50 100
acceleration in m/s2 Class 2
0.00 0.04 0.08 0.12 0.16 0.20
time in s -150
-100 -50 0 50 100
acceleration in m/s2 Class 3
0.00 0.04 0.08 0.12 0.16 0.20
time in s -150
-100 -50 0 50 100
acceleration in m/s2 Class 4
0.00 0.04 0.08 0.12 0.16 0.20
time in s -150
-100 -50 0 50 100
acceleration in m/s2 Class 5
Fig. 10.Typical waveforms of accelerometer signals for each class defined by unsupervised algorithms k-means and SOM: Different manifestations of stick–slip phenomena.
of inputs within each class, that was not the case for classes 3 and 4. Here, SOM achieved better results and more evenly distributed neurons. The distance of the closest neuron (neuron 2 for Class 3 and neuron 5 for Class 4) was larger compared to k-means centroids, which directly corresponds to a more robust Class 3 and 4 classification.
In some cases, the spread of inputs within one class was big enough that some of the vectors were placed in the vicinity of adjacent centroids and neurons: e.g. SOM’s inputs in Class 0 to neuron 1 (and vice-versa). The same can be said for Class 2 and neuron 1, while with k-means this only occurred in Class 3 with centroid 2. All in all, the distances of classified inputs seem to be far enough away from other centroids and neurons, which again corresponds well with the results in Table 4. It can be observed that the misclassified inputs are only one centroid or neuron away.
If we observe distances between individual centroids/neurons from a wider perspective, the conclusion of a strong “linearity” of the input space can be made. Centroids and neurons of classes 0 and 5 are always the furthest apart, no matter which class we examine.
Also, the ordering is maintained throughout all centroids and neurons. This again proves the robustness of the whole procedure: not only the ability to clearly differentiate “no creep groan” from severe (Class 5) creep groan, but also to detect its different manifestations.
Although both algorithms performed the classification exceptionally well, it was shown that k-means was overally better, while SOM’s advantage can be assigned to classes 3 and 4. Theoretically, SOM’s performance could have been reduced by choosing the same number of classes as for k-means because of SOM’s ability to converge to solutions with empty classes as well. Although the adding of a few classes might help with allowing greater distances between filled classes (Fig. 9), it is plausible that the lesser performance of SOM compared to the k-Means algorithm is due to the fact that a link is kept between non-stick–slip and stick–slip inputs by SOM’s neighbourhood function.
-0.1 -0.05 0 0.05 0.1
displacement in mm -0.04
-0.02 0 0.02 0.04
velocity in m/s
Class 0
-0.1 -0.05 0 0.05 0.1
displacement in mm -0.04
-0.02 0 0.02 0.04
velocity in m/s
Class 1
-0.1 -0.05 0 0.05 0.1
displacement in mm -0.04
-0.02 0 0.02 0.04
velocity in m/s
Class 2
-0.1 -0.05 0 0.05 0.1
displacement in mm -0.04
-0.02 0 0.02 0.04
velocity in m/s
Class 3
-0.1 -0.05 0 0.05 0.1
displacement in mm -0.04
-0.02 0 0.02 0.04
velocity in m/s
Class 4
-0.1 -0.05 0 0.05 0.1
displacement in mm -0.04
-0.02 0 0.02 0.04
velocity in m/s
Class 5
Fig. 11.Associated phase plots of one vibration cycle for each class defined by unsupervised algorithms k-means and SOM: Different manifestations of stick–slip phenomena.
4. Discussion
Based on the classification results of Section 3.1, typical waveforms of each class’ vibration signals were identified and are shown in Fig. 10. In the given order, the different signal snippets reveal the fundamental stick–slip interaction during different manifestations of creep groan.
Class 0 shows near-zero amplitudes of harmonic oscillation and no stick–slip interactions, probably related to a standstill of the vehicle. Class 1 shows a composition of harmonic oscillations, which certainly leads to higher signal levels but still lacks the presence of stick–slip transitions. Class 2 shows low-frequency groan with one distinct stick–slip transition per repetition cycle – therefore the low frequency with high regularity was found. Class 3 shows already some kind of transition groan: One can again observe a low-frequency repetition of the basic pattern. However, a second, slightly less intensive stick–slip transition occurs shortly after every first stick–slip transition. Class 4 shows again transition groan but now with groups of three stick–slip transitions. Class 5 finally shows the fully developed high-frequency groan. Please note that the time delay between two or three stick–slip transitions in Class 3 and 4 correspond to the basic repetition frequency of Class 5 high-frequency creep groan.
Fig. 11 shows phase plots of velocity over displacement for the signals of Fig. 10. Therefore, the original acceleration signals were high-pass filtered (forward/backward filtering with Chebyshev Type II, passband frequ. 18 Hz), integrated twice and detrended.
Afterwards, one full cycle of each phenomenon was chosen and plotted as a black-to-orange scatter plot.
Phase plots of Class 0 and Class 1 show rather low amplitudes, with a slightly “shakier” signature of Class 1 due to more high- frequency contents. Both of these phase plots show no signs of stick–slip effects and are therefore not related to groan but to stand- still and normal braking action.
Class 2 in Fig. 11 starts with the beginning of the slip phase (high acceleration and “separated” black dots), leading to a first significant circle. At some point afterwards, most likely when the maximum velocity is reached, the brake pads reattach to the brake disc. Part of a low-frequency oscillation (similar to Class 0) concludes the cycle until the next stick–slip transition occurs.
Class 3 in Fig. 11 starts with a stick-line, leading again to a rapid acceleration (“separated” black dots) of the first stick–slip transition. After a presumably short reattachment when reaching high speed, “separated” brown dots at maximum speed indicate the second stick–slip transition. A kink in the curve again shows the next reattachment, followed by the last, rather harmonic part of the vibration.
Class 4 in Fig. 11 shows similar behaviour but with three stick–slip transitions, indicated by three zones of “separated” dots. Stick lines can be found rather clearly at positive speeds, directly before the stick–slip transitions. The third of these ellipsoidal structures shows clearly lower amplitudes.
Class 5 in Fig. 11 shows high-frequency creep groan with only one stick–slip transition. Again, the stick line is found at positive maximum velocities.
Classification results from Fig. 10 and Fig. 11 give new insight in the bifurcation behaviour of creep groan phenomena. In all creep groan classes, the stick–slip events have the same basic low-frequency period (approx. 23 Hz) with superimposed events of roughly three times higher basic period (approx. 76 Hz). It is possible that all groan events, transition or not, have the same root cause in the form of an interaction between a longitudinal, low-frequency axle mode and a high-frequency torsion mode of wheel carrier, brake calliper and pads. During the transition from pure low- to high-frequency creep groan (Class 2 to Class 5), additional stick–slip transitions close the gap of the long pause during damped oscillation: two or three stick–slip transitions with Class 3 and Class 4. Due to the frequency relations between low- and high-frequency groan, transition groan with four stick–slip transitions cannot occur in the evaluated setup. Instead, fully developed high-frequency groan can be found. This hypothesis, in accordance with the vibration patterns shown in [20], will be addressed in future work.
Also, the average dissipation per repetition cycle could be of interest. Depending on operational parameters such as brake pressure, friction and vehicle speed, different amounts of energy need to be dissipated during one slip phase of a stable limit cycle. Signal levels, relating to these three parameters, were found to increase monotonously from class to class (Fig. 8a and 8d), indicating the link be- tween signal level (or, consequently, harshness), bifurcation and dissipation. Again, this could prove to be a valuable approach for further investigations.
For tests on a full vehicle system, such a detailed classification of creep groan manifestations was neither discovered nor expected by subjective evaluations yet. Unsupervised classification proved to be useful, logical, and confirms findings on bifurcations of fre- quencies. Based on the analysis of the results, we can conclude that three features of the vibration signal based on the EB signals are significantly relevant for the detection of creep groan and its different manifestations in real-time:
1. Level of the original accelerometer signal, describing the harshness of the creep groan
2. Number of EB peaks (within selected time intervals), defining the fundamental frequency of the creep groan 3. Relative standard deviation of EB peak frequency, describing the regularity of the creep groan.
What is more, these classification results can be used to reduce the necessity for laborious and time-consuming subjective eval- uations of creep groan. Based on the presented classification using k-means and SOM algorithm, the correlation with the subjective test was analysed, results are shown in Fig. 12. With a correlation R2 =0.77, the method indicates to be effective and valuable. On top of that, possible human errors during subjective rating can be reduced.
