The Dynamic Defect Models for Rotor Mechanical Assemblies of Rolling Stock

Abstract

The dynamic and static defect models for rotor mechanical assemblies have been compared in the article. The article gives reasons for the significance of dynamic modeling to define remaining assembly life. The article points out that it is feasible to use empiric data as a foundation for creating models. The article gives the example of a small gear defect dynamic modeling for a complete wheelset. The approximate amount of work on creating dynamic models has been defined in the article. The article proposes the methods for implementing dynamic defect models.

1 Introduction

Using the vibration diagnostic equipment (VDE) to determine the technical condition of the rotor mechanical assemblies (RMA) of rolling stock (roller and friction bearings, toothed-wheel gearing) is regulated by the standardized documents of Joint Stock Company “Russian Railways” (JSCo “RZD”) [1, 2]. Such equipment is used for the incoming and outgoing inspection.

2 Static Defect Models in the Time and Frequency Domains

2.1 Time Parameters and Characteristic Frequencies

For trouble shooting and identifying the degree of defect growth all VDE manufacturers use static models as a set of vibration signal parameters in the time and frequency domains.

Such parameters in the time domain are:

Root-mean-square value:

$$ X_{rmsv} = \sqrt {\frac{1}{T}\int\limits_{{t_{0} }}^{{t_{0} }} {\left[ {X\left( t \right)} \right]^{2} dt} } $$

(1)

Peak factor: it is defined as the ratio of the maximum (peak) signal value to RMA value of the vibration level:

$$ K_{peak} = \frac{{X_{\max } }}{{X_{rmsv} }}, $$

(2)

Kurtosis factor (KF) [3]: It can be calculated using the following formula:

$$ F_{K} = \frac{{\int\limits_{{t_{0} }}^{{t_{0} + T}} {\left( {X - X} \right)^{2} P\left( x \right)dx} }}{{\sigma^{4} }}, $$

(3)

where X(t) – signal amplitude from the vibration sensor (time signal), P(x) – probability function of the random value (time signal), T – monitoring period, t₀ – time of the monitoring starting, Xcp – root-mean-square deviation of the time signal.

The prime tool for identifying RMA defects is the spectral analysis for vibration signals which are received from the rotating assemblies during diagnostics.

2.2 The RMA Model

In general, RMA model of the frequency domain can be described as two groups of components [4]: the first group involves specific periodic components (including subhar-monics and superharmonics) generated by the individual bearing or toothed-wheel gearing elements during their operation; the second one involves all other components, including “background”, noise, impulse noise and the components generated by other elements during their operation.

$$ P\left( f \right) = P_{1} \left( f \right) + P_{2} \left( f \right) + P_{3} \left( f \right) + P_{4} \left( f \right) + Pz $$

(4)

where, P₁ – harmonic components which are multiple to the characteristic outer race frequency; P₂ – harmonic components which are multiple to the characteristic inner race frequency; P₃ – harmonic components which are multiple to the characteristic cage frequency; P₄ – harmonic components which are multiple to the characteristic roller frequency; P_z – noise terms.

Fig. 1.

The example of the assembly defect static model in the frequency domain (intensely developed small gear defect in a complete wheelset) - the direct spectrum and the envelope spectrum.

Formulas for calculating P₁; P₂; P₃ and P₄ (respectively: f_H; f_B; f_C; f_TК) are given, for example, in the literature [5].

Harmonics with frequencies which are multiple to the roller pass frequency outer race kf_H, where

$$ kf_{H} = \frac{1}{2}kf_{BP} \left( {1 - \frac{{d_{TK} }}{{d_{c} }}\cos \left( \alpha \right)} \right)z, $$

(5)

where d_TК – roller diameter; d_C – cage diameter; α – contact angle of bearing rollers and bearing races; z – number of rollers in one row of the bearing; k – harmonic number (whole number).

Harmonics with frequencies which are multiple to the roller pass frequency inner race kf_B, where

$$ kf_{B} = \frac{1}{2}kf_{BP} \left( {1 + \frac{{d_{TK} }}{{d_{c} }}\cos \left( \alpha \right)} \right)z, $$

(6)

Harmonics with frequencies which are multiple to the cage frequency kfc, where

$$ kf_{C} = \frac{1}{2}kf_{BP} \left( {1 - \frac{{d_{TK} }}{{d_{c} }}\cos \left( \alpha \right)} \right), $$

(7)

Harmonics with frequencies which are multiple to the roller frequency kf_TК, where

(8)

The number of additional groups of harmonics to be considered could be quite large - more than 10.

To identify individual RMA defects that the models obtained after the vibration signals processing in the frequency domain are compared with the reference models. According to the comparison of the results, a conclusion is made about the RMA technical condition.

2.3 The Static (Single-Dimensional) in Time Model

In this case, the technical condition is evaluated according to the static (single-dimensional) in time RMA model in the frequency domain (this is how all the VDE operate at Russian Railways). Static models do not provide essential information for predicting the remaining life of the investigated assembly. In order to perform that, it is necessary to know how each of the typical defects develops through time. With regard to rolling stock assemblies, it is reasonable to determine how the the defect growth depends on the distance run. Therefore, it is necessary to develop dynamic models.

3 Dynamic Defect Models in the Time and Frequency Domains

3.1 The Concept of Dynamic Model

In [6], it is suggested to introduce a multi-parameter vibration analysis including the function of time. Such an approach allows one to estimate the dynamics of individual spectral components related to diagnosing the evidence of defects. The trend of the envelope spectrum component of high-frequency vibration of the bearing assembly is given as an example. The envelope spectrum component is responsible for such defect as an uneven roller and cage wear. This trend was obtained for a specific bearing in a three-year time span.

In order to introduce digitalization and predictive analytics at Russian Railways it is proposed to develop and put into diagnosing practice the defect growth dynamic models in the significant RMA of the rolling stock.

3.2 The Options for Creating Models and Experiments

The dynamic model in this case should be understood as the dependence of the degree growth of every identified defect on time or, more preferably, on the distance run.

There are two ways to obtain dynamic models - analytical methods and using empirical data basis. According to the authors the first way is not acceptable due to a number of factors with unpredictable influence on the device under test during its operation (for example, impact loads that occur when there is a flat spot on the running surface of the wheel, expansion joint gaps, the violation of loading rules for freight cars, etc.) and the need to introduce a large number of assumptions.

The analysis of open information sources showed the absence of publications on the dynamic modelling for the developing rolling stock RMA. The authors in cooperation with their colleagues attempted to refine a technique for creating dynamic defect models based on the results of diagnostic and future disassembly analysis of the complete wheelsets of the motor-cars in the ED4M electric multiple unit.

During the experiments with the help of the vibration diagnostic equipment “Prognoz” the bearings and gear defects of the complete wheelsets were detected. The capabilities of the VDE allow to detect up to 12 defect types and the degree of each defect growth without disassembling. The degree of each detected defect growth is determined by the modulation depth of the largest characteristic harmonic component according to formulas 5–8 [7].

The target of the study was the wheel-motor drive unit of the ED4M electric multiple unit with an average daily run of 1012 km (operation on the West-Siberian railway).

Regular monitoring of the detected defect growth degree allowed us to develop a dynamic defect growth model depending on time and run.

Figures 2 and 3 demonstrate the defect dynamics in the small gear of the gear unit respectively depending on time and run.

Fig. 2.

The value of the defect growth depending on time.

Fig. 3.

The value of the defect growth depending on run.

The defect from the “light” scale turns into the “unacceptable” scale after 25 thousand kilometers of the distance run. Or, like in this particular case, in 25 days. From the graph shown in Fig. 1, it can be seen that the defect had been detected long before the moment when its growth degree required to roll out the complete wheelset and replace the gear. The gear with the incipient defect ran 60 thousand km and only after that the rapid defect growth began. The run of the electric multiple unit from the last large-scale regular maintenance to putting a complete wheelset out of operation due to the intensive toothed-wheel gear defect growth was 160336 km.

3.3 Complexity of Experiments

In order to establish the typical dependences of the defects growth on time or distance run, it is necessary to have data on a large number of the results of diagnosing assemblies with detected defects and the results of disassembling these assemblies. Each breakdown requires at least 10 selection. For 12 typical defects, the number of the selections will be 120. This is only for one of the complete wheelset elements. The complete wheelset contains five various types of rotating elements (small gear, large gear, axle bearings, motor-anchor bearings, small gear bearing). So, for the empirical defects growth modelling of one type of the complete wheelset it is necessary to conduct about 3000 diagnostic sessions and 600 disassemblies taking into account that each model will be built according to five reference points in time or distance run.

For the car wheel pairs, the amount of experiments will be far less.

As a result of experimental work the generalized dynamic of defect models can be obtained. It should be considered that even for the same type elements operating in different conditions, dynamic models possess a different nature. The operating conditions that affect the nature of the model include:

• grading and condition of the track in field;

• climatic conditions of the region of operation;

• the nature of the goods transported and loading technology.

In this regard, the generalized dynamic models can be adjusted (in terms of the defect growth rate), taking into account local conditions.

4 Dynamic Models Based on the Wavelet Transform

4.1 Fourier Transform and Wavelet Transform

Another trend in RMA defect dynamic modeling is using the wavelet transform of the vibration signals. At the beginning of the article it was already mentioned the static in the time RMA model in the frequency domain which represents the amplitude-frequency spectrum - the result of Fourier transforming a single time sample of the vibration signal.

Fourier transform can be presented as a sum of harmonic components, each of them has its own amplitude, frequency and phase shift.

$$ f\left( {\omega t} \right) = A_{0} + A_{1} \sin \left[ {\omega t + \varphi_{1} } \right] + A_{2} \sin \left[ {2\omega t + \varphi_{2} } \right] + A_{3} \sin \left[ {3\omega t + \varphi_{3} } \right] + ... $$

(9)

where A₀ – constant component amplitude; A₁ – sin(ωt + φ1) – fundamental harmonic; A₂, A₃, A₄,…. – the corresponding harmonic amplitudes.

One-dimensional signal wavelet transform f(x) is a two-dimentional function:

$$ W_{\psi } \left( {a,b} \right)f\left( x \right) = \frac{1}{{\sqrt {C_{\psi } } }}\int\limits_{ - \infty }^{ + \infty } {\frac{1}{{\sqrt {\left| a \right|} }}\psi \left( {\frac{b - x}{a}} \right)} f\left( x \right)dx $$

(10)

where the kernel Ψ is called a wavelet, b – a shift, a – a scale or a bar.

The normalization factor is equal to:

$$ C_{\psi } = 2\pi \int\limits_{ - \infty }^{ + \infty } {\frac{{\left| {\psi^{\prime}\left( \omega \right)} \right|^{2} }}{{\sqrt {\left| a \right|} }}d\omega } < \infty $$

(11)

where Ψ(ω) – Wavelet transform Ψ made with the help of Fourier transform.

When converting a time vibration signal using the fast Fourier transform (FFT), the obtained amplitude-frequency spectrum does not contain information on the harmonic component phases (Fig. 1). This is the static (point) RMA model in the frequency domain for a particular point in time (specific time sample).

Using the wavelet transform allows to trace the change of the spectral model throughout the entire time sample. In other words, it is possible to trace changes in the spectral components of the amplitude-frequency spectrum and the analysis of the RMA condition could be carried out in the three-dimensional space (the three-dimensional spectrogram, Fig. 4).

Fig. 4.

Example of a three-dimensional spectrogram as a result of wavelet transform.

Wavelet transform (WT) of a regular signal is a generalized Fourier series according to the system of basic functions. The continuous (integral) wavelet transform is the s(t) signal scalar product of the two-parameter wavelet function Ψ_{a, b} (t) of the selected type.

The integral function transformation of the function s(t) takes the form:

$$ S_{\psi } \left( {a,b} \right) = \int\limits_{ - \infty }^{ + \infty } {S\left( t \right)\psi_{a,b} \left( t \right)dt} $$

(12)

where, a – time scale parameter, inversely proportional to the frequency and responsible for the wavelet width; b – shift parameter determining the wavelet position on the time axis.

Wavelet function Ψa,b (t) of the affected set is obtained from one maternal function Ψ by stretch or compression and subsequent shift

$$ \psi_{a,b} \left( t \right) = \frac{1}{\sqrt a }\psi \frac{{\left( {t - b} \right)}}{a} $$

(13)

The multiplier ${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt a }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sqrt a }$}}$ determines that the integral energy of each wavelet Ψa, b (t) does not depend on a.

The function with two parameters S_Ψ(a,b) gives the information on the change in the relative contribution of components of different scales in time and is called the spectrum of wavelet transform coefficients. The scale is similar in meaning to the concept of frequency in Fourier transform [8].

4.2 Dynamic Model at a Particular Point in Time

Continuous wavelet transform is conducted in a particular time interval. For a vibration signal the result of the transformation will be a three-dimensional spectrogram, which will represent a three-dimensional model in the amplitude - frequency - time period. Such model is a dynamic model at a particular point in time, it represents the technical condition of the investigated assembly in a very short time period and it will be the model “at the point” about the defect growth time interval.

5 Time Samples Joining

5.1 Three-Dimensional Defect Growth Model in Time Period

It is proposed to create a three-dimensional defect growth model in time period with the help of wavelet transform, i.e. the investigated time interval should cover the time from the beginning of the defect growth to its transition to an unacceptable defect. It is almost impossible to record, save and process a continuous time signal within an interval equal to the time of full defect growth. It takes an extended period of time. To create a wavelet defect growth model in time period it seems advisable to use four or five time samples of a vibration signal reflecting the condition of a defect-free assembly, an assembly with an incipient, moderate, severe and unacceptable defect. Next, it is proposed to join these time samples into a unified time function. The necessary condition for joining is to ensure the smoothness and continuity of the function at the place of joining. Such condition is the continuity of the derivative at the position of the joint - on the interval between two local maxima of the opposite sign (the interval ab at the position of the joint of the two functions):

5.2 The Smoothness of the Function

The function f is smooth on (a, b) if it is continuous on the segment (a, b) and has such continuous derivative f (x) that there are limits f (a + 0) = A, f (b − 0) = B [9].

It makes sense to join at the points where f (Δt1; Δt2; Δt3; Δt4; Δt5) = 0.

According to the first Bolzano – Cauchy theorem such a point must exist on the segment ab of the smooth f (t) function:

If the function f (t) is continuous on the segment ab, the function values b at the ends of the segment have different signs f(a) > 0, f (b) < 0 or f(a) < 0, f(b) > 0, then there is a point ξ ∈ (a, b), in which the value of the function is zero f (ξ) = 0 [10].

5.3 Joining Functions

Figures 5a and 5b show the examples of the incorrect joining. Figure 5c shows the example of the desired joining.

Incorrect joining of time samples will cause appearing a powerful stray noise in the high-frequency domain after the Fourier transform in the amplitude-frequency spectrum,. This fact will crucially distort the dynamic defect growth model in time period. The authors do not issue the challenge to describe the algorithm for implementing the desired joining - this is a particular mathematical problem which probably has more than one alternative solution.

6 Joining of Three-Dimensional Spectrograms

Theoretically, there is another approach to implement a dynamic defect growth model in time period using wavelet transform. As it was already suggested above, four or five time samples of the vibration signal are used and the wavelet transform is conducted separately for each time sample. As a result, three-dimensional spectrograms are obtained, one of the versions is shown in Fig. 4 (some scientific works on the wavelet transform describe other methods of graphical interpretation). Then it is assumed that the three-dimensional platforms are joined separately. The joining conditions require a solid mathematical study and their determination is not the issue of the current discussion. According to the preliminary estimates the joining of three-dimensional spectrograms will be connected with additional restrictions that can lead to data corruption or data loss and it will require much more complex implementation algorithms.

Fig. 5.

a) Incorrect joining - f(t) function is not smooth on the ab segment; b) Incorrect joining - f(t) function is not smooth on the ab interval; c) desired joining - f(t) function is smooth on the ab segment.

7 Conclusions

• 1. The expediency of using vibration diagnostics for the RMA dynamic defect growth models is established.

• 2. The practicability of using empirical data as a basis for modeling is shown.

• 3. The example and the method of small gear defect dynamic modeling for the complete wheelset are given.

• 4. The estimated amount of work on dynamic modeling is determined.

• 5. Dynamic defect models will make it possible to forecast of the remaining life of the complete wheelset if only a nascent defect is detected.