# Research on Numerical Simulation Method of Nonstationary Random Vibration Signal Sensor in Railway Transportation

Authors: Wenchao Zhang; Bo Zhang; Fei Xu; Mohammad Asif
Journal: Journal of Sensors
 (2022)

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## Abstract

During railway transportation, due to various factors such as road conditions and operating conditions and produced vibrations and shocks, this kind of vibration environment may cause fatigue damage to on-board equipment and transported goods. The authors propose a research on the numerical simulation method of the nonstationary random vibration signal sensor of railway transportation; first, they establish the mathematical model of the railway nonstationary random vibration signal sensor and then introduce the method of reconstructing the railway nonstationary random vibration signal sensor. For railway nonstationary non-Gaussian random vibration reconstruction signal, compare the time-domain characteristics of the sampled signal, and for railway nonstationary non-Gaussian random vibration reconstruction signal, compare the frequency domain characteristics of the sampled signal. The results show that the relative error of the RMSM function is within 6%, the relative error of the sliding bias function is within 10%, and the relative error of the sliding kurtosis function is within 8%. The energy distribution of the edge Hilbert amplitude spectrum is very similar, with absolute error less than 6%. The energy fluctuations are similar in each band, with absolute error rates less than 4% in most bands. The method proposed in this article, suitable for reconstruction of railway nonstationary Gaussian random vibration and nonstationary non-Gaussian vibration signal sensor, verifies the effectiveness and feasibility of the signal reconstruction method. The model and signal reconstruction method proposed in this paper are applied to the railway nonstationary Gaussian and nonstationary non-Gaussian random vibration sampling signals.


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## Body

## 1. Introduction

Railway is one of China’s important infrastructures; it is a popular means of transportation and is the aorta of the national economy; railway transportation plays a very important role in the modern transportation system; it is the backbone of the transportation industry. Currently, there are two major trends in the development of railways in various countries around the world [1], namely, “freight heavy load” and “passenger high speed,” and my country’s railway transportation industry is also developing in the direction of high speed and heavy load [2]. In the course of the development of high-speed and heavy-duty railways, vehicle transportation safety and material transportation reliability are the focus and foothold of railway transportation to complete the huge transportation task [3]. In the process of railway “heavy load” and “high speed” development, due to the increase in cargo transportation and the increase in operating speed of trains, it will inevitably cause the vibration and shock conditions in the environment of railway vehicles to be more severe [4]. In order to ensure all kinds of vehicle equipment, various equipment for vehicle transportation, the strength, and reliability of the cargo, it is necessary to simulate the vibration and shock in the actual railway transportation environment [5]. Under the complex operating conditions of the railway, especially in the development of high-speed and heavy-duty railways, the vibration and shock of the railway transportation environment often have non-Gaussian distribution characteristics [6]. Therefore, Gaussian random vibration technology with controllable spectrum is no longer suitable for monitoring, evaluation, and simulation of loading conditions under special operating conditions [7]. The structure in the project is often subjected to vibration excitation caused by environmental loads such as earthquakes, wind, and waves. Figure 1 is a time-frequency domain hybrid adaptive filter active control process of structural vibration [8, 9]. In a series of non-Gaussian structural response analysis methods, due to the increase in computational efficiency, the simulation method is more and more widely used [10, 11]. In the simulation method, it is necessary to simulate the load time history with specified statistical characteristics and spectral characteristics. When the system excitation deviates significantly from the Gaussian distribution, the non-Gaussian characteristics of vibration must be accurately simulated [12]. Because of the extreme response of the system, especially sensitive to the high peak excitation of the load and the degree to which the response deviates from the Gaussian distribution, it will seriously affect the accumulation speed of system vibration fatigue damage [13]. The ARMA method is based on linear difference equations and simple calculation, but it cannot be displayed on irregular intervals; it has the characteristics of extremely large-amplitude pulse signals; therefore, it is not completely suitable for simulating non-Gaussian random time series [14]. Vibration test is a basic technical method in modern engineering technology; it uses specific test equipment to simulate and generate products and vibration excitation during use, transportation, and storage, under laboratory conditions, and synthesize various natural vibration environments and induced vibration environments [15]. The current vibration control system can only simulate the stable random vibration environment, and the use of the stable random vibration environment to approximate the nonstable random vibration environment is easy to cause undertest or overtest and cannot meet the requirements of environment adaptability and reliability test.Figure 1
Active control flow of structural vibration based on hybrid adaptive filtering in time-frequency domain.


## 2. Literature Review

In response to this research question, Kang et al. adopted the fast Fourier transform method and the ARMA model and generated Gaussian time history [16]. Rastehkenari and Ghadiri have done a lot of research work on the simulation of non-Gaussian stochastic processes [17]. Hache et al. used a multiparameter exponential peak autoregressive model (EPAR) and carried out one-dimensional or multidimensional variable non-Gaussian wind pressure time-history simulation research [18]. According to Liu et al., based on FFT technology, the exponential peak (EP) model with simplified parameters is used to simulate the large peak component in the random process and then simulate one-dimensional univariate non-Gaussian wind pressure time series, and it has been applied to the wind-vibration analysis of engineering roofs [19]. Che and Pang proposed a new static conversion method, used to simulate the time history of non-Gaussian wind pressure, but a sample generated by this method does not simulate the specified skewness and kurtosis well and needs to average multiple samples [20]. Majd and Mobayen have carried out research on power spectrum-related simulation technology [21]. Jannoun et al. simulated a non-Gaussian stochastic process based on the Johnson transformation system [22]. Jiang et al. used cubic polynomials; the conversion relationship between Gaussian stochastic process and non-Gaussian stochastic process is established; it is also used in the non-Gaussian wind pressure time-history simulation of the engineering layer cover structure [23]. Wang et al. rigorously deduced the relationship formulas of non-Gaussian features—kurtosis, skewness, and Fourier phase, and proposed the secondary phase modulation method; however, since the phase angle appears repeatedly in multiple terms in the skewness and kurtosis formulas, its algorithm is not easy to achieve simultaneous control of skewness and kurtosis [24]. Xia et al. proposed an amplitude modulation method, in order to obtain a reconstructed phase with non-Gaussian properties; this method can only be used to simulate sub-Gaussian signals [25]. Tong et al. generate Gaussian time series through FFT or ARMA model, use the method of nonlinear static transform to map to the non-Gaussian sample function, and generate a multidimensional non-Gaussian random uniform field [26]. Jin et al. also proposed some methods for simulating non-Gaussian stochastic processes, such as static transformation method and memory transformation (transform with memory) method; it was further improved to the spectra correction simulation method. With the help of this type of simulation method, univariate, multivariate non-Gaussian random processes are generated and used to describe the wind speed/pressure time-history acting on the roof of the building; it is in good agreement with the measured results [27]. Jiang et al. proposed using Karhunen-Loeve polynomial extension, in order to simulate a strong non-Gaussian process; it is relatively late to carry out the simulation research of non-Gaussian stochastic process in China [28]. Fan and Wu proposed a super-Gaussian random vibration simulation method based on secondary phase modulation and time-domain randomization [29]. Khanduri et al. are based on Johnson conversion system and digital filtering theory; a method that can quickly and effectively generate specified skewness, kurtosis, and power spectrum non-Gaussian fluctuating wind pressure is proposed [30]. On the basis of current research, the authors propose a research on the numerical simulation method of the nonstationary random vibration signal sensor of railway transportation; first, they establish the mathematical model of the railway nonstationary random vibration signal sensor and then introduce the method of reconstructing the railway nonstationary random vibration signal sensor. Compare the time-domain characteristics of the railway nonstationary non-Gaussian random vibration reconstruction signal and the sampled signal, and compare the frequency domain characteristics of the railway non-stationary non-Gaussian random vibration reconstruction signal and the sampled signal; the results show that the relative error of the sliding root mean square function is within ±6%, the relative error of the sliding skewness function is within ±10%, and the relative error of the sliding kurtosis function is within ±8%. The energy distribution of the edge Hilbert amplitude spectrum is very similar, and the absolute error is less than 6%. The energy fluctuations of each frequency band are very similar, and the absolute error rate in most frequency bands is less than 4%.


## 3. Methods

### 3.1. Mathematical Model of Railway Nonstationary Random Vibration Signal Sensor

On the basis of the mathematics of Hilbert-Huang transform describing nonstationary signals, introduce random elementφk to represent nonstationary random signal xt, as shown in formula (1): xt can be composed of n component signals ckt and residual rt, and among them, akt represents the time-varying amplitude function, ψkt represents the phase function, ϕkt represents the time-related part of ψkt, and φk is a random phase angle between 0 and 2π that obeys an independent uniform distribution, that is, φk∈U0,2π. Here, the random element φk is used to characterize the randomness of the railway vibration signal sensor and the time-varying nature of the instantaneous frequency of the component signal [31].
(1)xt=∑k=1nckt+rt=RE∑k=1naktejψkt+rt=RE∑k=1naktejϕkt+φk+rt=∑k=1naktcosϕkt+φk+rt.Based on formula (1), the following two assumptions are established:
(1)
Assume that the time-varying amplitudeakt obeys the probability distribution family Pkθk, such as formula (2), where P represents the probability distribution type, θ represents the distribution parameter vector, and Θ represents the parameter space [32](2)akt∈Pkθk:θk∈ΘAccording to this assumption, the time-varying amplitudeakt can be regarded as a random variable implicitly related to time. For convenience, we use ak to represent the time-varying amplitude [33].
(2)
Assuming the center frequencyωk0 of each component signal ckt, both are constants and different from each other, as shown in formula (3)(3)dψktdt=dϕkt+φkdt=dϕktdt=ωk0According to formula (3), the phase function ψkt can be based on the center frequency ωk0; it is obtained through mathematical integration, as shown in formula (4). Among them, C represents any constant generated in the indefinite integral process.
(4)ψkt=ϕkt+φk=∫ωk0dt+φk=ωk0t+φk+C.Because the cosine function is periodic, the constantC in formula (4) has little effect on the value of the cosine function, and it has less influence on the value of component signal ckt; therefore, without loss of generality and in order to express conciseness, the constant c=0 is set later in this article [34].According to formula (1) and the above two assumptions, the authors establish the following mathematical model to characterize the railway nonstationary random vibration signal sensor. Nonstationary random signal xt can be composed of n component signals ckt and residual rt, and among them, the time-varying amplitude ak of each component signal ckt obeys the probability distribution family Pkθk, and the center frequency ωk0 is constant and different from each other. Therefore, formula (1) can be simplified to the following:
(5)xt=∑k=1nckt+rt=∑k=1nakcosωk0t+φk+rt.


### 3.2. Method of Reconstructing Railway Nonstationary Random Vibration Signal Sensor

Combine architecture reconstruction signal time-varying amplitude and simulated reconstruction signal phase function in the known sampled signalst. Under the conditions, this section proposes a method of reconstructing the railway nonstationary random vibration signal sensor. The specific implementation steps are as follows:Method of rebuilding railway nonstationary random vibration signal sensorStep 1. Data preparation(i) Solve the Hilbert spectrumHsω,t=as,i,jm×n of the sampled signal st; (ii) solve the statistical moment function Mvst of the sampled signal st.Step 2. Architectural reconstruction of the Hilbert spectrum of the signalWhen using the framework to reconstruct the signal, change the amplitude to get the reconstructed signal, and the simulation time changes the amplitude and constructs its Hilbert spectrumHxω,t=a~x,i,jm×n.Step 3. Simulate the phase function of the reconstructed signal.Use simulation to reconstruct signal phase function simulation, and obtain the phase functionψ of the reconstructed signal.Step 4. Generate the reconstructed time series.Combine the reconstructed Hilbert spectrum of the signal obtained at step 2Hxω,t and the reconstructed signal phase function obtained in step 3 ψ. The reconstructed time series was generated using Equation (5) xt.


## 3.1. Mathematical Model of Railway Nonstationary Random Vibration Signal Sensor

On the basis of the mathematics of Hilbert-Huang transform describing nonstationary signals, introduce random elementφk to represent nonstationary random signal xt, as shown in formula (1): xt can be composed of n component signals ckt and residual rt, and among them, akt represents the time-varying amplitude function, ψkt represents the phase function, ϕkt represents the time-related part of ψkt, and φk is a random phase angle between 0 and 2π that obeys an independent uniform distribution, that is, φk∈U0,2π. Here, the random element φk is used to characterize the randomness of the railway vibration signal sensor and the time-varying nature of the instantaneous frequency of the component signal [31].
(1)xt=∑k=1nckt+rt=RE∑k=1naktejψkt+rt=RE∑k=1naktejϕkt+φk+rt=∑k=1naktcosϕkt+φk+rt.Based on formula (1), the following two assumptions are established:
(1)
Assume that the time-varying amplitudeakt obeys the probability distribution family Pkθk, such as formula (2), where P represents the probability distribution type, θ represents the distribution parameter vector, and Θ represents the parameter space [32](2)akt∈Pkθk:θk∈ΘAccording to this assumption, the time-varying amplitudeakt can be regarded as a random variable implicitly related to time. For convenience, we use ak to represent the time-varying amplitude [33].
(2)
Assuming the center frequencyωk0 of each component signal ckt, both are constants and different from each other, as shown in formula (3)(3)dψktdt=dϕkt+φkdt=dϕktdt=ωk0According to formula (3), the phase function ψkt can be based on the center frequency ωk0; it is obtained through mathematical integration, as shown in formula (4). Among them, C represents any constant generated in the indefinite integral process.
(4)ψkt=ϕkt+φk=∫ωk0dt+φk=ωk0t+φk+C.Because the cosine function is periodic, the constantC in formula (4) has little effect on the value of the cosine function, and it has less influence on the value of component signal ckt; therefore, without loss of generality and in order to express conciseness, the constant c=0 is set later in this article [34].According to formula (1) and the above two assumptions, the authors establish the following mathematical model to characterize the railway nonstationary random vibration signal sensor. Nonstationary random signal xt can be composed of n component signals ckt and residual rt, and among them, the time-varying amplitude ak of each component signal ckt obeys the probability distribution family Pkθk, and the center frequency ωk0 is constant and different from each other. Therefore, formula (1) can be simplified to the following:
(5)xt=∑k=1nckt+rt=∑k=1nakcosωk0t+φk+rt.


## 3.2. Method of Reconstructing Railway Nonstationary Random Vibration Signal Sensor

Combine architecture reconstruction signal time-varying amplitude and simulated reconstruction signal phase function in the known sampled signalst. Under the conditions, this section proposes a method of reconstructing the railway nonstationary random vibration signal sensor. The specific implementation steps are as follows:Method of rebuilding railway nonstationary random vibration signal sensorStep 1. Data preparation(i) Solve the Hilbert spectrumHsω,t=as,i,jm×n of the sampled signal st; (ii) solve the statistical moment function Mvst of the sampled signal st.Step 2. Architectural reconstruction of the Hilbert spectrum of the signalWhen using the framework to reconstruct the signal, change the amplitude to get the reconstructed signal, and the simulation time changes the amplitude and constructs its Hilbert spectrumHxω,t=a~x,i,jm×n.Step 3. Simulate the phase function of the reconstructed signal.Use simulation to reconstruct signal phase function simulation, and obtain the phase functionψ of the reconstructed signal.Step 4. Generate the reconstructed time series.Combine the reconstructed Hilbert spectrum of the signal obtained at step 2Hxω,t and the reconstructed signal phase function obtained in step 3 ψ. The reconstructed time series was generated using Equation (5) xt.


## 4. Results and Analysis

### 4.1. Comparison of Time-Domain Characteristics of Railway Nonstationary Non-Gaussian Random Vibration Reconstruction Signal and Sampled Signal

This section mainly compares the amplitude probability density function and sliding statistical moment function ofsnt and xnt.
(1)
Comparison of amplitude probability density functionsThe amplitude probability density functions ofsnt and xnt are shown in Figure 2; through comparison, it can be found that the difference between the amplitude probability density function of xnt and snt is very small, and the two have very similar peak and thick tail characteristics [35].
(2)
Comparison of sliding statistical moment functionsFigure 2
Comparison of the amplitude probability density of the nonstationary non-Gaussian reconstructed signal and the sampled signal.Calculate the sliding root mean square functionsRMSsn,r and RMSxn,r of the sampled signal and the reconstructed signal, respectively, sliding skewness functions λsn,r and λxn,r, and sliding kurtosis functions γsn,r and γxn,r. Railway nonstationary non-Gaussian random vibration reconstruction signal and the relative error of the sampled signal, respectively, are drawn in Figures 3–5. Observation found that the relative error of the sliding root mean square function of the reconstructed signal xnt is within ±6%, the relative error of the sliding skewness function is within ±10%, and the relative error of the sliding kurtosis function is within ±8%. The nonstationary non-Gaussian random signal sliding statistical moment function fluctuates more strongly than the railway nonstationary Gaussian random vibration signal, so that the error generated in the nonstationary non-Gaussian signal reconstruction will also be larger than that of the nonstationary Gaussian random signal. In general, the reconstructed signal xnt obtained in this example is also a sliding statistical moment function with similar time-varying characteristics to the sampled signal snt.Figure 3
Comparison of nonstationary non-Gaussian reconstruction signal and sampling signal sliding root mean square.Figure 4
Comparison of sliding skewness between nonstationary non-Gaussian reconstruction signal and sampled signal.Figure 5
Comparison of sliding kurtosis of nonstationary non-Gaussian reconstruction signal and sampled signal.


### 4.2. Comparison of Frequency Domain Characteristics of Railway Nonstationary Non-Gaussian Random Vibration Reconstruction Signal and Sampled Signal

(1)
Comparison of edge Hilbert amplitude spectrumCalculate the edge Hilbert amplitude spectraMHASsnω and MHASxnω of snt and xnt, respectively, and compare the two, as shown in Figures 6 and 7. Observations found that for snt and xnt, the energy distribution of the edge Hilbert amplitude spectrum is very similar, with an absolute error of less than 6%.
(2)
Comparison of nonstationarity of Hilbert amplitude spectrumFigure 6
Edge Hilbert amplitude spectrum.Figure 7
The relative error of the nonstationary non-Gaussian reconstruction signal and the edge Hilbert spectrum of the sampled signal.As shown in Figure8, observation found that the energy fluctuations of snt and xnt in each frequency band are very similar, and in most frequency bands, the absolute error rate is less than 4%.Figure 8
The relative error of the nonstationary degree of the nonstationary non-Gaussian reconstruction signal and the sampled signal Hilbert spectrum.


## 4.1. Comparison of Time-Domain Characteristics of Railway Nonstationary Non-Gaussian Random Vibration Reconstruction Signal and Sampled Signal

This section mainly compares the amplitude probability density function and sliding statistical moment function ofsnt and xnt.
(1)
Comparison of amplitude probability density functionsThe amplitude probability density functions ofsnt and xnt are shown in Figure 2; through comparison, it can be found that the difference between the amplitude probability density function of xnt and snt is very small, and the two have very similar peak and thick tail characteristics [35].
(2)
Comparison of sliding statistical moment functionsFigure 2
Comparison of the amplitude probability density of the nonstationary non-Gaussian reconstructed signal and the sampled signal.Calculate the sliding root mean square functionsRMSsn,r and RMSxn,r of the sampled signal and the reconstructed signal, respectively, sliding skewness functions λsn,r and λxn,r, and sliding kurtosis functions γsn,r and γxn,r. Railway nonstationary non-Gaussian random vibration reconstruction signal and the relative error of the sampled signal, respectively, are drawn in Figures 3–5. Observation found that the relative error of the sliding root mean square function of the reconstructed signal xnt is within ±6%, the relative error of the sliding skewness function is within ±10%, and the relative error of the sliding kurtosis function is within ±8%. The nonstationary non-Gaussian random signal sliding statistical moment function fluctuates more strongly than the railway nonstationary Gaussian random vibration signal, so that the error generated in the nonstationary non-Gaussian signal reconstruction will also be larger than that of the nonstationary Gaussian random signal. In general, the reconstructed signal xnt obtained in this example is also a sliding statistical moment function with similar time-varying characteristics to the sampled signal snt.Figure 3
Comparison of nonstationary non-Gaussian reconstruction signal and sampling signal sliding root mean square.Figure 4
Comparison of sliding skewness between nonstationary non-Gaussian reconstruction signal and sampled signal.Figure 5
Comparison of sliding kurtosis of nonstationary non-Gaussian reconstruction signal and sampled signal.


## 4.2. Comparison of Frequency Domain Characteristics of Railway Nonstationary Non-Gaussian Random Vibration Reconstruction Signal and Sampled Signal

(1)
Comparison of edge Hilbert amplitude spectrumCalculate the edge Hilbert amplitude spectraMHASsnω and MHASxnω of snt and xnt, respectively, and compare the two, as shown in Figures 6 and 7. Observations found that for snt and xnt, the energy distribution of the edge Hilbert amplitude spectrum is very similar, with an absolute error of less than 6%.
(2)
Comparison of nonstationarity of Hilbert amplitude spectrumFigure 6
Edge Hilbert amplitude spectrum.Figure 7
The relative error of the nonstationary non-Gaussian reconstruction signal and the edge Hilbert spectrum of the sampled signal.As shown in Figure8, observation found that the energy fluctuations of snt and xnt in each frequency band are very similar, and in most frequency bands, the absolute error rate is less than 4%.Figure 8
The relative error of the nonstationary degree of the nonstationary non-Gaussian reconstruction signal and the sampled signal Hilbert spectrum.


## 5. Conclusion

The authors propose a numerical simulation method for the nonstationary random vibration signal sensor of railway transportation. The signal is reconstructed by the railway nonstationary non-Gaussian random vibration, and when compared with the frequency domain characteristics of the sampled signal, it can be concluded that using the method proposed by the authors, the railway nonstationary non-Gaussian random vibration signal sensor with similar characteristics in time domain and frequency domain can be reconstructed. Taking the model and signal reconstruction method proposed by the authors, applied to railway nonstationary Gaussian and nonstationary non-Gaussian random vibration sampling signals under other working conditions, the effect is also very good. Future simulation techniques are needed for the non-Gaussian vibrational environment, investigating the Gaussian and ultra-Gaussian response based on random stress under the Monte-Carlo method.



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