Perspective Chapter: Predicting Vehicle-Track Interaction with Recurrence Plots

Abstract

This chapter presents a method for identifying the dynamic response of vehicles (railcars) running on a rail track. The method is based on the recurrence plots developed from the phase diagram (phase plane). The phase plane is constructed using Hamilton’s principle for a single-mass system. The state variables are estimated from acceleration measurements registered in an experimental test rig. The measurements were recorded with accelerometers and gyroscopes mounted on the railcar that circulated on a closed-loop track. The acceleration data were integrated using the empirical mode decomposition method and the time delay principle. The acceleration data were separated into two data vectors: rigid body motion and vibration modes. The recurrence plots were built for both data vectors, and it was found that the vibration modes were more sensitive to track defects and curvature changes. The recurrence plots were analyzed with the recurrence quantification analysis, and it was found that the Recurrence Plots can determine the location and type of defects.

1. Introduction

Many researchers work on the life prediction of nonlinear systems. Although the topic has been under research for decades, there are still many uncertainties and doubts, and still, it is an open issue from a practical point of view. There are different alternatives for modeling and analyzing nonlinear systems. This chapter presents the application of recurrence plots to predict defects in rails and railcars.

Recurrence plots are based on Poincaré’s concepts. Eckmann et al. [1] worked further on Poincaré’s principles and defined the basic procedure for constructing recurrence plots from a phase plane (phase space). Marwan and Weber [2] and Webber et al. [3] represented different dynamic systems using the recurrence plot procedure; their primary contribution is that the trajectory along a phase plane could be quasi-stationary.

The work presented by Eckmann et al. [1] described the application of recurrence plots to determine the time constancy of dynamic systems. They were the first to distinguish that the recurrence plots can measure the entropy of the phase plane, the dimension spectrum, or other information dimensions. They constructed phase plane orbits and estimated the repeatability of each cycle; then, they quantified the number of times that a point appeared in different cycles and proposed a method for finding time correlations in a signal. They distinguished two characteristics in the recurrence plots: large-scale forms “topologies,” and small-scale forms “textures.” They illustrated their results with experimental and numerical data. A detailed description of “topologies” and “textures” are presented in the following sections.

Many publications deal with the application of recurrence plots to single-frequency signals. For this kind of signal, the phase plane can be constructed by using the shifting process. But, in most cases, the dynamic response combines different frequencies, transient responses, and nonlinear effects. Torres et al. [4] analyzed the error caused by applying the shifting process to nonlinear signals. To avoid this error, Torres et al. [5] proposed a different alternative. This procedure is further discussed in the following sections.

Recurrence plots have been applied to electroencephalogram signals (EEG) [6, 7, 8, 9]. They also have been applied to direct current discharge plasma and the identification of the geodesic distance on Gaussian manifolds for chaotic systems [10]. Kwuimy and Kadji [11] and Kwuimy et al. [12, 13, 14] applied the recurrence plots to two Van der Pol type oscillators coupled by a nonlinear spring. They estimated the synchronization using two Recurrence Plots. Similar results are obtained with the Kuramoto’s parameter. Jana et al. [15] represented a food chain system as nonlinear ordinary differential equations, and they applied the Recurrence Plot for identifying the dynamic parameters.

Several researchers have analyzed data generated with a Rössler system. Thiel et al. [16] embedded Gaussian noise into the Rössler model and identified the noise with the recurrence plot. Kiss et al. [17] identified synchronization on a set of Rössler oscillators using recurrence plots and determined the synchronization by calculating the cross-correlation and the probability of two-state positions coinciding in the same phase plane after a certain period. Prakash and Roy [18] represented a chaotic electric system with a Rössler model.

Recurrence Plots have been used to explain the dynamic characteristics of bubbles within a water flow [19] and to flow measurements [20]. Xiong et al. [21] compared the empirical mode decomposition and the Recurrence Plots for the analysis of traffic flow, and Tang et al. [22] proposed an intelligent traffic control system based on the Recurrence Plots. Vlahogianni and Karlaftis [23] determined the complexity of traffic flow time series using the Recurrence Plots. Ukherjee et al. [24] identified the dynamic behavior of wireless network traffic by utilizing the Quantification analysis of Recurrence Plots. Syta and Litak [25] identified cutting parameters in a machining process with Recurrence Plots. In a similar work, Elias and Namboothiri [26] identified chatter in a turning process for constructing the phase plane and found the time delay by using the average mutual information function.

The recurrence plots show patterns with specific topologies and textures that reflect the system’s dynamic behavior. The analysis of these patterns relates the topologies and textures to the system’s response. In this context, Leonardi [27] measured the entropy of the Recurrence Plot to signals without noise. Spiegel et al. [28] described different measures and analyses for Recurrence Plots. Belaire and Contreras [29] classified signal data into a set of embedded vectors separated by a time delay. This concept is only valid for time series that have a single frequency. Pham and Yan [30] proposed the sample entropy to measure Recurrence Plots irregularities. Girault [31] measured the symmetry in Recurrence Plots to identify the system’s dynamic behavior. Sipers et al. [32] defined a procedure for retrieving a signal from a Recurrence Plot, but their method is valid only for embedded signals. Bot et al. [33] used Recurrence Plots to detect unknown signals embedded in white noise.

The life prediction of rails and railcars depends on the ability to determine failures at the rail, the wheel, and the rotating elements of the railcar. The difference from other systems is that these variables are a function of time and location, and the wheel-rail interface determines the dynamic condition. The rail’s imperfections and train speed are the dominant factors affecting the dynamic load at the wheel-rail contact point. Therefore, it is necessary to identify the dynamic load and its location along the track.

Ngamkhanong et al. [34] reviewed different models for describing the wheel-track interaction and the complexity of the elastic interaction between the rotating elements and the substructure. Most models assume that the rail behaves as an elastic beam supported by individual springs (sleepers) [35]. Ciotlaus et al. [36] defined the interaction based on the rolling contact, fatigue, and wear. These models confirmed the fatigue failures described by Smith [37].

This chapter presents the application of the Recurrence Plots to identify the dynamic effect of a rail on a railcar. The data were obtained in an experimental rig, and they consisted of acceleration and velocity measurements registered at the railcar. The acceleration was integrated using the empirical mode decomposition method and the shifting property of periodic functions [5]. Results can be extrapolated to real measurements.

The Recurrence Plots were produced from acceleration data obtained with a triaxle MEMs sensor and a triaxle gyroscope. It was possible to reproduce the railcar movements in every direction (6 degrees of freedom) with these data. Since the railcar travels in a close circuit, the acceleration data recorded rigid body and vibration motions. The data were regrouped into nonperiodic motion (rigid body motion) and periodic motion (vibrations) to separate the two types of motions. The new data sets were processed with the empirical mode decomposition method.

2. Recurrence plot

The basis for constructing recurrence plots is the phase diagram or phase plane. The phase plane [38] describes the stability of a dynamic system. It determines the relationship between the potential and kinetic energies at a given time interval and predicts the evolution of the energy balance. The recurrence plot is constructed from the analysis of the evolution of a phase plane. The dynamic system evolves following different paths represented in phase planes; these planes are built at intervals defined by the fundamental period. Then the recurrence plot identifies the number of times that a phase plane vector (the state variables that describe the system dynamics) repeats or recurs at every fundamental period. Before describing the construction of a recurrence plot, the following section introduces the basis of the phase plane.

2.1 Phase plane

A phase plane describes the mass trajectory along a two-dimensional energy field. (Two-dimensional state variables). Representing the dynamic behavior of a linear system as a differential equation system:

$$\dot{\mathit{x}}=\mathit{Bx}$$E1

Matrix $\mathit{B}$ contains the system’s parameters, and it is derived from Hamilton’s principle:

$$H\left(p,q\right)=T\left(p\right)+V\left(q\right)$$E2

Or

$$T\left(p\right)=\frac{p^2}{2m}$$E3

Hamilton’s principle states that the equilibrium of the system is obtained when:

$$\dot{q}=\frac{\partial H}{\partial p}$$E4

And

$$\dot{p}=-\frac{\partial H}{\partial q}$$E5

Therefore, at any instant, there is a function $\varphi \left(p,q\right)$ such that

$$\frac{\mathit{d\varphi }}{\mathit{dt}}=\frac{\partial \varphi }{\partial q}\frac{\partial H}{\partial p}-\frac{\partial \varphi }{\partial p}\frac{\partial H}{\partial q}$$E6

The dynamic stability and the evolution of the phase plane can be derived from Eq. (6). If, in the phase plane, the function $\varphi$ is constant in any trajectory, then the system is stable (Liouville’s theorem).

$$\frac{\mathit{dH}}{\mathit{dt}}=\frac{\partial H}{\partial q}\dot{q}+\frac{\partial H}{\partial p}\dot{p}=0$$E7

Eq. (7) implies that the phase plane has a constant volume.

If the mass is constant, the linear momentum only depends on the mass velocity. In linear systems, the potential energy is proportional to the displacement; therefore, the phase plane can be represented with the state variables velocity and position. Figure 1 shows the phase plane of a single degree of freedom system with a harmonic response ($\dot{p}+\mathit{kq}=0\mathit{\text{and}}\dot{q}=\frac{p}{m})$. It is clear that the trajectory is constant and smooth; meanwhile, a nonlinear system will show irregular patterns. These irregularities are the basis for studying nonlinear systems with the recurrence plots.

Figure 1.

Phase plane of a linear harmonic system.

2.2 Definition of the recurrence plot

The phase plane function can be defined as a vector that depends on a parameter (time) and defines the state of the dynamic system (Figure 1):

$$\overline{x}\left(t\right)=\left[{\overline{x}}_1,{\overline{x}}_2,\dots .,{\overline{x}}_n\right]$$E8

The dynamic evolution in time is represented in Figure 2. If the dynamic response is steady, then ${\overline{x}}_1\left(t\right)={\overline{x}}_n\left(t+\mathit{n\tau }\right)$, where $\tau$ is the period of a harmonic system and $n$ is an integer.

Figure 2.

Evolution of the state variables of a linear harmonic system.

A system recurs if two subsequent states are equal and it repeats every period $\tau$. According to Eckmann, Kamphorst, and Ruelle [1], a recurrence plot is a matrix representation of the similarity of two consecutive state conditions:

$$R_{\mathit{ij}}=\left\{\begin{array}{c}1:{\overline{x}}_i={\overline{x}}_j\\ 0:\text{otherwise}\end{array}\right.i,j=1,\dots ,N$$E9

The procedure applies to a continuous and discrete set of data. But, in general, data are discrete vectors; thus, $N$ is the number of state vectors in the time array, divided by loops of period $\tau$. Data contain noise and truncation errors; therefore, two vectors cannot have the same magnitude, and Eq. (9) is modified as:

$$R_{\mathit{ij}}=\left\{\begin{array}{c}1:\left|{\overline{x}}_i-{\overline{x}}_j\right|<\epsilon ,\\ 0:\left|{\overline{x}}_i-{\overline{x}}_j\right|>\epsilon ,\end{array}i,j\right.=1,\dots ,N$$E10

where $\epsilon$ is a tolerance value. The tolerance should be less than 10% of the mean diameter of the phase plane or five times larger than the standard deviation of the observational noise.

The orbits in a phase plane describe the system’s dynamics. Orbits can see growth due to the entropy, and the number of orbits depends on the characteristic frequencies, damping, and the presence of nonlinear behavior. These characteristics modify the recurrence plots in a way that they can describe the system’s behavior and predict future states. There are several characteristics of a recurrence plot that are classified as topology and texture. These characteristics are related to the system’s dynamics.

Since the recurrence plots depend on the phase plane, it is important to construct a proper phase plane. The phase shift principle helps to build the phase plane of single-frequency signals; but for mechanical systems, this procedure introduces artificial noise that corrupts the phase plane [4]. The following sections describe the procedure for constructing the phase plane from acceleration measurements.

2.3 Topology and texture

In a Recurrence Plot, the main diagonal is always present, and parallel diagonals occur only in periodic or quasi-periodic systems. The distance between the diagonals is the fundamental period, and the diagonal lengths define if a system has a predictable behavior (steady condition). If the system presents recurrence but at different frequencies, then the diagonals will be shifted from the main diagonal.

Eckmann et al. [3] presented a summary of different topologies:

• A stationary system produces a homogeneous graph.

• If only some states change slowly or stop at a singularity, the graph has horizontal and vertical lines.

• Nonpersistent or fluctuating states produce isolated points. These points also mean that the states have continuous variations.

• Systems cause drift with slowly varying parameters; for example, nonstationary systems display a drift in the graphs.

• The main diagonal is interrupted when there is a sudden change, and the graph shows white areas.

• The graph shows dark and pale areas at the corners if the system has slow dynamic changes or slow varying parameters.

The topology depends on the tolerance, if $\epsilon$ is small, the Recurrence Plot erases most of the similar vectors, and it will be almost empty. Too large tolerance will produce many artificial figures that have no relation to the dynamic behavior.

2.4 Recurrence plot characteristics

Besides the topology and texture, a recurrence plot has several characteristics related to the system’s dynamic response [39]. The following table (Table 1) summarizes the main features classified as recurrence quantification analysis.

RQA	Equation	Description
Recurrence Rate	$\mathit{RR}=\frac{1}{N^2}{\sum }_{i,j}^N{R}_{\mathit{ij}}$	Measures the average number of recurrence points
Determinism	$\mathit{DET}=\frac{{\sum }_{l=l_{\mathit{min}}}^N\mathit{lP}\left(l\right)}{{\sum }_{l=1}^N\mathit{lP}\left(l\right)}$	Measures the percentage of recurrence points that lay on a diagonal. $l$ is the index of the corresponding diagonal, $P\left(l\right)$ is the histogram of the diagonal. A chaotic system has short diagonals, and a periodic system forms regular diagonals, parallel to the main diagonal.
Longest Diagonal	$L_M=max\left\{l_i\right\}$	Measures the length (in points) of the longest diagonal.
Longest Vertical Line	$V_M=max\left\{v_i\right\}$	Measures the length of the longest vertical line.
Laminarity	$\mathit{LAM}=\frac{{\sum }_{v=v_{\mathit{min}}}^N\mathit{vP}\left(v\right)}{{\sum }_{l=1}^N\mathit{vP}\left(v\right)}$	Measures the percentage of recurrence points that create vertical lines. $v$ is the index of the corresponding vertical line, $P\left(v\right)$ is the histogram of the vertical line
Shannon Entropy	$E=-{\sum }_1^N{P}_i\mathit{Ln}\left(P_i\right)$	Measures the disorder of the phase plane. For a linear and periodic system, the Shannon entropy is almost cero. For a non-deterministic and noisy system, the Shannon entropy will be almost infinite. $P_i=P\left(x_j=x_i\right)$ is the relative frequency, it is calculated as the ratio of diagonal with non-recurrence points divided by the number of recurrent points.

Table 1.

Recurrence quantification parameters.

The first step in constructing a recurrence plot is to build the phase plane. The problem with creating the phase plane using acceleration data is the integration of the state variables. The data are time series vectors $\ddot{x}$, and the phase plane is built with the state variables $\dot{x}$ and $x$. When the signal has only one frequency, the integration can be achieved by normalizing the original data and shifting them to one-quarter of the period. The following section describes the procedure for the integration of time series with any pattern.

3. Integration

The numerical integration has a fundamental problem because it integrates between limits, whereas for constructing the phase plane, the integration has to be indefinite. Torres and Jauregui demonstrated that numerical integrations introduce artificial errors in the phase plane [4]. They proposed a different integration procedure to avoid these errors that combine the one-quarter shifting method and the empirical mode decomposition (EMD) [5]. The EMD adapts the time domain to process nonstationary and nonlinear time series. The EMD method decomposes a signal into intrinsic mode functions by applying Hilbert’s transform. The time series is approximated as [40].

$$a\left(t\right)=\sum _1^n{\mathit{IMF}}_i\left(t\right)+r\left(t\right)$$E11

The integration method begins by separating the acceleration signal into a set of intrinsic mode functions (IMF). The first iteration starts by identifying the points that determine the local maxima and minima. These sets of points define the lower and upper envelopes. Both groups of points are connected with a cubic spline. Then, the mean value function is the first IMF. The second step in the algorithm is to subtract the mean function from the original data. The following steps are finding the maxima and minima of the residual values, finding the mean function, and removing from the residual until the difference tends to minimum difference. The procedure is summarized in Figure 3.

Figure 3.

EMD procedure.

Each intrinsic mode functions (IMF) is a smooth time series with variable amplitude and one or more frequencies, while the Fourier transform divides the acceleration signal into a set of harmonic functions with constant amplitude and frequency.

Once the acceleration data is divided into a set of IMF, the integration is based on the following principle.

Assume that the acceleration function is represented by:

$$a\left(t\right)={\mathrm{a}}_{1}cos\left({\omega }_{1}t\right)+{\mathrm{a}}_{2}cos\left({\omega }_{2}t\right)+..+{\mathrm{a}}_{n}cos\left({\omega }_{n}t\right)$$E12

The indefinite integral is:

$$\int a\left(t\right)\mathit{dt}=-\frac{{\mathrm{a}}_1}{{\omega }_1}\mathrm{sen}\left({\omega }_{1}t\right)-\frac{{\mathrm{a}}_2}{{\omega }_2}\mathrm{sen}\left({\omega }_{2}t\right)+..-\frac{{\mathrm{a}}_n}{{\omega }_n}\mathrm{sen}\left({\omega }_{n}t\right)$$E13

Or

$$\int a\left(t\right)\mathit{dt}=\frac{{\mathrm{a}}_1}{{\omega }_1^2}cos\left({\omega }_{1}t+{\tau }_1\right)+\frac{{\mathrm{a}}_2}{{\omega }_2^2}cos\left({\omega }_{2}t+{\tau }_2\right)+..+\frac{{\mathrm{a}}_n}{{\omega }_n^2}cos\left({\omega }_{n}t+{\tau }_n\right)$$E14

Combined with the EMD method, the shifting process (time delay) produces a better representation of the acceleration signal than the Fourier transform. Figure 4 shows the application of the shifting procedure to one of the intrinsic mode functions (Mode 5). The integrated signal is obtained by adding all the shifted intrinsic mode functions.

Figure 4.

Application of the shifting process to the intrinsic mode functions 5.

Figure 5 shows the result of adding the shifted modes compared to the original signal.

Figure 5.

Original and shifted signals produced by the EMD and shifting process.

The method based on the EMD and the shifting process was applied to measurements of a scaled-down experimental train.

4. Experiments

The scaled-down experimental railcar (Figure 6) was mounted on a closed-circuit track (Figure 7). The track has curves that make the railcar rotate clock and counterclockwise. The railcar has two bogies, with a suspension and a platform that can be loaded with different weights and containers. One of the bogies is powered by a servomotor that can vary the railcar speed, and it is remotely controlled.

Figure 6.

Experimental railcar.

Figure 7.

Railcar mounted on the track.

The railcar is instrumented with an encoder (for measuring the wheel’s speed) with 500 PPR (pulses per revolution), three orthogonal accelerometers, and three gyroscopes (MEMs LSM6DS3) with a resolution of ±4 g and the gyroscopes ±143 DPS (degrees per second). As the railcar moved, a light sensor TCRT5000 counted the sleepers and determined the speed. The system had a data acquisition system with a 1000 Hz sampling rate.

The railcar ran freely around the track, but only the data from a segment was considered for the analysis. Figure 8 shows the trajectory considered for the study.

Figure 8.

Trajectory for data analysis.

The gyroscope data were used to locate the curvature changes within the data vectors. The Recurrence Plots can be built with any acceleration data; nevertheless, it was decided to use the vertical acceleration since it contains the higher dynamic responses and is more sensitive to tracking defects.

The following section describes the analysis of the measured data and the corresponding Recurrence Plots.

5. Results

Figure 9 shows data obtained with the gyroscope (yaw rotation). This figure indicates the instants when the railcar entered the two types of curves. The data was segmented into short vectors whose length contained data generated when the railcar passed over an average of six sleepers. The data length is equivalent to the fundamental period of the vibration signals and gives a better representation of the dynamic behavior than more extensive data that can contain nosy values. The reason was to reduce the noise on the phase plane. The analysis consisted on:

1. Determining the IMF modes for the acceleration in the vertical direction (segmented vectors)

2. The IMFs were grouped into modes defining the rigid body and vibration motion modes. The rigid body motion modes were added to form a single signal named low frequency, and the vibration modes were added to create a signal called high frequency

3. The new signals were integrated following the shifting process (Eq. 14).

4. Constructing the Recurrence Plots for the low and high-frequency modes

Figure 9.

Gyroscope measurements along the study segment.

The following paragraphs discuss the results based on the type of signal. The acceleration data were divided into the rigid motion modes, namely, low-frequency modes (intrinsic mode functions), and the vibration modes or high-frequency modes (intrinsic mode functions). The distinction came from the results obtained with the Empirical Mode Decomposition method.

Although the Recurrence Plots for produced for the entire trajectory, only the significant results are included in this chapter. Table 2 provides the definition of the segments.

Segment	Description
1	Entrance to the first curve (Figure 10)
2	Segment without defects (Figure 11)
3	Joint defect (Figure 12)
4	Substructure defect (Figure 13)
5	Curvature change (Figure 14)

Table 2.

Description of the segments included in the results.

The first analysis corresponds to the curvature change from a straight segment into a large curve. Figure 10 shows a picture of the track, identifying the beginning of the curvature change. The first sleeper in this segment has the tag “1.” Figure 15 describes the Recurrence Plots for the low- and high-frequency vectors. The low-frequency vector shows a high concentration of points along the diagonal and perpendicular diagonals, whereas the high-frequency vector presents diagonals and vertical and horizontal lines.

Figure 10.

Detail of the track at the beginning of the curve.

The following data correspond to a segment without defects (between sleepers 24 and 30). Figure 11 shows a picture of this segment, and Figure 16 presents the corresponding Recurrence Plots. In this case, both graphs show high concentrations along the main diagonal.

Figure 11.

Detail of the track in a segment without defects.

The defects on the track’s joints produced impacts load on the railcar. These impacts occurred at several places along the track. An example of these defects is shown in Figure 12; the defects are located at numbers 52 and 54. The corresponding recurrence plots are presented in Figure 17. The low-frequency vector shows a concentration along the main diagonal; meanwhile, the high-frequency vector shows a horizontal and a vertical line that connects at a cluster of points in the main diagonal.

Figure 12.

Detail of the track with joint defect.

The discontinuity on the table that holds the track is assumed to be a substructure defect (Figure 13 between sleeper 70 and 71). The dynamic response of the fault produced the recurrence plots shown in Figure 18. The low-frequency vector shows the main diagonal and a second perpendicular diagonal. The high-frequency vector shows perpendicular diagonals and a circular cluster of points that only appear in this plot.

Figure 13.

Detail of the track with joint and substructure defect.

Another dynamic condition related to the track design is the changes in curvature. Although the changes in curvature cannot be considered faults, they impose a different dynamic condition compared to straight or curved segments. Figure 14 shows a picture of the curvature change; in this part, there are no joints or discontinuities in the track. Figure 19 presents the recurrence plots when the railcar passed over this segment. The low-frequency vector has the main diagonal and a second perpendicular one with a large area in the intersegment. The high-frequency vector shows the main diagonal with a cluster of points at one extreme and two lines at the lower and upper corners. This pattern is different from other patterns along with the entire trajectory.

Figure 14.

Detail of the track with a curvature change.

Figure 15.

Change of curvature, a) low-frequency modes (rigid motion), b) high-frequency modes (vibration modes).

Figure 16.

Segment without defects, a) low-frequency modes (rigid motion), b) high-frequency modes (vibration modes).

Figure 17.

Segment with a track discontinuity, a) low-frequency modes (rigid motion), b) high-frequency modes (vibration modes).

Figure 18.

Segment with a track and substructure defects, a) low-frequency modes (rigid motion), b) high-frequency modes (vibration modes).

Figure 19.

Segment with a change in curvature, a) low-frequency modes (rigid motion), b) high-frequency modes (vibration modes).

Other analyses can also predict or detect defects on the track. The Lyapunov exponent could be used to identify chaotic responses. The Fourier transform identifies the dominant frequencies; these analyses were considered for future work where the sensitivity of different analysis techniques could be compared. In this work, only the recurrence quantification analysis was used for comparison.

The recurrence quantification analysis was applied to the five segments. The analysis was applied to the high-frequency vectors since they showed the Recurrence Plots with higher variations. Table 3 shows the results.

Segment	Recurrence rate	Determinism (%)	Longest vertical	LAM (%)	Shannon entropy
1	1668	8.1342	24	16	0.011682
2	2262	10.792	39	26	−0.02885
3	1786	8.6622	25	16.667	−0.006406
4	1792	8.6890	27	18	−0.005191
5	1730	8.4116	26	17.333	0.015306

Table 3.

Recurrence quantification analysis (RQA) (high-frequency vector).

The same analysis for the low-frequency vectors is described in Table 4.

Segment	Recurrence rate	Determinism (%)	Longest vertical	LAM (%)	Shannon entropy
1	2602	12.313	35	23.333	0.0022262
2	2690	12.707	31	20.667	0.0036516
3	2128	10.192	28	18.667	−0.0004128
4	3154	14.783	40	26.667	0.0041652
5	3522	16.430	38	25.333	0.0059131

Table 4.

Recurrence quantification analysis (RQA) (low-frequency vector).

The RQA numerical values describe the dynamic system behavior. The Recurrence Plots obtained from the high-frequency showed more significant variations than the low-frequency data. Segment 2 data is a condition without multiple excitations; thus, it can be considered a reference. It had the lowest Shannon Entropy coefficient and the most considerable recurrence rate, determinism, longest vertical, and LAM.

The results presented in Table 4 are more homogenous, except for Segment 3, which has lower values than the other segments.

The topology cannot be qualitatively analyzed. Thus, the recurrence plots need to be compared to each other. Further work will complement these results.

6. Conclusions

The application of recurrence plots to the dynamic analysis of vehicles, in this case, a scaled-down train, showed that variations on the track condition modified the graphical representation, and the recurrence quantification analysis demonstrated that these variations produced significant changes in the quantification parameters.

For producing the recurrence plots, acceleration measurements were recorded on the train’s platform while traveling along a closed circuit. The vertical acceleration data were divided into the rigid motion modes (identified as low-frequency vectors) and the vibration modes (identified as high-frequency vectors).

The train’s trajectory was divided into short segments to have cleaner graphs, approximately every six sleepers. This segmentation allowed the location of the defects within the time series (acceleration data). It was easier to locate the changes in curvature using the gyroscope data since it precisely showed the instants when the railcar changed direction and entered a curved section.

The recurrence plots were produced from the phase plane of each segment. The phase planes were calculated by separating the acceleration data into intrinsic mode functions using the empirical mode decomposition, applying the delay principle (wave shifting) to each intrinsic mode functions, and reconstructing the signal by adding them again. Before the addition, the intrinsic mode functions were grouped into two, one for the intrinsic mode functions associated with the rigid motion accelerations and the second for high-frequency vibration modes.

The high-frequency vector showed higher sensitivity to excitations. The track had several track joints, changes of curvature, and defects on the base plate. The recurrence plots showed different topologies that were considered “signatures” in all the cases.

The parameter that has a higher sensitivity for identifying defects on the track is the Shannon entropy. It measured the disorder on the recurrence matrices and provided a measurement tool for monitoring variation on the track or the railcar.

Further work should focus on automating the topology analysis and the definition of methodologies for predicting faults and defects using intelligent sensors on the railcar.

Nomenclature

$\mathit{B}$	Matrix containing the system parameters
$\mathit{x}$	Vector containing the state variables
$T$	Kinetic energy
$p$	Linear momentum$p=m\dot{q}$
$V$	Potential energy
$q$	Displacement
$m$	Mass
$\overline{x}\left(t\right)$	Vector of any point in the phase plane
$R_{\mathit{ij}}$	Element of the Recurrence Plot matrix
$\epsilon$	Tolerance value
$N$	Number of points$N=\frac{n\left(n-1\right)}{2}$
$n$	Number of rows (or columns) of the Recurrence Plot matrix
$a$	Acceleration signal
$t$	Time intervals
$r$	Residuals
${\mathit{IMF}}_i$	Intrinsic Mode Functions (IMF)