**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 singlefrequency 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) [6–9]. 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. [12–14] 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 crosscorrelation 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

### *Perspective Chapter: Predicting Vehicle-Track Interaction with Recurrence Plots DOI: http://dx.doi.org/10.5772/intechopen.105752*

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.
