**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:


**Figure 9.** *Gyroscope measurements along the study segment.*

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

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

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.

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.

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.


#### **Table 2.**

*Description of the segments included in the results.*

**Figure 10.** *Detail of the track at the beginning of the curve.*

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

**Figure 11.** *Detail of the track in a segment without defects.*

**Figure 12.** *Detail of the track with joint defect.*

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.

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.

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

**Figure 13.** *Detail of the track with joint and substructure defect.*

**Figure 14.** *Detail of the track with a curvature change.*

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.

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

**Figure 15.** *Change of curvature, a) low-frequency modes (rigid motion), b) high-frequency modes (vibration modes).*

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

#### **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).*


**Table 3.** *Recurrence quantification analysis (RQA) (high-frequency vector).*


#### **Table 4.**

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

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.

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

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.

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

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.
