**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].

$$a(t) = \sum\_{1}^{n} \text{IMF}\_{i}(t) + r(t) \tag{11}$$

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**.

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:

$$\mathbf{a}(t) = \mathbf{a}\_1 \cos \left(\alpha\_1 t\right) + \mathbf{a}\_2 \cos \left(\alpha\_2 t\right) + \dots + \mathbf{a}\_n \cos \left(\alpha\_n t\right) \tag{12}$$

The indefinite integral is:

$$\int a(t)dt = -\frac{\mathbf{a}\_1}{a\alpha\_1}\text{sen}(a\rho\_1 t) - \frac{\mathbf{a}\_2}{a\alpha\_2}\text{sen}(a\rho\_2 t) + \dots - \frac{\mathbf{a}\_n}{a\alpha\_n}\text{sen}(a\rho\_n t) \tag{13}$$

Or

$$\int a(t)dt = \frac{\mathbf{a\_1}}{a\_1^2}\cos\left(a\_1t + \tau\_1\right) + \frac{\mathbf{a\_2}}{a\_2^2}\cos\left(a\_2t + \tau\_2\right) + ... + \frac{\mathbf{a\_n}}{a\_n^2}\cos\left(a\_nt + \tau\_n\right) \tag{14}$$

**Figure 3.** *EMD procedure.*

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

#### **Figure 4.**

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

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

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

**Figure 5.** *Original and shifted signals produced by the EMD and shifting process.*
