**5. Track condition diagnosis using time-frequency analysis**

#### **5.1 Effect of track faults on time-frequency plane**

#### *5.1.1 Continuous wavelet transform (CWT)*

The wavelet transform is well-known technique for analyzing nonstationary signals [26, 27]. A CWT gives simultaneous detection of the frequency and time characteristics for a nonstationary signals using a wavelet *ψ*, which is a function of zero average:

$$\int\_{-\infty}^{\infty} \varphi(t)dt = 0.\tag{2}$$

The CWT is calculated using the mother wavelet *ψ*ð Þ*t* as

$$\mathcal{W}\_{\Psi}(a,b) = \int\_{-\infty}^{\infty} \frac{1}{\sqrt{a}} \boldsymbol{\mu}^\* \left(\frac{t-b}{a}\right) \boldsymbol{\varkappa}(t) dt,\tag{3}$$

where *a* and *b* correspond to the dilatation and location parameters, respectively.

Eq. (3) translates a source signal *x*(*t*) using the mother wavelet transformed by a time shift *b* in time, and by 1/*a* in frequency. *ψ* <sup>∗</sup> indicates the complex conjugate of *ψ*.

In this study, the *Morlet* wavelet, which has a good performance between localization of time and frequency, was used [5, 28].

The CWT is subject to the uncertainty principle on time-frequency domain. In case of fault detection using CWT, if we are focusing on frequency related on the fault, the time when the fault occurred will be vague. If we are focusing on the time when the fault occurred, the frequency will be spread widely on the time-frequency plane.

#### *5.1.2 Hilbert-Huang transform (HHT)*

The Hilbert-Huang transform (HHT) has been proposed for analyzing nonlinear and nonstationary data by Huang *et al.* [29]. This method is not subject to the uncertainty principle on time-frequency domain mentioned above. Thus, more localized fault detection is possible.

The HHT consists of two operations. The first operation is the empirical mode decomposition (EMD) and the second operation is Hilbert transform.

The EMD operation breaks time domain data into intrinsic oscillatory modes called intrinsic mode functions (IMFs). The second operation is the Hilbert transform. Instantaneous amplitude, instantaneous phase, and instantaneous frequency of the IMFs are obtained by the Hilbert transform.

An IMF must satisfy the following requirements: (1) the number of local extrema and the number of zero crossings must either equal or differ by at most one. (2) the mean value of the envelopes of local maxima and local minima is zero at any point.

*Track Condition Monitoring Based on In-Service Train Vibration Data Using Smartphones DOI: http://dx.doi.org/10.5772/intechopen.111703*

For extracting IMFs from the original signal, the iterative sifting process is applied. Once the first IMF is calculated, it is subtracted from the original signal to obtain a residual value. The EMD operation is applied again to the residual. This process repeats until the residual no longer contains any oscillation modes.

The original signal, *s*(*t*), can be expressed by EMD operation as:

$$\varkappa(t) = \sum\_{i=1}^{m} \varkappa\_i(t) + R(t),\tag{4}$$

where *xi*ð Þ*t* is the *i*th IMF and *R*(*t*) is a residual.

Followed by the EMD operation, the analytical signal *zi*ð Þ*t* is constructed on each IMFs component by:

$$
\varpi\_i(t) = \varkappa\_i(t) + j\wp\_i(t) = \mathfrak{a}\_i(t)e^{j\theta\_i(t)},\tag{5}
$$

where *yi* ð Þ*t* is a Hilbert transform of *xi*ð Þ*t* calculated by:

$$y\_i(t) = \frac{1}{\pi} \text{PV} \int\_{-\infty}^{\infty} \frac{\varkappa\_i(t)}{t - \tau} d\tau,\tag{6}$$

where PV shows Cauchy principal value.

Instantaneous amplitude, *ai*ð Þ*t* , and instantaneous frequency, *ωi*ð Þ*t* , can be obtained from the analytical signal *zi*ð Þ*t* as:

$$a\_i(t) = \sqrt{\varkappa\_i(t)^2 + \jmath\_i(t)^2},\tag{7}$$

$$o\_i(t) = \frac{d\theta\_i(t)}{dt},\tag{8}$$

where

$$\theta\_i(t) = \tan^{-1} \left( \frac{\mathcal{y}\_i(t)}{\mathcal{x}\_i(t)} \right). \tag{9}$$

This data-driven method is highly adaptive. However, intrinsic mode functions (IMFs) obtained by EMD strongly depend on the data itself. Thus, a small change in the data will appear on different decomposition level.

#### **5.2 Track condition diagnosis for regional railway lines**

#### *5.2.1 Regional railway A*

Time-frequency analysis was performed on the measured data to identify and evaluate the detailed location and type of track fault. When a train runs on a track where a fault exists, characteristic vibration corresponding to the type of track fault occurs. Therefore, one could identify the type of track fault and location of its occurrence by analyzing the time-frequency plane of measured car-body vertical acceleration.

**Figure 11.** *Time-frequency analysis of date measured in December 23, 2021 in railway A.*

**Figure 11** shows the time-frequency analyses, CWT and HHT of data measured in December 23, 2021 in Railway A (line length: 30.5 km, Stations:17, Max. speed: 85 km/ h). The data used for this analysis are data measured using Device G.

It can be seen from **Figure 11** that a high-frequency vibration appeared at 27000 m, which was caused by the joint depression [4]. Whereas a large vibration can be seen in low frequency in a 27,015–27,025 m section. This is caused by longitudinallevel track irregularities.

#### *5.2.2 Regional railway B*

**Figures 12** and **13** show the time-frequency analyses, CWT and HHT of data measured in June and October 2022, respectively. In June 2022, vibrations due to longitudinal-level irregularity were detected at 1–2 Hz between 600 and 700 m but were no longer detected in October 2022 due to track irregularity correction. The data used for this analysis are data measured using Device G on Regional Railway B (line length: 6.4 km, Stations: 8, Max. speed: 40 km/h) in June 2022 and October 2022.

**Figure 14** displays a photograph of the track section between 600 and 700 m in October 2022; the ballast was newly replenished, line maintenance work was carried out, and the longitudinal-level irregularity was eliminated. Thus, by performing timefrequency analysis using data measured by a smartphone, the type and location of track fault can be identified, and the effects of track irregularity correction can be confirmed.

*Track Condition Monitoring Based on In-Service Train Vibration Data Using Smartphones DOI: http://dx.doi.org/10.5772/intechopen.111703*

**Figure 12.**

*Time-frequency analysis of date measured in June 2022 in railway B.*

**Figure 13.** *Time-frequency analysis of date measured in October 2022 in railway B.*

**Figure 14.** *Section where track maintenance work done.*
