1. Introduction
Rail flaw detection is crucial due to the increased likelihood of rail damage from external loads as the service life extends. An accident can lead to significant loss of life and property. Currently, ultrasonic detection is the most widely employed technology for rail flaw detection. The fundamental operation of ultrasonic detection involves using a transducer to excite ultrasonic pulses that characterize and locate internal rail flaws through measured quantities such as the amplitude of the echo signal and time. However, a limitation of ultrasonic testing is that it requires point-to-point scanning of the rail, leading to low detection efficiency and a blind spot at the rail base.
UGW inspection is an emerging NDT technology [
1]. Due to the dispersive, multimodal, and attenuating characteristics of UGWs in long range detection, actual sampled guided wave signals often appear as weak signals against a background of strong noise. Scholars have extensively researched signal processing methods for UGWs, including time–frequency analysis [
2], such as short-time Fourier transform [
3], 2D Fourier transform [
4], wavelet transform [
5,
6], Hilbert–Huang transform [
7], empirical modal decomposition [
8], Wigner–Ville distribution [
9], and artificial neural networks [
10], as well as dispersion compensation methods [
11], time inversion focusing methods [
12], etc. Most of the above methods use noise suppression techniques to reduce the noise of the target signal and the noise signal superimposed on the overlapped signal, which can reduce the sensitivity of damage detection. Furthermore, some state-of-the-art fault detection methods [
13,
14,
15] have been proposed using training data collected with ambient noise in industrial processes.
With the development of nonlinear science, some scholars have begun to study weak signal detection methods based on nonlinear systems, and one of the most representative methods is chaos detection based on chaos theory. Chaos theory discusses the unity of complexity, randomness, and certainty that prevails in nature. Lorenz identified the following three characteristics of chaos [
16]: (1) an appearance of randomness, with the actual behavior determined by precise laws; (2) a sensitive dependence on initial conditions; and (3) a sensitive dependence on the intrinsic variability in initial conditions.
Typical models of chaotic dynamics include the Duffing equation [
17], the Van-der-pol system [
18], logistic mapping [
19], and the Loren attractor [
20], among which the Duffing equation is a typical model of chaotic dynamics that has garnered significant attention in the field of signal detection, due to its inclusion of a periodic excitation term. Jalilvand [
21] examined the impact of frequency, phase, and noise on a weak signal in a Duffing oscillator. Nohara [
22] researched the response of the Duffing system when subjected to square wave excitation, assessing its potential for square wave detection. Srinivasan [
23] delved into the dynamics of the Duffing equation under sawtooth wave excitation. As research progressed, scholars started incorporating chaotic determination indices into UGW signal detection, Cheng [
24] utilized the Poincaré map as a chaos indicator to identify pipe damage using the Duffing oscillator. Acknowledging the subjective nature of qualitative chaos indices in determining system motion states, some scientists began exploring quantitative chaos indexes. Zhang [
25] examined the effectiveness of an enhanced Duffing system for UGW detection in pipelines by altering the nonlinear term of the Duffing equation. Hu [
26,
27] detected weak second harmonic signals in plates due to micro-cracks by assessing the maximum Lyapunov exponent of the Duffing equation. Wu [
28] carried out simulation and experimental studies on the UGW detection of pipeline defects using the maximum Lyapunov exponent and the Lyapunov fractional dimensions as phase determination indexes, respectively. Ng [
29] identified hole defects in rails using the maximum Lyapunov exponent of the Duffing equation. Additionally, Cheng [
30] developed a pipeline damage detection method based on the double Duffing equation for detecting weak defect echo signals caused by pipeline defects.
Scholars have conducted extensive research on utilizing chaotic oscillators for weak signal detection. However, challenges persist in applying these methods to detect UGW signals:
The difficulty in the quantitative determination of the system parameters of a chaotic oscillator when it is in the critical state between chaotic and periodic states;
The commonly employed quantitative measures of chaos, such as maximum Lyapunov exponents and Lyapunov dimensions, necessitate constant re-orthogonalization during calculations, leading to computational inefficiency;
Currently, only qualitative assessment and localization of damage can be achieved, as it is difficult to quantitatively characterize defects using chaotic oscillators.
Otherwise, piezoelectric acoustic transducers are commonly used in UGW excitation and reception technology for current rail detection. Although the efficiency of the transducer is high, it is strongly influenced by the coupling conditions, which limits its engineering applicability.Thus, it is important to develop a non-contact UGW transducer.
Given the challenges that current studies have struggled to address, this paper aimed to achieve the following research objectives:
- 1.
To develop a quantitative method to determine the system parameters of a chaotic oscillator in the critical state;
- 2.
To develop a computationally efficient quantitative characterization of chaos;
- 3.
To develop a method for quantitative characterization of rail flaws;
- 4.
To design a non-contact UGW transducer.
The outline of this paper is as follows: In
Section 2, a UGW signal identification model based on the chaotic oscillator is introduced. The model incorporates a method for calculating Kolmogorov entropy (
) through orthogonal triangular decomposition. Within this framework,
is employed as a quantitative index that characterizes the motion state of the chaotic oscillator system. In
Section 3, an electromagnetic transducer is designed which can achieve unidirectional excitation for UGWs at the rail base and rail head, and experimental verification confirmed that the EMAT successfully amplified forward mode signals and suppressed reverse mode signals. In
Section 4, experiments demonstrated that the conventional wavelet transform method is incapable of detecting weak UGW signals reflected by small-size defects. Furthermore, this study’s proposal to use the Kolmogorov entropy of the Duffing oscillator for identifying rail damages was experimentally validated, highlighting its effectiveness in damage identification.
Section 5 provides a summary of this study.
3. Design of an Electromagnetic Acoustic Transducer for UGWs
The ultrasonic acoustic transducers in NDT based on UGWs mainly include piezoelectric, electromagnetic, air-coupled, and laser acoustic transducers. The piezoelectric acoustic transducer is currently the most widely used type, but the disadvantage is that it is greatly affected by the coupling conditions, thus limiting its applicability to engineering sites. Therefore, designing a non-contact type transducer is of great importance for the realization of rail flaw detection.
Electromagnetic acoustic transducer(EMAT) has the advantage of good design-ability and lower material costs compared to the lower conversion efficiency of piezoelectric transducers.The principle of the excitation of the Lorentz force-based EMAT is shown in
Figure 6.
Due to the complexity of the rail cross-section, it is difficult to excite a single guided wave mode in the rail, and when the guided wave encounters the boundary, the reflection will undergo a complicated mode conversion. If the guided wave propagates from both sides in the rail, the transducer will receive the reflected wave from both sides, which will greatly increase the difficulty of the subsequent signal identification and feature extraction. Therefore, in this study, a unidirectional excitation EMAT is designed to achieve enhancement of the forward guided waves modal and suppression of the reverse modal. The basic components of the EMAT are shown in
Figure 7. Amplification in the forward direction and suppression in the backward direction of guided-wave signals is achieved by the arrangement of two meander coils spaced apart from each other and fed with electrical pulses with a time delay, and the arrangement of the two coils is shown in
Figure 8. As is shown in
Figure 9, the distance between the adjacent wires of the coil is
(where
is the wavelength of the guided waves), so that the guided waves generated by each wire gain each other. The distance between coil 1 and coil 2 is
, and a reverse current is applied with a time delay of
(where
T is the cycle of the guided waves) to achieve amplification of the forward UGW signals and suppression of the reverse signals.
The effectiveness of the EMAT in exciting UGWs was verified through an experiment at the head and base of the rail. The experimental scheme is shown in
Figure 10. The effect of the unidirectional excitation of the double meander coils was verified by comparing the guided wave signals received at equidistant positions on both sides of the EMAT, and the experimental results are shown in
Figure 11. The experimental results showed that the EMAT designed in this study could achieve the amplification of the forward guided wave signals and the suppression of the backward guided wave signals at the rail head and at the rail base.
Thus, an EMAT was developed to enable unidirectional excitation of UGW signals at both the rail base and rail head. The strategic placement of two meander coils, separated at a specific distance, allows the amplification of guided-wave signals in the forward direction, while simultaneously suppressing them in the backward direction. This effect is achieved through feeding the coils with electrical pulses that are intentionally time-delayed.
5. Conclusions and Discussion
This study addressed the challenge of identifying weak UGW signals in a strong noise background in rail flaw detection by proposing a damage identification method based on chaotic oscillators. Initially, a mathematical model for detecting UGW signals using the Duffing oscillator was introduced. The motion state of the Duffing system was characterized by the Kolmogorov entropy, and a formula for calculating this entropy was established. Subsequently, an electromagnetic UGW transducer was developed to amplify forward UGW modes and suppress unidirectional UGW modes. The efficacy of the proposed model and transducer in rail damage detection was then validated through experimental testing. The main conclusions of this study are analyzed as follows:
- 1.
A UGW signal identification model based on the chaotic oscillator was established. The approach integrates the UGW response into the critical state of the Duffing system to serve as a disturbance control variable. This incorporation leads to alterations in the system’s motion state through the exploitation of the parameter disturbance sensitivity characteristic of chaotic systems and the traversal of chaotic motion. By evaluating the system’s motion state both pre- and post-introduction of the UGW response, the identification of ultrasonic guided wave signals can be realized. This methodology encapsulates the fundamental concept of employing chaotic systems for discerning faint guided wave signals in NDT applications centered on UGWs;
- 2.
A method for calculating Kolmogorov entropy based on orthogonal triangular decomposition was proposed. Kolmogorov entropy is a measure that describes the degree of chaos in a dynamical system and represents the average information growth of the system. can be used as a quantitative characterization factor of the motion state of the chaotic oscillator system. When , the system is in a state of periodic motion, and when , the system is in a chaotic state. This method eliminates the need for reconstructing the phase space, thereby improving the efficiency of calculating Kolmogorov entropy.
- 3.
An electromagnetic transducer was designed that can achieve unidirectional excitation for UGWs at the rail base and rail head. Amplification in the forward direction and suppression in the backward direction of guided-wave signals was achieved though the arrangement of two meander coils spaced apart from each other and fed with electrical pulses with a time delay.The distance between adjacent wires of the coil was , so that the UGW generated by each wire gained each other. The distance between coil 1 and coil 2 was , and a reverse current was applied with a time delay of to achieve amplification of the forward guided wave signal and suppression of the reverse signal. Experimental verification confirmed the effectiveness of the EMAT in producing the desired effects mentioned above;
- 4.
The experimental results indicated the challenge in effectively identifying the weak UGW echoes caused by small sized damage using time-domain signals. Although the traditional signal processing method based on wavelet transform showed improved denoising capabilities, it continued to struggle in effectively distinguishing the weak UGW signals.
- 5.
The width of notches at both the rail base and rail head were directly proportional to the , hence Kolmogorov entropy can serve as a quantitative characterization index of rail damage. The experimental results demonstrated the effectiveness of the detection system based on a chaotic oscillator in detecting weak UGW signals. Specifically, the Duffing oscillator system was capable of detecting a 0.46 mm notch at the rail base and a 1.78 mm notch at the rail head.
In summary, this study proposed a method for detecting rail flaws using the Kolmogorov entropy of a chaotic oscillator based on UGWs. This method aims to accurately locate and quantitatively characterize defects at the rail base and rail head to enhance the sensitivity of rail flaw detection. However, in engineering applications, the method described above may encounter limitations, particularly when dealing with large rail damage. In such cases, the guided wave within the damaged area may undergo mode conversion. The presence of multiple damages on the rail further complicates the situation, making it challenging to differentiate between the modal conversion signal of the initial damage and the reflection signal produced by subsequent damages. Hence, the study of the specific interaction between rail flaws and UGWs remains a key research direction for the future application of the method proposed in this paper.