## 3 Methodology  
The reciprocating compressor mechanism is a complex system with multiple sources of excitation, leading to a scenario where pertinent information about the piston rod and its fracture condition is obscured by signals from the operation of other components. To overcome this challenge, signal processing techniques are used to first identify the signal band associated with piston rod fractures. The relevant signal within this band is then extracted and the parameters which are sensitive to crack growth are calculated. Figure 8 outlines the research methodology used in this work to identify cracks in the piston rod.  
Figure 7: Acquisition of signals. Installation of the AE sensor **(a)**; monitoring of the TDC signal **(b)**.  
First, the AE signal is segmented into the crank angle domain from 0 to 360\({}^{\circ}\) based on the TDC signal. Next, kinetic and time-frequency domain analyses are performed to identify the frequency bands and events correlated with the crack. The continuous wavelet transform [53] is used to analyse the signal in both the time and frequency domains to determine the sensitive frequency band and crack-associated events. The sub-band AE signal is then extracted by bandpass filtering the original signal.  
The AE is recorded as a series of voltage amplitudes at uniform time intervals, which presents a challenge to direct analysis. Consequently, several characteristic parameters are used to describe the AE signal. Parameters describing the characteristics of the AE waveform include the amplitude, rise time, duration, counts/ringing counts, etc., as shown in Figure 9.  
The envelope is an indicator of variations in the signal amplitude and waveform. The energy of the AE signal can be determined by calculating the area enclosed by the envelope  
Figure 8: Schematic diagram of the research methodology.  
Figure 9: Variation curve of the envelope margin coefficient of the AE signal.  
waveform and the coordinate axis. The Hilbert amplitude demodulation [54] method is then integrated to obtain the envelope of the corresponding sub-band signal.  
Assuming that the measured signal is \(x(t)\), then, from the Hilbert transform, we can know:  
\[x(t)=-\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{\hat{x}(t)}{t-\tau}d\tau=- \frac{1}{\pi}\hat{x}(t), \tag{4}\]  
\[\hat{x}(t)=-\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{x(t)}{t-\tau}d\tau= \frac{1}{\pi t}x(t), \tag{5}\]  
Equations (4) and (5) are Hilbert transform pairs, so the analytical signal of \(x(t)\) is:  
\[g(t)=x(t)+j\hat{x}(t), \tag{6}\]  
The envelope of the signal \(x(t)\) is represented by the amplitude of the analytical signal \(g(t)\). The AE energy is then calculated according to Equation (7).  
\[X_{E}=\int_{0}^{T}g(t)dt, \tag{7}\]  
Finally, considering the inclusion of crack growth produces high-frequency signals of short duration and high amplitude. After obtaining the signal envelope for one cycle, the impulse factor is calculated by Equations (8)-(10) to reflect the mutations in the signal and identify the crack.  
\[X_{I}=\frac{x_{p}}{\overline{x}}, \tag{8}\]  
where \(x_{p}\) is the max value of the signal and \(\overline{x}\) is the average value of the signal.  
\[x_{p}=max|x(n)|, \tag{9}\]  
\[\overline{x}=\frac{1}{N}\sum_{i=1}^{n}x(n), \tag{10}\]  
The peak-to-peak values, \(X_{p}\), are calculated as follows.  
\[X_{p}=max|x(n)|-min|x(n)|, \tag{11}\]  
The AE count, \(X_{c}\), is obtained by counting the number of times the signal exceeds the threshold in a cycle. Based on the above calculations, a four-dimensional feature vector consisting of the AE energy, counts, peak-to-peak values, and impulse factor \(\left\{X_{E},X_{c},X_{p},X_{I}\right\}\) can be developed. Finally, based on the relationship between the feature parameters and the crack growth rate, the evaluation mode of the crack propagation rate can be established.