# Impact Features Extracting Method for a Reciprocating Compressor Based on the ABC-SGMD Model
## 3 Numerical Simulation Verification  
In order to primarily validate the superiority of the ABC-SGMD model, a typical simulated signal is constructed and processed. A simulated impact signal of a reciprocating air compressor under normal operation is generated, and strong white noise is added to simulate the actual signal [27]. The expression of the simulated signal is as follows:  
\[\left\{ \begin{array}{l}s(t)=\sum\limits_{i}A_{i}h(t-iT)+n(t)\\ h(t)=e^{-Ct}\cos(2\pi f_{n}t)\\ A_{i}=1+A_{0}\cos(2\pi f_{r}t)\end{array} \right. \tag{12}\]  
The signal _s(t)_ represents the total signal of the air compressor faults, where \(A_{i}\) is the amplitude of the _i_-th signal, and _h(t)_ is an exponentially decaying sine wave representing the characteristic vibration of the air compressor valve. \(T\) is the signal period, and _n(t)_ is additive Gaussian white noise representing random noise components in the signal. _h(t)_ is a simulated signal of a single compressor valve fault, where \(e^{-Ct}\) is the decaying exponential part, and \(cos(2\pi f_{n}t)\) is a cosine wave with frequency \(f_{n}\) representing the vibration mode of the valve. _A_\(i\) represents the variation of amplitude over time, where \(A_{0}\) is the baseline amplitude, and \(cos(2\pi f_{r}t)\) is a cosine wave with frequency \(f_{r}\) representing the periodic varia  
Figure 2: Algorithm flowchart. tion of the amplitude, possibly due to changes in the vibration modes caused by the air compressor faults. The specific data are shown in Table 2.  
The components and the synthesized simulation signal are shown in Figure 3a, and the spectrum of the components is shown in Figure 3b. Secondly, an added noise component with an amplitude of 0.2 is introduced into the simulated signal to mimic the environmental interference under actual working conditions, and a comparison between the noise-added signal and the original signal is shown in Figure 3c,d. First, the signal is decomposed using the SGMD method, and the corresponding spectrum distribution is displaced individually with Fourier transform, from which the matching relevance of the characteristic frequencies of the decomposed signal and the \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Data** \\ \hline
**Rotational Frequency, \(f_{r}\)** & 30 Hz \\
**Attenuation Coefficient, \(C\)** & 700 \\
**Resonance Frequency, \(f_{n}\)** & 5000 Hz \\
**Maximum Amplitude, \(A_{o}\)** & 0.3 \\
**Repetition Period, \(T\)** & 1/120 s \\
**Sampling Time, \(t\)** & 20 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specific parameters of the analog signal. Figure 3: Time-frequency graph of the analog signal and its components. original frequencies is observed. Additionally, the signal is decomposed using the EMD method, with both methods set to a decomposition order of 6. The decomposition results are compared, as shown in Figure 3. It can be observed that both methods successfully decompose the characteristic frequencies of the simulated signal at 6, 26, and 51 Hz. However, compared to SGMD, EMD does not perform as well when decomposing signals with noise. The 6 Hz signal is submerged in noise, as shown in Figure 3(a). Compared to EMD, SGMD provides clearer decomposition components. However, the number of decomposition modes in SGMD is manually set, and it can be observed from Figure 3(b) that multiple redundant components are decomposed, which greatly affects the efficiency and results of the decomposition. Therefore, the Artificial Bee Colony-optimized Sinusoidal Geometric Mode Decomposition, or ABC-SGMD, is proposed. The number of bee individuals is set to 50, the search range is set to 10, the search step size is 0.1, and the maximum number of iterations is 50.