## 4 The Analysis of Multi-Time-Scale Chaotic Characteristics Based on IVMD  
The vibration characteristics of the pumping station pipeline are different from those of the general pipeline, which is mainly reflected in the influence of the pumping station unit on the vibration of the connecting pipeline. The vibration sources are primarily composed of low-frequency water pulsations caused by the pipeline flow and blade frequency, rotation frequency and frequency doubling produced by the unit's operation [23]. Taking the vibration response of the specific point (point 1 Z-axis vibration under condition 4) as an example, the spectrum analysis is shown in Figure 10. Concerning the author's previous article [23; 24], 20, 40 and 60 Hz are the blade frequency, the rotation frequency and the frequency doubling, respectively, and 0.5 Hz is the low-frequency water pulsation. Spectrum analysis shows that the frequency band of the vibration excitation caused by water pulsation (0.5 Hz) is relatively wide. The wide-peak power spectrum is the typical characteristic of the chaotic system [25; 26]. The pipeline vibration excitation produced by the unit's operation (20 Hz, 40 Hz, and 60 Hz) corresponds to the peak power spectrum and has high periodicity. Therefore, it is speculated that the chaotic characteristics of the pipeline are mainly caused by water pulsation, while the unit vibration masks the chaotic characteristics of the pump station pipeline. The excitation components of different time scales must be effectively separated to clarify the vibration excitation with chaotic characteristics. As a new signal decomposition method, IVMD can adaptively decompose a signal into a series of IMFs with different scale characteristics. Therefore, the IVMD method is used to identify the vibration excitation that causes the chaotic characteristics of the pipeline. The multi-time scale chaotic vibration response characteristics of the specific point (point 1 Z-axis vibration under condition 4) are analysed. The modal parameters \(K\) of IVMD are determined as 4 by the MI method. Four IMFs are obtained by the IVMD decomposition of point 1 Z-axis vibration response under condition 4. Figure 11 is the time history of decomposed IMFs. Mutual information value: IMF1 is 1.000, IMF2 is 0.025, IMF3 is 0.038, IMF4 is 0.0661. It can be seen that the normalized mutual information values of the IMFs are all above the threshold of 0.02, which meets the decomposition requirements. Figure 11 shows that IVMD can sequentially decompose the original vibrational response to obtain four IMFs with increasing frequency. The frequencies from IMF1 to IMF4 correspond to four major frequency bands in the original response spectrum: 0.5, 20, 40 and 60 Hz, respectively, and the decomposition effect is improved. Then the chaotic characteristics of the decomposed IMFs are analysed using the saturation correlation dimension and the largest Lyapunov exponent. The calculation process of typical IMF chaotic eigenvalues is shown in Figure 12. From Figure 12a,b the saturation correlation dimension \(D_{2}\) of IMF1 is 1.115, and the largest Lyapunov exponent \(\lambda_{1}\) is 0.0774, indicating that IMF1 has prominent chaotic characteristics. The near-linear region of the \(D_{2}\) logarithmic curve cannot be found from IMF2 to IMF4; these components have no chaotic characteristics. Due to space limitations, only the slope of the IMF2 double logarithmic curve is given in Figure 12c. Figure 11: Time histories of IMFs decomposed by IVMD. Figure 10: Spectrogram of point 1 Z-axis vibration response under condition 4 (**a**) Holistic drawing, and (**b**) Partial enlarged drawing. By comparing the results of the chaotic eigenvalues of the IMFs with those of the vibrational response before decomposition, it can be concluded that:  
1. IMF1, which represents the water pulsation excitation, the saturated correlation dimension 1.115 is a fractal dimension, and the largest Lyapunov exponent is 0.0774 greater than zero, has prominent chaotic characteristics.