## 3 Chaotic Characteristics Analysis of Pipeline Vibration Response  
The pipe material is stainless steel. Model 891-2 vibration sensors are used in the test, which are divided into four grades: small speed, medium speed, large speed and acceleration. So speed sensors are used in this test. Taking the No. 2 pressure pipeline of the Jingdian Project pumping station No. 3 as a research objective, the No. 4 and No. 5 units of a 12005-56 horizontal centrifugal pump are connected with the branch pipe. Six measurement points are selected on the main pipe and two branches of the pipeline. Each point is equipped with vibration sensors in X, Y and Z directions. The measuring points are arranged as shown in Figure 2.  
In the prototype test, four working conditions were selected to collect the vibration responses of the pipeline. The descriptions of each working condition, sampling time and sampling frequency are shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multirow{2}{*}{**Description of Working Conditions**} & \multirow{2}{*}{**Sampling Time/s**} & \multirow{2}{*}{
\begin{tabular}{c} **Sampling** \\ **Frequency/Hz** \\ \end{tabular} } \\  & & \\ \hline
1 & No. 4 unit stable operating & 900 & 512 \\
2 & No. 4 unit opening & 1800 & 512 \\
3 & No. 4 unit closing & 1800 & 512 \\
4 & No. 4 and 5 units stable operating & 900 & 180 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Four working conditions. Figure 2: Layout of pipeline measuring points (**a**) Field test of pipeline, and (**b**) Measuring points layout (Note: 1-18 is the sensor number). The velocity-time history of points under typical conditions is shown in Figure 3. The chaotic characteristic analysis of the vibration responses under different conditions is carried out as follows:  
First, the reconstruction of the phase space of the time series is performed, that is, the calculation of the time delay \(\tau\) and embedding dimension \(m\). The CAO method essentially uses the minimum error method to determine the embedding dimension, which was proposed by [PERSON] in 1997. This paper calculates \(\tau\) by the autocorrelation function method and chooses the CAO method to obtain \(m\)[21]. The calculation process of \(\tau\) and \(m\) is illustrated by taking the Z-axis vibration of point 1 under condition 4 as an example. In the process of calculating \(\tau\) by the autocorrelation function method, when the value drops to 1-1/e of the initial value, the corresponding time delay is \(\tau\). The result of the autocorrelation function is shown in Figure 4.  
After obtaining \(\tau\), the embedding dimension is determined by the CAO method. \(E_{1}(m)\) represents the minimum embedding dimension. \(E_{2}(m)\) represents the characteristics of time series. When \(E_{1}(m)\) obviously no longer changes with the increase, and the \(E_{2}(m)\) value tends towards 1, the corresponding \(m\) is the optimal embedding dimension. From Figure 5, we can see that the optimal embedding dimension \(m\) of point 1 Z-axis vibration under condition 4 is 11. Figure 4: \(\tau\) Calculation of point 1 Z-axis vibration under condition 4. Figure 3: Velocity time history of points under typical conditions (**a**) Z-axis vibration of point 1 under condition 2, and (**b**) Z-axis vibration of point 1 under condition 4. The G-P algorithm [22] and the small data sets are chosen to calculate the saturation correlation dimension and the largest Lyapunov exponent. Two types of indexes are used to analyse the chaotic characteristics of time series. The G-P algorithm is a chaotic eigenvalue calculation method proposed by [PERSON] and [PERSON] to calculate the saturation correlation dimension \(D_{2}\). The embedding dimension is selected as \(m=2,4,6,\cdots,20\) and the \(\tau\) has been calculated above. According to the correlation function relation in Equation (3), the \(\log_{2}C(r)\sim\log_{2}r\) double logarithmic relation graph of different \(m\) is plotted, respectively.