Two-dimensional joint kurtosis analysis method for vibration detection of mixed-flow pump

Abstract

The invention discloses a two-dimensional joint kurtosis analysis method for vibration detection of a mixed-flow pump, and the method comprises the specific steps: collecting a radial vibration displacement signal of a main shaft and the pressure pulsation of an impeller outlet through a laser vibration meter and a pressure pulsation sensor, and solving the two-dimensional joint kurtosis based on the two collected signals; and evaluating the running state of the mixed-flow pump by using the standard normal distribution and the two-dimensional joint kurtosis reference value. According to the method, the problems of low precision and poor reliability when a traditional vibration detection method is used for recognizing a complex fault mode are solved.

Description

Technical Field

[0001] The invention belongs to the technical field of water jet propulsion and relates to a two-dimensional joint kurtosis analysis method for vibration detection of a mixed flow pump. Background Art

[0002] In water jet propulsion, mixed flow pumps are widely used due to their excellent performance. The operating stability of this type of pump is a key factor in ensuring the performance of water jet propulsion systems in terms of noise control and overall stability. The causes of vibration problems in mixed flow pumps are complex and diverse, including mechanical structural imbalance, fluid dynamics factors, and changes in the operating environment. Existing vibration detection methods mainly rely on signal analysis of a single measuring point, such as time domain analysis, frequency domain analysis, and time-frequency domain analysis. These methods can provide effective diagnostic information under certain simple working conditions, but their accuracy and reliability are insufficient when dealing with complex working conditions and nonlinear vibration problems. Especially in the vibration detection of transition processes such as mixed flow pump startup and speed change, the fluid characteristics and mechanical characteristics of the pump system are coupled with each other, making the vibration signal present a high degree of nonlinearity and non-stationary characteristics. Traditional single-measurement-point signal analysis methods are difficult to accurately distinguish and diagnose these complex vibration problems. The two-dimensional joint kurtosis analysis method can more comprehensively reflect the characteristics of the vibration signal by combining the time domain signals of two measuring points, and is particularly suitable for processing vibration signals under complex working conditions such as mixed flow pumps. Therefore, developing a mixed flow pump vibration detection technology based on the two-dimensional joint kurtosis analysis method has important practical significance and application value for improving the operating stability and reliability of the pump system. Summary of the invention

[0003] The purpose of the present invention is to provide a two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection, which solves the problems of low accuracy and poor reliability of traditional vibration detection methods when identifying complex fault modes.

[0004] The technical solution adopted by the present invention is a two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection. The specific process is as follows: a laser vibrometer and a pressure pulsation sensor are used to collect the radial vibration displacement signal of the main shaft and the impeller outlet pressure pulsation, and the two-dimensional joint kurtosis is solved based on the two collected signals; the standard normal distribution and the two-dimensional joint kurtosis reference value are used to evaluate the operating status of the mixed flow pump.

[0005] The present invention is also characterized in that:

[0006] The specific steps include:

[0007] Step 1, start the mixed flow pump vibration data acquisition system, and the water flows through the inlet flow channel, impeller, guide vane and outlet flow channel in sequence;

[0008] Step 2: Use a laser vibrometer to collect the main shaft vibration displacement signal, and use a pressure pulsation sensor to collect the outlet pressure pulsation signal of the impeller, so as to obtain the time series x1 of the main shaft vibration displacement signal and the outlet pressure pulsation signal x2 of the impeller;

[0009] Step 3, using a two-dimensional kernel density function to calculate the joint probability density function p(x1, x2) of the time series x1 of the vibration displacement signal and the impeller outlet pressure pulsation signal x2;

[0010] Step 4, integrate p(x1, x2) with respect to x2 in the range of (-∞, +∞) to obtain the marginal probability density P(x1) of x1; integrate p(x1, x2) with respect to x1 in the range of (-∞, +∞) to obtain the marginal probability density P(x2) of x2;

[0011] Step 5, the individual fourth-order moments of x1 and x2 and the mixed fourth-order moment between them;

[0012] Step 6, based on step 4 and step 5, calculate the two-dimensional joint kurtosis of x1 and x2;

[0013] Step 7: Calculate the maximum information coefficient of the time series x1 of the vibration displacement signal and the impeller outlet pressure pulsation signal x2, and determine the reference value K of the two-dimensional joint kurtosis through the correlation coefficient. ab ;

[0014] Step 8: Obtain the two-dimensional joint kurtosis reference value K of signals x1 and x2 ab After that, it will exceed the reference value K ab The two-dimensional joint kurtosis is named positive joint kurtosis, which is lower than the reference value K ab The joint kurtosis is negative.

[0015] In step 4, the marginal probability densities P(x1) and P(x2) are calculated using the following formulas (1) and (2):

[0016]

[0017]

[0018] In step 5, the fourth-order moments of x1 and x2 and the mixed fourth-order moments between the two are calculated by the following formulas (3) to (7);

[0019] μ1=∫∫(x1-E(x1)) 4 p(x1,x2)dx1dx2 (3)

[0020] μ2=∫∫(x2-E(x2)) 4 p(x1,x2)dx1dx2 (4)

[0021] μ3=∫∫(x1-E(x1)) 4 (x2-E(x2)) 4 p(x1,x2)dx1dx2 (5)

[0022]

[0023]

[0024] Among them, E(x1) and E(x2) represent the means of x1 and x2 respectively, μ1 and μ2 represent the individual fourth-order moments of x1 and x2 respectively, and μ3 represents the mixed fourth-order moment between x1 and x2.

[0025] In step 6, the two-dimensional joint kurtosis of x1 and x2 is calculated by the following formulas (8) to (10);

[0026]

[0027]

[0028]

[0029] in, and They represent the variance of x1 and x2 respectively, and K represents the two-dimensional joint kurtosis between them.

[0030] The specific process of step 7 is:

[0031] Step 7.1: Divide the time series signals x1 and x2 into interval sets {I1, I2, ..., I k} and {J1, J2, ..., J k}, each interval I i and J j They represent the division of signals x1 and x2 within a specific range of values, calculate the probability of each interval, and construct a joint distribution table based on the calculated probabilities, as follows:

[0032]

[0033]

[0034]

[0035] Among them, the probability P(I i ) means x1 falls within interval I i Possibility; probability P(J j ) means x2 falls within the interval J j The possibility of joint probability P(I i , Jj ) means x1 falls within interval I i And x2 falls in interval J j possibility;

[0036] Step 7.3, using the joint distribution table constructed in step 7.2 to calculate the mutual information I(x1; x2) between the two signals, as follows:

[0037]

[0038]

[0039]

[0040] I(x1;x2)=H(x1)+H(x2)-H(x1,x2) (17)

[0041] Among them, H(x1) represents the entropy of signal x1, which is used to measure x1 in each interval I i The uncertainty of the probability distribution on ; H(x2) represents the entropy of the signal x2, which is used to measure the probability distribution of x2 in each interval J j The uncertainty of the probability distribution on ; H(x1,x2) represents the joint entropy of the two signals, which measures the uncertainty of the combined distribution of the two signals;

[0042] Step 7.4: According to the normal distribution characteristics, the signals x1 and x2 are divided into different intervals {k1, k2, …, k m}, for different interval partitioning schemes {k1, k2, …, k m}, execute step 7.3 to calculate different mutual information values Finally, the maximum mutual information value is selected as the maximum information coefficient, and the maximum information coefficient is used as the correlation coefficient δ of the time series signals x1 and x2;

[0043]

[0044] Step 7.5, generate two independent standard normal distribution signals z1 and z2, and then transform these two signals into standard normal distribution signals x3 and x4 with specified correlation δ by linear combination method:

[0045] x3=z1 (19)

[0046]

[0047] Step 7.6, replace x1 with the standard normal distribution signal x3, and replace x2 with the standard normal distribution signal x4, and then execute steps 4 to 6 to obtain the two-dimensional joint kurtosis K between the two standard normal distribution signals x3 and x4.ab , K ab It is the two-dimensional joint kurtosis reference value of the time series x1 of the main shaft vibration displacement signal and the impeller outlet pressure pulsation signal x4.

[0048] The beneficial effect of the present invention is that the present invention improves the accuracy and reliability of mixed flow pump vibration detection by adopting a two-dimensional joint kurtosis analysis method based on the time domain signals of two measuring points. The maximum information coefficient is used to effectively capture the nonlinear and linear relationships between signals, making the identification of abnormal vibration characteristics more accurate. The operating status of the mixed flow pump is evaluated using the standard normal distribution and the two-dimensional joint kurtosis reference value, providing a reliable diagnostic basis, which helps to timely discover potential problems and perform maintenance and optimization. This method is not only suitable for vibration detection of mixed flow pump systems, but can also be extended to vibration analysis of other complex mechanical systems. It has strong versatility and practicality, thereby improving the operating stability and service life of the equipment and reducing maintenance costs. By jointly analyzing the data of the mixed flow pump main shaft vibration measuring point and the impeller outlet pressure pulsation measuring point, the present invention can improve the accuracy and sensitivity of fault detection, timely identify and prevent potential faults, ensure the normal operation of the equipment and extend its service life. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 It is a structural schematic diagram of a mixed flow pump vibration and pressure pulsation data acquisition test system used in a two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection of the present invention;

[0050] Figure 2 It is a position relationship diagram of the laser vibrometer and the pressure pulsation sensor respectively used in the two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection of the present invention and the measuring point A and the measuring point B;

[0051] Figure 3 It is a flow chart of a two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection according to the present invention;

[0052] FIG4(a) is a vibration displacement signal of measuring point A collected by a laser vibrometer in the two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection of the present invention.

[0053] FIG4( b ) is a pressure pulsation signal of the impeller outlet at the measuring point B collected by the pressure pulsation sensor in the two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection of the present invention;

[0054] Figure 5 is a two-dimensional joint kurtosis analysis result in a two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection of the present invention;

[0055] Figure 6It is the relationship between the two-dimensional joint kurtosis reference value and the correlation coefficient in the two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection of the present invention.

[0056] In the figure, 1. inlet flow channel, 2. impeller, 3. guide vane, 4. outlet flow channel, 5. pressure pulsation sensor, 6. drive motor, 7. laser vibrometer, 8. main shaft, 9. data acquisition card, 10. computer. DETAILED DESCRIPTION

[0057] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0058] Example 1

[0059] The two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection of the present invention adopts a mixed flow pump vibration and pressure pulsation data acquisition test system such as Figure 1 As shown, the system includes, in accordance with the water flow direction, an inlet flow channel 1, an impeller 2, a guide vane 3, and an outlet flow channel 4. Figure 2 As shown, two measuring points, namely measuring point A and measuring point B, are set in the mixed flow pump test system, and their real-time vibration signals and pressure pulsation signals are collected respectively for analysis. The mixed flow pump is controlled by a driving motor 6. For any mixed flow pump device including an inlet flow channel 1, an impeller 2, a guide vane 3 and an outlet flow channel 4, two visualization windows are opened near the shaft head of the mixed flow pump for the laser vibrometer 7 to measure the vibration of the main shaft 8. A pressure pulsation measuring point is set at the outlet of the mixed flow pump impeller, and the pressure pulsation at the outlet of the mixed flow pump is measured by a pressure pulsation sensor 5. The laser vibrometer 7 and the pressure pulsation sensor 5 are both connected to a data acquisition card 9 through a data transmission line, and the data acquisition card 9 is connected to a computer 10.

[0060] Example 2

[0061] like Figure 3 As shown, the two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection of the present invention is specifically implemented according to the following steps:

[0062] Step 1, start the mixed flow pump vibration data acquisition system, and the water flows through the inlet flow channel 1, impeller 2, guide vane 3 and outlet flow channel 4 in sequence. The length of the inlet flow channel 1 and the outlet flow channel 4 is twice the length of the impeller 2 and guide vane 3 flow channels to ensure that the flow is fully developed and avoid backflow;

[0063] Step 2, using a Doppler laser vibrometer 7 to collect the vibration displacement signal of the main shaft 8, and using a pressure pulsation sensor 5 to collect the outlet pressure pulsation signal of the impeller 2. The collected signals are collected by a data acquisition card and transmitted to a computer 10 via a data line. Thus, the time series x1 of the vibration displacement signal of the main shaft 8 and the outlet pressure pulsation signal x2 of the impeller 2 are obtained;

[0064] Step 3, using a two-dimensional kernel density function to calculate the joint probability density function p(x1, x2) of the time series x1 of the vibration displacement signal and the impeller outlet pressure pulsation signal x2. Since the Gaussian kernel function can effectively smooth the data distribution, the Gaussian kernel function is selected for kernel density estimation to provide an accurate joint probability density estimate;

[0065] Step 4: Integrate p(x1, x2) with respect to x2 in the range of (-∞, +∞) to obtain the marginal probability density P(x1) of x1. In addition, integrate p(x1, x2) with respect to x1 in the range of (-∞, +∞) to obtain the marginal probability density P(x2) of x2, where the marginal probability densities P(x1) and P(x2) are calculated as follows:

[0066]

[0067]

[0068] Step 5, based on step 4, calculate the individual fourth-order moments of x1 and x2 and the mixed fourth-order moment between the two through equations (3) to (7);

[0069] μ1=∫∫(x1-E(x1)) 4 p(x1,x2)dx1dx2 (3)

[0070] μ2=∫∫(x2-E(x2)) 4 p(x1,x2)dx1dx2 (4)

[0071] μ3=∫∫(x1-E(x1)) 4 (x2-E(x2)) 4 p(x1,x2)dx1dx2 (5)

[0072]

[0073]

[0074] Among them, E(x1) and E(x2) represent the means of x1 and x2 respectively, μ1 and μ2 represent the individual fourth-order moments of x1 and x2 respectively, and μ3 represents the mixed fourth-order moment between x1 and x2.

[0075] Step 6: Based on Step 4 and Step 5, calculate the two-dimensional joint kurtosis of x1 and x2 by using equations (8) to (10);

[0076]

[0077]

[0078]

[0079] in, and They represent the variance of x1 and x2 respectively, and K represents the two-dimensional joint kurtosis between them.

[0080] Step 7, calculate the maximum information coefficient of the time series x1 of the vibration displacement signal and the impeller outlet pressure pulsation signal x2, and determine the reference value of the two-dimensional joint kurtosis through their correlation coefficients.

[0081] Step 7.1, discretize the time series signals x1 and x2 into a series of finite intervals. Let x1 and x2 be two time series signals, and assume that x1 and x2 have N sample points. Divide the time series x1 and x2 into k intervals, and get the interval set {I1,I2,……,I k} and {J1,J2,……,J k}, where each interval I i represents the dynamic division of the vibration displacement signal x1 within its value range, and each interval J j Represents the dynamic division of the impeller outlet pressure pulsation signal x2 within its numerical range. The division here is not a simple equal division, but an adaptive adjustment based on the data distribution and the need to maximize mutual information to reveal the potential nonlinear relationship between x1 and x2;

[0082] Step 7.2, calculate the distribution frequency of the signal value in each interval and construct a joint distribution table. Divide the time series signals x1 and x2 into interval sets {I1, I2, ..., I k} and {J1,J2,……,J k}, each interval I i and J j Represents the division of signals x1 and x2 into specific numerical ranges. Calculate the frequency of each interval, that is, the number of samples whose signal value falls into this interval, and normalize it into probability. i ) means x1 falls in interval I i Possibility; probability P(J j ) means x2 falls within the interval J j The possibility of the joint probability P(I i ,J j ) means x1 falls in interval I i And x2 falls in interval J j These probabilities are used to construct a joint distribution table, which shows the probability distribution of two signals in all possible interval combinations to reveal their relationship;

[0083]

[0084]

[0085]

[0086] Step 7.3, use the constructed joint distribution table to calculate the mutual information I(x1; x2) between the two signals. The joint distribution table is a two-dimensional matrix that lists all possible interval combinations (I i , J j ) and its corresponding joint probability P(I i , J j ), which is used to describe the probability distribution of signals x1 and x2 falling in their respective intervals at the same time. Mutual information is a quantitative indicator to measure the degree of information sharing between two signals, and is defined as the difference between the joint entropy H(x1,x2) of the two signals and the sum of their respective entropies;

[0087]

[0088]

[0089]

[0090] I(x1;x2)=H(x1)+H(x2)-H(x1,x2) (17)

[0091] Among them, H(x1) represents the entropy of signal x1, which is used to measure x1 in each interval I i The uncertainty of the probability distribution on ; H(x2) represents the entropy of the signal x2, which is used to measure the probability distribution of x2 in each interval J j H(x1,x2) represents the joint entropy of the two signals, which measures the uncertainty of the combined distribution of the two signals; I(x1;x2) represents the mutual information between signals x1 and x2, which reflects the mutual dependence and degree of information sharing between x1 and x2.

[0092] Step 7.4: According to the normal distribution characteristics, the signals x1 and x2 are divided into different intervals {k1, k2, …, k m}. For different interval partitioning schemes {k1, k2, …, k m}, execute step 7.3 to calculate different mutual information values Finally, the maximum mutual information value is selected as the maximum information coefficient, and it is used as the correlation coefficient δ of the time series signals x1 and x2;

[0093]

[0094] Step 7.5, construct two standard normal distribution signals x3 and x4 with a correlation of δ. First, generate two independent standard normal distribution signals z1 and z2, and then transform these two signals into standard normal distribution signals x3 and x4 with a specified correlation δ by linear combination;

[0095] x3=z1 (19)

[0096]

[0097] Step 7.6, in steps 4, 5, and 6, replace x1 with the standard normal distribution signal x3, and replace x2 with the standard normal distribution signal x4, then execute steps 4 to 6 to calculate the two-dimensional joint kurtosis K between the two standard normal distribution signals x3 and x4 ab (K ab The calculation formula is the process of calculating K in formula (10), K ab It is the two-dimensional joint kurtosis reference value of the time series x1 of the vibration displacement signal of the main shaft 8 and the outlet pressure pulsation signal x4 of the impeller 2.

[0098] Step 8: Obtain the two-dimensional joint kurtosis reference value K of signals x1 and x2 ab After that, the two-dimensional joint kurtosis exceeding the benchmark value is named positive joint kurtosis, and the two-dimensional joint kurtosis below the benchmark value is named negative joint kurtosis. The larger the positive joint kurtosis, the greater the impact and the more the operating state deviates from its normal state.

[0099] Example 3

[0100] The present invention collects the main shaft vibration displacement signal of measuring point A and the impeller outlet pressure pulsation signal of measuring point B during the operation of the mixed flow pump by means of a laser vibrometer and a pressure pulsation sensor, respectively. The original signals are shown in FIG4(a) and FIG4(b). FIG4(a) is the vibration displacement signal of measuring point A collected by the laser vibrometer; FIG4(b) is the impeller outlet pressure pulsation signal of measuring point B collected by the pressure pulsation sensor; the two-dimensional joint kurtosis K is calculated by steps 3, 4, 5 and 6, and the two-dimensional joint kurtosis reference value K is calculated by steps 7 and 8. ab ,like Figure 5 As shown. Figure 5 It can be seen that the positive joint kurtosis accounts for 0.08% during the operation of the mixed flow pump, and the positive joint kurtosis reaches a maximum value of 2.98668 at 1.04s. This shows that at 1.04s, the mixed flow pump has the largest impact in the entire operation, and the operating state deviates from its normal state. Figure 6 The relationship between the correlation coefficient and the two-dimensional joint kurtosis reference value is demonstrated. The reference value of the two-dimensional joint kurtosis can be directly determined by calculating the correlation between the signals, so as to facilitate online monitoring during the operation of the unit.

[0101] The present invention can be used for, but not limited to, vibration monitoring of single-stage mixed-flow pumps, and can also be used for vibration analysis of other types of pump systems and complex mechanical systems, such as multi-stage centrifugal pumps, axial-flow pumps, reciprocating pumps, etc. In these applications, by collecting vibration signals and pressure pulsation signals from multiple measuring points and using a two-dimensional joint kurtosis analysis method, the operating status and vibration characteristics of the equipment can be more comprehensively reflected, thereby improving the accuracy and reliability of vibration detection, identifying potential faults in a timely manner, and performing effective maintenance and optimization, ultimately improving the operating stability and service life of the equipment and reducing maintenance costs.

[0102] The two-dimensional joint kurtosis analysis method for vibration detection of mixed flow pumps of the present invention utilizes a laser vibrometer and a pressure pulsation sensor to collect the radial vibration displacement signal of the main shaft and the pressure pulsation of the impeller outlet, and solves the two-dimensional joint kurtosis based on the two collected signals. The maximum information coefficient is used to effectively capture the nonlinear and linear relationships between the signals, making the identification of abnormal vibration characteristics more accurate. The operating status of the mixed flow pump is evaluated using the standard normal distribution and the two-dimensional joint kurtosis reference value, providing a reliable diagnostic basis, which helps to timely discover potential problems and perform maintenance and optimization. This method is not only suitable for vibration detection of mixed flow pump systems, but can also be extended to vibration analysis of other complex mechanical systems. It has strong versatility and practicality, thereby improving the operating stability and service life of the equipment and reducing maintenance costs.

Claims

1. A two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection, characterized in that: The specific process is as follows: the radial vibration displacement signal of the main shaft and the pressure pulsation of the impeller outlet are collected using a laser vibrometer and a pressure pulsation sensor, and the two-dimensional joint kurtosis is solved based on the two collected signals; the operating status of the mixed flow pump is evaluated using the standard normal distribution and the two-dimensional joint kurtosis benchmark value.

2. The two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection according to claim 1 is characterized in that: The specific steps include: Step 1, start the mixed flow pump vibration data acquisition system, and the water flows through the inlet flow channel (1), the impeller (2), the guide vane (3) and the outlet flow channel (4) in sequence; Step 2, using a laser vibrometer (7) to collect the vibration displacement signal of the main shaft (8), and using a pressure pulsation sensor (5) to collect the outlet pressure pulsation signal of the impeller (2), thereby obtaining a time series x1 of the vibration displacement signal of the main shaft (8) and a pressure pulsation signal x2 of the outlet of the impeller (2); Step 3, using a two-dimensional kernel density function to calculate the joint probability density function p(x1, x2) of the time series x1 of the vibration displacement signal and the impeller outlet pressure pulsation signal x2; Step 4, integrate p(x1, x2) with respect to x2 in the range of (-∞, +∞) to obtain the marginal probability density P(x1) of x1; integrate p(x1, x2) with respect to x1 in the range of (-∞, +∞) to obtain the marginal probability density P(x2) of x2; Step 5, the individual fourth-order moments of x1 and x2 and the mixed fourth-order moment between them; Step 6, based on step 4 and step 5, calculate the two-dimensional joint kurtosis of x1 and x2; Step 7: Calculate the maximum information coefficient of the time series x1 of the vibration displacement signal and the impeller outlet pressure pulsation signal x2, and determine the reference value K of the two-dimensional joint kurtosis through the correlation coefficient. ab ; Step 8: Obtain the two-dimensional joint kurtosis reference value K of signals x1 and x2 ab After that, it will exceed the reference value K ab The two-dimensional joint kurtosis is named positive joint kurtosis, which is lower than the reference value K ab The joint kurtosis is negative.

3. The two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection according to claim 2 is characterized in that: In step 4, the calculation method of the edge probability density P(x1) and P(x2) is as follows:

4. The two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection according to claim 3 is characterized in that: In step 5, the fourth-order moments of x1 and x2 and the mixed fourth-order moments between the two are calculated by the following formulas (3) to (7); μ1=∫∫(x1-E(x1)) 4 p(x1,x2)dx1dx2 (3) μ2=∫∫(x2-E(x2)) 4 p(x1,x2)dx1dx2 (4) μ3=∫∫(x1-E(x1)) 4 (x2-E(x2)) 4 p(x1,x2)dx1dx2 (5) Among them, E(x1) and E(x2) represent the means of x1 and x2 respectively, μ1 and μ2 represent the individual fourth-order moments of x1 and x2 respectively, and μ3 represents the mixed fourth-order moment between x1 and x2.

5. The two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection according to claim 4 is characterized in that: In step 6, the two-dimensional joint kurtosis of x1 and x2 is calculated by the following formulas (8) to (10); in, and They represent the variance of x1 and x2 respectively, and K represents the two-dimensional joint kurtosis between them.

6. The two-dimensional joint kurtosis analysis method for mixed flow pump vibration detection according to claim 5 is characterized in that: The specific process of step 7 is as follows: Step 7.1: Divide the time series signals x1 and x2 into interval sets {I1, I2, ..., I k } and {J1, J2, ..., J k }, each interval I i and J j They represent the division of signals x1 and x2 within a specific range of values, calculate the probability of each interval, and construct a joint distribution table based on the calculated probabilities, as follows: Among them, the probability P(I i ) means x1 falls within interval I i Possibility; probability P(J j ) means x2 falls within the interval J j The possibility of joint probability P(I i , J j ) means x1 falls within interval I i And x2 falls in interval J j possibility; Step 7.3, using the joint distribution table constructed in step 7.2 to calculate the mutual information I(x1; x2) between the two signals, as follows: I(x1;x2)=H(x1)+H(x2)-H(x1,x2) (17) Among them, H(x1) represents the entropy of signal x1, which is used to measure the entropy of x1 in each interval I i The uncertainty of the probability distribution on J; H(x2) represents the entropy of signal x2, which is used to measure the probability distribution of x2 in each interval J. j The uncertainty of the probability distribution on ; H(x1,x2) represents the joint entropy of the two signals, which measures the uncertainty of the combined distribution of the two signals; Step 7.4: According to the normal distribution characteristics, the signals x1 and x2 are divided into different intervals {k1, k2, ..., k m }, for different interval partitioning schemes {k1, k2, …, k m }, execute step 7.3 to calculate different mutual information values Finally, the maximum mutual information value is selected as the maximum information coefficient, and the maximum information coefficient is used as the correlation coefficient δ of the time series signals x1 and x2; Step 7.5, generate two independent standard normal distribution signals z1 and z2, and then transform these two signals into standard normal distribution signals x3 and x4 with specified correlation δ by linear combination method: x3=z1 (19) Step 7.6, replace x1 with the standard normal distribution signal x3, and replace x2 with the standard normal distribution signal x4, and then execute steps 4 to 6 to obtain the two-dimensional joint kurtosis K between the two standard normal distribution signals x3 and x4. ab , K ab It is a two-dimensional joint kurtosis reference value of the time series x1 of the vibration displacement signal of the main shaft (8) and the outlet pressure pulsation signal x4 of the impeller (2).

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