A pressure vessel health monitoring method based on intelligent algorithm

By acquiring pressure and stress data through a sensor array, calculating gradients and curve morphologies, and identifying damage boundaries and phase delays, the problem of insufficient damage identification accuracy in traditional methods is solved, and efficient health monitoring and life prediction of pressure vessels are achieved.

CN120632641BActive Publication Date: 2025-10-10NANTONG UNIV
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Patent Information

Application Number
CN202511113603.7
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2025-08-11
Publication Date
2025-10-10
Estimated Expiration
2045-08-11

AI Technical Summary

Technical Problem

Traditional pressure vessel health monitoring methods are unable to effectively capture early subtle damage characteristics, resulting in insufficient damage identification accuracy, insufficient fault detection real-time and life prediction accuracy, and cannot meet the monitoring needs of high-risk scenarios.

Method used

The pressure value, stress distribution and temperature gradient data are acquired in real time through the sensor array, the pressure gradient and equivalent stress curve morphology between adjacent monitoring points are calculated, the damage boundary coordinates and abnormal curvature area are extracted, and the phase delay increment and rate growth deviation of the pressure signal and stress response are combined to dynamically analyze the health status and establish a fault event detection and life prediction model.

Benefits of technology

It achieves precise identification of minor damage, improves the real-time performance of fault detection and the accuracy of life prediction, and enhances the overall assessment capability of the health status of pressure vessels.

✦ Generated by Eureka AI based on patent content.

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Abstract

The present application relates to the technical field of intelligent diagnosis, in particular to a pressure vessel health monitoring method based on intelligent algorithm, comprising the following steps: acquiring pressure, stress and temperature data in real time by using a sensor array, calculating trajectory evolution parameter set, identifying damage area, detecting fault event, predicting health state and residual service life, establishing damage grade classification framework, and obtaining damage grade classification result.In the present application, through real-time collection of multi-source data and spatial gradient refinement processing, the coupling relationship between pressure and stress distribution is utilized to realize accurate damage positioning and improve abnormal detection sensitivity, mark damage boundary coordinates and quantify the expansion rate of adjacent damage area, determine the phase delay in combination with multi-period pressure trajectory, enhance the stability of fault identification and optimize the accuracy of abnormal event classification, adopt material fatigue accumulation and pressure uniformity correlation means, and strengthen health score decay observation, improve health state quantization depth and expand the reliability of life prediction.
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Description

Technical Field

[0001] The present invention relates to the field of intelligent diagnosis technology, and in particular to a pressure vessel health monitoring method based on an intelligent algorithm. Background Art

[0002] The field of intelligent diagnostic technology includes the use of computer algorithms, mathematical modeling, and data analysis methods to monitor the state, predict faults, and assess the health of physical systems. The core content of this technology includes obtaining system operating parameters through sensors, extracting feature information using data analysis and pattern recognition methods, and implementing fault detection and trend prediction based on statistical models or artificial intelligence algorithms. It covers multiple links such as data acquisition, signal processing, feature extraction, health assessment, fault diagnosis, and life prediction. It is applied to multiple fields such as industrial equipment, aerospace, and medical health, aiming to improve equipment reliability and reduce maintenance costs through intelligent analysis.

[0003] Among them, a pressure vessel health monitoring method based on intelligent algorithms refers to the use of intelligent computing methods to perform data analysis and health assessment on the operating status of pressure vessels. The method targets multiple technical issues such as internal stress distribution, material fatigue, and corrosion damage of pressure vessels. Real-time operation data is collected through stress sensors, and pressure, temperature, stress changes and various key feature information are extracted by combining time series data analysis methods. A health assessment framework is constructed based on a deep learning model, and time series prediction is performed using a long short-term memory network. An adaptive filtering algorithm is used to optimize data input and improve prediction accuracy. Fault discrimination rules based on probability statistics are used in combination with a support vector machine classification model to realize abnormal state identification of pressure vessels. A fault development trend analysis model is established through a Bayesian network to realize dynamic monitoring of the health status of pressure vessels and fault prediction.

[0004] Traditional pressure vessel health monitoring methods identify damage through simple changes in pressure or stress values, ignoring pressure trajectory changes and spatial gradient information. This makes it difficult to effectively capture early subtle damage characteristics of the vessel, resulting in insufficient damage identification accuracy and an inability to meet the accuracy requirements of vessel health monitoring in high-risk scenarios. Abnormal fault detection usually relies on signal amplitude threshold judgment, ignoring the phase delay characteristics of pressure and structural stress responses, resulting in limited real-time fault identification capabilities and prone to delayed fault detection. Health assessment is static and independent, without combining pressure uniformity and material damage accumulation trends. There is a lack of synchronization analysis between dynamic health status and life prediction, resulting in life prediction errors, reducing the operational safety of the vessel, and leading to failure to detect early damage to the vessel in a timely manner, resulting in premature failure of the equipment. Summary of the Invention

[0005] The purpose of the present invention is to solve the shortcomings of the prior art and to propose a pressure vessel health monitoring method based on an intelligent algorithm.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: a pressure vessel health monitoring method based on an intelligent algorithm, comprising the following steps:

[0007] S1: Using the sensor array, we acquire real-time pressure values, stress distribution, and temperature gradient data. By calculating the pressure gradient between adjacent monitoring points, we analyze the pressure trajectory changes, calculate the equivalent stress curve shape and deformation rate, and obtain the trajectory evolution parameter set.

[0008] S2: calling the trajectory evolution parameter set, extracting the displacement vector change of the equivalent stress curve in each time window, calculating the unit length offset rate and curvature change trend, marking the abnormal curvature area, extracting the damage boundary coordinates and analyzing the distribution pattern of adjacent damage points to obtain the damage impact range;

[0009] S3: Calling the trajectory evolution parameter set and damage impact range, obtaining time series data of pressure signals and stress response signals, calculating phase delay increment, calculating rate growth deviation, detecting abnormal time periods, combining pressure-stress-rate difference, identifying fault event types, and obtaining fault event detection records;

[0010] S4: Call the damage impact range and fault event detection records, analyze the pressure uniformity change, calculate the health status score based on the damage accumulation rate of the material, and analyze the decay rate of the health score to predict the container life and obtain the health status dynamic score.

[0011] As a further solution of the present invention, the trajectory evolution parameter set includes the pressure gradient change rate, the equivalent stress curve morphological parameters, and the stress trajectory deformation rate; the damage impact range includes the damage boundary coordinate set, the abnormal curvature area label, and the spatial distribution pattern of the damage point; the fault event detection record includes the phase delay increment sequence, the rate growth deviation value, and the pressure stress rate difference; the health status dynamic score includes the pressure uniformity variation coefficient, the damage accumulation rate index, and the health score decay rate.

[0012] As a further solution of the present invention, the steps of obtaining the trajectory evolution parameter set are specifically as follows:

[0013] S111: Call the sensor array to obtain pressure values, stress distribution, and temperature gradient data in real time, extract pressure field data sequences within multiple time windows, calculate pressure gradient values ​​between adjacent monitoring points, and construct a time-synchronized pressure data set;

[0014] S112: Calling the time-synchronized pressure data set, calculating the pressure gradient change rate of adjacent monitoring points, comparing the gradient change trends in multiple time windows, and calculating the equivalent stress curve morphological parameters using the formula:

[0015] ;

[0016] Calculate the trajectory morphology change rate, calculate the trajectory deformation trend of multiple time windows, and obtain the trajectory morphology deformation rate matrix;

[0017] in, is the trajectory deformation rate, For the The pressure value of each monitoring point, For the The pressure value of each monitoring point, For the The pressure gradient value of each monitoring point, For the The pressure gradient value of each monitoring point, For the The change in trajectory shape of a monitoring point between two adjacent moments, For the The pressure gradient change rate of each monitoring point at adjacent moments is is the total number of monitoring points, Indicates the starting number of the monitoring point sequence, For the The next monitoring point number of the monitoring point;

[0018] S113: calling the trajectory morphology deformation rate matrix, calculating the trajectory deformation trend curves of multiple time windows, analyzing the trajectory deformation rate distribution between multiple monitoring points, and obtaining a trajectory evolution parameter set.

[0019] As a further solution of the present invention, the step of obtaining the damage impact range is specifically as follows:

[0020] S211: calling the trajectory evolution parameter set, extracting the coordinates of the equivalent stress curves of adjacent monitoring points, calculating the vector of each point on the curve, and obtaining the displacement vector change;

[0021] S212: Calculating a unit length offset rate based on the displacement vector change, and identifying and acquiring an abnormal curvature region by comparing the calculated offset rate with a preset offset rate threshold;

[0022] S213: Call the abnormal curvature area and analyze the curvature change trend of each monitoring point in the area using the formula:

[0023] ;

[0024] The discrete degree of curvature change is obtained by calculation, the coordinates of the damage boundary are extracted, and the distribution of boundary points is analyzed to obtain the damage impact range;

[0025] in, is the curvature variation discreteness, For the The curvature change of each monitoring point, For the The curvature change of each monitoring point, For the Monitoring point and The distance between monitoring points, For the The stress concentration factor of each monitoring point is is the total number of monitoring points, is the monitoring point index number, Index number of adjacent monitoring points.

[0026] As a further solution of the present invention, the steps of obtaining the fault event detection record are specifically as follows:

[0027] S311: calling the trajectory evolution parameter set and the damage impact range, obtaining time series data of the pressure signal and the stress response signal in the damage area, and calculating the time difference to generate a phase delay increment;

[0028] S312: Calculate the rate difference between the pressure signal and the stress response signal based on the phase delay increment using the formula:

[0029] ;

[0030] Calculate the acquisition rate growth deviation, identify abnormal periods, and establish abnormal period sequences;

[0031] in, is the rate growth deviation, For the The pressure signal value at each sampling moment, For the The pressure signal value at each sampling moment, For the The stress response signal value at each sampling moment, For the The stress response signal value at each sampling moment, For the The sampling moment corresponds to time, For the The sampling moment corresponds to time, is the total number of sampling points, The index number of the sampling point;

[0032] S313: calling the abnormal period sequence, extracting the pressure stress rate difference feature in each abnormal period, comparing it with the preset fault type rate difference feature interval, identifying the fault type, and establishing a fault event detection record.

[0033] As a further solution of the present invention, the steps for obtaining the dynamic health status score are specifically as follows:

[0034] S411: Calling the damage impact range and fault event detection records, obtaining the real-time pressure value of each monitoring point in the damage area, calculating the difference between the pressure value of each monitoring point and the average pressure value of the area, and establishing a pressure uniformity variation coefficient;

[0035] S412: Call the pressure uniformity variation coefficient, combine it with the material damage accumulation rate, and use the formula:

[0036] ;

[0037] Calculate the health score of pressure vessels;

[0038] in, Represents the container health status score, Representative The pressure uniformity variation coefficient of each damaged area is Representative The material damage accumulation rate in each damaged area is represents the average value of the pressure uniformity variation coefficient of all damaged areas, represents the total number of damaged areas, is the index number of the damaged area;

[0039] S413: The health status score of the pressure vessel is called, the decay rate of the health score is calculated, the remaining life of the pressure vessel is predicted, and a dynamic health status score is established.

[0040] As a further embodiment of the present invention, the method further comprises:

[0041] S5: Calling the health status dynamic score to obtain the stress trajectory change trend of the damage area, calculating the trajectory bifurcation point increments in multiple windows, obtaining the bifurcation point spatial distribution of each damage path area, calculating the bifurcation angle change rate, extracting the damage trajectory pattern, establishing a fault level classification framework, and obtaining the damage level classification result;

[0042] The damage level classification results are specifically the number of trajectory bifurcation points, the rate of change of bifurcation angles, and the structural pattern of the damaged trajectory.

[0043] As a further solution of the present invention, the steps for obtaining the damage level classification result are specifically as follows:

[0044] S511: calling the health status dynamic score, obtaining the stress trajectory coordinate sequence of each monitoring point in the damage area in a continuous time window, calculating the coordinate difference of the trajectories in adjacent time windows, and obtaining the stress trajectory change trend matrix;

[0045] S512: extracting the coordinates of bifurcation points of the trajectories of adjacent time windows based on the stress trajectory change trend matrix, and obtaining trajectory bifurcation point increments by calculating the difference in the number of bifurcation points in adjacent time windows;

[0046] S513: Call the trajectory bifurcation point increment value, calculate the spatial distribution characteristics between consecutive bifurcation points in each damage path area, analyze the bifurcation angle change characteristics, and use the formula:

[0047] ;

[0048] Calculate the rate of change of bifurcation angles, extract damage trajectory patterns, establish a fault level classification framework, and obtain damage level classification results;

[0049] in, is the rate of change of bifurcation angle, For the The bifurcation angle value of each trajectory bifurcation point, For the The bifurcation angle value of each trajectory bifurcation point, For the The trajectory bifurcation point to the The spatial straight-line distance between the bifurcation points of the trajectories, For the The local trajectory curvature of the trajectory bifurcation point, For the The local trajectory curvature of the trajectory bifurcation point, is the total number of trajectory bifurcation points, The index number of the trajectory bifurcation point.

[0050] Compared with the prior art, the advantages and positive effects of the present invention are:

[0051] In the present invention, by acquiring the pressure value, stress distribution and temperature gradient of the monitoring point in real time, the pressure gradient between adjacent monitoring points is finely normalized on a spatial scale, and the displacement vector change of the equivalent stress curve and the unit length offset rate are combined to accurately extract the damage boundary coordinates and the abnormal curvature area, thereby achieving fine identification of minor damage and improving the positioning accuracy of the damaged area. The phase delay increment and rate growth deviation between the pressure signal and the structural stress response are dynamically analyzed to identify subtle fault events and effectively improve the real-time performance of fault detection. By analyzing the change in pressure uniformity and integrating the material damage accumulation rate, a dynamic health scoring index is established, and the structural degradation state is dynamically quantified, effectively improving the accuracy of the container life prediction. According to the trajectory bifurcation point increment and the bifurcation angle change rate, the damaged area is classified into multiple levels to achieve a fine division of the damage evolution trend and enhance the overall assessment capability of the container health state. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a schematic diagram of the main steps of the present invention;

[0053] Figure 2 A flow chart for obtaining a trajectory evolution parameter set of the present invention;

[0054] Figure 3 Obtaining a flow chart for the damage impact range of the present invention;

[0055] Figure 4 A flow chart for obtaining fault event detection records of the present invention;

[0056] Figure 5 A flowchart for obtaining a dynamic health status score of the present invention;

[0057] Figure 6 The flowchart for obtaining the damage level classification result of the present invention is shown. DETAILED DESCRIPTION

[0058] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0059] In the description of the present application, it should be understood that the terms "length", "width", "upper", "lower", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer" and the like indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present application and simplifying the description, and do not indicate or imply that the devices or elements referred to must have a particular orientation, be constructed and operated in a particular orientation, and therefore cannot be understood as limiting the present application. In addition, in the description of the present application, the meaning of "a plurality of" is two or more, unless otherwise explicitly and specifically limited.

[0060] Please refer to Figure 1 The present application provides a technical solution: a pressure vessel health monitoring method based on intelligent algorithm, comprising the following steps:

[0061] S1: using a sensor array, real-time acquisition of pressure value, stress distribution and temperature gradient data, calculation of pressure gradient between adjacent monitoring points, analysis of pressure trajectory change, calculation of equivalent stress curve form and deformation rate, and obtaining of trajectory evolution parameter set;

[0062] S2: calling the trajectory evolution parameter set, extracting the displacement vector change of the equivalent stress curve in each time window, calculating the unit length offset rate and curvature change trend, marking the abnormal curvature area, extracting the damage boundary coordinates and analyzing the distribution mode of adjacent damage points, and obtaining the damage influence range;

[0063] S3: calling the trajectory evolution parameter set and the damage influence range, obtaining the time sequence data of the pressure signal and the stress response signal, calculating the phase delay increment, calculating the rate growth deviation, detecting the abnormal period, combining the pressure stress rate difference, identifying the fault event type, and obtaining the fault event detection record;

[0064] S4: calling the damage influence range and the fault event detection record, analyzing the pressure uniformity change, combining the material damage accumulation rate, calculating the health state score, analyzing the health score decay rate, predicting the container life, and obtaining the health state dynamic score;

[0065] S5: calling the health state dynamic score, obtaining the stress trajectory change trend of the damage area, calculating the trajectory bifurcation point increment in multiple windows, obtaining the bifurcation point spatial distribution of each damage path area, calculating the bifurcation angle change rate, extracting the damage trajectory mode, establishing a fault level classification framework, and obtaining the damage level classification result.

[0066] The trajectory evolution parameter set includes the pressure gradient change rate, the equivalent stress curve morphological parameters, and the stress trajectory deformation rate. The damage impact range includes the damage boundary coordinate set, the abnormal curvature area label, and the spatial distribution pattern of the damage point. The fault event detection record includes the phase delay increment sequence, the rate growth deviation value, and the pressure-stress rate difference. The health status dynamic score includes the pressure uniformity variation coefficient, the damage accumulation rate index, and the health score attenuation rate. The damage level classification results are specifically the number of trajectory bifurcation points, the bifurcation angle change rate, and the damage trajectory structure pattern.

[0067] See also Figure 2 , the specific steps for obtaining the trajectory evolution parameter set are:

[0068] S111: Call the sensor array to obtain pressure values, stress distribution, and temperature gradient data in real time, extract pressure field data sequences within multiple time windows, calculate pressure gradient values ​​between adjacent monitoring points, and construct a time-synchronized pressure data set;

[0069] The sensor array is called, which consists of pressure sensors, strain sensors, and temperature sensors. It is assumed that there are 10 monitoring points on the inner wall of the pressure vessel, and each monitoring point is installed with one of the above sensors. The real-time collected data is shown in Table 1:

[0070] Table 1 Real-time data collection at monitoring points

[0071] Monitoring point number Pressure value (MPa) Strain value #timg#(με) Temperature gradient value #timg# (℃ / m) 1 2.05 156 3.2 2 2.13 149 3.0 3 2.20 168 3.3

[0072] As shown in Table 1, taking monitoring point 1 and monitoring point 2 as an example, the pressure gradient values ​​of adjacent monitoring points are calculated. , the formula is:

[0073] ;

[0074] in, For monitoring points and The pressure gradient between the two is in MPa / m. and are the pressure values ​​of adjacent monitoring points (unit: MPa), is the spatial distance between adjacent monitoring points (set to 0.5m).

[0075] Substituting the data of monitoring points 1 and 2, we get:

[0076] ;

[0077] Similarly, calculate monitoring points 2 and 3:

[0078] ;

[0079] By analogy, the pressure gradient values ​​of all monitoring points are obtained, and then all monitoring data are time-series synchronized according to the data collection interval of every 10 seconds to generate a time-series synchronized pressure data set.

[0080] S112: Call the time-synchronized pressure data set, calculate the pressure gradient change rate of adjacent monitoring points, compare the gradient change trends in multiple time windows, and calculate the equivalent stress curve morphological parameters using the formula:

[0081] ;

[0082] Calculate the trajectory morphology change rate, calculate the trajectory deformation trend of multiple time windows, and obtain the trajectory morphology deformation rate matrix;

[0083] in, is the trajectory deformation rate, For the The pressure value of each monitoring point, For the The pressure value of each monitoring point, For the The pressure gradient value of each monitoring point, For the The pressure gradient value of each monitoring point, For the The change in trajectory shape of a monitoring point between two adjacent moments, For the The pressure gradient change rate of each monitoring point at adjacent moments is is the total number of monitoring points, Indicates the starting number of the monitoring point sequence, Relative to the The next monitoring point number of the monitoring point;

[0084] Call the time series synchronized pressure data set, select multiple consecutive time windows (such as each window is 5 minutes), take a single window as an example, and calculate the pressure gradient change rate of adjacent monitoring points in the window , the calculation formula is:

[0085] ;

[0086] in, For the The pressure gradient change rate of each monitoring point, unit: MPa / (m·s); and are the pressure gradient values ​​of adjacent monitoring points at two adjacent sampling moments (such as 0 minutes and 5 minutes), is the time difference between the two sampling moments (300 seconds in this example).

[0087] Set the initial time (0 minutes) gradient between monitoring points 1 and 2 to MPa / m, and the gradient increases to MPa / m, then:

[0088] ;

[0089] The result of 0.00027 MPa / (m·s) is within the set stability range of [0.0001, 0.0005] MPa / (m·s), and the trend is determined to be stable. The pressure gradient change rate at other monitoring points is calculated in a similar manner.

[0090] Further call strain data to calculate the trajectory morphological changes of adjacent monitoring points , the calculation formula is:

[0091] ;

[0092] Assuming the strain values ​​of monitoring points 1 and 2 are 156με and 149με respectively, the trajectory morphology change is:

[0093] ;

[0094] The trajectory deformation rate is calculated using the following formula: :

[0095] ;

[0096] in, is the trajectory deformation rate, 、 are the pressure values ​​of adjacent monitoring points, 、 are the pressure gradient values ​​of adjacent monitoring points, is the trajectory morphology change, is the pressure gradient change rate, is the number of monitoring points, From 1 to .

[0097] ;

[0098] ;

[0099] ;

[0100] Accumulate points Value, obtain trajectory deformation rate Value, construct the trajectory morphological deformation rate matrix.

[0101] S113: Call the trajectory morphological deformation rate matrix, calculate the trajectory deformation trend curve of each time window, analyze the trajectory deformation rate distribution between multiple monitoring points, and obtain the trajectory evolution parameter set;

[0102] Call the trajectory morphological deformation rate matrix, calculate the trajectory deformation trend curve of each time window, and use the data of three consecutive time windows, as shown in Table 2:

[0103] Table 2 Trajectory morphological deformation rate matrix

[0104] Time window #timg# value 1 16.27 2 17.55 3 19.82

[0105] By value, the trajectory deformation trend change rate is calculated , the formula is:

[0106] ;

[0107] Among them, is the trajectory deformation trend change rate, with a unit of %, and are the values in the adjacent two time windows.

[0108] ;

[0109] ;

[0110] The calculation results show that the trajectory morphological deformation rate growth amplitude of the second to third window significantly exceeds that of the first to second window, and it can be determined that the trajectory deformation trend is intensified; further analyze the trajectory deformation rate distribution of each monitoring point, and divide the rate interval according to , such as 1.60-1.70 defined as a low deformation rate interval, 1.71-1.80 defined as a medium deformation rate interval, and 1.81-1.90 defined as a high deformation rate interval, and then calculate the trajectory deformation rate of each monitoring point, such as the rate of monitoring point 1.63 in the low interval, the rate of monitoring point 2.75 in the medium interval, and the rate of monitoring point 3.88 in the high interval. In this way, the spatial distribution characteristics of the trajectory deformation trend of all monitoring points are determined, and the trajectory evolution parameter set is finally obtained by comprehensively calculating the above.

[0111] Please refer to Figure 3 , and the steps for obtaining the damage influence range are as follows:

[0112] S211: Call the trajectory evolution parameter set, extract the equivalent stress curve coordinates of adjacent monitoring points, calculate the vector of each point on the curve, and obtain the displacement vector change amount;

[0113] Call the trajectory evolution parameter set, extract the coordinates of the equivalent stress curves of adjacent monitoring points, calculate the displacement vector of each point on the curve, and obtain the displacement vector change. The calculation formula for the displacement vector change is:

[0114] ;

[0115] in, For the The displacement vector of each monitoring point, and Monitoring points and The horizontal axis, and Monitoring points and The vertical coordinate of the monitoring points is calculated using this formula, and the displacement change between the monitoring points is finally obtained. First, the coordinates of the equivalent stress curves of the adjacent monitoring points are extracted. Five monitoring points are arranged along the circumference of the inner wall of the pressure vessel. The coordinate data of the monitoring points are shown in Table 3:

[0116] Table 3 Coordinate table of equivalent stress curves at monitoring points

[0117] Monitoring point number Initial coordinates (mm) Coordinate after change (mm) 1 (0,0) (0.12,0.09) 2 (100,0) (100.15,-0.03) 3 (200,0) (200.40,0.20) 4 (300,0) (300.45,-0.25) 5 (400,0) (400.22,0.35)

[0118] Refer to Table 3. Taking monitoring point 1 as an example, based on the initial coordinates (0, 0) and the changed coordinates (0.12, 0.09), the point displacement vector is calculated. Its horizontal coordinate displacement is 0.12mm and the vertical coordinate displacement is 0.09mm. The displacement vector modulus of monitoring point 1 is:

[0119] ;

[0120] The displacement vectors of other monitoring points are calculated in the same way. For example, the displacement vector modulus of monitoring point 2 is 0.15 mm, the displacement vector modulus of monitoring point 3 is 0.45 mm, and so on, and the displacement vector change of each monitoring point is obtained.

[0121] S212: Calculating a unit length offset rate based on the displacement vector change, and identifying and obtaining an abnormal curvature region by comparing the calculated offset rate with a preset offset rate threshold;

[0122] Based on the displacement vector change, the unit length offset rate is further calculated. The calculation formula of the unit length offset rate is:

[0123] ;

[0124] in, For the With the The unit length deviation rate between monitoring points, and Respectively Hedi The displacement vector of the monitoring point, For the Hedi The spatial distance between monitoring points, taking monitoring points 1 and 2 as an example, the initial distance is 100mm, and the displacement vector moduli of the two points are 0.15mm and 0.15mm respectively. The unit length offset rate is:

[0125] ;

[0126] Taking monitoring points 2 and 3 as an example, the displacement vector moduli are 0.15mm and 0.45mm respectively, so the unit length offset rate is:

[0127] ;

[0128] The offset rate threshold is set to 0.002mm / mm, which is determined by the actual measurement data of the pre-experimental pressure vessel. The maximum offset rate of 0.0015mm / mm under normal operating conditions of the vessel is selected, and a certain margin is added to obtain it. 0.002mm / mm is used as a benchmark for comparison. If the unit length offset rate of 0.003mm / mm between monitoring points 2 and 3 calculated above is greater than the set threshold of 0.002mm / mm, it is determined that abnormal curvature exists in the monitoring point 2-3 area, and the areas between other monitoring points are determined in turn, and finally the abnormal curvature area is obtained.

[0129] S213: Call the abnormal curvature area and analyze the curvature change trend of each monitoring point in the area using the formula:

[0130] ;

[0131] The discrete degree of curvature change is obtained by calculation, the coordinates of the damage boundary are extracted, and the distribution of boundary points is analyzed to obtain the damage impact range;

[0132] in, is the curvature variation discreteness, For the The curvature change of each monitoring point, For the The curvature change of each monitoring point, For the Monitoring point and The distance between monitoring points, For the The stress concentration factor of each monitoring point is is the total number of monitoring points, is the monitoring point index number, Index numbers for adjacent monitoring points;

[0133] The abnormal curvature area is called to analyze the curvature change trend of each monitoring point in the area. Taking the three monitoring points in the monitoring point 2-4 area as an example, the curvature change data of the monitoring points are obtained as shown in Table 4:

[0134] Table 4 Curvature change of monitoring points in abnormal areas

[0135] Monitoring point number (j) Curvature change #timg# (mm⁻¹) Stress concentration factor #timg# 2 0.0052 1.10 3 0.0067 1.25 4 0.0095 1.30

[0136] Distance between monitoring points is the initial monitoring point distance (fixed value 100mm), the data is determined by the design layout; substitute the above parameters into the formula for calculation:

[0137] ;

[0138] ;

[0139] ;

[0140] ;

[0141] in, is the dispersion of curvature change, which indicates the concentration of regional curvature change. For monitoring points The curvature change is obtained by measuring the displacement change of the monitoring point and calculating it in mm⁻¹. For monitoring points The stress concentration factor is obtained through finite element analysis and ranges from 1.0 to 1.5. The distance between adjacent monitoring points is fixed at 100mm. is the total number of monitoring points in the area. , Indicates the monitoring point index number, which is 2 to , Number adjacent monitoring points.

[0142] Calculated curvature variation dispersion , compared with the set discreteness reference value of 0.003, the discreteness reference value is determined by statistical analysis of a large number of sample curvature changes, and the actual measured discreteness upper limit of 0.0027 is increased by 11%. When it is higher than the benchmark value of 0.003, it indicates that there is an obvious damage trend in the area. This result indicates that there is a concentration of curvature changes in the region. Further, based on the dispersion calculation results, the coordinates (200.40, 0.20) centered at monitoring point 3 were selected as the damage boundary coordinates. Combined with the curvature change trends of adjacent monitoring points, a distribution analysis was performed, with the coordinates (100.15, -0.03) and (300.45, -0.25) as boundary endpoints to determine the damage impact range. The formula is beneficial in that it introduces curvature change and stress concentration factors to weight the differences in curvature changes, highlighting the local concentration of damage within the region.

[0143] See also Figure 4 ,The specific steps for obtaining fault event detection records are:

[0144] S311: Calling the trajectory evolution parameter set and the damage impact range, obtaining the time series data of the pressure signal and the stress response signal in the damage area, and calculating the time difference to generate a phase delay increment;

[0145] The trajectory evolution parameter set and the damage impact range are called. First, the time series data of the pressure signal and stress response signal in multiple time windows in the damage area are obtained. The sampling time is synchronously adjusted to eliminate the time lag error introduced by different sampling devices. Then, the pressure signal curve and stress response signal curve in each time window are segmented, and the time interval between adjacent sampling times is calculated. The relative phase delay of each time is calculated. The phase delay increment sequence is constructed based on the calculated phase delay data. For the abnormal jump points in the phase delay increment sequence, the interval whose growth rate is greater than the set threshold is screened. The calculation formula is used:

[0146] ;

[0147] in, For the The phase delay increment at the moment, For the The time of the sampling moment, For the The time of the sampling moment, For the The stress response signal at each sampling moment, For the The stress response signal at each sampling moment.

[0148] Set in The time corresponding to the pressure signal , the stress response signal is ;exist The time corresponding to the pressure signal , stress response signal . Substitute the data into the formula to calculate:

[0149] ;

[0150] The operation obtains the phase delay increment.

[0151] S312: Based on the phase delay increment, the rate difference between the pressure signal and the stress response signal is calculated using the formula:

[0152] ;

[0153] Calculate the acquisition rate growth deviation, identify abnormal periods, and establish abnormal period sequences;

[0154] in, is the rate growth deviation, For the The pressure signal value at each sampling moment, For the The pressure signal value at each sampling moment, For the The stress response signal value at each sampling moment, For the The stress response signal value at each sampling moment, For the The sampling moment corresponds to time, For the The sampling moment corresponds to time, is the total number of sampling points, The index number of the sampling point;

[0155] Based on the phase delay increment, the rate difference between the pressure signal and the stress response signal is calculated. First, the time derivative calculation is performed on the pressure signal and stress signal of adjacent sampling points in each time window to obtain the instantaneous rate value at adjacent moments. Then, the change amplitude of the instantaneous rate is calculated and normalized using the standardization method to eliminate the influence of the numerical amplitude under different pressure levels. The calculation formula is:

[0156] ;

[0157] in, is the rate growth deviation, For the The pressure signal value at each sampling moment, For the The pressure signal value at each sampling moment, For the The stress response signal value at each sampling moment, For the The stress response signal value at each sampling moment, For the The time of the sampling moment, For the The time of the sampling moment, is the total number of sampling points.

[0158] There are 4 sampling points in total, and the pressure signal value 、 、 、 , stress response signal value 、 、 、 , sampling time 、 、 、 . Substitute the data into the formula to calculate:

[0159] ;

[0160] The rate growth deviation is obtained by calculation, and then the intervals where the rate growth deviation exceeds the dynamic threshold are screened and marked as abnormal periods, and an abnormal period sequence is established.

[0161] S313: Calling the abnormal period sequence, extracting the pressure stress rate difference feature within each abnormal period, comparing it with the preset fault type rate difference feature interval, identifying the fault type, and establishing a fault event detection record;

[0162] Call the abnormal period sequence, extract the pressure stress rate difference characteristics within each abnormal period, count the rate growth deviations of all sampling points within the abnormal period, and calculate the mean and standard deviation of the rate deviation within the period to distinguish the stability and mutation of the rate anomaly. Then match the mean rate deviation with the fault type rate characteristic interval using the calculation formula:

[0163] ;

[0164] in, is the mean rate deviation, For the The rate growth deviation of the sampling points, ' is the total number of sampling points during the abnormal period.

[0165] Assume that there are 3 sampling points in a certain abnormal period, and their rate growth deviations are 、 、 , calculate the mean of its rate deviation:

[0166] ;

[0167] This value can be used to compare with the set fault rate threshold interval to determine whether the abnormal period belongs to a specific fault type, and ultimately establish a fault event detection record.

[0168] See also Figure 5 ,The specific steps for obtaining the dynamic health status score are:

[0169] S411: Call the damage impact range and fault event detection records, obtain the real-time pressure value of each monitoring point in the damage area, calculate the difference between the pressure value of each monitoring point and the average pressure value of the area, and establish the pressure uniformity variation coefficient;

[0170] The damage impact range and fault event detection records are called up. First, real-time pressure values ​​are collected at all monitoring points within the damage area. For example, there are 9 monitoring points in three damage areas of a pressure vessel. The sensor array is placed on the inner wall of the vessel to obtain the following real-time pressure data (unit: MPa):

[0171] Table 5 Real-time pressure data

[0172] area Real-time pressure (MPa) 1 5.2 1 5.0 1 5.3 2 6.1 2 6.3 2 6.2 3 4.8 3 4.9 3 5.0

[0173] Referring to Table 5, based on the above real-time pressure data, the mean pressure in each damaged area is calculated. For example, the mean pressure in area 1 is calculated as:

[0174] ;

[0175] Then, calculate the difference between each monitoring point and the average pressure of the area. For example, the pressure deviation of monitoring point 1 is:

[0176] ;

[0177] Calculate the pressure deviation of all monitoring points and take the average value to get the pressure uniformity variation coefficient. The result for area 1 is:

[0178] ;

[0179] The pressure uniformity variation coefficients of regions 2 and 3 are calculated in the same way.

[0180] S412: Call the pressure uniformity variation coefficient, combined with the material damage accumulation rate, using the formula:

[0181] ;

[0182] Calculate the health score of pressure vessels;

[0183] in, Represents the container health status score, Representative The pressure uniformity variation coefficient of each damaged area is Representative The material damage accumulation rate in each damaged area is represents the average value of the pressure uniformity variation coefficient of all damaged areas, represents the total number of damaged areas, is the index number of the damaged area;

[0184] The pressure uniformity variation coefficient is called, and combined with the material damage accumulation rate, the health status score is calculated using the following formula ( ):

[0185] ;

[0186] in, Represents the container health status score, For the The pressure uniformity variation coefficient of each damaged area is For the The material damage accumulation rate of each damaged area is is the average value of the pressure uniformity variation coefficient of all damaged areas, is the total number of damaged areas, is the index number of the damaged area.

[0187] Assuming the material damage accumulation rate , , , pressure uniformity variation coefficient of each region , , , then: First calculate the average value of the pressure uniformity variation coefficient:

[0188] ;

[0189] Then, substitute the formula to calculate the health status score :

[0190] ;

[0191] ;

[0192] ;

[0193] This means the current container's health score is , and in good health.

[0194] S413: call the health state score of the pressure vessel, calculate the decay rate of the health score, predict the remaining life of the pressure vessel, and establish a dynamic health state score;

[0195] Call the health state score of the pressure vessel, obtain the health score value in different time windows through continuous monitoring, for example, assuming that the health scores obtained through continuous 4 cycles of monitoring are 、 、 、 Calculate the decay rate of the health score, and the health score decay rate formula is:

[0196] ;

[0197] Among them, represents the health score decay rate, represents the health score of the adjacent next cycle, represents the health score of the current cycle, is the time interval between adjacent monitoring cycles.

[0198] In this example, the cycle interval is set to months, and the 3rd cycle and the 4th cycle are taken as an example for calculation:

[0199] per month;

[0200] The calculated health score decay rate is used for container remaining life prediction, and the container remaining life calculation formula is:

[0201] ;

[0202] Among them, represents the remaining life, is the failure threshold, which is set to , is the current health state score, is the average value of the health score decay rate of each cycle.

[0203] Calculate :

[0204] ;

[0205] per month;

[0206] Put in the current score , the predicted remaining life is:

[0207] months;

[0208] This result indicates that the remaining life of the container is approximately months.

[0209] See also Figure 6 , the specific steps for obtaining the damage level classification results are as follows:

[0210] S511: Call the health status dynamic score to obtain the stress trajectory coordinate sequence of each monitoring point in the damage area in the continuous time window, calculate the coordinate difference of the trajectory of adjacent time windows, and obtain the stress trajectory change trend matrix;

[0211] Call the health status dynamic score to obtain the stress trajectory coordinate sequence of each monitoring point in the damage area under multiple continuous time windows. For the same monitoring point, extract the three-dimensional coordinate values ​​corresponding to adjacent time windows and calculate the Euclidean coordinate difference to measure the dynamic change trend in the trajectory evolution process. For any two adjacent time periods and , and its corresponding coordinate point is and , using the formula:

[0212] ;

[0213] in, Indicates the The coordinate offset between time windows, Represents a time window The three-dimensional coordinate values ​​under Represents a time window The three-dimensional coordinate values ​​under Indicates the window index number. The coordinates of monitoring point A in two consecutive windows are and Perform the operation:

[0214] ;

[0215] This value represents the spatial offset degree of the stress trajectory of monitoring point A between two adjacent windows, and then the corresponding coordinate differences of all monitoring points are integrated to form a time series matrix and establish the stress trajectory change trend matrix.

[0216] S512: Based on the stress trajectory change trend matrix, the coordinates of the bifurcation points of the trajectories in adjacent time windows are extracted, and the incremental values ​​of the trajectory bifurcation points are obtained by calculating the difference in the number of bifurcation points in adjacent time windows;

[0217] The stress trajectory change trend matrix is ​​called to extract the trajectory mutation points of each group of monitoring points in the continuous time window, and the positions where the trajectory turns and the curvature mutation occurs in a certain time period are marked as bifurcation points. The number of bifurcation points in each time window is counted, and then the number of bifurcation points in any two adjacent time windows is calculated. and The number of bifurcation points within and To calculate the difference, use the following formula:

[0218] ;

[0219] in, Indicates the The increment value of the trajectory bifurcation point between time windows, Represents a time window The number of bifurcation points extracted in Indicates the next time window The number of bifurcation points extracted in and represents two consecutive time windows, is the window index number. Taking monitoring point area A as an example, if in the time window Internal identification A bifurcation point, Internal identification bifurcation points, then the bifurcation point increment value is:

[0220] ;

[0221] This value represents the scale of change in the trajectory structure mutation of the monitoring point area A between the two windows. The incremental value of all monitoring points Sort the time dimension to form a sequence of incremental values ​​of trajectory bifurcation points.

[0222] S513: Call the trajectory bifurcation point increment value, calculate the spatial distribution characteristics between consecutive bifurcation points in each damage path area, analyze the bifurcation angle change characteristics, and use the formula:

[0223] ;

[0224] Calculate the rate of change of bifurcation angles, extract damage trajectory patterns, establish a fault level classification framework, and obtain damage level classification results;

[0225] in, is the rate of change of bifurcation angle, For the The bifurcation angle value of each trajectory bifurcation point, For the The bifurcation angle value of each trajectory bifurcation point, For the The trajectory bifurcation point to the The spatial straight-line distance between the bifurcation points of the trajectories, For the local trajectory curvatures of the trajectory branch points, the total number of trajectory branch points, the local trajectory curvatures of the trajectory branch points, the total number of trajectory branch points, the index number of the trajectory branch points;

[0226] The coordinate information and the corresponding branch angles between the plurality of trajectory branch points are obtained by calling the trajectory branch point increment value, and the local curvatures and the spatial position relationship thereof are extracted. The branch angle difference value and the spatial distance therebetween are calculated for two adjacent branch points in the order of the trajectory path, and the local trajectory curvatures of the adjacent branch points are called and standardized. The branch angle change rate is calculated by using the following formula:

[0227] ;

[0228] The four trajectory branch points monitored in the damage area of a certain container are set, and the angle values thereof are , , , The distances between adjacent points are 3.0, 4.2, and 2.8, respectively, and the local curvature values are 0.8, 1.2, 1.0, and 1.3, respectively. The formula is substituted:

[0229] ;

[0230] ;

[0231] ;

[0232] ;

[0233] The branch angle change rate is used to evaluate the degree of rapid change of the damage trajectory trend in the trajectory. If the reference preset change rate discrimination threshold range is 30-45, the result 52.24 indicates that the branch trend change in the trajectory is intense. The result is finally used to construct a trajectory turning feature set, extract a damage trajectory pattern, and match the damage trajectory pattern with a fault level classification standard to establish a fault level classification framework and obtain a damage level classification result.

[0234] The above is only a preferred embodiment of the present application, and does not limit the present application in other forms. Any person skilled in the art can modify or change the above disclosed technical content to equivalent embodiments applied to other fields, but any simple modification, equivalent change and modification made according to the technical essence of the present application to the above embodiments still belongs to the protection scope of the technical solution of the present application.

Claims

1. A pressure vessel health monitoring method based on intelligent algorithm, characterized in that: The following steps are involved: S1: Using the sensor array, we acquire real-time pressure values, stress distribution, and temperature gradient data. By calculating the pressure gradient between adjacent monitoring points, we analyze the pressure trajectory changes, calculate the equivalent stress curve shape and deformation rate, and obtain the trajectory evolution parameter set. S2: calling the trajectory evolution parameter set, extracting the displacement vector change of the equivalent stress curve in each time window, calculating the unit length offset rate and curvature change trend, marking the abnormal curvature area, extracting the damage boundary coordinates and analyzing the distribution pattern of adjacent damage points to obtain the damage impact range; S3: Calling the trajectory evolution parameter set and damage impact range, obtaining time series data of pressure signals and stress response signals, calculating phase delay increment, calculating rate growth deviation, detecting abnormal time periods, combining pressure-stress-rate difference, identifying fault event types, and obtaining fault event detection records; S4: Recalling the damage impact range and fault event detection records, analyzing the pressure uniformity change, combining the damage accumulation rate of the material, calculating the health status score, and analyzing the decay rate of the health score to predict the container life and obtain a dynamic health status score; S5: Calling the health status dynamic score to obtain the stress trajectory change trend of the damage area, calculating the trajectory bifurcation point increments in multiple windows, obtaining the bifurcation point spatial distribution of each damage path area, calculating the bifurcation angle change rate, extracting the damage trajectory pattern, establishing a fault level classification framework, and obtaining the damage level classification result; The damage level classification results are specifically the number of trajectory bifurcation points, the rate of change of bifurcation angles, and the structural pattern of the damaged trajectory.

2. The pressure vessel health monitoring method based on intelligent algorithm according to claim 1 is characterized in that: The trajectory evolution parameter set includes the pressure gradient change rate, the equivalent stress curve morphological parameters, and the stress trajectory deformation rate; the damage impact range includes the damage boundary coordinate set, the abnormal curvature area label, and the spatial distribution pattern of the damage point; the fault event detection record includes the phase delay increment sequence, the rate growth deviation value, and the pressure stress rate difference; the health status dynamic score includes the pressure uniformity variation coefficient, the damage accumulation rate index, and the health score decay rate.

3. The pressure vessel health monitoring method based on intelligent algorithm according to claim 1 is characterized in that: The steps for obtaining the trajectory evolution parameter set are specifically as follows: S111: Call the sensor array to obtain pressure values, stress distribution, and temperature gradient data in real time, extract pressure field data sequences within multiple time windows, calculate pressure gradient values ​​between adjacent monitoring points, and construct a time-synchronized pressure data set; S112: Calling the time-synchronized pressure data set, calculating the pressure gradient change rate of adjacent monitoring points, comparing the gradient change trends in multiple time windows, and calculating the equivalent stress curve morphological parameters using the formula: ; Calculate the trajectory morphology change rate, calculate the trajectory deformation trend of multiple time windows, and obtain the trajectory morphology deformation rate matrix; in, is the trajectory deformation rate, For the The pressure value of each monitoring point, For the The pressure value of each monitoring point, For the The pressure gradient value of each monitoring point, For the The pressure gradient value of each monitoring point, For the The change in trajectory shape of a monitoring point between two adjacent moments, For the The pressure gradient change rate of each monitoring point at adjacent moments is is the total number of monitoring points, Indicates the starting number of the monitoring point sequence, Relative to the The next monitoring point number of the monitoring point; S113: calling the trajectory morphology deformation rate matrix, calculating the trajectory deformation trend curves of multiple time windows, analyzing the trajectory deformation rate distribution between multiple monitoring points, and obtaining a trajectory evolution parameter set.

4. The pressure vessel health monitoring method based on intelligent algorithm according to claim 1 is characterized in that: The steps for obtaining the damage impact range are specifically as follows: S211: calling the trajectory evolution parameter set, extracting the coordinates of the equivalent stress curves of adjacent monitoring points, calculating the vector of each point on the curve, and obtaining the displacement vector change; S212: Calculating a unit length offset rate based on the displacement vector change, and identifying and acquiring an abnormal curvature region by comparing the calculated offset rate with a preset offset rate threshold; S213: Call the abnormal curvature area and analyze the curvature change trend of each monitoring point in the area using the formula: ; The discrete degree of curvature change is obtained by calculation, the coordinates of the damage boundary are extracted, and the distribution of boundary points is analyzed to obtain the damage impact range; in, is the curvature variation discreteness, For the The curvature change of each monitoring point, For the The curvature change of each monitoring point, For the Monitoring point and The distance between monitoring points, For the The stress concentration factor of each monitoring point is is the total number of monitoring points, is the monitoring point index number, Index number of adjacent monitoring points.

5. The pressure vessel health monitoring method based on intelligent algorithm according to claim 1 is characterized in that: The steps for obtaining the fault event detection record are specifically as follows: S311: calling the trajectory evolution parameter set and the damage impact range, obtaining time series data of the pressure signal and the stress response signal in the damage area, and calculating the time difference to generate a phase delay increment; S312: Calculate the rate difference between the pressure signal and the stress response signal based on the phase delay increment using the formula: ; Calculate the acquisition rate growth deviation, identify abnormal periods, and establish abnormal period sequences; in, is the rate growth deviation, For the The pressure signal value at each sampling moment, For the The pressure signal value at each sampling moment, For the The stress response signal value at each sampling moment, For the The stress response signal value at each sampling moment, For the The sampling moment corresponds to time, For the The sampling moment corresponds to time, is the total number of sampling points, The index number of the sampling point; S313: calling the abnormal period sequence, extracting the pressure stress rate difference feature in each abnormal period, comparing it with the preset fault type rate difference feature interval, identifying the fault type, and establishing a fault event detection record.

6. The pressure vessel health monitoring method based on intelligent algorithm according to claim 1 is characterized in that: The steps for obtaining the dynamic health status score are specifically as follows: S411: Calling the damage impact range and fault event detection records, obtaining the real-time pressure value of each monitoring point in the damage area, calculating the difference between the pressure value of each monitoring point and the average pressure value of the area, and establishing a pressure uniformity variation coefficient; S412: Call the pressure uniformity variation coefficient, combine it with the material damage accumulation rate, and use the formula: ; Calculate the health score of pressure vessels; in, Represents the container health status score, Representative The pressure uniformity variation coefficient of each damaged area is Representative The material damage accumulation rate in each damaged area is represents the average value of the pressure uniformity variation coefficient of all damaged areas, represents the total number of damaged areas, is the index number of the damaged area; S413: The health status score of the pressure vessel is called, the decay rate of the health score is calculated, the remaining life of the pressure vessel is predicted, and a dynamic health status score is established.

7. The pressure vessel health monitoring method based on intelligent algorithm according to claim 1 is characterized in that: The steps for obtaining the damage level classification result are specifically as follows: S511: calling the health status dynamic score, obtaining the stress trajectory coordinate sequence of each monitoring point in the damage area in a continuous time window, calculating the coordinate difference of the trajectories in adjacent time windows, and obtaining the stress trajectory change trend matrix; S512: extracting the coordinates of bifurcation points of the trajectories of adjacent time windows based on the stress trajectory change trend matrix, and obtaining trajectory bifurcation point increments by calculating the difference in the number of bifurcation points in adjacent time windows; S513: Call the trajectory bifurcation point increment value, calculate the spatial distribution characteristics between consecutive bifurcation points in each damage path area, analyze the bifurcation angle change characteristics, and use the formula: ; Calculate the rate of change of bifurcation angles, extract damage trajectory patterns, establish a fault level classification framework, and obtain damage level classification results; in, is the rate of change of bifurcation angle, For the The bifurcation angle value of the bifurcation point of the trajectory, For the The bifurcation angle value of the bifurcation point of the trajectory, For the The trajectory bifurcation point to the The spatial straight-line distance between the bifurcation points of the trajectories, For the The local trajectory curvature of the trajectory bifurcation point, For the The local trajectory curvature of the trajectory bifurcation point, is the total number of trajectory bifurcation points, The index number of the trajectory bifurcation point.

Citation Information

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