Aero-engine main shaft inner ring whirl failure identification method and system

By employing an order analysis method that synchronously acquires vibration and rotational speed data, combined with BPFI/BPFO exclusive verification and operating condition premise verification, the problem of early identification of vortex faults in the inner ring of the aero-engine main shaft has been solved. This has enabled highly reliable early warning and maintenance guidance, reducing the misjudgment rate and damage risk.

CN121720716BActive Publication Date: 2026-05-08杭州华翊科技有限公司
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Patent Information

Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
杭州华翊科技有限公司
Filing Date
2026-02-10
Publication Date
2026-05-08

AI Technical Summary

Technical Problem

Existing technologies are insufficient to effectively identify self-excited vortex faults in the inner ring of the main shaft of aero-engines, especially under clearance fit and light load conditions. This leads to difficulties in early diagnosis, a high misjudgment rate, and an inability to provide timely warnings of potential fretting wear and bearing damage.

Method used

By employing an order analysis method that simultaneously acquires vibration and rotational speed data, and extracting the 0.41×shaft frequency subsynchronous order feature, combined with BPFI/BPFO exclusive verification and operating condition premise verification, a reliable early warning for inner ring vortex is formed, reducing the false judgment rate and providing maintenance suggestions.

Benefits of technology

It enables early and reliable identification of whirl faults in the inner ring of the aero-engine main shaft, reduces the risk of false alarms, improves the credibility and reliability of diagnosis, guides timely maintenance measures, and reduces main shaft damage and downtime costs.

✦ Generated by Eureka AI based on patent content.

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Abstract

The present application relates to the technical field of aero-engine main shaft detection, and particularly relates to a method and system for identifying inner ring whirl failure of aero-engine main shaft. The method synchronously collects vibration acceleration signals and rotational speed signals, generates order spectrum by using order analysis method, extracts 0.41 order energy as a feature, combines order stability and exclusive failure checking, and diagnoses the inner ring whirl failure. The method accurately identifies the whirl failure by judging whether the 0.41 order energy exceeds a preset threshold, whether the order is stable, and whether common failures such as pitting and peeling are excluded. In addition, the reliability of the failure diagnosis is ensured by verifying the gap fit, light load working condition and the prerequisite of no sliding bearing. Compared with the traditional failure diagnosis method, the present application can effectively improve the early identification ability of the whirl failure, avoid false positives and false negatives, and has strong engineering application value.
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Description

Technical Field

[0001] This invention relates to the field of aero-engine main shaft testing technology, and in particular to a method and system for identifying whirl faults in the inner ring of an aero-engine main shaft. Background Technology

[0002] The main shaft system of an aero-engine typically operates in high-speed, high-temperature, complex-load, and strong-vibration environments. As a critical component for support and positioning, the rolling bearings of the main shaft directly affect the engine's safety and lifespan. To ensure load transfer and fretting resistance of the rolling bearings, engineering practices generally require an interference fit or transition fit between the bearing inner ring and the main shaft. This suppresses slippage and creep of the inner ring relative to the main shaft, thereby reducing fretting wear, heat accumulation, and cage impact risks at the mating surfaces. However, in some small aero-engines or structures with high requirements for ease of disassembly and assembly, the bearing inner ring and shaft may use a clearance fit (e.g., similar to H7 / h6 grades) or develop an equivalent clearance due to fretting wear after long-term service. Under conditions of high speed, light load, and easy penetration of lubricating oil into the clearance, such clearance-fitted rolling bearings may form a localized hydrodynamic oil film in the shaft-inner ring clearance. This causes asynchronous self-excited movement of the inner ring relative to the shaft, leading to abnormal vibration modes different from traditional pitting / spasting. Since this type of failure often does not manifest as obvious raceway impact pulses, early diagnosis is difficult. However, its consequences may continue to accumulate and eventually cause spindle surface wear, abnormal impact of bearing cage, or even shaft scrap. Therefore, it is urgent to sort out the existing diagnostic approaches and shortcomings at the background technology level.

[0003] Among existing rolling bearing fault diagnosis technologies, the most widely used approach is the spectral criterion and feature extraction method based on rolling element-raceway contact defects (such as pitting, spalling, and cracks). A typical approach involves calculating fault characteristic frequencies based on bearing geometry and rotational speed, such as the inner ring pass frequency (BPFI), outer ring pass frequency (BPFO), their harmonics, and sidebands. Corresponding peak values ​​are then identified in the frequency domain, envelope spectrum, or order spectrum of the vibration signal to determine the fault type and severity. Meanwhile, for variable speed operating conditions, the industry commonly employs order tracking, time-frequency analysis, and energy spectrum methods to map the vibration response to an order domain based on the spindle rotational frequency, improving the distinguishability of fault components under fluctuating speed conditions. For example, Chinese patent CN106769033B discloses an identification framework for variable speed rolling bearings, which typically uses synchronous resampling or order analysis of the rotational speed signal and extracts fault energy and ridge features in the order / time-frequency domain for typical rolling bearing fault identification and classification. This type of technology is effective for periodic impacts / modulations caused by local damage to the raceway, but its focus is still on impact-type defect mechanisms such as BPFI / BPFO. It lacks specific mechanism constraints and discrimination structures for non-contact fit failures caused by fit clearance (such as self-excited whirl of the inner ring relative to the shaft), which can easily lead to missed detections or misjudgments. On the one hand, if a typical impact does not occur, there may be no obvious peak value of the passing frequency in the envelope spectrum. On the other hand, abnormal subsynchronous components may be misattributed to other rotor dynamics problems or external interference.

[0004] Besides traditional raceway damage diagnosis, some existing technologies also attempt to monitor bearing slippage / loose fit / relative rotation of the inner ring. Chinese patent CN112393907B discloses an automatic diagnostic scheme based on frequency sweep analysis / spectrum comparison, which can extract target frequency points from the equipment vibration response and compare them with a preset set of fault frequency points to identify and alarm for bearing faults or slippage. The advantage of this type of scheme lies in its good automation and engineering adaptability, but its essence still leans towards frequency point matching—empirical rule path: it usually requires prior setting of fault frequency points or characteristic ranges, and mostly targets rolling element defects or general anomalies, without establishing a specific discrimination chain for specific causes such as clearance fit, light load, and oil film hydrodynamic instability. In other words, when the source of the anomaly is self-excited whirling of the inner ring under the action of oil film force, rather than raceway impact or simple mechanical loosening, relying solely on general frequency sweep matching may not be able to effectively distinguish it from other subsynchronous vibration sources, thus causing false alarms or unreliable diagnoses.

[0005] Other technologies approach the issue from the perspective of the kinematic difference between the inner ring and the shaft, determining whether the inner ring is rotating relative to the shaft by measuring the difference in period or angular velocity. For example, Japanese Patent JP2003294035A discloses an inner ring creep detection device. Its approach involves detecting the rotation period of the inner ring and the rotation period of the shaft separately and calculating the difference. When the difference exceeds a threshold, it is determined that there is relative motion in the inner ring. Theoretically, this method can directly address the phenomenon of relative motion in the inner ring. However, its implementation often relies on direct or indirect measurement of the inner ring's rotational state. For enclosed structures like aero-engine main bearings and in high-temperature, high-speed environments, there are challenges in sensor placement space, signal stability, and reliability. Furthermore, this type of method focuses on the kinematic judgment of whether relative rotation has occurred, without further addressing the dynamic type of relative motion. Therefore, it is difficult to form engineering criteria for the more characteristic fault mode of inner ring whirl, and it cannot be naturally integrated into vibration diagnostic systems.

[0006] In summary, existing technologies have at least the following shortcomings, limiting their early and reliable identification of whirl faults in the inner ring of aero-engine main shaft: 1) Bias in diagnostic targets: Mainstream methods primarily target impact defects such as raceway pitting / stripping, relying on passing frequency characteristics such as BPFI / BPFO; however, inner ring whirl is a self-excited subsynchronous motion caused by the coupling of the fit clearance and lubricating oil film, and often does not exhibit typical impact envelope characteristics in the early stages, making traditional BPFI / BPFO criteria insensitive. 2) Lack of multi-condition discrimination with mechanistic constraints: Schemes similar to Chinese patent CN112393907B emphasize spectrum point matching and automatic alarm, but fail to organize conditions such as clearance fit, light load, lack of typical passing frequency, and order stability into a mutually constraining discrimination system, making it difficult to elevate empirical subsynchronous peak values ​​into interpretable and verifiable fault modes, thus posing a risk of false alarms. 3) Insufficient engineering feasibility or insufficient information granularity: For example, Japanese patent JP2003294035A directly judges the relative motion of the inner ring through the period difference, corresponding to the existence detection of running / crawling, but the implementation threshold is high under the conditions of high speed and closed structure of aero-engine; moreover, this type of judgment lacks the characterization of the subsynchronous vibration mechanism, cannot effectively distinguish the inner ring vortex from other relative slip forms, and is difficult to support the setting of reliable threshold for early warning.

[0007] Therefore, from an engineering perspective, there is still a need for a method that can be used to address the specific structural premise of clearance-fit rolling bearings. This method should integrate speed synchronization, order domain analysis, and exclusiveness verification within an engineering-feasible sensing and computing framework, and take into account operating conditions such as light loads. This would allow for the development of an interpretable identification method for non-contact fit failures such as inner ring whirl, in order to achieve early warning of potential major damage to the main shaft of aero-engines. Summary of the Invention

[0008] The technical objective of this invention is to provide a fault identification method based on synchronous acquisition of vibration and rotational speed, order analysis, and multi-condition discrimination for the non-contact failure mode of inner ring self-excited whirling relative to the main shaft of rolling bearings in aero-engine main shafts, which may occur under conditions of clearance fit, light load, and easy lubricant infiltration. By capturing stable 0.41×shaft frequency subsynchronous order characteristics and combining BPFI / BPFO exclusive verification and operating condition / structural premise verification, an early and reliable warning of inner ring whirling can be achieved, thereby avoiding major damage caused by main shaft fretting wear, temperature rise, and abnormal impact of the cage.

[0009] Firstly, in order to achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0010] A method for identifying whirl faults in the inner ring of an aero-engine main shaft, the method comprising the following steps:

[0011] S1. Synchronously acquire vibration acceleration signals at the target rolling bearing. With spindle speed signal Wherein, the spindle frequency is defined as the ratio of the spindle speed to 60, and the ratio of any frequency to the spindle frequency is defined as the order;

[0012] S2, regarding the vibration acceleration signal Perform based on spindle speed signal Order analysis to generate order spectrum ,in For order, , The vibration frequency, For the corresponding time point The amplitude of the next-order spectrum;

[0013] S3, in the order spectrum In the process, the intensity index of the target order component is extracted, where the central order of the target order component is... The strength index is: Level energy The calculation formula is:

[0014] ;

[0015] in, For order half bandwidth, The amplitude of the order spectrum;

[0016] S4. Extract the energy ratio of the BPFI / BPFO region and calculate the order corresponding to the frequencies of BPFI and BPFO. , And calculate the energy in that region. and They are defined as follows:

[0017] ;

[0018] ;

[0019] in, For bandwidth;

[0020] S5, when Exceeding the preset threshold, and the order stability index When the preset stability conditions are met and the exclusive fault verification results do not trigger pitting / peeling faults, an inner ring vortex fault warning is output.

[0021] Preferably, step S3 also includes:

[0022] ;

[0023] in, It is the order half-bandwidth. Let be the amplitude of the order spectrum, and It should manifest as a sustained increase in energy over multiple consecutive time windows.

[0024] Preferably, the frequencies of BPFI and BPFO in step S4 are calculated as follows:

[0025] ;

[0026] ;

[0027] in, The number of rolling elements. The diameter of the rolling element, The diameter of the pitch circle. Contact angle, Main spindle frequency.

[0028] As a preferred option, the order stability index in step S5 By calculating the peak order within multiple time windows It is achieved through standard deviation, and the specific calculation formula is as follows: in, Extracted within a time window The order value corresponding to the order. The standard deviation is denoted as .

[0029] Preferably, step S4 also includes calculating the energy ratio of the BPFI and BPFO regions. and Defined as:

[0030] ;

[0031] in, The total energy over the entire frequency spectrum. and This refers to the energy within the BPFI and BPFO regions.

[0032] Preferably, the precondition verification in step S5 includes:

[0033] 1) Verify whether the clearance between the shaft and the inner ring meets the geometric conditions for the occurrence of whirl faults, and set the clearance threshold as follows: ,when When the condition is met, it is determined that the eddy current condition is satisfied.

[0034] 2) Verify whether the operating condition is a light load condition and set the load factor. threshold ,when At that time, it was determined to be a light load condition;

[0035] 3) Verify whether the system contains sliding bearings. If sliding bearings are present, do not perform inner ring whirl determination.

[0036] And / or, the output of step S5 includes:

[0037] 1) When outputting inner ring eddy warning information, additional maintenance suggestions are also included;

[0038] 2) The maintenance recommendations include at least checking the shaft surface for wear marks, verifying the fit clearance, and adjusting the position of the oil supply system or the viscosity of the oil.

[0039] Secondly, the present invention also provides an aero-engine main shaft inner ring vortex fault identification system, which is used to implement the method, including:

[0040] Data acquisition module: used to synchronously acquire vibration acceleration signals and spindle speed signals;

[0041] Order analysis module: used to perform order analysis on the vibration acceleration signal based on the spindle speed signal and generate an order spectrum; Feature extraction module: used to extract the 0.41st order energy. And calculate the order stability index. ;

[0042] Exclusionary fault verification module: used to calculate the energy ratio of the BPFI and BPFO regions and exclude pitting / stripping faults;

[0043] Prerequisite verification module: used to verify whether the shaft-inner ring clearance, operating load and structural configuration meet the prerequisites for whirl fault.

[0044] Judgment and Output Module: Used to output early warning information of inner ring vortex fault based on multi-condition judgment logic.

[0045] Preferably, the feature extraction module is further used to calculate the time window within each time window based on the order spectrum. Level energy It determines whether to trigger a vortex fault warning based on a set threshold.

[0046] Preferably, the exclusive fault verification module calculates the energy ratio between the BPFI and BPFO regions. and It compares the results with a preset threshold to exclude pitting / peeling faults.

[0047] Preferably, the discrimination and output module further provides maintenance suggestions such as shaft surface inspection, fit clearance verification, and oil supply system adjustment based on the output inner ring whirl fault warning information.

[0048] Thirdly, the present invention also provides a computer-readable storage medium having a computer program or instructions stored thereon, which, when executed by a processor, implement the steps of the method.

[0049] Fourthly, the present invention also provides a computer program product, including a computer program or instructions that, when executed by a processor, implement the steps of the method.

[0050] The technical advantages of this invention are as follows: By introducing the specific structural-operating condition of clearance fit, light load, and lubricating oil film hydrodynamic instability into the fault diagnosis process, and based on the order analysis of vibration and rotational speed synchronous acquisition, it can stably extract the 0.41×shaft frequency subsynchronous order component that varies proportionally with the spindle rotational frequency under variable speed or rotational speed fluctuation conditions. This transforms the subsynchronous peak value, which was previously easily regarded as noise or resonance interference, into a repeatable and quantifiable inner ring eddy criterion. At the same time, by introducing exclusive indicators such as the BPFI / BPFO region energy ratio, it significantly reduces the misjudgment of pitting / stripping impact faults as inner ring faults. By combining the probability of eddy current with multiple constraints such as non-slip bearing interference, order stability, light load threshold, and equivalent clearance threshold, a highly reliable closed-loop diagnosis of non-contact fit failures is achieved. In engineering applications, this invention can trigger early warnings before obvious annular bright bands appear on the shaft surface, fretting wear expands, or abnormal impacts occur on the cage. This guides maintenance personnel to take timely measures such as checking fit clearances, adjusting assembly tolerances, optimizing oil supply position / viscosity, or modifying fit forms, thereby reducing the risk of spindle scrapping, reducing sudden downtime and maintenance costs, and improving the safety, maintainability, and life consistency of the aero-engine spindle system. Attached Figure Description

[0051] Figure 1This is a waterfall plot of rotor test acceleration.

[0052] Figure 2 This is a frequency domain analysis diagram of the bearing inner ring vortex moment.

[0053] Figure 3 This is a diagram showing the traces left on the shaft by the whirling of the bearing's inner ring.

[0054] Figure 4 This is a system structure block diagram of the present invention.

[0055] Figure 5 This is a flowchart of the method of the present invention. Detailed Implementation

[0056] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the protection scope of the present invention.

[0057] I. Terminology Explanation

[0058] 1. Main shaft / rotor: The rotating shaft system of an aircraft engine (which may include rotors, couplings, etc.), whose rotational speed is measured by a speed sensor. The rotational angular velocity of the main shaft is denoted as... The unit is .

[0059] 2. Rotation speed Spindle revolutions per unit time, in units .

[0060] 3. Spindle speed Frequency derived from spindle speed, in units ,satisfy: in, Rotational speed ( ).

[0061] 4. Order Any frequency component With spindle speed The ratio satisfies The order can change over time and is often used to characterize the spectral stability of variable speed scenarios.

[0062] 5.0.41 × shaft frequency / 0.41 order: refers to The subsynchronous component, i.e., the frequency is approximately The vibration components.

[0063] 6. Inner ring whirl: When there is a gap (or equivalent gap) between the shaft and the inner ring, and lubricating oil enters the gap to form a local dynamic pressure oil film, the inner ring generates self-excited orbital motion under the tangential leading force of the oil film. This manifests as a stable subsynchronous order component (often falling into the engineering category). (near the level).

[0064] 7. Clearance fit / equivalent clearance : Radial clearance formed by the outer diameter of the shaft and the inner ring diameter of the bearing (or equivalent clearance formed by wear, loosening, etc.), unit .

[0065] 8. Light load condition: The bearing is subjected to a load that is relatively low compared to its rated load. To facilitate engineering implementation, a load factor is introduced. : in, For actual load, For rated load; when If the load is less than the preset threshold, it is considered a light load.

[0066] 9. BPFI / BPFO: The inner ring pass frequency and outer ring pass frequency of a rolling bearing, used to characterize the impact repetition frequency caused by local damage to the raceway. Its calculation can be performed using:

[0067] ;

[0068] ;

[0069] in, The number of rolling elements; The diameter of the rolling element; The diameter of the pitch circle; Contact angle; Main spindle frequency.

[0070] 10. Order stability: refers to the small degree of drift of the target peak order within a continuous time window, reflecting the characteristics of a fixed order caused by the self-excitation mechanism and a proportional change with the rotational speed.

[0071] II. System Structure of the Invention

[0072] To implement the method of this invention, a system for identifying whirl faults in the inner ring of an aero-engine main shaft can be constructed. This system can function as a standalone portable diagnostic device or be integrated into an engine health management system (EHM) or test bench monitoring system. Figure 4 As shown, the system includes at least the following modules (functional modules can interact with each other via bus, network port, serial port or shared memory):

[0073] 1. Data Acquisition Module

[0074] Used for synchronous acquisition of vibration acceleration signals With speed signal The vibration sensor is preferably a piezoelectric accelerometer, mounted on the outer surface of the casing / bearing housing near the target main bearing. The mounting direction can be radial, axial, or triaxial. The speed sensor can be a magnetoelectric / Hall effect / photoelectric speed probe, outputting pulses to calculate instantaneous speed. To ensure... Frequency band analysis accuracy, sampling frequency Optional And configure anti-aliasing filtering.

[0075] 2. Signal preprocessing module

[0076] right Processing includes DC removal, detrending, bandpass filtering, segmented windowing, and amplitude calibration; Pulse shaping, dejittering, and interpolation are performed to obtain a continuous instantaneous frequency. or instantaneous angular velocity .

[0077] 3. Order Analysis Module

[0078] based on right Perform equal-angle resampling to obtain equal-angle sequences. Then perform FFT or short-time FFT on each angular window to obtain the order spectrum. A diagram of tiered waterfalls.

[0079] 4. Feature Extraction Module

[0080] extract The parameters include order energy, peak amplitude, peak significance, order stability index, and the BPFI / BPFO region energy ratio used to exclude pitting / stripping.

[0081] 5. Prerequisite Verification Module

[0082] Read assembly information (tolerance zone or measured dimensions of shaft and inner ring), operating condition information (load / oil temperature / oil supply pressure, etc.) and structural information (whether the system contains sliding bearings) to verify the clearance fit, light load, and absence of sliding bearing interference.

[0083] 6. Judgment and Output Module

[0084] Execute multi-condition discrimination logic, output inner ring vortex warning / alarm and confidence level, and provide maintenance suggestions (such as checking the annular bright band on the shaft surface, verifying the fit clearance, adjusting the oil supply position or viscosity, etc.).

[0085] III. Specific technical routes for implementing the method of the present invention (e.g.) Figure 5 (As shown)

[0086] Step S1, Data Acquisition

[0087] 1. Sensor Configuration and Installation

[0088] Vibration sensor selection: The measurement range should cover the engine test vibration level, and the frequency response should at least cover... If the main focus is The frequency response is in The internal amplitude and phase error should be kept small.

[0089] Installation location: It is preferred to install it in the target main bearing housing or in the adjacent casing to improve the sensitivity to the transmission of bearing-related vibrations; at least one radial channel can be provided, and an axial channel can be added if necessary to help distinguish structural modes.

[0090] Speed ​​sensor: Outputs pulses when aligned with the gear plate or marked point; the number of pulses is recorded as the number of revolutions per second. One (commonly one or more teeth).

[0091] 2. Synchronous acquisition and clock consistency

[0092] Vibration signal With speed signal Sampling must be performed on the same acquisition card or at the same time reference to ensure phase consistency during subsequent equal-angle resampling.

[0093] Sampling frequency Selection: To avoid aliasing, the following conditions must be met. ,in To analyze the highest frequency; in engineering, it is advisable to... To cover It also allows for filtering margin.

[0094] 3. Data segmentation

[0095] The continuously collected data is segmented according to a time window, with each segment having a length of... Desirable The overlap rate can be set for adjacent time windows. (like To balance time resolution and frequency resolution. Define the first... The start and end times of the segment are .

[0096] Step S2, Order Analysis

[0097] The goal of order analysis is to map the vibration signal from the time domain to a domain with rotation angle as the independent variable when the rotation speed may fluctuate, so that the component that changes proportionally with the rotation speed exhibits a stable peak value in the order domain, thereby reliably capturing the 0.41st order subsynchronous component and distinguishing it from other asynchronous interferences.

[0098] S2.1 Rotational speed calculation and instantaneous frequency acquisition

[0099] 1. Pulse to Rotation Speed ​​Conversion

[0100] Let the arrival time sequence of the output pulses from the speed sensor be as follows: The interval between adjacent pulses is .

[0101] If the output per revolution One pulse, then instantaneous speed It can be estimated as follows:

[0102] ;

[0103] in, The unit is rpm. The unit is seconds (s).

[0104] Corresponding instantaneous spindle speed:

[0105] .

[0106] 2. Speed ​​sequence smoothing and interpolation

[0107] To reduce order drift caused by pulse jitter, one can... Perform median filtering or low-pass filtering; the filter window length can be 3-11 pulses.

[0108] To achieve a continuous rotational speed function consistent with the vibration sampling points, interpolation can be performed on the discrete sequence to obtain... or angular velocity The interpolation method can be linear interpolation, spline interpolation, or interpolation that maintains constant values ​​within the segment; in test scenarios where the speed change is relatively gradual, linear interpolation can meet the engineering requirements.

[0109] S2.2 Equal Angle Resampling

[0110] Since FFT is based on equal-time sampling by default, components of the same order will spread or drift in the frequency domain if the rotation speed changes. To stabilize the order components in the spectrum, this invention preferably uses equal-angle resampling:

[0111] 1. Angle Calculation

[0112] Integrating the angular velocity yields the rotation angle: in, The turning angle is measured in rad. ω is the angular velocity (rad / s).

[0113] 2. Constructing an equal-angle sampling sequence

[0114] Let the step size of the equal angle be... (For example, each transfer) If there are 1 point, then Constructing an isoangular sequence. And by reverse calculation (The corresponding time can be obtained by looking up a table / interpolating the numerical values, and then from the original vibration signal) Interpolation obtained Obtain equal-angle signals. After that, any periodic component related to the rotation angle will exhibit a fixed order rather than a frequency that drifts with the rotational speed.

[0115] 3. Corner window segmentation

[0116] In the corner domain pair Segmented by a fixed number of revolutions window, for example, each segment contains If the angle is rotated, then the length of each segment is... This approach ensures that each segment contains the same number of revolutions, making it easier to compare order energy changes under different operating conditions.

[0117] S2.3 Order Spectrum and Waterfall Plot Generation

[0118] 1. Obtain the order spectrum using angular domain FFT.

[0119] Segment the signal for each corner domain Perform an FFT to obtain the magnitude in the order domain. ,in For order index, This refers to the time window or segment number. The order resolution is determined by the window size. Decision. Generally speaking, The larger the value, the higher the order resolution, but the lower the temporal resolution. This is an optional feature in engineering. The system is designed to balance peak localization at the 0.41st order with time-varying characteristics.

[0120] 2. Waterfall / Time-Stage Chart Display

[0121] Will Arranging the data according to a time window sequence yields a waterfall plot or time-order plot, used to observe whether the energy within the 0.38-0.44 order bandwidth exhibits a continuous, stable, and patchy appearance. This visual characteristic is related to... Figure 1 The waterfall diagram shown exhibits a consistent energy level at a certain fixed order.

[0122] In engineering implementation, the horizontal axis of order can be limited to 0-5 (or converted to a frequency axis of 0-5000Hz), and the bandwidth of interest can be highlighted for quick identification.

[0123] The key characteristic of inner-ring vortex is its fixed order: its frequency changes proportionally with the rotational speed, i.e. Therefore, if the rotational speed fluctuates when observed directly in the frequency domain, It will drift around 400-450Hz and may be mistaken for noise; however, in the order domain, the 0.41st order will remain stable, facilitating the determination of stability of subsequent orders. This invention utilizes this point to transform the empirically determined 0.41× peak into a repeatable, verifiable, and condition-linked diagnostic evidence chain.

[0124] Step S3, Eddy Candidate Feature Extraction

[0125] The objective of step S3 is to quantify the 0.41st-order subsynchronous component in the order spectrum, forming a feature set that can be used for threshold discrimination, trend warning, and confidence assessment. Preferably, it includes at least: 0.41st-order energy / amplitude, peak significance, order stability, and first-order amplitude fluctuation characteristics related to asynchrony.

[0126] S3.1 Definition and Calculation of the 0.41st Order Strength Index

[0127] 1. Bandwidth settings

[0128] Due to engineering noise and resolution limitations, the actual peak value will not strictly fall within the range of... Single point. For this purpose, a bandwidth half-width is defined. ,For example The bandwidth to be considered is... In addition, to enhance robustness, a wider candidate band [0.38, 0.44] can be set for peak search.

[0129] 2. Energy-type indicators

[0130] In the Within a time window, the 0.41st order energy can be defined as:

[0131] ;

[0132] in, For the first The order spectral amplitude function of the window; It is the order half bandwidth.

[0133] If discrete spectral points are used, a summation approximation can be used:

[0134] ;

[0135] in, It is the set of discrete order points that fall within the 0.41 order bandwidth.

[0136] 3. Amplitude-type indicators

[0137] Peak amplitude can also be extracted directly:

[0138] ;

[0139] in, This corresponds to the maximum peak amplitude within the order range of 0.38 to 0.44. To ensure that the peak value indeed falls near the 0.41st order, additional constraints can be applied. ,in The order corresponding to the peak value.

[0140] 4. Peak significance

[0141] To avoid false alarms caused by an increase in the overall vibration level, a significance index of the peak value relative to the background can be introduced. :

[0142] ;

[0143] The denominator is the median of the spectral amplitude in the range of 0.30 to 0.60, which is used to characterize the background level. The larger the value, the more prominent the 0.41st order peak is.

[0144] S3.2 order stability index

[0145] Another key piece of evidence for inner-circle vortexes is order stability: the peak order remains constant across multiple consecutive time windows. It should fluctuate slightly around 0.41, rather than going through a random walk.

[0146] 1. Peak order sequence extraction

[0147] For each time window Searching for peak points within the range [0.38, 0.44] yields the peak order. Peak amplitude .

[0148] 2. Stability Measurement

[0149] In continuous Calculate the standard deviation over a time window:

[0150] ;

[0151] in, The length of the statistical window (e.g., K=5~30). When Less than the threshold When the order is stable, it is determined.

[0152] It can also calculate the peak amplitude persistence (duty cycle) indicator. :

[0153] ;

[0154] in, For indicator functions; This is the amplitude threshold. The higher the value, the more persistent the 0.41 order.

[0155] S3.3 First-order amplitude fluctuation coefficient

[0156] This invention proposes using first-order amplitude fluctuation as one of the auxiliary features to quantify the effective time-varying eccentricity and asynchronous effects caused by eddy currents. The implementation is as follows:

[0157] 1. Extract the first-order amplitude sequence

[0158] In each time window Extract First-order amplitude near the order For example in Take the maximum value within the bandwidth:

[0159] ;

[0160] in, It is a first-order bandwidth half-width.

[0161] Calculate the volatility coefficient:

[0162] ;

[0163] in, and These are the standard deviation and the mean, respectively. This is the length of the statistical window. The larger the value, the more obvious the fluctuation of the first-order amplitude, indicating that the system has a non-synchronous or time-varying eccentric mechanism. The continuous occurrence of the 0.41st order can enhance the confidence of eddy current discrimination.

[0164] S3.4 combination Figure 2 Feature Explanation

[0165] Figure 2 The frequency domain analysis at a certain moment is given: when the spindle speed is approximately 58920 rpm, ,but As can be seen in the picture, it is close to The significant peak value; at the same time in A peak value also exists at the axis frequency (approximately). In this invention, the peak value, after being mapped to the order domain, should exhibit the following behavior: A stable peak value, and a small peak value within multiple consecutive windows. .

[0166] Step S4, Exclusive Fault Verification

[0167] The purpose of step S4 is to avoid misdiagnosing common raceway pitting / scraping failures as inner ring whirl. Since pitting / scraping is an impact-type failure, its energy is usually significant at the passing frequency (BPFI / BPFO) and its harmonics and sidebands. Inner ring whirl is a self-excited orbital motion of non-contact fit failure, and it usually does not produce a typical impact envelope in the early stages. Therefore, the energy ratio in the passing frequency region can be used for exclusion verification.

[0168] S4.1 through frequency calculation and region setting

[0169] 1. Through frequency calculation

[0170] Calculate according to the aforementioned formula , To be used in the order domain, the corresponding order can be defined:

[0171] ;

[0172] in, Main spindle frequency.

[0173] 2. Regional bandwidth

[0174] Let the half-width of the frequency region be... Then the BPFI region is The same applies to the BPFO region.

[0175] S4.2 Regional Energy Ratio and Criteria

[0176] 1. Regional Energy

[0177] ;

[0178] ;

[0179] in, The order spectral amplitude function, It is the half-width of the frequency range.

[0180] 2. Energy ratio

[0181] To compensate for changes in the overall vibration level, an energy ratio can be defined:

[0182] ;

[0183] in, For a preset order range (e.g.) The total energy within the (order).

[0184] 3. Exclusivity determination

[0185] when or Exceeding the pitting / peeling trigger threshold (The threshold can be set by baseline statistics or experience) When the threshold is not exceeded, the system tends to prioritize impact failures and does not proceed to the inner-loop vortex conclusion; when neither of these thresholds is exceeded... If no obvious peak value of the passing frequency or its harmonic sideband is observed, step S4 determines that the passing frequency is not significant, thus providing support for the inner circle vortex discrimination.

[0186] Step S5, Verification of Operating Conditions and Structural Prerequisites

[0187] This invention emphasizes that inner-ring vortex does not occur under arbitrary conditions, but requires the simultaneous fulfillment of prerequisites such as clearance, light load, oil film conditions, and structural exclusion. Step S5 is used to transform the mechanistic prerequisites into verification conditions that can be read / calculated in engineering, thereby significantly reducing false alarms.

[0188] S5.1 Gap Condition Verification

[0189] 1. Gap acquisition method

[0190] Method A: Tolerance Zone Calculation: Read the tolerance zones of the shaft's outer diameter and inner ring diameter, estimate the maximum / minimum clearance based on the nominal dimensions, and take the equivalent clearance. This is a representative value for statistical significance.

[0191] Method B: Calculation based on actual measured dimensions: Measure the outer diameter of the shaft. With inner ring diameter Then the diameter gap is The radial clearance is approximately: in, , The unit can be ,but Need to be converted to .

[0192] 2. Threshold Determination

[0193] Let the gap threshold be ,when If the geometric conditions for forming an effective dynamic pressure wedge are met, it is determined that the conditions are met; otherwise, it indicates that the risk of eddy currents is low or that further judgment is needed based on other evidence.

[0194] S5.2 Light Load Condition Verification

[0195] 1. Load data source

[0196] load It can be obtained from test bench loading commands, torque estimation, engine controller output, or empirical model calculations. If it cannot be obtained directly... Alternatively, alternative indicators can be used, such as the ratio of the load valve opening to the output power and the estimated torque.

[0197] 2. Load factor determination

[0198] Calculate according to the aforementioned formula Set a light load threshold. ,when The load is judged to be light. The significance of the light load is that the rolling element load is small, the drag force of the rolling element on the inner ring is insufficient to keep the inner ring synchronized, and the oil film continuity is easier to maintain, thus making self-excited whirl more likely to occur.

[0199] S5.3 Structural exclusion condition: No interference from sliding bearings

[0200] Classical oil film whirl is commonly found in sliding bearing systems and can also generate subsynchronous components of 0.4~0.48 × shaft frequency. To avoid misdiagnosing sliding bearing oil film whirl as rolling bearing inner ring whirl, this invention requires verification of the presence of sliding bearings in the target rotor support structure. The implementation includes: 1) reading the configuration table from the engine structure BOM or health management system and marking the support type (rolling / sliding); 2) if sliding bearings are present in the system, further determining the source of the subsynchronous component (through multi-point comparison, coherence analysis, or structural transmission path determination); if it cannot be ruled out, reducing the confidence level of the inner ring whirl conclusion or not outputting this conclusion.

[0201] Step S6, multi-condition discrimination output

[0202] Step S6 organizes the information extracted / verified in steps S3 to S5 into an evidence chain, forming an interpretable, verifiable, and engineering-applicable judgment rule. Unlike alarms based solely on a single peak value, this invention employs a combination of logic including core criteria, exclusive verification, prerequisite verification, and stability constraints.

[0203] S6.1 Core Criterion: 0.41-order intensity exceeds the threshold

[0204] 1. Fixed threshold method

[0205] set up The first energy threshold is Or peak threshold is .when or And continue If the above is above the window, then the candidate for vortex is determined to be valid.

[0206] 2. Adaptive threshold method

[0207] Data collected under normal baseline operating conditions. Window sample, calculation Average energy of the first order with standard deviation , build:

[0208] ;

[0209] in, For threshold coefficients (e.g.) ).when The anomaly was considered significant at the time. This method is adaptable to different engines, different mounting points, and different sensor sensitivity differences.

[0210] S6.2 Stability Constraint: Order stable and persistent

[0211] when (like )and (like When the subsynchronous component is considered to have self-excited vortex characteristics, rather than occasional interference or short-term resonance amplification, it is believed that the component has these characteristics.

[0212] S6.3 Exclusivity constraint: Passing frequency is not significant

[0213] when , All below the threshold And in When no significant peak value or harmonic sideband appears in the neighborhood, pitting / stripping impact failures are excluded as the main cause, thereby enhancing the consistency of the explanation for non-contact mating failures.

[0214] S6.4 Prerequisite Constraints:

[0215] when , Furthermore, if the structure is confirmed to be free from interference from sliding bearings, the chain is judged to be consistent with the physical mechanism, and an inner ring eddy warning is allowed to be output.

[0216] S6.5 Discrimination of Output Format and Maintenance Guidelines

[0217] An inner-circle eddy warning will be output when the following conditions are met simultaneously: 0.41st order intensity anomaly ( or (Beyond the threshold); order stability satisfies () And persistence Satisfied); BPFI / BPFO exclusivity verification passed (energy ratio in the frequency range did not trigger impact fault); prerequisite verification passed (gap, light load, and structural exclusion).

[0218] Output may include: alert level (warning / alarm); key evidence (e.g., current situation). , , , , , );

[0219] Maintenance recommendations: Verify the fit between the shaft and inner ring, check the oil supply position, oil temperature and viscosity, and if necessary, stop the machine to check whether there is an annular bright band or friction band on the spindle surface.

[0220] IV. Specific Application Examples

[0221] The following application examples were all performed on a laboratory engine spindle bearing test bench. The test bench enables closed-loop speed control, oil injection lubrication supply, and adjustable load (no load / light load / medium load), and vibration acceleration sensors and speed pulse sensors are arranged on the outer wall of the bearing housing. Each example uses the same data acquisition and analysis process, controlling variables only in assembly fit, load, and fault mode to verify the invention's early identification capability, false alarm suppression capability, and mechanism consistency regarding spindle inner ring whirl faults.

[0222] 1. Test objects and assembly / lubrication conditions

[0223] Bearing type: angular contact ball bearing (the experimental piece is a 7202C type structure).

[0224] Assembly fit: Spindle tolerance zone is The tolerance zone of the bearing inner ring bore is This forms a clearance fit; the measured equivalent radial clearance .

[0225] Lubrication method: oil spray lubrication, with the oil outlet located near the inner ring end face of the bearing, making it easier for the lubricating oil to enter the shaft-inner ring clearance.

[0226] Load condition: No-load test run, load percentage less than (Light load).

[0227] Operating speed: stable at .

[0228] Based on the above rotational speeds, the spindle frequency can be obtained:

[0229] ;

[0230] in, spindle speed ( ).

[0231] 2. Phenomenon and Data Characteristics

[0232] (1) Stable subsynchronous components appear in the waterfall plot

[0233] like Figure 1 As shown in the acceleration waterfall plot, a relatively stable and continuous energy band appears on the frequency axis during the latter part of the test time window. The vernier reading shows the frequency near a typical moment. There is a considerable amplitude at this point ( Figure 1Vernier mark This frequency component is not a random scattered point, but rather exhibits continuity and repeatability over time, consistent with self-excited vibration rather than accidental impact.

[0234] (2) The 0.41× axis frequency component and the 1× axis frequency component are both significant.

[0235] like Figure 1 (3D waterfall diagram) and Figure 2 As shown in the (frequency domain signal), there is a significant peak at 407 Hz, and also a significant peak at 982 Hz (i.e., 1 × axis frequency). Figure 2 The text indicates the rotor's 0.41st harmonic (approximately 407 Hz) and 1st harmonic (approximately 982 Hz) frequencies, and provides the vernier readings of the peak amplitudes.

[0236] The amplitude is approximately ;

[0237] The amplitude is approximately .

[0238] To verify their order relationship, the corresponding order is calculated:

[0239] ;

[0240] in, This represents the order corresponding to the subsynchronous peak. Its value falls within... Within the typical order range of eddy currents, and consistent with the mechanism described in this invention where 0.41× is often used as the criterion in engineering.

[0241] 3. Diagnostic process according to the method of the present invention (1) Order analysis and candidate bandwidth focusing

[0242] right Perform speed synchronization order analysis and generate an order waterfall plot (corresponding to...) Figure 1 / Figure 2 (The way it is displayed). Within the order range, stable peak points were continuously captured. nearby.

[0243] (2) 0.41 order intensity index overthreshold and persistence

[0244] Calculation within a continuous time window ,Discover It rises significantly relative to the baseline and exhibits a continuous pattern; meanwhile, the peak amplitude... It remains above the threshold across multiple windows. This result indicates that the subsynchronous component is not random noise.

[0245] (3) Order stability verification

[0246] In continuous Within a time window, the peak order around Slight fluctuations, If the condition of being less than the stability threshold is met, it indicates that the component changes proportionally with the rotational speed and remains stable in order, which is consistent with the typical characteristics of self-excited eddy caused by oil film hydrodynamic instability.

[0247] (4) Exclusivity check: BPFI / BPFO region is not significant

[0248] Verification of the frequency neighborhood energy revealed no significant enhancement of BPFI / BPFO or its harmonics and sidebands; combined with Figure 3 The overall spectrum structure shows that, except for the 1×axis frequency and 0.41×axis frequency, the typical characteristics of a pass-through frequency main peak + broadband rise + modulation sideband combination of impulsive faults are not observed. This exclusive result supports the judgment of non-impulsive coordination failure.

[0249] (5) Prerequisite verification:

[0250] Assembly information display The gap threshold condition is met; the load percentage is less than [a certain value]. The test object met the light load conditions; its support structure was a rolling bearing system with no interference from sliding bearings. The preconditions and mechanism were perfectly matched.

[0251] (6) Multi-condition discrimination output

[0252] Based on the comprehensive conditions of significant intensity of 0.41, stable order, frequency exclusion, and fulfillment of preconditions, the system outputs an early warning of eddy current in the inner ring of the main shaft.

[0253] 4. Shutdown, disassembly, inspection, and verification (corresponding to) Figure 3 Shaft surface marks)

[0254] After the test, stop the machine and disassemble to inspect the mating surfaces of the spindle and bearing inner ring. For example... Figure 3 As shown, a distinct annular bright band / wear mark appears on the spindle surface, and its morphological characteristics are consistent with the asynchronous orbital motion of the inner ring relative to the spindle (inner ring running / fretting wear). This physical evidence and the diagnostic results form a closed loop verification, demonstrating that the present invention can reliably identify inner ring whirl without relying on direct measurement of the inner ring rotational speed, solely through vibration-rotational speed synchronization and multi-condition discrimination.

[0255] The foregoing description of embodiments of the present invention, through which those skilled in the art are able to implement or use the present invention, will be readily apparent to those skilled in the art. Various modifications to these embodiments will be readily apparent to those skilled in the art. The general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the present invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novelty disclosed herein.

[0256] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0257] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0258] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0259] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0260] In a typical configuration, a computing device includes one or more processors (CPU), input / output interfaces, network interfaces, and memory.

[0261] Memory may include non-persistent memory in computer-readable media, such as random access memory (RAM) and / or non-volatile memory, such as read-only memory (ROM) or flash RAM. Memory is an example of computer-readable media.

[0262] Computer-readable media includes both permanent and non-permanent, removable and non-removable media that can store information using any method or technology. Information can be computer-readable instructions, data structures, modules of programs, or other data. Examples of computer storage media include, but are not limited to, phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, CD-ROM, digital versatile optical disc (DVD) or other optical storage, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other non-transferable medium that can be used to store information accessible by a computing device. As defined herein, computer-readable media does not include transient computer-readable media, such as modulated data signals and carrier waves.

Claims

1. A method for identifying whirl faults in the inner ring of an aero-engine main shaft, characterized in that, The method includes the following steps: S1. Synchronously acquire vibration acceleration signals at the target rolling bearing. With spindle speed signal Wherein, the spindle frequency is defined as the ratio of the spindle speed to 60, and the ratio of any frequency to the spindle frequency is defined as the order; S2, regarding the vibration acceleration signal Perform based on spindle speed signal Order analysis to generate order spectrum ,in For order, , The vibration frequency, For the corresponding time point Lower order spectrum Main spindle frequency; S3, in the order spectrum In the process, the intensity index of the target order component is extracted, where the central order of the target order component is... The strength index is: Tier Energy The calculation formula is: ; in, For order half bandwidth, The amplitude of the order spectrum; S4. Extract the energy ratio of the BPFI / BPFO region and calculate the order corresponding to the frequencies of BPFI and BPFO. , And calculate the energy in that region. and They are defined as follows: ; ; in, For bandwidth; S5, when Exceeding the preset threshold, and the order stability index When the preset stability conditions are met and the exclusive fault verification results do not trigger pitting / peeling faults, an inner ring vortex fault warning is output.

2. The method according to claim 1, characterized in that: Step S3 also includes: ; in, It is the order half-bandwidth. Let be the amplitude of the order spectrum, and It manifests as a sustained increase in energy over multiple consecutive time windows.

3. The method according to claim 1, characterized in that: The frequencies of BPFI and BPFO in step S4 are calculated as follows: ; ; in, The number of rolling elements. The diameter of the rolling element, The diameter of the pitch circle. Contact angle, Main spindle frequency.

4. The method according to claim 1, characterized in that: In step S5, the order stability index By calculating the peak order within multiple time windows It is achieved through standard deviation, and the specific calculation formula is as follows: in, Extracted within a time window The order value corresponding to the order. The standard deviation is denoted as .

5. The method according to claim 1, characterized in that: Step S4 also includes calculating the energy ratio of the BPFI and BPFO regions. and Defined as: ; in, The total energy over the entire frequency spectrum. and This refers to the energy within the BPFI and BPFO regions.

6. The method according to claim 1, characterized in that: The prerequisite verification in step S5 includes: 1) Verify whether the clearance between the shaft and the inner ring meets the geometric conditions for the occurrence of whirl faults, and set the clearance threshold as follows: ,when When the condition is met, c is the equivalent clearance; 2) Verify whether the operating condition is a light load condition and set the load factor. threshold ,when At that time, it was determined to be a light load condition; 3) Verify whether the system contains sliding bearings. If sliding bearings are present, do not perform inner ring whirl determination. And / or, the output of step S5 includes: 1) When outputting the "inner ring vortex" warning message, additional maintenance suggestions are also included; 2) The maintenance recommendations include at least checking the shaft surface for wear marks, verifying the fit clearance, and adjusting the position of the oil supply system or the viscosity of the oil.

7. A fault identification system for the inner ring of an aero-engine main shaft, characterized in that, The system is used to implement the method according to any one of claims 1-6, comprising: Data acquisition module: used to synchronously acquire vibration acceleration signals and spindle speed signals; Order analysis module: used to perform order analysis on the vibration acceleration signals based on the spindle speed signals and generate an order spectrum; Feature extraction module: used to extract the 0.41st order energy. And calculate the order stability index. Exclusionary fault verification module: used to calculate the energy ratio of the BPFI and BPFO regions and exclude pitting / stripping faults; Prerequisite verification module: used to verify whether the shaft-inner ring clearance, operating load and structural configuration meet the prerequisites for whirl faults; Judgment and output module: used to output inner ring whirl fault warning information based on multi-condition judgment logic.

8. The system according to claim 7, characterized in that: The feature extraction module is further used to calculate the value within each time window based on the order spectrum. Level energy And based on the set threshold, determine whether to trigger a vortex fault warning; And / or, the exclusive fault verification module calculates the energy ratio of the BPFI and BPFO regions. and It is compared with a preset threshold to exclude pitting / peeling faults; And / or, based on the output "inner ring whirl fault" warning information, the discrimination and output module further provides maintenance suggestions for shaft surface inspection, fit clearance verification, and oil supply system adjustment.

9. A computer-readable storage medium having a computer program or instructions stored thereon, characterized in that, When the computer program or instructions are executed by a processor, they implement the steps of the method according to any one of claims 1-6.

10. A computer program product, comprising a computer program or instructions, characterized in that, When the computer program or instructions are executed by a processor, they implement the steps of the method according to any one of claims 1-6.

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