An Automated Method for Identifying Near-Frequency Modal Parameters Based on Joint Diagonalization

Abstract

This invention discloses an automated method for identifying near-frequency modal parameters based on joint diagonalization of spectral density. The method includes acquiring the response power spectral density matrix of the system under test and determining the frequency peak points. When dense near-frequency components exist and the frequency peak points belong to the resonant frequency band, the natural frequency and damping ratio are determined based on the rotation matrix determined by the response power spectral density matrix. This invention utilizes joint diagonalization technology to achieve automated identification of near-frequency modal parameters, while improving the accuracy of mode shape identification. By using joint diagonalization to process the power spectral density matrix of near-frequency modes, the natural frequency and damping ratio of the system under test are automatically extracted. Based on the natural frequency and damping ratio, the mode shape of the system under test can be determined. Thus, efficient identification of near-frequency modes is achieved automatically through algorithms, reducing human intervention and significantly improving analysis efficiency. This method is suitable for online monitoring and modal analysis of large-scale structures. This invention has wide applications in the field of control technology.

Description

Technical Field

[0001] This invention relates to the field of control technology, and in particular to an automated method for identifying near-frequency modal parameters of spectral density based on joint diagonalization. Background Technology

[0002] Modal parameter identification is a key technology in structural dynamics and health monitoring. Its goal is to extract the natural frequencies (natural frequencies), damping ratios, and mode shapes of a structure from its vibration response. These parameters can effectively characterize the dynamic performance of a structure and provide important information for fault diagnosis, fatigue analysis, and life assessment.

[0003] Current modal parameter identification (MTI) techniques generally utilize frequency domain data from Fourier transforms for MPI identification. This method requires assumptions about the initial damping ratio, which is extremely difficult to estimate in practical engineering, often making accurate estimation impossible. The accuracy of the initial damping ratio directly impacts the identification results. Furthermore, current MPI techniques typically require manual determination of the frequency bandwidth of dense near-frequency modes. This process relies on observing the resonance frequency band of the spectral density image near the near-frequency modes. However, in actual measurements, the coupling of multiple modal frequencies can make the resonance bandwidth of near-frequency modes indistinct on the spectral density image, increasing the difficulty of manually selecting frequency bands. In addition, the number of dense near-frequency modes also needs to be determined manually, placing high demands on non-professional operators or the analysis of complex systems, potentially leading to errors or inefficiency. On the other hand, due to the small intervals between frequencies, dense near-frequency modes can cause modal responses to overlap in the frequency domain, further increasing the difficulty of MPI identification and making it difficult to distinguish near-frequency modes. Summary of the Invention

[0004] To address the technical problems encountered in modal parameter identification, such as difficulties in estimating damping ratio, inefficiency and error-proneness of relying on manual observation, and difficulty in distinguishing near-frequency modes, the present invention aims to provide an automated method for identifying near-frequency modal parameters based on joint diagonalization of spectral density.

[0005] This invention includes an automated method for identifying spectral density near-frequency modal parameters based on joint diagonalization. The automated method for identifying spectral density near-frequency modal parameters based on joint diagonalization includes the following steps:

[0006] Acquire the kinematic information of the system under test;

[0007] Based on the kinematic information, determine the response power spectral density matrix;

[0008] Based on the response power spectral density matrix, determine the frequency peak points and the presence state of dense near frequencies;

[0009] When the presence of dense near frequencies indicates the existence of dense near frequencies, determine whether the frequency peak point belongs to the resonant frequency band.

[0010] When the frequency peak point belongs to the resonant frequency band, the rotation matrix is ​​determined according to the response power spectral density matrix.

[0011] Based on the rotation matrix, the natural frequency and damping ratio of the system under test are determined.

[0012] Furthermore, acquiring the kinematic information of the detected system includes:

[0013] The system under test is detected by an accelerometer sensor to obtain acceleration response data;

[0014] The acceleration response data is used as the kinematic information.

[0015] Further, determining the response power spectral density matrix based on the kinematic information includes:

[0016] According to the formula

[0017]

[0018] Perform calculations; among which, The response power spectral density matrix, The total number of sampling points for the kinematic information. The sampling time interval for the kinematic information is... and This refers to the acceleration response data.

[0019] Further, determining the frequency peak point and the presence state of dense near frequencies based on the response power spectral density matrix includes:

[0020] According to the formula

[0021]

[0022] Singular value decomposition is performed on the response power spectral density matrix;

[0023] The frequency peak points are extracted using an automatic peak picking algorithm. .

[0024] Further, determining the frequency peak point and the presence state of dense near frequencies based on the response power spectral density matrix includes:

[0025] Based on the results of singular value decomposition of the response power spectral density matrix, the local extrema of the third-order singular value, the local extrema of the second-order singular value, and the maximum value of the first-order singular value are obtained.

[0026] The first ratio is determined by comparing the local extreme points of the third-order singular value with the maximum value of the first-order singular value;

[0027] When the first ratio is greater than the first threshold, it is determined that there are 3 dense near frequencies; otherwise, the second ratio is determined by comparing the local extreme point of the second-order singular value with the maximum value of the first-order singular value.

[0028] When the second ratio is greater than the second threshold, it is determined that there are two dense near frequencies; otherwise, it is determined that there are no dense near frequencies.

[0029] Further, determining whether the frequency peak point belongs to the resonant frequency band includes:

[0030] The frequency peak point is obtained based on the result of singular value decomposition of the response power spectral density matrix. The first singular value vector at ... Second-order singular value vector ;

[0031] According to the formula

[0032]

[0033] Determine the transformation matrix ;in, yes The conjugate matrix, yes The conjugate matrix;

[0034] According to the formula

[0035]

[0036] Sure and ;

[0037] According to the formula

[0038]

[0039] calculate ;

[0040] when If the frequency peak value is greater than the third threshold, the peak value is determined to belong to the resonant frequency band; otherwise, the peak value is determined not to belong to the resonant frequency band.

[0041] Further, determining the rotation matrix based on the response power spectral density matrix includes:

[0042] The rotation matrix is ​​determined using the Jacobi rotation algorithm.

[0043] Further, determining the rotation matrix using the Jacobi rotation algorithm includes:

[0044] According to the formula

[0045]

[0046] Determine the input matrix ,in These are the singular values ​​obtained by performing singular value decomposition on the response power spectral density matrix;

[0047] Set threshold ;

[0048] Set the initial value of the rotation matrix. ;

[0049] make ,when At that time, the following steps (1)-(3) are executed repeatedly:

[0050] (1) Let Repeat the following steps (101)-(105):

[0051] (101) According to the formula

[0052] ;

[0053] Perform calculations;

[0054] (102) According to the formula

[0055] ;

[0056] Perform calculations;

[0057] (103) Take The normalized eigenvector corresponding to the largest eigenvalue ,in ;

[0058] (104) Calculate the rotation parameters ;

[0059] (105) According to the formula

[0060]

[0061] Construct rotation matrix ;

[0062] (2) According to the formula Update;

[0063] (3) According to the formula Update;

[0064] Output the results obtained from the last execution of steps (1)-(3) , as the rotation matrix.

[0065] Further, determining the natural frequency and damping ratio of the system under test based on the rotation matrix includes:

[0066] According to the formula

[0067]

[0068] Perform calculations to determine the matrix. ;

[0069] According to the matrix The natural frequency and the damping ratio are determined.

[0070] Furthermore, the matrix Determining the natural frequency and the damping ratio includes:

[0071] According to the formula

[0072]

[0073] Determine the natural frequency and the damping ratio .

[0074] The beneficial effects of this invention are as follows: The automated method for identifying near-frequency modal parameters based on joint diagonalization in the embodiments utilizes joint diagonalization technology to achieve automated identification of near-frequency modal parameters, while improving the accuracy of mode shape identification. Specifically, the power spectral density matrix of near-frequency modes is processed using joint diagonalization technology, enabling automated extraction of the natural frequency and damping ratio of the system under test. Based on the natural frequency and damping ratio, the mode shape of the system under test can be determined, thereby achieving efficient identification of near-frequency modes through automated algorithm, reducing human intervention, significantly improving analysis efficiency, and is suitable for online monitoring and modal analysis of large-scale structures. Attached Figure Description

[0075] Figure 1 This is a schematic diagram illustrating the steps of the automated method for identifying near-frequency modal parameters of spectral density based on joint diagonalization in the embodiment;

[0076] Figure 2 This is a flowchart illustrating the automated method for identifying near-frequency modal parameters of spectral density based on joint diagonalization in the embodiments.

[0077] Figure 3 This is a schematic diagram illustrating the principle of steps S3-S6 in the embodiment. Detailed Implementation

[0078] Terminology Explanation:

[0079] Joint diagonalization: A linear algebraic technique used to simultaneously transform multiple sets of matrices into approximately diagonal forms. This method can be applied to signal processing, blind source separation, modal analysis, and other fields requiring the extraction of useful information from matrix features. The mathematical basis of this technique is finding an optimal solution that, after a unified transformation, makes the multiple sets of matrices as close as possible to a diagonal matrix, while preserving the inherent properties of the data structure.

[0080] Power spectral density: A quantitative description of a signal or system in the frequency domain, typically used to analyze the distribution of a signal's energy or power at different frequencies.

[0081] Modal parameter identification (MTI) is a technique used to analyze and extract the dynamic characteristics of structures or systems. By studying the input-output relationship of a system, it determines parameters such as natural frequencies, damping ratios, and mode shapes. These parameters describe the vibration behavior of a system under dynamic loading and are commonly used in engineering vibration analysis, structural health monitoring, and mechanical system design. MPI effectively assesses the dynamic characteristics of structures or systems and plays a crucial role in dynamic design optimization, fault diagnosis, vibration control, and system reliability assessment.

[0082] Automation: Utilizing automated tools or algorithms to make the modal parameter identification process more efficient and intelligent, reducing human intervention.

[0083] Dense near-frequency: This refers to a phenomenon in modal parameter identification where the natural frequencies of different modes are very close to each other. This situation typically occurs in complex structures or systems, such as mechanical, aerospace, and civil engineering structures with high symmetry or strong coupling.

[0084] In this embodiment, refer to Figure 1 An automated method for identifying near-frequency modal parameters based on joint diagonalization of spectral density includes the following steps:

[0085] S1. Obtain the kinematic information of the system being tested;

[0086] S2. Determine the response power spectral density matrix based on kinematic information;

[0087] S3. Determine the frequency peak points and the existence state of dense near frequencies based on the response power spectral density matrix;

[0088] S4. When the presence of dense near frequencies indicates the existence of dense near frequencies, determine whether the frequency peak point belongs to the resonant frequency band.

[0089] S5. When the frequency peak point belongs to the resonant frequency band, determine the rotation matrix based on the response power spectral density matrix;

[0090] S6. Determine the natural frequency and damping ratio of the system under test based on the rotation matrix.

[0091] Steps S1-S6 can be performed using a computer. The process of steps S1-S6 is as follows: Figure 2 As shown.

[0092] In step S1, the system being tested can be a structure that may vibrate in fields such as machinery, aerospace and civil engineering, such as gears and shafts.

[0093] In step S1, the kinematic information to be acquired can be the displacement, velocity, or acceleration of the system being tested. In this embodiment, the acceleration response data of the system being tested is used as an example of kinematic information. Specifically, when performing step S1, an accelerometer sensor can be used to detect the system being tested and obtain acceleration response data. This serves as the kinematic information of the system being detected. Indicates time.

[0094] In step S2, the Welch method can be used based on kinematic information. The response power spectral density matrix was calculated. Specifically, the calculation can be performed using the following formula:

[0095]

[0096]

[0097] in, In response to the power spectral density matrix, The total number of sampling points for kinematic information. The sampling time interval for kinematic information, and These are the parameters used in the calculation process. and All are acceleration response data The function form, through and To distinguish different variables.

[0098] In step S3, the response power spectral density matrix is... Singular value decomposition can be used to obtain frequency peaks. Specifically, this can be achieved using the formula...

[0099]

[0100] For the response power spectral density matrix Perform singular value decomposition to obtain , and Identifying singular values ​​and using an automatic peak picking algorithm to extract frequency peak points. .

[0101] In step S3, the response power spectral density matrix can also be used as a reference. Determine the presence of dense near frequencies, such as whether there are dense near frequencies in the vibration spectrum of the system being tested, and if so, the number of dense near frequencies.

[0102] The number of near-frequency modes can be determined by observing singular values ​​to identify the presence and quantity of dense near-frequency modes. For example, when there are no near-frequency modes, the first-order singular value is significantly larger than other-order singular values. Similarly, when dense near-frequency modes exist, the number of larger singular values ​​corresponds to the number of dense near-frequency modes. This process can be implemented manually. In this embodiment, based on the characteristics of the above process, a criterion for automatically detecting the number of near-frequency modes is designed, which is then applied to automatically detect the number of dense near-frequency modes near a specific frequency when executing step S3.

[0103] In this embodiment, the principle of steps S3-S6 is as follows: Figure 3 As shown.

[0104] Specifically, in step S3, for any specific frequency in the vibration spectrum of the system being tested:

[0105] The first step is to find the power spectral density matrix of the response. Find the local extrema of the third-order singular value at this specific frequency among the decomposed singular values. Local extrema of the second-order singular values and the maximum value of the first-order singular value ;

[0106] The second step is to use the local extrema of the third-order singular values. The maximum value of the first-order singular value In comparison, determine the first ratio. ;

[0107] The third step is to determine the first ratio. The size, if If the value is greater than the first threshold (specifically 0.2), then it can be determined that there are three dense near-frequency modes near this specific frequency in the vibration spectrum of the detected system. If the value is less than or equal to the first threshold, then further judgment can be made;

[0108] The fourth step is to use the local extrema of the second-order singular values. The maximum value of the first-order singular value In comparison, determine the second ratio. ;

[0109] Fifth step, determine the second ratio. The size, if If the value is greater than the second threshold (specifically, it could be 0.2), then it can be determined that there are two dense near-frequency modes near this specific frequency in the vibration spectrum of the detected system. If the frequency is less than or equal to the second threshold, then it can be determined that there are no dense near-frequency modes near this specific frequency point.

[0110] In step S3, it is determined that there are dense near frequencies (e.g., frequency points). If there are three or two densely packed near-frequency frequencies in its vicinity, proceed to step S4 to determine the frequency peak point. Does it belong to the resonant frequency band, that is, is it located at the frequency peak point in the spectrum? Are the nearby frequency points within the resonant frequency band?

[0111] In the absence of near-frequency modes, the resonant frequency bandwidth can be automatically selected during modal parameter identification using the Modal Assurance Criterion (MAC). However, when near-frequency modes are present, calculating the MAC value is no longer suitable for automatically selecting the resonant frequency bandwidth. Step S4 is performed precisely in the case of dense near-frequency modes, therefore, other techniques are needed to determine the frequency peak point. Does it belong to the resonant frequency band?

[0112] In cases involving near-frequency frequencies, near the frequency peak point There will always be

[0113]

[0114] in, It is obtained based on the result of singular value decomposition of the response power spectral density matrix, with the frequency peak point... The first singular value vector at that point, It is the frequency peak point The second-order singular value vector at that point, yes The conjugate matrix, yes The conjugate matrix. The transformation matrix can be calculated. Moreover, the transformation matrix satisfy

[0115]

[0116] in Frequency point The first singular value vector at that point, Frequency point The second-order singular value vector at that point, There are dense near-frequency points in the vicinity, which allows us to determine the frequency based on the transformation matrix. Solving for the results and Next, through the formula

[0117]

[0118] It can be calculated and judge The size. In this embodiment, when If the value is greater than the third threshold (which could be 0.2), then it can be determined that the peak value is near. The point belongs to the resonant frequency band.

[0119] When near frequencies exist, the mode shape can be expressed as: ,in, In frequency At singular values, therefore, construct the matrix. The Jacobi rotation algorithm can be used to determine the response power spectral density matrix. Sure The modal shape can be obtained by taking the value of .

[0120] Specifically, when using the Jacobi rotation algorithm to determine the rotation matrix, the following steps can be performed:

[0121] S501. According to the formula

[0122]

[0123] Determine the input matrix ,in These are the singular values ​​obtained by performing singular value decomposition on the response power spectral density matrix. The maximum frequency point in the spectrum;

[0124] S502. Set threshold (Specifically, it can be 10) -8 );

[0125] S503. Set the initial value of the rotation matrix. ;in for An identity matrix of order 1;

[0126] S504. Order ,when At that time, the following steps (1)-(3) are executed repeatedly (i.e., the loop variable) The range is 1 to -1, after each execution of step (1)-(3) then Increase by 1, then execute the next step (1)-(3), when Exceed -1 or (End of loop)

[0127] (1) Let The following steps (101)-(105) are executed repeatedly (i.e., the loop variable) The range is +1 to After each execution of steps (101)-(105) then Increase by 1, then execute the next step (101)-(105), when Exceed (End of loop)

[0128] (101) According to the formula

[0129] ;

[0130] Perform calculations;

[0131] (102) According to the formula

[0132] ;

[0133] Perform calculations;

[0134] (103) Take The normalized eigenvector corresponding to the largest eigenvalue ,in ;

[0135] (104) Calculate the rotation parameters ;

[0136] (105) According to the formula

[0137]

[0138] Construct rotation matrix ;

[0139] This concludes one round of steps (101)-(105).

[0140] (2) According to the formula Update;

[0141] (3) According to the formula Update;

[0142] Output the results obtained from the last execution of steps (1)-(3) , as a rotation matrix.

[0143] The rotation matrix is ​​obtained by performing step S5. Next, when performing step S6, which involves determining the natural frequency and damping ratio of the system under test based on the rotation matrix, you can first use the formula...

[0144]

[0145] Perform calculations to determine the matrix. .in and Same. Note that the natural frequency and damping ratio are included in the matrix. In, its form is

[0146]

[0147] in For the damping ratio, It is the natural frequency.

[0148] Multiply both sides of the above equation by... ,get

[0149]

[0150] in It is a constant. For the frequency with damping, based on the damping ratio With natural frequency Relationship

[0151]

[0152] It can be concluded that

[0153]

[0154] At this point, the natural frequency can be calculated using the formula. Damping ratio expression

[0155]

[0156] That is, matrix calculations are performed on the right side of the equation to obtain a matrix in the form of the left side of the equation, from which the natural frequency is extracted. Damping ratio .

[0157] The automated method for identifying near-frequency modal parameters based on joint diagonalization in this embodiment utilizes joint diagonalization technology to achieve automated identification of near-frequency modal parameters and improve the accuracy of mode shape identification. Specifically, joint diagonalization technology is used to process the power spectral density matrix of near-frequency modes, enabling automated extraction of the natural frequencies and damping ratios of the system under test. Based on the natural frequencies and damping ratios, the mode shapes of the system under test can be determined. Thus, efficient identification of near-frequency modes is achieved through automated algorithms, reducing human intervention and significantly improving analysis efficiency. This method is suitable for online monitoring and modal analysis of large-scale structures.

[0158] The automated method for identifying near-frequency modal parameters of spectral density based on joint diagonalization in this embodiment has the following advantages:

[0159] 1. Automated identification: This method avoids subjective errors caused by manually determining the frequency bandwidth of dense near-frequency bands and reduces the difficulty of manually selecting frequency bands. Furthermore, a criterion for automatically identifying near-frequency bands is proposed, improving the efficiency of the method.

[0160] 2. Identification of complex near frequencies: The joint diagonalization technique is adopted, which solves the problem that previous modal parameter identification methods could only identify orthogonal near frequencies, making it possible to identify non-orthogonal near frequencies.

[0161] 3. Efficiently estimating modal parameters: Existing modal parameter identification methods often estimate modal parameters by performing inverse Fourier transform of frequency domain data into time domain data. This invention directly uses spectral density data for modal parameter identification, which improves the identification efficiency.

[0162] The same technical effect as the automated method for identifying spectral density near-frequency modal parameters based on joint diagonalization in this embodiment can be achieved by writing a computer program to execute the automated method for identifying spectral density near-frequency modal parameters based on joint diagonalization in this embodiment.

[0163] It should be noted that, unless otherwise specified, when a feature is referred to as "fixed" or "connected" to another feature, it can be directly fixed or connected to the other feature, or indirectly fixed or connected to the other feature. Furthermore, the descriptions of "upper," "lower," "left," and "right" used in this disclosure are only relative to the relative positional relationships of the components of this disclosure in the accompanying drawings. The singular forms "a," "an," and "the" used in this disclosure are also intended to include the plural forms, unless the context clearly indicates otherwise. Moreover, unless otherwise defined, all technical and scientific terms used in this embodiment have the same meaning as commonly understood by one of ordinary skill in the art. The terminology used in this embodiment specification is only for describing particular embodiments and is not intended to limit the invention. The term "and / or" as used in this embodiment includes any combination of one or more of the associated listed items.

[0164] It should be understood that although the terms first, second, third, etc., may be used to describe various elements in this disclosure, these elements should not be limited to these terms. These terms are only used to distinguish elements of the same type from each other. For example, a first element may also be referred to as a second element without departing from the scope of this disclosure, and similarly, a second element may also be referred to as a first element. The use of any and all instances or exemplary language (“e.g.,” “such as,” etc.) provided in this embodiment is intended only to better illustrate embodiments of the invention and, unless otherwise required, does not impose a limitation on the scope of the invention.

[0165] It should be recognized that embodiments of the present invention can be implemented or carried out by computer hardware, a combination of hardware and software, or by computer instructions stored in a non-transitory computer-readable storage medium. The method can be implemented using standard programming techniques—including a non-transitory computer-readable storage medium configured with a computer program, wherein such a storage medium causes the computer to operate in a specific and predefined manner—according to the methods and drawings described in the specific embodiments. Each program can be implemented in a high-level procedural or object-oriented programming language to communicate with the computer system. However, if desired, the program can be implemented in assembly or machine language. In any case, the language can be a compiled or interpreted language. Furthermore, for this purpose, the program can run on a programmed application-specific integrated circuit (ASIC).

[0166] Furthermore, the procedures described in this embodiment can be performed in any suitable order unless otherwise indicated by this embodiment or clearly contradicted by the context. The procedures (or variations and / or combinations thereof) described in this embodiment can be executed under the control of one or more computer systems configured with executable instructions, and can be implemented by hardware or a combination thereof as code (e.g., executable instructions, one or more computer programs, or one or more applications) that commonly executes on one or more processors. A computer program includes multiple instructions executable by one or more processors.

[0167] Furthermore, the method can be implemented in any suitable type of computing platform, including but not limited to personal computers, minicomputers, mainframes, workstations, networked or distributed computing environments, standalone or integrated computer platforms, or in communication with charged particle tools or other imaging devices, etc. Aspects of the invention can be implemented as machine-readable code stored on a non-transitory storage medium or device, whether removable or integrated into a computing platform, such as a hard disk, optical read and / or write storage medium, RAM, ROM, etc., such that it is readable by a programmable computer, and when the storage medium or device is read by the computer, it can be used to configure and operate the computer to perform the processes described herein. Furthermore, the machine-readable code, or portions thereof, can be transmitted via wired or wireless networks. The invention of this embodiment includes these and other different types of non-transitory computer-readable storage media when such media comprises instructions or programs that implement the steps above in conjunction with a microprocessor or other data processor. When programmed according to the methods and techniques of the invention, the invention also includes the computer itself.

[0168] A computer program can be applied to input data to perform the functions of this embodiment, thereby transforming the input data to generate output data stored in non-volatile memory. The output information can also be applied to one or more output devices, such as a display. In a preferred embodiment of the invention, the transformed data represents physical and tangible objects, including specific visual depictions of physical and tangible objects generated on the display.

[0169] The above are merely preferred embodiments of the present invention. The present invention is not limited to the above-described embodiments. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention, as long as they achieve the technical effects of the present invention by the same means, should be included within the scope of protection of the present invention. Within the scope of protection of the present invention, the technical solutions and / or implementation methods can have various modifications and variations.

Claims

1. An automated method for identifying near-frequency modal parameters of spectral density based on joint diagonalization, characterized in that, The automated method for identifying spectral density near-frequency modal parameters based on joint diagonalization includes: The system under test is detected by an accelerometer sensor to obtain acceleration response data; The acceleration response data is used as kinematic information; According to the formula Perform calculations; among which, In response to the power spectral density matrix, The total number of sampling points for the kinematic information. The sampling time interval for the kinematic information is... For the acceleration response data in the variable The function form at time, For the acceleration response data in the variable The function form at time, and These are parameters used in the calculation process. This represents the frequency peak point; Based on the response power spectral density matrix, determine the frequency peak points and the presence state of dense near frequencies; When the presence of dense near frequencies indicates the existence of dense near frequencies, determine whether the frequency peak point belongs to the resonant frequency band. When the frequency peak point belongs to the resonant frequency band, the rotation matrix is ​​determined according to the response power spectral density matrix. Based on the rotation matrix, the natural frequency and damping ratio of the system under test are determined.

2. The automated method for identifying near-frequency modal parameters of spectral density based on joint diagonalization according to claim 1, characterized in that, The step of determining the frequency peak point and the existence state of dense near frequencies based on the response power spectral density matrix includes: Singular value decomposition is performed on the response power spectral density matrix; The frequency peak points are extracted using an automatic peak picking algorithm. .

3. The automated method for identifying near-frequency modal parameters of spectral density based on joint diagonalization according to claim 2, characterized in that, The step of determining the frequency peak point and the existence state of dense near frequencies based on the response power spectral density matrix includes: Based on the results of singular value decomposition of the response power spectral density matrix, the local extrema of the third-order singular value, the local extrema of the second-order singular value, and the maximum value of the first-order singular value are obtained. The first ratio is determined by comparing the local extreme points of the third-order singular value with the maximum value of the first-order singular value; When the first ratio is greater than the first threshold, it is determined that there are 3 dense near frequencies; otherwise, the second ratio is determined by comparing the local extreme point of the second-order singular value with the maximum value of the first-order singular value. When the second ratio is greater than the second threshold, it is determined that there are two dense near frequencies; otherwise, it is determined that there are no dense near frequencies.

4. The automated method for identifying near-frequency modal parameters of spectral density based on joint diagonalization according to claim 1, characterized in that, Determining the rotation matrix based on the response power spectral density matrix includes: The rotation matrix is ​​determined using the Jacobi rotation algorithm.

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