A method for structural modal parameter identification based on dynamic modal decomposition

Abstract

The present invention proposes a structural modal parameter identification method based on dynamic modal decomposition. According to the dynamic modal decomposition method, the method selects the displacement and velocity vibration signals of each response point in the measured structure to construct a snapshot matrix, obtains the eigenvalues ​​and eigenvectors of the system under modal truncation through singular value decomposition and eigenvalue analysis, transforms different modal truncation orders and solves the modal frequencies under the corresponding orders, constructs a relationship diagram between the truncation order and the solved modal frequencies, and uses the property that the structural modal frequency is independent of the singular value truncation order to eliminate false modes and obtain modal parameters such as the true modal frequency, modal damping ratio and modal vibration shape of the measured structure. The method for constructing a stability diagram proposed by the present invention can eliminate a large number of false modes appearing in the dynamic modal decomposition method, improves the analysis process of the dynamic modal decomposition method in the structural modal identification problem, and the modal parameters of the structure can be obtained through the present invention.

Description

Technical Field

[0001] The invention relates to the field of structural vibration modal testing, and in particular to a structural modal parameter identification method based on dynamic modal decomposition. Background Art

[0002] Identifying the dynamic model of a structure through data processing is an important aspect of structural dynamics research and an important means of solving the design of complex structural dynamic characteristics. By applying the modal parameter identification method, the complex actual structure can be simplified into a so-called modal model (including modal frequency, modal damping ratio and modal vibration shape), and then the system response can be calculated, which can greatly reduce the amount of calculation.

[0003] According to the type of experimental data, the structural modal parameter identification methods include two categories: time domain method and frequency domain method. The time domain method is based on the time domain response signal to identify the modal parameters. Commonly used methods include the time domain least squares iteration method, the time domain complex exponential fitting method, the Ibrahim time domain method (ITD method) and the characteristic system realization algorithm (ERA). The frequency domain method is generally based on the frequency response function or the power spectrum density function to identify the modal parameters. Common methods include the admittance circle method, the Levy method and the least squares frequency domain method.

[0004] Dynamic mode decomposition (DMD) is a linear system decomposition method proposed by Schmid et al. based on the Koopman operation. It was first used to analyze the characteristics of unsteady flow fields. Since dynamic mode decomposition can not only separate the spatial characteristics of the dynamic system, but also obtain the temporal characteristics of the system in transient changes. Therefore, applying this method to the analysis of structural vibration signals will obtain the modal parameters of the structure, such as modal frequency, modal damping ratio and modal vibration shape. In essence, this modal parameter identification method based on dynamic mode decomposition is a data-driven modeling method and can be regarded as a new type of time domain method for structural modal parameter identification. However, what kind of time domain signal to use for analysis and how to eliminate false modes to obtain the actual modal parameters of the structure are problems that need to be solved for structural modal parameter identification based on the dynamic mode decomposition method. Summary of the invention

[0005] In order to solve the problems of what kind of time domain signal to use for analysis and how to eliminate false modes in structural modal parameter identification based on dynamic modal decomposition method, and then obtain the actual modal parameters of the structure, the present invention proposes a structural modal parameter identification method based on dynamic modal decomposition.

[0006] The technical solution of the present invention is:

[0007] A structural modal parameter identification method based on dynamic modal decomposition comprises the following steps:

[0008] Step 1: Obtain the vibration response signal of the structure under test, and construct a column vector consisting of the displacement and velocity response at each sampling moment for each measuring point, and organize the column vectors into a snapshot matrix in time sequence;

[0009] Step 2: Perform singular value decomposition on the snapshot matrix obtained in step 1 to obtain a singular value matrix and a unitary matrix; truncate and select the first r singular values, and use the corresponding truncated singular value matrix and unitary matrix as the approximate system matrix; obtain the eigenvalues ​​and eigenvectors of the approximate system matrix through eigenvalue analysis;

[0010] Step 3: By changing the truncation order r of the singular value, repeat step 2 to obtain the eigenvalues ​​corresponding to different truncation orders; use the imaginary part of the eigenvalue and the corresponding singular value truncation order r to draw a stability diagram; eliminate the false modes in the stability diagram to obtain the true structural modal frequency and its corresponding modal damping ratio and modal vibration shape, which are the true modal parameters of the measured structure.

[0011] Furthermore, in step 3, according to the property that the structural modal frequency is independent of the singular value cutoff order, the false modes are eliminated in the stability diagram.

[0012] Furthermore, the specific process of step 2 is:

[0013] Assume that the snapshots at two adjacent moments have the following relationship:

[0014] x i+1 =Ax i

[0015] where x i represents the column vector of displacement and velocity response at the i-th sampling moment; then the snapshot matrix X of two adjacent moments is [x1 x2 … x m-1 ] and Y=[x2 x3 … x m ]There is a relationship between:

[0016] Y=AX

[0017] Perform singular value decomposition on the snapshot matrix X:

[0018] X=U∑V H

[0019] in: is the singular value matrix of the snapshot matrix; σ1 … σ m are the singular values ​​arranged from large to small; U and V are unitary matrices; () H represents conjugate transpose;

[0020] The singular value matrix corresponding to the first r singular values ​​is The remaining singular values ​​form ∑ rem , the first r columns of the corresponding unitary matrix U are The rest is denoted as U rem , the first r rows of the corresponding unitary matrix V are The rest is denoted as V rem ;

[0021] but

[0022] Then get Take the approximate relationship as right Solving the eigenvalues ​​yields , where μ i and w i are the i-th order eigenvalue and eigenvector respectively;

[0023] Thus, the i-th order eigenvalue λ of the structure is obtained i and the mode shape Φ i They are:

[0024] λ i =log(μ i ) / Δt

[0025]

[0026] make and Represents the real part and imaginary part operation respectively, then the modal frequency and modal damping ratio of the i-th order structure are:

[0027]

[0028]

[0029] A computer-readable storage medium stores computer-executable instructions, which are used to implement the method described above when executed.

[0030] A computer system comprises: one or more processors, and a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement the method as described above.

[0031] A computer program comprises computer executable instructions, which are used to implement the method described above when the instructions are executed.

[0032] Beneficial Effects

[0033] As a data-driven modeling method, the present invention can process a large amount of vibration signal data generated by large and complex structures. The present invention has a simple principle, obvious effect, high data processing efficiency, and provides a new idea for the engineering application of structural modal parameter identification in the future, and has broad application prospects in structural model correction and response prediction.

[0034] Additional aspects and advantages of the present invention will be given in part in the following description and in part will be obvious from the following description, or will be learned through practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] The above and / or additional aspects and advantages of the present invention will become apparent and easily understood from the description of the embodiments in conjunction with the following drawings, in which:

[0036] Figure 1 is a schematic diagram of the process of constructing a snapshot matrix in the present invention;

[0037] Figure 2 It is a schematic diagram of a stability diagram in one state of the present invention. DETAILED DESCRIPTION

[0038] The embodiments described below with reference to the accompanying drawings are exemplary and are intended to be used to explain the present invention, but should not be construed as limiting the present invention.

[0039] First, if Figure 1 As shown in Figure 1, an initial disturbance is applied at a certain position of the structure under test, and the structural vibration response is obtained through experiments or numerical simulation calculations. N sampling points are taken on the structural model to collect the structural response. The collected response includes the displacement and velocity of each sampling point, which is denoted as w i and Where i = 1, ..., N, and the response column vector is formed according to the position number Sampling is performed with a step size of Δt, using x i The column vector representing the i-th sampling moment is called a snapshot. A total of m snapshots are collected, namely x1 x2 … x m These snapshots can be used to form a snapshot matrix X = [x1 x2 … x m-1 ] and Y=[x2 x3 … x m ].

[0040] Then, the modal parameters are identified based on the dynamic modal decomposition method. In the dynamic modal decomposition method, it is assumed that the snapshots at two adjacent moments have the following relationship:

[0041] x i+1 =Ax i (1)

[0042] Therefore, the snapshot matrices at two adjacent moments have the following relationship:

[0043] Y=AX (2)

[0044] Perform singular value decomposition on X, that is:

[0045]

[0046] in: is the singular value matrix of the snapshot matrix; σ1 … σ m are the singular values ​​arranged from large to small; U and V are unitary matrices; () H represents the conjugate transpose.

[0047] The singular value matrix corresponding to the first r singular values ​​is (The remaining singular values ​​form ∑ rem ), the first r columns of the corresponding unitary matrix U are (The rest is denoted as U rem ), the first r rows of the corresponding unitary matrix V are (The rest is denoted as V rem ). For the convenience of description, r is called the singular value cutoff order.

[0048] Substituting formula (3) into formula (2) yields:

[0049]

[0050] To avoid the matrix dimension being too high, the approximate relationship is as follows:

[0051]

[0052] You can use Approximate A, and Solving the eigenvalues, we can get:

[0053]

[0054] Where: μ i and w i are the i-th order eigenvalue and eigenvector respectively.

[0055] Then, the i-th order eigenvalue λ of the structure i and the mode shape Φ i They are:

[0056]

[0057] If you and Represents the real part and imaginary part operation respectively, then the modal frequency and modal damping ratio of the i-th order structure are:

[0058]

[0059]

[0060] Due to numerical noise and other reasons, the modes obtained in the above process contain a large number of false modes that need to be eliminated.

[0061] Finally, a stability diagram is constructed to remove false modes. By changing the singular value cutoff order r, repeating the analysis process from formula (3) to formula (8) above, the modal frequencies ω of each order under the corresponding cutoff order are calculated. i , damping ratio ζ i and the mode shape Φ i . The singular value cutoff order r and each modal frequency ω i Plotted in the same coordinate system, a stable diagram is formed, such as Figure 2 As shown. Since the structural modal frequency is an inherent characteristic of the structure, it has nothing to do with the singular value truncation order. According to this property, the false modes related to the modal frequency and modal truncation order in the stability diagram are eliminated, and the remaining ones are the true modal frequencies obtained by the present invention. For the same order mode, the connecting line is perpendicular to the horizontal axis of the coordinate system, such as Figure 2 The retained modal frequency and the corresponding modal damping ratio and modal vibration shape constitute the true modal parameters of the measured structure.

[0062] Although the embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention without departing from the principles and intent of the present invention.

Claims

1. A structural modal parameter identification method based on dynamic modal decomposition, characterized in that: The following steps are involved: Step 1: Obtain the vibration response signal of the structure under test, and construct a column vector consisting of the displacement and velocity response at each sampling moment for each measuring point, and organize the column vectors into a snapshot matrix in time sequence; Step 2: Perform singular value decomposition on the snapshot matrix obtained in step 1 to obtain a singular value matrix and a unitary matrix; truncate and select the first r singular values, and use the corresponding truncated singular value matrix and unitary matrix as the approximate system matrix; obtain the eigenvalues ​​and eigenvectors of the approximate system matrix through eigenvalue analysis; The specific process is: Assume that the snapshots at two adjacent moments have the following relationship: x i+1 =Ax i where x i represents the column vector of displacement and velocity response at the i-th sampling moment; then the snapshot matrix X of two adjacent moments is [x1x2…x m-1 ] and Y = [x2x3…x m ]There is a relationship between: Y=AX Perform singular value decomposition on the snapshot matrix X: X=U∑V H in: is the singular value matrix of the snapshot matrix; σ1…σ m are the singular values ​​arranged from large to small; U and V are unitary matrices; () H represents conjugate transpose; The singular value matrix corresponding to the first r singular values ​​is The remaining singular values ​​form ∑ rem , the first r columns of the corresponding unitary matrix U are The rest is denoted as U rem , the first r rows of the corresponding unitary matrix V are The rest is denoted as V rem ; but Then get Take the approximate relationship as right Solving the eigenvalues ​​yields where μ i and w i are the i-th order eigenvalue and eigenvector respectively; Thus, the i-th order eigenvalue λ of the structure is obtained i and the mode shape Φ i They are: l i =log(μ i ) / Δt make and Represents the real part and imaginary part operation respectively, then the modal frequency and modal damping ratio of the i-th order structure are: Step 3: By changing the truncation order r of the singular value, repeat step 2 to obtain the eigenvalues ​​corresponding to different truncation orders; use the imaginary part of the eigenvalue and the corresponding singular value truncation order r to draw a stability diagram; eliminate the false modes in the stability diagram to obtain the true structural modal frequency and its corresponding modal damping ratio and modal vibration shape, which are the true modal parameters of the measured structure.

2. The structural modal parameter identification method based on dynamic modal decomposition according to claim 1, characterized in that: In step 3, based on the property that the structural modal frequency is independent of the singular value cutoff order, the false modes are eliminated from the stability diagram.

3. A computer-readable storage medium storing computer-executable instructions, wherein the instructions are used to implement the method according to any one of claims 1 to 2 when executed.

4. A computer system comprising: One or more processors, the computer-readable storage medium as described in claim 3, is used to store one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement any one of the methods described in claims 1 to 2.

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