CN108614926A - A kind of modal parameters discrimination method being combined with Hilbert-Huang transform based on manifold learning - Google Patents

Abstract

The present invention discloses a kind of modal parameters discrimination method being combined with Hilbert-Huang transform based on manifold learning, includes the following steps：Step 1: acquiring the time domain response data of measuring point in structure；Step 2: being handled using manifold learning arithmetic the time domain response data of step 1 acquisition, the vibration shape and intrinsic frequency of structure are obtained；Step 3: being handled using Hilbert-Huang transform method the time domain response data of step 1 acquisition, the damping ratio of structure is obtained.Compared with the existing technology, the invention has the advantages that：First, when the method that two kinds of algorithms combine in using the present invention carries out modal parameter extraction, in the case of structural material parameter and unknown experimental condition, it is only necessary to response data, you can obtain the vibration shape with degree of precision and intrinsic frequency, damping ratio；Second, the method for the present invention can be used for handling nonlinear data, the non-linearity manifold of structure can be retained.

Description

A Structural Modal Parameter Based on the Combination of Manifold Learning and Hilbert-Huang Transform number identification method

technical field

The invention belongs to the technical field of structural dynamic parameter identification, and in particular relates to a structural mode parameter identification method.

Background technique

The key to structural modal analysis is the identification of modal parameters, including modal frequencies, modal shapes, and damping ratios. These parameters are generally obtained through modal tests. However, due to the complex environment and technical limitations, it is often difficult to implement effective modal excitation and excitation force measurement, resulting in difficulties in obtaining the modal parameters of complex or large-scale structures. In comparison, response data are easier to obtain from experiments. In order to overcome this problem, many methods for modal parameter identification based only on response data have been proposed in the prior art. The traditional signal processing method is mainly based on Fourier transform, which uses the superposition of complex sine components of different frequencies to fit the original function, and the Fourier spectrum is scattered on the frequency axis, which cannot reflect the non-stationary random signal statistics over time. In addition, some traditional modal parameter identification methods (such as peak picking method, frequency domain decomposition method, etc.) have problems such as low damping ratio identification accuracy.

Although the structural response data are mostly in high-dimensional spaces, the internal manifolds of these high-dimensional spaces are actually very simple, so many dimensionality reduction methods have been proposed for parameter identification, such as principal component analysis (PCA), blind source separation Analysis method (BSS). However, these algorithms are all linear dimensionality reduction methods, which can only discover the global Euclidean distance of the structure but cannot discover the intrinsic submanifold structure. However, since the nonlinear response is mostly located in the submanifold of the outer space, many nonlinear manifold learning algorithms have been proposed, but none of them have been applied in the field of modal parameter identification.

Contents of the invention

The object of the present invention is to provide a kind of structural mode parameter identification method based on the local linear embedding algorithm (LLE) in the manifold learning and the Hilbert-Huang Transform (HHT) combination, utilize the combination of the LLE and the HHT algorithm The method realizes the identification of the modal parameters of the structure based only on the nonlinear response data, so as to solve the above technical problems.

In order to achieve the above object, the present invention adopts the following technical solutions:

The identification of modal frequency and mode shape by LLE algorithm is realized through the following technical solutions:

A structural modal parameter identification method based on the combination of manifold learning and Hilbert-Huang transform, comprising the following steps:

Step 1, collecting time-domain response data of measuring points in the structure;

Step 2. Process the time-domain response data collected in step 1 with a manifold learning algorithm to obtain the mode shape and natural frequency of the structure;

Step 3: The time-domain response data collected in step 1 is processed by the Hilbert-Huang transform method to obtain the damping ratio of the structure.

Further, the time-domain response data of the measuring point in the collection structure in step 1 is X(x, t), x represents the response of the sampling point, and t represents the sampling time;

Step two specifically includes:

2.1): Determine the number of neighborhood points and find the neighborhood

For the test sample X(x,t) is a matrix of D×N, D is the total number of sampling points, N is the maximum number of samples of the same sampling point; calculate the data point x _i and other data points x _j of the same sampling point The Euclidean distance between them, find the k nearest neighbor points to x _i , and the program automatically selects the k value corresponding to the minimum reconstruction error; i=1,2,...,N; j=1,2,. ..,N;

2.2): Calculate the reconstruction weight W

The local reconstruction weight matrix of each sample point is calculated from the neighbor points of each sample point, so that the reconstruction error of the sample point is minimized, that is, the following optimal problem is solved:

Among them: W _ij is the weight between x _i and x _j ; the following two constraints are met: ① When a data point x _j does not belong to the neighbor data point of the reconstructed data point x _i , the weight W _ij =0; ②The sum of elements in each row in the weight matrix is equal to 1, that is

2.3): Calculate the low-dimensional embedding vector Y

The dimension d of the low-dimensional embedding vector Y is the number of modes concerned by the structure; by minimizing the embedding cost function equation (10), the low-dimensional reconstruction error ε(Y) is minimized. At this time, the low-dimensional embedding vector is M The smallest 2nd to d+1th eigenvectors;

where M = (IW) ^T (IW), and

2.4): Obtain the structural mode shape and natural frequency

According to the theory of structural dynamics, the response of the system is expressed as a linear combination of intrinsic modes; in the process of data acquisition, the three-dimensional system is discretized into D measuring points, and the time is discretized into N sampling points. The state order d, the response is expressed by equation (3):

x _D×N ＝Φ _D×d ·η _d×N (3)

Among them, x _D×N is the original time domain response, Φ _D×d is the mode shape matrix, η _d×N is the modal coordinate;

The eigenvector Y obtained by reducing the dimensionality of the LLE algorithm is the modal coordinate η _d×N in the modal analysis, and then through the following formula:

The mode shape matrix Φ is calculated, and the natural frequency is obtained by Fourier transform of the modal coordinates.

Further, step three specifically includes:

3.1): NExT extracts free decay response

Filter the collected time-domain acceleration response data, select a measuring point as a reference point, and calculate the cross-correlation function between a response point and the reference point. For linear structures, the cross-correlation function and impulse function between white noise responses Consistent, for the signal of free decay, expressed as:

Among them, R _ji is the cross-correlation function between two measuring points; τ is time; ψ _jr is the jth element of the rth order mode shape; G _ir is a constant only related to i, r; m _r ,ξ _r , ω _r , ω _dr are the r-th order modal mass, damping ratio, undamped natural frequency and damped natural frequency of the structure respectively; θ _k is the k-th order phase angle;

3.2): Empirical mode decomposition decomposition to obtain a stationary random signal

Perform EMD decomposition on the cross-correlation function R _ji to obtain a finite number of intrinsic mode functions, which are used as the input of the Hilbert transform;

3.3): HT analysis

Hilbert transform is performed on each order free decay signal, and the analytical signal z(τ) is constructed:

Then the amplitude A(τ) and phase θ(τ) are expressed as:

Calculate the natural logarithm and derivative of the amplitude and phase of the above formula respectively, and get:

Use the least squares method to fit the amplitude spectrum and phase spectrum respectively, and obtain ξ _r ω _r and ω _dr respectively;

Compared with the prior art, the present invention has the following beneficial effects:

First, when using the method of combining two algorithms in the present invention to extract modal parameters, in the case of unknown structural material parameters and test conditions, only response data is needed to obtain mode shapes and intrinsic frequency, damping ratio;

Second, the method of the present invention can be used to process nonlinear data, and can preserve the nonlinear manifold of the structure.

Description of drawings

Fig. 1 is method flowchart of the present invention;

Figure 2 is a schematic diagram of the finite element model of the plate and the positions of 12 response points;

Fig. 3 is cross-correlation function signal;

Fig. 4 is the free attenuation signal of the first order mode;

Fig. 5 is a logarithmic amplitude curve and its least squares fitting schematic diagram;

Fig. 6 is the instantaneous frequency schematic diagram of the first-order modal response;

Detailed ways

The technical scheme of the present invention will be further described below in conjunction with the accompanying drawings.

The invention is a structural mode parameter identification method based on the combination of manifold learning and Hilbert-Huang transformation. When the LLE algorithm is used for mode parameter identification, the response data is regarded as a high-dimensional data set. From the perspective of geometric feature extraction, mode shapes are considered as inherent properties of high-dimensional datasets. Using the LLE algorithm to reduce the dimensionality of the high-dimensional response data set, the mode shape and natural frequency can be obtained.

The data distributed on the high-dimensional manifold can be approximately regarded as distributed on a low-dimensional hyperplane in a small local area. In this neighborhood, it can be assumed that there is a linear mapping between the high-dimensional data and the low-dimensional embedding. Therefore, for a new sample point, first find its neighborhood in the original space, then construct a linear map from high-dimensional to low-dimensional in this neighborhood, and finally realize the generalization of the new sample through the linear map.

As shown in Figure 1, assume that a response data set X _D×N =[x ₁ ,x ₂ ,…,x _N ]∈R ^D×N contains N column vectors, each column dimension is D, and it needs to be reduced to d dimension, The steps of LLE algorithm identification are as follows:

1) Determine the number k of neighborhood points: calculate the data points x _i (i=1,2,...,N) and other data points x _j (j=1,2,...,N) of the same sampling point ) to find the k nearest neighbor points to x _i , and the program automatically selects the k value corresponding to the minimum reconstruction error.

2) Calculation of reconstruction weight W: Calculate the local reconstruction weight matrix of each sample point from the neighboring points of each sample point, so as to minimize the reconstruction error of the sample point, that is, to solve the following optimal problem:

Among them: W _ij is the weight between x _i and x _j . In order to obtain the optimal weight, the following two constraints need to be met: ① When a data point x _j does not belong to the neighboring data point of the reconstructed data point x _i , the weight W _ij = 0; ② The weight matrix The sum of the elements of each row in is equal to 1, that is

3) Calculate the low-dimensional embedding vector Y: By minimizing the embedding cost function, the low-dimensional reconstruction error ε(Y) is minimized. At this time, the low-dimensional embedding is the second to d+1th eigenvectors with the smallest M.

where M = (IW) ^T (IW), and

According to the theory of structural dynamics, the response of the system can be expressed as a linear combination of natural modes. In the process of data collection, the 3D system will be discretized into D measuring points, and the time will be discretized into N sampling points. The system response can be expressed by equation (3):

x _D×N ＝Φ _D×d ·η _d×N (3)

where x _D*N is the response, Φ _D*d is the mode shape matrix, and η _d*N is the modal coordinate vector.

In the process of using the LLE algorithm for modal analysis, the principal coordinate matrix η _d×N is first obtained through the feature extraction of x _D*N , and then through Calculate the mode shape matrix Φ _D*d .

The identification of modal frequency and damping ratio by HHT and Natural Excitation Technology (NExT) is achieved through the following technical solutions:

Assuming that x(t) is a structural response time series signal obtained under environmental excitation, band-pass filter x(t) and use NExT to obtain the corresponding free decay response, and then perform empirical mode decomposition (EMD). The natural frequency of the structure is firstly estimated according to the Fourier spectrum, and then the band-pass filter is carried out. The first-order intrinsic mode function (IMF) obtained by performing empirical mode decomposition on the filtered time series signal is generally very close to the modal response of the structure. Specific steps are as follows:

1) Filter the time-domain response data to obtain the input signal of the NExT method, and calculate the cross-correlation function between two points. The cross-correlation function between two points is expressed as:

Among them, R _ji is the cross-correlation function between two measuring points; τ is time; ψ _jr is the jth element of the rth order vibration shape; G _ir is a constant related only to i, r; m _r ,ξ _r , ω _r , ω _dr are the rth order modal mass, damping ratio, undamped natural frequency and damped natural frequency of the structure respectively; θ _k is the kth order phase angle.

2) The cross-correlation function is subjected to EMD processing to obtain the first-order modal response of the structure, which is a stationary random signal;

3) Use HT transform to perform Hilbert transform on the signal obtained by EMD decomposition, and construct the analytical signal:

Then the amplitude A(τ) and phase θ(τ) can be expressed as:

Calculate the natural logarithm and derivative of the amplitude and phase of the above formula respectively, and you can get:

Using the least squares method to fit the amplitude spectrum and phase spectrum to obtain ξ _r ω _r and ω _dr respectively, and In this way, ω _r and ξ _r can be obtained.

Taking an orthotropic composite plate as an object, the plate is a composite material plate with 13 layers. The material parameters of the two adjacent layers are shown in Table 1. The response data is under the four-sided free condition under the excitation of white noise. 12 points evenly distributed on the board, calculate the time-domain acceleration response X(x,t) of each point, where x represents the time-domain acceleration response, and t is the corresponding sampling time. The finite element model of the board and the positions of the 12 response points are shown in Figure 2, and the sampling frequency is 1024Hz. Since the structural material is a composite material, there is nonlinearity in the structural response under white noise excitation. That is, D=12, N=1024.

Table 1 Material parameters of the plate

The application proposed by the present invention is to use the LLE algorithm to obtain the modal parameters of the plate only based on the nonlinear time domain response data X(x, t) of the plate, including the modal frequency and mode shape, and to obtain Damping ratio vs. modal frequency. Then compare the correlation value between the mode shape extracted by the algorithm and the finite element FEM mode shape:

V _LLE represents the mode shape matrix obtained by the LLE algorithm, and V _FE represents the mode shape matrix extracted by the finite element method. MAC _LLE,FE represents the correlation value between two matrices. When MAC _{LLE, FE} is greater than or equal to 0.8, it means that the mode shape matrix extracted by LLE algorithm is consistent with the mode shape matrix extracted by finite element method, and when MAC _{LLE, FE} is less than 0.2, it means that the two are orthogonal.

Perform the following steps on the data sample through the LLE algorithm, as shown in Figure 1:

Step 1: Determine the number of neighborhood points and find the neighborhood

For the test sample X(x,t) is a 12×1024 matrix, calculate the data points x _i (i=1,2,...,1024) and other data points x _j (j=1,2 ,...,1024), find the k nearest neighbor points to x _i , and the program automatically selects the k value corresponding to the minimum reconstruction error.

Step 2: Calculate the reconstruction weight W

The local reconstruction weight matrix of each sample point is calculated from the neighbor points of each sample point to minimize the reconstruction error of the sample point, that is, to find the following optimal problem

Among them: W _ij is the weight between x _i and x _j . In order to obtain the optimal weight value, the following two constraints must be met: ① When a data point x _j does not belong to the neighboring data point of the reconstructed data point x _i , the weight value W _ij = 0; ② weight matrix The sum of the elements of each row in is equal to 1, that is

Step 3: Calculate the low-dimensional embedding vector Y

Among them, the dimension d of the low-dimensional embedding vector Y is the number of modes concerned by the structure. By minimizing the embedding cost function equation (11), the low-dimensional reconstruction error ε(Y) is minimized. At this time, the low-dimensional embedding vector is the second to d+1th eigenvectors with the smallest M.

where M = (IW) ^T (IW), and

Step 4: Calculate the structural mode shapes and natural frequencies

According to the theory of structural dynamics, the response of the system can be expressed as a linear combination of natural modes. In the process of data acquisition, the three-dimensional system will be discretized into 12 measuring points, and the time will be discretized into 1024 sampling points. The modal order of concern is d=3, and the response can be expressed by equation (12):

x _12×1024 ＝Φ _12×3 ·η _3×1024 (12)

Among them, x _12×1024 is the original time domain response, Φ _12×3 is the mode shape matrix, and η _3×1024 is the mode coordinate.

The eigenvector Y obtained by dimensionality reduction through the LLE algorithm is the modal coordinate η _{3 × 1024} in the modal analysis, and then through the following formula:

The mode shape matrix Φ is calculated, and the natural frequency is obtained by Fourier transform of the modal coordinates. The dimensionality reduction results are shown in Table 2. The second row in the table shows the first three vibration modes and natural frequencies identified by the finite element FEM method, and the third row shows the first three vibration modes and natural frequencies of the structure identified by the LLE algorithm. Natural frequency; the natural frequency analyzed by the two methods and the correlation MAC value of frequency are listed in Table 3. It can be seen from the data in the table that the maximum difference between the first three order frequencies identified by the LLE algorithm and the natural frequency identified by the FEM method is 1, about 1.72%; the correlation MAC values of the first three order modes of the two are respectively 0.9286, 0.8743, 0.9961, all greater than 0.8, indicating that the mode shapes extracted by the two are consistent.

Table 2 Comparison of modal parameters identified by FEM and LLE algorithms

Table 3 Comparison of results between FEM and LLE

The specific steps to identify the damping ratio and frequency using the method of combining HHT and NExT are as follows:

Step 1: NExT extracts free decay response

Filter the collected time-domain acceleration response data, select a measuring point as a reference point, and calculate the cross-correlation function between a response point and the reference point. For linear structures, the cross-correlation function and impulse function between white noise responses Consistently, for a signal that decays freely, it can be expressed as:

Among them, R _ji is the cross-correlation function between two measurement points; τ is time; ψ _jr is the jth element of the rth order mode shape; G _ir is a constant related only to i,r; m _r ,ξ _r , ω _r , ω _dr are the rth order modal mass, damping ratio, undamped natural frequency and damped natural frequency of the structure respectively; θ _k is the kth order phase angle.

In the embodiment of the present invention, measuring point 3 is selected as a reference point, and the cross-correlation function between measuring point 4 and reference point 3 is calculated to obtain a free attenuation response, and the attenuation signal is as shown in Figure 3;

Step 2: Empirical mode decomposition to obtain a stationary random signal

The empirical mode EMD decomposition of the cross-correlation function R _ji is performed to obtain a limited number of intrinsic mode functions IMF, which are used as the input of the Hilbert transform, and the first-order free decay signal is shown in Figure 4;

Step 3: HT analysis

Hilbert transform is performed on each order free decay signal, and the analytical signal z(τ) is constructed:

Then the amplitude A(τ) and phase θ(τ) can be expressed as:

Calculate the natural logarithm and derivative of the amplitude and phase of the above formula respectively, and you can get:

Use the least squares method to fit the amplitude spectrum and phase spectrum respectively, and obtain ξ _r ω _r and ω _dr respectively. Among them, the logarithmic amplitude spectrum and its least squares fitting line are shown in Figure 5, and its phase spectrum is shown in Figure 6 ,again:

In this way, ω _r and ξ _r can be obtained. The first two-order results extracted by HHT are compared with the FFT identification results, LLE algorithm extraction results, and finite element results, as shown in Table 4. It can be seen from the results that the two-order natural frequencies obtained by using the HHT, FFT, and LLE algorithms are basically consistent with the results of the finite element analysis, but there is a certain difference in the damping ratio.

Table 4 Comparison of recognition results of each method

It can be seen from the comprehensive analysis that the method of the present invention can obtain the characteristics of the main modes of each order in the frequency range where the structure is susceptible to influence only based on the response data, which can be used for the analysis of the vibration characteristics of the structural system, the diagnosis and prediction of vibration faults, and the analysis of the dynamic characteristics of the structure. It provides a basis for optimal design and also provides a new method for system identification.

Claims (2)

1. A structural modal parameter identification method based on manifold learning combined with Hilbert-Huang transform, characterized in that, comprising the following steps: Step 1, collecting time-domain response data of measuring points in the structure; Step 2. Process the time-domain response data collected in step 1 with a manifold learning algorithm to obtain the mode shape and natural frequency of the structure; Step 3: The time-domain response data collected in step 1 is processed by the Hilbert-Huang transform method to obtain the damping ratio of the structure. 2. a kind of structural modal parameter identification method based on manifold learning and Hilbert-Huang transform combination according to claim 1, it is characterized in that, in the step 1, the time domain response data of measuring point in the collection structure is X(x,t), x represents the sampling point response, and t represents the sampling time; Step two specifically includes: 2.1): Determine the number k of neighborhood points, and find the neighborhood For the test sample X(x,t) is a matrix of D×N, D is the total number of sampling points, N is the maximum number of samples of the same sampling point; calculate the data point x _i and other data points x _j of the same sampling point The Euclidean distance between them, find the k nearest neighbor points to x _i , and the program automatically selects the k value corresponding to the minimum reconstruction error; i=1,2,...,N; j=1,2,. .., N; 2.2): Calculate the reconstruction weight W The local reconstruction weight matrix of each sample point is calculated from the neighbor points of each sample point, so that the reconstruction error of the sample point is minimized, that is, the following optimal problem is solved: Among them: W _ij is the weight between x _i and x _j ; the following two constraints are met: ① When a data point x _j does not belong to the neighbor data point of the reconstructed data point x _i , the weight W _ij =0; ②The sum of elements in each row in the weight matrix is equal to 1, that is 2.3): Calculate the low-dimensional embedding vector Y The dimension d of the low-dimensional embedding vector Y is the number of modes concerned by the structure; by minimizing the embedding cost function equation (2), the low-dimensional reconstruction error ε(Y) is minimized. At this time, the low-dimensional embedding vector is M The smallest 2nd to d+1th eigenvectors; where M = (IW) ^T (IW), and 2.4): Obtain the structural mode shape and natural frequency According to the theory of structural dynamics, the response of the system is expressed as a linear combination of intrinsic modes; in the process of data acquisition, the three-dimensional system is discretized into D measuring points, and the time is discretized into N sampling points. The state order is d, and the response is expressed by equation (3): x _D×N ＝Φ _D×d ·η _d×N (3) Among them, x _D×N is the original time domain response, Φ _D×d is the mode shape matrix, η _d×N is the modal coordinate; The eigenvector Y obtained by reducing the dimensionality of the LLE algorithm is the modal coordinate η _d×N in the modal analysis, and then through the following formula: Calculate the mode shape matrix Φ, and obtain the natural frequency by Fourier transforming the modal coordinates; Step three specifically includes: 3.1): NExT extracts free decay response Filter the collected time-domain acceleration response data, select a measuring point as a reference point, and calculate the cross-correlation function between a response point and the reference point. For linear structures, the cross-correlation function and impulse function between white noise responses Consistently, it is a signal that decays freely, expressed as: Among them, R _ji is the cross-correlation function between two measuring points; τ is time; ψ _jr is the jth element of the rth order mode shape; G _ir is a constant only related to i, r; m _r ,ξ _r , ω _r , ω _dr are the r-th order modal mass, damping ratio, undamped natural frequency and damped natural frequency of the structure respectively; θ _k is the k-th order phase angle; 3.2): Empirical mode decomposition to obtain a stationary random signal Perform empirical mode decomposition EMD on the cross-correlation function R _ji to obtain a limited number of intrinsic mode functions, which are used as the input of the Hilbert transform; 3.3): HT analysis Hilbert transform is performed on each order free decay signal, and the analytical signal z(τ) is constructed: Then the amplitude A(τ) and phase θ(τ) are expressed as: Calculate the natural logarithm and derivative of the amplitude and phase of the above formula respectively, and get: Use the least squares method to fit the amplitude spectrum and phase spectrum respectively, and obtain ξ _r ω _r and ω _dr respectively;