CN108614926A - A kind of modal parameters discrimination method being combined with Hilbert-Huang transform based on manifold learning - Google Patents
A kind of modal parameters discrimination method being combined with Hilbert-Huang transform based on manifold learning Download PDFInfo
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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
Technical field
The invention belongs to structural dynamic parameter identification technology field, more particularly to a kind of modal parameters identification side
Method.
Background technology
The key of Constructional Modal Analysis is the identification of modal parameter, including modal frequency, Mode Shape and damping ratio.These
Parameter is generally obtained by modal test.However, since environment is complicated and technology restriction, it tends to be difficult to implement effective mode and swash
Encourage and measured with exciting force, cause complicated or large-scale structure modal parameter acquisition difficult, in comparison, response data compared with
Easily obtained from experiment.This in order to overcome the problems, such as, the prior art proposes many and is based only upon response data progress modal parameter
Know method for distinguishing.Traditional signal processing method is mainly based upon Fourier transformation, it uses each multiple sinusoidal component of different frequency
Superposition be fitted original function, Fourier spectrum spread on the frequency axis, cannot reflect nonstationary random signal statistic at any time
Variation；In addition, there are damping ratios for some traditional Modal Parameters Identifications (such as peak picking method, frequency domain decomposition method etc.)
The problems such as accuracy of identification is not high.
Since though structural response data many places are in higher dimensional space, the inherent manifold of these practical higher dimensional spaces is very simple,
Therefore it also proposed many dimension reduction methods to identify for parameter, for example, Principal Component Analysis (PCA), blind source separating analytic approach
(BSS).However these algorithms are linear dimension reduction methods, can only find that the global Euclidean distance of structure can not but find inherent son
Manifold structure.But since nonlinear response multidigit is in the submanifold of the external space, just propose many non-linearity manifolds
Practise algorithm, but it is not deployed its Modal Parameter Identification field application.
Invention content
The purpose of the present invention is to provide a kind of Local Liner Predictions (LLE) and Martin Hilb based in manifold learning
The modal parameters discrimination method that spy-Huang (HHT) is combined, the method realization pair being combined using LLE and HHT algorithms
Structure is based only upon the Modal Parameter Identification of nonlinear response data, to solve the above technical problems.
To achieve the goals above, the present invention adopts the following technical scheme that：
LLE algorithms are identified by following technical scheme realization to modal frequency and the vibration shape：
A kind of modal parameters discrimination method being combined with Hilbert-Huang transform based on manifold learning, including with
Lower step：
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, structure is obtained
The vibration shape and intrinsic frequency；
Step 3: being handled using Hilbert-Huang transform method the time domain response data of step 1 acquisition, obtain
The damping ratio of structure.
Further, the time domain response data that measuring point in structure is acquired in step 1 are X (x, t), and x indicates that sampled point is rung
It answers, t indicates the sampling time；
Step 2 specifically includes：
2.1)：It determines neighborhood point number, finds neighborhood
For the matrix that test sample X (x, t) is D × N, D is sampled point total number, and N is that the maximum of same sampled point is adopted
Sample number；Calculate the data point x of same sampled pointiWith other data points xjBetween Euclidean distance, find and xiAt a distance of nearest k
A neighborhood point, the k values corresponding to reconstruction error minimum are chosen by program automatically；I=1,2 ..., N；J=1,2 ..., N；
2.2)：It calculates and rebuilds weights W
The partial reconstruction weight matrix that the sample point is calculated by the Neighbor Points of each sample point makes the reconstruction of sample point miss
It is poor minimum, that is, seek following optimal problem：
Wherein：WijIt is xiAnd xjBetween weights；Meet following two restrictive conditions：1. as some data point xjIt is not belonging to
Reconstructed data point xiNeighbour's data point when, weights Wij=0；2. the sum of element often capable is equal to 1 in weight matrix, i.e.,
2.3)：Calculate low-dimensional insertion vector Y
The dimension d that low-dimensional is embedded in vector Y is structure mode number of concern；By minimizing embedded cost functional equation
(10), make low-dimensional reconstructed error ε (Y) minimum, at this point, low-dimensional insertion vector is the 2nd of M minimums to the d+1 feature vector；
Wherein, M=(I-W)T(I-W),And
2.4)：Obtain structural eigenvector and intrinsic frequency
According to the theory of Structural Dynamics, the response of system is expressed as the linear combination of natural mode of vibration；In data acquisition
In the process, three dimension system by it is discrete be D measuring point, the time can by it is discrete be N number of sampled point, rank number of mode d of concern, respond
It is indicated by equation (3)：
xD×N=ΦD×d·ηd×N (3)
Wherein, xD×NIt is original time domain response, ΦD×dIt is vibration shape matrix, ηd×NIt is modal coordinate；
Feature vector Y as obtained by LLE algorithm dimensionality reductions is the modal coordinate η in model analysisd×N, then pass through following formula：
Vibration shape matrix Φ is calculated, intrinsic frequency is obtained by the Fourier transformation to modal coordinate.
Further, step 3 specifically includes：
3.1)：NExT is extracted from by convergent response
The time domain acceleration responsive data of acquisition are filtered, it is reference point optionally to take a measuring point, calculates a sound
The cross-correlation function between reference point should be put, for linear structure, the cross-correlation function between white noise acoustic response and impulse function one
It causes, is free damped signal, is expressed as：
Wherein, RjiFor the cross-correlation function between two measuring points；τ is the time；ψjrIt is j-th of element of r first order modes；GirIt is
Only with i, the relevant constants of r；mr,ξr,ωr,ωdrRespectively the r ranks modal mass of structure, damping ratio, undamped natural frequency of a mechanical system
And there is damped natural frequency；θkFor the phase angle of kth rank；
3.2)：Empirical mode decomposition decomposes to obtain stationary random signal
To cross-correlation function RjiIt carries out EMD to decompose to obtain limited a intrinsic mode function, as the defeated of Hilbert transform
Enter；
3.3)：HT is analyzed
Hilbert transformation, and tectonic knot signal z (τ) are carried out to every single order free damping signal：
Then amplitude A (τ) and phase theta (τ) are expressed as：
Above formula amplitude and phase are carried out respectively to seek natural logrithm and differentiated, is obtained：
It is utilized respectively least square fitting amplitude spectrum and phase spectrum, acquires ξ respectivelyrωrAnd ωdr；
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, join in structural material
In the case that number and experimental condition are unknown, it is only necessary to response data, you can the vibration shape with degree of precision and intrinsic frequency are obtained,
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 of the drawings
Fig. 1 is flow chart of the method for the present invention；
Fig. 2 is the finite element model of plate and the position view of 12 response points；
Fig. 3 is cross-correlation function signal；
Fig. 4 is first step mode free damping signal；
Fig. 5 is Logarithmic magnitude curve and its least square fitting schematic diagram；
Fig. 6 is the instantaneous frequency schematic diagram of first step mode response；
Specific implementation mode
Technical scheme of the present invention is further described below in conjunction with the accompanying drawings.
The present invention is a kind of modal parameters identification side being combined with Hilbert-Huang transform based on manifold learning
Method, when carrying out Modal Parameter Identification using LLE algorithms, response data is seen as a High Dimensional Data Set.It is carried from geometric properties
From the perspective of taking, the vibration shape is considered as the inherent characteristic of High Dimensional Data Set.Higher-dimension response data sets are carried out using LLE algorithms
Dimension-reduction treatment, you can obtain the vibration shape and intrinsic frequency.
The data being distributed on manifold of higher dimension a very little regional area can approximation regard that be distributed in a low-dimensional super as
In plane, in this neighborhood, it will be assumed that there are a Linear Mappings between high dimensional data and low-dimensional insertion.Therefore, for one
A new sample point, finds its neighborhood in luv space first, and one is then built in this neighborhood from higher-dimension to low
The Linear Mapping of dimension is realized finally by Linear Mapping to the extensive of new samples.
As shown in Figure 1, it is assumed that a response data sets XD×N=[x1,x2,…,xN]∈RD×NIncluding N column vectors, each column dimension
The step of degree is D, to drop to d dimensions, and LLE algorithms identify is as follows：
1) neighborhood point number k is determined：Calculate the data point x of same sampled pointi(i=1,2 ..., N) and other data points
xjEuclidean distance between (j=1,2 ..., N), finds and xiAt a distance of k nearest neighborhood point, chooses reconstruction automatically by program and miss
K values corresponding to poor minimum.
2) it calculates and rebuilds weights W：The partial reconstruction weight matrix of the sample point is calculated by the Neighbor Points of each sample point,
Keep the reconstruction error of sample point minimum, that is, seeks following optimal problem：
Wherein：WijIt is xiAnd xjBetween weights.In order to obtain optimal weights, following two restrictive conditions need to be met：
1. as some data point xjIt is not belonging to reconstructed data point xiNeighbour's data point when, weights Wij=0；2. often going in weight matrix
The sum of element be equal to 1, i.e.,
3) low-dimensional insertion vector Y is calculated：By minimum embedded cost function, keep low-dimensional reconstructed error ε (Y) minimum, this
When, low-dimensional insertion is M minimum the 2nd to the d+1 feature vector.
Wherein, M=(I-W)T(I-W),And
According to the theory of Structural Dynamics, the response of system can be expressed as the linear combination of natural mode of vibration.It is adopted in data
During collection, three dimension system can by it is discrete be D measuring point, the time can be by discrete for N number of sampled point.System response can be by equation
(3) it indicates：
xD×N=ΦD×d·ηd×N (3)
Wherein, xD*NIt is in response to, ΦD*dIt is vibration shape matrix, ηd*NIt is modal coordinate vector.
During carrying out model analysis using LLE algorithms, first pass through to xD*NFeature extraction obtain principal coordinate matrix
ηd×N, then pass throughCalculate vibration shape matrix ΦD*d。
HHT and natural excitation technique (NExT) realize the following technical scheme that is identified by of modal frequency and damping ratio
's：
It is assumed that x (t) is the structural response time series signal obtained under an environmental excitation, bandpass filtering is carried out to x (t)
Corresponding free damping is obtained with NExT to respond, then carry out empirical mode decomposition (EMD) again afterwards.First according to Fourier spectrum
Structural natural frequencies are obtained according to a preliminary estimate, then carry out bandpass filtering.Empirical modal point is carried out to filtered time series signal
Solution, obtained the first rank intrinsic mode function (Intrinsic mode function, IMF), general just very close structure
Modal response.It is as follows：
1) time domain response data are filtered, obtain the input signal of NExT methods, calculate the cross-correlation of point-to-point transmission
Function, point-to-point transmission cross-correlation function are expressed as：
Wherein, RjiFor the cross-correlation function between two measuring points；τ is the time；ψjrIt is j-th of element of r first order modes；GirIt is
Only with i, the relevant constants of r；mr,ξr,ωr,ωdrRespectively the r ranks modal mass of structure, damping ratio, undamped natural frequency of a mechanical system
And there is damped natural frequency；θkFor the phase angle of kth rank.
2) cross-correlation function is subjected to EMD processing, obtains the response of structure first step mode, as stationary random signal；
3) it is converted using HT and Hilbert transformation, and tectonic knot signal is carried out to the signal that EMD is decomposed：
Then amplitude A (τ) and phase theta (τ) can be expressed as：
Above formula amplitude and phase are carried out respectively to seek natural logrithm and differentiated, can be obtained：
Using least square fitting amplitude spectrum and phase spectrum, ξ is acquired respectivelyrωrAnd ωdr, againThis
Sample ωrAnd ξrIt can find out and.
Using an Orthotropic Composite plate as object, which is 13 layers of composite panel, the material of adjacent two layers
Expect that parameter is as shown in table 1, response data is to choose 12 points be distributed on plate under white-noise excitation under four side free conditions,
Calculate the time domain acceleration responsive X (x, t) of each point, wherein x indicates that time domain acceleration responsive, t are corresponding sampling instant.Plate
Finite element model and 12 response point positions as shown in Fig. 2, sample frequency is 1024Hz.Since construction material properties are compound
Material, there are non-linear for the response of white-noise excitation lower structure.That is D=12, N=1024.
The material parameter of 1 plate of table
Application proposed by the present invention is to obtain plate only in accordance with the nonlinear time-domain response data X (x, t) of plate using LLE algorithms
Modal parameter, including modal frequency and the vibration shape, and damping ratio and mode frequency are obtained with the NExT methods being combined using HHT
Rate.Then the relevance values between the vibration shape and the finite element FEM vibration shapes of comparison algorithm extraction：
VLLERepresent the vibration shape matrix obtained by LLE algorithms, VFERepresent the vibration shape matrix of finite element method extraction.
MACLLE,FERepresent the relevance values between two matrixes.Work as MACLLE,FEExpression is extracted by LLE algorithms when more than or equal to 0.8
Vibration shape matrix is consistent with the vibration shape matrix that finite element is extracted, and works as MACLLE,FEWhen less than 0.2, show that the two is orthogonal.
Following step, such as Fig. 1 are executed to data sample by LLE algorithms：
Step 1：It determines neighborhood point number, finds neighborhood
The matrix for being 12 × 1024 for test sample X (x, t), calculates the data point x of same sampled pointi(i=1,
2 ..., 1024) and other data points xj(j=1,2 ..., 1024) between Euclidean distance, find and xiIt is adjacent at a distance of nearest k
Domain point is chosen the k values corresponding to reconstruction error minimum by program automatically.
Step 2：It calculates and rebuilds weights W
The partial reconstruction weight matrix that the sample point is calculated by the Neighbor Points of each sample point makes the reconstruction of sample point miss
It is poor minimum, that is, seek following optimal problem
Wherein：WijIt is xiAnd xjBetween weights.In order to obtain optimal weights, following two restrictive conditions need to be met：
1. as some data point xjIt is not belonging to reconstructed data point xiNeighbour's data point when, weights Wij=0；2. often going in weight matrix
The sum of element be equal to 1, i.e.,
Step 3：Calculate low-dimensional insertion vector Y
Wherein, the dimension d of low-dimensional insertion vector Y is structure mode number of concern.By minimizing embedded cost function
Equation (11) keeps low-dimensional reconstructed error ε (Y) minimum, at this point, low-dimensional insertion vector is the 2nd of M minimums to the d+1 feature
Vector.
Wherein, M=(I-W)T(I-W),And
Step 4：Calculate structural eigenvector and intrinsic frequency
According to the theory of Structural Dynamics, the response of system can be expressed as the linear combination of natural mode of vibration.It is adopted in data
During collection, three dimension system can by it is discrete be 12 measuring points, the time can by it is discrete be 1024 sampled points, mode of concern
Exponent number d=3, response can be indicated by equation (12)：
x12×1024=Φ12×3·η3×1024 (12)
Wherein, x12×1024It is original time domain response, Φ12×3It is vibration shape matrix, η3×1024It is modal coordinate.
Feature vector Y as obtained by LLE algorithm dimensionality reductions is the modal coordinate η in model analysis3×1024, then under passing through
Formula：
Vibration shape matrix Φ is calculated, intrinsic frequency is obtained by the Fourier transformation to modal coordinate.Dimensionality reduction result such as table
2, the second row shown in table is first three first order mode and intrinsic frequency that finite element FEM methods identify, the third line show LLE calculations
First three first order mode and intrinsic frequency for the structure that method identifies；Listed in table 3 intrinsic frequency that two methods are analyzed and
The correlation MAC value of frequency.It can see by data in table, what first three order frequency that LLE algorithms identify was identified with FEM methods
The maximum difference of intrinsic frequency is 1, about 1.72%；The correlation MAC value of first three first order mode of the two is respectively 0.9286,
0.8743,0.9961, it is all higher than 0.8, shows that the vibration shape that the two is extracted is consistent.
2 FEM of table and the modal parameter of LLE algorithms identification compare
The Comparative result of table 3 FEM and LLE methods
It is as follows with the method identification damping ratio and frequency that NExT is combined using HHT：
Step 1：NExT is extracted from by convergent response
The time domain acceleration responsive data of acquisition are filtered, it is reference point optionally to take a measuring point, calculates a sound
The cross-correlation function between reference point should be put, for linear structure, the cross-correlation function between white noise acoustic response and impulse function one
It causes, is free damped signal, can be expressed as：
Wherein, RjiFor the cross-correlation function between two measuring points；τ is the time；ψjrIt is j-th of element of r first order modes；GirIt is
Only with i, the relevant constants of r；mr,ξr,ωr,ωdrRespectively the r ranks modal mass of structure, damping ratio, the intrinsic frequency of undamped
Rate and there is damped natural frequency；θkFor the phase angle of kth rank.
It is reference point that measuring point 3 is chosen in the embodiment of the present invention, calculates the cross-correlation function between measuring point 4 and reference point 3, obtains
Free damping responds, the following Fig. 3 of deamplification；
Step 2：Empirical mode decomposition obtains stationary random signal
To cross-correlation function RjiIt carries out empirical modal EMD to decompose to obtain limited a intrinsic mode function IMF, as Martin Hilb
The input of spy's transformation, first following Fig. 4 of rank free damping signal；
Step 3：HT is analyzed
Hilbert transformation, and tectonic knot signal z (τ) are carried out to every single order free damping signal：
Then amplitude A (τ) and phase theta (τ) can be expressed as：
Above formula amplitude and phase are carried out respectively to seek natural logrithm and differentiated, can be obtained：
It is utilized respectively least square fitting amplitude spectrum and phase spectrum, acquires ξ respectivelyrωrAnd ωdr, wherein Logarithmic magnitude
Spectrum and its following Fig. 5 of least square fitting straight line, phase spectrum such as Fig. 6, again：
Such ωrAnd ξrIt can find out and.Two rank results are extracted with FFT recognition results, LLE algorithms before HHT is extracted
As a result, finite element result is compared, such as table 4.By result it can be seen that using HHT, FFT, LLE algorithm to two ranks it is intrinsic
Frequency is almost the same with Finite element analysis results, and damping ratio then has certain difference.
The comparison of 4 each method recognition result of table
Comprehensive analysis can obtain the susceptible frequency of structure it is found that using method of the invention only in accordance with response data
The characteristic of each rank primary modal within the scope of rate is Analysis of Vibration Characteristic, vibrating failure diagnosis and forecast and the knot of structural system
The optimization design of structure dynamic characteristics provides foundation, and new method is equally also provided for System Discrimination.
Claims (2)
1. a kind of modal parameters discrimination method being combined with Hilbert-Huang transform based on manifold learning, feature are existed
In including 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 of structure is obtained
And intrinsic frequency；
Step 3: being handled using Hilbert-Huang transform method the time domain response data of step 1 acquisition, structure is obtained
Damping ratio.
2. a kind of structural modal ginseng being combined with Hilbert-Huang transform based on manifold learning according to claim 1
Number discrimination method, which is characterized in that the time domain response data that measuring point in structure is acquired in step 1 are X (x, t), and x indicates sampling
Point response, t indicate the sampling time；
Step 2 specifically includes：
2.1)：It determines neighborhood point number k, finds neighborhood
For the matrix that test sample X (x, t) is D × N, D is sampled point total number, and N is the maximum sampling of same sampled point
Number；Calculate the data point x of same sampled pointiWith other data points xjBetween Euclidean distance, find and xiIt is adjacent at a distance of nearest k
Domain point is chosen the k values corresponding to reconstruction error minimum by program automatically；I=1,2 ..., N；J=1,2 ..., N；
2.2)：It calculates and rebuilds weights W
The partial reconstruction weight matrix that the sample point is calculated by the Neighbor Points of each sample point, makes the reconstruction error of sample point most
It is small, that is, seek following optimal problem：
Wherein：WijIt is xiAnd xjBetween weights；Meet following two restrictive conditions：1. as some data point xjIt is not belonging to weigh
Structure data point xiNeighbour's data point when, weights Wij=0；2. the sum of element often capable is equal to 1 in weight matrix, i.e.,
2.3)：Calculate low-dimensional insertion vector Y
The dimension d that low-dimensional is embedded in vector Y is structure mode number of concern；By minimizing embedded cost functional equation (2), make
Low-dimensional reconstructed error ε (Y) is minimum, at this point, low-dimensional insertion vector is the 2nd of M minimums to the d+1 feature vector；
Wherein, M=(I-W)T(I-W),And
2.4)：Obtain structural eigenvector and intrinsic frequency
According to the theory of Structural Dynamics, the response of system is expressed as the linear combination of natural mode of vibration；In the process of data acquisition
In, three dimension system by it is discrete be D measuring point, the time can by it is discrete be N number of sampled point, rank number of mode of concern be d, respond by
Equation (3) indicates：
xD×N=ΦD×d·ηd×N (3)
Wherein, xD×NIt is original time domain response, ΦD×dIt is vibration shape matrix, ηd×NIt is modal coordinate；
Feature vector Y as obtained by LLE algorithm dimensionality reductions is the modal coordinate η in model analysisd×N, then pass through following formula：
Vibration shape matrix Φ is calculated, intrinsic frequency is obtained by the Fourier transformation to modal coordinate；
Step 3 specifically includes：
3.1)：NExT is extracted from by convergent response
The time domain acceleration responsive data of acquisition are filtered, it is reference point optionally to take a measuring point, calculates a response point
Cross-correlation function between reference point, for linear structure, the cross-correlation function between white noise acoustic response is consistent with impulse function, is
Free damped signal, is expressed as：
Wherein, RjiFor the cross-correlation function between two measuring points；τ is the time；ψjrIt is j-th of element of r first order modes；GirBe only with
The relevant constant of i, r；mr,ξr,ωr,ωdrRespectively the r ranks modal mass of structure, damping ratio, undamped natural frequency of a mechanical system and have
Damped natural frequency；θkFor the phase angle of kth rank；
3.2)：Empirical mode decomposition obtains stationary random signal
To cross-correlation function RjiIt carries out empirical mode decomposition EMD and obtains limited a intrinsic mode function, as Hilbert transform
Input；
3.3)：HT is analyzed
Hilbert transformation, and tectonic knot signal z (τ) are carried out to every single order free damping signal：
Then amplitude A (τ) and phase theta (τ) are expressed as：
Above formula amplitude and phase are carried out respectively to seek natural logrithm and differentiated, is obtained：
It is utilized respectively least square fitting amplitude spectrum and phase spectrum, acquires ξ respectivelyrωrAnd ωdr；
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| CN115579090A (en) * | 2022-11-09 | 2023-01-06 | 陕西科技大学 | Shell structure material classification method based on low-dimensional admittance characteristics |
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