CN109254321B - Method for identifying rapid Bayesian modal parameters under seismic excitation - Google Patents

Method for identifying rapid Bayesian modal parameters under seismic excitation Download PDF

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CN109254321B
CN109254321B CN201810842715.XA CN201810842715A CN109254321B CN 109254321 B CN109254321 B CN 109254321B CN 201810842715 A CN201810842715 A CN 201810842715A CN 109254321 B CN109254321 B CN 109254321B
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张凤亮
倪艳春
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Tongji University
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    • G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
    • G01V1/28—Processing seismic data, e.g. for interpretation or for event detection
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Abstract

The invention relates to a rapid Bayesian modal parameter identification method under seismic excitation, which comprises the following steps: s1, acquiring seismic input and structural response data under known seismic excitation; s2, selecting a frequency domain segment, obtaining an initial value of the natural frequency from the singular value spectrum of the data in the step S1, and setting an initial value of the damping ratio; s3, obtaining the optimal values of the natural frequency and the damping ratio by minimizing the objective function obtained by the Bayesian formula; and S4, obtaining the modal contribution factor through the optimal values of the natural frequency and the damping ratio. Compared with the prior art, the method has the advantages that the use process is convenient, the calculation can be directly carried out only by selecting the frequency domain section and then inputting the initial frequency and the damping ratio, the experience analysis of professionals is not needed, the calculation speed is higher than that of the traditional method, and the method can be directly used in the test process only within a few seconds after the analysis is usually completed.

Description

Method for identifying rapid Bayesian modal parameters under seismic excitation
Technical Field
The invention relates to a modal parameter identification technology under seismic excitation, in particular to a rapid Bayesian modal parameter identification method under seismic excitation.
Background
Under seismic excitation, parameters for modal parameter identification based on the acquired structural vibration response mainly include the natural frequency, damping ratio and mode shape of the structure. The three parameters are inherent properties of the structure, generally remain substantially unchanged, and if changes occur, structural damage may be caused, so that the parameters play an important role in structural model modification, damage identification and structural health monitoring. Modal identification under seismic excitation can help to know the energy consumption capability of the structure under larger vibration amplitude, and meanwhile, an effective method is provided for the analysis of the experiment data of the vibrating table.
The prior art has the following two problems. The first problem is that the existing identification method has complicated steps, usually needs professionals to analyze data, has slow calculation process and cannot analyze data in time on a test site. The second problem is that because the input seismic excitation is random load, certain uncertainty necessarily exists in the output modal parameters, however, at present, no effective method can be used for realizing the accuracy evaluation of the modal parameters under the seismic excitation, and a frame needs to be built to provide a platform for the modal parameter evaluation.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a rapid Bayesian modal parameter identification method under seismic excitation.
The purpose of the invention can be realized by the following technical scheme:
a fast Bayesian modal parameter identification method under seismic excitation comprises the following steps:
s1, acquiring seismic input and structural response data under known seismic excitation;
s2, selecting a frequency domain segment, obtaining an initial value of the natural frequency from the singular value spectrum of the data in the step S1, and setting an initial value of the damping ratio;
s3, obtaining the optimal values of the natural frequency and the damping ratio by minimizing the objective function obtained by the Bayesian formula;
and S4, obtaining the modal contribution factor through the optimal values of the natural frequency and the damping ratio.
Preferably, the objective function obtained by the bayesian formula is:
Figure BDA0001745952150000021
wherein f isiShowing the natural frequency, ζ, of the ith modeiRepresenting the damping ratio of the ith order mode,
Figure BDA0001745952150000022
a fast fourier transform, k 2,3, N, representing the measured acceleration response of the kth time domain sampleq，Nq＝int[N/2]+1，int[N/2]Indicating a rounding down to the nearest integer for N/2, N representing the number of samples measured in the time domain,
Figure BDA0001745952150000023
to represent
Figure BDA0001745952150000024
The conjugate transpose of (a) is performed,
Figure BDA0001745952150000025
wherein f isk(k-1)/N Δ t, Δ t representing the time interval between sampling points, Fg(fk) Representing known seismic excitations
Figure BDA0001745952150000026
The fast fourier transform of (a) the fast fourier transform,
Figure BDA0001745952150000027
represents the kronecker product, Re represents the vector real part in the middle bracket, InRepresenting an n × n identity matrix, n representing the number of measured structural degrees of freedom, Rnm×nmRepresenting a real matrix of order nm × nm,
Figure BDA0001745952150000028
represents hkConjugate transpose of (i), hkComprises the following steps:
hk＝[h1k,h2k,...,hmk]∈R1×m
wherein h isikExpressed at a frequency fkI is more than or equal to 1 and less than or equal to m, m represents the number of contributing modes,
Figure BDA0001745952150000029
wherein the content of the first and second substances,
Figure BDA00017459521500000210
is represented by Fg(fk) Conjugate matrix of RnmRepresenting a nm × 1 order real matrix.
Preferably, the modal contribution factor is:
γi＝||Φγ(i)||
wherein, | Φγ(i) I represents phi normalized to 1γ(i)，Φγ(i) By slave (phi)γAnd (b) is extracted from the optimal values of (a) and (b), wherein:
Figure BDA00017459521500000211
said (phi)γBy the optimal values of P and Q
Figure BDA00017459521500000212
And
Figure BDA00017459521500000213
thus obtaining the product.
Preferably, said (Φ)γThe optimal values of the following components:
Figure BDA00017459521500000214
wherein the content of the first and second substances,
Figure BDA00017459521500000215
p and Q, respectively, at the time of minimization of the objective function.
Preferably, the fast fourier transform of the measured acceleration response of the kth time-domain sample
Figure BDA0001745952150000031
Comprises the following steps:
Figure BDA0001745952150000032
wherein, Fk(theta) fast Fourier transform of theoretical values of acceleration response, FekIs the fast fourier transform of the prediction error:
Figure BDA0001745952150000033
wherein S iseMagnitude of power spectral density, Z, representing prediction error1kAnd Z2kGaussian real vectors, i, representing two criteria2＝-1。
Preferably, said at frequency fkEquation h for the conversion of the ith order mode ofikComprises the following steps:
hik＝[(βik 2-1)+i(2ζiβik)]-1
wherein, βik＝fi/fk，i2Is-1. Compared with the prior art, the invention has the following advantages:
1. the method has the advantages that the use process is convenient, only the frequency domain section needs to be selected, then the initial frequency and the damping ratio can be directly calculated, a professional does not need to perform experience analysis, the calculation speed is higher than that of a traditional method, and the method can be directly used in the test process as long as several seconds are required for completing the analysis.
2. The method can simultaneously analyze the vibration table experiment data and the data under the earthquake response measured by the real structure, and has better robustness.
3. Based on the framework provided by the method, a platform is provided for modal parameter evaluation, and the uncertainty evaluation of modal parameters which cannot be done in the prior art can be realized.
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FIG. 1 is a schematic flow diagram of the process of the present invention.
Detailed Description
The invention is described in detail below with reference to the figures and specific embodiments. The present embodiment is implemented on the premise of the technical solution of the present invention, and a detailed implementation manner and a specific operation process are given, but the scope of the present invention is not limited to the following embodiments.
The application provides a method for carrying out rapid modal parameter identification based on known seismic excitation and structural response under a Bayesian theory framework. Firstly, based on the acquired structural vibration response and the input seismic motion, including acceleration, speed, displacement and the like, according to the basic principle of structural dynamics, a likelihood function and a prior probability density function under a Bayes frame are constructed, a posterior probability density function is constructed, finally, a negative log-likelihood function is obtained, and the modal parameter optimal value can be rapidly identified by derivation of the negative log-likelihood function and an iterative optimization algorithm. The Bayesian framework developed by the method can also provide a platform for later-stage modal parameter evaluation.
Examples
As shown in fig. 1, a fast bayesian modal parameter identification method under seismic excitation includes the following steps:
s1, acquiring seismic input and structural response data under known seismic excitation;
s2, selecting a frequency domain segment and obtaining inherent data from the singular value spectrum of the data in the step S1Frequency fiSet damping ratio ζiMay be set to 1%, and a single mode or a plurality of modes may be selected in the frequency domain segment;
s3, obtaining the optimal values of the natural frequency and the damping ratio by minimizing the objective function obtained by the Bayesian formula;
and S4, obtaining the modal contribution factor through the optimal values of the natural frequency and the damping ratio.
Application of acceleration response of n degrees of freedom of structure measured in step S1
Figure BDA0001745952150000041
Where N denotes the number of samples measured in the time domain, for convenience, it is simplified to
Figure BDA0001745952150000042
The jth measured acceleration response may be expressed as:
Figure BDA0001745952150000043
in the formula
Figure BDA0001745952150000044
Is a theoretical value of the acceleration response, ej∈RnThe prediction error represents the difference between the measured acceleration value and the theoretical value, and is composed of model error, noise and the like.
In the range of the frequency domain,
Figure BDA0001745952150000045
the Fast Fourier Transform (FFT) of (a) can be defined as:
Figure BDA0001745952150000046
in the formula i2-1; Δ t represents the time interval between sampling points; k 2,3qCorresponding to a frequency fkFFT data of (k-1)/N Δ t, Nq＝int[N/2]+1. Only is provided withFFT data in a selected frequency band is used for mode identification.
Fourier transforms are performed on both sides of equation (1),
Figure BDA0001745952150000047
the fourier transform of (a) can be expressed as:
Figure BDA0001745952150000048
wherein, Fk(theta) FFT, F representing theoretical values of acceleration responseekIs the FFT of the prediction error. In general, the prediction error can be modeled as white gaussian noise, and thus the power spectral density of the prediction error can be assumed to be constant and of magnitude S in a selected frequency domain segmente，FekCan be expressed as:
Figure BDA0001745952150000049
wherein Z is1kAnd Z2kRepresenting two standard gaussian real vectors whose values are independent.
The process of obtaining the objective function by the Bayesian formula is as follows:
consider a structure under known seismic excitation, represented as
Figure BDA0001745952150000051
Assuming that the structural response due to seismic excitation dominates, the response due to environmental excitation can be modeled as a prediction error. Assuming classical damping, the acceleration response of a linear structure can be expressed as:
Figure BDA0001745952150000052
where m represents the number of contributing modalities; phii∈RnRepresenting a mode shape vector corresponding to the test degree of freedom;
Figure BDA0001745952150000053
a modal response representing an ith order mode that satisfies the following equation:
Figure BDA0001745952150000054
in the formula of omegai＝2πfi，fiA natural frequency representing the i-th order mode; zetaiRepresents the damping ratio of the ith order; p is a radical ofi(t) represents the ith order modal force, which under seismic excitation, can be expressed as:
Figure BDA0001745952150000055
wherein the content of the first and second substances,
Figure BDA0001745952150000056
representing the mode shape containing all degrees of freedom of the structure, M being the mass matrix, 1 representing the n × 1 vector, all elements inside it being equal to 1. define:
Figure BDA0001745952150000057
for modal contribution factors, then one can get:
Figure BDA0001745952150000058
fourier transform and rearrangement are carried out on the formula (9), and modal acceleration response is carried out
Figure BDA00017459521500000517
The FFT of (a) can be expressed as:
Figure BDA0001745952150000059
it can be obtained by substituting the following two relationships:
Figure BDA00017459521500000510
Figure BDA00017459521500000511
in the formula, Fg(fk) Is that
Figure BDA00017459521500000512
FFT of (2);
Figure BDA00017459521500000513
and
Figure BDA00017459521500000514
are respectively
Figure BDA00017459521500000515
And ηiFFT of (t); and is
hik＝[(βik 2-1)+i(2ζiβik)]-1(13)
Is a complex number which is represented at a frequency fkβ transformation equation of the ith order modeik＝fi/fkThe ratio of the frequencies is indicated.
Substituting equation (10) into (5), the FFT of the modal response can be expressed as:
Figure BDA00017459521500000516
let D denote the FFT data in a selected frequency domain segment, which contains the modality we are interested in. Based on the bayesian theory, the posterior probability density function of the modal parameter θ to be identified can be expressed as:
Figure BDA0001745952150000061
where p (θ) is a prior probability density function; p (D | theta) represents a likelihood function; p (d) can be seen as a constant.
P (D | θ) changes faster than p (θ) due to the influence of data D. Thus, assuming a single prior, the a posteriori probability density function is proportional to the likelihood function, that is,
p(θ|D)∝p(D|θ) (16)
the likelihood function p (D | θ) can be constructed by the following derivation.
Definition of
Figure BDA0001745952150000062
Is a vector consisting of the real and imaginary parts of the FFT data;
Figure BDA0001745952150000063
including selecting FFT data within a frequency domain segment. It can be shown that for large N and small deltat,
Figure BDA0001745952150000064
are independent at different frequencies and follow a gaussian distribution. Based on this fact, the likelihood function p (D | θ) can be expressed as:
Figure BDA0001745952150000065
in the formula CkTo represent
Figure BDA0001745952150000066
The covariance matrix of (a); det (·) denotes determinant; mu.sk＝[ReFk(θ)+ImFk(θ)]Is that
Figure BDA0001745952150000067
Theoretical value of (2), here Fk(θ) is a complex vector that can be expressed as:
Fk(θ)＝ReFk(θ)+iImFk(θ) (18)
ReFk(theta) and ImFk(theta) each represents FkReal and imaginary parts of (θ).
According to the formula (4),
Figure BDA0001745952150000068
is equal to Se2, the likelihood function can therefore be expressed as:
Figure BDA0001745952150000069
simultaneously:
Figure BDA00017459521500000610
thus, the above equation can be written as:
Figure BDA00017459521500000611
where the prime symbol of a character indicates the conjugate transpose of the corresponding character.
For optimization convenience, we convert the maximum problem to the minimum problem with a negative log-likelihood function, as follows:
p(θ|D)∝exp(-L(θ)) (22)
in the above formula
Figure BDA00017459521500000612
Theoretically, the optimal value of the modal parameter can be achieved by minimizing the negative log-likelihood function according to equation (23). However, performing the optimization directly would be very time consuming, and the computational efficiency would be greatly reduced with the number of degrees of freedom of the test and the number of modes within the selected frequency domain segment. To improve computational efficiency, some parameters may be obtained by analytical solutions, which are expressed as a function of other modal parameters, and the remaining modal parameters may be obtained by optimization.
In the formula (23), SeIndependent of other modal parameters, a parameter set θ ═ f is definedi,ζi,γiΦ (i): i ═ 1.., m }, where γi∈R，Φ(i)∈Rn(ii) a Theta does not include Se. Defining an objective function:
Figure BDA0001745952150000071
thus:
Figure BDA0001745952150000072
wherein term that do not depend on SeIs represented by the formulaeIrrelevant parameters, since the form of alnx + b/x has a unique minimum value at x ═ b/a, SeThe optimum value of (c) can be obtained from the following equation:
Figure BDA0001745952150000073
in equation (25), when J (θ) reaches its minimum value, L (θ) will also reach the minimum value. Therefore, J (θ) is independent of SeThe optimum value of θ can be obtained by minimizing J (θ).
For mode shapes within θ, a normalized constraint is usually required, that is, | | Φi1. These constraints are also to be reasonably considered when performing the optimization. In the formula (14), γiAlways heel phiiAppear together, thus defining a new variable:
Φγ(i)＝γiΦi(27)
this is an unconstrained vector. Equation (14) can thus be written as:
Figure BDA0001745952150000074
due to phiγ(i) Is an n × m matrix, when m > 1, J (theta) to phiγ(i) The derivative solution of (a) will be very difficult. To solve this problem, equation (28) can be reconstructed as:
Figure BDA0001745952150000075
in the formula InRepresenting an n × n identity matrix,
Figure BDA0001745952150000076
which represents the kronecker product of,
hk＝[h1k,h2k,...,hmk]∈R1×m(30)
and:
Figure BDA0001745952150000077
substituting equation (29) into J (θ) in equation (24) can result in:
Figure BDA0001745952150000078
in the formula (I), the compound is shown in the specification,
Figure BDA0001745952150000081
Figure BDA0001745952150000082
wherein R isnm×nmRepresenting a real matrix of order nm × nm,
Figure BDA0001745952150000083
represents hkRe denotes the vector real part in the middle brackets,
Figure BDA0001745952150000084
representing seismic excitations
Figure BDA0001745952150000085
By conjugate transpose of the fast Fourier transform of (1), RnmThe representation represents a real matrix of order nm × 1.
J (theta) to phiγFirst derivative and let the derivative equal zero, can obtain (phi)γOptimal values of (1):
Figure BDA0001745952150000086
wherein the content of the first and second substances,
Figure BDA0001745952150000087
are optimal for P and Q, respectively, since P and Q depend only on fi,ζiTherefore (phi)γThat is, by substituting it into equation (32), the objective function can be expressed as:
Figure BDA0001745952150000088
Figure BDA0001745952150000089
to represent
Figure BDA00017459521500000810
The conjugate transpose of (c).
{fi,ζiThe optimal value of J ({ f) } can be determined by minimizing J ({ f)i,ζi}) and, once obtained, can yield
Figure BDA00017459521500000811
Figure BDA00017459521500000812
(ΦγThat is, can be obtained by the formula (35). In step S4,. phi.γ(i) Can be selected from (phi)γIs extracted from) so that the modal contribution factor gammai＝||Φγ(i) I can be determined by setting phiiThe regularization results in 1.
For large amounts of data, typically the optimization process is globally identifiable. For problems that are not globally identifiable, some more advanced tools, such as Markov Chain Monte Carlo (MCMC), may be used to solve these problems, which is not within the scope of the present invention.

Claims (5)

1. A fast Bayesian modal parameter identification method under seismic excitation is characterized by comprising the following steps:
s1, acquiring seismic input and structural response data under known seismic excitation;
s2, selecting a frequency domain segment, obtaining an initial value of the natural frequency from the singular value spectrum of the data in the step S1, and setting an initial value of the damping ratio;
s3, obtaining the optimal values of the natural frequency and the damping ratio by minimizing the objective function obtained by the Bayesian formula;
s4, obtaining modal contribution factors through the optimal values of the natural frequency and the damping ratio;
the objective function obtained by the Bayesian formula is as follows:
Figure FDA0002308565420000011
wherein f isiShowing the natural frequency, ζ, of the ith modeiRepresenting the damping ratio of the ith order mode,
Figure FDA0002308565420000012
a fast fourier transform, k 2,3, N, representing the measured acceleration response of the kth time domain sampleq，Nq＝int[N/2]+1，int[N/2]Indicating a rounding down to the nearest integer for N/2, N representing the number of samples measured in the time domain,
Figure FDA0002308565420000013
to represent
Figure FDA0002308565420000014
The conjugate transpose of (a) is performed,
Figure FDA0002308565420000015
wherein f isk(k-1)/N Δ t, Δ t representing the time interval between sampling points, Fg(fk) Representing known seismic excitations
Figure FDA0002308565420000016
The fast fourier transform of (a) the fast fourier transform,
Figure FDA0002308565420000017
represents the kronecker product, Re represents the vector real part in the middle bracket, InRepresenting an n × n identity matrix, n representing the number of measured structural degrees of freedom, Rnm×nmRepresenting a real matrix of order nm × nm,
Figure FDA0002308565420000018
represents hkConjugate transpose of (i), hkComprises the following steps:
hk＝[h1k,h2k,...,hmk]∈R1×m
wherein h isikExpressed at a frequency fkI is more than or equal to 1 and less than or equal to m, m represents the number of contributing modes,
Figure FDA0002308565420000019
wherein the content of the first and second substances,
Figure FDA00023085654200000110
is represented by Fg(fk) Conjugate matrix of RnmRepresenting a nm × 1 order real matrix.
2. The method for identifying Bayesian modal parameters under seismic excitation according to claim 1, wherein the modal contribution factors are:
γi＝||Φγ(i)||
wherein, | Φγ(i) I represents phi normalized to 1γ(i)，Φγ(i) By slave (phi)γAnd (b) is extracted from the optimal values of (a) and (b), wherein:
Figure FDA0002308565420000021
said (phi)γBy the optimal values of P and Q
Figure FDA0002308565420000022
And
Figure FDA0002308565420000023
thus obtaining the product.
3. The method for identifying Bayesian modal parameters under seismic excitation as recited in claim 2, wherein (Φ) isγThe optimal values of the following components:
Figure FDA0002308565420000024
wherein the content of the first and second substances,
Figure FDA0002308565420000025
p and Q, respectively, at the time of minimization of the objective function.
4. The method according to claim 1, wherein the fast fourier transform of the acceleration response measured from the kth time domain sample is used to identify the bayesian modal parameters under seismic excitation
Figure FDA0002308565420000026
Comprises the following steps:
Figure FDA0002308565420000027
wherein, Fk(theta) fast Fourier transform of theoretical values of acceleration response, FekIs the fast fourier transform of the prediction error:
Figure FDA0002308565420000028
wherein S iseMagnitude of power spectral density, Z, representing prediction error1kAnd Z2kGaussian real vectors representing two criteria, j2＝-1。
5. The method according to claim 1, wherein the Bayesian modal parameters are identified at a frequency fkEquation h for the conversion of the ith order mode ofikComprises the following steps:
hik＝[(βik 2-1)+j(2ζiβik)]-1
wherein, βik＝fi/fk，j2＝-1。
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