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CN112798253A - A Structural Modal Parameter Identification Method Considering the Influence of Non-white Environmental Loads - Google Patents

A Structural Modal Parameter Identification Method Considering the Influence of Non-white Environmental Loads Download PDF

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CN112798253A
CN112798253A CN202110073487.6A CN202110073487A CN112798253A CN 112798253 A CN112798253 A CN 112798253A CN 202110073487 A CN202110073487 A CN 202110073487A CN 112798253 A CN112798253 A CN 112798253A
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康杰
居鹤华
余萌
韩亮亮
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Nanjing University of Aeronautics and Astronautics
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Abstract

本发明公开了一种考虑非白环境载荷影响的结构模态参数辨识方法，该方法首先测量工程结构的响应信号并计算功率谱密度函数矩阵。采用频域分解方法，根据功率谱密度函数矩阵的奇异值曲线初步辨识出工程结构的模态参数，包括模态频率和模态振型。随后构造功率谱密度传递率矩阵，根据功率谱密度传递率矩阵奇异值的变化规律剔除由于非白噪声载荷导致的虚假模态参数。最后在剩余的模态频率处对工程结构响应信号进行带通滤波，计算滤波信号的超值峰度系数，根据超值峰度系数进一步剔除谐波载荷导致的虚假模态参数，保留工程结构真实模态参数并作为最终辨识结果。本发明能同时消除非白噪声载荷和谐波载荷的不利影响，具有较大的工程应用意义。

Figure 202110073487

The invention discloses a structural modal parameter identification method considering the influence of non-white environmental loads. The method first measures the response signal of the engineering structure and calculates the power spectral density function matrix. Using the frequency domain decomposition method, the modal parameters of the engineering structure, including modal frequency and modal shape, are preliminarily identified according to the singular value curve of the power spectral density function matrix. Then the power spectral density transmissibility matrix is constructed, and the spurious modal parameters caused by the non-white noise load are eliminated according to the variation law of the singular value of the power spectral density transmissibility matrix. Finally, band-pass filtering is performed on the response signal of the engineering structure at the remaining modal frequencies, and the excess kurtosis coefficient of the filtered signal is calculated. According to the excess kurtosis coefficient, the false modal parameters caused by harmonic loads are further eliminated, and the true engineering structure is preserved. The modal parameters are used as the final identification result. The invention can simultaneously eliminate the adverse effects of non-white noise loads and harmonic loads, and has great engineering application significance.

Figure 202110073487

Description

Structural modal parameter identification method considering non-white environment load influence
Technical Field
The invention relates to a structural modal parameter identification method, in particular to a structural modal parameter identification method considering non-white environment load influence, and belongs to the technical field of modal parameter identification.
Background
A large number of engineering structures exist in daily life and industrial production, and the engineering structures in real working states are excited by various types of excitation, so that structural vibration is generated. In order to avoid the large-amplitude vibration of the engineering structure in the operating state and ensure the safety of the engineering structure, the dynamic characteristic analysis design needs to be carried out on the engineering structure in the design stage, and the dynamic characteristic analysis is carried out on the engineering structure by measuring the structural vibration response signal of the engineering structure in the working stage of the engineering structure so as to realize the monitoring of the structural dynamic characteristic. As an important characterization of the structure dynamics, the structural modal parameters are very important in the engineering structure design, development and maintenance stages.
As an important method and way for obtaining structural modal parameters, structural modal parameter identification research becomes one of the research focuses in the field of structural dynamics. The structure modal parameter identification method can extract the modal parameters of the structure by measuring the load acting on the structure and the response signal of the structure, and mainly comprises the modal frequency, the modal shape and the modal damping ratio of the structure. Since the load applied to the structure in the working state is generally environmental load, such as wind load, earthquake load, tide, etc., it is difficult to accurately measure the load applied to each position. Therefore, the structure modal parameter identification method based on the engineering structure response signal only becomes the most effective way for extracting the engineering structure modal parameters. The method for identifying the structural modal parameters based on the engineering structure response signals is also called as an identification method for identifying the output-only structural modal parameters or an identification method for identifying the operating mode parameters.
Currently, the identification method of only outputting structural modal parameters is mainly divided into a frequency domain method and a time domain method. The Frequency domain method generally estimates a power spectral density function of the response signal by Welch and the like, and then performs modal parameter identification according to the estimated power spectral density function, and the common Frequency domain methods include a Frequency-domain decomposition (FDD) method, an Enhanced Frequency Domain Decomposition (EFDD) method, a PolyMAX method, and the like. The time domain method directly constructs an identification model according to a time domain signal, calculates structural modal parameters after estimating the parameters of the model, and has the advantages of no energy leakage and high frequency resolution, and common time domain methods comprise an Ibrahim Time Domain (ITD) method based on a state space model, a random subspace (SSI) method, a time series model-based method and the like.
However, the identification methods of the modal parameters of the output-only structure, including the identification methods of FDD, EFDD, PolyMAX, ITD, SSI, etc., all assume that the load on the engineering structure is broadband white noise, and the power spectrum thereof should be flat. Under this assumption, the peak values of the power spectrum of the engineering structure response signal are all caused by the resonant frequency of the engineering structure, so the modal parameters of the engineering structure can be identified only through the engineering structure response signal. However, in the actual working process, the loads on the engineering structure are complex and various, and the engineering structure does not meet the white noise assumption in many cases and has obvious non-white characteristics. For example: tests of Ariane 5, Titan II and the like show that low-frequency oscillation also exists in the thrust of the carrier rocket engine, and the vibration main frequency (the minimum dozen or odd hertz) of the carrier rocket engine is located in the concerned structural frequency band; the transverse Delauton (Dryden) gust power spectrum commonly used in gust load analysis also does not have the broadband flat spectrum characteristic; harmonic loads are generated by rotating parts such as rotating shafts and fans in engineering machinery and liquid shaking.
The non-white environmental loads on the engineering structure can be divided into two types: one is a non-white noise load, with randomness, such as engine thrust oscillations, gusts, etc. belonging to this category; the other is harmonic loading, with periodicity such as that caused by rotating shafts, liquid sloshing, etc. When the load borne by the engineering structure belongs to a non-white environment load, a false mode which cannot be eliminated is caused, the reliability of the identification result of the structural mode parameters is influenced, the evaluation result of the dynamic characteristics of the engineering structure is wrong, and great risk is caused in the application process.
In short, the method for identifying the structural modal parameters only by outputting in the prior art is basically only suitable for loads in a white noise form, and cannot effectively solve the problem of identifying the structural modal parameters under the combined action of non-white noise loads and harmonic loads. In engineering application, the non-white environment load brings great difficulty to the modal parameter identification of an engineering structure, and an effective and feasible solution is lacked at present.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method is characterized in that a frequency domain decomposition method in the prior art is expanded, modal parameters identified by the frequency domain decomposition method are screened by adopting a power spectral density transfer rate and an excess kurtosis coefficient, false modes caused by non-white noise loads and harmonic loads are eliminated, and the technical problem that the prior art cannot realize structural modal parameter identification under the combined action of the non-white noise loads and the harmonic loads is solved.
The invention adopts the following technical scheme for solving the technical problems:
a structural modal parameter identification method considering non-white environment load influence comprises the following steps:
step 1, measuring and recording an acceleration or speed or displacement response signal of an engineering structure, and calculating a power spectral density function matrix of the response signal;
step 2, performing singular value decomposition on the power spectral density function matrix obtained by calculation in the step 1, drawing a singular value curve chart of the power spectral density function matrix by adopting a frequency domain decomposition method, judging the modal order contained in the engineering structure response signal according to the singular value curve chart, and preliminarily identifying the modal parameters of the engineering structure, including modal frequency and modal vibration mode;
step 3, constructing a power spectral density transfer rate matrix according to the power spectral density function matrix, eliminating false modal parameters caused by non-white noise loads in the modal parameters obtained in the step 2 according to the power spectral density transfer rate matrix, and reserving real modal parameters of the engineering structure and false modal parameters caused by harmonic loads;
and 4, eliminating false modal parameters caused by harmonic loads by adopting a band-pass filtering method and a harmonic detection method, and reserving real modal parameters of the engineering structure as a final identification result.
Compared with the prior art, the invention adopting the technical scheme has the following technical effects:
1. the invention expands the frequency domain decomposition method in the prior art, adopts the power spectral density transfer rate to directly eliminate the false mode caused by non-white noise load, adopts the band-pass filtering and the over-peak kurtosis coefficient to directly eliminate the false mode caused by harmonic load, is not limited by the form of environmental load, and expands the application range of the structure mode parameter identification method in the prior art.
2. The method does not need prior knowledge of the load form borne by the structure, and compared with the method adopting signal pre-treatment or post-treatment in the prior art, the method adopts the power spectral density transfer rate and the over-peak kurtosis coefficient to identify and eliminate the false mode caused by the non-white environment load, directly eliminates the influence of the non-white environment load theoretically, does not need to manually participate in the mode screening process, is easier to operate, and has low requirements on the professional knowledge background and the working experience of engineering technicians.
3. The method takes the engineering structure modal parameters identified by the frequency domain decomposition method as an initial result, eliminates the false modes in the initial result according to the singular value change rule of the power spectral density transfer rate matrix and the over-value kurtosis coefficient of the band-pass filtering signal, not only fully utilizes the advantages of high identification precision and strong robustness to noise of the frequency domain decomposition method, but also utilizes the advantages of the power spectral density transfer rate capable of eliminating the false modes caused by non-white noise loads and the over-value kurtosis coefficient capable of eliminating the false modes caused by harmonic loads, has the characteristics of high precision and strong robustness to noise and load forms, and improves the quality of the engineering structure modal parameter identification.
4. The structure modal parameters finally identified by the method are not influenced by the load form of the structure at all, belong to the structure real modal parameters, can fully reflect the real dynamic characteristics of the structure, and avoid the risk caused by false modes when the structure modal parameters are used.
Drawings
Fig. 1 is a flowchart of a structural modal parameter identification method considering non-white environmental load influence according to the present invention.
Fig. 2 is a schematic view of a truss structure in an embodiment of the invention.
Fig. 3 is a plot of the magnitude of the diagonal elements of the power spectral density function matrix of the truss displacement response signal in an embodiment of the present invention.
Fig. 4 is a power spectral density function matrix singular value curve of a truss displacement response signal in an embodiment of the invention.
Fig. 5 is a plot of singular values of a power spectral density transfer rate matrix in an embodiment of the present invention.
Fig. 6 is a schematic diagram of a MAC matrix between a finally identified truss structure mode shape and a theoretical mode shape in the embodiment of the present invention.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
Aiming at the technical problems existing when the existing identification method only outputs structural modal parameters is applied to an actual engineering structure, the invention provides the identification method of the structural modal parameters considering the influence of non-white environmental loads, the frequency domain decomposition method in the prior art is expanded, the modal parameters identified by the frequency domain decomposition method are screened by adopting the power spectral density transfer rate and the over-peak kurtosis coefficient, the false modes caused by the non-white noise loads and the harmonic loads are eliminated, the technical problem that the identification of the structural modal parameters under the combined action of the non-white noise loads and the harmonic loads cannot be realized in the prior art is solved, meanwhile, the requirements on the professional knowledge background and the actual engineering experience of a user are reduced, and the identification method has great engineering application significance.
As shown in fig. 1, a flowchart of a structural modal parameter identification method considering non-white environmental load influence according to the present invention includes the following steps:
step 1: and measuring an acceleration or speed or displacement response signal of the engineering structure through a sensor, recording the response signal through a signal acquisition system, and calculating a power spectral density function matrix of the response signal.
Acquiring a response signal of the engineering structure by using a signal acquisition system, and recording the response signal as x (t), wherein x (t) is a dimension NoX 1, and the variable "t" is a time variable, indicating that the response signal is time-varying. Calculating a power spectral density function matrix S (omega) of the response signal x (t) by using a Welch method, wherein S (omega) is a dimension No×NoThe variable "ω" is a frequency variable that represents the power spectral density function matrix as a function of frequency.
Step 2: and (3) identifying modal parameters of the structure by adopting a frequency domain decomposition method (FDD) according to the power spectral density function matrix S (omega) calculated in the step (1).
Step 2.1: performing singular value decomposition on the power spectral density function matrix S (omega) obtained in the step 1, wherein the equation (1) is as follows:
S(ω)＝U(ω)Σ(ω)VH(ω) (1)
in the formula (1), a matrix U (ω) and a matrix V (ω) respectively represent a left singular matrix and a right singular matrix of the power spectral density function matrix, the matrix Σ (ω) is a diagonal singular value matrix, and a superscript "H" of the matrix V (ω) represents a conjugate transpose of the matrix V (ω). Since the matrix Σ (ω) is a diagonal matrix, it is expressed as shown in equation (2):
Figure BDA0002906755320000051
in the formula (2), σ1(ω) represents the first order singular values, σ, of the matrix S (ω)2(omega) represents the second order singular values of the matrix S (omega), and so on, and satisfies the magnitude relationship
Figure BDA0002906755320000052
Since the singular values of the orders of the matrix S (ω) are all frequency ω -dependent, the order will vary from σ1(omega) to
Figure BDA0002906755320000053
The curve of all order singular values with frequency change is drawn on a graph to complete the processAnd (5) carrying out the following steps.
Step 2.2: judging the modal orders contained in the whole frequency band of the engineering structure response signal x (t) according to the singular value curve graph drawn in the step 2.1
Figure BDA0002906755320000054
And identifies
Figure BDA0002906755320000055
A modal parameter of the order structure.
Step 2.2.1: identifying first order singular values sigma1(ω) corresponding modal parameters.
Observing first order singular values sigma in a plot of singular values1(omega), finding out the peak position of the curve, and recording the number n of peak values1Each peak frequency (in order from small to large
Figure BDA0002906755320000061
) And the first column of the left singular matrix U (ω) at each peak frequency (sequentially marked as frequency from small to large)
Figure BDA0002906755320000062
Figure BDA0002906755320000063
)。n1Is sigma1(ω) the modal order of the identification,
Figure BDA0002906755320000064
is sigma1(ω) the identified modal frequencies,
Figure BDA0002906755320000065
is sigma1(ω) identified mode shape.
Step 2.2.2: identifying second order singular values sigma2(ω) to NthoSingular value of order
Figure BDA0002906755320000066
Corresponding modal parameters.
For second order singularitiesValue sigma2(ω) to NthoSingular value of order
Figure BDA0002906755320000067
The following operations are carried out in sequence:
for the ith order singular value (i ═ 2,3, …, No) Observing the ith order singular value sigma in the plot of singular valuesi(ω), if σi(ω) singular values σ in order i-1i-1(ω) the peak value occurs at the position where the peak value occurs, the ith order singular value σi(ω) there is also a first order mode at this position; recording the ith order singular value sigmaiNumber of peaks n of (ω)iEach peak frequency (in order from small to large
Figure BDA0002906755320000068
) And the ith column (sequentially marked as frequency from small to large) of the left singular matrix U (omega) at each peak frequency
Figure BDA0002906755320000069
Figure BDA00029067553200000610
) In subscripts
Figure BDA00029067553200000611
The sum symbol represents the total modal order of the first i-1 order singular value identification; n isiI.e. the ith order singular value sigmai(ω) the modal order of the identification,
Figure BDA00029067553200000612
i.e. the ith order singular value sigmai(ω) the identified modal frequencies,
Figure BDA00029067553200000613
is sigmai(ω) identified mode shape.
For second order singular value sigma2(ω) to NthoSingular value of order
Figure BDA00029067553200000614
Repeating the above operations, the modal order finally identified by the FDD method is
Figure BDA00029067553200000615
Identified modal frequency of
Figure BDA00029067553200000616
The identified mode shape is
Figure BDA00029067553200000617
And step 3: and eliminating false modes caused by non-white noise loads according to the power spectral density transmissibility matrix, and reserving real modes of the engineering structure and false modes caused by harmonic loads.
Step 3.1: and constructing a power spectral density transfer rate matrix according to the power spectral density function matrix S (omega) calculated in the step 1.
Constructing a power spectral density transmissibility matrix as shown in formula (3):
Figure BDA0002906755320000071
in the formula (3), the reaction mixture is,
Figure BDA0002906755320000072
is represented by formula (4):
Figure BDA0002906755320000073
in the formula (4)
Figure BDA0002906755320000074
Represents the u (th) of the power spectral density function matrix S (omega) in step 1dLine zcThe elements of the column are,
Figure BDA0002906755320000075
denotes the power spectral density function matrix S (ω), the r-th in step 1eLine zcThe elements of the column. a. b and p are positive integers, ua、rbAnd zpAre all positive integers and are all less than or equal to dimension N of response signal x (t) in step 1oIn practical application, a ═ p ═ N may be takeno，b＝1。
Step 3.2: and (4) carrying out singular value decomposition on the matrix T (omega) in the formula (3) in the step 3.1, and drawing a singular value curve graph.
Step 3.1 the dimensionality of the matrix T (omega) in formula (3) is ab x p, and the number N of singular values thereofsThe smaller of ab and p. All N aresThe order singular values are arranged from large to small, and since the matrix T (ω) varies with the frequency ω, NsThe order singular value is also variable with frequency omega, all NsThe order singular values are plotted on a graph to obtain a singular value graph.
Step 3.3: observing the singular value curve chart drawn in the step 3.2, and if all N are in the singular value curve chartsThe k-th order modal frequency identified in step 2 by only the first order singular value in the order singular values
Figure BDA0002906755320000076
(wherein
Figure BDA0002906755320000077
) If no wave trough appears and the second order singular value appears, the k order modal frequency obtained in the step 2
Figure BDA0002906755320000078
And corresponding k-th order mode shape
Figure BDA0002906755320000079
Belongs to a real mode of an engineering structure or a false mode caused by harmonic load; if all NsIf no trough appears in the singular values from the second order to the higher order in the order singular values, the modal frequency of the kth order identified in the step 2
Figure BDA0002906755320000081
And corresponding k-th order mode shape
Figure BDA00029067553200000816
And (3) eliminating false modes caused by non-white noise load from the identification result obtained in the step (2).
After eliminating false mode caused by non-white noise load, recording the order of residual structural mode parameter as
Figure BDA0002906755320000082
Identified modal frequency of
Figure BDA0002906755320000083
The identified mode shape is
Figure BDA0002906755320000084
Figure BDA0002906755320000085
And 4, step 4: and further eliminating false modes caused by harmonic excitation by adopting a band-pass filtering method and a harmonic detection method.
Step 4.1: the spectrum x (ω) of the response signal x (t) acquired in step 1 is calculated using fourier transform.
Step 4.2: determining bandwidth
Figure BDA0002906755320000086
Wherein f issAnd NtRespectively the sampling rate and the total point number of the engineering structure response signals collected in the step 1. Each modal frequency identified in step 3
Figure BDA0002906755320000087
(wherein
Figure BDA0002906755320000088
) Selecting a frequency band
Figure BDA0002906755320000089
Spectral data within, guaranteed bandwidth Δ ω such that
Figure BDA00029067553200000810
The frequency band does not contain any wave trough of the frequency spectrum amplitude curve, and the frequency band is converted by inverse Fourier transform
Figure BDA00029067553200000811
The spectral data in (b) is converted to a time domain signal. From the modal frequency of order I
Figure BDA00029067553200000812
The time domain signal obtained by conversion is marked as x[l](t)。
Step 4.3: all time domain signals x in step 4.2 are calculated[l](t) (wherein
Figure BDA00029067553200000813
) The coefficient of the peak value over the peak value is shown as the formula (5):
Figure BDA00029067553200000814
in the formula (5), the reaction mixture is,
Figure BDA00029067553200000815
representing a time domain signal x[l](t) fourth order central moment, σ[l]Representing a time domain signal x[l](t) standard deviation.
Step 4.4: if the above-mentioned value of the coefficient of kurtosis K [ is ] calculated in step 4.3l]In the range of [ -0.5,0.5 [)]In range, then the time domain signal x[l](t) the corresponding modal frequency is a real structural mode and needs to be reserved; if the over-peak kurtosis coefficient K calculated in the step 4.3[l]In the range of [ -2, -1 [, C ]]In range, then the time domain signal x[l]And (t) the corresponding modal frequency is caused by harmonic load, belongs to a false mode, and is removed.
For all modal frequencies identified in step 3
Figure BDA0002906755320000091
All steps are carried out in step 4.4, all false modes caused by harmonic loads are removed, and the true modes of the structure are reserved.
Rejection of harmonic load causesAfter the false mode, the residual structural mode parameter order is recorded as NmIdentified modal frequency of
Figure BDA0002906755320000092
The identified mode shape is
Figure BDA0002906755320000093
The modal parameters of the complete engineering structure identified by the invention comprise modal frequency and modal shape.
Identifying complete engineering structure modal parameters
Figure BDA0002906755320000094
And
Figure BDA0002906755320000095
the method is applied to the field of engineering structure design. The difference between the natural frequency of the engineering structure and the working environment load frequency band range is analyzed, the damping characteristic of the engineering structure is evaluated, whether the dynamic performance of the engineering structure design meets the requirements of the engineering structure application standard, vibration isolation and the like is detected, the reliability of the structural design is improved, and the redundancy of the dynamic characteristic design is reduced.
Identifying complete engineering structure modal parameters
Figure BDA0002906755320000096
And
Figure BDA0002906755320000097
the method is applied to the field of engineering structure dynamic model correction. And comparing the identified complete engineering structure modal parameters with the established engineering structure dynamic model to complete sensitivity analysis of the dynamic model parameters, thereby realizing correction of the dynamic model parameters, improving modeling and analyzing capabilities of the complex engineering structure and further promoting improvement of the engineering structure design level.
Identifying complete engineering structure modal parameters
Figure BDA0002906755320000098
And
Figure BDA0002906755320000099
the method is applied to the field of engineering structure state monitoring. And comparing the identified complete engineering structure modal parameters with the engineering structure modal parameters identified in the healthy state, detecting whether the structure fails or not, evaluating the failure degree of the engineering structure according to the difference of the two parameters, and improving the real-time performance and reliability of the monitoring of the engineering structure state.
Examples
The truss structure of this embodiment is shown in fig. 2 and is composed of a total of 10 rods having a circular cross section, which are indicated by the numerals 1,2, …,10 in fig. 2 and the letter aqThe cross-sectional area of the q-th rod piece (where q is 1,2, …,10) is indicated, and L is a dimensional parameter and has a value of 9.144 m. The truss material is aluminum, the Young modulus is 69.8Gpa, and the material density is 2770 kg.m-3454kg of concentrated mass is added to the nodes I, II, III and IV respectively, the damping matrix is in direct proportion to the mass matrix, the damping ratio of the first-order mode is 5% by the proportional factor, and the undamped natural frequencies of the eighth-order modes of the truss structure are 5.89 Hz, 14.97 Hz, 17.15 Hz, 20.80 Hz, 26.81 Hz, 31.52 Hz, 43.68 Hz and 47.54Hz respectively. The truss structure contains a total of 4 unconstrained nodes, each node having two degrees of freedom along the x-axis and the y-axis, for a total of 8 degrees of freedom. The degrees of freedom of the truss structure are numbered in the order of node number, x-axis first and y-axis second, for example, degrees of freedom 3 and 4 represent the translation degrees of freedom of the node along the x-axis and the y-axis respectively.
The 4 unconstrained nodes of the truss are subjected to the action of non-white noise load in the x direction and the y direction, wherein the non-white noise load is colored noise with a main frequency of 10Hz and a damping ratio of 3%. In addition, the node (r) also acts a harmonic load with a frequency of 40Hz in the x-direction. The displacement response signal sampling rate is 120Hz, and the total time length is 500 s. The response power spectrum function with 8 degrees of freedom is shown in fig. 3, it can be seen that peak values exist at the non-white noise load main frequency 10Hz and the harmonic load frequency 40Hz in the displacement response, and if the modal parameters of the truss structure are identified by using the modal parameter identification methods such as FDD, SSI and the like in the prior art, the modal frequencies of 10Hz and 40Hz are identified, and the modal is a false mode.
The embodiment of the invention discloses a structural modal parameter identification method considering non-white environment load influence for parameter identification, which comprises the following steps:
step 1: and calculating a power spectral density function matrix of the response signal by taking the calculated displacement response signal as an engineering structure response signal acquired by the signal acquisition system.
In the present embodiment, all 8-order modal parameters of the truss structure are identified by using 8-degree-of-freedom responses of the truss structure, so that the response signal dimension is a vector of 8 × 1, that is, N o8. The power spectral density function matrix S (ω) of the response signal x (t) is calculated using the Welch method, where S (ω) has dimensions of 8 × 8.
Step 2: and (3) identifying modal parameters of the structure by adopting a frequency domain decomposition method (FDD) according to the power spectral density function matrix S (omega) calculated in the step (1).
Step 2.1: singular value decomposition is carried out on the power spectral density function matrix S (omega) obtained in the step 1, the dimension of the power spectral density function matrix S (omega) is 8 multiplied by 8, the number of singular values is 8, and the sigma is calculated1(omega) to sigma8The plot of (ω) versus frequency is plotted on a graph, as shown in fig. 4, where the 8 th order singular values are arranged from large to small, and the gray solid dot positions represent the first order modes recognized here by the frequency domain decomposition method.
Step 2.2: judging the modal orders contained in the whole frequency band of the engineering structure response signal x (t) according to the singular value curve graph drawn in the step 2.1
Figure BDA0002906755320000101
And identifies
Figure BDA0002906755320000102
A modal parameter of the order structure.
Step 2.2.1: identifying first order singular values sigma1(ω) corresponding modal parameters.
Observing first order singular values sigma in a plot of singular values1(ω) 10 peaks can be found, the number of peaks n110, record eachThe horizontal axis frequency value corresponding to the peak value and the first column of the left singular matrix U (omega) at the frequency value are respectively the 10-order modal frequency and the modal shape finally identified
Figure BDA0002906755320000111
And
Figure BDA0002906755320000112
Figure BDA0002906755320000113
step 2.2.2: identifying second order singular values sigma2Singular values of the order (omega) to 8 sigma8(ω) corresponding modal parameters.
The second order singular value σ is used in the embodiment2(ω) As an example, the second-order singular value σ is explained2Singular values of the order (omega) to 8 sigma8(ω) identifying modal parameters. Observing the second-order singular value curve, and judging sigma2(ω) singular values of the upper order (i.e. first order singular values σ)1(ω)) whether a peak also occurs at the position where the peak occurs. The embodiment has singular values of sigma in the first order1At the position of the (ω) peak, the second order singular value σ2(ω) peaks again only around 10Hz, so the second order singular value σ2(ω) only one order mode, n, can be identified2Recording the horizontal axis frequency value corresponding to the peak value and the second row of the left singular matrix U (omega) at the frequency value, and finally identifying 1-order modal frequency and modal shape as 1
Figure BDA0002906755320000114
And
Figure BDA0002906755320000115
for the third order singular value sigma3Singular values of the order (omega) to 8 sigma8(ω) repeat the above operation, σ in this example3(omega) to sigma8All the (ω) can only identify the first-order mode, and all the peaks are marked by gray solid circles in fig. 4, so that the modal order finally identified by the FDD method is
Figure BDA0002906755320000116
Identified modal frequency of
Figure BDA0002906755320000117
Figure BDA0002906755320000118
The identified mode shape is
Figure BDA0002906755320000119
The modal frequencies are shown in table 1.
TABLE 1 truss structure modal frequencies identified by frequency domain decomposition method
Serial number 1 2 3 4 5 6 7 8 9
frequency/Hz 5.80 9.96 15.00 17.17 20.92 26.89 31.52 40.02 43.59
Table 1 shows the modal frequencies of the truss structure identified by the frequency domain decomposition method
Serial number 10 11 12 13 14 15 16 17
frequency/Hz 47.40 10.02 9.96 10.02 10.08 9.90 10.08 10.02
And step 3: and eliminating false modes caused by non-white noise loads according to the power spectral density transmissibility matrix, and reserving real modes of the engineering structure and false modes caused by harmonic loads.
Step 3.1: and constructing a power spectral density transfer rate matrix according to the power spectral density function matrix S (omega) calculated in the step 1.
In the present embodiment, with the displacement response of degree of freedom 4 as the reference channel, let a be p be N o8, b is 1, i.e. z in formula (3)pAnd uaEach corresponding to a degree of freedom 8, constructing a power spectral density transfer rate matrix as shown in formula (6):
Figure BDA0002906755320000121
step 3.2: and (3) carrying out singular value decomposition on the matrix T (omega) in the formula (6) in the step 3.1, and drawing a singular value curve graph.
The dimension of the matrix T (omega) in the step 3.1 formula is 8 multiplied by 8, and the number N of singular valuessWhen 8, the singular values are arranged from large to small, a singular value graph is drawn as shown in fig. 5, and the singular values of the 8 th order are arranged from large to small.
Step 3.3: observing the singular value curve chart drawn in the step 3.2, in this embodiment, no trough appears from the second-order singular value to the eighth-order singular value in the vicinity of 10Hz, so that the 8-order modes with the sequence numbers of 2, 11 to 17 in table 1 all belong to the false modes caused by the non-white noise load, and need to be removed. A total of 9 modes with numbers 1, 3 to 10 are reserved, thus
Figure BDA0002906755320000122
The identified modal frequency is
Figure BDA0002906755320000123
Figure BDA0002906755320000124
The identified mode shape is
Figure BDA0002906755320000125
The modal frequencies are shown in table 2.
TABLE 2 truss structure modal frequency after power spectral density transmissibility method screening
Serial number 1 2 3 4 5 6 7 8 9
frequency/Hz 5.80 15.00 17.17 20.92 26.89 31.52 40.02 43.59 47.40
And 4, step 4: and further eliminating false modes caused by harmonic excitation by adopting a band-pass filtering method and a harmonic detection method.
Step 4.1: and calculating the frequency spectrum x (omega) of the truss structure displacement response signal x (t) by using Fourier transform.
Step 4.2: in this embodiment, the bandwidth Δ ω is 0.1Hz, and each modal frequency identified in step 3 is used
Figure BDA0002906755320000126
(where l is 1,2, …,9), selecting a frequency band
Figure BDA0002906755320000127
The frequency spectrum data in Hz is converted into time domain signals by adopting inverse Fourier transform to obtain time domain signals x corresponding to all modal frequencies[1](t) to x[9](t)。
Step 4.3: all time domain signals x in step 4.2 are calculated[1](t) to x[9](t) the coefficient of the peak value over time.
Step 4.4: step 4.3 in x[1](t) to x[9]The coefficient of the peak intensity of the excess value of (t) is-0.06, 0.07, -0.03, 0.33, -0.34, -1.41, 0.05 and 0.05 respectively. It can be seen that there is only a time domain signal x[7](t) the coefficient of the peak value of excess lies in [ -2, -1 []In the range, the over-peak kurtosis coefficients of the rest 8 time domain signals are all close to 0, so that the modal frequency can be judged
Figure BDA0002906755320000131
The method is characterized in that the method is caused by harmonic loads, belongs to a false mode, and is eliminated, and the rest mode frequencies are structural true mode frequencies and are reserved.
In this embodiment, the remaining structure modal parameter order NmThe identified modal frequency is 8
Figure BDA0002906755320000132
Figure BDA0002906755320000133
The identified mode shape is
Figure BDA0002906755320000134
The modal parameters of the complete engineering structure identified by the invention comprise modal frequency and modal shape.
The finally identified modal frequency and percentage error of the truss structure are shown in table 3, and it can be seen that all the modal frequencies of the truss structure are high in identification precision, and the percentage error of the first-order modal frequency is the largest and is only 1.53%.
TABLE 3 finally identified truss structure modal frequencies and percentage errors
Serial number 1 2 3 4 5 6 7 8
frequency/Hz 5.80 15.00 17.17 20.92 26.89 31.52 43.59 47.40
Error/%) 1.53 0.20 0.12 0.58 0.30 0.00 0.21 0.29
Calculating the identified modal shape
Figure BDA0002906755320000135
And a modal confidence criterion (MAC) matrix between the modal shape and the theoretical modal shape, wherein the more the diagonal line of the MAC matrix is close to 1, the more the off-diagonal line of the MAC matrix is close to 0, and the higher the modal shape identification precision is shown. The MAC matrix calculated in this embodiment is shown in fig. 6. It can be seen that the diagonal line of the MAC matrix is very close to 1, and the off-diagonal element is very close to 0, so that the identified mode shape has high accuracy.
Complete engineering structure model to be recognizedState parameter
Figure BDA0002906755320000136
And
Figure BDA0002906755320000137
the method is applied to the field of engineering structure dynamics to solve practical engineering problems and mainly comprises the fields of engineering structure design, dynamics model correction and state monitoring.
Therefore, the structural modal parameter identification method considering the influence of the non-white environmental load has the advantages that the obtained modal parameters are high in precision in practical application, false modes caused by the non-white environmental load can be eliminated, the reliability is high, and the method has a more guiding significance for the next work.
The above embodiments are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modifications made on the basis of the technical scheme according to the technical idea of the present invention fall within the protection scope of the present invention.

Claims (4)

1. A structural modal parameter identification method considering non-white environment load influence is characterized by comprising the following steps:
step 1, measuring and recording an acceleration or speed or displacement response signal of an engineering structure, and calculating a power spectral density function matrix of the response signal;
step 2, performing singular value decomposition on the power spectral density function matrix obtained by calculation in the step 1, drawing a singular value curve chart of the power spectral density function matrix by adopting a frequency domain decomposition method, judging the modal order contained in the engineering structure response signal according to the singular value curve chart, and preliminarily identifying the modal parameters of the engineering structure, including modal frequency and modal vibration mode;
step 3, constructing a power spectral density transfer rate matrix according to the power spectral density function matrix, eliminating false modal parameters caused by non-white noise loads in the modal parameters obtained in the step 2 according to the power spectral density transfer rate matrix, and reserving real modal parameters of the engineering structure and false modal parameters caused by harmonic loads;
and 4, eliminating false modal parameters caused by harmonic loads by adopting a band-pass filtering method and a harmonic detection method, and reserving real modal parameters of the engineering structure as a final identification result.
2. The method for identifying structural modal parameters considering non-white environmental load influence according to claim 1, wherein the specific process of the step 2 is as follows:
step 2.1, performing singular value decomposition on the power spectral density function matrix S (ω) calculated in step 1 as follows:
S(ω)＝U(ω)Σ(ω)VH(ω)
wherein ω is frequency, U (ω) and V (ω) respectively represent a left singular matrix and a right singular matrix of the power spectral density function matrix, superscript H represents a conjugate transpose, Σ (ω) is a diagonal singular value matrix, and Σ (ω) specifically is:
Figure FDA0002906755310000011
wherein σ1(ω)、σ2(ω)、
Figure FDA0002906755310000012
Respectively representing the first, second and Nth order of the matrix S (omega)oSingular value of order, and satisfy
Figure FDA0002906755310000013
NoDimension for the engineered structure response signal x (t);
will be from σ1(omega) to
Figure FDA0002906755310000021
The curves of all orders of singular values changing along with the frequency are drawn on a graph to obtain a singular value curve graph of a power spectral density function matrix;
step 2.2, judging the modal orders contained in the whole frequency band of the engineering structure response signal according to the singular value curve graph in the step 2.1, and preliminarily identifying the modal parameters of the engineering structure;
step 2.2.1, identify first order singular value sigma1(ω) a corresponding modal parameter;
observing first order singular value sigma in singular value curve chart1Finding out the peak position of the curve corresponding to the curve (omega), and recording the number n of the peak values1The frequency corresponding to each peak is recorded from small to large in sequence
Figure FDA0002906755310000022
Figure FDA0002906755310000023
The first column of the left singular matrix U (omega) at the frequency corresponding to each peak value is sequentially marked as the frequency from small to large
Figure FDA0002906755310000024
n1Is sigma1(ω) the modal order of the identification,
Figure FDA0002906755310000025
Figure FDA0002906755310000026
is sigma1(ω) the identified modal frequencies,
Figure FDA0002906755310000027
is sigma1(ω) identified mode shape;
step 2.2.2, identify second order singular value sigma2(ω) to NthoSingular value of order
Figure FDA0002906755310000028
Corresponding modal parameters;
for the ith order singular value σi(ω) i ═ 2,3, …, NoObservation of σ in the plot of singular valuesi(ω) corresponding to the curve if σi(omega) in the secondSingular value sigma of order i-1i-1(ω) the peak value occurs at the position where the peak value occurs, the ith order singular value σi(ω) there is also a first order mode at this position; recording the ith order singular value sigmaiNumber of peaks n of (ω)iThe frequency corresponding to each peak is recorded from small to large in sequence
Figure FDA0002906755310000029
The ith column of the left singular matrix U (omega) at the frequency corresponding to each peak value is sequentially marked as the frequency from small to large
Figure FDA00029067553100000210
Figure FDA00029067553100000211
Figure FDA00029067553100000212
Representing the total modal order of the first i-1 order singular value identification; n isiIs sigmai(ω) the modal order of the identification,
Figure FDA00029067553100000213
is sigmai(ω) the identified modal frequencies,
Figure FDA00029067553100000214
Figure FDA00029067553100000215
is sigmai(ω) identified mode shape;
the modal order contained in the entire frequency band of the engineered structure response signal is
Figure FDA00029067553100000216
Identified modal frequency of
Figure FDA0002906755310000031
The identified mode shape is
Figure FDA0002906755310000032
3. The method for identifying structural modal parameters considering non-white environmental load influence according to claim 1, wherein the specific process of the step 3 is as follows:
step 3.1, constructing the following power spectral density transfer rate matrix T (omega) according to the power spectral density function matrix S (omega):
Figure FDA0002906755310000033
wherein,
Figure FDA0002906755310000034
Figure FDA0002906755310000035
represents the u-th of the matrix S (ω)dLine zcThe elements of the column are,
Figure FDA0002906755310000036
denotes the r-th of the matrix S (ω)eLine zcElements of columns, a, b, p, ua、rbAnd zpAre all positive integers, and ua、rbAnd zpAre all less than or equal to dimension N of response signal x (t) in step 1o；
Step 3.2, carrying out singular value decomposition on the matrix T (omega), wherein the dimensionality of the T (omega) is ab multiplied by p, and the number N of singular values issFor the smaller of ab and p, all N will besArranging the order singular values from large to small, and drawing the order singular values on a graph to obtain a singular value curve graph;
step 3.3, observing the singular value curve chart drawn in the step 3.2, and if all N are in the singular value curve chartsThe k-th order modal frequency identified in step 2 by only the first order singular value in the order singular values
Figure FDA0002906755310000037
If no wave trough appears and the second order singular value appears, the k order modal frequency obtained in the step 2
Figure FDA0002906755310000038
And corresponding k-th order mode shape phik FDDBelongs to a real mode of an engineering structure or a false mode caused by harmonic load,
Figure FDA0002906755310000039
Figure FDA00029067553100000310
the modal order contained in the whole frequency band of the engineering structure response signal; if all NsSecond to Nth order among the order singular valuessIf no wave trough appears in the order singular value, the k-th order modal frequency identified in the step 2
Figure FDA0002906755310000041
And corresponding k-th order mode shape phik FDDThe false mode caused by the non-white noise load is removed from the mode parameters obtained in the step 2;
after eliminating false mode caused by non-white noise load, recording the residual mode order as
Figure FDA0002906755310000042
The remaining modal frequency is
Figure FDA0002906755310000043
The remaining mode shape is
Figure FDA0002906755310000044
4. The method for identifying structural modal parameters considering non-white environmental load influence according to claim 1, wherein the specific process of the step 4 is as follows:
step 4.1, calculating the frequency spectrum x (omega) of the engineering structure response signal x (t) obtained in the step 1 by adopting Fourier transform;
step 4.2, determine the bandwidth Δ ω, and
Figure FDA0002906755310000045
fsand NtRespectively obtaining the sampling rate and the total point number of the engineering structure response signal obtained in the step 1 and each modal frequency obtained in the step 3
Figure FDA0002906755310000046
Selecting a frequency band
Figure FDA0002906755310000047
Spectral data within, guaranteed bandwidth Δ ω such that
Figure FDA0002906755310000048
The frequency band does not contain any wave trough of the frequency spectrum amplitude curve, and the frequency band is converted by inverse Fourier transform
Figure FDA0002906755310000049
The spectrum data in the spectrum is converted into time domain signal with the I-order modal frequency
Figure FDA00029067553100000410
The time domain signal obtained by conversion is marked as x[l](t)，
Figure FDA00029067553100000411
Step 4.3, calculating all time domain signals x[l](t) coefficient of kurtosis over the value K[l]：
Figure FDA00029067553100000412
Wherein,
Figure FDA00029067553100000413
representing a time domain signal x[l](t) fourth order central moment, σ[l]Representing a time domain signal x[l](t) standard deviation;
step 4.4, if the above-mentioned excessive kurtosis coefficient K [ 2 ] calculated in step 4.3l]In the range of [ -0.5,0.5 [)]In range, then the time domain signal x[l](t) the corresponding modal frequency is a real structural mode and needs to be reserved; if the over-peak kurtosis coefficient K calculated in the step 4.3[l]In the range of [ -2, -1 [, C ]]In range, then the time domain signal x[l](t) the corresponding modal frequency is caused by harmonic load, belongs to a false mode, and is removed;
after eliminating false modes caused by harmonic loads, recording the remaining mode order as NmThe remaining modal frequency is
Figure FDA0002906755310000051
The remaining mode shape is
Figure FDA0002906755310000052
The final recognition result is obtained.
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Application publication date: 20210514