Underdetermined working modal parameter identification method and system based on tensor decomposition recursion

Abstract

The invention discloses an underdetermined working mode parameter identification method and system based on tensor decomposition recursion, and relates to the technical field of working mode parameter identification, and the method comprises the steps: obtaining a linear time-varying structure vibration response signal of a sensor measurement point under the excitation of a set environment, and extracting a principal component score of the response signal through PCA; by calculating the autocorrelation of the principal component score and combining a peak value in an autocorrelation function, determining a proper hysteresis period, and constructing the proper hysteresis period and a response signal into a third-order tensor; under a time-varying condition, a weighted observation matrix is established through recursive decomposition and a dynamic forgetting factor selection mechanism, and a decomposed result is subjected to recursive updating, so that a more accurate modal shape and inherent frequency are obtained. According to the method, the recognition precision, stability and robustness under the complex time-varying working mode are remarkably improved, and the method is suitable for health monitoring of large civil engineering structures such as bridges, buildings and wind power generation towers and helps to find structural damage in time.

Description

Technical Field

[0001] The present invention relates to the technical field of working modal parameter identification, and in particular to an underdetermined working modal parameter identification method and system based on tensor decomposition recursion. Background Art

[0002] Working modal parameter identification is a blind identification method that identifies the modal parameters of a system by relying only on the measured structural response. Compared with traditional experimental modal parameter identification, working modal parameter identification has the following advantages: only the time domain vibration output data needs to be obtained, and there is no need to measure the input load; the measured data comes directly from the actual operating state of the structure, so the identification results are more in line with the actual boundary conditions and working conditions. Therefore, working modal parameter identification is a powerful tool, especially suitable for the dynamic characteristics analysis of large and complex structures. Because it uses data under actual working conditions for modal parameter identification, the identification results are more accurate and closer to the actual situation. More importantly, working modal parameter identification can also realize online modal analysis of the structure and monitor the dynamic performance of the structure in real time. Because working modal analysis has strong practicality and can be analyzed online, it has been widely used in the analysis of large engineering structures such as aircraft, automobiles, and civil engineering.

[0003] As more large and high-speed engineering structures emerge, the dynamic challenges of linear systems involving time-varying mass, stiffness, and damping increase. Therefore, accurately predicting the vibration response of structures becomes more critical.

[0004] At present, although the working modal parameter identification theory tends to be mature, it is mainly based on the assumption that the system is linearly time-varying, which has certain limitations in some cases. The working modal parameter identification theory of time-varying systems is still in the initial development stage. In the real world, the structural parameters (such as mass, stiffness, damping, etc.) of many engineering structures will change over time during actual operation, thus showing time-varying characteristics. For example, in the model of a train crossing a bridge, since the train itself has mass, the dynamic characteristics of the vehicle-bridge coupling system will continue to change with the movement of the train; during the launch of a rocket or missile, the change in system mass caused by fuel consumption will also affect its dynamic characteristics. In addition, in the fields of energy, ships, etc., more and more structural time-varying characteristics are also appearing. For example, the rotor blades of a wind turbine rotate during operation, which will change the dynamic characteristics of the entire structural system. The current development of industry has put forward new requirements for structural dynamics analysis. Therefore, the problem of parameter identification of time-varying structures has become more and more urgent. Summary of the invention

[0005] In view of the problems of the prior art, the purpose of the present invention is to provide a method and system for identifying underdetermined working modal parameters based on tensor decomposition recursion, wherein the obtained signal matrix is ​​optimized and constructed as a third-order tensor through the PCA (autocorrelation optimized principal component) method, and the constructed tensor is simultaneously diagonalized and recursively decomposed, and the decomposed results are analyzed to obtain the corresponding modal vibration shapes and natural frequencies; without destroying the original connection of the signal, the modal vibration shapes and natural frequencies under time-varying conditions are better identified, thereby improving the recognition accuracy and recognition stability.

[0006] The present invention adopts the following technical solution:

[0007] On the one hand, a method for identifying underdetermined operating modal parameters based on tensor decomposition recursion includes:

[0008] The vibration response signal acquisition step is to acquire the vibration response signal of the time-varying structure collected at the selected sensor measurement point at time t under the set environmental excitation. ;in, , is the number of sensors, N(t) is the number of time samples;

[0009] The principal component score calculation step is to process the acquired vibration response signal using principal component analysis (PCA) to calculate the principal component score;

[0010] The lag determination step is to calculate the autocorrelation function of the principal component scores and determine the lag by detecting the peak in the autocorrelation function. ;

[0011] The third-order tensor construction step is based on the vibration response signal at time t Decomposed mode shape matrix , modal response matrix and lag period Constructed into a dimension ) The third-order tensor of ;

[0012] The third-order tensor update step, the third-order tensor at time t Add new data slices in the second dimension to construct a third-order tensor at time t+1 ;

[0013] The weighted observation matrix construction step is based on the third-order tensor at time t+1 , introduce the dynamic forgetting factor to construct the weighted observation matrix at time t+1 , and perform windowing and weighting processing on the observation matrix;

[0014] The factor matrix recursive update step is based on the weighted observation matrix at time t Two factorizations of, diagonalization of connection matrix, singular value tracking and modal response matrix and The common blocks between them are recursively updated to obtain the modal vibration matrix at time t+1 , modal response matrix The sum signal contribution matrix to the mode Θ(t+1);

[0015] The working modal parameter identification step is to identify the modal vibration matrix , modal response matrix The modal contribution matrix Θ(t+1) of the sum signal is decomposed to identify the time-varying working modal parameters.

[0016] Preferably, the principal component score calculation step specifically includes:

[0017] The collected vibration response signal Perform standardization to obtain a standardized matrix ,as follows:

[0018] ;

[0019] in, is the mean of the original data, is the standard deviation;

[0020] Standardized matrix Calculate the covariance matrix , for the covariance matrix Diagonalize to obtain a matrix of eigenvectors ,as follows:

[0021] ;

[0022] Where T is the transpose of the matrix, , is a diagonal matrix, ;

[0023] Based on the standardized matrix and the matrix consisting of eigenvectors Get the principal component score matrix ,as follows:

[0024] .

[0025] Preferably, the hysteresis period determination step specifically includes:

[0026] Calculate the autocorrelation function of the principal component scores ,as follows:

[0027] ;

[0028] Among them, i is the index of a selected principal component, which represents the principal component score matrix The i-th column of Represents the value of the i-th principal component score at time t; express The value of the i-th principal component score at time; represents the mean of the i-th principal component score, that is, , N is the length of the data, l is the number of lags;

[0029] Using the autocorrelation function , determine the appropriate hysteresis period by peak detection to determine the periodic characteristics :

[0030] ;

[0031] ;

[0032] Where k represents the peak index; len( ) represents the autocorrelation function The length of , that is, the number of autocorrelation function values; represents the peak array; Indicates the number of peaks.

[0033] Preferably, based on the vibration response signal at time t Decomposed mode shape matrix , modal response matrix and lag period Constructed into a dimension ) The third-order tensor of , which is expressed as follows:

[0034] ;

[0035] Where R is the rank of CP decomposition; For tensors The vector in the first dimension, the mode shape matrix The rth mode shape vector of ; For tensors The vector in the second dimension, the modal response matrix The rth modal response vector of ; For tensors The vector in the third dimension is the rth vector of the signal contribution matrix Θ(t); operator Represents the vector outer product.

[0036] Preferably, the third-order tensor at time t is Add new data slices in the second dimension to construct a third-order tensor at time t+1 , which is expressed as follows:

[0037] ;

[0038] in, Represents a new data slice.

[0039] Preferably, the weighted observation matrix construction step specifically includes:

[0040] Calculate the data change rate change_rate of the vibration response signal input at the current time point, and The forgetting factor of the moment Dynamic Adjustment The forgetting factor of the moment ,as follows:

[0041] ;

[0042] in, represents the initial value of the forgetting factor;

[0043] Use dynamically adjusted forgetting factor Weight the historical data and construct a new weighted observation matrix ,as follows:

[0044] (t+1);

[0045] in, is to convert the tensor The matrix after expansion along the first dimension has dimensions ) , to extract modal information from the data using matrix decomposition techniques; (t+1) is the weighted matrix, which is composed of the current forgetting factor, as follows:

[0046] ;

[0047] in, To use the forgetting factor Constructed diagonal matrix;

[0048] The weighted observation matrix Decompose to get the initial mode shape matrix , modal response matrix The sum signal contribution matrix Θ(t+1) to the modal, let ,get Loading matrix at time ; Operator represents the Hadamard product.

[0049] Preferably, the factor matrix recursive updating step specifically includes:

[0050] The weighted observation matrix at time The two factorizations of are as follows:

[0051] ;

[0052] The first decomposition is achieved by replacing the formula (t) Replace it with the result obtained by its CP decomposition, ; The second decomposition is The economic scale SVD decomposition of , and ; is a combination of singular vectors of the tensor in the first and third dimensions; To include Singular values ​​in the decomposition; are the singular vectors of the containing tensor in the second dimension;

[0053] There exists a non-singular connectivity matrix , so that:

[0054] ;

[0055] in, , is the auxiliary matrix; connect the two equations It is found by solving the simultaneous diagonalization of a set of matrices of size R × R;

[0056] Combining the above two formulas, we can calculate , , and ; for The inverse matrix of

[0057] Loading matrix Perform singular value decomposition as follows:

[0058] ;

[0059] in, express The rth vector of ; is a matrix The left singular vector matrix of The eigenvector of represents the orthogonal basis vector of the data in the row space; represents the diagonal matrix of singular values, containing The singular values ​​of reflect the strength of the matrix in each orthogonal direction; is a matrix The right singular vector matrix of The eigenvector of represents the orthogonal basis vector of the data in the column space; the updated factor matrix and Used to adjust the tensor decomposition results in the recursive process;

[0060] Based on the singular value decomposition results, we get exist Update of the moment and exist Update of the moment ,as follows:

[0061] Assign the left singular vector matrix to Update , assign the decomposed singular values ​​to Update ;based on ,get exist Update of the moment ;

[0062] Assign the right singular vector matrix to Update ;

[0063] At time t+1, the formula The conversion is as follows:

[0064]

[0065] use The time-shift structure connects time t and time t+1, where represents the auxiliary matrix at time t+1; represents the non-singular connection matrix at time t+1; for The inverse matrix of

[0066] Due to the time-varying structure and There is a common block, denoted as ; and The dimensions are N(t)×R and N(t+1)×R respectively, so, and The same is true for the dimensions of ; in this case, the common block yes itself;

[0067] Define the following matrix based on the set window:

[0068] ;

[0069] represents the No changes;

[0070] It means to intercept the first N(t) rows from the L(t+1) matrix to form ;

[0071] It means extracting the N(t+1)th row of L(t+1), forming ;

[0072] By common block get:

[0073] ;

[0074] in, represents the square root of the forgetting factor, which is used to weight historical data;

[0075] get:

[0076] ;

[0077] ;

[0078] in, represents the pseudo-inverse of a matrix; is an identity matrix, so ;

[0079] and Update: Based on the formula ,accomplish and Updates to get and ;

[0080] Update: Based on the formula renew , and finally appended to To build ;

[0081] judge Relative to Is the change of less than the preset second threshold? If so, further determine whether the factor matrices of all ranks have been updated. If so, output the updated , and .

[0082] In another aspect, a system for identifying under-timed variable operating modal parameters includes:

[0083] A motor system and a support platform for providing mechanical vibration and structural support;

[0084] Measurement and analysis systems, including force and motion sensors, for collecting vibration response signals of time-varying structures;

[0085] An excitation system, used to apply control excitations on the time-varying structure to simulate the real working environment;

[0086] The processing unit analyzes the collected vibration response signal and identifies the time-varying working modal parameters based on the underdetermined working modal parameter identification method based on tensor decomposition recursion.

[0087] Compared with the prior art, the present invention has the following beneficial effects:

[0088] The present invention obtains the linear time-varying structural vibration response signal of the sensor measuring point to construct a tensor, and can more accurately capture the modal parameters in the time-varying structure through the adaptive tensor decomposition technology, thereby significantly improving the recognition accuracy; by combining the simultaneous diagonalization tracking technology and the dynamic selection of forgetting factor technology, it is possible to maintain high stability and robustness when facing complex working modes; the method of the present invention is particularly suitable for use in the health monitoring of large civil engineering structures (such as bridges, buildings, wind power towers, etc.), similar neighborhoods where the modal parameters of the structure (such as frequency, damping ratio and modal shape) change with time and external environment, help monitor the health status of the structure and promptly discover potential structural damage. BRIEF DESCRIPTION OF THE DRAWINGS

[0089] Figure 1 It is a flow chart of an underdetermined operating modal parameter identification method based on tensor decomposition recursion according to an embodiment of the present invention;

[0090] Figure 2 A diagram of a time-varying three-degree-of-freedom spring oscillator model provided by an embodiment of the present invention;

[0091] FIG3 (a) is a diagram of a Gaussian white noise excitation simulation signal provided by an embodiment of the present invention;

[0092] FIG3( b ) is a simulation signal diagram of a displacement response signal 1 ( m1 ) provided in an embodiment of the present invention;

[0093] FIG3 (c) is a simulation signal diagram of a displacement response signal 2 (m2) provided in an embodiment of the present invention;

[0094] FIG3( d ) is a simulation signal diagram of a displacement response signal 3 ( m3 ) provided in an embodiment of the present invention;

[0095] Figure 4 Comparison between the experimental method provided in the embodiment of the present invention for identifying the theoretical natural frequency of a three-degree-of-freedom time-varying structure and the natural frequency variation curve;

[0096] Figure 5 The experimental method provided by the embodiment of the present invention identifies the MAC value change curve of the three-degree-of-freedom time-varying system;

[0097] FIG6 (a) is a comparison of the natural vibration mode of the three-degree-of-freedom time-varying system at 38.125 s by the experimental method provided by an embodiment of the present invention and the instantaneous vibration mode identified by the proposed method;

[0098] FIG6 (b) is a comparison between the natural vibration mode of the three-degree-of-freedom time-varying system at 450 s using the experimental method provided by an embodiment of the present invention and the instantaneous vibration mode identified by the proposed method;

[0099] FIG6 (c) is a comparison of the natural vibration mode of the three-degree-of-freedom time-varying system in 900 s using the experimental method provided by an embodiment of the present invention and the instantaneous vibration mode identified by the proposed method;

[0100] FIG6 (d) is a comparison of the natural vibration mode of the three-degree-of-freedom time-varying system at 1500 s using the experimental method provided by an embodiment of the present invention and the instantaneous vibration mode identified by the proposed method;

[0101] Figure 7 A structural schematic diagram of an under-timed variable working modal parameter identification system provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0102] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the invention claimed for protection, but merely represents the selected embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0103] like Figure 1 As shown, the present invention provides an underdetermined operating modal parameter identification method based on tensor decomposition recursion, which includes the following steps.

[0104] S1, a vibration response signal acquisition step, which acquires the vibration response signal of the time-varying structure collected at the selected sensor measurement point at time t under the set environmental excitation ;in, , is the number of sensors, and N(t) is the number of time samples.

[0105] Specifically, multiple sensors are arranged on the time-varying structure to be monitored, and these sensors are used to collect the vibration response signals of the structure in real time. Assume that the structure has These sensors record vibration signals at a fixed sampling rate to form a vibration response matrix at time t. .

[0106] S2, a principal component score calculation step, uses principal component analysis PCA to process the acquired vibration response signal and calculates the principal component score.

[0107] Specifically, the collected vibration response signal Perform standardization to obtain a standardized matrix ,as follows:

[0108] ;

[0109] in, is the mean of the original data, is the standard deviation;

[0110] Standardize the matrix Calculate the covariance matrix , for the covariance matrix Diagonalize to obtain a matrix of eigenvectors ,as follows:

[0111] ;

[0112] Where T is the transpose of the matrix, , is a diagonal matrix, ;

[0113] Based on the standardized matrix and the matrix consisting of eigenvectors Get the principal component score matrix ,as follows:

[0114] .

[0115] S3, lag determination step, calculates the autocorrelation function of the principal component scores and determines the lag by detecting the peak in the autocorrelation function .

[0116] Specifically, the autocorrelation function of the principal component scores is calculated ,as follows:

[0117] ;

[0118] Among them, i is the index of a selected principal component, which represents the principal component score matrix The i-th column of Represents the value of the i-th principal component score at time t; express The value of the i-th principal component score at time; represents the mean of the i-th principal component score, that is, , N is the length of the data, l is the number of lags;

[0119] Using the autocorrelation function , determine the appropriate hysteresis period by peak detection to determine the periodic characteristics :

[0120] ;

[0121] ;

[0122] Where k represents the peak index; len( ) represents the autocorrelation function The length of , that is, the number of autocorrelation function values; represents the peak array; Indicates the number of peaks.

[0123] S4, the third-order tensor construction step, based on the vibration response signal at time t Decomposed mode shape matrix , modal response matrix and lag period Constructed into a dimension ) The third-order tensor of .

[0124] Specifically, the third-order tensor , which is expressed as follows:

[0125] ;

[0126] Where R is the rank of CP decomposition; For tensors The vector in the first dimension, the mode shape matrix The rth mode shape vector of ; For tensors The vector in the second dimension, the modal response matrix The rth modal response vector of ; For tensors The vector in the third dimension is the rth vector of the signal contribution matrix Θ(t).

[0127] S5, third-order tensor update step, the third-order tensor at time t Add new data slices in the second dimension to construct a third-order tensor at time t+1 .

[0128] Specifically, the third-order tensor at time t+1 is , which is expressed as follows:

[0129] ;

[0130] in, Represents a new data slice.

[0131] S6, weighted observation matrix construction step, based on the third-order tensor at time t+1 , introduce the dynamic forgetting factor to construct the weighted observation matrix at time t+1 , and perform windowing and weighting processing on the observation matrix.

[0132] Specifically, in the initial stage of the algorithm, the forgetting factor is set The initial value of , and use the initial data sample number M to build the initial model.

[0133] Calculate the data change rate change_rate of the vibration response signal input at the current time point, and adjust according to the change rate The forgetting factor of the moment ,as follows:

[0134] ;

[0135] Use dynamically adjusted forgetting factor Weight the historical data and construct a new weighted observation matrix :

[0136] (t+1);

[0137] in, is to convert the tensor After the matrix is ​​expanded along the first dimension, its dimension is ) ,The purpose of this processing is to use matrix decomposition ,technique to extract modal information from the data. (t+1) is the weighted matrix, which is composed of the current forgetting factor:

[0138] ;

[0139] in, To use the forgetting factor Construct the diagonal matrix.

[0140] The weighted observation matrix Decompose to get the initial factor matrix , and .

[0141] Factor Matrix Perform recursive updates and estimate by least squares method time Initial estimate of :

[0142] ;

[0143] in, is the loading matrix at the new moment, To load the matrix The pseudo-inverse matrix of .

[0144] S7, factor matrix recursive update step, based on the weighted observation matrix at time t Two factorizations of, diagonalization of connection matrix, singular value tracking and modal response matrix and The common blocks between them are recursively updated to obtain the modal vibration matrix at time t+1 , modal response matrix The sum of the signal contribution to the modal matrix Θ(t+1).

[0145] Specifically, consider The weighted observation matrix at time The two factorizations of are as follows:

[0146] ;

[0147] The first decomposition is achieved by replacing the formula (t) Replace it with the result obtained by its CP decomposition, ; The second decomposition is The economic scale SVD decomposition of , and ; is a combination of singular vectors of the tensor in the first and third dimensions; To include Singular values ​​in the decomposition; contains the singular vectors of the tensor along the second dimension.

[0148] There exists a non-singular connectivity matrix , so that:

[0149] ;

[0150] in, , is the auxiliary matrix; connect the two equations It is found by solving the simultaneous diagonalization of a set of matrices of size R×R.

[0151] Combining the above two formulas, we can calculate , , and ; for The inverse matrix of .

[0152] Next, SVD tracking is performed. First, both the left subspace and the right subspace (the left subspace is the left singular vectors of the modal vibration matrix, and the right subspace is the right singular vectors of the modal response matrix) must be tracked. In addition, the orthogonal (orthogonalized right singular vectors) right subspace basis vectors need to be calculated, which will be used to derive the Recursive update of With growing dimensions, use a dual-iteration technique to update and . Represents a matrix obtained in the simultaneous diagonalization recursion, specifically a part of the right singular vector matrix.

[0153] When tracking specifically, the loading matrix Perform singular value decomposition as follows:

[0154] ;

[0155] in, express The rth vector of ; is a matrix The left singular vector matrix of The eigenvector of represents the orthogonal basis vector of the data in the row space; represents the diagonal matrix of singular values, containing The singular values ​​of reflect the strength of the matrix in each orthogonal direction; is a matrix The matrix composed of the right singular vectors of The eigenvector of represents the orthogonal basis vector of the data in the column space; the updated factor matrix and Used to adjust the tensor decomposition results in the recursive process; and represents the modal components used for updating in each recursive iteration. In the recursive process, they are iteratively updated by the least squares method to ensure that the model can gradually approach the real modal characteristics. Based on the singular value decomposition results, we get exist Update of the moment and exist Update of the moment ,as follows:

[0156] Assign the left singular vector matrix to Update , assign the decomposed singular values ​​to Update ;based on ,get exist Update of the moment ;

[0157] Assign the right singular vector matrix part to Update .

[0158] At time t+1, the formula The conversion is as follows:

[0159]

[0160] use The time-shift structure connects time t and time t+1, where express Auxiliary matrix of time; represents the non-singular connection matrix at time t+1; for The inverse matrix of .

[0161] Due to the time-varying structure and There is a common block, denoted as ; and The dimensions are N(t)×R and N(t+1)×R respectively, so, and The same is true for the dimensions of ; in this case, the common block yes itself.

[0162] Define the following matrix based on the set window:

[0163] ;

[0164] represents the No changes;

[0165] It means to intercept the first N(t) rows from the L(t+1) matrix to form ;

[0166] It means extracting the N(t+1)th row of L(t+1), forming .

[0167] By common block get:

[0168] ;

[0169] in, , used to weight the historical data

[0170] Specifically, Based on the forgetting factor The attenuation of makes the weighted decrease smoother.

[0171] get:

[0172] ;

[0173] ;

[0174] in, represents the pseudo-inverse of a matrix; is an identity matrix, so .

[0175] and Update: Based on the formula ,accomplish and Updates to get and .

[0176] Update: Based on the formula renew , and finally appended to To build .

[0177] judge Relative to Is the change of less than the preset second threshold? If so, further determine whether the factor matrices of all ranks have been updated. If so, output the updated , and .

[0178] S8, working modal parameter identification step, modal vibration matrix , modal response matrix The modal contribution matrix Θ(t+1) of the sum signal is decomposed to identify the time-varying working modal parameters, such as modal vibration shape and natural frequency.

[0179] Through the above embodiments, the under-timed variable working modal parameter identification method based on tensor decomposition with forgotten aging factor and diagonalization recursion proposed by the present invention can be adaptively recursively updated. Modal formation matrix, The modal response matrix and The equal weight matrix can effectively handle the identification of the working modal parameters of the time-varying structure.

[0180] In addition, in another preferred embodiment of the present invention, a linear time-varying three-degree-of-freedom spring oscillator system is used to verify the modal parameter identification method proposed in the present invention.

[0181] like Figure 2 The linear time-varying three-degree-of-freedom spring oscillator system shown.

[0182] Set the sampling frequency The sampling interval is 40Hz. The sampling time is 0.025s and the sampling time is t=2000s.

[0183] Assume that the initial conditions of the three degrees of freedom displacement of the system are all zero, apply Gaussian white noise excitation to block 1, and set the mass to:

[0184] ;

[0185] 1kg, stiffness k1=1000N / m, k2=1000N / m, k3=1000N / m; damping c1=0.01Ns / m, c2=0.01Ns / m, c3=0.01Ns / m.

[0186] Then the three-degree-of-freedom spring oscillator was tested. The response signal is obtained by simulation. The sampling frequency of the signal is 40Hz, the sampling time is t=2000s, and the sampling interval is 0.025s. The applied Gaussian white noise excitation and the simulated response signals are shown in Figures 3(a), 3(b), 3(c), and 3(d).

[0187] like Figure 4 As shown, this embodiment compares and evaluates the working modal parameter identification methods of linear time-varying structures through the modal assurance criterion (MAC) and the relative error of the modal natural frequency.

[0188] The formula of the modal confidence criterion is as follows:

[0189] ;

[0190] in, and Respectively represent The MAC value varies between 0 and 1. The larger the MAC value (closer to 1), the better the vibration shape.

[0191] The formula for the natural frequency relative error evaluation method is as follows:

[0192] 00%

[0193] in, and Respectively represent The estimated modal natural frequencies and theoretical natural frequencies are The closer the value is to 0, the more accurate the identified natural frequency is.

[0194] The MAC value recognition results of the three-degree-of-freedom spring oscillator obtained in this experiment at the instant of 900s are shown in Table 1. The natural frequency error is shown in Table 2. The MAC value change curve of the three-degree-of-freedom time-varying system identified by the experimental method provided in the embodiment of the present invention is shown in Figure 5 shown.

[0195] Table 1. Confidence criteria of each modal at 900s;

[0196]

[0197] Table 2. Natural frequency errors of each order at 900s;

[0198]

[0199] 6(a), 6(b), 6(c) and 6(d) are comparisons of the natural vibration modes of the three-degree-of-freedom time-varying system at different times of 38.125s, 450s, 900s and 1500s using the experimental method provided by an embodiment of the present invention and the instantaneous vibration modes identified by the proposed method.

[0200] It can be seen from Table 1, Table 2, Figures 6(a), 6(b), 6(c) and 6(d) that the method proposed in the present invention can well realize the identification of modal parameters under under-time variation conditions.

[0201] The global symbols, meanings and specifications in the present invention are listed in Table 3.

[0202] Table 3. Global symbol table;

[0203]

[0204]

[0205] like Figure 7 As shown, an embodiment of the present invention further provides a system for identifying under-time variable working modal parameters, including:

[0206] Motor system and support platform (DC micromotor motion controller) to provide mechanical vibration and structural support;

[0207] Measurement and analysis systems, including force and motion sensors, to collect vibration response signals of time-varying structures;

[0208] Excitation system (exciter), used to apply controlled excitation to simulate the real working environment;

[0209] The processing unit (computer) analyzes the collected signals and identifies the time-varying modal parameters based on the underdetermined working modal parameter identification method based on tensor decomposition recursion.

[0210] Furthermore, the present invention also provides an under-determined operating modal parameter identification device, which includes a memory and a processor, wherein the memory stores a computer program, and the computer program can be executed by the processor to implement the under-determined operating modal parameter identification method based on tensor decomposition recursion as described above.

[0211] Furthermore, the present invention also provides a computer-readable storage medium, which includes computer-readable instructions stored on the computer-readable storage medium, and when the computer-readable instructions are executed by a processor of a device where the computer-readable storage medium is located, the underdetermined operating modal parameter identification method based on tensor decomposition recursion as described above is implemented.

[0212] In several embodiments provided in the embodiments of the present invention, it should be understood that the disclosed apparatus and method can also be implemented in other ways. The apparatus and method embodiments described above are merely schematic. For example, the flowcharts in the accompanying drawings show the possible architecture, functions and operations of the apparatus, method and computer program product according to multiple embodiments of the present invention. In this regard, each box in the flowchart or block diagram can represent a module, a program segment or a part of a code, and the module, program segment or a part of the code contains one or more executable instructions for implementing the specified logical function. It should also be noted that in some alternative implementations, the functions marked in the box can also occur in a different order from the order marked in the accompanying drawings. For example, two consecutive boxes can actually be executed substantially in parallel, and they can sometimes be executed in the opposite order, depending on the functions involved. It should also be noted that each box in the block diagram and / or flow chart, and the combination of boxes in the block diagram and / or flow chart can be implemented with a dedicated hardware-based system that performs a specified function or action, or can be implemented with a combination of dedicated hardware and computer instructions.

[0213] In addition, the functional modules in the various embodiments of the present invention may be integrated together to form an independent part, or each module may exist independently, or two or more modules may be integrated to form an independent part.

[0214] If the function is implemented in the form of a software function module and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention is essentially or the part that contributes to the prior art or the part of the technical solution can be embodied in the form of a software product, which is stored in a storage medium and includes several instructions to enable a computer device (which can be a personal computer, an electronic device, or a network device, etc.) to perform all or part of the steps of the method described in each embodiment of the present invention. The aforementioned storage medium includes: U disk, mobile hard disk, read-only memory (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), disk or optical disk and other media that can store program code. It should be noted that in this article, the term "include", "include" or any other variant thereof is intended to cover non-exclusive inclusion, so that the process, method, article or device including a series of elements includes not only those elements, but also includes other elements that are not explicitly listed, or also includes elements inherent to such process, method, article or device. Without more constraints, an element defined by the phrase "comprising a..." does not exclude the existence of other identical elements in the process, method, article or apparatus comprising the element.

[0215] The terms used in the embodiments of the present invention are only for the purpose of describing specific embodiments, and are not intended to limit the present invention. The singular forms "a", "said" and "the" used in the embodiments of the present invention and the appended claims are also intended to include plural forms, unless the context clearly indicates other meanings.

[0216] It should be understood that the term "and / or" used in this article is only a description of the association relationship of associated objects, indicating that there can be three relationships. For example, A and / or B can represent: A exists alone, A and B exist at the same time, and B exists alone. In addition, the character " / " in this article generally indicates that the associated objects before and after are in an "or" relationship.

[0217] The word "if" as used herein may be interpreted as "at the time of" or "when" or "in response to determining" or "in response to detecting", depending on the context. Similarly, the phrases "if it is determined" or "if (stated condition or event) is detected" may be interpreted as "when it is determined" or "in response to determining" or "when detecting (stated condition or event)" or "in response to detecting (stated condition or event)", depending on the context.

[0218] The "first\second" mentioned in the embodiments is only to distinguish similar objects, and does not represent a specific order for the objects. It is understandable that the "first\second" can be interchanged with the specific order or sequence where permitted. It should be understood that the objects distinguished by "first\second" can be interchanged where appropriate, so that the embodiments described herein can be implemented in an order other than those illustrated or described herein.

[0219] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for identifying underdetermined operating modal parameters based on tensor decomposition recursion, characterized in that: include: The vibration response signal acquisition step is to acquire the vibration response signal of the time-varying structure collected at the selected sensor measurement point at time t under the set environmental excitation. ;in, , is the number of sensors, N(t) is the number of time samples; The principal component score calculation step is to process the acquired vibration response signal using principal component analysis (PCA) to calculate the principal component score; The lag determination step is to calculate the autocorrelation function of the principal component scores and determine the lag by detecting the peak in the autocorrelation function. ; The third-order tensor construction step is based on the vibration response signal at time t Decomposed mode shape matrix , modal response matrix and lag period Constructed into a dimension ) The third-order tensor of ; The third-order tensor update step, the third-order tensor at time t Add new data slices in the second dimension to construct a third-order tensor at time t+1 ; The weighted observation matrix construction step is based on the third-order tensor at time t+1 , introduce the dynamic forgetting factor to construct the weighted observation matrix at time t+1 , and perform windowing and weighting processing on the observation matrix; The factor matrix recursive update step is based on the weighted observation matrix at time t Two factorizations of, diagonalization of connection matrix, singular value tracking and modal response matrix and The common blocks between them are recursively updated to obtain the modal vibration matrix at time t+1 , modal response matrix The sum signal contribution matrix to the mode Θ(t+1); The working modal parameter identification step is to identify the modal vibration matrix , modal response matrix The modal contribution matrix Θ(t+1) of the sum signal is decomposed to identify the time-varying working modal parameters.

2. The underdetermined operating modal parameter identification method based on tensor decomposition recursion according to claim 1 is characterized in that: The principal component score calculation step specifically includes: The collected vibration response signal Perform standardization to obtain a standardized matrix ,as follows: ； in, is the mean of the original data, is the standard deviation; Standardized matrix Calculate the covariance matrix , for the covariance matrix Diagonalize to obtain a matrix of eigenvectors ,as follows: ； Where T is the transpose of the matrix, , is a diagonal matrix, ; Based on the standardized matrix and the matrix consisting of eigenvectors Get the principal component score matrix ,as follows: 。 3. The underdetermined operating modal parameter identification method based on tensor decomposition recursion according to claim 1 is characterized in that: The hysteresis period determination step specifically includes: Calculate the autocorrelation function of the principal component scores ,as follows: ； Among them, i is the index of a selected principal component, which represents the principal component score matrix The i-th column of Represents the value of the i-th principal component score at time t; express The value of the i-th principal component score at time; represents the mean of the i-th principal component score, that is, , N is the length of the data, l is the number of lags; Using the autocorrelation function , determine the appropriate hysteresis period by peak detection to determine the periodic characteristics : ； ； Where k represents the peak index; len( ) represents the autocorrelation function The length of , that is, the number of autocorrelation function values; represents the peak array; Indicates the number of peaks.

4. The underdetermined operating modal parameter identification method based on tensor decomposition recursion according to claim 1 is characterized in that: Based on the vibration response signal at time t Decomposed mode shape matrix , modal response matrix and lag period Constructed into a dimension ) The third-order tensor of , which is expressed as follows: ； Where R is the rank of CP decomposition; For tensors The vector in the first dimension, the mode shape matrix The rth mode shape vector of ; For tensors The vector in the second dimension, the modal response matrix The rth modal response vector of ; For tensors The vector in the third dimension is the rth vector of the signal contribution matrix Θ(t); operator Represents the vector outer product.

5. The underdetermined operating modal parameter identification method based on tensor decomposition recursion according to claim 4 is characterized in that: The third-order tensor at time t Add new data slices in the second dimension to construct a third-order tensor at time t+1 , which is expressed as follows: ； in, Represents a new data slice.

6. The underdetermined operating modal parameter identification method based on tensor decomposition recursion according to claim 5 is characterized in that: The weighted observation matrix construction step specifically includes: Calculate the data change rate change_rate of the vibration response signal input at the current time point, and time The forgetting factor Dynamic Adjustment The forgetting factor of the moment ,as follows: ； in, represents the initial value of the forgetting factor; Use dynamically adjusted forgetting factor Weight the historical data and construct a new weighted observation matrix ,as follows: (t+1)； in, is to convert the tensor The matrix after expansion along the first dimension has dimensions ) , to extract modal information from the data using matrix decomposition techniques; (t+1) is the weighted matrix, which is composed of the current forgetting factor, as follows: ； in, To use the forgetting factor Constructed diagonal matrix; The weighted observation matrix Decompose to get the initial mode shape matrix , modal response matrix The sum signal contribution matrix Θ(t+1) to the modal, let ,get Loading matrix at time ; Operator represents the Hadamard product.

7. The underdetermined operating modal parameter identification method based on tensor decomposition recursion according to claim 6 is characterized in that: The factor matrix recursive updating step specifically includes: The weighted observation matrix at time The two factorizations of are as follows: ； The first decomposition is achieved by replacing the formula (t) Replace it with the result obtained by its CP decomposition, ; The second decomposition is The economic scale SVD decomposition of , and ; is a combination of singular vectors of the tensor in the first and third dimensions; To include Singular values ​​in the decomposition; are the singular vectors of the containing tensor in the second dimension; There exists a non-singular connectivity matrix , so that: ； in, , is the auxiliary matrix; connect the two equations It is found by solving the simultaneous diagonalization of a set of matrices of size R × R; Combining the above two formulas, we can calculate , , and ; for The inverse matrix of Loading matrix Perform singular value decomposition as follows: ； in, express The rth vector of ; is a matrix The left singular vector matrix of The eigenvector of represents the orthogonal basis vector of the data in the row space; represents the diagonal matrix of singular values, containing The singular values ​​of reflect the strength of the matrix in each orthogonal direction; is a matrix The right singular vector matrix of The eigenvector of represents the orthogonal basis vector of the data in the column space; the updated factor matrix and Used to adjust the tensor decomposition results in the recursive process; Based on the singular value decomposition results, we get exist Update of the moment and exist Update of the moment ,as follows: Assign the left singular vector matrix to Update , assign the decomposed singular values ​​to Update ;based on ,get exist Update of the moment ; Assign the right singular vector matrix to Update ; At time t+1, the formula The conversion is as follows: ； use The time-shift structure connects time t and time t+1, where represents the auxiliary matrix at time t+1; represents the non-singular connection matrix at time t+1; for The inverse matrix of Due to the time-varying structure and There is a common block, denoted as ; and The dimensions are N(t)×R and N(t+1)×R respectively, so, and The same is true for the dimensions of ; in this case, the common block yes itself; Define the following matrix based on the set window: ； represents the No changes; It means to intercept the first N(t) rows from the L(t+1) matrix to form ; It means extracting the N(t+1)th row of L(t+1), forming ; By common block get: ； in, represents the square root of the forgetting factor, which is used to weight historical data; get: ； ； in, represents the pseudo-inverse of a matrix; is an identity matrix, so ; and Update: Based on the formula ,accomplish and Updates to get and ; Update: Based on the formula renew , and finally appended to To build ; judge Relative to Is the change of less than the preset second threshold? If so, further determine whether the factor matrices of all ranks have been updated. If so, output the updated , and .

8. An under-timed variable working modal parameter identification system, characterized in that: include: A motor system and a support platform for providing mechanical vibration and structural support; Measurement and analysis systems, including force and motion sensors, for collecting vibration response signals of time-varying structures; An excitation system, used to apply control excitations on the time-varying structure to simulate the real working environment; The processing unit analyzes the collected vibration response signal and identifies the time-varying working modal parameters based on the method according to any one of claims 1 to 7.

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