A Time-Varying Cross-Mode Identification Method Based on Nonlinear Frequency Modulation Modal Decomposition

Abstract

The present invention belongs to the technical field of civil engineering structural health monitoring data analysis, and relates to a time-varying cross-modal identification method based on non-linear frequency modulation modal decomposition. First, the time-frequency distribution of the collected acceleration response is calculated by using the short-time Fourier transform, and the ridge line of the time-frequency distribution is extracted; secondly, the time modal coefficients at different frequencies are calculated by using the time-frequency distributions at the positions of each sensor; then, the time modal correlation coefficients at the time-frequency points of the ridge line are calculated, and the initial center frequency of the cross-modal is determined by combining the ridge line frequencies; finally, the instantaneous frequency of the cross-modal is extracted by the non-linear frequency modulation modal decomposition method. The present invention uses the time modal correlation coefficient to determine the initial center frequency, and can accurately identify the modal parameters of each order of the time-varying structure even in the case of the existence of cross-modal in the time-varying structure.

Description

Technical Field

[0001] The present invention belongs to the technical field of civil engineering structural health monitoring data analysis, and relates to a modal identification method for cross-modal of time-varying structures, specifically a time-varying cross-modal identification method based on non-linear frequency modulation modal decomposition. Background Art

[0002] Structural modal parameters include modal frequency, mode shape, damping ratio, etc., which can reflect the dynamic characteristics of the structure. Under the combined action of environment and time, the structural modal parameters will change accordingly. Therefore, accurately identifying the time-varying modal parameters of the structure is crucial. Traditional modal parameter identification methods assume that the structure does not change with time. However, the dynamic characteristics of civil engineering structures in operation often change with time, which makes the modal parameters show time-varying characteristics, and traditional time-invariant modal identification methods are no longer applicable.

[0003] Time-frequency decomposition methods can capture the time-varying frequency in the signal, so they are applicable to the time-varying of time-varying modal parameters. Empirical mode decomposition method and ensemble empirical mode decomposition method are classical time-frequency decomposition methods, but this method lacks a mathematical theoretical basis and has problems such as modal aliasing and end effect. In addition, the variational mode decomposition algorithm realizes the separation of modes by solving the optimal solution of the variational constraint problem, but it cannot directly obtain the time-varying structural parameters. The non-linear frequency modulation modal decomposition method can be used for signal decomposition when the frequency change range is large, but there is no relevant application for modal identification.

[0004] The modal parameters of the actual structure will change greatly with the change of the environment or structural damage. During the change process, there may be a phenomenon of time crossing between different modes, resulting in the confusion and inseparability of different modes. Therefore, studying a time-varying cross-modal parameter identification method has important engineering significance. Summary of the Invention

[0005] According to the above-mentioned technical problems, a time-varying cross-modal identification method based on non-linear frequency modulation modal decomposition is provided. The technical means adopted by the present invention are as follows:

[0006] A time-varying cross-modal identification method based on non-linear frequency modulation modal decomposition includes the following steps:

[0007] Step 1: Perform short-time Fourier transform on the acceleration responses collected by each acceleration sensor on the structure to obtain the time-frequency distribution;

[0008] Step 2: Calculate the partial derivative of the time-frequency distribution with respect to time, and further calculate the frequency F corresponding to the time-frequency ridge line obtained at the i sensor positions i(t, f), where \(i = 1, 2, \cdots, n\) and \(n\) is the number of sensors, calculate the average frequency of the time-frequency ridge frequencies corresponding to all sensor positions And mark the time-frequency point position corresponding to the average frequency as

[0009] Step 3: Take the time-frequency distribution \(X_1(t, f)\) at the first sensor position as a reference, and calculate the time-mode coefficients \(\varphi\) at the \(k\)-th frequency \(f\) at the \(i = 1, 2, \cdots, n\) positions k at i (t, f k );

[0010] Step 4: Extract the time-mode coefficients corresponding to the time-frequency point position and construct them into a mode vector in the order of sensor positions: Denote the time at \(t = 0\) as \(t_0\), and extract the ridge frequency corresponding to the moment \(t_0\) where m is the number of initial center frequencies. Calculate the time-mode correlation coefficient \(M\) of each ridge frequency , and select the time-frequency points corresponding to \(M\) l ≥ \(1 - e\) as the time-frequency points for determining the initial center frequencies of cross modes, where \(e\) is the error value. The initial center frequencies are determined as functions of time, i.e., l

[0011] Step 5: The initial center frequencies are Calculate the modulation and demodulation operators required in the mode decomposition process and Calculate the initial values of the two demodulated signals and ;

[0012] Step 6: Give the minimization objective function The constraint condition is Update \(u\) l , \(v\) l , \(f\) l (t) in the objective function using an optimization algorithm, and take the finally obtained \(f\) l (t), \(l = 1, 2, \cdots, m\) as the instantaneous frequency estimates of time-varying cross modes. (t), \(l = 1, 2, \cdots, m\) as the instantaneous frequency estimates of time-varying cross modes.

[0013] Furthermore, in Step 2, calculate the frequency corresponding to the time-frequency ridge obtained at the \(i\)-th sensor position in the following way:

[0014]

[0015] where \(X\) i(t,f) represents the time-frequency distribution of the acceleration response at the i-th sensor position, represents X i the partial derivative of (t,f) with respect to time t, represents taking the partial derivative with respect to t, and j represents the unit complex number;

[0016] In step 3, the time mode shape coefficient at the k-th frequency f at the i-th position is calculated in the following manner: k at the location:

[0017]

[0018] where φ represents the mode shape coefficient, Re[] represents extracting the real part of the signal, and X i (t,f k ) represents the time-frequency distribution coefficient corresponding to time t and frequency f k ;

[0019] In step 4, the time mode shape correlation coefficient of each ridge frequency is calculated in the following manner:

[0020]

[0021] where M l represents the time mode shape correlation coefficient of the ridge frequency , || represents taking the absolute value, and || || represents calculating the 2-norm;

[0022] In step 5, the modulation and demodulation operators and the initial values of the demodulated signals are calculated in the following manner:

[0023]

[0024]

[0025]

[0026]

[0027] where, and respectively represent the initial modulation and demodulation operators, diag[] represents taking the diagonal elements, where α is a given penalty factor and Ξ is a second-order difference factor: is the initial center frequency.

[0028] Advantages of the present invention:

[0029] The present invention extracts the initial center frequency of time-varying modes with cross-modalities by using the time-frequency distribution of acceleration response and the time mode correlation coefficient, and uses the initial center frequency as the input parameter of the non-linear frequency modulation mode decomposition to accurately identify the time-varying instantaneous frequency with cross-modalities. Specific implementation manner

[0030] To make the objectives, 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 embodiments. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0031] The embodiments of the present invention disclose a method for identifying time-varying cross-modalities based on non-linear frequency modulation mode decomposition, including the following steps:

[0032] Step 1: Represent the acceleration response vectors at n sensor positions of the structure as x = [x1, x2,..., x n , and use the short-time Fourier transform to calculate the time-frequency distribution of x i , and its expression is X i (t, f), where i = 1, 2,..., n represents the sensor position, t represents time, and f represents frequency;

[0033] Step 2: Calculate the partial derivative of the time-frequency distribution X i (t, f) with respect to time, denoted as where represents the partial derivative with respect to t, and further calculate the frequency corresponding to the time-frequency ridge line: where F i represents the time-frequency ridge line frequency obtained at the i-th sensor position, and j represents the unit complex number. Calculate the average frequency of the time-frequency ridge line frequencies corresponding to all sensor positions, and mark the time-frequency point position corresponding to the average frequency as

[0034] Step 3: Take the time-frequency distribution X1(t, f) at the first sensor position as a reference, and calculate the time mode coefficient at the i-th (i = 1, 2,..., n) position at the k-th frequency f k : where φ represents the mode coefficient, Re[] represents the real part of the extracted signal, and X i (t, f k ) represents the time-frequency distribution coefficient corresponding to time t and frequency f k ;

[0035] Step 4: Extract the time-frequency point position The corresponding time mode coefficient And construct it into a mode vector in the order of the sensor positions: Record the time when t = 0 as t0, and extract the ridge frequency corresponding to the moment of t0 Where m is the number of initial center frequencies. Calculate the time mode correlation coefficient of each ridge frequency: Where M l Represents the time mode correlation coefficient of the ridge frequency , | | represents taking the absolute value, and || || represents calculating the 2-norm. Select the time-frequency points corresponding to M l ≥ 1 - e as the time-frequency points for determining the initial center frequency of the cross mode, where e is the error value. The initial center frequency is determined as a function of time, that is

[0036] Step 5, the initial center frequency is Calculate the parameters required in the modal decomposition process And Where And Represent the initial modulation and demodulation operators respectively, and diag[] represents taking the diagonal elements. Calculate the initial values of the two demodulated signals And : Where α is a pre-given penalty factor, and Ξ is a second-order difference factor:

[0037] Step 6, give the minimization objective function The constraint condition is Where Represent the modulation and demodulation operators of the l-th order modal component respectively, From t0 to t N-1 Represents discretizing the continuous time axis t into N time points, and f l (t) represents the instantaneous frequency of the l-th order mode, and u l , v l Represent the demodulated signals, and ε is the allowable error. Use an optimization algorithm to update u l , v l , f l (t) in the objective function, and take the finally obtained f l (t), l = 1, 2,..., m as the estimated instantaneous frequency of the time-varying cross mode.

[0038] Embodiment

[0039] ​The simulation data analysis of a simply supported beam bridge is used for illustration. The rectangular cross-section size of the simply supported beam bridge is 40 cm in width, 85 cm in height, and 10 m in length. Four vertical acceleration sensors are respectively fixed at the positions of the beam top at 2 m, 4 m, 6 m, and 8 m. The time-varying damage simulation of the elastic modulus of the local beam end between 2 m and 4 m is carried out, and the elastic modulus linearly decreases from 3×10 4 Mpa to 0 after 1400 s and then returns to 3×10 4 Mpa and continues for 600 s. The total time length of generating acceleration is 2000 s. The excitation form is white noise, and the sampling frequency is 200 Hz. The simulated structure has five-order modes. During the simulation of damage, the variation form of the third-order mode frequency with time is that it drops from 60 Hz to 20 Hz from 0 to 1400 s and remains at 60 Hz from 1401 s to 2000 s. The variation form of the fourth-order mode frequency with time is that it drops from 100 Hz to 30 Hz from 0 to 1400 s and remains at 100 Hz from 1401 s to 2000 s. Therefore, the third-order mode frequency changes into the range of the fourth-order mode frequency at 1401 s - 2000 s, and a crossover occurs, making it difficult to distinguish which order of mode the frequency of 60 Hz belongs to.

[0040] The acceleration response vectors at the 4 sensor positions of the structure are expressed as x = [x1, x2, x3, x4], and the short-time Fourier transform is used to calculate the time-frequency distribution of x i , and its expression is X i (t, f), where i = 1, 2, 3, 4 represents the sensor position, t represents time, and f represents frequency;

[0041] Calculate the partial derivative of the time-frequency distribution X i (t, f) with respect to time, which is expressed as where represents taking the partial derivative with respect to t, and further calculate the frequency corresponding to the time-frequency ridge line: where F i represents the time-frequency ridge line frequency obtained at the i-th sensor position, and j represents the unit complex number. Calculate the average frequency of the time-frequency ridge line frequencies corresponding to all sensor positions and mark the time-frequency point position corresponding to the average frequency as

[0042] Taking the time-frequency distribution X1(t, f) at the first sensor position as a reference, calculate the time modal coefficient at the k-th frequency f k for the i = 1, 2, 3, 4 positions: where φ represents the modal coefficient, Re[] represents extracting the real part of the signal, and X i (t, f k ) represents the time t and frequency f kThe corresponding time-frequency distribution coefficient;

[0043] Extract the time-frequency point position The corresponding time mode coefficient And construct a mode vector in the order of the sensor positions: Record the time when t = 0 as t0, and extract the ridge frequency corresponding to the moment of t0 Where Calculate the time mode correlation coefficient of each ridge frequency: Where M l Represents the ridge frequency Of the time mode correlation coefficient, || represents taking the absolute value, and |||| represents calculating the 2-norm. Select M l ≥ 1 - e corresponding time-frequency points As the time-frequency points for determining the initial center frequency of the cross mode, take e = 0.001, where the initial center frequency is determined as a function of time, that is

[0044] The initial center frequency is Calculate the parameters required in the mode decomposition process And Where And Represent the initial modulation and demodulation operators respectively, and diag[] represents taking the diagonal elements. Calculate the initial values of the two demodulated signals And : Where α is a pre-given penalty factor, and Ξ is a second-order difference factor:

[0045] Give the minimized objective function The constraint condition is Where Represent the modulation and demodulation operators of the l-th order mode component respectively, From t0 to t N-1 Represents discretizing the continuous time axis t into N time points, N = 200 × 2000, a total of 4 × 10 5 Data points, f l (t) represents the instantaneous frequency of the l-th order mode, u l , v l Represent the demodulated signals. Use an optimization algorithm to update u l , v l , f l (t) in the objective function, and take the finally obtained f l (t), l = 1, 2, 3, 4, 5 as the instantaneous frequency estimation of the time-varying cross mode.

Claims

1. A time-varying cross-modal identification method based on non-linear frequency modulation modal decomposition, characterized in that Including the following steps: Step 1, perform short-time Fourier transform on the acceleration responses collected by each acceleration sensor on the structure to obtain the time-frequency distribution; Step 2: Calculate the partial derivative of the time-frequency distribution with respect to time, and further calculate the frequency F corresponding to the time-frequency ridge obtained at the positions of i sensors, where i = 1, 2, ..., n, and n is the number of sensors. Calculate the average frequency of the time-frequency ridge frequencies corresponding to all sensor positions i (t,f), and mark the position of the time-frequency point corresponding to the average frequency as ​ Step 3: Taking the time-frequency distribution X1(t,f) at the first sensor position as a reference, calculate the time mode coefficient φ k (t,f i ) at the k-th frequency f for the i = 1, 2, ..., n-th positions; k ) Step 4: Extract the time-frequency point positions The corresponding time mode coefficient And construct a mode vector in the order of the sensor positions: Record the time when t = 0 as t0, and extract the ridge frequency corresponding to the moment of t0 Where m is the number of initial center frequencies. Calculate the time mode correlation coefficient M of each ridge frequency l , select M l ≥ 1 - e corresponding time-frequency points As the time-frequency points for determining the initial center frequencies of the cross modes, e is the error value, where the initial center frequencies are determined as a function of time, that is Step 5, with the initial center frequency being calculate the modulation and demodulation operators required in the modal decomposition process and calculate the initial values of two demodulated signals and ; Step 6: Give the minimized objective function The constraint conditions are Use an optimization algorithm to update u l , v l , f l (t), and take the finally obtained f l (t), l = 1, 2,..., m as the instantaneous frequency estimate of the time-varying cross-modal 2. The time-varying cross-modal identification method based on non-linear frequency modulation modal decomposition according to claim 1, wherein In step 2, calculate the frequency corresponding to the time-frequency ridge obtained at the position of the i-th sensor in the following manner: where X i (t, f) represents the time-frequency distribution of the acceleration response at the i-th sensor position, represents X i the partial derivative of (t, f) with respect to time t, represents taking the partial derivative with respect to t, and j represents the unit complex number; In the step 3, the time mode coefficient at the i-th position and the k-th frequency f k is calculated in the following manner: where φ represents the mode shape coefficient, Re[] represents the real part of the extracted signal, and X i (t, f k ) represents the time-frequency distribution coefficient corresponding to time t and frequency f k ; In step 4, calculate the time-mode correlation coefficient of each ridge frequency in the following manner: Among them, M l represents the ridge line frequency of the time mode correlation coefficient. | | represents taking the absolute value, and || || represents calculating the 2-norm; In step 5, calculate the modulation and demodulation operators and the initial values of the demodulated signals in the following manner: Among them, and respectively represent the initial modulation and demodulation operators, diag[] represents taking the diagonal elements, where α is a given penalty factor and Ξ is a second-order difference factor: is the initial center frequency.

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