Method for modal parameter identification of super high-rise structure based on improved sine geometry modal decomposition-natural excitation technique and direct interpolation method

Abstract

A modal parameter identification method for super high-rise structures based on improved symplectic geometric mode decomposition (SGM), natural excitation technique, and direct interpolation is proposed. The method utilizes the Hankel matrix and kurtosis theory to obtain the matrix dimension during decomposition; Pearson correlation coefficients are used to reconstruct the symplectic geometric components; and energy entropy theory is used to obtain the number of components. An improved SGM method is obtained through these techniques. The measured non-stationary acceleration response signal x(t) of a super high-rise structure is decomposed using this improved SGM method to obtain a series of single-component signals, namely the symplectic geometric components (SGCs). Natural excitation technique is used to analyze the SGCs, obtaining their free decay response signals. Direct interpolation and curve fitting are used to obtain the structure's natural frequencies and damping ratios. This invention solves the defects of existing technologies, such as mode aliasing, endpoint effects, and spurious modes.

Description

Technical Field

[0001] This invention relates to the field of structural modal parameter identification technology, and in particular to a method for identifying modal parameters of super high-rise structures based on an improved symplectic geometric mode decomposition-natural excitation technique and direct interpolation method. Background Technology

[0002] Modal parameter identification technology for super high-rise structures under environmental excitation has become a key aspect of health monitoring for these structures. The measured dynamic response signals of super high-rise structures are typically low-amplitude vibration data, exhibiting nonlinearity, non-stationarity, and high noise levels, making accurate identification of natural frequencies and damping ratios using such measurements extremely difficult. Currently, there are many methods for modal parameter identification of structures under typhoon conditions. Among them, time-frequency methods such as the Hilbert-Huang Transform (HHT) and Empirical Wavelet Transform (EWT) have been widely applied in the field of structural modal parameter identification. The first step of the HHT transform is the Empirical Wavelet Transform (EMD), an adaptive analysis method capable of handling nonlinear and non-stationary signals. However, the EMD method suffers from problems such as mode aliasing, endpoint effects, and spurious modes. The Empirical Wavelet Transform (EWT), proposed by Gilles in 2013, is a frequency-domain-based adaptive signal decomposition method. This method combines the advantages of wavelet transform and EMD, possesses a complete mathematical foundation, and can obtain better decomposition results. However, while EWT can overcome some noise interference and eliminate spurious modes, making it suitable for analyzing non-stationary, nonlinear, and noisy signals, it still cannot address frequency aliasing. In summary, existing modal parameter identification methods have limited capabilities and cannot accurately identify the modal parameters of ultra-high-rise structures. Therefore, researching an effective and accurate modal parameter identification method for ultra-high-rise structures is an urgent task. Summary of the Invention

[0003] To address the shortcomings of existing technologies such as modal aliasing, endpoint effects, and spurious modes, this invention proposes a modal parameter identification method for super high-rise structures based on an improved symplectic geometric mode decomposition-natural excitation technique and direct interpolation.

[0004] The present invention adopts the following technical solution:

[0005] A modal parameter identification method for super high-rise structures based on improved symplectic geometric mode decomposition-natural excitation technology and direct interpolation method is characterized by the following steps:

[0006] A1. Improved Symplectic Geometric Mode Decomposition Method

[0007] (1) Using the phase space reconstruction principle, the measured non-stationary acceleration response signal x(t) is reconstructed into a Hankel matrix, and the minimum matrix dimension of the phase space reconstruction is determined by using kurtosis theory.

[0008] (2) Using symplectic orthogonal matrix decomposition, the Hankel matrix in step (1) is reconstructed into a new matrix. Using diagonal mean transformation, the new matrix is ​​converted into a series of single-component signals. By calculating the Pearson correlation coefficient between each component, the obtained components are recombined, and the same periodic components are merged to obtain a series of recombined components.

[0009] (3) Using the energy entropy theory, extract the components containing structural mode information from a series of recombined components, namely the symplectic geometric components (SGCs);

[0010] A2. The measured non-stationary acceleration response signal x(t) of a super high-rise building is decomposed using the improved symplectic geometric mode decomposition method to obtain a series of single-component signals, namely symplectic geometric components SGCs.

[0011] A3. Analyze SGCs using natural excitation techniques to obtain the free decay response signal of SGCs;

[0012] A4. Analyze the free decay response signal of SGCs using the direct interpolation method to obtain the natural frequency of the super high-rise structure.

[0013] A5. Combining the natural frequency of the super high-rise structure obtained in step A4, the damping ratio of the super high-rise structure is obtained by curve fitting using the least squares method.

[0014] Preferably, in step (1), the formula for calculating kurtosis using kurtosis theory is:

[0015]

[0016] Where: x(i) is the recombined signal, μ(x(i)) is the average value of x(i), σ(x(i)) is the standard deviation of x(i), and kp is the kurtosis value of x(i).

[0017] Preferably, in step (2), the formula for calculating the Pearson correlation coefficient is:

[0018]

[0019] Where: A and B are the components of the initial decomposition, μ A and μ B σ is the average value. A and σ B Let be the standard deviation and R be the correlation coefficient. When R > 0.8, A and B can be reconstructed by combination.

[0020] Preferably, in step (3), the formula for calculating energy entropy using the theory is:

[0021] p j =Ej / E;

[0022] Where: E j Let E be the energy of the j-th SGC, E be the sum of the energies of all components, and H be the energy entropy value.

[0023] Based on the SGC reorganization process, when the entropy value of a certain event is much smaller than the previous entropy value, it indicates that the SGC contains less information. At this point, the reorganization stops, and the number of reorganized events is the total number of SGCs before the end of this SGC.

[0024] Preferably, in step A3, the free decay signal of SGCs is obtained by analyzing the natural excitation technique. The essence of the natural excitation technique is that under the excitation of an environment with approximate white noise, the cross-correlation function of the response between two points of the system has an approximate mathematical expression with the impulse response function of the structure. Therefore, the cross-correlation function of the response between any two points can be used to replace the impulse response function of the structure, so that the cross-correlation function of the response can be used to replace the impulse response function in the time domain for modal parameter identification.

[0025] Preferably, in step A4, the natural frequency is calculated using the direct interpolation method, which includes the following steps:

[0026] (1) Calculate the frequency interpolation coordinates after finding the extreme points of each SGC;

[0027] (2) Add boundary points for frequency interpolation coordinates based on linear interpolation;

[0028] (3) Based on cubic spline interpolation, the curve f(t) is obtained;

[0029] (4) The instantaneous frequency curve can be obtained by calculating the maximum value of {0,f(t)}, and the natural frequency can be obtained by calculating the mean value of the instantaneous frequency curve.

[0030] Preferably, in step A5, the formula for calculating the curve fitting is:

[0031] f(t) = Ce bt b = -2πωζ;

[0032]

[0033] Where: C is the amplitude of the fitted envelope, b is the power value of the exponential decay function, ξ is the damping ratio, ω is the natural frequency, and t is the time variable.

[0034] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0035] (1) This invention improves the existing symplectic geometric mode decomposition method by using the Hankel matrix and kurtosis theory to determine the matrix dimension, using the Pearson correlation coefficient to reconstruct the symplectic geometric components (SGC), and using the energy entropy theory to obtain the number of SGC. The above three improvements solve the problems of mode aliasing and over-decomposition in the original symplectic geometric mode decomposition method for decomposing the measured response signal of ultra-high-rise structures, thereby providing the optimal single component signal and ensuring the accuracy of subsequent identification of modal parameters.

[0036] (2) In this invention, the most effective method for extracting modal parameters of super high-rise structures is given by combining natural excitation technology, direct interpolation method and curve fitting. Attached Figure Description

[0037] Figure 1 This is a flowchart of the method for identifying modal parameters of super high-rise structures based on improved symplectic geometric mode decomposition-natural excitation technology and direct interpolation method in Embodiment 1 of the present invention;

[0038] Figure 2 This is a model diagram of a two-story reinforced concrete frame in Embodiment 2 of the present invention;

[0039] Figure 3 This is the spectrum of the original signal in Embodiment 2 of the present invention;

[0040] Figure 4 The spectrum diagram of each component SGCs obtained by using the improved symplectic geometric mode decomposition in Embodiment 2 of the present invention;

[0041] Figure 5 The spectrum diagram of each component IMF is obtained by using the Complementary Set Empirical Mode Decomposition (CEEMD) method in Embodiment 2 of the present invention.

[0042] Figure 6 The instantaneous frequency curves of components SGCs and IMFs in Embodiment 2 of the present invention (two-story reinforced concrete frame model);

[0043] Figure 7 This is a diagram showing the non-stationary acceleration response of a super high-rise office building in the X and Y directions during a typhoon, as measured in Embodiment 3 of the present invention.

[0044] Figure 8 The spectrum of the measured response in Embodiment 3 of the present invention.

[0045] Figure 9 The spectrum diagram of each component SGCs obtained by using the improved symplectic geometric mode decomposition in Embodiment 3 of the present invention;

[0046] Figure 10The spectrum of each component IMFs obtained by using the Complementary Set Empirical Mode Decomposition (CEEMD) method in Embodiment 3 of the present invention;

[0047] Figure 11 This is a graph showing the instantaneous frequency curves of components SGCs and IMFs in Embodiment 3 of the present invention (a super high-rise office building). Detailed Implementation

[0048] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0049] Example 1

[0050] like Figure 1 As shown, a method for identifying modal parameters of super high-rise structures based on improved symplectic geometric mode decomposition-natural excitation technology and direct interpolation includes the following steps:

[0051] A1. Using the phase space reconstruction principle, the measured non-stationary acceleration response signal x(t) = x1, x2, ..., x n Reconstructed into a Hankel matrix:

[0052]

[0053] Where d is the matrix dimension, m = n - d + 1, n is the signal length, and m is the number of rows of the X matrix calculated from n.

[0054] The minimum matrix dimension for phase space reconstruction is determined using kurtosis theory, and the calculation formula is as follows:

[0055]

[0056] Where: x(i) is the recombined signal, μ(x(i)) is the average value of x(i), σ(x(i)) is the standard deviation of x(i), and kp is the kurtosis value of x(i).

[0057] A2. Using the symplectic orthogonal matrix QR decomposition, reconstruct matrix X into matrix W:

[0058]

[0059] Among them, G i Let X be a matrix T X is the eigenvector corresponding to X, and W is the reconstruction matrix of the original phase space.

[0060] Using the diagonal mean transformation, matrix Wk (1≤k≤d) is converted into a set of time series components Y of length n. k (1≤k≤d):

[0061]

[0062] Where, d * =min(m,d), m * =max(m,d).

[0063] The above-mentioned d-components can be obtained using the symplectic orthogonal matrix QR decomposition and diagonal mean transformation. Since the components may share the same periodicity, the obtained components are reorganized by calculating the Pearson correlation coefficient between each component, merging components with the same periodicity. The formula for calculating the Pearson correlation coefficient is:

[0064]

[0065] Where: A and B are the components obtained using the symplectic orthogonal matrix QR decomposition and diagonal average transformation, σ A and σ B σ is the average value. A and σ B Let be the standard deviation, and R be the correlation coefficient. When R > 0.8, A and B can be combined and recombined.

[0066] A3. Using energy entropy theory, extract components containing structural mode information from a series of recombined components. The calculation formula for energy entropy theory is:

[0067] p j =E j / E;

[0068] Where: E j Let E be the energy of the j-th SGC, E be the sum of the energies of all components, and H be the energy entropy value.

[0069] In conjunction with the recombination process of symplectic geometric components, when the entropy value of a certain step is much smaller than the previous entropy value, it indicates that the symplectic geometric component contains less information. At this point, the recombination stops, and the number of recombinations is all the symplectic geometric components before the end of this symplectic geometric component.

[0070] A4. Using the improved symplectic geometric mode decomposition method described above, the measured non-stationary acceleration response signal x(t) of the super high-rise structure is decomposed to obtain a series of single-component signals, namely symplectic geometric components (SGCs).

[0071] A5. The free decay signal of SGCs can be obtained by analyzing the natural excitation technique. The essence of the natural excitation technique is that under near-white noise environmental excitation, the cross-correlation function of the response between two points in the system has an approximate mathematical expression with the impulse response function of the structure. Therefore, the cross-correlation function between any two points can be used to replace the impulse response function of the structure. Thus, the natural excitation technique can be used to analyze SGCs and obtain their free decay response signal.

[0072] A6. Using the direct interpolation method, analyze the free decay response signal of SGCs to obtain the natural frequency. The direct interpolation method mainly consists of the following four steps:

[0073] (1) After finding the extreme point of each SGC, calculate the interval between two adjacent maximum points and adjacent minimum points and the midpoint of the corresponding time between two adjacent maximum points and adjacent minimum points;

[0074] (2) The interval and the midpoint of time in (1) above are in a one-to-one correspondence. Let the interval be the period and take its reciprocal to get the frequency.

[0075] (3) Perform cubic spline interpolation on the frequency points in (2) above to obtain a smooth instantaneous frequency curve f(t);

[0076] (4) The natural frequency can be obtained by calculating the mean of the instantaneous frequency curve.

[0077] A7. The damping ratio is obtained using curve fitting. The formula for calculating the curve fitting is:

[0078] f(t) = Ce bt b = -2πωζ;

[0079]

[0080] Where: C is the amplitude of the fitted envelope, b is the power value of the exponentially decaying function, and ξ is the damping ratio.

[0081] Example 2

[0082] In this embodiment, a two-story reinforced concrete frame model was generated using Matlab software, such as... Figure 2 As shown. The mass of each floor of the frame is 20,000 kg, the stiffness is 30 kN / m, and the damping is 0.5 kN·s / m. A non-stationary excitation is applied to the top floor, and the acceleration response of the frame structure is obtained using the Newmark-β method.

[0083] The acceleration response was decomposed using the improved symplectic geometric mode decomposition (SGMD) method proposed in Example 1, and spectral analysis was performed on each component. Figure 3 The image shown is the original signal spectrum. Figure 4 To obtain the spectrum of each component SGCs using the improved symplectic geometric mode decomposition, Figure 5 The images show the spectra of the individual IMFs obtained using the Complementary Set Empirical Mode Decomposition (CEEMD) method. Comparing the three images, it can be seen that the spectrum of each SGC coincides with the spectrum of the original signal, while the IMFs (the components obtained using CEEMD) do not. This indicates that the improved Symplectic Geometric Mode Decomposition (SGMD) outperforms the CEEMD algorithm in decomposition.

[0084] Using the direct interpolation method proposed in Example 1, combined with natural excitation techniques and curve fitting, the instantaneous frequency curves of the components SGCs and IMFs were calculated as follows: Figure 6 As shown in Table 1, the natural frequencies and damping values ​​are also presented. The results indicate that the instantaneous frequency curves identified by the method proposed in Embodiment 1 of this invention exhibit very small fluctuations in all modes and are consistent with theoretical values; the natural frequencies identified by both methods are essentially consistent with theoretical values. However, the damping identified by the method proposed in Embodiment 1 of this invention is more accurate than that based on CEEMD.

[0085] Table 1. Modal parameter identification results of the two-layer frame model

[0086]

[0087] To verify the noise resistance of the method proposed in this invention, Table 2 lists the identification results of the method proposed in Embodiment 1 of this invention after adding noise with different signal-to-noise ratios. Clearly, the identification error of the natural frequency is negligible, and the maximum identification error of the damping ratio is less than 9%. This indicates that even with high noise effects, the method proposed in Embodiment 1 of this invention can accurately identify the modal parameters of the structure. Therefore, the method proposed in Embodiment 1 of this invention is robust to noise.

[0088] Table 2. Modal parameter identification results under different noise levels.

[0089]

[0090] Example 3

[0091] Identification based on measured non-stationary acceleration response signals of a super high-rise office building:

[0092] In this embodiment, two sets of instruments are installed on the roof of the office building at 420, each set containing two accelerometers with a sampling frequency of 20Hz. They measure acceleration data in the X and Y directions, respectively.

[0093] Step 1: The measured non-stationary acceleration response signal is decomposed into several components using the improved symplectic geometric mode decomposition proposed in Example 1.

[0094] The measured acceleration response data were obtained during a typhoon. Non-stationary acceleration response data in the X and Y directions were selected using the rotation testing method, such as... Figure 7 As shown in the figure. Then, the response data in the X and Y directions are decomposed using an improved symplectic geometric mode decomposition, and spectral analysis is performed on each component. Figure 8 The image shown is the original signal spectrum. Figure 9 To obtain the spectrum of each component SGCs using the improved symplectic geometric mode decomposition, Figure 10 To obtain the spectrograms of each component IMF using the Complementary Set Empirical Mode Decomposition (CEEMD) method. From Figure 8-9 As can be seen, the first two SGCs were successfully decomposed without any overlap. Furthermore, the spectrum of each SGC is consistent with the spectrum of the corresponding original signal. This means that the improved symplectic geometric mode decomposition (SGMD) can appropriately decompose non-stationary responses into single-component signals. Figure 8-10 It can be seen that the first two IMF components are not correctly decomposed, and each IMF contains more noise. The comparison shows that the improved symplectic geometric mode decomposition (SGMD) algorithm has stronger noise resistance.

[0095] Step 2: Using the direct interpolation method proposed in Embodiment 1 of this invention, combined with natural excitation techniques and curve fitting, the instantaneous frequency curves of the components SGCs and IMFs are calculated respectively, as shown below. Figure 11 As shown. The results indicate that the instantaneous frequency identified by the method proposed in this invention remains almost constant in both directions under the two-mode conditions, while the CEEMD-based method exhibits significant oscillations in the second-mode condition in both directions, making it impossible to accurately assess the frequency.

[0096] Table 3 shows the identification results of natural frequencies and damping ratios for the two methods. It can be seen that the identification results of the first-order natural frequencies by both methods are completely consistent. This is because the noise component in the first-order mode is small, and frequency aliasing does not occur. The second-order natural frequencies identified by the two methods are also very close. However, the identification error of the damping ratio is relatively large. The CEEMD-based method can almost not identify the second-order damping ratio because there is significant noise in the second-order mode, which greatly affects the fitted damping ratio. Therefore, the method proposed in this invention is a high-precision, noise-resistant modal parameter identification method for ultra-high-rise buildings, which can be used for modal parameter identification under strong noise and non-stationary excitation.

[0097] Table 3. Modal parameter identification results of a super high-rise office building

[0098]

[0099] The above description is merely an example and illustration of the structure of the present invention. Those skilled in the art can make various modifications or additions to the specific embodiments described, or use similar methods to replace them, as long as they do not deviate from the structure of the present invention or exceed the scope defined in the claims, all of which should fall within the protection scope of the present invention.

Claims

1. A method for modal parameter identification of super high-rise structures based on improved Sine Geometry Modal Decomposition-Natural Excitation Technique and direct interpolation method, characterized in that, Includes the following steps: A1. Improved Symplectic Geometric Mode Decomposition Method (1) Using the phase space reconstruction principle, the measured non-stationary acceleration response signal x(t) is reconstructed into a Hankel matrix, and the minimum matrix dimension of the phase space reconstruction is determined by using kurtosis theory. (2) Using symplectic orthogonal matrix decomposition, the Hankel matrix in step (1) is reconstructed into a new matrix. Using diagonal mean transformation, the new matrix is ​​converted into a series of single-component signals. By calculating the Pearson correlation coefficient between each component, the obtained components are recombined, and the same periodic components are merged to obtain a series of recombined components. (3) Using the energy entropy theory, extract the components containing structural mode information from a series of recombined components, namely the symplectic geometric components (SGCs); A2. The measured non-stationary acceleration response signal x(t) of a super high-rise building is decomposed using the improved symplectic geometric mode decomposition method to obtain a series of single-component signals, namely symplectic geometric components SGCs. A3. Analyze SGCs using natural excitation techniques to obtain the free decay response signal of SGCs; A4. Analyze the free decay response signal of SGCs using the direct interpolation method to obtain the natural frequency of the super high-rise structure. A5. Combining the natural frequency of the super high-rise structure obtained in step A4, the damping ratio of the super high-rise structure is obtained by curve fitting using the least squares method.

2. The method of claim 1, wherein the method is characterized by, In step (1), the formula for calculating kurtosis according to kurtosis theory is: Where: x(i) is the recombined signal, μ(x(i)) is the average value of x(i), σ(x(i)) is the standard deviation of x(i), and kp is the kurtosis value of x(i).

3. The method of claim 1, wherein the method is characterized by, In step (2), the formula for calculating the Pearson correlation coefficient is: Where: A and B are the components of the initial decomposition, μ A and μ B σ is the average value. A and σ B Let be the standard deviation and R be the correlation coefficient. When R > 0.8, A and B can be reconstructed by combination.

4. The method of claim 1, wherein the method is characterized by, In step (3), the formula for calculating energy entropy using the theory is: Where: E j Let E be the energy of the j-th SGC, E be the sum of the energies of all components, and H be the energy entropy value. Based on the SGC reorganization process, when the entropy value of a certain event is much smaller than the previous entropy value, it indicates that the SGC contains less information. At this point, the reorganization stops, and the number of reorganized events is the total number of SGCs before the end of this SGC.

5. The method of claim 1, wherein the method is characterized by: In step A3, the free decay signal of SGCs is obtained by analyzing the natural excitation technique. The essence of the natural excitation technique is that under the excitation of an environment with approximate white noise, the cross-correlation function of the response between two points of the system has an approximate mathematical expression with the impulse response function of the structure. Therefore, the cross-correlation function of the response between any two points can be used to replace the impulse response function of the structure, so that the cross-correlation function of the response can be used to replace the impulse response function in the time domain for modal parameter identification.

6. The method of claim 1, wherein the method is characterized by, In step A4, the natural frequency is calculated using the direct interpolation method, which includes the following steps: (1) Calculate the frequency interpolation coordinates after finding the extreme points of each SGC; (2) Add boundary points for frequency interpolation coordinates based on linear interpolation; (3) Based on cubic spline interpolation, the curve f(t) is obtained; (4) The instantaneous frequency curve can be obtained by calculating the maximum value of {0,f(t)}, and the natural frequency can be obtained by calculating the mean value of the instantaneous frequency curve.

7. The method of claim 1, wherein the method is characterized by, In step A5, the formula for calculating the curve fitting is: f(t) = Ce bt b = -2pcoz; Where: C is the amplitude of the fitted envelope, b is the power value of the exponentially decaying function, ξ is the damping ratio, ω is the natural frequency, and t is the time variable.

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