A structural mode identification method based on principal component analysis with narrowband filtering

Abstract

This invention discloses a structural mode shape identification method based on principal component analysis using narrowband filtering, comprising the following steps: deploying accelerometers on the structure under test to record the structural vibration response; estimating the frequency range of each order of the structure using a three-dimensional energy-time-frequency diagram; defining specific narrow-band frequencies containing only single-order frequencies; performing bandpass filtering on the data from each measurement point according to the upper and lower cutoff frequencies of each band; and finally performing principal component analysis to obtain the first principal component with the largest contribution, which is then the structural mode shape contained in that frequency band. This method can effectively identify the structural mode shapes corresponding to each order of frequency without requiring particularly precise identification of the structure's natural frequencies. It is computationally simple, has good noise robustness, and possesses good engineering applicability.

Description

Technical Field

[0001] This invention relates to the field of structural health monitoring technology, and specifically to a structural mode identification method based on principal component analysis using narrowband filtering. Background Technology

[0002] With the development of the times, various large-scale building structures, such as high-rise buildings, bridges, and offshore platforms, are not only being constructed at a faster pace and in greater quantities, but also facing increasingly stringent functional requirements. During their service life, they need to cope with increasingly severe construction conditions and operating environments, such as typhoons, earthquakes, waves, and temperature changes. Structural failure can occur at any time throughout the entire design life of the structure. Therefore, structural health monitoring is essential.

[0003] In recent decades, modal parameter-based health monitoring methods have been a major area of ​​research and development, with mode shapes being one of the important structural modal parameters. However, traditional methods for mode shape identification require the precise identification of the structure's natural frequencies, which presents several problems in actual engineering structures. First, modal identification inevitably introduces subjective errors and power spectrum leakage, leading to the inability to obtain accurate natural frequency values ​​and thus inaccurate mode shape identification. Second, time-domain damage identification methods based on natural frequencies suffer from system order determination and mode loss issues. Third, some methods require manual intervention, introducing randomness and making them unsuitable for automated online analysis and health monitoring of massive amounts of continuous monitoring data. Therefore, there is an urgent need to propose methods for structural mode shape identification based on a defined natural frequency range. Summary of the Invention

[0004] The purpose of this invention is to overcome the aforementioned deficiencies in the prior art and provide a structural mode shape identification method based on principal component analysis using narrowband filtering. This method effectively obtains the corresponding structural mode shape without requiring precise identification of a specific frequency. When a structure vibrates in a single mode at only one natural frequency, the mode shape is equal to the ratio of the accelerations at each measuring point. Therefore, the vibration test data can be filtered to produce a response history dominated by a specific natural frequency, thus allowing identification of the mode shape corresponding to that frequency. Power spectrum analysis can roughly obtain the range of natural frequencies of the structure. Then, bandpass filtering within this range is performed to retain only the structural vibration response at a specific frequency. Principal component analysis is then used to obtain the corresponding mode shape. This process requires no structural analysis model, is computationally simple, and exhibits good noise robustness, making it well-suited for health monitoring of practical engineering structures.

[0005] The objective of this invention can be achieved by adopting the following technical solutions:

[0006] A structural mode identification method based on principal component analysis with narrowband filtering, the structural mode identification method comprising the following steps:

[0007] S1. Arrange accelerometers at different positions on the structure under test to test the acceleration response;

[0008] S2. Apply power spectrum analysis to the acceleration response data at each measuring point to obtain a three-dimensional energy time-frequency diagram and determine the natural frequency range of each order of the structure under test.

[0009] S3. Based on the ridge distribution of the three-dimensional energy time-frequency diagram, calculate the narrow frequency band containing only a certain order frequency;

[0010] S4. Bandpass filtering is performed on the acceleration response data using the upper and lower cutoff frequencies of the narrow frequency band as parameters.

[0011] S5. Merge the acceleration response data after bandpass filtering into a response matrix, perform principal component analysis, and arrange the column vectors of the feature matrix according to the corresponding eigenvalues; select the eigenvector corresponding to the first principal component as the mode shape of the structure under test, where the eigenvector corresponding to the first principal component corresponds to the first column of the feature matrix.

[0012] Furthermore, in step S2, power spectrum analysis is performed by combining acceleration response data at different locations on the structure under test to obtain a three-dimensional energy time-frequency diagram. The range of each natural frequency of the structure under test is determined by the three-dimensional energy time-frequency diagram. The process is as follows:

[0013] The acceleration response signal of the m-th measuring point for a pre-specified time period (e.g., within one hour or half an hour) is divided into L segments. The segmented data is then windowed by multiplying it by a window function, where the window function is a rectangular window or a Hanning window. The acceleration response signal of the j-th segment has a length of n. t1, t2, ..., t k , ..., t n For n time points when acquiring acceleration response signals, assume the window function is w(t1), w(t2), ..., w(t... k ),…,w(t n ), adding a window yields a new signal y j (t k k = 1, 2, ... ,n For j = 1, 2, ..., L, calculate using the following formula:

[0014] Acceleration response signal y to the windowed function j (t k Perform the following Fourier transform:

[0015]

[0016] In equation (2), ω is the angular frequency variable, and i is a complex number. Averaging the power spectrum of the L-segment of the acceleration response signal with a predetermined time length can reduce the power of the excitation and ambient frequencies, while the power of the structure's natural frequencies will become more prominent, becoming one of the few extreme points on the power spectrum curve. The average power spectrum is calculated as follows:

[0017]

[0018] Plot the acceleration response power spectrum of all different time periods into a three-dimensional graph with the x-axis representing time, the y-axis representing frequency, and the z-axis representing vibrational energy. Since the power value is the highest at the natural frequency, a ridge will be formed along the time axis. Therefore, the range of each natural frequency can be determined from this ridge.

[0019] Furthermore, in step S3, a specific narrow frequency band containing only single-order frequencies is set, as follows:

[0020] Based on the ridge distribution, select a frequency that contains only the first-order frequency f1 or any k-th-order frequency f. k Specific narrow frequency bands, and determine the upper and lower cutoff frequencies of each narrow frequency band.

[0021] If the ridges in the three-dimensional energy-time frequency plot reveal that the frequency distribution of each order of the structure is sparse, that is:

[0022] f k -f k-1 ≥2 (4)

[0023] Then the upper and lower cutoff frequencies f of the narrow frequency band t f d Calculate according to the following formula:

[0024]

[0025] If the frequency distribution of each order in the structure is dense, the upper and lower cutoff frequencies f need to be calculated according to the following formula. t f d :

[0026]

[0027] When k=1, calculate the lower cutoff frequency f. d When, f k-1 =0.

[0028] Based on the above situation and calculation method, calculate and record the corresponding narrow-domain frequency band [f] for each order. d ,f t ].

[0029] Furthermore, in step S4, the acceleration response data collected by each channel is bandpass filtered according to the upper and lower cutoff frequencies of a specific narrow frequency band to obtain filtered acceleration response data.

[0030] Furthermore, S5 can perform principal component analysis on the filtered acceleration response data of each channel to determine the mode shapes, as follows:

[0031] After applying bandpass filtering to the acceleration responses at p measurement points for each specific narrow frequency band, the filtered acceleration responses at p measurement points are obtained. t = t1, t2, ..., t n Let n represent the n times when the acceleration response signal is acquired, and establish the response matrix X.

[0032]

[0033] in:

[0034] The response matrix X is standardized, then principal component analysis is performed to calculate R = XXX. T The eigenvalues ​​and eigenvectors are used, and the eigenvector corresponding to the largest eigenvalue is taken as the first principal component, which is the structural mode shape corresponding to this frequency.

[0035] The theoretical basis is that when the filtered vibration response signal from p measuring points contains only one frequency component, such as A k For amplitude, Let ε(t) be the phase, ε(t) be Gaussian noise with a mean of 0, and the covariance R of the response be...

[0036]

[0037] Because of the following equation,

[0038]

[0039] Therefore, the eigenvalues ​​of the covariance matrix R are The first principal component is [A1 A2 … A… p ] T .

[0040] Due to the acceleration response at the m-th measuring point for,

[0041]

[0042] In the formula, F(ω) is the Fourier transform of the excitation, and M... r C r and K rLet represent the modal mass, modal damping, and modal stiffness of the r-th order, respectively, and N be the total number of modal orders. and φ mr These are the m-th component of the r-th mode shape and the r-th mode shape, respectively.

[0043] When the vibration signal contains only the r-th single mode, then

[0044]

[0045] Comparing equations (9) and (11), explain the first principal component [A1 A2 … A p ] T This is the mode shape [φ] corresponding to that frequency component. 1r ,φ 2r ,…,φ pr ] T .

[0046] Therefore, after performing bandpass filtering on the acceleration response at each measuring point at a certain frequency, and then performing principal component analysis, the first principal component obtained is the structural mode shape corresponding to that frequency.

[0047] The present invention has the following advantages and effects compared with the prior art:

[0048] 1) This invention only needs to identify the natural frequency range of each order of the structure, and then perform principal component analysis on the acceleration response after narrowband filtering to identify the mode shape corresponding to the frequency. This avoids the errors caused by resolution and energy leakage when identifying frequencies in traditional identification methods, and can identify more accurate mode shapes.

[0049] 2) The identification method of the present invention is simple to operate. It only requires filtering and principal component analysis, which can be completed in the time domain without complex transformation calculations. It has strong anti-noise ability, high identification accuracy, and no human intervention. It is suitable for automatic online analysis of massive continuous monitoring data. Attached Figure Description

[0050] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0051] Figure 1 This is a schematic diagram of the implementation process of a mode shape identification method based on principal component analysis using narrowband filtering disclosed in this invention;

[0052] Figure 2 This is a diagram showing the installation positions of the four-layer frame sensor in Embodiment 1 of the present invention;

[0053] Figure 3This is a diagram showing the location of damaged members under various working conditions in Embodiment 1 of the present invention;

[0054] Figure 4 This is the three-dimensional energy time-frequency diagram obtained by power spectrum analysis in Embodiment 1 of the present invention;

[0055] Figure 5 This is a comparison diagram of the first-order vibration modes obtained by the method of the present invention and eigenvalue analysis calculation under four structural conditions in Embodiment 1 of the present invention;

[0056] Figure 6 This is a comparison diagram of the second-order vibration modes obtained by the method of the present invention and eigenvalue analysis calculation under four structural conditions in Embodiment 1 of the present invention;

[0057] Figure 7 This is a comparison diagram of the third-order vibration modes obtained by the method and eigenvalue analysis of the present invention under four structural conditions in Embodiment 1 of the present invention;

[0058] Figure 8 This is a diagram showing the steel beam model, sensor installation location, damage setting location, and excitation location in Embodiment 2 of the present invention;

[0059] Figure 9 This is the three-dimensional energy time-frequency diagram obtained by power spectrum analysis in Embodiment 2 of the present invention;

[0060] Figure 10 This is a comparison diagram of the first five vibration modes of the steel beam in five states calculated using the method of the present invention in Embodiment 2 of the present invention. Detailed Implementation

[0061] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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 some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0062] Example 1

[0063] The main implementation process of mode identification in this invention is as follows: Figure 1 As shown, the specific steps of a mode shape identification method based on principal component analysis using narrowband filtering are as follows:

[0064] T1. Arrange accelerometers at different locations on the structure under test to test the acceleration response;

[0065] T2. Perform power spectrum analysis on the acceleration response at each measuring point to obtain a three-dimensional energy time-frequency diagram. Determine the frequency range of each order from the ridge line in the time-frequency diagram.

[0066] T3. Calculate the upper and lower cutoff frequencies of the narrow-domain frequency band corresponding to each natural frequency.

[0067] T4. Bandpass filtering is applied to the acceleration response to obtain a vibration response containing only the first-order natural frequency.

[0068] T5. Perform principal component analysis on the filtered vibration response and obtain the corresponding mode shape from the first principal component.

[0069] Taking a four-story frame structure as an example, the structure is a scaled-down model of a 4-story, 2×2 span steel frame, with a planar dimension of 2.5m×2.5m and a height of 3.6m. The frame components are made of hot-rolled 300W grade steel (nominal yield strength 300MPa), with scaled-down cross-sections as shown in Table 1. The frame columns have a cross-section of B100×39, and the beams have a cross-section of S75×11. Beams and columns are fixed together, while supports and the structure are flexibly connected, allowing for free disassembly and installation according to the researchers' needs. To make the mass distribution more realistic, the first, second, and third floor slabs weigh 1000kg, and the fourth floor slab weighs 750kg. Additionally, the lower floor slab serves as a working platform for the upper floor, adding an extra 35kg.

[0070] Table 1. Structural Component Parameter Table

[0071] characteristic column Liang support Section category B100*9 S75*11 L25*25*3 <![CDATA[Cross-sectional area A (m 2 )]]> <![CDATA[1.133*10 -3 ]]> <![CDATA[1.43*10 -3 ]]> <![CDATA[0.141*10 -3 ]]> <![CDATA[Moment of inertia I about the y-axis y (m 4 )]]> <![CDATA[1.97*10 -6 ]]> <![CDATA[1.22*10 -6 ]]> 0 <![CDATA[Moment of inertia I about the z-axis z (m 4 )]]> <![CDATA[0.664*10 -6 ]]> <![CDATA[0.249*10 -6 ]]> 0 <![CDATA[Torque J (m 4 )]]> <![CDATA[8.01*10 -9 ]]> <![CDATA[38.2*10 -9 ]]> 0 Young's modulus E (Pa) <![CDATA[2*10 11 ]]> <![CDATA[2*10 11 ]]> <![CDATA[2*10 11 ]]> Shear modulus G (Pa) E / 2.6 E / 2.6 E / 2.6 <![CDATA[Body density ρ (kg / m 3 )]]> 7800 7800 7800

[0072] Structural model and sensor installation locations, as follows Figure 2 As shown, four different working conditions of the structure were simulated. The first condition is a damage-free state. The second condition is when the stiffness of a single diagonal brace on the first floor is reduced to two-thirds of its original value. Figure 3 The first layer of thick black bars shown is denoted as BMD1; the third type is the removal of... Figure 3 The first layer of thick, black diagonal bracing members is designated BMD2; the fourth type consists of the first and third layers with one diagonal bracing member removed, i.e., removing... Figure 3 The two thick black bars in the image are designated BMD3.

[0073] The specific process of mode shape identification based on narrowband filtering is as follows:

[0074] (1) As Figure 2 Sensors are placed to collect acceleration data in the Y-axis direction. In this embodiment, the acceleration response of each sensor is simulated and tested by structural finite element modeling and numerical simulation calculation.

[0075] (2) Select the acceleration response data from one of the measurement points, calculate the power spectrum curve of the acceleration response data every 10 minutes for a total of 480 hours, and obtain the three-dimensional energy-time frequency diagram as shown below. Figure 4As shown in the figure, there are three very obvious ridges, indicating that the vibration response of the sensor contains three natural frequencies. If the natural frequencies are accurately identified by these ridges, the accuracy of the identified frequencies will be reduced due to the effects of resolution and energy dispersion, which will in turn affect the subsequent mode shape identification. However, the range of each natural frequency can be easily determined by these ridges.

[0076] (3) Based on the distribution of the ridge line and natural frequency, apply equations (4) to (6) to calculate the upper and lower cutoff frequencies of a specific narrow band for each frequency order. The results are shown in Table 2.

[0077] (4) Bandpass filter the acceleration response data of each channel to obtain a narrowband response signal containing only one natural frequency.

[0078] (5) Perform principal component analysis on the bandpass filtered signals at each measurement point. The first principal component is the mode shape corresponding to that frequency of the structure. The mode shape diagrams obtained by performing three filters and principal component analysis are shown below. Figure 5 , Figure 6 and Figure 7 As shown by the solid line, the theoretical mode shape of the structure is obtained through finite element modeling and eigenvalue analysis. Figure 5 , Figure 6 and Figure 7 As shown by the dashed lines, it was found that for the four structural states, including the undamaged structure and the three damage conditions BMD1, BMD2 and BMD3, the vibration modes represented by the solid and dashed lines are very close. This indicates that the vibration mode results obtained by the theoretical eigenvalue analysis and the method of this invention are consistent, proving that this method can effectively and accurately identify each vibration mode.

[0079] Table 2. Structural frequency range determined by three-dimensional energy time-frequency diagram (Hz)

[0080] order Upper cutoff frequency Lower cutoff frequency First stage 7.18 9.18 Second stage 14.15 16.15 Third stage 21.37 23.37

[0081] Example 2

[0082] For example Figure 8 The laboratory simply supported steel beam shown was analyzed to further demonstrate the mode shape identification method proposed in this invention. Different depths of notches (0%, 10%, 20%, 30%, and 40%) were set in the thick black area of ​​the steel beam to simulate different structural conditions. The structural parameters are: Young's modulus E = 200 GPa; material density ρ = 7.85 × 10⁻⁶. 3 kg / m 3 Poisson's ratio υ = 0.28; the beam length is 1.2m, both ends are hinged, and the cross-section is 30mm × 10mm.

[0083] The specific implementation steps for structural mode shape identification are as follows:

[0084] R1, such as Figure 8 As shown, accelerometers were installed at the junctions of each equally divided section, a total of 9. A modal force hammer with PCB model 086D05 was used to apply vertical impact excitation to the beam. At a sampling frequency of 2000Hz, the vertical acceleration response of the upper surface of the 9 nodes of the beam was recorded by the TMR-200 data acquisition system for 6 seconds.

[0085] R2. Power spectrum analysis of the acceleration response from the third measuring point is performed to obtain the three-dimensional energy-time frequency diagram as shown below. Figure 9 As shown in the figure, the five ridges in the figure indicate that the vibration response contains five natural frequencies. Using equations (4) to (6) to analyze the ridges and frequency distribution, the upper and lower cutoff frequencies of the narrow frequency bands of each order are determined as shown in Table 3.

[0086] Table 3. Structural frequency range determined by three-dimensional energy time-frequency diagram (Hz)

[0087] order Upper cutoff frequency Lower cutoff frequency First stage 14.12 16.12 Second stage 56.65 58.65 Third stage 128.10 130.10 Fourth stage 231.10 233.54 Fifth stage 355.19 357.19

[0088] R3. Based on each narrow frequency band, bandpass filtering is performed on the acceleration response signal of each measurement point. Since there are 5 frequency bands, bandpass filtering is performed five times respectively.

[0089] R4. Perform principal component analysis on the bandpass-filtered acceleration response data. The first principal component obtained is the corresponding mode shape. The first five mode shapes obtained through narrowband-filtered principal component analysis for the undamaged structural state (0% depth notch) are shown in Table 4. Figure 10 As shown.

[0090] Table 4. List of the first five vibration modes obtained by the method of this invention and the peak method when the structure is in a damage-free state.

[0091]

[0092]

[0093] Since this structure is a measured structure, it is not suitable for finite element modeling and feature analysis to identify theoretical mode shapes. To evaluate the effectiveness of the mode shape identification method proposed in this invention, the first five mode shapes obtained by analyzing the vibration acceleration response of the structure using power spectral density curves and peak picking are shown in Table 4. The results in this table show that the mode shapes identified by the method of this invention and the peak picking method match well. However, the peak picking method requires accurate identification of the structure's natural frequencies, which is difficult to achieve for some structures. The first five mode shapes of the other four states of this steel beam structure identified by the method of this invention are also shown in Table 4. Figure 10 As shown, this method can correctly identify the vibration modes of the structure.

[0094] In summary, analysis of the numerical simulation data from Example 1 and the measured data from Example 2 reveals the following characteristics of the mode shape identification method of the present invention: Signals are easily measured; precise frequency identification is unnecessary, only the range of natural frequencies needs to be determined by the ridge line; filtering and principal component analysis in the time domain accurately identify the corresponding mode shape of the structure; calculation is simple and requires no manual intervention; comparison with mode shapes obtained through theoretical analysis or other methods shows that it can accurately identify each order of mode shape; and it provides a more effective identification method for mode shape identification of practical engineering structures.

[0095] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A structural mode identification method based on principal component analysis using narrowband filtering, characterized in that, The structural mode identification method includes the following steps: S1. Arrange accelerometers at different positions on the structure under test to test the acceleration response; S2. Apply power spectrum analysis to the acceleration response data at each measuring point to determine the natural frequency range of each order of the structure under test. S3. Based on the ridge line and natural frequency distribution, calculate the narrow frequency band containing only a certain order frequency; S4. Bandpass filtering is performed on the acceleration response data using the upper and lower cutoff frequencies of the narrow frequency band as parameters. S5. Merge the acceleration response data after bandpass filtering into a response matrix, perform principal component analysis, and arrange the column vectors of the feature matrix according to the corresponding eigenvalues; select the eigenvector corresponding to the first principal component as the mode shape of the structure under test, where the eigenvector corresponding to the first principal component corresponds to the first column of the feature matrix.

2. The structural mode identification method based on principal component analysis with narrowband filtering according to claim 1, characterized in that, In step S2, power spectrum analysis is performed by combining acceleration response data at different locations on the structure under test to obtain a three-dimensional energy time-frequency diagram. The natural frequency ranges of each order of the structure under test are then determined using the three-dimensional energy time-frequency diagram. The process is as follows: The acceleration response signal of the m-th measuring point is divided into L segments of a pre-specified time length. The segmented data is then windowed by multiplying it by a window function, where the window function is either a rectangular window or a Hanning window. The j-th segment of the acceleration response signal... t1,t2,…,t k ,…,t n For n time points when acquiring acceleration response signals, assume the window function is w(t1), w(t2), ..., w(t... k ),…,w(t n ), windowing yields a new signal y j (t k Given k = 1, 2, ..., n, j = 1, 2, ..., L, calculate using the following formula: Acceleration response signal y to the windowed function j (t k Perform the following Fourier transform: In equation (2), ω is the angular frequency variable, and i is a complex number. The average power spectrum of the L-segment of the acceleration response signal with a predetermined time length is averaged as follows: The acceleration response power spectrum of all different time periods is plotted as a three-dimensional graph with time on the x-axis, frequency on the y-axis, and vibration energy on the z-axis. The ridge line of this three-dimensional graph determines the range of natural frequencies of each order.

3. The structural mode identification method based on principal component analysis with narrowband filtering according to claim 1, characterized in that, In step S3, a specific narrow frequency band containing only single-order frequencies is set, and the process is as follows: Based on the ridge distribution, select a frequency that contains only the first-order frequency f1, the second-order frequency f2, or the k-th-order frequency f. k Specific narrow frequency bands, and determine the upper and lower cutoff frequencies of each narrow frequency band.

4. The structural mode identification method based on principal component analysis with narrowband filtering according to claim 3, characterized in that, In step S3 If the ridges in the three-dimensional energy-time frequency plot reveal that the frequency distribution of each order of the structure is sparse, that is: f k -f k-1 ≥2 (4) Then the upper and lower cutoff frequencies f of the narrow frequency band t f d Calculate according to the following formula: If the frequency distribution of each order in the structure is dense, then the upper and lower cutoff frequencies f are calculated according to the following formula. t f d : When k=1, calculate the lower cutoff frequency f. d When, f k-1 =0; Calculate and record the narrow-domain frequency band [f] corresponding to each order. d ,f t ].

5. The structural mode identification method based on principal component analysis with narrowband filtering according to claim 1, characterized in that, In step S4, the acceleration response data collected by each channel is bandpass filtered according to the upper and lower cutoff frequencies of a specific narrow frequency band to obtain filtered acceleration response data.

6. The structural mode identification method based on principal component analysis with narrowband filtering according to claim 1, characterized in that, The process of step S5 is as follows: After applying bandpass filtering to the acceleration responses at p measurement points in specific narrow frequency bands, the filtered acceleration responses at p measurement points are obtained. t = t1, t2, ..., t n Let n represent the n times when the acceleration response signal is acquired, and establish the response matrix X. in: The response matrix X is standardized, then principal component analysis is performed to calculate R = XXX. T The eigenvalues ​​and eigenvectors are used, and the eigenvector corresponding to the largest eigenvalue is taken as the first principal component, which is the structural mode shape corresponding to the current order frequency.

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