Modal Analysis Method and System Based on Piecewise Iteration

Abstract

The present invention discloses a modal analysis method and system based on segmented iteration, belonging to the technical field of modal analysis. The measured frequency response is grouped according to different reference points, and a multi-modal peak function is selected as the modal indicator function according to the grouping result; the peak-valley points are identified from the modal indicator function, and the frequency domain is segmented; the frequency points to be calculated are sampled between the peak-valley points to construct a frequency response sequence to be calculated; the orthogonal polynomial method of introducing a weighted matrix is used for modal analysis, parameter identification is carried out for the segmentation, and the coefficients of the polynomial are obtained; the fitting frequency response is calculated according to the polynomial coefficients of the segmentation, the fitting frequency response is subtracted within the frequency band of the frequency response to be calculated, and modal analysis is re-performed on the new frequency response to be calculated to obtain the polynomial coefficients within the segmentation; according to the polynomial coefficients calculated for each segmentation, polynomials of the frequency response within each segmentation are constructed, and finally the structural inherent characteristics are obtained. The present invention solves the calculation error caused by the mutual influence between noise and frequency bands in modal analysis.

Description

Technical Field

[0001] The present invention belongs to the technical field of modal analysis, and particularly relates to a modal analysis method and system based on segmented iteration. Background Art

[0002] The structural modal characteristics usually include structural inherent characteristics such as natural frequency, damping ratio, and modal shape. Experimental modal analysis is a conventional means to obtain structural modal characteristics. It uses experimental means to obtain vibration data, and then conducts modal analysis on the vibration data to obtain structural modal characteristics. The rational fraction orthogonal polynomial algorithm is a commonly used algorithm in modal analysis. This method measures the frequency response or cross-power spectrum between the excitation and response points according to the vibration experiment. First, it calculates the modal indicator function, evaluates the order of the polynomial to be calculated, and then constructs the rational fraction orthogonal polynomial to calculate the system poles and zeros, so as to finally obtain the system natural frequency, damping ratio, and modal shape. In order to distinguish the natural frequencies with close frequencies and obtain more modal orders, a higher sampling frequency is usually used during vibration testing to obtain higher frequency domain resolution. And due to the inevitable introduction of noise signals during the testing process, false modes will appear during modal analysis, which will also cause calculation deviations of the system poles and zeros, and ultimately cause large calculation errors in the system natural frequency, damping ratio, and modal shape. Especially when the calculation frequency bandwidth is wider, the number of modal orders is more, and the frequency domain resolution is higher, the more false modes are obtained, and the greater the calculation error is. Summary of the Invention

[0003] In order to solve the deficiencies of the prior art and achieve the purpose of effectively reducing the influence of noise and improving the accuracy and efficiency of modal parameter identification, the present invention adopts the following technical solutions:

[0004] A modal analysis method based on segmented iteration includes the following steps:

[0005] Step 1: Group the measured frequency response according to different reference points, and select a multi-modal peak function as the modal indicator function according to the grouping result;

[0006] Step 2: Identify the peak and valley points from the modal indicator function, and segment the frequency domain;

[0007] Step 3: Sample the frequency points to be calculated between the peak and valley points to construct a frequency response sequence to be calculated;

[0008] Step 4: Introduce the orthogonal polynomial method with a weighted matrix for modal analysis, and conduct parameter identification for the segmentation to improve the calculation accuracy and obtain the coefficients of the polynomial;

[0009] Step 5: Calculate the fitting frequency response according to the coefficients of the segmented polynomials, subtract the fitting frequency response within the frequency band where the frequency response to be calculated is located to update the frequency response to be calculated, so as to reduce the influence of other frequency bands on this frequency band, improve the calculation accuracy, perform modal analysis on the new frequency response to be calculated again, and obtain the polynomial coefficients within the segment;

[0010] Step 6: Construct polynomials of the frequency response within each segment according to the polynomial coefficients calculated for each segment, and finally obtain the structural inherent characteristics.

[0011] Further, the multi-modal peak function in the above Step 1 is as follows:

[0012]

[0013] where ω represents the frequency of the measured frequency response, represents the i-th measured frequency response with reference to the j-th point, and Cal(·) represents the conversion function of. When calculating, select the amplitude, real part or imaginary part of as the conversion function, M represents the number of reference points, and N represents the number of measured frequency responses with reference to the reference point j.

[0014] Further, Step 2 specifically includes the following steps:

[0015] Step 2.1: Identify the local maximum values in the modal indicator function as peak points, and the minimum value points between two peak points as valley points;

[0016] Step 2.2: Segment the frequency domain according to the distances between the peak points, and group the adjacent peak points into the same segment.

[0017] Further, in Step 2.2, normalize the frequencies of each peak point to calculate the distances between the peak points, select a segmentation threshold (usually 2 times the average distance), and when the distance between two adjacent peak points is greater than this threshold, establish a segmentation at the valley point position between the two peak points.

[0018] Further, the generation method of the frequency response sequence in Step 3 is as follows:

[0019] Obtain the peak points of the modal indicator function, find the adjacent valley points on its left and right, calculate the nearest frequency points within a certain range below the peak points, and include all the frequencies between the peak points and the nearest frequency points on both sides into the frequency points to be calculated;

[0020] Perform interval sampling between the nearest frequency points and the valley points, and set the number of sampling points according to the number of all frequencies between the peak points and the nearest frequency points on both sides, and include the sampled frequency points into the frequency points to be calculated;

[0021] When the value of the nearest frequency point is less than the valley point or the number of frequency points between the peak and valley is less than the point threshold, then all the frequency points between that peak and valley are included in the frequency points to be calculated;

[0022] Sort out the frequency points obtained by each modal indicator function to form a frequency sequence. At the same time, remove duplicate points and sort them according to the frequency size to obtain the frequency response sequence to be calculated.

[0023] Resample the frequency response function through this sequence as the frequency response to be analyzed, reducing the number of points involved in the calculation, improving the calculation efficiency, and at the same time reducing the influence of noise.

[0024] Furthermore, perform equally spaced sampling between the nearest frequency point and the valley point. The number of sampling points is the same as the number of all frequencies between the peak point and the nearest frequency points on the left and right sides. The sampled frequency points are also included in the frequency points to be calculated.

[0025] Furthermore, in step 4, construct a frequency response function matrix:

[0026]

[0027] Among them, represents the frequency response function matrix (with p response points and q excitation points), represents the numerator polynomial matrix, represents the denominator polynomial matrix. For simplicity of expression, the denominator represents matrix inversion;

[0028] For the o-th row of the frequency response matrix, it is independently expressed as:

[0029]

[0030] Among them, represents the frequency response function matrix of the o-th row and q columns, represents the numerator polynomial matrix of the o-th row and q columns, represents the corresponding denominator polynomial matrix of q rows and q columns;

[0031] The row vector decomposition form of the numerator polynomial matrix of the o-th row and q columns is as follows:

[0032]

[0033] Among them, m represents the highest order of the numerator polynomial matrix of, represents the basis function of the numerator polynomial matrix, whose highest order is r, represents the polynomial coefficient;

[0034] The decomposition form of the denominator polynomial matrix is as follows:

[0035]

[0036] Among them, n represents the highest order of the denominator polynomial matrix , represents the basis function of the denominator polynomial matrix, and its highest order is r, represents the polynomial coefficient;

[0037] Using the least squares method, the polynomial coefficients α and β are identified from the measured frequency response function matrix ;

[0038] By introducing the weighting matrix , the following calculation formula is obtained:

[0039]

[0040] Among them, represents the polynomial coefficient of the o-th row, represents the matrix expansion of the polynomial coefficient α;

[0041]

[0042]

[0043]

[0044]

[0045]

[0046]

[0047]

[0048] Among them, represents the number of frequency points of the frequency response function, Re(•) represents taking the real part of the complex value, k represents the k-th data point of the frequency response function, represents the weighting coefficient of the frequency response function, represents the frequency of the k-th data point of the frequency response function, H represents the conjugate transpose of the matrix, represents the measured frequency response of the o-th row.

[0049] Furthermore, step 5 specifically includes the following steps:

[0050] Step 5.1: Calculate the fitting frequency response according to the calculated segmented polynomial coefficients;

[0051] Step 5.2: Subtract the fitting frequency response within the frequency response band to be calculated, and update the frequency response to be calculated;

[0052] Step 5.3: For the new frequency response to be calculated, execute the said Step 4 to complete the calculation of this segment and obtain the polynomial coefficients within the segment.

[0053] Step 5.4: Conduct multiple loop iterations within the entire investigation range until the change in each calculation result remains within a certain range.

[0054] Further, in the said Step 6, according to the polynomial coefficients obtained from each segment calculation, construct the numerator and denominator polynomials of the frequency response within each segment, calculate the system poles and zeros, and finally obtain the structural natural frequency, damping ratio, and mode shape.

[0055] The modal analysis system based on segment iteration includes a memory and one or more processors. The memory stores executable code. When the one or more processors execute the executable code, it is used to implement the modal analysis method based on segment iteration.

[0056] The advantages and beneficial effects of the present invention are as follows:

[0057] The present invention uses a peak indication function to obtain peak points and valley points, obtains a frequency sequence for modal analysis, and resamples the frequency response function with this sequence as the frequency response to be analyzed, reducing the number of points involved in the calculation, improving the calculation efficiency, and reducing the influence of noise at the same time; by introducing the orthogonal polynomial method with a weighting matrix to perform modal analysis, the calculation accuracy is improved; according to the modal indication function, the frequency response to be analyzed is segmented, and only the frequency response within the segment is analyzed each time, while subtracting the modal influence of the already calculated frequency band, reducing the influence of other frequency bands on this frequency band, and improving the calculation accuracy. Description of the Drawings

[0058] Figure 1 is the flowchart of the method in the embodiment of the present invention.

[0059] Figure 2 is the segmentation schematic diagram based on the complex modal indication function in the embodiment of the present invention.

[0060] Figure 3 is the schematic diagram of generating the frequency response sequence based on the complex modal indication function in the embodiment of the present invention.

[0061] Figure 4a is the comparison diagram between the fitted frequency response obtained by the traditional modal analysis method and the original frequency response.

[0062] Figure 4b is the comparison diagram between the fitted frequency response obtained by the modal analysis method in the embodiment of the present invention and the original frequency response. Detailed Embodiments

[0063] The specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are only for the purpose of illustrating and explaining the present invention, and are not intended to limit the present invention.

[0064] As Figure 1 shown, the modal analysis method based on segmented iteration includes the following steps:

[0065] Step 1: Calculate the modal indicator function. In the calculation, first group the measured frequency responses according to different reference points, and then select the multi-modal peak function as the modal indicator function according to the results.

[0066] The calculation method of the multi-modal peak function is as follows:

[0067]

[0068] Among them, The optional amplitude, real part or imaginary part method.

[0069] Step 2: Identify the peak and valley points and segment the frequency domain. The process includes two steps:

[0070] (1) Identify the local maximum in the modal indicator function as the peak point, and the minimum point between two peak points as the valley point.

[0071] (2) Segment the calculated frequency domain according to the distance between the peak points, and group the adjacent peak points into the same segment, as Figure 2 shown.

[0072] The grouping method can be specified manually or automatically calculated as follows:

[0073] Normalize the frequencies of each peak point, calculate the distance between each peak point, select the segmentation threshold (usually 2 times the average distance), and when the distance between two adjacent peak points is greater than the threshold, establish a segmentation at the valley point between the two peak points.

[0074] Step 3: Generate a frequency sequence, sample the original frequency response to generate a new frequency response sequence to be calculated

[0075] The determination method of the frequency sequence is as follows:

[0076] (1) Obtain the peak points of the modal indicator function, find the adjacent valley points on its left and right, calculate the 3dB value below the peak point, and find its nearest frequency point. Include all the frequencies within the 3dB frequency range on both sides of the peak point in the frequency points to be calculated.

[0077] (2) Perform equally spaced sampling between the 3dB point and the valley point. The number of sampling points should be the same as the number of points between the adjacent 3dB point and the peak point. The sampled frequency points are included in the calculated frequency points.

[0078] (3) When the 3 dB point is less than the valley point or the number of frequency points between the peak and valley is less than 10 points, all are included in the calculated frequency points.

[0079] (4) Organize the frequency points obtained from each modal indicator function to form a frequency sequence. At the same time, remove duplicate points and sort them according to the frequency magnitude to obtain the calculated frequency point sequence, as Figure 3 shown.

[0080] Step 4: Conduct parameter identification for each segment.

[0081] Use the orthogonal polynomial method with a weighted matrix for parameter identification. The identification algorithm is as follows:

[0082] First, the frequency response function matrix can be expressed as a right matrix fraction:

[0083]

[0084] where, is the frequency response function matrix with p rows and q columns (there are p response points and q excitation points), is the numerator polynomial matrix, is the denominator polynomial matrix. For simplicity of expression, the denominator represents matrix inversion;

[0085] The o-th row of the frequency response matrix can be independently expressed as:

[0086]

[0087] The row vector of the numerator polynomial matrix can be written in a decomposed form:

[0088]

[0089] where, m is the highest order of the numerator polynomial matrix; is the basis function of the numerator polynomial matrix, and its highest order is r; is the o-th row of; are the polynomial coefficients.

[0090] Similarly, the denominator polynomial matrix can be expressed as:

[0091]

[0092] where, is the basis function of the denominator polynomial matrix, with the highest order of r; are the polynomial coefficients.

[0093] The next problem evolves into using the least squares method to identify the system coefficients from the measured frequency response function matrix ​ and . Further, the least squares method is used to deduce the calculation process, and a weighted matrix is introduced. Finally, the calculation formula of the system can be obtained as follows:

[0094]

[0095] where

[0096]

[0097]

[0098]

[0099]

[0100]

[0101]

[0102]

[0103] Step 5: Remove the modal influence of other segments and calculate the next segment. The calculation method is as follows:

[0104] (1) Calculate the fitting frequency response according to the polynomial coefficients and of the already calculated segment.

[0105] (2) Subtract the fitting frequency response within the frequency range to be calculated, and update the frequency response to be calculated.

[0106] (3) For the new frequency response to be calculated, execute Step 4 to complete the calculation of this segment and obtain the polynomial coefficients and within the segment.

[0107] (4) Perform multiple loop iterations within the entire investigation range until the change in each calculation result meets a certain range.

[0108] Step 6: Collect the calculation results of each segment and calculate the system mode.

[0109] According to the polynomial coefficients and calculated for each segment, construct the numerator and denominator polynomials of the frequency response within each segment, calculate the system poles and zeros, and finally obtain the system natural frequency, damping ratio, and mode shape. To verify whether the modal analysis is accurate, a fitting frequency response can be generated based on the system natural frequency, damping ratio, and mode shape, and compared with the original input frequency response. If the two are relatively close, it indicates that the modal analysis is accurate.

[0110] Actual verification shows that, compared with the fitting frequency response obtained by the traditional modal analysis method, the fitting frequency response obtained by using the modal analysis method based on piecewise iteration described in the present invention is more in line with the original frequency response, and the obtained modal characteristics are more accurate, as Figure 4a , Figure 4b shown.

[0111] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A modal analysis method based on segmented iteration, characterized in that It includes the following steps: Step 1: Group the measured frequency responses according to different reference points, and select a multi-modal peak function as the modal indicator function based on the grouping results; Step 2: Identify the peak and valley points from the modal indicator function, and segment the frequency domain, including the following steps: Step 2.1: Identify the local maximum values in the modal indicator function as peak points, and the minimum value points between two peak points as valley points; Step 2.2: Segment the frequency domain according to the distances between the peak points, and group the adjacent peak points into the same segment; Step 3: Sample the frequency points to be calculated between the peak and valley points to construct a sequence of frequency responses to be calculated; Step 4: Introduce the orthogonal polynomial method with a weighted matrix for modal analysis, perform parameter identification for the segmented parts, and obtain the coefficients of the polynomials; Step 5: Calculate the fitted frequency response according to the polynomial coefficients of the segments, subtract the fitted frequency response within the frequency band of the frequency response to be calculated, perform modal analysis on the new frequency response to be calculated again, and obtain the polynomial coefficients within the segment; Step 6: Construct polynomials of the frequency responses within each segment according to the polynomial coefficients calculated for each segment, and finally obtain the structural inherent characteristics.

2. The modal analysis method based on segmented iteration according to claim 1, characterized in that: The multi-modal peak function in Step 1 is as follows: ， where ω represents the frequency response frequency to be measured, represents the i-th frequency response to be measured with the j-th point as the reference, and Cal(·) represents the conversion function. During calculation, select the amplitude, real part or imaginary part of as the conversion function according to the actual effect. M represents the number of reference points, and N represents the number of frequency responses to be measured with the reference point j as the reference.

3. The modal analysis method based on segmented iteration according to claim 1, wherein: In Step 2.2, normalize the frequencies of the peak points to calculate the distances between the peak points, select a segmentation threshold, and when the distance between two adjacent peak points is greater than the threshold, establish a segmentation at the valley point position in the middle of the two peak points.

4. The modal analysis method based on segmented iteration according to claim 1, characterized in that: The generation method of the frequency response sequence in Step 3 is as follows: Obtain the peak points of the modal indicator function, find the adjacent valley points on its left and right, calculate the nearest frequency points within a certain range below the peak points, and include all the frequencies between the peak points and the nearest frequency points on both sides into the frequency points to be calculated; Perform interval sampling between the nearest frequency points and the valley points, and set the number of sampling points according to the number of points of all the frequencies between the peak points and the nearest frequency points on both sides, and also include the sampled frequency points into the frequency points to be calculated; When the value of the nearest frequency point is less than the valley point or the number of frequency points between the peak and valley is less than the point threshold, then include all the frequency points between that peak and valley into the frequency points to be calculated; Sort out the frequency points obtained from each modal indicator function to obtain a sequence of frequency responses to be calculated.

5. The modal analysis method based on segmented iteration according to claim 4, wherein: Perform equally spaced sampling between the nearest frequency points and the valley points, and set the number of sampling points to be the same as the number of points of all the frequencies between the peak points and the nearest frequency points on both sides, and also include the sampled frequency points into the frequency points to be calculated.

6. The modal analysis method based on segmented iteration according to claim 1, wherein: In Step 4, construct a frequency response function matrix: ， Among them, represents the frequency response function matrix, represents the numerator polynomial matrix, represents the denominator polynomial matrix; For the o-th row of the frequency response matrix, it is independently expressed as: ， Among them, represents the frequency response function matrix of o rows and q columns, represents the numerator polynomial matrix of o rows and q columns, represents the corresponding denominator polynomial matrix of q rows and q columns; The row vector decomposition form of the numerator polynomial matrix of the o-th row and q-th column is as follows: ， where m represents the highest order of the molecular polynomial matrix , denotes the basis function of the molecular polynomial matrix, whose highest order is r, represents the polynomial coefficient; The denominator polynomial matrix decomposition form is as follows: ， where n represents the highest order of the denominator polynomial matrix , represents the basis function of the denominator polynomial matrix, and its highest order is r, represents the polynomial coefficient; Identify the polynomial coefficients α and β using the least squares method from the measured frequency response function matrix ; By introducing a weighted matrix , the following calculation formula is obtained: ， Among them, represents the polynomial coefficient of the $o$-th row, represents the matrix expansion of the polynomial coefficient $\alpha$. ， ， ， ， ， ， ， Among them, represents the number of frequency points of the frequency response function, Re(•) represents taking the real part of a complex value, and k represents the k-th data point of the frequency response function. represents the weighting coefficient of the frequency response function. represents the frequency of the k-th data point of the frequency response function, and H represents the conjugate transpose of the matrix. represents the measured frequency response of the o-th row.

7. The modal analysis method based on segmented iteration according to claim 1, wherein: Step 5 specifically includes the following steps: Step 5.1: Calculate the fitted frequency response according to the polynomial coefficients of the already calculated segments; Step 5.2: Subtract the fitted frequency response within the frequency band of the frequency response to be calculated to update the frequency response to be calculated; Step 5.3: For the new frequency response to be calculated, execute Step 4 to complete the calculation of this segment and obtain the polynomial coefficients within the segment; Step 5.4: Perform multiple cyclic iterations within the entire investigation range until the change in each calculation result remains within a certain range.

8. The modal analysis method based on segmented iteration according to claim 1, wherein: In step 6, according to the polynomial coefficients obtained by each segmentation calculation, construct the numerator and denominator polynomials of the frequency response within each segmentation, calculate the system poles and zeros, and finally obtain the structural natural frequency, damping ratio, and mode shape.

9. A modal analysis system based on segmented iteration, characterized in that, It includes a memory and one or more processors. Executable code is stored in the memory. When the one or more processors execute the executable code, it is used to implement the modal analysis method based on segmented iteration described in any one of claims 1-8.

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