A statistical analysis method for flutter eigenvalues ​​based on similar samples

Abstract

This invention discloses a statistical analysis method for flutter eigenvalues ​​based on similar samples, belonging to the field of flutter signal processing. The proposed method includes a method for obtaining similar samples, a method for obtaining flutter eigenvalues, and a method for nonparametric estimation of confidence intervals for flutter eigenvalues. This statistical analysis method based on similar samples overcomes the deficiency of insufficient flutter experimental data; the nonparametric estimation of confidence intervals for flutter eigenvalues ​​quantifies the stability of the flutter eigenvalues. Furthermore, the nonparametric estimation method does not require the assumption of a mathematical model, avoiding large errors caused by incorrect assumptions about the population.

Description

Technical Field

[0001] This invention belongs to the field of flutter signal processing, specifically relating to a statistical analysis method for flutter feature values ​​based on similar samples. Background Art

[0002] Actual wind tunnel flutter test data is very limited. Considering factors such as manpower and cost, it is not possible to obtain sufficient test data through a large number of repeated tests to conduct statistical analysis of flutter characteristic values ​​based on wind tunnel flutter tests.

[0003] To address these issues, traditional statistical analysis methods first identify modal parameters from a small amount of experimental data, assuming the identification results follow a certain probability distribution. Then, they calculate the mean and variance based on a limited set of observations, and finally perform statistical analysis using the Monte Carlo method. This approach directly assumes the probability distribution of the modal parameter identification results, and the mean and variance obtained from a small number of observations are unreliable, thus easily influencing the statistical analysis results. Summary of the Invention

[0004] This invention addresses the problems of limited experimental data, especially high-quality experimental data, and the need for prior assumptions about probability distribution types in statistical analysis by proposing a statistical analysis method that combines flutter characteristic values ​​from similar samples.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] The present invention proposes a statistical analysis method for flutter eigenvalues ​​based on similar samples, including a method for obtaining similar samples, a method for obtaining flutter eigenvalues, and a method for estimating confidence intervals for flutter eigenvalues ​​using nonparametric estimation.

[0007] Step 1: First, for the structural response signal, repeatedly use the iterative amplitude adjustment Fourier algorithm to generate similar samples;

[0008] Step 2: After preprocessing similar samples and identifying modal parameters, flutter feature values ​​are obtained to get the initial samples;

[0009] Step 3: Finally, the confidence interval is obtained by nonparametric estimation and statistical analysis is performed in combination with the initial sample mean.

[0010] Furthermore, the method for obtaining similar samples is as follows: For the existing flutter test data, the Fourier algorithm with iterative amplitude adjustment is used multiple times to generate similar samples to enrich the sample quantity of the flutter test data. This mainly consists of the following five steps.

[0011] Step 1.1: First, record the sequence number of the original experimental data, apply Fourier transform to the original experimental data and calculate its amplitude value.

[0012] Step 1.2: Use the amplitude-adjusted Fourier algorithm to calculate preliminary data with the same probability distribution.

[0013] Step 1.3: Perform a Fourier transform on the obtained preliminary data, and replace the amplitude value with the amplitude value of the Fourier transform of the original experimental data while keeping the phase unchanged.

[0014] Step 1.4: Perform an inverse Fourier transform on the data obtained in Step 1.3 and rearrange it according to the original data sequence number.

[0015] Step 1.5: Repeat steps 1.3 and 1.4 until the obtained data has a power spectrum similar to the original experimental data.

[0016] Furthermore, the method for obtaining flutter eigenvalues ​​is as follows: after obtaining an appropriate amount of sample data, flutter eigenvalues ​​are obtained for each group of data. In this invention, flutter margin is selected as the flutter eigenvalue, which is mainly divided into the following three steps.

[0017] Step 2.1: For the sample composed of all the data obtained above, wavelet transform is used for noise reduction to increase the signal-to-noise ratio and thus reduce the impact of noise on modal parameter identification.

[0018] Step 2.2: Use Empirical Mode Decomposition (EMD) to perform mode decomposition on the preprocessed data samples. Then, use natural excitation techniques to extract the free decay signals of each decomposed single mode. Finally, use the matrix bundle method to identify the mode parameters of the single-mode free decay signals extracted by natural excitation techniques.

[0019] Step 2.3: Substitute the obtained modal parameters pairwise into the flutter margin formula:

[0020]

[0021] Where F M Here, ω1 and ω2 are the frequencies of the two modes, and ζ1 and ζ2 are the damping ratios of the two modes;

[0022] The flutter margin obtained from the mode combination with a clear downward trend is selected as the flutter characteristic value.

[0023] Furthermore, the nonparametric method for estimating confidence intervals is as follows: After obtaining the flutter feature values, the confidence intervals of the flutter feature values ​​are estimated nonparametrically. First, a set of flutter feature values ​​already obtained is used as the initial sample. Then, a random sampling method with replacement is used to draw sample values ​​from the initial sample to form a new sample. The size of the new sample is consistent with that of the initial sample. The above method is used cyclically to obtain M sets of samples. The following five steps are mainly used to calculate the estimated confidence intervals of all samples:

[0024] Step 3.1: First, calculate the sample mean of the initial sample. and sample standard deviation

[0025] Step 3.2: Calculate the sample mean of the remaining sample groups. Sample standard deviation Calculate pivot quantity

[0026] Step 3.3: Arrange the calculated M pivot quantities in ascending order: w1≤w2≤…≤w M .

[0027] Step 3.4: Find the confidence interval with a confidence level of 1-α, let... w k1 w k2 respectively as w α / 2 w 1-α / 2 The estimate.

[0028] Step 3.5: From The confidence interval with a confidence level of 1-α is obtained:

[0029] The flutter margin confidence intervals were repeatedly obtained at multiple wind speeds. The deviation range between the initial sample mean and the confidence interval was calculated. The relationship between the confidence interval and the wind speed was statistically analyzed, and the characteristics of the flutter margin eigenvalues ​​were analyzed.

[0030] The advantages of this invention over the prior art are as follows:

[0031] The statistical analysis method for flutter eigenvalues ​​based on similar samples proposed in this invention overcomes the deficiency of insufficient flutter test data; it quantifies the stability of flutter eigenvalues ​​by nonparametrically estimating the confidence intervals; and the nonparametric estimation method does not require the assumption of a mathematical model, thus avoiding large errors caused by incorrect assumptions about the population.

[0032] This invention proposes a statistical analysis method for flutter eigenvalues ​​based on similar samples. The basic idea of ​​this method is to introduce a method to increase the sample size before statistical analysis, ensuring that the added samples have the same probability distribution as the experimental data while also maintaining a high degree of similarity in power spectrum; these samples are called similar samples. After obtaining a suitable number of similar samples, a non-parametric estimation method is used for statistical analysis. This method does not require assumptions about the probability distribution of flutter characteristics, effectively avoiding large errors caused by inappropriate assumptions in the probability distribution model. Attached Figure Description

[0033] Figure 1 This invention is a statistical analysis method for flutter feature values ​​of similar samples.

[0034] Figure 2 This is a comparison of the time-domain signals of the original data at a wind speed of 70 m / s with those of similar samples in this embodiment of the invention;

[0035] Figure 3 This is a comparison of the power spectrum of the original data at a wind speed of 70 m / s with that of similar samples in this embodiment of the invention;

[0036] Figure 4 This refers to the pivotal quantity of all samples arranged at a wind speed of 70 m / s in this embodiment of the invention.

[0037] Figure 5 This is a trend chart of the initial sample mean and confidence interval under the corresponding wind speed in the embodiments of the present invention. Detailed Implementation

[0038] To make the objectives, technical solutions, and effects of this invention clearer and more explicit, the following examples provide a more detailed description of the invention. It should be noted that the specific embodiments described herein are merely illustrative and not intended to limit the scope of the invention.

[0039] This invention addresses the problems of limited experimental data, especially high-quality experimental data, and the need for prior assumptions about probability distribution types in statistical analysis by proposing a statistical analysis method that combines flutter characteristic values ​​from similar samples.

[0040] To achieve the above objectives, the present invention adopts the following technical solution:

[0041] The present invention proposes a statistical analysis method for flutter eigenvalues ​​based on similar samples, including a method for obtaining similar samples, a method for obtaining flutter eigenvalues, and a method for nonparametric estimation of confidence intervals for flutter eigenvalues. The method flow is as follows: Figure 1As shown, firstly, the iterative amplitude adjustment Fourier algorithm is repeatedly used to generate similar samples for the structural response signal; after preprocessing and modal parameter identification of the similar samples, flutter eigenvalues ​​are obtained to obtain the initial samples; finally, confidence intervals are obtained by nonparametric estimation and statistical analysis is performed in conjunction with the mean of the initial samples. Specific data examples are provided below to illustrate this.

[0042] A high-quality vibration response signal under the wind speed condition of 70 m / s was selected as the original experimental data, and its time-domain plot is shown below. Figure 2 The frequency spectrum of the similar sample curves shown below is... Figure 3 The similar sample curves are shown below. First, record the sequence number of the original experimental data, which we will denote as sequence number a. Then, generate a segment of Gaussian white noise data x, with the same data length as the original experimental data. Rearrange the generated Gaussian white noise according to sequence number a to obtain a new Gaussian white noise signal y. Perform a Fourier transform on the new Gaussian white noise y and randomize its phase to obtain signal z. Record the sequence number of signal z, which we will denote as sequence number b. Finally, rearrange the original experimental data according to sequence number b to obtain preliminary data X with the same probability distribution as the original experimental data.

[0043] The preliminary data X is then iteratively processed to make its power spectrum approximate the power spectrum of the original experimental data:

[0044] Step 1: First, apply a Fourier transform to the raw experimental data and calculate its amplitude value |Y|. i ||.

[0045] Step 2: Perform a Fourier transform on the obtained preliminary data, keeping its phase unchanged while adjusting its amplitude value |X|. i |Amplitude values ​​of the Fourier transform using the original experimental data|Y i | Replace.

[0046] Step 3: Perform an inverse Fourier transform on the obtained data and rearrange it according to the original data sequence.

[0047] Step 4: Repeat steps 2 and 3 until the obtained data Z has a power spectrum similar to the original experimental data.

[0048] Repeat the above operations to obtain 1200 sets of sample data. The next step is to obtain the flutter feature value. After spectral analysis, the 1200 sets of data are subjected to bandpass filtering for noise reduction. Then, empirical mode decomposition is used to extract single modes, and the free decay signal is extracted by natural excitation technology. The modal parameters of the single-mode free decay signal extracted by natural excitation technology are then identified by matrix beam method. Finally, the flutter margin is obtained as the flutter feature value using the flutter margin method. The flutter margin has a quadratic relationship with dynamic pressure and can be predicted by polynomial fitting.

[0049] After obtaining the flutter eigenvalues, the confidence intervals of these eigenvalues ​​are estimated nonparametrically. First, the 1200 obtained flutter eigenvalues ​​are used as the initial sample. Then, random sampling with replacement is used to draw sample values ​​from the initial sample, maintaining the same sample size as the initial sample. This process is repeated to obtain 10,000 samples. The pivot values ​​of these 10,000 samples are then calculated.

[0050] First, calculate the sample mean of the initial sample. Sample standard deviation Calculate the sample mean of the remaining sample groups. Sample standard deviation Calculate pivot quantity Arrange the calculated 10,000 pivot values ​​in ascending order: w1≤w2≤…≤w 10000 The arrangement results are as follows Figure 4 As shown. Next, we calculate the confidence interval with a 95% confidence level, letting... w k1 w k2 respectively as w 0.05 / 2 w 1-0.05 / 2 The estimate yields a 95% confidence interval: (4.01 × 10⁻⁶). 11 5.00×10 11 ).

[0051] The same process was then applied to the remaining eight sets of experimental data at wind speeds of 50-130 m / s, ultimately yielding the confidence boundaries for flutter eigenvalues ​​within the 50-130 m / s wind speed range, as follows: Figure 5 As shown.

[0052] Error statistics were performed on the obtained confidence intervals and the initial sample mean, and the results are shown in Table 1. As can be seen from the error analysis in Table 1, the 95% confidence interval obtained by the flutter eigenvalue statistical analysis method based on similar samples proposed in this invention has a large deviation in the upper and lower bounds within the 50m / s-80m / s wind speed range, with a deviation of about 10% from the initial sample mean. In the 80m / s-130m / s wind speed range, the deviation of the upper and lower bounds of the 95% confidence interval shows a significant decreasing trend, and the accuracy of the confidence interval improves rapidly. This statistical analysis result indicates that the flutter margin eigenvalue tends to a stable value as wind speed increases. This is consistent with the theoretical analysis results of flutter margin as a flutter criterion, thus proving the effectiveness of the flutter eigenvalue statistical analysis method based on similar samples proposed in this invention.

[0053] Table 1. Error Analysis Table Between Confidence Intervals and Initial Sample Mean

[0054]

[0055] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements can be made without departing from the principle of the present invention, and these improvements should also be considered within the scope of protection of the present invention.

Claims

1. A statistical analysis method for flutter eigenvalues ​​based on similar samples, characterized in that, The method is as follows: Step 1: Repeatedly use the iterative amplitude adjustment Fourier algorithm to generate similar samples for the structural response signal; Step 2: After preprocessing similar samples and identifying modal parameters, flutter feature values ​​are obtained to get the initial samples; Step 3: Obtain confidence intervals from nonparametric estimates and perform statistical analysis in conjunction with the initial sample mean; The method for obtaining the flutter feature value in step two is as follows: After obtaining an appropriate amount of sample data, the flutter feature value is obtained for each group of data, and the flutter margin is selected as the flutter feature value. This is specifically divided into the following steps: 2.1 For the sample composed of all the data obtained above, wavelet transform is used for noise reduction to increase the signal-to-noise ratio and thus reduce the impact of noise on modal parameter identification. 2.

2. Empirical mode decomposition is used to decompose the preprocessed data samples. Then, natural excitation technique is used to extract the free decay signal of each decomposed single mode. Finally, the matrix bundle method is used to identify the mode parameters of the free decay signal of the single mode extracted by natural excitation technique. 2.3 Substitute the obtained modal parameters in pairs into the flutter margin formula: Among them, F M For flutter margin, ω1 and ω2 are the angular frequencies of the two modes, and ζ1 and ζ2 are the damping ratios of the two modes; The flutter margin obtained from the mode combination with a clear downward trend is selected as the flutter characteristic value.

2. The method for statistical analysis of flutter eigenvalues ​​based on similar samples according to claim 1, characterized in that, The method for obtaining similar samples in step one is as follows: For existing flutter test data, the Fourier algorithm with iterative amplitude adjustment is used multiple times to generate similar samples, thereby enriching the sample size of the flutter test data. This is specifically divided into the following steps: 1.1 First, record the sequence number of the original experimental data, apply Fourier transform to the original experimental data and calculate its amplitude value; 1.

2. Preliminary data with the same probability distribution were obtained by using the amplitude-adjusted Fourier algorithm; 1.3 Perform a Fourier transform on the obtained preliminary data, and replace the amplitude value with the amplitude value of the Fourier transform of the original experimental data while keeping the phase unchanged; 1.4 Perform an inverse Fourier transform on the data obtained in step 1.3, and rearrange the data according to the original sequence number; 1.5 Repeat steps 1.3 and 1.4 until the obtained data has a power spectrum similar to the original experimental data.

3. The statistical analysis method for flutter feature values ​​based on similar samples according to claim 1, characterized in that the nonparametric estimation of confidence intervals in step three is as follows: after obtaining the flutter feature values, the confidence intervals of the flutter feature values ​​are nonparametrically estimated. First, a set of flutter feature values ​​already obtained is used as the initial sample. Then, a random sampling method with replacement is used to extract sample values ​​from the initial sample to form a new sample. The size of the new sample is consistent with that of the initial sample. The above method is used cyclically to obtain M sets of samples. The confidence intervals estimated for all samples are calculated, specifically in the following steps: 3.1 First, calculate the sample mean of the initial sample. and sample standard deviation 3.2 Calculate the sample mean of the remaining sample groups. Sample standard deviation Calculate pivot quantity 3.3 Arrange the calculated M pivot quantities in ascending order: w1≤w2≤…≤w M ; 3.

4. Find the confidence interval with a confidence level of 1-α, let... w k1 w k2 respectively as w α / 2 w 1-α / 2 The estimate; 3.5, by The confidence interval with a confidence level of 1-α is obtained: The flutter margin confidence intervals were repeatedly obtained at multiple wind speeds. The deviation range between the initial sample mean and the confidence interval was calculated. The relationship between the confidence interval and the wind speed was statistically analyzed, and the characteristics of the flutter margin eigenvalues ​​were analyzed.

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