An automatic judgment method for sampling frequency setting considering spectral aliasing error

Abstract

The present invention belongs to the technical field of experimental modal analysis data acquisition, and discloses an automatic judgment method for sampling frequency setting considering spectral aliasing error. By collecting the pulse excitation force and displacement response, the measured frequency response function of the structure is calculated. The modal parameters of the structure are identified from the measured frequency response function by the half-power bandwidth method, and the equivalent flexibility is calculated. The natural frequency ratio and equivalent flexibility ratio are calculated for each order of mode in turn, and the weighted equivalent flexibility ratio and weighted natural frequency ratio are formed based on their weights. The measured spectral aliasing error and theoretical spectral aliasing error of the structure are calculated respectively to judge whether the sampling frequency setting meets the requirements of data analysis. When the requirements are met, the original sampling frequency is retained; when the requirements are not met, a new sampling frequency is calculated according to the simplified equation formed by the measured spectral aliasing error. The present invention utilizes the characteristic of analogy between the measured spectral aliasing error and theoretical spectral aliasing error of a certain order of mode in the frequency response function of the structure to complete the automatic judgment of the sampling frequency setting.

Description

Technical Field

[0001] The present invention relates to the technical field of experimental modal analysis data acquisition, and particularly to an automatic judgment method for sampling frequency setting considering spectral aliasing error. Background Art

[0002] The sampling frequency is an important parameter in the measurement of the structural frequency response function. Correct sampling frequency setting can ensure the accuracy and reliability of the frequency response function measurement. If the sampling frequency is set too low, the discrete displacement response of the structure will be insufficient to reflect the waveform characteristics of the original continuous signal, leading to serious spectral aliasing errors, resulting in poor accuracy of the measured frequency response function and reduced accuracy of the structural modal parameter identification results. On the other hand, if the sampling frequency is set too high, it will occupy a large amount of storage space of the data acquisition device and the computing time of data processing, and it is also easy to introduce high-frequency noise, making the amplitude of the measured frequency response function unstable. Therefore, judging whether the sampling frequency selection is reasonable has become one of the key issues in the setting of the data acquisition system in shock and vibration tests.

[0003] Currently, laboratory tests and field tests mainly select the appropriate sampling frequency based on the work experience of the test personnel or the test results of similar systems. Among them, Wang Jigang et al. in "Introduction to Modal Test and Analysis Method" believe that the sampling frequency should be determined according to the type of the analyzed signal, the highest analysis frequency of the structure, and the hardware conditions of the acquisition device, etc. For transient data, it is usually the highest analysis frequency within the range of five to ten times. M.A. Hadianfard et al. in "Analysis of modal frequencies estimated from frequency domain decomposition method" verified through numerical examples that only by satisfying the Nyquist theorem, that is, the sampling frequency is greater than twice the maximum modal natural frequency, the identification error of the modal frequency tends to be stable. However, the sampling frequency setting method based on experience has poor applicability, specifically manifested in that for the frequency response functions of different structures, their natural frequency distributions, modal damping ratio characteristics, and peak amplitude ratios vary greatly, and cannot meet the automatic judgment requirements of sampling frequency setting.

[0004] Therefore, it is of great significance to calculate the influence of the sampling frequency on the spectral aliasing error of the structural frequency response function by using the basic equations of structural dynamics and signal analysis theory, and then propose an automatic judgment method for sampling frequency setting considering spectral aliasing error. Summary of the Invention

[0005] The present invention aims to provide an automatic judgment method for sampling frequency setting considering spectral aliasing error, which is used for the automatic judgment of sampling frequency setting of the data acquisition system in the impulse hammer excitation test, and solves the problem that the poor applicability of the sampling frequency setting method causes an increase in spectral aliasing error, and further leads to the inconsistency between the measured frequency response function of the structure and the theoretical frequency response function.

[0006] The technical solution of the present invention is as follows: An automatic judgment method for sampling frequency setting considering spectral aliasing error calculates the measured frequency response function of the structure through the collected impulse excitation force and displacement response, and identifies the modal parameters of the structure from the measured frequency response function by the half-power bandwidth method, including natural frequency, modal damping ratio and peak amplitude, and calculates the equivalent flexibility; successively select each order of mode to calculate the natural frequency ratio and equivalent flexibility ratio, and form the weighted equivalent flexibility ratio and weighted natural frequency ratio according to their weights; respectively calculate the measured spectral aliasing error and theoretical spectral aliasing error of the structure to judge whether the sampling frequency setting meets the data analysis requirements; when the requirements are met, retain the original sampling frequency; when the requirements are not met, calculate the new sampling frequency according to the simplified equation composed of the measured spectral aliasing error.

[0007] The automatic judgment method for sampling frequency setting considering spectral aliasing error specifically includes the following steps:

[0008] Step 1: Arrange N o displacement sensors at each measuring point position on the structure, perform impulse hammer excitation at a certain measuring point position, set the sampling frequency F s , and collect the time history data of the displacement response of each measuring point and the time history data of the impulse excitation force.

[0009] Step 2: Calculate the measured frequency response function of the structure through the collected data:

[0010]

[0011] In the formula, H pl (ω) is the measured frequency response function between the response measuring point p and the excitation measuring point l of the structure; the cross-power spectral density function the auto-power spectral density function The symbol "*" represents complex conjugate; M t is the number of impulse hammer excitation tests; X pr (ω) is the fast Fourier transform value of the displacement response at the response measuring point p of the structure collected for the rth time; F lr (ω) is the fast Fourier transform value of the impulse excitation force at the excitation measuring point l collected for the rth time;

[0012] Step 3: Identify the modal parameters of each order of the structure from the measured frequency response function H pl (ω) by the half-power bandwidth method, including the natural frequency f = [f1, f2,..., fN 1. The modal damping ratio ζ = [ζ1, ζ2, …, ζ N 1. The peak amplitude A = [A1, A2, …, A N 1. Where N is the total number of modal orders participating in the impulse hammer excitation vibration; the natural frequency f is the abscissa frequency corresponding to the peak amplitude A of each order of mode;

[0013] Step 4. Calculate the equivalent flexibility, natural frequency ratio, and equivalent flexibility ratio according to the modal damping ratio ζ and the peak amplitude A:

[0014] c = [c1, c2, …, c i , …, c N = [2A1ξ1, 2A2ξ2, …, 2A N ξ N

[0015] In the formula, c i represents the equivalent flexibility of the i-th order;

[0016] Calculate the natural frequency ratio and equivalent flexibility ratio of the k-th order mode:

[0017]

[0018]

[0019] In the formula, R f,ik is the ratio of the natural frequency of the k-th order mode to the natural frequency of the i-th order mode; R c,ki is the ratio of the equivalent flexibility of the i-th order to the equivalent flexibility of the k-th order;

[0020] Step 5. Calculate the weighted natural frequency ratio of the k-th order mode to be weighted:

[0021]

[0022] In the formula, min(·) represents taking the minimum value; calculate the weighted natural frequency ratio of the k-th order mode according to the weighted natural frequency ratio of the k-th order mode to be weighted:

[0023]

[0024] Calculate the weighted equivalent flexibility ratio of the k-th order mode:

[0025]

[0026] Step 6. Calculate the measured spectral aliasing error corresponding to the k-th order mode in the measured frequency response function of the structure:

[0027]

[0028] ​Step 7: Calculate the theoretical sampling frequency ratio and theoretical spectral aliasing error of the single-degree-of-freedom frequency response function corresponding to the k-th mode:

[0029]

[0030] where ζ k is the damping ratio of the k-th mode, in percentage;

[0031] Calculate the theoretical spectral aliasing error of the k-th mode:

[0032]

[0033] Step 8: When the measured spectral aliasing error of the k-th mode is less than or equal to the theoretical spectral aliasing error, i.e., where k = 1, 2, …, N, it indicates that the sampling frequency set to F s meets the basic requirements for data analysis;

[0034] Conversely, when the measured spectral aliasing error of the k-th mode is greater than the theoretical spectral aliasing error, i.e., it means that the sampling frequency set to F s does not meet the requirements, and a new sampling frequency needs to be set;

[0035] Construct a simplified quadratic equation from the measured spectral aliasing error and extract the coefficient terms:

[0036]

[0037]

[0038]

[0039] where a k is the quadratic coefficient; b k is the linear coefficient; d k is the constant term; calculate the new sampling frequency based on the coefficient terms:

[0040]

[0041] F′ s,k = ceil(α′ k f k )

[0042] where α′ k is the ratio of the new sampling frequency to the natural frequency of the k-th mode; F′ s,k is the new sampling frequency corresponding to the k-th mode; ceil(·) represents rounding up to positive infinity.

[0043] An automatic judgment device for sampling frequency setting considering spectral aliasing error, comprising:

[0044] An acquisition module, configured to obtain the time history data of the displacement response at each measurement point and the time history data of the pulse excitation force;

[0045] A memory, configured to store the time history data of the displacement response at each measurement point, the time history data of the pulse excitation force, and a computer program obtained;

[0046] A processor, configured to execute the computer program stored in the memory. When the computer program is executed, the processor is configured to:

[0047] Read the stored time history data of the displacement response and the time history data of the pulse excitation force. The time history data of the displacement response and the time history data of the pulse excitation force are collected and stored at a preset sampling frequency; calculate the measured frequency response function of the structure according to the obtained time history data of the displacement response and the time history data of the pulse excitation force; identify each order modal parameter of the structure from the measured frequency response function, and calculate and obtain the equivalent flexibility, natural frequency ratio, and equivalent flexibility ratio; calculate the weighted natural frequency ratio and the weighted equivalent flexibility ratio of the k-th order mode respectively, obtain the measured spectral aliasing error and the theoretical spectral aliasing error and make a judgment; maintain the original sampling frequency or calculate a new sampling frequency according to the judgment result.

[0048] Advantageous effects of the present invention: Utilize the characteristic of comparing the measured spectral aliasing error and the theoretical spectral aliasing error of a certain order mode in the structural frequency response function to complete the automatic judgment of the sampling frequency setting. This method makes full use of the distribution characteristics of the natural frequency, modal damping ratio, and peak amplitude of the measured frequency response function of the structure, making the judgment method for sampling frequency setting more applicable and effective. Description of the Drawings

[0049] Figure 1 Is a flowchart of the method of the present invention.

[0050] Figure 2 Is a schematic diagram of identifying structural modal parameters by the half-power bandwidth method. Detailed Embodiments

[0051] The following combines the technical solutions to clarify the implementation manners of the present invention.

[0052] An automatic judgment device for sampling frequency setting considering spectral aliasing error, comprising:

[0053] An acquisition module, configured to obtain the time history data of the displacement response at each measurement point and the time history data of the pulse excitation force;

[0054] A memory, configured to store the time history data of the displacement response at each measurement point, the time history data of the pulse excitation force, and a computer program obtained;

[0055] A processor for executing a computer program stored in the memory, which, when executed, is configured to:

[0056] Read the stored displacement response time history data and pulse excitation force time history data, which are collected and stored at a preset sampling frequency; calculate the measured frequency response function of the structure based on the obtained displacement response time history data and pulse excitation force time history data; identify the modal parameters of each order of the structure from the measured frequency response function, and calculate the equivalent flexibility, natural frequency ratio, and equivalent flexibility ratio; calculate the weighted natural frequency ratio and weighted equivalent flexibility ratio of the k-th order mode respectively, obtain the measured spectrum aliasing error and the theoretical spectrum aliasing error and make a judgment; maintain the original sampling frequency or calculate a new sampling frequency according to the judgment result.

[0057] Taking a three-degree-of-freedom structure as an example, perform an impact vibration test on the structure and automatically judge the sampling frequency setting. As Figure 1 shown, the specific implementation is as follows:

[0058] (1) Arrange 3 displacement sensors at the corresponding measuring point positions on the structure, perform pulse hammer excitation at the second measuring point position, and set the sampling frequency to 180 Hz to collect the displacement response and pulse excitation force time history data of each measuring point.

[0059] (2) Calculate the measured frequency response function of the structure from the collected data:

[0060]

[0061] (3) Use the half-power bandwidth method to identify the modal parameters of each order of the structure from the measured frequency response function H pl (ω), as Figure 2 shown, including the natural frequencies f = [f1, f2, f3] = [49.83, 55.08, 60.36] Hz and the modal damping ratios ζ = [ζ1, ζ2, ζ3] = [2.0267, 1.8513, 2.3064]×10 -2 , and the peak amplitudes

[0062] A = [A1, A2, A3] = [5.4445, 4.1723, 3.2314]×10 -4 .

[0063] (4) Calculate the equivalent flexibility according to ζ and A;

[0064] c = [c1, c2, c3] = [2.207, 1.545, 1.491]×10 -5

[0065] Then, calculate the natural frequency ratio and equivalent flexibility ratio of the third order mode:

[0066] R f,3 = [R f,13 , R f,23 , R f,33 = [1.2113, 1.0959, 1]

[0067] R c,3 = [R c,31 , R c,32 , R c,33 = [1.4805, 1.0364, 1].

[0068] (5) Calculate the 3rd - order modal natural frequency ratio R' to be weighted f,3 = 1.0959; then, calculate the weighted natural frequency ratio R'3 of the 3rd - order mode according to the above formula: R'3 = 0.5479; then calculate the weighted equivalent flexibility ratio of the 3rd - order mode:

[0069] R' c,3 = [R' c,31 , R' c,32 , R' c,33 = [2.1920, 1.0741, 1].

[0070] (6) Calculate the measured spectrum aliasing error corresponding to the 3rd - order mode in the measured frequency response function of the structure

[0071] (7) Calculate the theoretical sampling frequency ratio and theoretical spectrum aliasing error of the single - degree - of - freedom frequency response function corresponding to the 3rd - order mode:

[0072] α3 = - 5.871×10 -3 ×2.3064 2 + 1.465×10 -1 ×2.3064 + 2.058 = 2.3647

[0073] Then, calculate the theoretical spectrum aliasing error of the k - th order mode

[0074] (8) When the measured spectrum aliasing error of the 3rd - order mode is greater than the theoretical spectrum aliasing error, that is it indicates that the sampling frequency set to 180Hz does not meet the requirements and a new sampling frequency needs to be set;

[0075] Construct a simplified quadratic equation with one variable through the measured spectrum aliasing error and extract the coefficient terms:

[0076]

[0077]

[0078]

[0079] Then, calculate the new sampling frequency according to the above coefficient terms:

[0080] F′ s,3 = ceil(α′3f3) = 220Hz

[0081] Finally, F′ s,3 is the new sampling frequency corresponding to the 3rd order mode.

Claims

1. An automatic judgment method for sampling frequency setting considering spectral aliasing error, characterized in that, Calculate the measured frequency response function of the structure from the pulse excitation force and displacement response obtained by acquisition. Identify the modal parameters of the structure from the measured frequency response function by the half-power bandwidth method, and calculate the equivalent flexibility. Select each order of mode in turn to calculate the natural frequency ratio and equivalent flexibility ratio, and form the weighted equivalent flexibility ratio and weighted natural frequency ratio based on the weights of the two. Calculate the measured spectrum aliasing error and theoretical spectrum aliasing error of the structure respectively to determine whether the sampling frequency setting meets the data analysis requirements. When the requirements are met, retain the original sampling frequency. When the requirements are not met, calculate the new sampling frequency according to the simplified equation composed of the measured spectrum aliasing error.

2. The automatic judgment method for sampling frequency setting considering spectral aliasing error according to claim 1, characterized in that Specifically, it includes the following steps: Step 1: Arrange N displacement sensors at the positions of each measuring point in the structure. Apply impulse hammer excitation at a certain measuring point position, and set the sampling frequency F o to collect the time history data of the displacement responses of each measuring point and the time history data of the impulse excitation force; s ​ Step 2: Calculate the measured frequency response function of the structure from the acquired data: Where, H pl (ω) is the measured frequency response function between the structural response measurement point p and the excitation measurement point l; the cross-power spectral density function the auto-power spectral density function The symbol "*" represents the complex conjugate; M t is the number of impulse hammer excitation tests; X pr (ω) is the fast Fourier transform value of the displacement response at the structural response measurement point p collected for the r-th time; F lr (ω) is the fast Fourier transform value of the impulse excitation force at the excitation measurement point l collected for the r-th time; Step 3: Identify the modal parameters of each order of the structure from the measured frequency response function H pl (ω) by the half-power bandwidth method, including the natural frequencies f = [f1, f2, …, f N , modal damping ratios ζ = [ζ1, ζ2, …, ζ N , and peak amplitudes A = [A1, A2, …, A N ; where N is the total number of modal orders participating in the vibration excited by the impulse hammer; the natural frequency f is the abscissa frequency corresponding to the peak amplitude A of each order of the mode Step 4: Calculate the equivalent flexibility, natural frequency ratio and equivalent flexibility ratio according to the modal damping ratio ζ and peak amplitude A: c = [c1, c2, …, c i , …, c N = [2A1ξ1, 2A2ξ2, …, 2A N ξ N ​ where c i represents the i-th order equivalent flexibility; Calculate the natural frequency ratio and equivalent flexibility ratio of the k-th order mode: where R f,ik is the ratio of the k-th order modal natural frequency to the i-th order modal natural frequency; R c,ki is the ratio of the i-th order equivalent flexibility to the k-th order equivalent flexibility; Step 5: Calculate the weighted natural frequency ratio of the k-th order mode to be weighted: In the formula, min(·) represents taking the minimum value; calculate the weighted natural frequency ratio of the k-th order mode according to the weighted natural frequency ratio of the k-th order mode to be weighted: Calculate the weighted equivalent flexibility ratio of the k-th order mode: Step 6: Calculate the measured spectrum aliasing error corresponding to the k-th order mode in the measured frequency response function of the structure: Step 7: Calculate the theoretical sampling frequency ratio and theoretical spectrum aliasing error of the single-degree-of-freedom frequency response function corresponding to the k-th order mode: where ζ k is the modal damping ratio of the k-th order, in percentage; Calculate the theoretical spectrum aliasing error of the k-th order mode: Step 8. When the measured spectrum aliasing error of the k-th order mode is less than or equal to the theoretical spectrum aliasing error, i.e., where k = 1, 2, …, N, it indicates that the sampling frequency is set to F in this impulse hammer excitation test s to meet the basic requirements for data analysis; On the contrary, when the measured spectrum aliasing error of the k-th order mode is greater than the theoretical spectrum aliasing error, that is it indicates that the sampling frequency is set to F s does not meet the requirements, and a new sampling frequency needs to be set again; Form a simplified unary quadratic equation through the measured spectrum aliasing error and extract the coefficient terms: where a k is the quadratic coefficient; b k is the linear coefficient; d k is the constant term; calculate the new sampling frequency according to the coefficient terms: F′ s,k = ceil(α′ k f k ) where α′ k is the ratio of the new sampling frequency to the natural frequency of the k-th order mode; F′ s,k is the new sampling frequency corresponding to the k-th order mode; ceil(·) represents rounding up to positive infinity.

3. An automatic judgment device for sampling frequency setting considering spectral aliasing error, characterized in that Including: An acquisition module for obtaining the time history data of the displacement response and the time history data of the pulse excitation force at each measurement point; A memory for storing the time history data of the displacement response and the time history data of the pulse excitation force obtained at each measurement point and the computer program; A processor for executing the computer program stored in the memory. When the computer program is executed, the processor is used for: Read the stored time history data of the displacement response and the time history data of the pulse excitation force. The time history data of the displacement response and the time history data of the pulse excitation force are acquired and stored at a preset sampling frequency. Calculate the measured frequency response function of the structure according to the acquired time history data of the displacement response and the time history data of the pulse excitation force. Identify each order of modal parameters of the structure from the measured frequency response function, and calculate and obtain the equivalent flexibility, natural frequency ratio and equivalent flexibility ratio. Calculate the weighted natural frequency ratio and weighted equivalent flexibility ratio of the k-th order mode respectively, obtain the measured spectrum aliasing error and theoretical spectrum aliasing error and make a judgment. Keep the original sampling frequency or calculate a new sampling frequency according to the judgment result.

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