Response kurtosis prediction method of single-degree-of-freedom system based on super Gaussian excitation

Abstract

The invention discloses a response kurtosis prediction method for a single-degree-of-freedom system based on super Gaussian excitation, and the method comprises the following prediction steps: (1) setting a cut-off frequency for a super Gaussian excitation signal, and setting a signal component exceeding the cut-off frequency to be 0; (2) calculating a parameter x; (3) calculating a kurtosis transfer rule model; and (4) calculating to obtain the response kurtosis. The cut-off frequency is introduced, so that the frequency characteristic of the super Gaussian excitation signal is consistent with the real working condition. Parameters such as the resonant frequency, the resonant gain and the cut-off frequency of the system are integrated through the parameter x, a kurtosis transfer rule model is established, an existing simple linear fitting model is replaced, the internal function relation between the parameters and kurtosis transfer is disclosed, and the prediction error of response kurtosis is greatly reduced. By adjusting the input power spectral density PSD of the super Gaussian excitation signal, the problem of kurtosis transmission failure caused by excitation spectrum distortion is avoided, and burst characteristics of high kurtosis excitation can be effectively transmitted to a system for response.

Description

Technical Field

[0001] This invention belongs to the field of mechanical vibration and signal processing technology, specifically relating to a method for predicting the kurtosis of a single-degree-of-freedom system response based on super-Gaussian excitation. Background Technology

[0002] During product transportation, service, and testing in complex environments, mechanical structures are often subjected to random vibration loads. Traditional random vibration tests typically simulate actual working conditions by generating Gaussian random signals based on power spectral density (PSD) control, implicitly assuming that the vibration excitation follows a Gaussian distribution. However, in real-world scenarios such as vehicles traveling on complex terrain, aerospace equipment encountering airflow disturbances, ships navigating in severe sea conditions, or buildings experiencing seismic loads, the measured vibration signals often exhibit significant non-Gaussian characteristics, manifested as numerous sudden peaks and asymmetric waveforms, i.e., super-Gaussian features.

[0003] This type of ultra-Gaussian vibration excitation generates greater peak stress in mechanical structures, significantly increasing the risk of fatigue damage. If shaking table tests are still conducted using the Gaussian distribution, the probability of high-amplitude events will be severely underestimated, leading to deviations in structural fatigue life predictions and potentially causing premature structural failure or even major safety accidents. For example, when a vehicle travels on complex terrain, the ultra-Gaussian vibration excitation transmitted from the ground, amplified by the suspension, may lead to serious safety hazards such as suspension fracture.

[0004] Existing techniques generate non-Gaussian excitation signals using amplitude modulation, which are then input into a single-degree-of-freedom (SDOF) system, and the system response is calculated through numerical simulation. To analyze the kurtosis propagation behavior, existing methods decompose the excitation signal using frequency decomposition and investigate the kurtosis of the decomposed signal. With system response kurtosis Based on the relationship between them, an empirical linear fitting model was established. .

[0005] However, the existing technology has obvious defects: it does not introduce the cutoff frequency parameter, while the cutoff frequency directly determines the time correlation and suddenness of the excitation, which has a significant impact on kurtosis propagation; at the same time, the existing model does not reveal the intrinsic functional relationship between kurtosis propagation and key parameters such as cutoff frequency and system resonant gain Q, resulting in excessive error in response kurtosis prediction, which cannot meet the needs of high-precision fatigue life assessment and reliability analysis in engineering practice. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a method for predicting the kurtosis of the response of a single-degree-of-freedom system based on super-Gaussian excitation, for predicting the kurtosis of the response of a single-degree-of-freedom system excited by a super-Gaussian excitation signal.

[0007] The technical solution adopted by this invention to solve the above-mentioned technical problems is: a response kurtosis prediction method for single-degree-of-freedom systems based on super-Gaussian excitation, wherein the super-Gaussian excitation signal is used as the input signal of the single-degree-of-freedom system. The steps for predicting the kurtosis of the response of the single-degree-of-freedom system after excitation by a super-Gaussian excitation signal are as follows: (1) The cutoff frequency Ucut of the super-Gaussian excitation signal is set, and the signal components exceeding the cutoff frequency Ucut are set to 0; (2) Calculate parameter x: ; in It is the resonant frequency of a single-degree-of-freedom system; Q value is the resonant gain; (3) Calculate the kurtosis propagation law model: ; Where a, b, c, and d are model parameters, and the optimal solutions for parameters a, b, c, and d are obtained by nonlinear least squares identification based on the response kurtosis fitting curve. (4) Calculate and obtain the response kurtosis kres: ; Where K is the reference value for steady-state vibration. It is the input kurtosis of a single-degree-of-freedom system.

[0008] Preferably, the super-Gaussian excitation signal Where G(t) is a stationary Gaussian signal and u(t) is a variable amplitude modulation signal.

[0009] Preferably, the pseudo-velocity transfer function of the single-degree-of-freedom system is expressed as: ; ; In the formula, S is the Laplace variable. It is the damping coefficient.

[0010] Preferably, K=3. Greater than 3.

[0011] Preferably, in step (3), the method for obtaining the optimal solution of parameters a, b, c, and d by nonlinear least squares identification based on the response kurtosis fitting curve is as follows: (11) Generate the required super-Gaussian excitation signal according to the given input power spectral density PSD, and load the super-Gaussian excitation signal into the single-degree-of-freedom system set in the simulation system; solve the response kurtosis by digital filtering method and store it in an array; (12) Change the resonant frequency Input kurtosis The cutoff frequency Ucut and resonant gain Q are used to obtain multiple response kurtosis values, which are then stored in an array. (13) Plot the response kurtosis values ​​in the array to obtain the response kurtosis fitting curve; (14) Using a four-parameter second-order model To approximate the response kurtosis fitting curve, the Nelder-Mead simplex algorithm is used to identify the model parameters using nonlinear least squares; the optimal parameter solution is obtained automatically through the MATLAB function fminsearch iteratively.

[0012] Preferably, the optimal solution for the model parameters is: a=1.4899; b=0.5955; c=1.3705; d=0.6008.

[0013] Preferably, step (1) further includes: adjusting the input power spectral density (PSD) of the super-Gaussian excitation signal, wherein the adjustment method is as follows: (S1) Set the initial super-Gaussian excitation signal y(t); An initial super-Gaussian excitation signal y(t) is applied to a single-degree-of-freedom system; (S2) Acquire the actual excitation signal applied to the single-degree-of-freedom system; (S3) Perform power spectral density (PSD) analysis on the acquired actual excitation signal to obtain the actual excitation PSD curve of the single-degree-of-freedom system; If the actual excitation PSD curve shows a peak in the region near the resonant frequency fn, then the energy value of the initial input power spectral density PSD near the resonant frequency fn is reduced to form the adjusted input power spectral density PSD. If the actual excitation PSD curve has a valley in the range near the resonant frequency fn, then the energy value of the initial input power spectral density PSD near the resonant frequency fn is increased to form the adjusted input power spectral density PSD.

[0014] Preferably, the range near the resonant frequency fn is: fn ± a predetermined frequency range.

[0015] The beneficial effects of this invention are: the introduction of a cutoff frequency Ucut, which determines the time correlation and burstiness of the excitation, so that the frequency characteristics of the super-Gaussian excitation signal are more consistent with the actual working conditions.

[0016] By integrating parameters such as the resonant frequency fn, resonant gain Q, and cutoff frequency Ucut of the integrated system, a kurtosis transfer law model is established to replace the existing simple linear fitting model. This model reveals the intrinsic functional relationship between each parameter and kurtosis transfer, significantly reducing the prediction error of response kurtosis. Based on the super-Gaussian excitation signal and the response kurtosis prediction results, it can accurately reflect the peak stress and fatigue damage risk experienced by the mechanical structure under actual working conditions.

[0017] By adjusting the input power spectral density (PSD) of the super-Gaussian excitation signal, the problem of kurtosis propagation failure caused by excitation spectrum distortion is avoided, and the burst characteristics of high kurtosis excitation can be effectively propagated to the system response. Attached Figure Description

[0018] Figure 1 This is the response kurtosis fitting curve obtained in step (13) of the method of the present invention; Figure 2 This is a schematic diagram of the installation of the cantilever beam and the shaking table in a cantilever beam experiment; Figure 3 This is a schematic diagram of the signal acquisition equipment; Figure 4 This is a curve comparison chart of the original input PSD and the actual excitation PSD in the cantilever beam experiment; Figure 5 It is a curve comparison chart of the adjusted input PSD and the actual excitation PSD; Figure 6 It is a comparison graph of the measured response kurtosis and the response kurtosis fitting curve in multi-condition experiments; Figure 7 This is the specific procedure for solving the response kurtosis using the digital filtering method; Figure 8 It is a program that automatically iterates to obtain the optimal parameter solution using the MATLAB function fminsearch; Among them, 1. cantilever beam; 2. vibration excitation device; 3. speedometer one; 4. speedometer two; 5. signal acquisition equipment. Detailed Implementation

[0019] To make the technical solution of the present invention clearer and easier to understand, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0020] A method for predicting the kurtosis of the response of a single-degree-of-freedom system based on super-Gaussian excitation, wherein the super-Gaussian excitation signal is used as the input signal of the single-degree-of-freedom system. The steps for predicting the kurtosis of the response of the single-degree-of-freedom system after excitation by a super-Gaussian excitation signal are as follows: (1) The cutoff frequency Ucut of the super-Gaussian excitation signal is set, and the signal components exceeding the cutoff frequency Ucut are set to 0; (2) Calculate parameter x: ; in It is the resonant frequency of a single-degree-of-freedom system; Q value is the resonant gain; (3) Calculate the kurtosis propagation law model: ; Where a, b, c, and d are model parameters, and the optimal solutions for parameters a, b, c, and d are obtained by nonlinear least squares identification based on the response kurtosis fitting curve. (4) Obtain the response kurtosis kres: ; Where K is the reference value for steady-state vibration. It is the input kurtosis of a single-degree-of-freedom system.

[0021] In one optional implementation, the super-Gaussian excitation signal Where G(t) is a stationary Gaussian signal and u(t) is a variable amplitude modulation signal.

[0022] In one optional implementation, the pseudo-velocity transfer function of the single-degree-of-freedom system is expressed as: ; ; In the formula, S is the Laplace variable. It is the damping coefficient.

[0023] In one alternative implementation, K=3. Greater than 3.

[0024] In one optional implementation, the method for obtaining the optimal solution for parameters a, b, c, and d in step 3 by nonlinear least squares identification based on the response kurtosis fitting curve is as follows: (11) Generate the required super-Gaussian excitation signal based on the given input power spectral density PSD, and load the super-Gaussian excitation signal into the single-degree-of-freedom system set in the simulation system; solve the response kurtosis by digital filtering and store it in an array; see the specific program for solving the response kurtosis by digital filtering. Figure 7 .

[0025] (12) Change the resonant frequency Input kurtosis The cutoff frequency Ucut and resonant gain Q are used to obtain multiple response kurtosis values, which are then stored in an array. For example, the values ​​are 5, 6, 7, and 8; the cutoff frequency Ucut has values ​​of 3, 7, 11, and 15; and the resonant gain Q has values ​​of 10, 15, 20, and 25. The value is [10, 4000].

[0026] (13) The response kurtosis fitting curve is obtained by plotting the response kurtosis values ​​in the array. One embodiment of the fitting curve is shown in [reference needed]. Figure 1 ; (14) Using a four-parameter second-order model To approximate the response kurtosis fitting curve, the Nelder-Mead simplex algorithm is used to perform nonlinear least-squares identification of the model parameters; the optimal parameter solution is obtained automatically through iteration using the MATLAB function fminsearch, as detailed in the program below. Figure 8 In one optional implementation, the optimal solution for the model parameters is: a=1.4899; b=0.5955; c=1.3705; d=0.6008.

[0027] To verify the prediction method proposed above, a cantilever beam was selected as a single-degree-of-freedom system for the experiment.

[0028] I. Cantilever Beam Experiment Test object: A cantilever beam was selected as a single-degree-of-freedom system. Preliminary tests yielded its resonant frequency (natural frequency) fn = 23.125 Hz, resonant gain Q = 40, and damping coefficient... =0.0125.

[0029] Equipment setup: according to Figure 2 , Figure 3 The experimental setup is shown. The first end of the cantilever beam 1 is fixed to the vibration output head of the vibration excitation device of the vibration table, while the second end of the cantilever beam 1 is suspended in the air. The vibration excitation device 2 of the vibration table is installed on the vibration table base. The signal generation device generates an excitation signal and outputs it to the vibration excitation device 2 to cause the vibration output head of the vibration excitation device 2 to vibrate accordingly. The excitation signal generated by the signal generation device is connected to channel two of the signal acquisition device, and the data from channel two of the signal acquisition device is recorded as the original input signal.

[0030] An accelerometer 3 is installed at the connection between the vibration output head and the first end of the cantilever beam 1. The output signal of the accelerometer 3 is connected to channel 1 of the signal acquisition device 5, and the actual excitation signal applied to the cantilever beam is acquired through the accelerometer 3.

[0031] A second accelerometer 4 is installed at the end of the cantilever beam 1. The output signal of the second accelerometer 4 is connected to channel 3 of the signal acquisition device 5, and the system response signal of the cantilever beam 1 is acquired through the second accelerometer 4.

[0032] Excitation signal parameter settings: The super-Gaussian excitation signal is synthesized using the amplitude modulation method. The input kurtosis kinp is set to 6, 7, and 8 (all satisfying the super-Gaussian characteristic kinp>3); the cutoff frequency Ucut is set to 0.0578Hz, 0.2Hz, 0.431Hz, 0.929Hz, 2.0Hz, 4.31Hz, 9.29Hz, and 20Hz, for a total of 8 values; the steady vibration reference value K=3.

[0033] II. PSD Input and Adjustment Process Initial super-Gaussian excitation signal y(t): An initial super-Gaussian excitation signal is generated according to the set parameters and applied to the cantilever beam through a vibration excitation device.

[0034] Record the original input signal, the actual excitation signal, and the system response signal. Actual excitation PSD analysis: Perform PSD analysis on the actual excitation signal acquired through channel 0 to obtain the actual excitation PSD curve. For example... Figure 4 As shown, there is a difference between the original input PSD and the actual excitation PSD. The actual excitation PSD has a spike near the cantilever beam resonant frequency fn=23.125Hz, indicating a distortion problem due to excessive energy.

[0035] Input PSD adjustment: For the resonant frequency fn=23.125Hz and its vicinity (set to 22Hz~24.2Hz, i.e. ±5%×fn), reduce the energy value of the initial input PSD to form the adjusted input PSD.

[0036] Iterative verification: Import the adjusted input PSD into the signal generation device, repeat the above signal output and acquisition process, and analyze the actual excitation PSD again. For example... Figure 5 As shown, the actual excitation PSD curve corresponding to the adjusted input PSD remains flat and stable near the resonant frequency fn, without obvious peaks or valleys, indicating that the PSD adjustment is complete.

[0037] III. Multi-condition test If multiple single-degree-of-freedom test specimens with different natural frequencies cannot be obtained in practical applications, response kurtosis prediction under multiple working conditions can be achieved through equivalent natural frequencies. Only one cantilever beam is needed. By adjusting the cutoff frequency of the super-Gaussian signal, the cantilever beam can be simulated as a single-degree-of-freedom test specimen with other different natural frequencies. The natural frequency of this simulated single-degree-of-freedom test specimen is calculated using the equivalent natural frequency formula.

[0038] The formula for equivalent natural frequency is: ,in, It is the equivalent natural frequency of the cantilever beam. This is the cutoff frequency (15Hz) used in the simulation experiment. The measured natural frequency of the cantilever beam is 23.125Hz. It is the cutoff frequency of the measured excitation signal.

[0039] For 8 different The value (i.e., Ucut) was experimentally determined, and the corresponding x-axis parameter is recorded in Table 1. The corresponding equivalent natural frequencies are recorded in Table 2; the Q-value of the cantilever beam used in the experiment was 40.

[0040] Table 1: Table 2: like Figure 6 As shown, the experimentally measured system response kurtosis (6, 7, 8) under different input kurtosis (6, 7, 8) and different parameter x values ​​(respectively...) Figure 6 The measured response kurtosis values ​​of the discrete markers for the pink, green, and blue color blocks in the data are shown in Table 3 for the corresponding 24 sets of data. These values ​​are basically close to... Figure 1 The corresponding response kurtosis fitting curve, i.e., the measured value, is close to the response kurtosis value obtained by the prediction method of the present invention, which verifies the accuracy of the prediction method of the present invention.

[0041] The method of this invention is applicable to various single-degree-of-freedom structures such as cantilever beams, simplified automotive suspension systems, and spring-mass-damping systems, and can be widely used in vibration testing and performance evaluation in multiple fields such as vehicle transportation and aerospace.

[0042] The above description is merely a specific embodiment of the present invention. Various examples and illustrations do not constitute a limitation on the substantive content of the present invention. Those skilled in the art can modify or transform the specific embodiments described above after reading the specification without departing from the essence and scope of the invention.

Claims

1. A method for predicting the kurtosis of a single-degree-of-freedom system based on super-Gaussian excitation, wherein the super-Gaussian excitation signal is used as the input signal of the single-degree-of-freedom system, characterized in that: The steps for predicting the kurtosis of the response of the single-degree-of-freedom system after excitation by a super-Gaussian excitation signal are as follows: (1) The cutoff frequency Ucut of the super-Gaussian excitation signal is set, and the signal components exceeding the cutoff frequency Ucut are set to 0; (2) Calculate parameter x: ; in It is the resonant frequency of a single-degree-of-freedom system; Q value is the resonant gain; (3) Calculate the kurtosis propagation law model: ; Where a, b, c, and d are model parameters, and the optimal solutions for parameters a, b, c, and d are obtained by nonlinear least squares identification based on the response kurtosis fitting curve. (4) Calculate and obtain the response kurtosis kres: ； Where K is the reference value for steady-state vibration. It is the input kurtosis of a single-degree-of-freedom system.

2. The method for predicting the kurtosis of a single-degree-of-freedom system based on super-Gaussian excitation according to claim 1, characterized in that: The supergaussian excitation signal Where G(t) is a stationary Gaussian signal and u(t) is a variable amplitude modulation signal.

3. The method for predicting the kurtosis of a single-degree-of-freedom system based on super-Gaussian excitation according to claim 1, characterized in that: The pseudo-velocity transfer function of the single-degree-of-freedom system is expressed as: ； ； In the formula, S is the Laplace variable. It is the damping coefficient.

4. The method for predicting the kurtosis of a single-degree-of-freedom system based on super-Gaussian excitation according to claim 1, characterized in that: K=3, Greater than 3.

5. The method for predicting the kurtosis of a single-degree-of-freedom system based on super-Gaussian excitation according to claim 1, characterized in that: In step (3), the method for obtaining the optimal solution of parameters a, b, c, and d by nonlinear least squares identification based on the response kurtosis fitting curve is as follows: (11) Generate the required super-Gaussian excitation signal according to the given input power spectral density PSD, and load the super-Gaussian excitation signal into the single-degree-of-freedom system set in the simulation system; solve the response kurtosis by digital filtering method and store it in an array; (12) Change the resonant frequency Input kurtosis The cutoff frequency Ucut and resonant gain Q are used to obtain multiple response kurtosis values, which are then stored in an array. (13) Plot the response kurtosis values ​​in the array to obtain the response kurtosis fitting curve; (14) Using a four-parameter second-order model To approximate the response kurtosis fitting curve, the Nelder-Mead simplex algorithm is used to identify the model parameters using nonlinear least squares; the optimal parameter solution is obtained automatically through the MATLAB function fminsearch iteratively.

6. The method for predicting the kurtosis of a single-degree-of-freedom system based on super-Gaussian excitation according to claim 1, characterized in that: The optimal solution for the model parameters is: a=1.4899; b=0.5955; c=1.3705; d=0.6008.

7. The method for predicting the kurtosis of a single-degree-of-freedom system based on super-Gaussian excitation according to claim 1 or 5, characterized in that: Step (1) further includes: adjusting the input power spectral density (PSD) of the super-Gaussian excitation signal, wherein the adjustment method is as follows: (S1) Set the initial super-Gaussian excitation signal y(t); An initial super-Gaussian excitation signal y(t) is applied to a single-degree-of-freedom system; (S2) Acquire the actual excitation signal applied to the single-degree-of-freedom system; (S3) Perform power spectral density (PSD) analysis on the acquired actual excitation signal to obtain the actual excitation PSD curve of the single-degree-of-freedom system; If the actual excitation PSD curve shows a peak in the region near the resonant frequency fn, then the energy value of the initial input power spectral density PSD near the resonant frequency fn is reduced to form the adjusted input power spectral density PSD. If the actual excitation PSD curve has a valley in the range near the resonant frequency fn, then the energy value of the initial input power spectral density PSD near the resonant frequency fn is increased to form the adjusted input power spectral density PSD.

8. The method for predicting the kurtosis of a single-degree-of-freedom system based on super-Gaussian excitation according to claim 7, characterized in that: The range near the resonant frequency fn is defined as: fn ± the set frequency range.

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