Method for calculating peak factor of vortex-induced vibration based on sinusoidal superposition gaussian process probability distribution

Abstract

The application discloses a vortex-induced vibration peak factor calculation method based on a sinusoidal superposition Gaussian process probability distribution. First, based on wind tunnel test, field measurement or numerical simulation data, the correlation statistics of the cross-wind displacement are obtained, including the kurtosis and bandwidth parameters; then, the probability distribution of the cross-wind displacement is determined according to the kurtosis, the probability density function of the cross-wind amplitude is calculated, and the probability distribution function is calculated; secondly, the probability distribution function of the cross-wind displacement extreme value is calculated through the crossing rate theory, and the influence of the narrow bandwidth characteristic is considered; finally, the peak factor is calculated according to the extreme value probability distribution function. The significant effect of the method is that the peak factor calculation formula is analytical and has high precision, and is convenient for application in actual engineering.

Description

Technical Field

[0001] This invention relates to the field of vortex-induced vibration of highly flexible structures, specifically a method for calculating the peak factor of vortex-induced vibration based on the probability distribution of sinusoidal superposition Gaussian processes. Background Technology

[0002] Due to regular vortex shedding and unique nonlinear aeroelastic effects, engineering structures in the atmospheric boundary layer may experience severe crosswind vibrations. Response extrema are key parameters for wind-resistant structural design. However, in practice, response samples are always very limited, rendering the extrema unreliable and thus preventing the acquisition of accurate extrema through direct statistics. To evaluate extrema using a small sample size, methods for assessing the extrema of stochastic processes need to be developed.

[0003] The extreme values ​​of the response are expressed as the product of the root mean square (RMS) and peak factor of the response. In this sense, the RMS and peak factor of the crosswind response should be accurately evaluated; the RMS can be determined with a small number of samples, so the determination of the extreme values ​​can be transformed into the determination of the peak factor. For Gaussian processes, the extreme value distribution is determined based on the Poisson approximation of the first crossing, and the closed form of its peak factor has been derived and widely used in the wind-resistant design of flexible structures. However, due to the influence of nonlinear self-excited forces on the crosswind response, it exhibits different statistical characteristics with changes in wind speed and structural damping, and its distribution form varies between Gaussian and harmonic processes, making it a typical hardened non-Gaussian process. To date, it has been difficult to find analytical solutions for the extreme value distribution of hardened non-Gaussian processes. For the study of extreme values ​​of hardened non-Gaussian processes, the transformation process method has been widely used. Its transformation function can be determined by the statistical moments of the non-Gaussian process, or by directly curve fitting the mapping between the cumulative distribution functions of the non-Gaussian process and the basic Gaussian process. The moment-based transformation process method has a closed-form solution, but it significantly overestimates the hardened non-Gaussian process with a kurtosis close to 1.5. Direct curve fitting can better predict the peak factor, but the translation function lacks a closed-form expression, which is inconvenient for engineering applications. Therefore, it is necessary to propose a method for estimating the extrema of hardened non-Gaussian processes that simultaneously guarantees analytical expression and high accuracy. Summary of the Invention

[0004] The purpose of this invention is to provide a method for calculating the peak factor of vortex-induced vibration of highly flexible structures. This method can estimate the peak factor of the unbiased hardening non-Gaussian response of highly flexible structures in the crosswind direction. This method has the advantages of both analytical expression and high accuracy, which is beneficial for practical engineering applications.

[0005] The main technical solutions adopted are as follows:

[0006] A method for calculating the peak factor of vortex-induced vibration based on the probability distribution of a sinusoidal superposition Gaussian process is key, and its main steps are as follows:

[0007] Step 1: Obtain crosswind response data;

[0008] Obtain the crosswind displacement sample set {y(t) when a highly flexible structure undergoes vortex-induced vibration. k )}, where y(t) k ) for in t k Crosswind displacement samples collected at all times, t k Let k be the kth sampling time, where k = 1, 2, ..., N, and N is the number of samples.

[0009] Step 2: Calculate the kurtosis and bandwidth coefficient of the displacement samples;

[0010] Calculate the kurtosis value α using the following formula ① 4y :

[0011]

[0012] In formula ①:

[0013] σ y This represents the root mean square value of the crosswind displacement sample;

[0014] Calculate the bandwidth coefficient q using the following formula ②:

[0015]

[0016] In formula ②:

[0017] λ0, λ1, and λ2 are the 0th, 1st, and 2nd order spectral moments of the crosswind displacement, respectively.

[0018] Step 3: Define the crosswind displacement as a composite process of a Gaussian process and a simple harmonic process, and use its kurtosis value α. 4y Determine the ratio R of the energy contained in the harmonic process and the Gaussian process, as shown in the following formula ③:

[0019]

[0020] Step 4: Calculate the probability density function f of the amplitude according to formula ④. A (A):

[0021]

[0022] In the above formula:

[0023] A represents the amplitude;

[0024] α and β are both functions of R;

[0025] J0(αβA) is a Bessel function of the first kind with respect to the zeroth order transformation of the variable αβA;

[0026] Step 5: Calculate the cumulative probability distribution function F of the amplitude according to formula ⑤. A (A);

[0027]

[0028] In the above formula:

[0029] J n (αβA) is the nth-order transformation of the Bessel function of the first kind with respect to the variable αβA;

[0030] Step 6: Calculate the cross-wind displacement rate υ(z) according to equation ⑥:

[0031]

[0032] In the above formula:

[0033] υ(z) represents the crossing rate of the threshold z through crosswind displacement;

[0034] f s The first natural frequency of the structure in the crosswind direction;

[0035] f A (z) and F A (z) represent the values ​​of the magnitude probability density function and the cumulative probability distribution function at the threshold z, respectively;

[0036] Step 7: Calculate the probability distribution function of the extreme values, as shown in the following formula ⑦;

[0037] F max (z max )=exp{-v(z max )T},Formula ⑦,

[0038] In the above formula:

[0039] F max (z max ) for z max The probability distribution function;

[0040] z max The extreme values ​​of the crosswind displacement after normalization by the root mean square value;

[0041] υ(z max ) represents the crosswind displacement with respect to the threshold z max The crossing rate is calculated according to formula ⑥;

[0042] T represents the duration of the response;

[0043] Step 8: Calculate the peak factor g using the following formula:

[0044]

[0045] In the above formula:

[0046] F represents max (z max The inverse function of (p) is the function value of the independent variable when p is taken;

[0047] p is the quantile of the probability distribution function corresponding to the extreme value average. Attached Figure Description

[0048] Figure 1 This shows the variation of kurtosis values ​​with wind speed obtained from displacement samples.

[0049] Figure 2 This shows the variation of the bandwidth parameter with wind speed obtained from displacement samples.

[0050] Figure 3 The displacement amplitude probability density function curves obtained using the method and samples presented in this paper;

[0051] Figure 4 The displacement amplitude probability distribution function curve obtained using the method and samples presented in this paper;

[0052] Figure 5 For comparison of the displacement peak factor obtained using the method in this paper and the sample. Detailed Implementation

[0053] The present invention will be further described below with reference to the embodiments and accompanying drawings.

[0054] A method for calculating the peak factor of vortex-induced vibration based on the probability distribution of a sinusoidal superposition Gaussian process is key, and its main steps are as follows:

[0055] Step 1: Obtain crosswind response data;

[0056] Obtain the crosswind displacement sample set {y(t) when a highly flexible structure undergoes vortex-induced vibration. k )}, where y(t) k ) for in t k Crosswind displacement samples collected at all times, t k Let k be the kth sampling time, where k = 1, 2, ..., N, and N is the number of samples.

[0057] Step 2: Calculate the kurtosis and bandwidth coefficient of the displacement samples;

[0058] Calculate the kurtosis value α using the following formula ① 4y :

[0059]

[0060] In formula ①:

[0061] σ y This represents the root mean square value of the crosswind displacement sample;

[0062] Calculate the bandwidth coefficient q using the following formula ②:

[0063]

[0064] In formula ②:

[0065] λ0, λ1, and λ2 are the 0th, 1st, and 2nd order spectral moments of the crosswind displacement, respectively.

[0066] Step 3: Define the crosswind displacement as a composite process of a Gaussian process and a simple harmonic process, and use its kurtosis value α. 4y Determine the ratio R of the energy contained in the harmonic process and the Gaussian process, as shown in the following formula ③:

[0067]

[0068] Step 4: Calculate the probability density function f of the amplitude according to formula ④. A (A):

[0069]

[0070] In the above formula:

[0071] A represents the amplitude;

[0072] α and β are both functions of R;

[0073] J0(αβA) is a Bessel function of the first kind with respect to the zeroth order transformation of the variable αβA;

[0074] Step 5: Calculate the cumulative probability distribution function F of the amplitude according to formula ⑤. A (A);

[0075]

[0076] In the above formula:

[0077] J n (αβA) is the nth-order transformation of the Bessel function of the first kind with respect to the variable αβA;

[0078] Step 6: Calculate the cross-wind displacement rate υ(z) according to equation ⑥:

[0079]

[0080] In the above formula:

[0081] υ(z) represents the crossing rate of the threshold z through crosswind displacement;

[0082] f s The first natural frequency of the structure in the crosswind direction;

[0083] f A (z) and F A (z) represent the values ​​of the magnitude probability density function and the cumulative probability distribution function at the threshold z, respectively;

[0084] Step 7: Calculate the probability distribution function of the extreme values, as shown in the following formula ⑦;

[0085] F max (z max )=exp{-υ(z max )T},Formula ⑦,

[0086] In the above formula:

[0087] F max (z max ) for z max The probability distribution function;

[0088] z max The extreme values ​​of the crosswind displacement after normalization by the root mean square value;

[0089] υ(z max ) represents the crosswind displacement with respect to the threshold z max The crossing rate is calculated according to formula ⑥;

[0090] T represents the duration of the response;

[0091] Step 8: Calculate the peak factor g using the following formula:

[0092]

[0093] In the above formula:

[0094] F represents max (z max The inverse function of (p) is the function value of the independent variable when p is taken;

[0095] p is the quantile of the probability distribution function corresponding to the extreme value average.

[0096] To explain the rationality of the crosswind extreme value calculation method for highly flexible structures based on the probability distribution of sinusoidal superposition Gaussian processes, a comparative analysis is conducted below by introducing Monte Carlo simulation results of the crosswind response of the structure.

[0097] The crosswind nonlinear motion equations established based on the time-varying aerodynamic damping model are used to generate sample time histories of displacement response through Monte Carlo simulation. Parameters such as kurtosis, bandwidth, and peak factor can be obtained, thereby evaluating the accuracy of the above scheme.

[0098] Figure 1 This shows the variation of kurtosis values ​​with converted wind speed, obtained from crosswind displacement samples.

[0099] Figure 2 The invention lists the bandwidth parameters for different equivalent wind speeds calculated from displacement samples. Through this invention, the peak factor can be estimated quickly and accurately using response kurtosis and bandwidth parameters.

[0100] Figures 3-5 The results show a comparison between the probability density function, probability distribution function, and peak factor of the crosswind displacement amplitude calculated using the above methods and the measured data. The results indicate that the probability density / distribution function calculated using kurtosis agrees well with the measured values ​​and effectively reflects the non-Gaussian nature of the crosswind displacement. The method proposed in this invention can quickly and accurately estimate the peak factor of the vortex-induced vibration response.

[0101] The technical solution of this invention has the following beneficial effects:

[0102] 1. An analytical probabilistic model is provided to describe the crosswind vortex-induced vibration response exhibiting hardened non-Gaussian characteristics, providing a theoretical basis for the calculation of the crosswind peak factor;

[0103] 2. The peak factor calculation method combines the advantages of analytical expression and high accuracy, which is beneficial for practical engineering applications.

[0104] Finally, it should be noted that the above description is merely a preferred embodiment of the present invention. Those skilled in the art, under the guidance of the present invention, can make various similar representations without departing from the spirit and claims of the present invention, and such modifications all fall within the protection scope of the present invention.

Claims

1. A method for calculating peak factor of vortex-induced vibration based on sinusoidal superposition Gaussian process probability distribution, characterized in that Follow these steps: Step 1: Obtain crosswind response data; Obtain a sample set of crosswind displacements when a highly flexible structure undergoes vortex-induced vibration. ,in In order to be in Crosswind displacement samples collected at all times For the first Each sampling time, , The number of samples; Step 2: Calculate the kurtosis and bandwidth coefficient of the displacement samples; Calculate the kurtosis value using the following formula ① : Formula ①; In formula ①: This represents the root mean square value of the crosswind displacement sample; Calculate the bandwidth factor using the following formula ② : , Official formula ②; In formula ②: These are the 0th, 1st, and 2nd order spectral moments of the crosswind displacement, respectively. Step 3: Define the crosswind displacement as a composite process of a Gaussian process and a simple harmonic process, and analyze its kurtosis value. Determine the ratio of the energy contained in the harmonic process and the Gaussian process. : Official formula ③; Step 4: Calculate the probability density function of the amplitude according to formula ④. : Official ④, In the above formula: Indicates amplitude; and All are about The function; For about variables The 0th-order deformation of the first-kind Bessel function; Step 5: Calculate the cumulative probability distribution function of the amplitude according to formula ⑤. ; Official ⑤, In the above formula: For about variables of The first-order Bessel function of the first kind; Step 6: Calculate the cross-wind displacement rate according to formula 6. : Official formula ⑥; In the above formula: To adjust the threshold by crosswind displacement Crossing rate; The first natural frequency of the structure in the crosswind direction; and These represent the magnitude probability density function and the cumulative probability distribution function at the threshold, respectively. The value of ; Step 7: Calculate the probability distribution function of the extreme values; Official 7, In the above formula: for The probability distribution function; The extreme values ​​of the crosswind displacement after normalization by the root mean square value; Indicates the crosswind displacement relative to extreme values The crossing rate is calculated according to formula ⑥; The duration of the response; Step 8: Calculate the peak factor Calculate using the following formula: Official ⑧; In the above formula: express The inverse function takes the value of . The function value at that time; The quantiles of the probability distribution function corresponding to the extreme value average are given.

2. The method for calculating the peak factor of vortex-induced vibration based on the probability distribution of a sinusoidal superposition Gaussian process according to claim 1, characterized in that: In step one, the crosswind displacement sample Obtained through wind tunnel testing, field measurements, or numerical simulation.

3. The method for calculating the peak factor of vortex-induced vibration based on the probability distribution of a sinusoidal superposition Gaussian process according to claim 1, characterized in that: In step two, the 0th, 1st, and 2nd order spectral moments of the crosswind displacement are... Calculate according to the following formula; ， ； in: Displacement power spectrum; For frequency.

4. The method for calculating the peak factor of vortex-induced vibration based on the probability distribution of a sinusoidal superposition Gaussian process according to claim 1, characterized in that: In step four, ； in: Indicates angle; For the order of summation.

5. The method for calculating the peak factor of vortex-induced vibration based on the probability distribution of a sinusoidal superposition Gaussian process according to claim 4, characterized in that: In step five, 。 6. The method for calculating the peak factor of vortex-induced vibration based on the probability distribution of a sinusoidal superposition Gaussian process according to claim 1, characterized in that: In step eight, the quantiles of the probability distribution function corresponding to the extreme value average are taken as follows: .

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