A method for sensitivity analysis of an aero-engine fuel pump sliding bearing

Abstract

The application discloses a sensitivity analysis method for an aero-engine fuel pump sliding bearing, comprising the following steps: step 1, determining random parameters affecting the bearing capacity of the sliding bearing to be analyzed, and distribution information and a probability density function of the random parameters; step 2, obtaining a data set of the random parameters of the sliding bearing and the bearing capacity thereof with a size of N t ; step 3, establishing a Gaussian process regression model to replace a real relationship between the bearing capacity of the sliding bearing to be analyzed and the random parameters; step 4, sampling the Gaussian process regression model established in step 3 to obtain a plurality of function samples; and step 5, for each function sample generated in step 4, calculating a corresponding sensitivity index sample, wherein each sensitivity index sample comprises a main index, a total index and a cross-influence index. The method efficiently solves the global sensitivity analysis problem of the existing sliding bearing.

Description

Technical Field

[0001] This invention belongs to the field of aviation fuel pump technology, and specifically relates to a sensitivity analysis method for aviation fuel pump sliding bearings. Background Technology

[0002] Sliding bearings are supporting and positioning components of aviation fuel pumps, and their random parameters affect bearing performance, further impacting the service life of the fuel pump. Therefore, sensitivity analysis of the sliding bearings in aviation engine fuel pumps to determine the main parameters affecting bearing performance is crucial.

[0003] In solving bearing reliability problems, existing techniques typically employ response surface methodology and neural network methods to establish the relationship between equivalent contact stress and function with random parameters, respectively, and calculate the sensitivity indices of the random parameter distribution parameters. Both methods calculate local sensitivity indices, reflecting the impact of fluctuations in uncertain parameters at the mean point on response performance, but failing to reflect the impact of the uncertainty of the uncertain parameters on response performance. Furthermore, both methods are specifically designed for rolling bearings. Besides using numerical methods for sensitivity analysis, sensors can be used to detect impact signals and calculate vibration transmission sensitivity. However, obtaining data through experimental methods is costly and does not consider the influence of random factors. Summary of the Invention

[0004] The purpose of this invention is to provide a sensitivity analysis method for sliding bearings in aviation fuel pumps, so as to efficiently solve the problem of global sensitivity analysis of existing sliding bearings.

[0005] This invention employs the following technical solution: a sensitivity analysis method for a sliding bearing of an aviation fuel pump, comprising the following steps:

[0006] Step 1: Determine the random parameters affecting the bearing capacity of the sliding bearing to be analyzed, as well as the distribution information and probability density function of the random parameters;

[0007] The random parameters include bearing width, gear shaft radius, radial clearance, rotational speed, medium viscosity, and eccentricity; the distribution information includes distribution type, mean, and coefficient of variation.

[0008] Step 2: Obtain a size of N t Dataset of random parameters and load-bearing capacity of sliding bearings

[0009] Based on the distribution information of the random parameters determined in step 1, N is randomly generated. t a sample set The oil film pressure distribution of the sliding bearing to be analyzed is determined for the given sample set; the bearing capacity of the sliding bearing to be analyzed is calculated based on the oil film pressure distribution.

[0010] Step 3: Establish a Gaussian process regression model To replace the actual relationship between the bearing capacity and random parameters of the sliding bearing to be analyzed, g (x) ;

[0011] Step 4: Apply the Gaussian process regression model established in Step 3. Sampling is performed to obtain multiple function samples;

[0012] Step 5: For each function sample generated in Step 4, calculate the corresponding sensitivity index sample. Each sensitivity index sample includes a main index, a total index, and a cross-influence index.

[0013] Furthermore, step 3 specifically involves:

[0014] Let y = g(x) represent the relationship between the actual bearing capacity of the sliding bearing to be analyzed and the random parameters, where g is a row vector composed of random parameters and y is the bearing capacity;

[0015] Based on the sample obtained in step 2

[0016] Suppose that y follows a Gaussian process, denoted as:

[0017]

[0018] Where m(x) is the preset prior mean, and κ(x, x′) is the preset prior covariance function of any two points x and x′;

[0019] The posterior expectation and posterior covariance function are derived as follows:

[0020]

[0021] and

[0022]

[0023] in, Let k be the covariance matrix of the sample, and let k(x, x′) be the i-th element. and function value at and They are respectively The i-th and j-th row vectors; Let be a row vector, and let κ(x, x′) be the i-th element. function value at Let k be a column vector, and let k(x, x′) be the i-th element. function value at Let m(x) be a column vector, and let m(x) be the i-th element. function value at

[0024] Furthermore, step 4 specifically involves:

[0025] S4.1. The Gaussian process regression model established in step 3 is equivalently transformed using the following formula:

[0026]

[0027] in, It is an unconditional Gaussian process model Let be a column vector, and its i-th element is exist A randomly generated sample.

[0028] S4.2. As can be seen from the above formula, by extraction The sample can be obtained Function sample:

[0029]

[0030] Furthermore, step 5 specifically involves:

[0031] S5.1 Extract matrices A and B, each with a sample size of M, from the probability density function of x. Matrices A and B are both of size M×n, where n is the number of random parameters contained in x.

[0032] S5.2, for any i-th element x of the random parameter x i Construct A by replacing the i-th column of matrix A with the i-th column of matrix B. Bi Matrix; construct B by replacing the i-th column of matrix B with the i-th column of matrix A. Ai ;

[0033] S5.3 For each sample extracted in step 4 x can be obtained from the following formula i Sensitivity samples, specifically including main indicators and overall indicators

[0034]

[0035] In the formula,

[0036]

[0037]

[0038]

[0039] In the formula, A (s) B (s) , and Let A, B, and A represent matrices respectively. Bi and B Ai The element in the s-th row; This indicates the computation of the k-th function sample. In x = A (s) The function value at that location; This indicates the computation of the k-th function sample. In x = B (s) The function value at that location; This indicates the computation of the k-th function sample. exist The function value at that location; This indicates the computation of the k-th function sample. exist The function value at that location;

[0040] S5.4, for any i-th element x of the random parameter x i and the j-th element x j Given that i ≠ j, construct matrix A by replacing the i-th and j-th columns of matrix B with the i-th and j-th columns of matrix B. Bij ; Construct B by replacing the i-th and j-th columns of matrix B with the i-th and j-th columns of matrix A. Aij ;

[0041] S5.5, For Each sample x can be obtained from the following formula i and x j A sample of cross-influence sensitivity indexes, namely, cross-influence indexes.

[0042]

[0043] in, and Let x represent the i-th element calculated in step S5.3. i and the j-th element x j The main indicator; and Represent matrix A respectively Bij and B Aij The element in the s-th row; This indicates the computation of the k-th function sample. exist The function value at that location; This indicates the computation of the k-th function sample. exist The function value at that location.

[0044] Furthermore, it also includes step 6: calculating the mean, standard deviation, and confidence interval of each of the sensitivity index samples obtained in step 5, where the mean represents the estimated value of the sensitivity index, and the standard deviation is the corresponding estimation error.

[0045] Furthermore, step 6 specifically involves:

[0046] N is obtained through step 5. g A sample of sensitivity indicators and Then, calculate the sample mean and standard deviation, i.e.:

[0047]

[0048] and

[0049] and

[0050] Where the mean represents the estimated value of sensitivity, and the standard deviation is the corresponding prediction error;

[0051] The confidence interval can be obtained from the mean and standard deviation.

[0052] The beneficial effects of this invention are as follows: This invention uses a variance-based global sensitivity index to measure the degree of influence of random variables, which can better reflect the overall influence of random variables on bearing capacity; it uses a Gaussian process regression model to establish the relationship between random variables and bearing capacity, which can not only give the estimated value at the unknown point of the random variable, but also give the estimation error; it uses the Gaussian process regression model sampling technique to extract samples of the Gaussian process regression model, which can obtain samples of the global sensitivity index, and further obtain the estimated value and confidence interval.

[0053] This method uses bearing load capacity as a bearing performance indicator and employs a global sensitivity analysis method to determine the magnitude and confidence intervals of the influence of bearing width, gear shaft radius, radial clearance, rotational speed, medium viscosity, and eccentricity on bearing load capacity. This includes primary, total, and cross-influence indicators. Based on the calculation method of this application, key influencing factors affecting bearing performance can be identified. Taking the calculation results of the embodiment as an example, bearing width, gear shaft radius, and eccentricity have the greatest impact on bearing load capacity. Therefore, in the bearing design process, these three parameters can be set as design variables, with bearing load capacity as the objective function, to find the optimal design values ​​of the parameters. It should be noted that this method remains applicable to different bearing performance indicators. Attached Figure Description

[0054] Figure 1 This is a comparison chart of the main sensitivity indices of various random parameters in embodiments of the present invention;

[0055] Figure 2 This is a comparison chart of the overall sensitivity indices of various random parameters in an embodiment of the present invention;

[0056] Figure 3 This is a comparison chart of the sensitivity cross-influence indices of various random parameters in embodiments of the present invention. Detailed Implementation

[0057] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0058] A sensitivity analysis method for a sliding bearing of an aviation fuel pump includes the following steps:

[0059] Step 1: Determine the random parameters affecting the bearing capacity of the sliding bearing to be analyzed, as well as the distribution information and probability density function of the random parameters;

[0060] The random parameters include bearing width, gear shaft radius, radial clearance, rotational speed, medium viscosity, and eccentricity; the distribution information includes distribution type, mean, and coefficient of variation.

[0061] Step 2: Obtain a size of N t Dataset of random parameters and load-bearing capacity of sliding bearings

[0062] Based on the distribution information of the random parameters determined in step 1, N is randomly generated. t a sample set The oil film pressure distribution of the sliding bearing to be analyzed is determined for the given sample set; the bearing capacity of the sliding bearing to be analyzed is calculated based on the oil film pressure distribution.

[0063] Step 3: Establish a Gaussian process regression model To replace the actual relationship between the bearing capacity and random parameters of the sliding bearing to be analyzed, g(x);

[0064] Step 4: Apply the Gaussian process regression model established in Step 3. Sampling is performed to obtain multiple function samples;

[0065] Step 5: For each function sample generated in Step 4, calculate the corresponding sensitivity index sample. Each sensitivity index sample includes a main index, a total index, and a cross-influence index.

[0066] In some embodiments, step 3 specifically includes:

[0067] Let y = g(x) represent the relationship between the actual bearing capacity of the sliding bearing to be analyzed and the random parameters, where x is a row vector composed of random parameters and y is the bearing capacity;

[0068] Based on the sample obtained in step 2

[0069] Suppose that y follows a Gaussian process, denoted as:

[0070]

[0071] in, Let be the covariance matrix of the sample, and let κ(x, x′) be the element of the sample. and function value at and They are respectively The i-th and j-th row vectors; Let k be a row vector, and let k(x, x′) be the i-th element. function value at Let k be a column vector, and let k(x, x′) be the i-th element. function value at Let m(x) be a column vector, and let m(x) be the i-th element. function value at In some embodiments, in order to calculate the global sensitivity index of random parameters, it is necessary to sample the posterior Gaussian process regression model. Generally speaking, this sample function is difficult to obtain because its posterior covariance matrix often has singularity, making it difficult to directly obtain posterior samples.

[0072] Therefore, step 4 of the present invention is specifically as follows:

[0073] S4.1. The Gaussian process regression model established in step 3 is equivalently transformed using the following formula:

[0074]

[0075] in, It is an unconditional Gaussian process model Let be a column vector, and its i-th element is exist A randomly generated sample.

[0076] S4.2. As can be seen from the above formula, by extraction The sample can be obtained Function sample:

[0077] The simulation was achieved using methods such as spectral decomposition, stochastic coherence function, and KL expansion.

[0078] In some embodiments, to globally analyze the impact of design variables on bearing capacity, a comprehensive index is needed, and variance-based model sensitivity analysis is currently the most commonly used method. The variance sensitivity index decomposes the structural response variance into each random input variable and its interaction terms to obtain the individual contribution, interaction contribution, and total contribution of each variable to the structural response variance.

[0079] Step 5 specifically involves:

[0080] S5.1 Extract matrices A and B, each with a sample size of M, from the probability density function of x. Matrices A and B are both of size M×n, where n is the number of random parameters contained in x.

[0081] S5.2, for any i-th element x of the random parameter x i Construct A by replacing the i-th column of matrix A with the i-th column of matrix B. Bi Matrix; construct B by replacing the i-th column of matrix B with the i-th column of matrix A. Ai ;

[0082] S5.3 For each sample extracted in step 4 x can be obtained from the following formula i Sensitivity samples, specifically including main indicators and overall indicators

[0083]

[0084] In the formula,

[0085]

[0086]

[0087]

[0088] In the formula, A (s) B (s) , and Let A, B, and A represent matrices respectively. Bi and B Ai The element in the s-th row; This indicates the computation of the k-th function sample. In x = A (s) The function value at that location; This indicates the computation of the k-th function sample. In x = B(s) The function value at that location; This indicates the computation of the k-th function sample. exist The function value at that location; This indicates the computation of the k-th function sample. exist The function value at that location;

[0089] S5.4, for any i-th element x of the random parameter x i and the j-th element x j Given that i ≠ j, construct matrix A by replacing the i-th and j-th columns of matrix B with the i-th and j-th columns of matrix B. Bij ; Construct B by replacing the i-th and j-th columns of matrix B with the i-th and j-th columns of matrix A. Aij ;

[0090] S5.5, For Each sample x can be obtained from the following formula i and x j A sample of cross-influence sensitivity indexes, namely, cross-influence indexes.

[0091]

[0092] in, and Let x represent the i-th element calculated in step S5.3. i and the j-th element x j The main indicator; and Represent matrix A respectively Bij and B Aij The element in the s-th row; This indicates the computation of the k-th function sample. exist The function value at that location; This indicates the computation of the k-th function sample. exist The function value at that location.

[0093] In some embodiments, the method further includes step 6: calculating the mean, standard deviation, and confidence interval of each of the sensitivity index samples obtained in step 5, wherein the mean represents the estimated value of the sensitivity index, and the standard deviation is the corresponding estimation error.

[0094] In some embodiments, step 6 specifically includes:

[0095] N is obtained through step 5. g A sample of sensitivity indicators and Then, calculate the sample mean and standard deviation, i.e.:

[0096]

[0097] and

[0098]

[0099] Where the mean represents the estimated value of sensitivity, and the standard deviation is the corresponding prediction error;

[0100] The confidence interval can be obtained from the mean and standard deviation. For example, the 90% confidence interval for sensitivity can be expressed as:

[0101] Example

[0102] 1. Simulation parameters

[0103] Taking the load-bearing capacity of a sliding bearing in a certain type of aircraft gear pump as an example, the bearing width, journal radius, radial clearance, rotational speed, viscosity, and eccentricity are random parameters, and their specific distribution information is shown in Table 1. Let x = (B, R, C, n, η, ε) be the input random parameters.

[0104] Table 2 shows the specific parameters of a certain type of sliding bearing for an aircraft gear pump.

[0105] Table 1. Distribution information of random parameters of sliding bearings in aircraft gear pumps.

[0106] Bearing width B (mm) normal distribution 37 0.02 Gear shaft radius R (mm) normal distribution 16 0.02 Radial gap C (mm) normal distribution 0.06 0.02 Rotational speed n (r / min) normal distribution 6000 0.02 Medium viscosity η (Pa·s) normal distribution <![CDATA[2.7372×10 -5 ]]> 0.02 Eccentricity ε normal distribution 0.7 0.02

[0107] Table 2 Specific parameters of sliding bearings for aircraft gear pumps

[0108] Bearing inner diameter R (mm) 32.12 Bearing thickness t (mm) 18.94 Bearing elastic modulus E (GPa) 75.8 The elastic modulus E (GPa) of the gear shaft 221 Poisson's ratio v 0.373

[0109] 2. Simulation Results

[0110] The axial direction is divided into 101 grid points, the circumferential direction into 201 grid points, and the radial direction into 101 grid points. Based on the sensitivity calculation method given in step five, the sensitivity index of the random parameters to the bearing load capacity is obtained, and the results are as follows. Figure 1 , Figure 2 and Figure 3 As shown.

[0111] 3. Results Analysis

[0112] Conclusion 1: The confidence intervals of the principal and total indices of the calculated random parameter x = (B, R, C, n, η, ε) are very small, that is, the estimation error is small, so the estimation can be considered accurate.

[0113] Conclusion 2: From Figure 1 and Figure 2 It can be seen that R and ε have a significant impact on the oil film bearing capacity, with R having the greatest impact and ε having the second greatest impact.

[0114] Conclusion 3: From Figure 3 It can be seen that there is no cross-influence between the random parameters, and the main index and the total index of the random parameters are basically the same, which proves the consistency of the results.

[0115] Conclusion 4: The above results show that the theory and method proposed in this invention can accurately and efficiently perform sensitivity analysis on sliding bearings.

[0116] In summary, this invention uses a variance-based global sensitivity index to measure the influence of random variables, which better reflects the overall impact of random variables on bearing capacity. It employs a Gaussian process regression model to establish the relationship between random variables and bearing capacity, providing not only estimates of the unknown points of the random variables but also estimation errors. Furthermore, by using Gaussian process regression model sampling techniques to extract samples from the model, samples of the global sensitivity index can be obtained, leading to further estimates and confidence intervals.

[0117] This method uses bearing load capacity as a bearing performance indicator and employs a global sensitivity analysis method to determine the magnitude and confidence intervals of the influence of bearing width, gear shaft radius, radial clearance, rotational speed, medium viscosity, and eccentricity on bearing load capacity. This includes primary, total, and cross-influence indicators. Based on the calculation method of this application, key influencing factors affecting bearing performance can be identified. Taking the calculation results of the embodiment as an example, bearing width, gear shaft radius, and eccentricity have the greatest impact on bearing load capacity. Therefore, in the bearing design process, these three parameters can be set as design variables, with bearing load capacity as the objective function, to find the optimal design values ​​of the parameters. It should be noted that this method remains applicable to different bearing performance indicators.

Claims

1. A sensitivity analysis method for a sliding bearing of an aviation fuel pump, characterized in that, Includes the following steps: Step 1: Determine the random parameters affecting the bearing capacity of the sliding bearing to be analyzed, as well as the distribution information and probability density function of the random parameters; The random parameters include bearing width, gear shaft radius, radial clearance, rotational speed, medium viscosity, and eccentricity; the distribution information includes distribution type, mean, and coefficient of variation. Step 2, obtain the size as Dataset of random parameters and load-bearing capacity of sliding bearings : Based on the distribution information of the random parameters determined in step 1, random generation is performed. a sample set The oil film pressure distribution of the sliding bearing to be analyzed is determined for the given sample set; the bearing capacity of the sliding bearing to be analyzed is calculated based on the oil film pressure distribution. ; Step 3: Establish a Gaussian process regression model To replace the actual relationship between the bearing capacity and random parameters of the sliding bearing to be analyzed. ; Step 4: Apply the Gaussian process regression model established in Step 3. Sampling was performed to obtain multiple function samples; specifically including: S4.

1. The Gaussian process regression model established in step 3 is equivalently transformed using the following formula: ， in, It is an unconditional Gaussian process model , ; Let be a column vector, and its first... The elements are exist Randomly generated samples; S4.2, from formula It can be seen that by extraction The sample can be obtained Function sample: ; Step 5: For each function sample generated in Step 4, calculate the corresponding sensitivity index sample. Each sensitivity index sample includes a main index, a total index, and a cross-influence index. The specific steps are as follows: S5.1 from The number of samples drawn in the probability density function is 1. matrix and , where the matrix and Size , for The number of random parameters included; S5.2, For random parameters any of the first element ,Will The first of the matrix Listed Matrix number Column replacement method, construct Matrix; will The first of the matrix Listed Matrix number Column replacement method, construct ; S5.3 For each sample extracted in step 4 It can be obtained through the following formula Sensitivity samples, specifically including main indicators and overall indicators : ， In the formula, ， ， ； In the formula, , , and Represent matrices respectively , , and The row element; Indicates the calculation of the first Function samples exist The function value at that location; Indicates the calculation of the first Function samples exist The function value at that location; Indicates the calculation of the first Function samples exist The function value at that location; Indicates the calculation of the first Function samples exist The function value at that location; S5.4, For random parameters any of the first element and the element ,satisfy ,Will The first of the matrix Listed Matrix number Column replacement method, construct ;Will The first of the matrix Listed Matrix number Column replacement method, construct ; S5.5, For Each sample It can be obtained through the following formula and A sample of cross-influence sensitivity indexes, namely, cross-influence indexes. : ， in, , and These represent the calculations in step S5.

3. element and the element The main indicator; and Represent matrices respectively and The row element; Indicates the calculation of the first Function samples exist The function value at that location; Indicates the calculation of the first Function samples exist The function value at that location.

2. The sensitivity analysis method for a sliding bearing of an aviation fuel pump as described in claim 1, characterized in that, Step 3 specifically involves: make This represents the relationship between the actual bearing capacity of the sliding bearing to be analyzed and the random parameters, where A row vector composed of random parameters. For bearing load capacity; in It follows a Gaussian process, denoted as: ， In the formula, The preset prior mean, For any two preset points and The prior covariance function; And based on the samples obtained in step 2 ,right The posterior expectation and posterior covariance function are derived as follows: ， and ， in, Let be the covariance matrix of the sample, and its th The elements are exist and function value at , and They are respectively The and the Row vectors; Let be a row vector, and its first row vector be the row vector. The elements are exist function value at ; Let be a column vector, and its first... The elements are exist function value at ; Let be a column vector, and its first... The elements are exist function value at .

3. The sensitivity analysis method for a sliding bearing of an aviation fuel pump as described in claim 2, characterized in that, It also includes step 6: calculating the mean, standard deviation, and confidence interval of each of the sensitivity index samples obtained in step 5, where the mean represents the estimated value of the sensitivity index and the standard deviation is the corresponding estimation error.

4. The sensitivity analysis method for a sliding bearing of an aviation fuel pump as described in claim 3, characterized in that, Step 6 specifically involves: Obtained through step 5 A sample of sensitivity indicators , and Then, calculate the sample mean and standard deviation, i.e.: ， and ， Where the mean represents the estimated value of sensitivity, and the standard deviation is the corresponding prediction error; The confidence interval can be obtained from the mean and standard deviation.

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