Gear pump tip clearance error analysis method based on chaos polynomial expansion theory

Abstract

The present application relates to a kind of gear pump top gap error analysis methods based on chaos polynomial expansion theory, including the shell inner circular face geometric uncertainty in the gear pump top gap profile is defined as the vector disturbance of random geometric parameter in left, right inner circular profile normal, based on Karhuben-Loève expansion method, the random geometric parameter in the gear pump gap profile is quantified, and gear pump working condition condition parameter is set to establish working condition parameter uncertainty polynomial chaos expansion proxy model, further by leave-one-cross validation, the prediction model capable of quantifying random geometric parameter and working condition condition parameter to the performance of the gear pump is obtained, the present application method is accurately influenced by using the uncertainty sensitivity analysis of random parameter and working condition parameter and prediction.

Description

Technical Field

[0001] This invention relates to the field of uncertainty analysis technology for aviation fuel gear pumps. Background Technology

[0002] A gear pump is a machine that converts the mechanical energy of a prime mover into the energy of a liquid, thereby increasing the potential energy, pressure energy, and kinetic energy of the liquid. The prime mover drives the gears to rotate via the pump shaft, performing work on the liquid and increasing its energy.

[0003] Figure 1 The diagram shows the internal fuel flow path of a gear pump. As can be seen, backflow occurs within the fuel flow, which is closely related to volumetric efficiency and flow pulsation coefficient. Tool path, machining accuracy, and cutting angle during manufacturing are the main causes of geometric uncertainty. Deviations in operating conditions lead to significant uncertainties in the operating status. Both geometric and operating uncertainty contribute to fluctuations in backflow, thus affecting performance parameters (volumetric efficiency and flow pulsation coefficient). With aero-engines placing increasingly stringent stability and reliability requirements on fuel pumps, it is necessary to quantify and control the impact of uncertainties on fuel gear pumps.

[0004] In the study of the impact of machining errors and operating condition fluctuations on the performance of aviation fuel gear pumps, traditional numerical modeling and simulation methods are all constructed under a defined mathematical model. They are deterministic studies conducted under defined mathematical models and boundary conditions, which cannot quantify the impact of uncertainties in geometric parameters and operating condition parameters on the performance and flow field characteristics of fuel gear pumps. Therefore, it is necessary to propose a complete method for uncertainty quantification and analysis.

[0005] Meanwhile, when constructing traditional proxy models, radial basis function (RBF) models, support vector machine (SVM) models, and Kriging models are usually used for numerical regression. However, these methods are only suitable for small, low-dimensional sample sets. When faced with a large number of flow field data samples, they cannot make approximate estimates and suffer from overfitting and high computational costs. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the prior art by providing a method for analyzing the top clearance error of a fuel gear pump based on chaotic polynomial expansion theory, which quantifies the uncertain influence of gear pump geometric parameters and operating parameters on the performance and flow field characteristics of the fuel gear pump and accurately judges the top clearance error of the gear pump.

[0007] To achieve the above objectives, the technical solution adopted by this invention is as follows: a method for analyzing the top clearance error of a gear pump based on chaotic polynomial expansion theory, comprising the following steps: Step 1: Define the geometric uncertainty of the inner circular surface of the housing within the top clearance profile of the gear pump as a vector perturbation of the random geometric parameters in the normal direction of the left and right inner circular profiles. Then, based on the Karhuben-Loève expansion method, quantify the random geometric parameters within the top clearance profile of the gear pump. Subsequently, construct a fluid domain model of the gear pump based on 3D modeling software. Meanwhile, based on the probabilistic modeling method, the operating condition parameters of the gear pump under uncertain operating conditions are modeled as a normal distribution, and the operating condition parameters fluctuate around the nominal value of the operating conditions. Step 2: Using the normal distribution of the aforementioned random geometric parameters and operating condition parameters as input, and volumetric efficiency and flow fluctuation coefficient as output, establish a polynomial chaotic expansion surrogate model for the uncertainty of the operating conditions parameters. Then, based on leave-one-out cross-validation, the error of the surrogate model is... ,but The following error conditions must be met: in This is the standard value for error judgment; Step 3: Based on the high-dimensional model and the surrogate model that satisfies the error conditions, a variance-based sensitivity analysis is performed to examine the influence of the quantized random geometric parameters and operating condition parameters on the volumetric efficiency and flow pulsation coefficient of the gear pump.

[0008] Furthermore, the step of quantifying the random geometric parameters within the top clearance profile of the gear pump in step one is as follows: Assume the random geometric parameters have zero mean and the covariance function shown in equation (1): (1), In the formula, It is the standard deviation. and It is the length of the top clearance profile of the gear pump. It is the maximum value among all gear pump tip clearance profile lengths. ; Based on the Karhuben-Loève expansion, the geometric uncertainty of the inner circular surface profiles of the left and right semi-circular shells is expressed as Equation (2): (2), In the formula, Curve coordinates The nominal coordinates at that location For random geometric parameters, It is the unit normal vector of the unit normal vector of the random perturbation on the profile of the gear pump top clearance. and These are the eigenvalues ​​and eigenfunctions of the covariance kernel, respectively. Then, the eigenvalues ​​and eigenfunctions of the covariance kernel are defined as eigenvalues ​​and eigenfunctions that satisfy equation (3): (3), Substituting equation (1) into equation (3), we can analytically obtain the eigenvalues ​​and eigenfunctions of the covariance kernel, which are expressed by equations (4) and (5) respectively: (4), (5), In the formula, , For rotational speed, The solution for the value of is shown in equation (6) of the transcendental equation: (6), As can be seen from equation (6), with The increase, Gradually decreasing, it is assumed that the characteristic value satisfying equation (7) is the one that describes the change in the gear pump tooth tip clearance profile caused by machining errors: (7), In the formula These are the eigenvalues ​​of the covariance kernel; Finally, the random geometric parameter values ​​of the left and right fluid semicircular arc profiles in the normal direction were calculated. Then there are d random geometric parameters upward from the left inner circle contour method and d random geometric parameters upward from the right inner circle contour method, that is, there are 2d random geometric parameters that together quantify the blade machining error in the top clearance contour of the gear pump.

[0009] Furthermore, by quantifying the geometric uncertainty of the inner circular surface of the semi-circular shell into vector perturbations of three random geometric parameters, the geometric uncertainty of the inner circular surfaces of the left and right semi-circular shells can be controlled collaboratively by six random geometric parameters. , , , , , This leads to at least 30 sets of random geometric parameter datasets.

[0010] Furthermore, the specific steps for setting the gear pump operating condition parameters in step one are as follows: The operating conditions include the speed, inlet pressure, and outlet pressure of the gear pump; the nominal values ​​of each operating condition are as follows: the nominal speed is 12000 rpm, the nominal inlet pressure is 0.57 MPa, and the nominal outlet pressure is 6 MPa, with a normal distribution and a coefficient of variation of 1%. The speed, inlet pressure, and outlet pressure of the gear pump always fluctuate around the nominal values.

[0011] Furthermore, the specific steps for cross-validating the proxy model using the leave-one-out method in step two are as follows: First, construct N chaotic polynomial expansion proxy models sequentially. , For the index of the proxy model, the first The training set input for each proxy model is The validation set input is At the same time, calculate the input of the validation set. The A proxy model response ; Then, the error between the model's predicted value and the actual value is calculated using equation (8), and it is determined whether the surrogate model's prediction accuracy meets the requirements: (8), In the formula, As the standard for judging errors; This represents the actual model response for the i-th sample. This represents the sample mean of the actual model. The calculation formula is: (9).

[0012] Furthermore, step two also includes an accuracy verification step for the proxy model, specifically: A graph is constructed with the actual values ​​of flow pulsation coefficient and volumetric efficiency on the horizontal axis and the zero-error line distribution of the predicted values ​​of flow pulsation coefficient and volumetric efficiency obtained by the surrogate model on the vertical axis. This graph is used to display the difference between the combined sample data points composed of the actual value coordinates and the predicted value coordinates and the zero-error reference line, thereby determining the accuracy of the chaotic polynomial expansion surrogate model. The actual values ​​of the flow pulsation coefficient and volumetric efficiency are obtained in fluid dynamics software using the gear pump fluid domain model from step one.

[0013] Furthermore, step three specifically involves: Based on the high-dimensional model, the variance-based sensitivity index and response function The high-dimensional model description expands to: (10), In the formula: (11), in For the model response function, This is the expected value; Taking the variance of both sides of equation (11) yields: (12), In the formula: (13), in For variance; Based on the decomposition of equation (13), the main effect index Total effect index Defined as: (14), and (15), in, For variance, Main effect index, The total effect index; Indicates division variable All remaining variables besides the main effect index Also known as the first-order effect index, it describes the variable For total variance Individual contribution, total effect index For variables Individual effect and with residual variables Cross-effects on the model response; The variance sensitivity index value is obtained by solving formulas (14) and (15).

[0014] Furthermore, using the aforementioned variance sensitivity index value, a sensitivity index diagram of volumetric efficiency and flow efficiency to parameters was established, with the horizontal axis representing random geometric parameters and operating condition parameters, and the vertical axis representing the main effect index and the total effect index.

[0015] Furthermore, the sensitivity index diagrams of volumetric efficiency and flow efficiency to parameters are specifically as follows: The horizontal axis consists of two geometric parameters and three operating condition parameters; the two random geometric parameters are respectively and ,in, The left semicircular contour of the control gear pump housing , and Variable composition, The right semicircular contour of the control gear pump housing , and The variables consist of three operating parameters: speed, etc. Import pressure and export pressure The vertical axis represents the main effect index and the total effect index of the volumetric efficiency sensitivity to different parameters using the Sobol parameter.

[0016] Furthermore, the housing of the gear pump is formed by a straight housing with an inlet and an outlet, and left and right semi-circular housings that are axially symmetrically arranged relative to the straight housing. Two meshing pump gears are matched and arranged in the left and right semi-circular housings and connected by the straight housing. The inner circular surfaces of the left and right semi-circular housings serve as the left and right inner circle contours.

[0017] The beneficial effects of this invention are as follows: To quantify the impact of uncertainties in geometric and operating parameters on the performance of a fuel gear pump, this invention, based on chaotic polynomial theory, comprehensively applies theoretical methods such as CFD flow field simulation models, KL expansion geometric decomposition methods, OMP orthogonal matching algorithms, and LOO cross-validation to construct a high-precision predictive surrogate model for gear pump performance. The model considers the uncertainties in geometric and operating parameters and, combined with the Sobol exponent, conducts parameter sensitivity analysis of performance parameters and flow field characteristics under multiple operating conditions. Specifically, this invention also includes the following beneficial effects: (1) After the geometric decomposition method of KL expansion is used to quantify the left and right semicircular contours on the shell symmetry plane, the geometric uncertainty is described as a random process disturbance of finite points on the contour, which is consistent with the actual manufacturing process of the shell. (2) The volumetric efficiency and flow pulsation coefficient PCE proxy model constructed in this invention predicts the true value of the validation sample with a low error level, and conducts parameter sensitivity analysis on the output performance based on the Sobol exponent. The uncertainty sensitivity analysis of the random parameters and operating parameters of this invention yielded relatively accurate results. The results show that both geometric parameters and operating parameters contribute to flow leakage and flow pulsation. Among them, the inner circle profile of the shell and the outlet pressure are the dominant factors in volumetric efficiency variation, while the rotational speed and inlet pressure have a greater impact on the flow pulsation coefficient. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the fuel flow path inside the gear pump of the present invention; Figure 2 This is a flowchart illustrating the steps of the uncertainty analysis of the random geometric parameters of the gear pump in this invention. Figure 3 These are schematic diagrams (a) and (b) of the covariance function eigenvalues ​​and eigenfunctions of the present invention. Figure 4 This is a schematic diagram of the unit normal vector of random geometric parameter perturbation on the contour line of the present invention; Figure 5 These are the random contour lines and designed contour lines of the shell of this invention; Figure 6 This is a schematic diagram of the fluid domain model of the aviation fuel gear pump of the present invention; Figure 7 This is a schematic diagram of the leave-one-out cross-validation principle of the present invention; Figure 8 This is a schematic diagram showing the distribution of predicted and actual values ​​of volumetric efficiency and flow pulsation coefficient under the design conditions of this invention. Figure 9 This is a schematic diagram of the Sobol sensitivity index of the fuel gear pump volumetric efficiency to parameters under the design conditions of this invention. Figure 10 This is a schematic diagram of the Sobol sensitivity index of the fuel gear pump flow pulsation coefficient to parameters under the design conditions of this invention. Detailed Implementation

[0019] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0020] To achieve the above objectives, the present invention provides the following specific embodiments: Example 1: As Figure 1 , Figure 2 As shown, the gear pump casing is formed by a straight casing with an inlet and an outlet, and left and right semi-circular casings that are axially symmetrically arranged relative to the straight casing. Two meshing pump gears are matched and arranged inside the left and right semi-circular casings and connected by the straight casing. The inner circular surfaces of the left and right semi-circular casings serve as the left and right inner circular contours. The inner circular surfaces of the left and right semi-circular casings serve as the semi-circular arc contours of the left and right fluid segments, so the fluid inside the pump forms an arc-shaped flow surface profile. The straight surfaces of the casing at the inlet and outlet ends serve as the straight contours of the upper and lower fluids, so the fluid inside the pump forms a straight flow surface profile.

[0021] Therefore, the gear pump top clearance error analysis method based on chaotic polynomial expansion theory includes the following steps: Step 1: Define the geometric uncertainty of the inner circular surface of the housing within the top clearance profile of the gear pump as a vector perturbation of the random geometric parameters in the normal direction of the left and right inner circular profiles. Then, based on the Karhuben-Loève expansion method, quantify the random geometric parameters within the top clearance profile of the gear pump. Subsequently, construct a fluid domain model of the gear pump based on 3D modeling software. The step of quantifying the random geometric parameters within the top clearance profile of the gear pump is as follows: Assume the random geometric parameters have zero mean and the covariance function shown in equation (1): (1), In the formula, It is the standard deviation. and It is the length of the top clearance profile of the gear pump. It is the maximum value among all gear pump tip clearance profile lengths. ; Based on the Karhuben-Loève expansion, the geometric uncertainty of the inner circular surface profiles of the left and right semi-circular shells is expressed as Equation (2): (2), In the formula, Curve coordinates The nominal coordinates at that location For random geometric parameters, It is the unit normal vector of the unit normal vector of the random perturbation on the profile of the gear pump top clearance. and These are the eigenvalues ​​and eigenfunctions of the covariance kernel, respectively. Then, the eigenvalues ​​and eigenfunctions of the covariance kernel are defined as eigenvalues ​​and eigenfunctions that satisfy equation (3): (3), Substituting equation (1) into equation (3), we can analytically obtain the eigenvalues ​​and eigenfunctions of the covariance kernel, which are expressed by equations (4) and (5) respectively: (4), (5), In the formula, , For rotational speed, The solution for the value of is shown in equation (6) of the transcendental equation: (6), As can be seen from equation (6), with The increase, Gradually decreasing, it is assumed that the characteristic value satisfying equation (7) is the one that describes the change in the gear pump tooth tip clearance profile caused by machining errors: (7), In the formula These are the eigenvalues ​​of the covariance kernel; Finally, the random geometric parameter values ​​of the left and right fluid semicircular arc profiles in the normal direction were calculated. Then there are d random geometric parameters upward from the left inner circle contour method and d random geometric parameters upward from the right inner circle contour method, that is, there are 2d random geometric parameters that together quantify the blade machining error in the top clearance contour of the gear pump.

[0022] Specifically, the geometric uncertainty of the inner circular surface of the semi-circular shell is quantified into vector perturbations of three random geometric parameters. Then, the geometric uncertainty of the inner circular surfaces of the left and right semi-circular shells can be controlled collaboratively by six random geometric parameters. , , , , , This leads to at least 30 sets of random geometric parameter datasets.

[0023] Meanwhile, this step is based on the working characteristics of the gear pump under specific operating conditions, where the fuel control system has a dynamic feedback adjustment effect on the gear pump. The speed, outlet pressure and inlet pressure of the gear pump are not constant at the nominal value, but fluctuate around it. Therefore, this invention is based on the probabilistic modeling method to model the operating condition parameters of the gear pump under uncertain operating conditions as a normal distribution, and the operating condition parameters fluctuate around the nominal value of the operating condition. Specifically, the parameters for setting the operating conditions of the gear pump are as follows: The operating conditions include the speed, inlet pressure, and outlet pressure of the gear pump; the nominal values ​​of each operating condition are as follows: the nominal speed is 12000 rpm, the nominal inlet pressure is 0.57 MPa, and the nominal outlet pressure is 6 MPa. The gear pump follows a normal distribution with a coefficient of variation of 1%, and the speed, inlet pressure, and outlet pressure of the gear pump always fluctuate around the nominal values.

[0024] Step 2: Using the normal distribution of the aforementioned random geometric parameters and operating condition parameters as input, and volumetric efficiency and flow fluctuation coefficient as output, establish a polynomial chaotic expansion surrogate model for the uncertainty of the operating conditions parameters. Then, based on leave-one-out cross-validation, the error of the surrogate model is... ,but The following error conditions must be met: in This is the standard value for error judgment; The specific steps of cross-validating the proxy model using the leave-one-out method are as follows: First, construct N chaotic polynomial expansion proxy models sequentially. , For the index of the proxy model, the first The training set input for each proxy model is The validation set input is At the same time, calculate the input of the validation set. The A proxy model response ; Then, the error between the model's predicted value and the actual value is calculated using equation (8), and it is determined whether the surrogate model's prediction accuracy meets the requirements: (8), In the formula, As the standard for judging errors; This represents the actual model response for the i-th sample. This represents the sample mean of the actual model. The calculation formula is: (9).

[0025] It also includes an accuracy verification step for the proxy model, specifically: A graph is constructed with the actual values ​​of flow pulsation coefficient and volumetric efficiency on the horizontal axis and the zero-error line distribution of the predicted values ​​of flow pulsation coefficient and volumetric efficiency obtained by the surrogate model on the vertical axis. This graph is used to display the difference between the combined sample data points composed of the actual value coordinates and the predicted value coordinates and the zero-error reference line, thereby determining the accuracy of the chaotic polynomial expansion surrogate model. The actual values ​​of the flow pulsation coefficient and volumetric efficiency are obtained in fluid dynamics software using the gear pump fluid domain model from step one.

[0026] Step 3: Based on the surrogate model and high-dimensional model that satisfy the aforementioned error conditions, a variance-based sensitivity analysis is performed to assess the impact of the quantized random geometric parameters and operating condition parameters on the volumetric efficiency and flow pulsation coefficient of the gear pump. Specifically: Based on the high-dimensional model, the variance-based sensitivity index and response function The high-dimensional model description expands to: (10), In the formula: (11), in For the model response function, This is the expected value; Taking the variance of both sides of equation (11) yields: (12), In the formula: (13), in For variance; Based on the decomposition of equation (13), the main effect index Total effect index Defined as: (14), and (15), in, For variance, Main effect index, The total effect index; Indicates division variable All remaining variables besides the main effect index Also known as the first-order effect index, it describes the variable For total variance Individual contribution, total effect index For variables Individual effect and with residual variables Cross-effects on the model response; The variance sensitivity index value was obtained by solving formulas (14) and (15). Based on the variance sensitivity index value, a sensitivity index diagram of volumetric efficiency and flow efficiency to parameters was established, with the horizontal axis representing random geometric parameters and operating condition parameters, and the vertical axis representing the main effect index and the total effect index.

[0027] The sensitivity index diagrams for volumetric efficiency and flow efficiency to parameters are as follows: The horizontal axis consists of two geometric parameters and three operating condition parameters; the two random geometric parameters are respectively and ,in, The left semicircular contour of the control gear pump housing , and Variable composition, The right semicircular contour of the control gear pump housing , and The variables consist of three operating parameters: speed, etc. Import pressure and export pressure The vertical axis represents the main effect index and the total effect index of the volumetric efficiency sensitivity to different parameters using the Sobol parameter.

[0028] Specific calculation example: Figure 3-10 As shown: 01. Given the gear pump tooth tip clearance profile, assume the random process has zero mean and the covariance function shown in equation (1): In the formula, It is the standard deviation. and It is the length of the top clearance profile of the gear pump. It is the maximum value among all gear pump tip clearance profile lengths. ; Based on the Karhuben-Loève expansion, the geometric uncertainty of the inner circular surface profiles of the left and right semi-circular shells is expressed as Equation (2): (2), In the formula, Curve coordinates The nominal coordinates at that location For random geometric parameters, It is the unit normal vector of the unit normal vector of the random perturbation on the profile of the gear pump top clearance. and These are the eigenvalues ​​and eigenfunctions of the covariance kernel, respectively. Then, the eigenvalues ​​and eigenfunctions of the covariance kernel are defined as eigenvalues ​​and eigenfunctions that satisfy equation (3): (3), Substituting equation (1) into equation (3), we can analytically obtain the eigenvalues ​​and eigenfunctions of the covariance kernel, which are expressed by equations (4) and (5) respectively: (4), (5), In the formula, , For rotational speed, The solution for the value of is shown in equation (6) of the transcendental equation: (6), From equation (6), and as Figure 3 As shown, it can be seen that with The increase, Gradually decreasing, it is assumed that the characteristic value satisfying equation (7) is the one that describes the change in the gear pump tooth tip clearance profile caused by machining errors: (7), In the formula These are the eigenvalues ​​of the covariance kernel; Finally, the random geometric parameter values ​​of the left and right fluid semicircular arc profiles in the normal direction were calculated. Then there are d random geometric parameters upward from the left inner circle contour method and d random geometric parameters upward from the right inner circle contour method, that is, there are 2d random geometric parameters that together quantify the blade machining error in the top clearance contour of the gear pump.

[0029] This calculation example quantifies the machining errors of the left and right circles of the gear pump tooth tip into vector disturbances of three random geometric parameters. In other words, the geometric uncertainty of the left and right contours of the gear pump housing can be controlled collaboratively by six random geometric parameters. , , , , , ; like Figure 4 , Figure 5 The figure shows the solution to the machining error under 50 different sets of random numbers. Simultaneously, a fluid domain model of the gear pump is constructed based on 3D modeling software. Figure 6 As shown (this image is one of 50 groups). Various 3D modeling software can be used, such as Solidworks, ProE, and UG.

[0030] Then, based on the working characteristics of aviation fuel gear pumps, under specific operating conditions, the fuel control system has a dynamic feedback regulation effect on the gear pump. The speed, inlet pressure and outlet pressure of the gear pump always fluctuate around the nominal value. The test shows that the variability of the fluctuation is basically within 1% of the nominal value. Based on this analysis, it can be assumed that the three uncertain operating condition parameters of speed, inlet pressure and outlet pressure follow a normal distribution with a mean of nominal value and a coefficient of variation of 1%, as shown in Table 1.

[0031] Table 1 Uncertainty Modeling of Fuel Gear Pump Operating Parameters Simultaneously, the fluid domain and operating conditions are imported into computational fluid dynamics (CFD) software to obtain the flow pulsation coefficient and volumetric efficiency. Various CFD software programs can be used, including CFX, Fluent, and Pumplinx.

[0032] 02. Using the aforementioned random geometric parameters and operating condition parameters as inputs, and volumetric efficiency and flow fluctuation coefficient as outputs, establish an OMP-PCE surrogate model for uncertain operating parameters using polynomial chaotic expansion. The random geometric parameters and operating condition parameters mentioned above are input datasets as shown in Table 2, and only 10 datasets are listed in Table 3: Table 2 The OMP-PCE surrogate model for uncertain operating parameters is established as shown in Table 3. The surrogate model is then stored in numerical software, which can be Matlab, C++, Python, or other similar software.

[0033] Table 3 Based on the construction of the OMP-PCE surrogate model, this study investigates cross-validation of the prediction model using the LOO method to evaluate the credibility of the trained model. Figure 7The principle of cross-validation using the LOO method is given, and the following specific method is adopted: First, construct N chaotic polynomial expansion proxy models sequentially. , For the index of the proxy model, the first The training set input for each proxy model is The validation set input is At the same time, calculate the input of the validation set. The A proxy model response ; Then, the error between the model's predicted value and the actual value is calculated using equation (8), and it is determined whether the surrogate model's prediction accuracy meets the requirements: (8), In the formula, As the standard for judging errors; This represents the actual model response for the i-th sample. This represents the sample mean of the actual model. The calculation formula is: (9); To quantify the accuracy of the surrogate model, it is necessary to construct a distribution chart of predicted and actual values ​​based on the zero error line, such as... Figure 8 As shown in the figure, the horizontal axis represents the actual values ​​obtained based on the gear pump fluid domain model, and the vertical axis represents the predicted values ​​calculated based on the OMP-PCE surrogate model. Data points represent combined sample data points composed of actual and predicted coordinates. The gray solid line is the zero-error reference line. Figure 8 As can be seen, the OMP-PCEE proxy model is well-constructed and has high accuracy.

[0034] 03. Based on the surrogate model and high-dimensional model that satisfy the aforementioned error conditions, a variance-based sensitivity analysis is conducted to examine the influence of the quantized random geometric parameters and operating condition parameters on the volumetric efficiency and flow pulsation coefficient of the gear pump, specifically as follows: Based on the high-dimensional model, the variance-based sensitivity index and response function The high-dimensional model description expands to: (10), In the formula: (11), in For the model response function, This is the expected value; Taking the variance of both sides of equation (11) yields: (12), In the formula: (13), in For variance; Based on the decomposition of equation (13), the main effect index Total effect index Defined as: (14), and (15), in, For variance, Main effect index, The total effect index; Indicates division variable All remaining variables besides the main effect index Also known as the first-order effect index, it describes the variable For total variance Individual contribution, total effect index For variables Individual effect and with residual variables Cross-effects on the model response; Based on the constructed OMP-PCE proxy model, the variance sensitivity index is obtained by solving equations (14, 15), thereby analyzing the impact of uncertainty parameters on performance (volume efficiency, flow pulsation coefficient). Figure 9 , Figure 10 The results of parameter sensitivity analysis for the volumetric efficiency and flow pulsation coefficient of the fuel gear pump under design operating conditions are presented. In the figure, the horizontal axis consists of two geometric parameters and three operating condition parameters. The two geometric parameters are... and ,in, The left semicircular contour of the control gear pump housing , and Variable composition, The right semicircular contour of the control gear pump housing , and The variables consist of three operating parameters: speed, etc. Import pressure and export pressure The vertical axis represents the Sobol sensitivity index of volumetric efficiency to different parameters, which is divided into the main effect index and the total effect index.

[0035] As shown in the figure, both geometric parameters and operating parameters contribute to the variation in volumetric efficiency. The geometric parameters of the left and right circular profiles of the shell have roughly the same impact on volumetric efficiency. Under maximum speed conditions, the sensitivity of volumetric efficiency to geometric parameters is the most dominant among all parameters.

[0036] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for analyzing the top clearance error of a gear pump based on chaotic polynomial expansion theory, characterized in that, Includes the following steps: Step 1: Define the geometric uncertainty of the inner circular surface of the housing within the top clearance profile of the gear pump as a vector perturbation of the random geometric parameters in the normal direction of the left and right inner circular profiles. Then, based on the Karhuben-Loève expansion method, quantify the random geometric parameters within the top clearance profile of the gear pump, specifically: Assume the random geometric parameters have zero mean and the covariance function shown in equation (1): (1)， In the formula, It is the standard deviation. and It is the length of the top clearance profile of the gear pump. It is the maximum value among all gear pump tip clearance profile lengths. ; Based on the Karhuben-Loève expansion, the geometric uncertainty of the inner circular surface profile of the left and right semi-circular shells is expressed as Equation (2): (2)， In the formula, Curve coordinates The nominal coordinates at that location For random geometric parameters, It is the unit normal vector of the unit normal vector of the random perturbation on the profile of the gear pump top clearance. and These are the eigenvalues ​​and eigenfunctions of the covariance kernel, respectively. Then, the eigenvalues ​​and eigenfunctions of the covariance kernel are defined as eigenvalues ​​and eigenfunctions that satisfy equation (3): (3)， Substituting equation (1) into equation (3), we can analytically obtain the eigenvalues ​​and eigenfunctions of the covariance kernel, which are expressed by equations (4) and (5) respectively: (4)， (5)， In the formula, , For rotational speed, The solution for the value of is shown in equation (6) of the transcendental equation: (6)， As can be seen from equation (6), with The increase, Gradually decreasing, it is assumed that the characteristic value satisfying equation (7) is the one that describes the change in the gear pump tooth tip clearance profile caused by machining errors: (7)， In the formula These are the eigenvalues ​​of the covariance kernel; Finally, the random geometric parameter values ​​of the left and right fluid semicircular arc profiles in the normal direction were calculated. Then there are d random geometric parameters upward from the left inner circle contour method and d random geometric parameters upward from the right inner circle contour method, that is, there are 2d random geometric parameters that together quantify the blade machining error in the top clearance contour of the gear pump. Furthermore, a fluid domain model of the gear pump is constructed based on 3D modeling software; at the same time, based on probabilistic modeling methods, the operating condition parameters of the gear pump under uncertain operating conditions are modeled as a normal distribution, and the operating condition parameters fluctuate around the nominal value of the operating conditions. Step 2: Using the normal distribution of the aforementioned random geometric parameters and operating condition parameters as input, and volumetric efficiency and flow fluctuation coefficient as output, establish a polynomial chaotic expansion surrogate model for the uncertainty of the operating conditions parameters. Then, based on leave-one-out cross-validation, the error of the surrogate model is... ,but The following error conditions must be met: ， in This is the standard value for error judgment; Step 3: Based on the high-dimensional model and the surrogate model that satisfies the error conditions, the influence of random geometric parameters and operating condition parameters on the volumetric efficiency and flow pulsation coefficient of the gear pump is quantified by variance-based sensitivity analysis. The operating conditions include the speed, inlet pressure, and outlet pressure of the gear pump; the nominal values ​​of each operating condition are as follows: the nominal speed is 12000 rpm, the nominal inlet pressure is 0.57 MPa, and the nominal outlet pressure is 6 MPa, with a normal distribution and a coefficient of variation of 1%. The speed, inlet pressure, and outlet pressure of the gear pump always fluctuate around the nominal values.

2. The gear pump top clearance error analysis method based on chaotic polynomial expansion theory as described in claim 1, characterized in that, The geometric uncertainty of the inner circular surface of the semicircular shell is quantified into vector perturbations of three random geometric parameters. Therefore, the geometric uncertainty of the inner circular surfaces of the left and right semicircular shells can be controlled collaboratively by six random geometric parameters. , , , , , This leads to at least 30 sets of random geometric parameter datasets.

3. The gear pump top clearance error analysis method based on chaotic polynomial expansion theory as described in claim 1, characterized in that, The specific steps for cross-validating the proxy model using the leave-one-out method in step two are as follows: First, construct N chaotic polynomial expansion proxy models sequentially. , For the index of the proxy model, the first The training set input for each proxy model is The validation set input is At the same time, calculate the input of the validation set. The A proxy model response ; Then, the error between the predicted value and the actual value of the surrogate model is calculated using equation (8), and it is determined whether the prediction accuracy of the surrogate model meets the requirements: (8)， In the formula, As the standard for judging errors; This represents the actual model response for the i-th sample. This represents the sample mean of the actual model. The calculation formula is: (9)。 4. The gear pump top clearance error analysis method based on chaotic polynomial expansion theory as described in claim 1, characterized in that, Step two further includes an accuracy verification step for the proxy model, specifically: A graph is constructed with the actual values ​​of flow pulsation coefficient and volumetric efficiency on the horizontal axis and the zero-error line prediction and actual values ​​of the predicted values ​​of flow pulsation coefficient and volumetric efficiency obtained by the surrogate model on the vertical axis. This graph is used to display the difference between the combined sample data points composed of the actual value coordinates and the predicted value coordinates and the zero-error reference line, thereby determining the accuracy of the chaotic polynomial expansion surrogate model. The actual values ​​of the flow pulsation coefficient and volumetric efficiency are obtained in fluid dynamics software using the gear pump fluid domain model from step one.

5. The gear pump top clearance error analysis method based on chaotic polynomial expansion theory as described in claim 1, characterized in that, Step three specifically refers to: Based on the high-dimensional model, the variance-based sensitivity index and response function The high-dimensional model description expands to: (10)， In the formula: (11)， in For the model response function, This is the expected value; Taking the variance of both sides of equation (11) yields: (12)， In the formula: (13)， in For variance; Based on the decomposition of equation (13), the main effect index Total effect index Defined as: (14)， and (15)， in, For variance, Main effect index, The total effect index; Indicates division variable All remaining variables besides the main effect index Also known as the first-order effect index, it describes the variable For total variance Individual contribution, total effect index For variables Individual effect and with residual variables Cross-effects on the model response; The variance sensitivity index value is obtained by solving formulas (14) and (15).

6. The gear pump top clearance error analysis method based on chaotic polynomial expansion theory as described in claim 5, characterized in that, Based on the aforementioned variance sensitivity index values, a sensitivity index diagram of volumetric efficiency and flow efficiency to parameters was established, with the horizontal axis representing random geometric parameters and operating condition parameters, and the vertical axis representing the main effect index and the total effect index.

7. The gear pump top clearance error analysis method based on chaotic polynomial expansion theory as described in claim 6, characterized in that, The sensitivity index diagrams of volumetric efficiency and flow efficiency to parameters are as follows: The horizontal axis consists of two geometric parameters and three operating condition parameters; the two random geometric parameters are respectively and ,in, Controlled by the left semicircular contour of the gear pump housing , and Variable composition, The right semicircular contour of the control gear pump housing , and The variables consist of three operating parameters: speed, etc. Import pressure and export pressure The vertical axis represents the main effect index and the total effect index of the volumetric efficiency sensitivity to different parameters using the Sobol parameter.

8. The gear pump top clearance error analysis method based on chaotic polynomial expansion theory as described in any one of claims 1-7, characterized in that, The housing of the gear pump is formed by a straight housing with an inlet and an outlet, and left and right semi-circular housings that are axially symmetrically arranged relative to the straight housing. Two meshing pump gears are matched and arranged in the left and right semi-circular housings and connected by the straight housing. The inner circular surfaces of the left and right semi-circular housings serve as the left and right inner circle contours.

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