A Method for Solving the Fatigue Failure Probability Function of Turbine Shafts Based on Extended Dimensional Reduction Integral Method
By extending the dimension reduction integral method, standardizing the input variables and selecting effective unit direction vectors, and using an interpolation strategy to calculate the fatigue failure probability function of the turbine shaft, the problems of high computational cost and low efficiency in traditional methods are solved, and efficient reliability optimization design is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NORTHWESTERN POLYTECHNICAL UNIV
- Filing Date
- 2023-10-10
- Publication Date
- 2026-06-30
AI Technical Summary
Traditional methods for solving the fatigue failure probability function of turbine shafts are computationally expensive and inefficient, making it difficult to achieve efficient decoupling in reliability optimization design.
We adopt an extended dimension reduction integral-based method to determine the fatigue failure limit state function by standardizing the input variables, selecting effective unit direction vectors, and using interpolation strategies and dimension reduction integrals to calculate the failure probability function value, thereby reducing the computational dimension and improving the solution efficiency.
The computational complexity of the turbine shaft fatigue failure probability function was reduced, the solution efficiency and accuracy were improved, and efficient reliability optimization design was achieved.
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Figure CN117350050B_ABST
Abstract
Description
Technical Field
[0001] This disclosure relates to the field of reliability optimization design technology, and more specifically, to a method for solving the fatigue failure probability function of a turbine shaft based on extended dimensionless integral. Background Technology
[0002] The failure probability function reflects the influence of the input variable distribution parameters on the structural failure probability. Solving the failure probability function can achieve complete decoupling in reliability optimization design, thereby improving the solution efficiency of reliability optimization design. The turbine shaft structure and its load are complex. The traditional two-layer method for solving the failure probability function requires repeated reliability analysis under different distribution parameters, which is computationally costly and inefficient.
[0003] It should be noted that the information disclosed in the background section above is only used to enhance the understanding of the background of this disclosure, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention
[0004] The purpose of this disclosure is to overcome the shortcomings of the prior art and provide a method for solving the fatigue failure probability function of turbine shaft based on extended dimension reduction integral, which can reduce computational cost and improve computational efficiency.
[0005] According to one aspect of this disclosure, a method for solving the fatigue failure probability function of a turbine shaft based on extended dimensionless integration is provided, including:
[0006] The input variables affecting the fatigue reliability of the turbine shaft are normalized under the first distribution parameters, and the fatigue failure limit state function corresponding to the turbine shaft in the standard normal space is determined.
[0007] Multiple uniformly distributed unit direction vectors are obtained in the standard normal space, and the first limit state surface of fatigue failure of the turbine shaft in the standard normal space is determined according to the fatigue failure limit state function.
[0008] The unit direction vector in the standard normal space corresponding to the first distribution parameter is filtered to obtain the initial sample on the first limiting state surface and the effective unit direction vector corresponding to the initial sample;
[0009] The initial sample is transformed to the second limiting state surface in the standard normal space corresponding to the second distribution parameter, wherein the second distribution parameter is located in the neighborhood of the first distribution parameter;
[0010] The distance between the origin of the coordinate system in the standard normal space corresponding to the second distribution parameter and the second limiting state surface along each effective unit direction vector is determined by using an interpolation strategy, and the distance is defined as the target distance.
[0011] The failure probability function value corresponding to the second distribution parameter is determined by using dimensionality reduction integral based on the target distance.
[0012] In one exemplary embodiment of this disclosure, the input variables affecting the fatigue reliability of the turbine shaft include the vent radius, the inner diameter of the splined turbine shaft, the maximum operating speed, the cruise operating speed, the idle operating speed, the elastic modulus, the Poisson's ratio, and the fatigue life dispersion parameter.
[0013] In one exemplary embodiment of this disclosure, the input variables affecting the fatigue reliability of the turbine shaft are standardized and normalized under a first distribution parameter, and the fatigue failure limit state function corresponding to the turbine shaft in the standardized normal space is determined, including:
[0014] Determine the fatigue failure limit state function of the turbine shaft in the original physical space;
[0015] The input variables affecting the fatigue reliability of the turbine shaft are normalized under the first distribution parameters to obtain standard normal input variables;
[0016] The fatigue failure limit state function of the turbine shaft in the standard normal space is determined based on the standard normal input variables and the fatigue failure limit state function of the turbine shaft in the original physical space.
[0017] In one exemplary embodiment of this disclosure, the fatigue failure limit state function of the turbine shaft in the original physical space is:
[0018]
[0019] Wherein, the N f (x) represents the cumulative damage fatigue life of the turbine shaft under multi-stage cyclic loads; This is the threshold for the low-cycle fatigue life of the turbine shaft.
[0020] In one exemplary embodiment of this disclosure, the first limit state surface is the surface corresponding to the value of the fatigue failure limit state function being 0 in the standard normal space corresponding to the first distribution parameter.
[0021] In one exemplary embodiment of this disclosure, the second limit state surface is the surface corresponding to the value of the fatigue failure limit state function being 0 in the standard normal space corresponding to the second distribution parameter.
[0022] In an exemplary embodiment of this disclosure, filtering is performed on each unit direction vector in the standard normal space corresponding to the first distribution parameter to obtain an initial sample on the first limiting state surface and an effective unit direction vector corresponding to the initial sample, including:
[0023] Calculate the limiting distance where the one-dimensional failure probability is 0 along the unit direction vector;
[0024] Each unit direction vector from the origin of the coordinate system along the unit direction vector to the first limit state surface is selected if the distance is greater than 0 and less than the limit distance. Each selected unit direction vector is used as a valid unit direction vector. The sample on the first limit state surface corresponding to each valid unit direction vector is used as an initial sample. The initial sample includes the distance information from the origin of the coordinate system along the valid unit direction vector to the first limit state surface.
[0025] In one exemplary embodiment of this disclosure, the limiting distance where the one-dimensional failure probability of the unit direction vector is 0 is calculated according to a first calculation formula, wherein the first calculation formula is:
[0026]
[0027] in, For n-degree-of-freedom chi-square distribution χ 2 The cumulative distribution function of (n); r lim Let be the limit distance where the one-dimensional failure probability along the unit direction vector is 0.
[0028] In one exemplary embodiment of this disclosure, determining the failure probability function value corresponding to the second distribution parameter based on the target distance using dimensionality reduction integral includes calculating the failure probability function value corresponding to the second distribution parameter using a second calculation formula, wherein the second calculation formula is:
[0029]
[0030] in, The value is the failure probability function; M is the number of effective unit direction vectors; For n-degree-of-freedom chi-square distribution χ 2 The cumulative distribution function of (n); r θ (a k (k = 1, 2, ..., M) represents the target distance.
[0031] In one exemplary embodiment of this disclosure, the solution method further includes:
[0032] The fatigue failure probability function of the turbine shaft is solved by a two-layer Monte Carlo simulation method, and the solution obtained by the two-layer Monte Carlo simulation method is compared with the failure probability function value corresponding to the second distribution parameter determined by dimension reduction integral based on the target distance.
[0033] This disclosed method for solving the fatigue failure probability function of turbine shafts based on extended dimension-reduction integrals firstly transforms the input variables into standard normal distributions with the same mean and variance by standardizing the input variables under different distribution parameters. This facilitates the transformation of samples in the standard normal space corresponding to different distribution parameters and the application of dimension-reduction integrals. Secondly, by transforming the initial samples from the first distribution parameter to the second limit state surface in the standard normal space corresponding to the second distribution parameter, the failure probability function value of the samples under the second distribution parameter can be accurately estimated. Simultaneously, the effective unit direction vector of the target distance determined by the interpolation strategy is used to ensure the accuracy of the calculated failure probability function value. Finally, the dimension-reduction integral method reduces the dimension of the input variable space of the high-dimensional problem by one dimension, thereby reducing computational complexity and improving the efficiency of solving the failure probability function value.
[0034] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and are not intended to limit this disclosure. Attached Figure Description
[0035] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this disclosure and, together with the description, serve to explain the principles of this disclosure. It is obvious that the drawings described below are merely some embodiments of this disclosure, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.
[0036] Figure 1 This is a flowchart of the method for solving the fatigue failure probability function of the turbine shaft based on extended dimensionless integral in the embodiments of this disclosure.
[0037] Figure 2 This is a schematic diagram of the turbine shaft in the embodiment of this disclosure.
[0038] Figure 3 This is a schematic diagram illustrating the transformation relationship of samples on the fatigue limit state surface of the turbine shaft in the standard normal space corresponding to different distribution parameters in an embodiment of this disclosure.
[0039] Figure 4 This is a schematic diagram illustrating the processing of sample information on the fatigue limit state surface of the turbine shaft using an interpolation strategy in an embodiment of this disclosure.
[0040] Figure 5This diagram illustrates a comparison between the failure probability function calculated by dimension reduction integral for the mean vent radius in this embodiment and the solution obtained by the two-layer Monte Carlo method.
[0041] Figure 6 This diagram illustrates a comparison between the failure probability function calculated by dimension reduction integration to determine the mean inner diameter of the spline-end turbine shaft in this embodiment and the solution obtained by the double-layer Monte Carlo method.
[0042] In the diagram: 1. Spline; 2. Spindle; 3. Flange. Detailed Implementation
[0043] Example embodiments will now be described more fully with reference to the accompanying drawings. However, example embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided to make this disclosure more comprehensive and complete, and to fully convey the concept of the example embodiments to those skilled in the art. The described features, structures, or characteristics can be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided to give a full understanding of embodiments of this disclosure. However, those skilled in the art will recognize that the technical solutions of this disclosure can be practiced with one or more of the specific details omitted, or other methods, components, apparatus, steps, etc., can be employed. In other instances, well-known technical solutions are not shown or described in detail to avoid obscuring various aspects of this disclosure.
[0044] Furthermore, the accompanying drawings are merely illustrative of this disclosure and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities may be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.
[0045] The terms "the" and "the" are used to indicate the existence of one or more elements / components / etc.; the term "including" is used to indicate an open-ended inclusion and means that there may be other elements / components / etc. in addition to the listed elements / components / etc. The terms "first" and "second" are used only as markers and are not a limitation on the number of objects.
[0046] The fatigue failure probability function solution method of this exemplary embodiment can be implemented by a server, that is, the server can execute each step of the solution method described below. In this case, the apparatus and modules corresponding to the solution method can be configured in the server. It should be understood that terminal devices (e.g., computers, mobile phones, etc.) can also implement the steps of the method described below, and the corresponding apparatus and modules can be configured in the terminal device.
[0047] This disclosure provides a method for solving the fatigue failure probability function of a turbine shaft based on extended dimensionless integral. Figure 1 The flowchart of the method for solving the turbine shaft fatigue failure probability function based on extended dimensionless integration disclosed herein is shown, as follows: Figure 1 As shown, the solution method includes steps S110-S160, wherein:
[0048] Step S110: Standardize the input variables affecting the fatigue reliability of the turbine shaft under the first distribution parameters, and determine the fatigue failure limit state function of the turbine shaft in the standard normal space.
[0049] Step S120: Obtain multiple uniformly distributed unit direction vectors in the standard normal space, and determine the first limit state surface of fatigue failure of the turbine shaft in the standard normal space according to the fatigue failure limit state function.
[0050] Step S130: Filter the unit direction vector in the standard normal space corresponding to the first distribution parameter to obtain the initial sample on the first limiting state surface and the effective unit direction vector corresponding to the initial sample.
[0051] Step S140: Transform the initial sample to the second limiting state surface in the standard normal space corresponding to the second distribution parameter, wherein the second distribution parameter is located in the neighborhood of the first distribution parameter;
[0052] Step S150: Use an interpolation strategy to determine the distance between the origin of the coordinate system in the standard normal space corresponding to the second distribution parameter and the second limiting state surface along each effective unit direction vector, and define the distance as the target distance;
[0053] Step S160: Use dimensionality reduction integral to determine the failure probability function value corresponding to the second distribution parameter based on the target distance.
[0054] This disclosed method for solving the fatigue failure probability function of turbine shafts based on extended dimension-reduction integrals firstly transforms the input variables into standard normal distributions with the same mean and variance by standardizing the input variables under different distribution parameters. This facilitates the transformation of samples in the standard normal space corresponding to different distribution parameters and the application of dimension-reduction integrals. Secondly, by transforming the initial samples from the first distribution parameter to the second limit state surface in the standard normal space corresponding to the second distribution parameter, the failure probability function value of the samples under the second distribution parameter can be accurately estimated. Simultaneously, the effective unit direction vector of the target distance determined by the interpolation strategy is used to ensure the accuracy of the calculated failure probability function value. Finally, the dimension-reduction integral method reduces the dimension of the input variable space of the high-dimensional problem by one dimension, thereby reducing computational complexity and improving the efficiency of solving the failure probability function value.
[0055] like Figure 2 As shown, the turbine shaft structure may include: a main shaft 2, a spline 1, and a flange 3, etc., which can be made of stainless steel, metal, or alloy materials. Of course, it can also be made of other materials with high rigidity, without special limitation. For example, it can be a turbine shaft structure made of high-temperature alloy (e.g., its model can be GH4169). With the increase of service time, the turbine shaft structure is prone to fatigue or damage under load and constraint, which can lead to partial structural failure and affect the normal operation of the engine. Therefore, it is very necessary to accurately predict the failure probability and reliability of the turbine shaft structure.
[0056] like Figure 1 As shown, in step S110, the input variables affecting the fatigue reliability of the turbine shaft are normalized under the first distribution parameters, and the fatigue failure limit state function corresponding to the turbine shaft in the standard normal space is determined.
[0057] The turbine shaft is a critical component of an aircraft, and low-cycle fatigue failure can occur under various operational profiles. In this disclosure, the aircraft operational profiles primarily include "start-maximum-start," "idle-maximum-idle," and "cruise-maximum-cruise." Under the "start-maximum-start" operational profile, the turbine shaft speed (in revolutions per minute) is 0-1124-0, with 1128 cycles; under the "idle-maximum-idle" operational profile, the turbine shaft speed (in revolutions per minute) is 360-1124-360, with 1805 cycles; and under the "cruise-maximum-cruise" operational profile, the turbine shaft speed (in revolutions per minute) is 1006-1124-1006, with 23046 cycles.
[0058] In some embodiments of this disclosure, finite element static strength analysis can be performed on the turbine shaft structure to determine the test components of the turbine shaft. Through analysis, it can be concluded that:
[0059] Under the "Start-Max-Start" profile, the turbine shaft cycle ratio is 1, the stress (in MPa) is 0-952.318-0, and the strain (in percentage) is 0-0.005298-0; under the "Idle-Max-Idle" profile, the turbine shaft cycle ratio is 1.6002, the stress (in MPa) is 102.771-952.318-102.771, and the strain... The stress (in megapascals) is 0.000570-0.005298-0.000570; under the "cruise-maximum-cruise" profile mission, the turbine shaft cycle ratio is 20.4309, the stress (in megapascals) is 871.968-952.318-871.968, and the strain (in percentages) is 0.004852-0.005298-0.004852.
[0060] According to the finite element analysis results, the stress and strain at the turbine shaft spline vent are the greatest under any task conditions. Therefore, this part can be used as the fatigue reliability test part.
[0061] According to Miner's linear fatigue cumulative damage theory, the fatigue cumulative damage of the turbine shaft within a composite load block is shown in equation (I).
[0062]
[0063] Where n1, n2, and n3 are the ratios of the number of iterations, and n1 = 1128 / 1128, n2 = 1805 / 1128, n3 = 23046 / 1128; N f1 Fatigue life under the "start-maximum-start" profile; N f2 Fatigue life under the "slow-maximum-slow" profile task; N f3 The fatigue life is defined as the "cruise-maximum-cruise" profile.
[0064] The Manson-Coffin formula was used as the prediction model for the low-cycle fatigue life of the turbine shaft. Low-cycle fatigue performance tests were conducted based on the specific materials of the turbine shaft. The fatigue life calculation formulas for the turbine shaft under three different mission profiles, derived from the experimental data, are shown in equations (II) to (V).
[0065]
[0066]
[0067]
[0068] Where, ε ZD ε represents the maximum strain at the turbine shaft spline vent hole under maximum conditions. MC ε represents the maximum strain at the turbine shaft spline vent hole under idle conditions. XH σ represents the maximum strain at the turbine shaft spline vent hole during cruise operation. ZD ; represents the maximum stress at the turbine shaft spline vent hole under maximum conditions; σ MC The maximum stress at the turbine shaft spline vent hole under slow speed conditions; σ XH denoted as the maximum stress at the turbine shaft spline vent hole during cruise mode; u represents the parameters of the life prediction model.
[0069] N can be calculated from equations (II) to (V) respectively. f1 N f2 and N f3 and N f1 N f2 and N f3 Substituting into equation (I), and based on the reciprocal relationship between fatigue life and damage, the probabilistic fatigue life N of the turbine shaft can be obtained. f :
[0070]
[0071] In an exemplary embodiment of this disclosure, standardizing the input variables affecting the fatigue reliability of the turbine shaft under the first distribution parameters and determining the fatigue failure limit state function corresponding to the turbine shaft in the standard normal space (i.e., step S110) may include steps S210-S230, wherein:
[0072] Step S210: Determine the fatigue failure limit state function of the turbine shaft in the original physical space.
[0073] By conducting reliability analysis on the low-cycle fatigue life of the turbine shaft under the influence of various random input variables, its fatigue failure limit state function in the original physical space can be:
[0074]
[0075] Where, N f (x) represents the cumulative damage fatigue life of the turbine shaft under multi-stage cyclic loads as shown in Equation (IV); is the threshold for the low-cycle fatigue life of the turbine shaft. x is a random input variable that affects the reliability of the low-cycle fatigue life of the turbine shaft.
[0076] In one exemplary embodiment of this disclosure, the input variables affecting the fatigue reliability of the turbine shaft may be multiple, such as the vent radius, the inner diameter of the splined turbine shaft, the maximum operating speed, the cruise operating speed, the idle operating speed, the elastic modulus, Poisson's ratio, fatigue life dispersion parameters, etc.
[0077] Step S220: Under the first distribution parameters, the input variables affecting the fatigue reliability of the turbine shaft are standardized and normalized to obtain standardized normal input variables.
[0078] The input variables may follow a normal distribution; the first distribution parameter may include the mean and standard deviation. Of course, this disclosure may also include input variables following other distributions, which are not specifically limited here. After standardizing the input variables affecting the fatigue reliability of the turbine shaft under the first distribution parameter, the mean μ1 corresponding to the vent radius r1 (mm) is 0.85, and the standard deviation σ1 is 0.00255; the mean μ2 corresponding to the spline end turbine shaft inner diameter r2 (mm) is 20, and the standard deviation σ2 is 0.004; the maximum state speed n... ZD The mean μ3 corresponding to (rad / s) is 1124, and the standard deviation σ3 is 33.72; the cruising speed n XH The mean μ4 corresponding to (rad / s) is 1006, and the standard deviation σ4 is 30.18; the idle speed n MC The mean μ5 for (rad / s) is 360, and the standard deviation σ5 is 10.8; the mean μ6 for elastic modulus E (GPa) is 181, and the standard deviation σ6 is 5.43; the mean μ7 for Poisson's ratio γ is 0.3, and the standard deviation σ7 is 0.006; the mean μ8 for fatigue life dispersion parameter u is 0, and the standard deviation σ8 is 1.
[0079] In this disclosure, the first distribution parameter can be denoted as (where n) θ Let x = {x1, x2, ..., x} be the number of distributed parameters. The input variables affecting the fatigue reliability of the turbine shaft are denoted as x = {x1, x2, ..., x}. n} (where n is the dimension of the input variable), the standard normalized input variable is denoted as u = {u1, u2, ..., u n}, where u={u1,u2,…,u8}=T XU (x)={(x1-μ1) / σ1, (x2-μ2) / σ2,…, (x8-μ8) / σ8}.
[0080] Step S230: Determine the fatigue failure limit state function of the turbine shaft in the standard normal space based on the standard normal input variables and the fatigue failure limit state function of the turbine shaft in the original physical space.
[0081] The inverse transformation of the standard normal input variable can be performed and substituted into y = g X (x) to obtain the first distribution parameter θ * Fatigue failure limit state function of turbine shaft in lower standard normal space
[0082] like Figure 1 As shown, in step S120, multiple uniformly distributed unit direction vectors are obtained in the standard normal space, and the first limit state surface of fatigue failure of the turbine shaft in the standard normal space is determined according to the fatigue failure limit state function.
[0083] like Figure 3 As shown, multiple uniformly distributed unit direction vector samples a can be obtained within a unit hypersphere centered at the origin in standard normal space. i (i = 1, 2, ..., N) where N is the number of uniformly distributed unit direction vector samples, and in some embodiments of this disclosure, N = 3000. In this process, a sufficient number of unit direction vectors can be selected to ensure the accuracy of the sampling effect.
[0084] For example, a standard normal variable x can be generated first in an n-dimensional input space. i (i = 1, 2, ..., N), then for x i Normalizing (i = 1, 2, ..., N) yields a sample of unit direction vectors a uniformly distributed within a unit hypersphere centered at the origin in standard normal space. i (i=1,2,…,N), that is, a i =x i / ||x i ||(i=1,2,…,N).
[0085] In some embodiments of this disclosure, the first limit state surface may be the surface corresponding to the fatigue failure limit state function being 0 in the standard normal space corresponding to the first distribution parameter θ. * In the corresponding standard normal space, from the origin along a i (i = 1, 2, ..., N) to the first limit state surface g U (u|θ * Distance of ) = 0 It can be obtained by solving the nonlinear equation in the polar coordinate system shown in the following equation.
[0086]
[0087] like Figure 1As shown, in step S130, the unit direction vector in the standard normal space corresponding to the first distribution parameter is filtered to obtain the initial sample on the first limiting state surface and the effective unit direction vector corresponding to the initial sample.
[0088] The initial samples can be a set of samples on the first limit state surface corresponding to each unit direction vector that satisfies preset conditions. In an exemplary embodiment of this disclosure, the unit direction vector a can be calculated first. i The limiting distance r where the one-dimensional failure probability is 0 lim Then, select the distances from the origin along the unit direction vector to the first limit state surface that satisfy both greater than 0 and less than the limit distance r. lim The unit direction vectors are selected, and the selected unit direction vectors are taken as valid unit direction vectors. The samples on the first limit state surface corresponding to each valid unit direction vector are taken as initial samples. These initial samples contain distance information from the origin along the valid unit direction vectors to the first limit state surface. For ease of distinction, the selected valid unit direction vectors can be denoted as a. k (k = 1, 2, ..., M), where M is the number of valid unit direction vectors retained, and M ≤ N; simultaneously, a k The distance between (k = 1, 2, ..., M) and the first limit state surface is denoted as . In some embodiments of this disclosure, M = 468.
[0089] In one exemplary embodiment of this disclosure, the limiting distance where the one-dimensional failure probability along a unit direction vector is 0 can be calculated according to a first calculation formula, which may be:
[0090]
[0091] in, For n-degree-of-freedom chi-square distribution χ 2 The cumulative distribution function of (n); r lim Let be the limit distance where the one-dimensional failure probability along the unit direction vector is 0.
[0092] like Figure 1 As shown, in step S140, the initial sample is transformed to the second limiting state surface in the standard normal space corresponding to the second distribution parameter, wherein the second distribution parameter is located in the neighborhood of the first distribution parameter.
[0093] Please continue reading Figure 3 As shown, the samples selected in step S130 can be transformed using spatial transformation relationships. The transformation proceeds to the second limit state surface in the standard normal space corresponding to the second distribution parameter θ. The second limit state surface can be defined as the fatigue failure limit state function in the standard normal space corresponding to the second distribution parameter having a value of 0 (i.e., g...). U The surface corresponding to (u|θ)=0). In one embodiment of this disclosure, the second distribution parameter differs from the first distribution parameter only at the realized value of the mean. In some embodiments of this disclosure, the second distribution parameter may also differ from the first distribution parameter at the realized value of the standard deviation.
[0094] In one exemplary embodiment of this disclosure, the original physical space can be used as an intermediary to filter the first limit state surface g selected in step S130. U (u|θ * Samples at ) = 0 Transform to the standard normal space corresponding to the second distribution parameter θ, and then obtain the second limit state surface g. U Samples at (u|θ)=0 The specific conversion relationship is as follows:
[0095]
[0096]
[0097]
[0098] in, For the sample From the original physical space to the first distribution parameter θ * Inverse transform of the corresponding standard normal space The limit state surface g in the obtained original physical space X Samples at (x) = 0; T XU (x k |θ) is the value of x k The transformation relationship from the original physical space to the standard normal space corresponding to the second distribution parameter θ; and x ki (i = 1, 2, ..., 8) represent respectively and x k The i-th dimension.
[0099] In this disclosure, the converted sample can be denoted as Wherein, the second distribution parameter θ is located at the first distribution parameter θ * The neighborhood of [θ] L ,θ U Within ], that is, θ∈[θ L ,θ U ](θ≠θ * ).
[0100] In one exemplary embodiment of this disclosure, the vent radius r1 and the splined turbine shaft inner diameter r2 can be considered as key factors. That is, the vent radius r1 and the splined turbine shaft inner diameter r2 are input variables affecting the fatigue reliability of the turbine shaft. A failure probability function with respect to the mean of these two input variables can be calculated to determine the corresponding distribution parameter region of interest.
[0101] like Figure 1 As shown, in step S150, the distance between the origin of the coordinate system in the standard normal space corresponding to the second distribution parameter and the second limit state surface along each effective unit direction vector is determined using an interpolation strategy, and the distance is defined as the target distance.
[0102] Interpolation strategies can be linear interpolation, polynomial interpolation, etc., without special limitations. For example... Figure 4 As shown, the samples obtained in step S140 can be transformed using an interpolation strategy. Post-processing is performed to obtain the standard normal space corresponding to the second distribution parameter θ, from the origin along a. k (k=1,2,…,M) to the second limit state surface g U The distance r between (u|θ)=0 θ (a k (k=1,2,…,M), that is, the second limit state surface g under the second distribution parameter θ. U Samples at (u|θ)=0 The following relationship must be satisfied;
[0103]
[0104] like Figure 1 As shown, in step S160, the failure probability function value corresponding to the second distribution parameter is determined by dimensionality reduction integral based on the target distance.
[0105] In some embodiments of this disclosure, the failure probability function value corresponding to the second distribution parameter can be determined based on the target distance using dimensionality reduction integral, including calculating the failure probability function value corresponding to the second distribution parameter using a second calculation formula, which may be:
[0106]
[0107] in, The value is the failure probability function; M is the number of effective unit direction vectors; For n-degree-of-freedom chi-square distribution χ 2 The cumulative distribution function of (n); r θ (a k (k = 1, 2, ..., M) represents the target distance.
[0108] In one exemplary embodiment of this disclosure, the failure probability function value corresponding to the second distribution parameter is calculated using a second calculation formula. The calculated mean value of the turbine shaft structure with respect to the vent radius r1 is... The solution to the failure probability function is as follows: Figure 5 As shown, the average inner diameter r2 of the splined end turbine shaft The solution to the failure probability function is as follows: Figure 6 As shown.
[0109] In one exemplary embodiment of this disclosure, please continue to refer to Figure 5 and Figure 6 As shown, the solution method of this disclosure further includes: solving the fatigue failure probability function of the turbine shaft using a two-layer Monte Carlo simulation method, and comparing the solution result of the two-layer Monte Carlo simulation method with the failure probability function value corresponding to the second distribution parameter determined by the target distance using dimension reduction integration. Under the condition that the accuracy of the obtained failure probability function results is similar, the extended dimension reduction integration method of this disclosure uses a sample size of 3000, and the two-layer Monte Carlo simulation method uses a sample size of 11×1×10. 5 The extended dimension reduction integral method disclosed herein involves 9621 function calls, while the two-layer Monte Carlo simulation method involves 11 × 1 × 10⁻⁶ function calls. 5 This demonstrates that the solution method disclosed in this paper significantly reduces the number of computational samples and substantially improves the efficiency of solving the turbine shaft fatigue failure probability function.
[0110] Other embodiments of this disclosure will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of this disclosure are indicated by the appended claims.
Claims
1. A method for solving the fatigue failure probability function of a turbine shaft based on extended dimensionless integration, characterized in that, include: The input variables affecting the fatigue reliability of the turbine shaft are normalized under the first distribution parameters, and the fatigue failure limit state function corresponding to the turbine shaft in the standard normal space is determined. Multiple uniformly distributed unit direction vectors are obtained in the standard normal space, and the first limit state surface of fatigue failure of the turbine shaft in the standard normal space is determined according to the fatigue failure limit state function. The unit direction vector in the standard normal space corresponding to the first distribution parameter is filtered to obtain the initial sample on the first limiting state surface and the effective unit direction vector corresponding to the initial sample; The initial sample is transformed to the second limiting state surface in the standard normal space corresponding to the second distribution parameter, wherein the second distribution parameter is located in the neighborhood of the first distribution parameter; The distance between the origin of the coordinate system in the standard normal space corresponding to the second distribution parameter and the second limiting state surface along each effective unit direction vector is determined by using an interpolation strategy, and the distance is defined as the target distance. The failure probability function value corresponding to the second distribution parameter is determined by using dimension reduction integral based on the target distance; Filtering the unit direction vectors in the standard normal space corresponding to the first distribution parameter to obtain initial samples on the first limiting state surface and the effective unit direction vectors corresponding to the initial samples includes: Calculate the limiting distance where the one-dimensional failure probability is 0 along the unit direction vector; Each unit direction vector from the origin of the coordinate system along the unit direction vector to the first limit state surface is selected if the distance is greater than 0 and less than the limit distance. Each selected unit direction vector is used as a valid unit direction vector. The sample on the first limit state surface corresponding to each valid unit direction vector is used as an initial sample. The initial sample includes the distance information from the origin of the coordinate system along the valid unit direction vector to the first limit state surface.
2. The solution method according to claim 1, characterized in that, The input variables affecting the fatigue reliability of the turbine shaft include the vent radius, the inner diameter of the splined turbine shaft, the maximum speed, the cruise speed, the idle speed, the elastic modulus, the Poisson's ratio, and the fatigue life dispersion parameter.
3. The solution method according to claim 1, characterized in that, The input variables affecting the fatigue reliability of the turbine shaft are standardized and normalized under the first distribution parameters, and the fatigue failure limit state function corresponding to the turbine shaft in the standard normal space is determined, including: Determine the fatigue failure limit state function of the turbine shaft in the original physical space; The input variables affecting the fatigue reliability of the turbine shaft are normalized under the first distribution parameters to obtain standard normal input variables; The fatigue failure limit state function of the turbine shaft in the standard normal space is determined based on the standard normal input variables and the fatigue failure limit state function of the turbine shaft in the original physical space.
4. The solution method according to claim 3, characterized in that, The fatigue failure limit state function of the turbine shaft in the original physical space is: Among them, the The cumulative damage fatigue life of the turbine shaft under multi-stage cyclic loads; This is the threshold for the low-cycle fatigue life of the turbine shaft.
5. The solution method according to claim 1, characterized in that, The first limit state surface is the surface corresponding to the value of the fatigue failure limit state function being 0 in the standard normal space corresponding to the first distribution parameter.
6. The solution method according to claim 1, characterized in that, The second limit state surface is the surface corresponding to the value of the fatigue failure limit state function being 0 in the standard normal space corresponding to the second distribution parameter.
7. The solution method according to claim 1, characterized in that, The limit distance where the one-dimensional failure probability is 0 along the unit direction vector is calculated according to the first calculation formula, which is: in, for Chi-square distribution of degrees of freedom The cumulative distribution function; Let be the limit distance where the one-dimensional failure probability along the unit direction vector is 0.
8. The solution method according to any one of claims 1-7, characterized in that, Determining the failure probability function value corresponding to the second distribution parameter based on the target distance using dimensionality reduction integral includes calculating the failure probability function value corresponding to the second distribution parameter using a second calculation formula, wherein the second calculation formula is: in, This represents the failure probability function value. M The number of effective unit direction vectors; for Chi-square distribution of degrees of freedom The cumulative distribution function; The target distance.
9. The solution method according to claim 8, characterized in that, The solution method also includes: The fatigue failure probability function of the turbine shaft is solved by a two-layer Monte Carlo simulation method, and the solution obtained by the two-layer Monte Carlo simulation method is compared with the failure probability function value corresponding to the second distribution parameter determined by dimension reduction integral based on the target distance.