A method for evaluating strength reliability of gear structure under multiple failure modes

Abstract

The application provides a gear structure strength reliability evaluation method under multiple failure modes, belongs to the field of evaluation of aircraft transmission system gear structure, and specifically establishes a finite element model of the aircraft transmission system gear structure, obtains a strength response parameter of the gear structure under the action of input load; according to the probability and statistics theory, a probability model of related parameters of the gear structure is established; a function function of the gear structure based on strength failure is established; according to three failure modes of the gear structure, and in combination with the strength failure function function, reliability function functions of the transmission system gear structure under each failure mode are respectively established; failure regions of the gear structure under each failure mode are obtained; a strength reliability analysis model of the transmission system gear structure is established by using a subset simulation method, and reliability analysis is performed on the transmission system gear structure to obtain a failure probability of the system. Through the processing scheme, the calculation amount can be greatly reduced, and the calculation efficiency can be improved.

Description

Technical Field

[0001] This application relates to the field of gear structure evaluation in aircraft transmission systems, and in particular to a method for evaluating the strength and reliability of gear structures under multiple failure modes. Background Technology

[0002] Gears are critical components in aircraft transmission systems. Failure of the gear structure in an aircraft transmission system can lead to transmission system failure, causing anything from minor aircraft malfunctions to catastrophic crashes. To extend service life and improve flight safety, it is essential to conduct strength and reliability assessments of the gear structure in aircraft transmission systems under multiple failure modes.

[0003] Currently, the reliability assessment method for gear structures in aircraft transmission systems generally employs a combination of theoretical analysis and Monte Carlo simulation. This involves theoretically analyzing the strength of the gear structure and then directly calculating the failure probability using the Monte Carlo method to assess the structure's reliability. While this method has achieved good results for single-failure-mode gear structures, it suffers from problems such as long computation time, narrow applicability, and difficulty in simulating the coupling effects of multiple failure modes. This is because the causes of gear structure failures are complex and varied. Therefore, it is essential to improve the traditional reliability assessment method for gear structures under multiple failure modes.

[0004] However, it should be noted that while the probabilistic model of Monte Carlo simulation has significant advantages, it has strict requirements for the amount of statistical information on parameters. For gear structures in transmission systems, obtaining data on the objective uncertainties of structural parameters is costly and time-consuming; due to differences in materials and manufacturing processes, this data is difficult to use. In practical engineering problems, information on the distribution density of structural parameters is often lacking, failing to meet the large sample size requirement of Monte Carlo simulation. Once the assumptions about the probability distribution of parameters do not match reality, using a stochastic model for structural reliability analysis becomes meaningless. Clearly, Monte Carlo simulation is powerless for low-probability problems with multiple failure modes. Summary of the Invention

[0005] In view of this, this application provides a method for evaluating the strength and reliability of gear structures under multiple failure modes, which solves the problems in the prior art and can greatly reduce the amount of calculation and improve the calculation efficiency.

[0006] This application provides a method for evaluating the strength and reliability of gear structures under multiple failure modes, which employs the following technical solution:

[0007] A method for assessing the strength and reliability of gear structures under multiple failure modes includes:

[0008] Step 1: Establish a finite element model of the gear structure of the aircraft transmission system and obtain the strength response parameter S of the gear structure under input load;

[0009] Step 2: Based on probability and statistics theory, establish a probabilistic model for the relevant parameters of the gear structure;

[0010] Step 3: Establish the function g = [S] - S based on the strength failure of the gear structure, where [S] is the allowable parameter of the material selected for the structure;

[0011] Step 4: Based on the three failure modes of the gear structure and in conjunction with the strength failure function, establish the reliability function g(x) of the transmission system gear structure under each failure mode:

[0012] Step 5: Obtain the failure region in each failure mode of the gear structure;

[0013] Step 6: Use the subset simulation method to establish a strength and reliability analysis model for the gear structure of the transmission system, and perform reliability analysis on the gear structure of the transmission system to obtain the failure probability of the system.

[0014] Optionally, in step 2, the gear structure-related parameters include the input load T, thickness h, tooth width b, relative roughness f, and elastic modulus E of the gear structure.

[0015] The joint probability density function of input load T, thickness h, tooth width b, relative roughness f and elastic modulus E is f(T,h,b,f,E); where input load T is an interval variable, T∈[T0,T1].

[0016] Optionally, in step 3, the function function of the gear structure based on strength failure includes both random variables (x1, x2, x3, x4) and interval variables (y1). Therefore, the function function of the gear structure based on strength failure is:

[0017] g = [S] - g(x,y)

[0018] = [S]-g(x1,x2,x3,x4,y1);

[0019] Where x1 is the thickness h of the gear structure, x2 is the tooth width b of the gear structure, x3 is the relative roughness f of the gear structure, x4 is the elastic modulus E of the gear structure, and y1 is the input load T of the gear structure.

[0020] Optionally, in step 4, the reliability function of the transmission system gear structure under the three failure modes are as follows:

[0021] The reliability function under strength failure caused by principal stress is g (σ) (x)=[σ]-σ(x);

[0022] The reliability function under strain-induced strength failure is g. (ε) (x)=[ε]-ε(x);

[0023] The reliability function under shear stress-induced strength failure is g. (τ) (x)=[τ]-τ(x).

[0024] Optionally, in step 5:

[0025] The failure regions for the three failure modes are as follows:

[0026] Failure region F under principal stress-induced strength failure (σ) ={(x,y):F=[σ]-g σ (x1,x2,x3,x4,y1)<0};

[0027] Failure region F under strain-induced strength failure (ε) ={(x,y):F=[ε]-g ε (x1,x2,x3,x4,y1)<0};

[0028] Failure region F under shear stress-induced strength failure (τ) ={(x,y):F=[τ]-g τ (x1,x2,x3,x4,y1)<0}.

[0029] Optionally, step 6 specifically includes:

[0030] The total failure domain F of the transmission system gear structure is obtained. The total failure domain F is the sum of the failure regions under the three modes: F = F0 (σ) ∪F (ε) ∪F (τ) ;

[0031] For the failure domain F, we introduce b1>b2>…>b m A series of critical values ​​b1, b2, ..., b = 0 m The introduced critical values ​​constitute a nested failure event F. k ={(x,y):g(x,y)≤b k}(k=1,2,…,m);

[0032] and Based on the multiplication theorem and the inclusion relationship of events, the following formula for calculating the failure probability can be obtained:

[0033]

[0034] Let P1 = P{P1}, P k =P{F k |F (k-1)}(k=2,3,…,m);

[0035] Finally, the failure probability of the gear structure in the transmission system is obtained.

[0036] In summary, this application includes the following beneficial technical effects:

[0037] To address the lack of distribution density information on gear structural parameters in aircraft transmission systems and the coexistence of multiple failure modes, a reliability analysis model based on the finite element method and subset simulation method is proposed. This model demonstrates better engineering applicability in handling issues such as insufficient statistical data on gear structural parameters, multiple failure modes, and computational time consumption, especially when some parameter information is scarce and the probability density function of the parameters cannot be accurately obtained. Attached Figure Description

[0038] To more clearly illustrate the technical solutions of the embodiments of this application, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0039] Figure 1 This is a flowchart illustrating the gear structure strength and reliability assessment method under multiple failure modes proposed in this application. Detailed Implementation

[0040] The embodiments of this application will now be described in detail with reference to the accompanying drawings.

[0041] The following specific examples illustrate the implementation of this application. Those skilled in the art can easily understand other advantages and effects of this application from the content disclosed in this specification. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them. This application can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of this application. It should be noted that, in the absence of conflict, the following embodiments and features in the embodiments can be combined with each other. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0042] It should be noted that various aspects of embodiments within the scope of the appended claims are described below. It will be apparent that the aspects described herein can be embodied in a wide variety of forms, and any particular structure and / or function described herein is merely illustrative. Based on this application, those skilled in the art will understand that one aspect described herein can be implemented independently of any other aspect, and two or more of these aspects can be combined in various ways. For example, any number of aspects set forth herein can be used to implement the device and / or practice the method. Additionally, this device and / or method can be implemented using structures and / or functionalities other than one or more of the aspects set forth herein.

[0043] It should also be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of this application. The illustrations only show the components related to this application and are not drawn according to the number, shape and size of the components in actual implementation. In actual implementation, the form, quantity and proportion of each component can be arbitrarily changed, and the layout of the components may also be more complex.

[0044] Furthermore, specific details are provided in the following description to facilitate a thorough understanding of the examples. However, those skilled in the art will understand that the described aspects can be practiced without these specific details.

[0045] This application provides a method for evaluating the strength and reliability of gear structures under multiple failure modes.

[0046] like Figure 1 As shown, a method for assessing the strength and reliability of gear structures under multiple failure modes includes the following steps:

[0047] Finite element software was used to establish a finite element model of the gear structure of the aircraft transmission system, and the strength response parameters S of the gear structure under input load were obtained: maximum principal stress σ, maximum elongation strain ε, and maximum shear stress τ.

[0048] Based on probability and statistics theory, a probabilistic model of the relevant parameters of the gear structure is established. The strength response parameters of the gear structure are functions of the input load T, thickness h, tooth width b, relative roughness f, and elastic modulus E of the gear structure. The joint probability density function of the five parameters is f(T,h,b,f,E); the load T is an interval variable, T∈[T0,T1].

[0049] Establish a functional function for the gear structure based on strength failure:

[0050] g = [S] - S (1);

[0051] In the formula, [S] represents the allowable parameter for the material selected for the structure;

[0052] S=f(σ,ε,τ) (2).

[0053] There are three failure modes of gear structures: 1) failure caused by the maximum principal stress σ being greater than the allowable stress of the material; 2) failure caused by the maximum elongation strain ε being greater than the allowable strain of the material; 3) failure caused by the maximum shear stress τ being greater than the allowable shear stress of the material.

[0054] Based on the three failure modes of gear structures and the strength failure function, the reliability function g(x) of the transmission system gear structure is established respectively: the reliability function under the strength failure fault caused by principal stress is g. (σ) (x)=[σ]-σ(x), the reliability function under strain-induced strength failure is g (ε) (x)=[ε]-ε(x), the reliability function under strength failure caused by shear stress is g (τ) (x)=[τ]-τ(x).

[0055] A strength and reliability analysis model for the gear structure of the transmission system is established using the subset simulation method. Reliability analysis of the gear structure is then performed to obtain the system's failure probability.

[0056] This application uses a gear in an aircraft transmission system as an example for illustration, and a finite element model of the gear structure is established using finite element software.

[0057] During the model processing, geometric parameters that do not affect the structural strength and stiffness, such as holes, chamfers, and small-sized grooves in non-critical areas designed to meet assembly, sealing, and production requirements, are deleted. In addition, the complex shapes of some non-critical areas are modified.

[0058] For complex parts, high-quality tetrahedral meshes are used for meshing. For non-critical parts, the mesh size is selected between 3 and 5 mm according to the meshing requirements. For critical parts, after multiple trial calculations from large to small, 2 mm is selected. The selection criterion is that the reduction of the mesh size no longer affects or has a minimal impact on the stress results of the critical parts.

[0059] The strength response parameters of the gear structure can be obtained through finite element analysis.

[0060] The strength response parameters of a gear structure are functions of the input load T, thickness h, tooth width b, relative roughness f, and elastic modulus E. The gear thickness h, tooth width b, relative roughness f, and elastic modulus E are random variables, while the load T is an interval variable.

[0061] In this embodiment, the gear structure is made of structural steel 30CrMnSiA. The gear material and related structural parameters are shown in Tables 1 and 2.

[0062] Table 1. Probability distribution and numerical characteristics of random variables

[0063]

[0064] Table 2. Relevant parameters for interval variables

[0065]

[0066] For equation (1), which simultaneously includes random variables (x1,x2,x3,x4) and interval variables (y1), it can be rewritten as:

[0067] g = [S] - g(x,y)

[0068] =[S]-g(x1,x2,x3,x4,y1) (3);

[0069] The failure region F of the gear structure can be expressed as Equation (4);

[0070] F={(x,y):F=[S]-g(x1,x2,x3,x4,y1)<0} (4);

[0071] Where x = (x1, x2, x3, x4) is a random vector, and y = (y1) is a one-dimensional interval variable. y 1 and These are the upper and lower bounds of the interval variables, respectively, and the joint probability density function is f. (x,y) (x,y).

[0072] There are three failure modes for gear structures in transmission systems: 1) failure caused by the maximum principal stress σ exceeding the allowable stress of the material; 2) failure caused by the maximum elongation strain ε exceeding the allowable strain of the material; and 3) failure caused by the maximum shear stress τ exceeding the allowable shear stress of the material. Based on the design criteria for the working strength of gear structures in transmission systems, the function g of the gear structure is established using principal stress, strain, and shear stress respectively. (σ) g (ε) and g (τ) The specific forms are shown in equations (5), (6) and (7), respectively.

[0073] Principal stress: g (σ) (x,y)=[σ]-σ(x1,x2,x3,x4,y1) (5);

[0074] Strain: g (ε) (x,y)=[ε]-ε(x1,x2,x3,x4,y1) (6);

[0075] Shear stress: g (τ)(x,y)=[τ]-τ(x1,x2,x3,x4,y1) (7).

[0076] Function g (σ) g (ε) and g (τ) The failure domains are F (σ) F (ε) and F (τ) As shown in equations (8), (9) and (10);

[0077] F (σ) ={(x,y):F=[σ]-g σ (x1,x2,x3,x4,y1)<0} (8);

[0079] F (ε) ={(x,y):F=[ε]-g ε (x1,x2,x3,x4,y1)<0} (9);

[0081] F (τ) ={(x,y):F=[τ]-g τ (x1,x2,x3,x4,y1)<0}

[0082] (10).

[0083] Let the failure probabilities of the principal stress, strain, and shear stress of the gear structure in the transmission system be P, respectively. f(σ) P f(ε) and P f(τ) The solution formulas are shown in equations (11), (12) and (13), respectively.

[0084]

[0085] It can be seen from equations (8), (9) and (10) that the intersection of the failure domains of principal stress, strain and shear stress in the gear structure of the transmission system is an empty set. Therefore, the total failure domain F of the gear structure of the transmission system is the sum of the three and satisfies equation (14).

[0086] F = F (σ) ∪F (ε) ∪F (τ) (14).

[0087] Therefore, the failure probability P of the gear structure strength failure in the transmission system is... f It can be expressed as equation (15);

[0088]

[0089]

[0090] As can be seen from equation (15), by obtaining the failure probability of the principal stress, strain and shear stress of the gear structure of the transmission system, the functional reliability of the gear structure under strength failure can be obtained.

[0091] The subset simulation method transforms a small failure probability into a product of a series of larger conditional probabilities by introducing reasonable intermediate failure events. Its basic steps start with the joint probability density function of the basic variables, construct the sampling density function on each subset step by step, and then obtain the sampling density function of the limit state equation to be obtained, and finally obtain the estimated value of the failure probability.

[0092] For the failure domain F shown in equation (4), we can introduce b1>b2>...>b m A series of critical values ​​b1, b2, ..., b = 0 m These introduced critical values ​​can constitute failure events with nested relationships, as shown in equation (16).

[0093] F k ={(x,y):g(x,y)≤b k}(k=1,2,…,m) (16)

[0094] At this time there is And there are Based on the multiplication theorem and the inclusion relationship of events, the following formula for calculating the failure probability can be obtained:

[0095]

[0096] Let P1 = P{F1}, P k =P{F k |F (k-1) If k = 2, 3, ..., m, then the above formula can be rewritten as:

[0097]

[0098] Clearly, as can be seen from the above derivation process, the reliability analysis of the subset simulation method transforms small probabilities into a product of larger conditional probabilities, thereby improving the efficiency of digital simulation calculations.

[0099] Substituting the principal stress, strain, and shear stress values ​​obtained from the finite element analysis into equations (11), (12), and (13), respectively, the functional functions of the gear structure of the transmission system under different failure modes can be obtained. Then, the functional reliability is calculated using the subset simulation method and the Monte Carlo method. The estimated value of the Monte Carlo method is used as the accurate value. The relative error of the subset simulation method in solving the failure probability estimate under strength failure is calculated. The calculation parameters and results are shown in Table 3 below.

[0100] Table 3 Reliability Analysis Results of Transmission System Gear Structure Strength Failure

[0101]

[0102] It can be seen that the estimated value obtained by the Monte Carlo method is accurate, while the relative error of the estimated failure probability of the transmission system gear structure under strength failure obtained by the subset simulation method is 4.44%. The subset simulation method not only meets the accuracy requirements, but also requires only 20% of the sample size of the Monte Carlo method. Based on the above analysis results, the functional reliability calculation results based on the subset simulation method in Table 3 are substituted into equation (15) to obtain the subset simulation method under a sampling of 2×10 4 The calculated failure probability of the gear structure in the transmission system is 0.002773.

[0103] This demonstrates that using the subset simulation method can significantly reduce the amount of computation and improve computational efficiency, which has great advantages in the practical engineering application of gear structures in transmission systems.

[0104] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A method for evaluating the strength and reliability of gear structures under multiple failure modes, characterized in that, include: Step 1: Establish a finite element model of the gear structure of the aircraft transmission system and obtain the strength response parameters of the gear structure under input load. ; Step 2: Based on probability and statistics theory, establish a probabilistic model for the relevant parameters of the gear structure; Step 3: Establish the functional function of the gear structure based on strength failure. In the formula, Allowable parameters for selecting materials for the structure. These are the strength response parameters of the gear structure under input load; Step 4: Based on the three failure modes of the gear structure and in conjunction with the strength failure function, establish reliability function functions for the transmission system gear structure under the failure modes of principal stress-induced strength failure, strain-induced strength failure, and shear stress-induced strength failure, respectively. , and ; Step 5: Obtain the failure region for each failure mode of the gear structure. The failure regions for the three failure modes are as follows: failure region under the principal stress-induced strength failure. Failure region under strain-induced strength failure Failure region under shear stress-induced strength failure ; Step 6: Use the subset simulation method to establish a strength and reliability analysis model for the gear structure of the transmission system, and perform reliability analysis on the gear structure of the transmission system to obtain the failure probability of the system. Step 6 specifically includes: Obtain the overall failure domain of the gear structure in the transmission system. It is the sum of the failure areas under the three modes: ; For the failure domain , Introduction A series of critical values The introduced critical values ​​constitute nested failure events. , ; ,and Based on the multiplication theorem and the inclusion relationship of events, the following formula for calculating the failure probability can be obtained: ； make , , ; Finally, the failure probability of the gear structure in the transmission system is obtained. .

2. The method for evaluating the strength and reliability of gear structures under multiple failure modes according to claim 1, characterized in that, In step 2, the gear structure-related parameters include the input load of the gear structure. ,thickness Tooth width Relative roughness and elastic modulus ; Input load ,thickness Tooth width Relative roughness and elastic modulus The joint probability density function is Among them, input load For interval variables, .

3. The method for evaluating the strength and reliability of gear structures under multiple failure modes according to claim 2, characterized in that, In step 3, the function of the gear structure based on strength failure also includes random variables. and interval variables The function of the gear structure based on strength failure is: ； in, For the thickness of the gear structure , For the tooth width of the gear structure , The relative roughness of the gear structure , The elastic modulus of the gear structure , Input load for gear structure .

4. The method for evaluating the strength and reliability of gear structures under multiple failure modes according to claim 3, characterized in that, In step 4, the reliability function of the transmission system gear structure under the three failure modes are as follows: The reliability function under strength failure caused by principal stress is: ; The reliability function under strain-induced strength failure is ; The reliability function under shear stress-induced strength failure is: .

5. The method for evaluating the strength and reliability of gear structures under multiple failure modes according to claim 4, characterized in that, In step 5: The failure regions for the three failure modes are as follows: Failure region under principal stress-induced strength failure ; Failure region under strain-induced strength failure ; Failure region under shear stress-induced strength failure .

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