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CN109918618B - Importance Sampling Method for Gear Reliability Design - Google Patents

Importance Sampling Method for Gear Reliability Design Download PDF

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CN109918618B
CN109918618B CN201910238421.0A CN201910238421A CN109918618B CN 109918618 B CN109918618 B CN 109918618B CN 201910238421 A CN201910238421 A CN 201910238421A CN 109918618 B CN109918618 B CN 109918618B
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gear
coefficient
importance sampling
reliability
density function
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莫文辉
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Hubei University of Automotive Technology
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Abstract

本发明提出了齿轮可靠性设计的重要性抽样方法，对齿轮进行可靠性设计，使齿轮的重量减轻，更加安全可靠，提高了产品质量。本发明考虑随机因素的影响,T1,d1,K,b,ZE,ZH,Zε被看着正态随机变量，其，中T1为小齿轮传递的转矩，d1为小齿轮分度圆直径，u是齿数比，b是齿宽，ZE是弹性系数，ZH是节点区域系数，Zε是重合度系数，K是载荷系数；应用可靠性计算的重要性抽样方法求出齿轮接触疲劳强度可靠度；考虑随机因素的影响,T1,d1,K,b,YFa,YSa,Yε,m被看着正态随机变量，其中m为模数，YFa为齿形系数，YSa为齿根应力修正系数，Yε为重合度系数，确定概率密度函数，重要性抽样概率密度函数，应用可靠性计算的重要性抽样方法求出齿轮弯曲疲劳强度可靠度。

Figure 201910238421

The invention proposes an importance sampling method for gear reliability design, and carries out reliability design on the gear, so that the weight of the gear is reduced, the gear is safer and more reliable, and the product quality is improved. The present invention considers the influence of random factors, T 1 , d 1 , K, b, Z E , Z H , Z ε are regarded as normal random variables, where T 1 is the torque transmitted by the pinion, and d 1 is Pinion pitch circle diameter, u is the gear ratio, b is the tooth width, Z E is the elastic coefficient, Z H is the node area coefficient, Z ε is the coincidence degree coefficient, K is the load coefficient; Importance sampling for application reliability calculation method to obtain the reliability of gear contact fatigue strength; considering the influence of random factors, T 1 ,d 1 ,K,b,Y Fa ,Y Sa ,Y ε ,m are regarded as normal random variables, where m is the modulus, Y Fa is the tooth shape coefficient, Y Sa is the tooth root stress correction coefficient, Y ε is the coincidence coefficient, determine the probability density function, the importance sampling probability density function, and apply the importance sampling method of reliability calculation to obtain the gear bending fatigue strength reliability.

Figure 201910238421

Description

Importance sampling method for gear reliability design
Technical Field
The invention relates to an importance sampling method for gear reliability design, belonging to the fields of mechanical design, mechanical reliability design and mechanical modern design methods.
Background
Gears are common, important mechanical parts. The failure of gears causes malfunctions or even major malfunctions of machine tools, engineering machines, metallurgical machines, mining machines, petroleum machines, agricultural machines, vehicles, etc. The gear is complex to manufacture, the working condition is complex, and the influence factors on the normal work of the gear are more. The mechanical reliability design treats some variables in the conventional design, such as load, material strength, geometric dimensions of parts and the like, as random variables, and considers the influence of working condition changes and various random factors. The domestic scholars propose a stress-intensity interference model method for gear reliability design, an HL-RF method for gear reliability design, a Monte Carlo method for gear reliability design, and the like.
At present, no importance sampling method for gear reliability design exists.
Disclosure of Invention
The invention provides an importance sampling method for the reliability design of gears, which is used for carrying out the reliability design on the gears, so that the weight of the gears is reduced, the gears are safer and more reliable, and the product quality is improved.
For this purpose, the technical scheme of the invention is as follows: the importance sampling method for the gear reliability design comprises the following steps:
(1) Determining gear contact fatigue strength reliability by using importance sampling method of reliability calculation
Considering the influence of random factors, T 1 ，d 1 ，K，b，Z E ，Z H ，Z ε Is looked at a normal random variable T 1 Torque transmitted for pinion, d 1 For the diameter of the indexing circle of the pinion, u is the gear ratio, b is the tooth width, Z E Is the elastic coefficient, Z H Is the node area coefficient, Z ε Is the overlap ratio coefficient, K is the load coefficient;
the limit state equation is:
Figure GDA0004251847410000011
wherein: x is x A Representing a plurality of normal random variables [ sigma ] H ]Representing allowable contact stress;
the probability density function is:
Figure GDA0004251847410000012
wherein,,
Figure GDA0004251847410000021
e () is the sign of the averaged value, C is the covariance matrix;
the failure probability is: .
Figure GDA0004251847410000022
Wherein I (g (x) i ) Let.ltoreq.0) =1 is an indicator function if x i 0 in the failure domain; psi (x) i ) Is an importance sampling probability density function;
the importance sampling probability density function is:
Figure GDA0004251847410000023
obtaining a converged calculation result through the calculation, and obtaining the failure probability of the contact fatigue strength of the gear;
(2) Determining gear bending fatigue strength reliability by using importance sampling method of reliability calculation
The formula of the bending stress of the gear is as follows:
Figure GDA0004251847410000024
wherein m is a modulus, Y Fa Is the tooth form coefficient, Y Sa For the root stress correction factor, Y ε Is the overlap ratio coefficient.
Considering the influence of random factors, T 1 ，d 1 ，K，b，Y Fa ，Y Sa ，Y ε M is looked at as a normal random variable;
the limit state equation is:
Figure GDA0004251847410000025
wherein: x is x B Representing a plurality of normal random variables [ sigma ] F ]Representing allowable bending stresses;
the probability density function is:
Figure GDA0004251847410000026
wherein,,
Figure GDA0004251847410000031
e () is the sign of the average value, C 1 Is a covariance matrix;
wherein: x is X 1 Represents a normal random variable, mu 11 Representing the average value of the normal random variable;
the failure probability is:
Figure GDA0004251847410000032
wherein I (g) 1 (x i ) Let.ltoreq.0) =1 is an indicator function ifx i 0 in the failure domain; psi (x) i ) Is an importance sampling probability density function;
the importance sampling probability density function is:
Figure GDA0004251847410000033
through the calculation, a convergent calculation result is obtained, and the failure probability of the bending fatigue strength of the gear is obtained.
The beneficial effects of the invention are as follows: according to the method, the gear contact fatigue strength and the bending fatigue strength are obtained through calculation of an importance sampling method.
Drawings
FIG. 1 is a schematic diagram of a method for sampling the importance of gear contact fatigue strength reliability.
FIG. 2 is a schematic diagram of a method for sampling the importance of gear bending fatigue strength reliability.
Detailed Description
The invention is further described below with reference to examples.
The importance sampling method for the gear reliability design comprises the following steps:
(1) Determining gear contact fatigue strength reliability by using importance sampling method of reliability calculation
The formula of the gear contact stress is as follows:
Figure GDA0004251847410000034
wherein T is 1 Torque transmitted for pinion, d 1 For the diameter of the indexing circle of the pinion, u is the gear ratio, b is the tooth width, Z E Is the elastic coefficient, Z H Is the node area coefficient, Z ε Is the overlap ratio coefficient, K is the load coefficient;
considering the influence of random factors, T 1 ，d 1 ，K，b，Z E ，Z H ，Z ε Is looked at a normal random variable;
the limit state equation is:
Figure GDA0004251847410000041
wherein: x is x A Representing a plurality of normal random variables [ sigma ] H ]Representing allowable contact stress;
the probability density function is:
Figure GDA0004251847410000042
wherein,,
Figure GDA0004251847410000043
e () is the sign of the averaged value, C is the covariance matrix;
the failure probability is: .
Figure GDA0004251847410000044
Wherein I (g (x) i ) Let.ltoreq.0) =1 is an indicator function if x i 0 in the failure domain; psi (x) i ) Is an importance sampling probability density function;
the importance sampling probability density function is:
Figure GDA0004251847410000045
obtaining a converged calculation result through the calculation, and obtaining the failure probability of the contact fatigue strength of the gear;
(2) Determining gear bending fatigue strength reliability by using importance sampling method of reliability calculation
The formula of the bending stress of the gear is as follows:
Figure GDA0004251847410000046
wherein m is a modulus, Y Fa Is the tooth form coefficient, Y Sa For the root stress correction factor, Y ε Is the coefficient of coincidence;
considering the influence of random factors, T 1 ，d 1 ，K，b，Y Fa ，Y Sa ，Y ε M is looked at as a normal random variable;
the limit state equation is:
Figure GDA0004251847410000047
wherein: x is x B Representing a plurality of normal random variables [ sigma ] F ]Representing allowable bending stresses;
the probability density function is:
Figure GDA0004251847410000051
wherein,,
Figure GDA0004251847410000052
e () is the sign of the average value, C 1 Is a covariance matrix;
wherein X is 1 Represents a normal random variable, mu 11 Representing the average value of the normal random variable;
the failure probability is:
Figure GDA0004251847410000053
wherein I (g) 1 (x i ) Let.ltoreq.0) =1 is an indicator function if xi is 0 in the failure domain; psi (x) i ) Is an importance sampling probability density function;
the importance sampling probability density function is:
Figure GDA0004251847410000054
through the calculation, a convergent calculation result is obtained, and the failure probability of the bending fatigue strength of the gear is obtained.
As shown in fig. 1, a flowchart of a method for sampling the importance of the reliability of the contact fatigue strength of gears is shown.
As shown in fig. 2, a flowchart of a method for sampling the importance of the reliability of the bending fatigue strength of gears is shown.

Claims (1)

1.齿轮可靠性设计的重要性抽样方法，包括如下步骤：1. Importance sampling method for gear reliability design, including the following steps: (1)应用可靠性计算的重要性抽样方法求出齿轮接触疲劳强度可靠度(1) Apply the importance sampling method of reliability calculation to obtain the reliability of gear contact fatigue strength 齿轮接触应力的公式如下：The formula for gear contact stress is as follows:
Figure QLYQS_1
Figure QLYQS_1
 式中，T1为小齿轮传递的转矩，d1为小齿轮分度圆直径，u是齿数比，b是齿宽，ZE是弹性系数，ZH是节点区域系数，Zε是重合度系数，K是载荷系数；In the formula, T 1 is the torque transmitted by the pinion, d 1 is the pitch circle diameter of the pinion, u is the gear ratio, b is the tooth width, Z E is the elastic coefficient, Z H is the node area coefficient, and Z ε is the coincidence degree coefficient, K is the load coefficient; 考虑随机因素的影响，T1，d1，K，b，ZE，ZH，Zε被看着正态随机变量；Considering the influence of random factors, T 1 , d 1 , K, b, Z E , Z H , Z ε are looked at as normal random variables; 极限状态方程为：The limit state equation is:
Figure QLYQS_2
Figure QLYQS_2
 式中：xA代表多个正态随机变量，[σH]代表许用接触应力；In the formula: x A represents multiple normal random variables, [σ H ] represents the allowable contact stress; 概率密度函数为：The probability density function is:
Figure QLYQS_3
Figure QLYQS_3
 其中，
Figure QLYQS_4
E()是求均值的符号，C是协方差矩阵；in,
Figure QLYQS_4
E() is the symbol for the mean value, and C is the covariance matrix; 失效概率为：.The probability of failure is: .
Figure QLYQS_5
Figure QLYQS_5
 其中，I(g(xi)≤0)＝1是指示函数，如果xi在失效领域为0；ψ(xi)是重要性抽样概率密度函数；Among them, I(g( xi )≤0)=1 is an indicator function, if xi is 0 in the failure domain; ψ( xi ) is an importance sampling probability density function; 重要性抽样概率密度函数为：The importance sampling probability density function is:
Figure QLYQS_6
Figure QLYQS_6
 通过上述计算，从而得到获得收敛的计算结果，得到齿轮接触疲劳强度的失效概率；Through the above calculation, the calculation result of convergence is obtained, and the failure probability of the contact fatigue strength of the gear is obtained; (2)应用可靠性计算的重要性抽样方法求出齿轮弯曲疲劳强度可靠度齿轮弯曲应力的公式为：(2) Applying the importance sampling method of reliability calculation to obtain the gear bending fatigue strength reliability formula of the gear bending stress is:
Figure QLYQS_7
Figure QLYQS_7
 其中，m为模数，YFa为齿形系数，YSa为齿根应力修正系数，Yε为重合度系数；Among them, m is the modulus, Y Fa is the tooth shape coefficient, Y Sa is the tooth root stress correction coefficient, Y ε is the coincidence degree coefficient; 考虑随机因素的影响，T1，d1，K，b，YFa，YSa，Yε，m被看着正态随机变量；Considering the influence of random factors, T 1 , d 1 , K, b, Y Fa , Y Sa , Y ε , m are looked at as normal random variables; 极限状态方程为：The limit state equation is:
Figure QLYQS_8
Figure QLYQS_8
 式中：xB代表多个正态随机变量，[σF]代表许用弯曲应力；In the formula: x B represents multiple normal random variables, [σ F ] represents allowable bending stress; 概率密度函数为：The probability density function is:
Figure QLYQS_9
Figure QLYQS_9
 其中，
Figure QLYQS_10
Figure QLYQS_11
E()是求均值的符号，C1是协方差矩阵；in,
Figure QLYQS_10
Figure QLYQS_11
E() is the symbol for the mean value, and C 1 is the covariance matrix; 式中，X1代表正态随机变量，μ11代表正态随机变量的均值；In the formula, X 1 represents a normal random variable, μ 11 represents the mean value of a normal random variable; 失效概率为：The probability of failure is:
Figure QLYQS_12
Figure QLYQS_12
 其中I(g1(xi)≤0)＝1是指示函数，如果xi在失效领域为0；ψ(xi)是重要性抽样概率密度函数；where I(g 1 ( xi )≤0)=1 is an indicator function, if xi is 0 in the failure domain; ψ( xi ) is an importance sampling probability density function; 重要性抽样概率密度函数为：The importance sampling probability density function is:
Figure QLYQS_13
Figure QLYQS_13
 通过上述计算，从而得到获得收敛的计算结果，得到齿轮弯曲疲劳强度的失效概率。Through the above calculation, a converged calculation result is obtained, and the failure probability of the gear bending fatigue strength is obtained.
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