CN109918618B - Importance sampling method for gear reliability design - Google Patents
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Abstract
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. The invention considers the influence of random factors, T 1 ,d 1 ,K,b,Z E ,Z H ,Z ε Is looked at a normal random variable, where 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 reliability of the contact fatigue strength of the gear is obtained by applying an importance sampling method of reliability calculation; considering the influence of random factors, T 1 ,d 1 ,K,b,Y Fa ,Y Sa ,Y ε M is seen as a normal random variable, where m is the modulus, Y Fa Is the tooth form coefficient, Y Sa For the root stress correction factor, Y ε And determining a probability density function for the coincidence ratio coefficient, sampling the probability density function by importance, and obtaining the reliability of the bending fatigue strength of the gear by applying an importance sampling method for reliability calculation.
Description
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:
wherein: x is x A Representing a plurality of normal random variables [ sigma ] H ]Representing allowable contact stress;
the probability density function is:
wherein,,
e () is the sign of the averaged value, C is the covariance matrix;
the failure probability is: .
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:
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:
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:
wherein: x is x B Representing a plurality of normal random variables [ sigma ] F ]Representing allowable bending stresses;
the probability density function is:
wherein,,
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:
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:
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:
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:
wherein: x is x A Representing a plurality of normal random variables [ sigma ] H ]Representing allowable contact stress;
the probability density function is:
wherein,,
e () is the sign of the averaged value, C is the covariance matrix;
the failure probability is: .
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:
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:
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:
wherein: x is x B Representing a plurality of normal random variables [ sigma ] F ]Representing allowable bending stresses;
the probability density function is:
wherein,,
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:
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:
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. 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:
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:
wherein: x is x A Representing a plurality of normal random variables [ sigma ] H ]Representing allowable contact stress;
the probability density function is:
wherein,,
e () is the sign of the averaged value, C is the covariance matrix;
the failure probability is: .
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:
obtaining a converged calculation result through the calculation, and obtaining the failure probability of the contact fatigue strength of the gear;
(2) The formula for calculating the gear bending stress of the gear bending fatigue strength reliability by applying the importance sampling method of reliability calculation is as follows:
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:
wherein: x is x B Representing a plurality of normal random variables [ sigma ] F ]Representing allowable bending stresses;
the probability density function is:
wherein,,
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:
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:
through the calculation, a convergent calculation result is obtained, and the failure probability of the bending fatigue strength of the gear is obtained.
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