CN114580292B - Method for analyzing reliability and sensitivity of plastic deformation of rolling bearing - Google Patents

Abstract

The invention discloses a reliability and sensitivity analysis method for the plastic deformation of a rolling bearing, which combines the Newton iteration method, Latin hypercube design, neural network method and high-order moment method, considers the randomness of the rolling bearing load parameters, geometric parameters and material parameters, and obtains neural network training samples by sampling using the Newton iteration method and Latin hypercube design on the basis of reasonably constructing the neural network structure, and obtains the mapping relationship between the function function and the random variable by learning and training the samples. The limit state function of the contact between the rolling element and the raceway is obtained according to the stress-strength interference model, and then the reliability index and reliability of the rolling bearing are obtained by using the high-order moment method, and the reliability sensitivity analysis of the plastic deformation of the rolling bearing is performed.

Description

A reliability and sensitivity analysis method for plastic deformation of rolling bearings

Technical Field

The invention belongs to the technical field of reliability analysis of rolling bearings, and in particular relates to a reliability and sensitivity analysis method of plastic deformation of rolling bearings.

Background Art

Rolling bearings are one of the most widely used components in industrial applications, and their failure is one of the most common causes of mechanical failure. If an excessive load is applied to a rolling bearing in a stationary state, the elastic deformation between the raceway and the rolling element of the rolling bearing will be converted into plastic deformation. After the plastic deformation produces indentations inside the rolling bearing, it will cause vibration, noise, and changes in friction torque during rotation, making the rolling bearing unable to work properly and even becoming the cause of early fatigue failure. Therefore, there is an urgent need to improve the reliability of plastic deformation of rolling bearings to prevent catastrophic mechanical failures.

The reliability design of rolling bearings mainly focuses on the structural reliability design. In the traditional structural design process, the design parameters usually adopt quantitative reliability design, and the method used is the probability design method. The reliability measurement index generally involves the reliability index β and the reliability R. Rolling bearings are one of the earliest mechanical products to adopt reliability design. For example, Lundberg and Palmgren gave a calculation method for the rated life and basic rated dynamic load with a reliability of 90% in 1947, which was included in the ISO international standard in 1962 and is still used today. Therefore, unlike other mechanical designs that regard reliability design as special design, modern advanced design, etc., the conventional design of rolling bearings is reliability design.

Since the characteristics and parameters of mechanical products (such as strength, stress, physical variables, geometric dimensions, etc.) have inherent randomness, the existing limit state function based on the stress-strength interference model usually treats the important design parameters of rolling bearings as basic random variables to further conduct reliability analysis of rolling bearings. When the design parameters are treated as basic random variables for reliability analysis, a large number of test samples are required to determine the probability distribution of the basic random variables and their probability statistical characteristics.

In the traditional rolling bearing design process, the external load, material strength and geometric dimensions of the rolling bearing are treated as fixed values. In fact, these parameters are highly random, which causes large deviations or even errors in the calculation results.

The existing reliability analysis of rolling bearing plastic deformation based on stress-strength interference model requires that the important design parameters of rolling bearings be processed as basic random variables. Due to the weak basic conditions for rolling bearing reliability design, complex working conditions, many influencing factors, and lack of effective data, it is usually difficult to accurately determine the probability distribution of rolling bearing design parameters in engineering practice. It will be very challenging to conduct reliability analysis and design in the absence of probability information. In recent years, with the development of rolling bearings in multiple directions such as high speed, heavy load, and high reliability, higher requirements have been placed on the reliability of rolling bearings.

Summary of the invention

In order to solve the shortcomings of the above-mentioned prior art, the present invention proposes a reliability and sensitivity analysis method for plastic deformation of a rolling bearing, which can analyze the reliability of plastic deformation of a rolling bearing in the absence of probability information. The specific technical solution of the present invention is as follows:

A reliability and sensitivity analysis method for plastic deformation of a rolling bearing comprises the following steps:

S1: Establish a mechanical model of rolling bearing plastic deformation, regard the rolling bearing load parameters, geometric parameters and material parameters as basic random variables, use Latin hypercube design to sample n groups of basic random variables, and bring the sampling results into the static equilibrium equation of the rolling bearing to obtain the corresponding n maximum contact loads Q;

S2: According to the Hertz contact theory, for the elliptical contact area formed by the rolling element and the raceway, the maximum contact stress is q _max . The fourth strength theory is used to obtain the equivalent stress q _eq . The n equivalent stresses q _eq are obtained through the n maximum contact loads Q.

S3: The relationship between the basic random variables and the equivalent stress q _eq is fitted by the BP neural network, the optimal initial weights and thresholds of the network are obtained by the particle swarm algorithm (PSO), and the limit state function of the plastic deformation of the rolling bearing is established according to the stress-strength interference model;

S4: Based on the high-order moment method, reliability and sensitivity analysis of the plastic deformation of rolling bearings are carried out to determine the reliability index and reliability of the plastic deformation of rolling bearings and the reliability change trend with the basic random variables, revealing the influence of the change of basic random variables on the reliability of the plastic deformation of rolling bearings.

Furthermore, the specific process of step S1 is as follows:

S101: The static equilibrium equation of a rolling bearing under the combined load of axial force _Fa , radial force _Fr and moment M is:

Where, _Kn is the stiffness coefficient, ^α0 is the initial contact angle, R _i is the radius of the inner raceway groove curvature center trajectory, ψ is the azimuth angle, d _m is the pitch circle diameter, δ _a is the relative axial displacement, δ _r is the relative radial displacement, and θ is the relative angular displacement;

A is the distance between the center of curvature of the inner and outer grooves at any position in the rolling bearing under no-load contact state:

A＝( _fi +fe _- 1) _Dw (4)

is the dimensionless displacement:

Where, _fi is the inner raceway groove curvature radius coefficient, _fe is the outer raceway groove curvature radius coefficient, and _Dw is the rolling element diameter;

S102: Equations (1)-(3) are a set of simultaneous nonlinear equations with δ _a , δ _r and θ as unknown variables. The values of δ _a , δ _r and θ are calculated using the Newton iteration method. At ψ = 0, the maximum contact load Q on the rolling element of the rolling bearing is obtained:

S103: Use Latin hypercube design to sample n groups of basic random variables, and bring the sampling results into the static equilibrium equation in groups to obtain the corresponding n maximum contact loads Q.

Furthermore, the specific process of step S2 is as follows:

S201: According to Hertz's basic theory, for the elliptical contact area formed by the contact between the rolling element and the raceway, the maximum contact stress q _max is:

Among them, the major semi-axis a and the minor semi-axis b of the contact area ellipse are:

Where E is the equivalent elastic modulus, ξ ₁ , ξ ₂ are the Poisson's ratios of the raceway and rolling element, E ₁ , E ₂ are the elastic moduli of the raceway and rolling element, ∑ρ is the curvature of the raceway and;

Π(e) is the complete elliptic integral of the second kind, e is the eccentricity of the ellipse, specifically,

e ² =1-1/k ² (11)

Where _Rx is the equivalent radius of curvature along the direction of motion at the contact point of the ball bearing, and _Ry is the equivalent radius of curvature perpendicular to the direction of motion at the contact point of the ball bearing;

S202: Under point contact conditions, the stress state at the contact center is a triaxial compressive stress state of σ ₁ , σ ₂ and σ ₃ , expressed using engineering principal stress symbols, i.e. σ ₁ ≥σ ₂ ≥σ ₃ :

σ ₁ =-q _max [0.505+0.255(b/a) ^0.6609 ] (13)

σ ₂ =-q _max [1.01-0.250(b/a) ^0.5797 ] (14)

σ ₃ = -q _max (15)

In the formula, the negative sign indicates that the principal stress is compressive stress;

For the static stress state under triaxial compression, the fourth strength theory is used, and the equivalent stress q _eq in the elliptical contact area is:

Substituting equations (13)-(15) into (16), we obtain:

S203: Substitute the n maximum contact loads Q obtained in step S1 into equations (7) and (17) respectively to obtain corresponding n equivalent stresses q _eq .

Furthermore, the specific process of step S3 is as follows:

S301: The n groups of basic random variables obtained in step S1 and the corresponding n equivalent stresses q _eq obtained in step S2 constitute n groups of data sets, and the data sets are divided into a training set, a test set and a validation set;

S302: Normalization and denormalization of data;

The random variable data of the training set is normalized by formula (18), where _xk is the original data and _yk is the corresponding normalized data:

The random variable data of the test set and validation set are normalized and denormalized by formula (19):

Where x _min is the minimum value in the data series, x _max is the maximum value in the data series, and the maximum and minimum values of y _k are recorded as y _max and y _min respectively;

S303: Assume that in a D-dimensional space, there are n particles, i.e., a population H = (H ₁ H ₂ … H _n ) composed of the aforementioned n sets of data sets, and the ith particle is represented by a D-dimensional vector H _i = (h _i1 ,h _i2 , …,h _iD ) ^T , which represents the position of the ith particle in the D-dimensional search space and also represents a potential solution to the problem; according to the objective function, the fitness value corresponding to each particle position H _i can be obtained, the speed of the ith particle is V _i = (V _i1 ,V _i2 , …,V _iD ) ^T , its individual extreme value is P _i = (P _i1 ,P _i2 , …,P _iD ) ^T , and the total extreme value of the population is P _g = (P _g1 ,P _g2 , …,P _gD ) ^T ;

In each iteration, the particle updates its own speed and position through individual extrema and global extrema. The update formula is:

Wherein, ω is the inertia weight; i = 1, 2, …, n, d = 1, 2, …, D; k is the current iteration number; V _id is the velocity of the particle; c ₁ and c ₂ are non-negative constants, which become acceleration factors; r ₁ and r ₂ are random numbers distributed between [0, 1]. To prevent blind search of particles, their position and velocity are limited to a certain interval of [-X _max , X _max ], [-V _max , V _max ];

The initial weights and thresholds of the BP neural network are obtained according to the individual. The system output is predicted after the BP neural network is trained with the training data. The absolute value of the error between the predicted output and the expected output and E are taken as the individual fitness value F. The calculation formula is:

In the formula, each particle constitutes a node, that is, n is the number of network output nodes in the neural network, _yi is the expected output of the ith node of the BP neural network, _oi is the predicted output of the ith node, and t is the coefficient;

S303: The geometric dimensions, loads and material strength of the rolling bearing are regarded as basic random parameters X = (x ₁ , x ₂ , ..., x _n ). The BP neural network is a one-way propagation forward network consisting of an input layer, a hidden layer and an output layer, wherein the number of nodes in the input layer is p, the number of nodes in the hidden layer is q, and the number of nodes in the output layer is r. The functional relationship of each internal layer is obtained by neuron transmission:

Where h _k is the output of the hidden layer, y _j is the output of the output layer, _xi is the input variable, w _ik and v _kj are weights, α _k and β _j are the thresholds of the hidden layer and the output layer, respectively, f ₁ (·) is the S-type nonlinear function tansig, and f ₂ (·) is the linear function purelin;

Through the neural network mapping relationship shown in equations (23) and (24), the input and output of the data are fitted within the allowable error range, and the fitting function relationship is:

According to the stress-strength interference model, the limit state function g(X) of the reliability of rolling bearing plastic deformation in the absence of probability information is:

Where, the allowable yield stress σ _s is used.

Furthermore, the specific process of step S4 is as follows:

S401: Reliability analysis of plastic deformation of rolling bearings;

According to the stress-strength interference model of structural reliability, solving the reliability requires establishing the limit state function g(X) of reliability analysis to represent the two states of reliability of rolling bearings during operation, namely

In the formula, the limit state function equation g(X) = 0 is an n-dimensional limit state surface;

The mean μ _g , standard deviation σ _g , third-order moment θ _g and fourth-order moment η _g of the limit state function:

Wherein, μ _X , C ₂ (X), C ₃ (X) and C ₄ (X) represent the mean matrix, variance and covariance matrix, third-order moment matrix and fourth-order moment matrix of the basic random parameter X respectively; whether the basic random parameter X is an independent variable or not, the diagonal term of the variance and covariance matrix C ₂ (X) is used as the standard deviation vector C ₂ (X) = diag(σ _X )[ρ]diag(σ _X );

When the probability distribution of the basic random variable vector X cannot be determined, if the mean vector E(X) = μ _X , the variance and covariance matrix C ₂ (X), the third-order matrix C ₃ (X) and the fourth-order matrix C ₄ (X) are known, the reliability index β _FM is defined as follows according to the higher-order moment method:

In the formula, is the reliability index when the basic random parameter vector X obeys the normal distribution, is the skewness coefficient of the limit state function g(X), is the peak coefficient of the limit state function g(X). After sorting, another expression of the reliability index of the high-order moment method is obtained:

The approximate estimated value of the reliability R _FM is further determined as:

R _FM =Φ(β _FM ) (34)

Where Φ(·) represents the standard normal distribution function;

S402: performing sensitivity analysis on the plastic deformation of the rolling bearing in the absence of probability information;

According to equations (33) and (34), the sensitivity of reliability R _FM to the mean vector μ _X and standard deviation vector σ _X of the basic random variable vector X is obtained as follows:

In the formula,

Where φ(·) represents the standard normal probability density function;

The derivatives of the reliability index with respect to the first four moments (μ _g , σ _g , θ _g and η _g ) of the limit state function are:

The derivatives of the first four moments of the limit state function with respect to the mean _μX of the basic random variable are:

The derivatives of the first four moments of the limit state function with respect to the standard deviation σ _X of the basic random variable are:

in, [ρ] is the correlation coefficient matrix, diag(·) is the diagonal matrix, which are expressed as:

[ρ ₃ ] is the third-order correlation coefficient matrix;

[ρ ₄ ] is the fourth-order correlation coefficient matrix;

Substituting the known conditions and relevant data into equations (35) and (36), we can obtain the reliability sensitivity and Numeric value;

The reliability sensitivity gradient is expressed as:

In the formula, gradient grad[·] represents the rate of change of reliability;

The dimensionless reliability sensitivity to the mean value τ _i and standard deviation sensitivity η _i of the basic random parameters are expressed as:

The dimensionless reliability sensitivity gradient _si is:

After standardizing s _{i ,} the sensitivity factor λ _i is:

The beneficial effects of the present invention are:

1. Based on the stress-strength interference model, the present invention proposes a theoretical method for reliability design and sensitivity analysis of plastic deformation of rolling bearings in the absence of probabilistic information on the allowable yield stress, taking into account random parameters such as the geometric dimensions and material properties of the rolling bearing itself, as well as the influence of the combined loads on the rolling bearing.

2. The present invention derives a matrix formula for reliability design of rolling bearing plastic deformation in the absence of probability information, and at the same time, based on the direct differential method of sensitivity analysis, derives a matrix formula for reliability differential sensitivity calculation. For reliability and reliability sensitivity in the absence of probability information, a matrix equation is derived. The matrix format has the advantages of clear expression and easy programming.

3. Due to the complexity of the static equilibrium equation of rolling bearings, the high-order moment method cannot be used directly. The present invention fits the relationship between basic random variables and equivalent stresses through BP neural network. At the same time, in order to solve the defects of local optimum and premature convergence in artificial neural networks, a particle swarm algorithm (PSO) is introduced into the penalty function to optimize the BP neural network, so that the optimized BP neural network can better predict the output of the function.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. By referring to the drawings, the features and advantages of the present invention will be more clearly understood. The drawings are schematic and should not be understood as limiting the present invention in any way. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work. Among them:

FIG1 is a flow chart of a reliability and sensitivity analysis method for plastic deformation of a rolling bearing according to the present invention;

FIG2 is a diagram showing the displacement of the inner ring of a rolling bearing under the combined load of radial force, axial force and moment (the outer ring is fixed);

FIG3 is a relative position diagram of the groove curvature center trajectory of the rolling bearing before and after the inner ring is displaced;

FIG4 is a flow chart of particle swarm algorithm optimization;

Fig. 5 is a topological structure diagram of the BP neural network;

FIG6 is a regression graph of the BP neural network, wherein FIG6(a) is a regression graph of the training set, FIG6(b) is a regression graph of the validation set, FIG6(c) is a regression graph of the test set, and FIG6(d) is a regression graph of the entire data set;

FIG7 is a diagram showing the reliability sensitivity analysis results of the basic random variables, wherein FIG7(a) is the sensitivity of the reliability to the mean of the basic random variables, FIG7(b) is the sensitivity of the reliability to the standard deviation of the basic random variables, FIG7(c) is the reliability sensitivity gradient, and FIG7(d) is the sensitivity factor level diagram;

FIG8 is a graph showing the relationship between reliability and rolling element diameter;

FIG. 9 is a graph showing the relationship between reliability and inner groove curvature radius.

DETAILED DESCRIPTION

In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the embodiments of the present invention and the features in the embodiments can be combined with each other without conflict.

In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those described herein. Therefore, the protection scope of the present invention is not limited to the specific embodiments disclosed below.

As shown in FIG1 , a reliability and sensitivity analysis method for plastic deformation of a rolling bearing comprises the following steps:

S1: Establish a mechanical model of rolling bearing plastic deformation, regard the rolling bearing load parameters, geometric parameters and material parameters as basic random variables, use Latin hypercube design to sample n groups of basic random variables, and bring the sampling results into the static equilibrium equation of the rolling bearing to obtain the corresponding n maximum contact loads Q; the specific process is:

S101: The static equilibrium equation of a rolling bearing under the combined load of axial force _Fa , radial force _Fr and moment M is:

Wherein, _Kn is the stiffness coefficient, ^α0 is the initial contact angle, R _i is the radius of the inner raceway groove curvature center trajectory, ψ is the azimuth angle, d _m is the pitch circle diameter, δ _a is the relative axial displacement, δ _r is the relative radial displacement, and θ is the relative angular displacement; the outer ring of the rolling bearing under the combined load of radial force, axial force and moment is fixed, and the displacement relationship of the inner ring is shown in Figure 2, and the relative position relationship of the groove curvature center trajectory of the rolling bearing before and after the inner ring displacement is shown in Figure 3;

A is the distance between the center of curvature of the inner and outer grooves at any position in the rolling bearing under no-load contact state:

A＝( _fi +fe _- 1) _Dw (59)

is the dimensionless displacement:

Where, _fi is the inner raceway groove curvature radius coefficient, _fe is the outer raceway groove curvature radius coefficient, and _Dw is the rolling element diameter;

S102: Equations (56)(1)-(3)(58) are a set of simultaneous nonlinear equations with δ _a , δ _r and θ as unknown variables. The values of δ _a , δ _r and θ are calculated using the Newton iteration method. At ψ = 0, the maximum contact load Q on the rolling element of the rolling bearing is obtained:

S103: Use Latin hypercube design to sample n groups of basic random variables, and bring the sampling results into the static equilibrium equation in groups to obtain the corresponding n maximum contact loads Q.

Among them, Latin hypercube design (LHD) has an effective space filling ability. Compared with orthogonal experiment, Latin hypercube design can study more combinations with the same number of points. LHD is to divide each dimension of the coordinate interval into n-dimensional space. Divide into m intervals evenly, each small interval is recorded as Randomly select m points to ensure that each level of a factor is studied only once, that is, to construct an n-dimensional space with a Latin hypercube method with a sample size of m, denoted as m×n LHD.

S2: According to Hertz contact theory, for the elliptical contact area formed by the rolling element and the raceway, the maximum contact stress is q _max . The fourth strength theory is used to obtain the equivalent stress q _eq . The n equivalent stresses q _eq are obtained through n maximum contact loads Q. The specific process is as follows:

S201: According to Hertz's basic theory, for the elliptical contact area formed by the contact between the rolling element and the raceway, the maximum contact stress q _max is:

Among them, the major semi-axis a and the minor semi-axis b of the contact area ellipse are:

Where E is the equivalent elastic modulus, ξ ₁ , ξ ₂ are the Poisson's ratios of the raceway and rolling element, E ₁ , E ₂ are the elastic moduli of the raceway and rolling element, ∑ρ is the curvature of the raceway and;

Π(e) is the second kind of complete elliptic integral, e is the eccentricity of the ellipse, and the approximate calculation formula is used to solve k, e, Γ(e), and Π(e). The error of the calculated results compared with the exact values is no more than 3%. Specifically,

e ² =1-1/k ² (66)

Where _Rx is the equivalent radius of curvature along the direction of motion at the contact point of the ball bearing, and _Ry is the equivalent radius of curvature perpendicular to the direction of motion at the contact point of the ball bearing;

S202: Under point contact conditions, the stress state at the contact center is a triaxial compressive stress state of σ ₁ , σ ₂ and σ ₃ , expressed using engineering principal stress symbols, i.e. σ ₁ ≥σ ₂ ≥σ ₃ :

σ ₁ =-q _max [0.505+0.255(b/a) ^0.6609 ] (68)

σ ₂ =-q _max [1.01-0.250(b/a) ^0.5797 ] (69)

σ ₃ = -q _max (70)

In the formula, the negative sign indicates that the principal stress is compressive stress;

For the static stress state under triaxial compression, the fourth strength theory is used, and the equivalent stress q _eq in the elliptical contact area is:

Substituting equations (68)-(70) into (71), we obtain:

S203: Substitute the n maximum contact loads Q obtained in step S1 into equations (62) and (72) respectively to obtain corresponding n equivalent stresses q _eq .

S3: The relationship between the basic random variables and the equivalent stress q _eq is fitted by the BP neural network, the optimal initial weights and thresholds of the network are obtained by the particle swarm algorithm (PSO), and the limit state function of the plastic deformation of the rolling bearing is established according to the stress-strength interference model; the specific process is as follows:

S301: The n groups of basic random variables obtained in step S1 and the corresponding n equivalent stresses q _eq obtained in step S2 constitute n groups of data sets, and the data sets are divided into a training set, a test set and a validation set;

S302: Normalization and denormalization of data;

In order to avoid the impact caused by the large difference in the order of magnitude between the input and output dimensions, the minimum and maximum values of the rows of the data matrix are mapped to [-1,1] through data normalization to process the matrix so that each indicator is at the same order of magnitude, which is suitable for comprehensive comparative evaluation. Secondly, normalization can also be used to further improve the convergence speed and accuracy of the model.

The random variable data of the training set is normalized by formula (73), where _xk is the original data and _yk is the corresponding normalized data:

The random variable data of the test set and validation set are normalized and denormalized using formula (74):

Where x _min is the minimum value in the data series, x _max is the maximum value in the data series, and the maximum and minimum values of y _k are recorded as y _max and y _min respectively;

S303: Assume that in a D-dimensional space, there are n particles, i.e., a population H = (H ₁ H ₂ … H _n ) composed of the aforementioned n sets of data sets, and the ith particle is represented by a D-dimensional vector H _i = (h _i1 ,h _i2 , …,h _iD ) ^T , which represents the position of the ith particle in the D-dimensional search space and also represents a potential solution to the problem; according to the objective function, the fitness value corresponding to each particle position H _i can be obtained, the speed of the ith particle is V _i = (V _i1 ,V _i2 , …,V _iD ) ^T , its individual extreme value is P _i = (P _i1 ,P _i2 , …,P _iD ) ^T , and the total extreme value of the population is P _g = (P _g1 ,P _g2 , …,P _gD ) ^T ;

In each iteration, the particle updates its own speed and position through individual extrema and global extrema. The update formula is:

Wherein, ω is the inertia weight; i = 1, 2, …, n, d = 1, 2, …, D; k is the current iteration number; V _id is the velocity of the particle; c ₁ and c ₂ are non-negative constants, which become acceleration factors; r ₁ and r ₂ are random numbers distributed between [0, 1]. To prevent blind search of particles, their position and velocity are limited to a certain interval of [-X _max , X _max ], [-V _max , V _max ];

The initial weights and thresholds of the BP neural network are obtained according to the individual. The system output is predicted after the BP neural network is trained with the training data. The absolute value of the error between the predicted output and the expected output and E are taken as the individual fitness value F. The calculation formula is:

In the formula, each particle constitutes a node, that is, n is the number of network output nodes in the neural network, _yi is the expected output of the ith node of the BP neural network, _oi is the predicted output of the ith node, and t is the coefficient;

In order to solve the defects of local optimum and premature convergence in artificial neural networks, the particle swarm algorithm (PSO) is introduced in the penalty function to optimize the BP neural network. Each particle represents the weight and threshold of the neural network. The optimal initial weight and threshold of the network are found through particle optimization, so that the optimized BP neural network can better predict the output of the function.

The process of the function extreme value optimization algorithm based on the PSO algorithm is shown in Figure 4. Among them, the particle and velocity initialization assigns random values to the initial particle position and particle velocity. According to the fitness function of formula (77), the BP neural network is trained with the training data, and the training data error is used as the individual fitness value F. The individual extreme value and the group extreme value are determined according to the initial particle fitness value. The particle velocity and position are updated according to formulas (75) and (76). The individual extreme value and the group extreme value are updated according to the particle fitness value in the new population.

S303: The geometric dimensions, loads and material strength of the rolling bearing are regarded as basic random parameters X = (x ₁ , x ₂ , ..., x _n ). The BP neural network is a one-way propagation forward network consisting of an input layer, a hidden layer and an output layer, wherein the number of nodes in the input layer is p, the number of nodes in the hidden layer is q, and the number of nodes in the output layer is r. The functional relationship of each internal layer is obtained by neuron transmission:

Where h _k is the output of the hidden layer, y _j is the output of the output layer, _xi is the input variable, w _ik and v _kj are weights, α _k and β _j are the thresholds of the hidden layer and the output layer, respectively, f ₁ (·) is the S-type nonlinear function tansig, and f ₂ (·) is the linear function purelin;

Through the neural network mapping relationship shown in equations (78) and (79), the input and output of the data are fitted within the allowable error range, and the fitting function relationship is:

According to the stress-strength interference model, the limit state function g(X) of the reliability of rolling bearing plastic deformation in the absence of probability information is:

Where, the allowable yield stress σ _s is used.

The force of rolling bearing is a complex implicit nonlinear system, which cannot be directly analyzed by high-order moment method. Therefore, BP neural network is used to explicitly express the relationship between basic random variables and equivalent stress q _eq .

S4: Based on the high-order moment method, reliability and sensitivity analysis of the plastic deformation of rolling bearings are carried out to determine the reliability index and reliability of the plastic deformation of rolling bearings and the reliability change trend with the basic random variables, and reveal the influence of the change of basic random variables on the reliability of the plastic deformation of rolling bearings; the specific process is as follows:

S401: Reliability analysis of plastic deformation of rolling bearings;

According to the stress-strength interference model of structural reliability, solving the reliability requires establishing the limit state function g(X) of reliability analysis to represent the two states of reliability of rolling bearings during operation, namely

In the formula, the limit state function equation g(X) = 0 is an n-dimensional limit state surface;

The mean μ _g , standard deviation σ _g , third-order moment θ _g and fourth-order moment η _g of the limit state function:

Wherein, μ _X , C ₂ (X), C ₃ (X) and C ₄ (X) represent the mean matrix, variance and covariance matrix, third-order moment matrix and fourth-order moment matrix of the basic random parameter X respectively; whether the basic random parameter X is an independent variable or not, the diagonal term of the variance and covariance matrix C ₂ (X) is used as the standard deviation vector C ₂ (X) = diag(σ _X )[ρ]diag(σ _X );

When the probability distribution of the basic random variable vector X cannot be determined, if the mean vector E(X) = μ _X , the variance and covariance matrix C ₂ (X), the third-order matrix C ₃ (X) and the fourth-order matrix C ₄ (X) are known, the reliability index β _FM is defined as follows according to the higher-order moment method:

In the formula, is the reliability index when the basic random parameter vector X obeys the normal distribution, is the skewness coefficient of the limit state function g(X), is the peak coefficient of the limit state function g(X). After sorting, another expression of the reliability index of the high-order moment method is obtained:

The approximate estimated value of the reliability R _FM is further determined as:

R _FM =Φ(β _FM ) (89)

Where Φ(·) represents the standard normal distribution function;

S402: performing sensitivity analysis on the plastic deformation of the rolling bearing in the absence of probability information;

According to equations (33) and (34), the sensitivity of reliability R _FM to the mean vector μ _X and standard deviation vector σ _X of the basic random variable vector X is obtained as follows:

In the formula,

Where φ(·) represents the standard normal probability density function;

The derivatives of the reliability index with respect to the first four moments (μ _g , σ _g , θ _g and η _g ) of the limit state function are:

The derivatives of the first four moments of the limit state function with respect to the mean _μX of the basic random variable are:

The derivatives of the first four moments of the limit state function with respect to the standard deviation of the basic random variable σ _X are:

in, [ρ] is the correlation coefficient matrix, diag(·) is the diagonal matrix, which are expressed as:

[ρ ₃ ] is the third-order correlation coefficient matrix;

[ρ ₄ ] is the fourth-order correlation coefficient matrix;

Substituting the known conditions and relevant data into equations (35) and (36), we can obtain the reliability sensitivity and Numeric value;

The reliability sensitivity gradient is expressed as:

In the formula, gradient grad[·] represents the rate of change of reliability;

The dimensionless reliability sensitivity to the mean value τ _i and standard deviation sensitivity η _i of the basic random parameters are expressed as:

The dimensionless reliability sensitivity gradient _si is:

After standardizing s _{i ,} the sensitivity factor λ _i is:

The dimensionless reliability sensitivity avoids the incomparability between reliability sensitivities caused by inconsistent units of random parameters.

In order to facilitate understanding of the above technical solutions of the present invention, the above technical solutions of the present invention are described in detail below through specific embodiments.

Example 1

Among many types of bearings, angular contact ball bearings can withstand radial and axial forces at the same time, and have good stability and good lubrication properties. They are widely used in machine tool spindles and electric main characteristics, which will affect the working performance of the rotating system.

In this embodiment, a certain type of angular contact rolling bearing is selected. It is known that the number of rolling elements Z = 16, the initial contact angle α = 40°, the Poisson's ratio and elastic modulus of the raceway and the rolling element are ξ ₁ = ξ ₂ = 0.3, and E ₁ = E ₂ = 208 GPa. In the basic random parameter vector X = (d _m D _w r _i r ₀ F _a F _r σ _s ) ^T , the allowable yield stress σ _s is the only random variable whose probability distribution is difficult to determine, but the first four moments of σ _s are known. Other basic random parameter variables all obey independent normal distribution, and their means and standard deviations are shown in Table 1.

Table 1 Basic random variables and statistical characteristics of rolling bearings

Latin hypercube design is used to sample d _m , D _w , r _i , r ₀ , F _a , and F _r according to normal distribution. The sampled data are grouped into the static equilibrium equation of the rolling bearing to obtain the equivalent stress q _eq , and 500 groups of data are obtained, 60% of the data are used for the training set, 20% for the validation set, and the remaining 20% for the test set. A 3-layer BP neural network is selected, which is a one-way propagation forward network consisting of an input layer, a hidden layer, and an output layer. The network structure connection is shown in Figure 5. Among them, the number of input layer nodes is 6, the number of hidden layer nodes is 10, and the number of output layer nodes is 1. According to the neural network mapping relationship, the training error is set to 10 ^-5 , and the input and output of the data are fitted within the error allowable range. The fitting function relationship is:

Where _xi is the input variable, _wik and _vkj are the weights between the three layers, _αk and _βj are the thresholds of the hidden layer and the output layer, respectively, _f1 (·) is the S-type nonlinear function sigmoid, and _f2 (·) is the linear function purelin.

The regression graphs of the BP neural network, as shown in Figures 6(a) to 6(d), show the correlation between the equivalent stress output by the neural network and all the data constituting the training set, validation set, and test set, as well as the mean value of the three graph sets R = 0.99995, indicating that the correlation between the output and the target is 99.995%, indicating that the trained BP neural network has a good fitting effect.

According to the stress-strength interference model, the reliability limit state function of rolling bearing plastic deformation in the absence of probability information can be expressed as:

According to the previously derived theory, by substituting the relevant data of the rolling bearing into the first four moment expressions (83)-(86) of the reliability limit state function, and then substituting them into the reliability index and reliability expressions (87) and (89) of the plastic deformation of the rolling bearing for reliability design calculation, the reliability index β _FM and reliability R _FM of the rolling bearing are obtained as follows:

β _FM = 2.7727

R _FM ＝0.99722

Monte Carlo simulation (MCS) has become a benchmark for calculating reliability because the error in the calculation is independent of the dimension of the problem and it does not need to discretize the continuous problem. The reliability obtained by MCS simulation 10 ⁵ times is R _MC :

R _MC ＝0.99805

The calculation results show that the results obtained by the method of the present invention are in good agreement with those obtained by Monte Carlo numerical simulation.

Table 2 Sensitivity results

According to equations (107)-(110), the sensitivity calculation results are shown in Table 2. Combined with Figures 8-9, the following conclusions are obtained:

(1) The analysis results of the reliability mean sensitivity are shown in Figure 7(a). It can be seen that the reliability of rolling bearing plastic deformation is positively sensitive to the mean values of d _m , D _w , _Fa and σ _s of the rolling bearing, indicating that the reliability of the rolling bearing increases as their mean values increase; in other words, the influence is positive. The mean sensitivity of _ri , _re and F _r is negative, indicating that their influence is negative. For example, in the engineering design of rolling bearings, improving the strength properties of the material can effectively reduce the occurrence of plastic deformation and increase the corresponding reliability; as _ri and _re increase, the fit of the rolling bearing will decrease, and the reliability of plastic deformation will also decrease.

(2) The analysis results of the reliability standard deviation sensitivity are shown in Figure 7(b). It can be seen that the reliability of the rolling bearing decreases with the increase of the standard deviation of the seven variables, and the result will tend to fail. The most sensitive are _Dw and _ri , and the other variables are not very sensitive. The larger the standard deviation of the basic random parameters, the greater the random dispersion of the parameters. The standard deviation of the basic random parameters will have a negative impact on the reliability of the plastic deformation of the rolling bearing, which is also consistent with the actual engineering design rules.

(3) The analysis results of the reliability gradient are shown in Figure 7(c). The gradients are ranked from high to low in the following order: ① rolling element diameter _Dw , ② inner groove curvature radius r _i , ③ outer groove curvature radius r _e , ④ allowable yield stress σ _s , ⑤ pitch circle diameter d _m , ⑥ radial force F _r , ⑦ axial force F _a . On this basis, the influence of the change of each basic random parameter on the reliability is evaluated. It can be used as an effective and practical tool for engineering modification design, re-analysis and re-design.

(4) The analysis results of the proportion of the reliability gradient of the unmodified rigidification are shown in Figure 7(d). From _si and λ _i, it can be seen that the three random parameters σ _s , F _r and D _w have a greater influence and are more sensitive, accounting for about 95.5% of the total, followed by r _i . Therefore, these four factors should be strictly controlled in the design, production and use of rolling bearings. It can also be seen from the results that the most important factors that determine whether the rolling bearing undergoes plastic deformation are the bearing material itself and the radial force acting on the rolling bearing, which is also consistent with the actual engineering situation.

The above analysis results show that improving the material properties of rolling bearing materials and reducing the magnitude of the force are the most effective ways to reduce the probability of failure. For the geometric dimensions of rolling bearings, the diameters _Dw and _ri of the rolling elements should be strictly controlled to prevent the occurrence of plastic deformation. Figures 8 and 9 show the changes in the reliability of plastic deformation of rolling bearings near the mean values of _Dw and _ri , so as to improve these two parameters during design. It can be seen that the relationship between reliability and basic random variables _Dw and _ri is consistent with the results of the sensitivity analysis of reliability.

The theory and practice of reliability engineering show that reliability indicators are closely related to statistical characteristic quantities such as the mean and standard deviation of basic random variables and are very sensitive. The calculation results obtained from the reliability theoretical model of rolling bearing plastic deformation in the absence of probability information established by the present invention are consistent with the actual engineering design rules. Furthermore, the theory established by the present invention quantitatively gives the influence of basic random variables on the reliability of rolling bearing plastic deformation, which is exactly where the reliability theoretical model and reliability sensitivity analysis established by the present invention reflect their value.

The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims (1)

1. A reliability and sensitivity analysis method for plastic deformation of a rolling bearing, characterized by comprising the following steps: S1: Establish a mechanical model of rolling bearing plastic deformation, regard the rolling bearing load parameters, geometric parameters and material parameters as basic random variables, use Latin hypercube design to sample n groups of basic random variables, and bring the sampling results into the static equilibrium equation of the rolling bearing to obtain the corresponding n maximum contact loads Q; S2: According to the Hertz contact theory, for the elliptical contact area formed by the rolling element and the raceway, the maximum contact stress is q _max . The fourth strength theory is used to obtain the equivalent stress q _eq . The n equivalent stresses q _eq are obtained through the n maximum contact loads Q. S3: The relationship between the basic random variables and the equivalent stress q _eq is fitted by the BP neural network, the optimal initial weights and thresholds of the network are obtained by the particle swarm algorithm (PSO), and the limit state function of the plastic deformation of the rolling bearing is established according to the stress-strength interference model; S4: Based on the high-order moment method, reliability and sensitivity analysis of the plastic deformation of rolling bearings are carried out to determine the reliability index and reliability of the plastic deformation of rolling bearings and the changing trend of reliability with basic random variables, and to reveal the influence of the change of basic random variables on the reliability of plastic deformation of rolling bearings; The specific process of step S1 is as follows: S101: The static equilibrium equation of a rolling bearing under the combined load of axial force _Fa , radial force _Fr and moment M is: Where, _Kn is the stiffness coefficient, ^α0 is the initial contact angle, R _i is the radius of the inner raceway groove curvature center trajectory, ψ is the azimuth angle, d _m is the pitch circle diameter, δ _a is the relative axial displacement, δ _r is the relative radial displacement, and θ is the relative angular displacement; A is the distance between the center of curvature of the inner and outer grooves at any position in the rolling bearing under no-load contact state: A＝( _fi +fe _- 1) _Dw (4) is the dimensionless displacement: Where, _fi is the inner raceway groove curvature radius coefficient, _fe is the outer raceway groove curvature radius coefficient, and _Dw is the rolling element diameter; S102: Equations (1)-(3) are a set of simultaneous nonlinear equations with δ _a , δ _r and θ as unknown variables. The values of δ _a , δ _r and θ are calculated using the Newton iteration method. At ψ = 0, the maximum contact load Q on the rolling element of the rolling bearing is obtained: S103: Use Latin hypercube design to sample n groups of basic random variables, and bring the sampling results into the static equilibrium equation to obtain the corresponding n maximum contact loads Q; The specific process of step S2 is: S201: According to Hertz's basic theory, for the elliptical contact area formed by the contact between the rolling element and the raceway, the maximum contact stress q _max is: Among them, the major semi-axis a and the minor semi-axis b of the contact area ellipse are: in, is the equivalent elastic modulus, ξ ₁ , ξ ₂ are the Poisson's ratios of the raceway and rolling element, E ₁ , E ₂ are the elastic moduli of the raceway and rolling element, ∑ρ is the curvature of the raceway and; Π(e) is the complete elliptic integral of the second kind, e is the eccentricity of the ellipse, specifically, e ² =1-1/k ² (11) Where _Rx is the equivalent radius of curvature along the direction of motion at the contact point of the ball bearing, and _Ry is the equivalent radius of curvature perpendicular to the direction of motion at the contact point of the ball bearing; S202: Under point contact conditions, the stress state at the contact center is a triaxial compressive stress state of σ ₁ , σ ₂ and σ ₃ , expressed using engineering principal stress symbols, i.e. σ ₁ ≥σ ₂ ≥σ ₃ : σ ₁ =-q _max [0.505+0.255(b/a) ^0.6609 ] (13) σ ₂ =-q _max [1.01-0.250(b/a) ^0.5797 ] (14) σ ₃ = -q _max (15) In the formula, the negative sign indicates that the principal stress is compressive stress; For the static stress state under triaxial compression, the fourth strength theory is used, and the equivalent stress q _eq in the elliptical contact area is: Substituting equations (13)-(15) into (16), we obtain: S203: Substitute the n maximum contact loads Q obtained in step S1 into equations (7) and (17) respectively to obtain corresponding n equivalent stresses q _eq ; The specific process of step S3 is as follows: S301: The n groups of basic random variables obtained in step S1 and the corresponding n equivalent stresses q _eq obtained in step S2 constitute n groups of data sets, and the data sets are divided into a training set, a test set and a validation set; S302: Normalization and denormalization of data; The random variable data of the training set is normalized by formula (18), where _xk is the original data and _yk is the corresponding normalized data: The random variable data of the test set and validation set are normalized and denormalized by formula (19): Where x _min is the minimum value in the data series, x _max is the maximum value in the data series, and the maximum and minimum values of y _k are recorded as y _max and y _min respectively; S303: Assume that in a D-dimensional space, there are n particles, i.e., a population H = (H ₁ H ₂ LH _n ) composed of the aforementioned n sets of data sets, and the ith particle is represented by a D-dimensional vector H _i = (h _i1 ,h _i2 ,L,h _iD ) ^T , which represents the position of the ith particle in the D-dimensional search space and also represents a potential solution to the problem; according to the objective function, the fitness value corresponding to each particle position H _i can be obtained, the speed of the ith particle is V _i = (V _i1 ,V _i2 ,L,V _iD ) ^T , its individual extreme value is P _i = (P _i1 ,P _i2 ,L,P _iD ) ^T , and the total extreme value of the population is P _g = (P _g1 ,P _g2 ,L,P _gD ) ^T ; In each iteration, the particle updates its own speed and position through individual extrema and global extrema. The update formula is: Wherein, ω is the inertia weight; i = 1, 2, L, n, d = 1, 2, L, D; k is the current iteration number; V _id is the velocity of the particle; c ₁ and c ₂ are non-negative constants, which become acceleration factors; r ₁ and r ₂ are random numbers distributed between [0, 1]. To prevent blind search of particles, their position and velocity are limited to a certain interval of [-X _max , X _max ], [-V _max , V _max ]; The initial weights and thresholds of the BP neural network are obtained according to the individual. The system output is predicted after the BP neural network is trained with the training data. The absolute value of the error between the predicted output and the expected output and E are taken as the individual fitness value F. The calculation formula is: In the formula, each particle constitutes a node, that is, n is the number of network output nodes in the neural network, _yi is the expected output of the ith node of the BP neural network, _oi is the predicted output of the ith node, and t is the coefficient; S303: The geometric dimensions, load and material strength of the rolling bearing are regarded as basic random parameters X = ( _x1 , _x2 , L, _xn ). The BP neural network is a one-way propagation forward network consisting of an input layer, a hidden layer and an output layer, wherein the number of nodes in the input layer is p, the number of nodes in the hidden layer is q, and the number of nodes in the output layer is r. The functional relationship of each internal layer is obtained by neuron transmission: Where h _k is the output of the hidden layer, y _j is the output of the output layer, _xi is the input variable, w _ik and v _kj are weights, α _k and β _j are the thresholds of the hidden layer and the output layer, respectively, f ₁ (g) is the S-type nonlinear function tansig, and f ₂ (g) is the linear function purelin; Through the neural network mapping relationship shown in equations (23) and (24), the input and output of the data are fitted within the allowable error range, and the fitting function relationship is: According to the stress-strength interference model, the limit state function g(X) of the reliability of rolling bearing plastic deformation in the absence of probability information is: Where, allowable yield stress σ _s ; The specific process of step S4 is as follows: S401: Reliability analysis of plastic deformation of rolling bearings; According to the stress-strength interference model of structural reliability, solving the reliability requires establishing the limit state function g(X) of reliability analysis to represent the two states of reliability of rolling bearings during operation, namely In the formula, the limit state function equation g(X) = 0 is an n-dimensional limit state surface; The mean μ _g , standard deviation σ _g , third-order moment θ _g and fourth-order moment η _g of the limit state function: Wherein, μ _X , C ₂ (X), C ₃ (X) and C ₄ (X) represent the mean matrix, variance and covariance matrix, third-order moment matrix and fourth-order moment matrix of the basic random parameter X respectively; whether the basic random parameter X is an independent variable or not, the diagonal term of the variance and covariance matrix C ₂ (X) is used as the standard deviation vector C ₂ (X) = diag(σ _X )[ρ]diag(σ _X ); When the probability distribution of the basic random variable vector X cannot be determined, if the mean vector E(X) = μ _X , the variance and covariance matrix C ₂ (X), the third-order matrix C ₃ (X) and the fourth-order matrix C ₄ (X) are known, the reliability index β _FM is defined as follows according to the higher-order moment method: In the formula, is the reliability index when the basic random parameter vector X obeys the normal distribution, is the skewness coefficient of the limit state function g(X), is the peak coefficient of the limit state function g(X). After sorting, another expression of the reliability index of the high-order moment method is obtained: The approximate estimated value of the reliability R _FM is further determined as: R _FM =Φ(β _FM ) (34) Where Φ(·) represents the standard normal distribution function; S402: performing sensitivity analysis on the plastic deformation of the rolling bearing in the absence of probability information; According to equations (33) and (34), the sensitivity of reliability R _FM to the mean vector μ _X and standard deviation vector σ _X of the basic random variable vector X is obtained as follows: In the formula, Where φ(·) represents the standard normal probability density function; The derivatives of the reliability index with respect to the first four moments (μ _g , σ _g , θ _g and η _g ) of the limit state function are: The derivatives of the first four moments of the limit state function with respect to the mean _μX of the basic random variable are: The derivatives of the first four moments of the limit state function with respect to the standard deviation σ _X of the basic random variable are: in, [ρ] is the correlation coefficient matrix, diag(·) is the diagonal matrix, which are expressed as: [ρ ₃ ] is the third-order correlation coefficient matrix; [ρ ₄ ] is the fourth-order correlation coefficient matrix; Substituting the known conditions and relevant data into equations (35) and (36), we can obtain the reliability sensitivity and Numeric value; The reliability sensitivity gradient is expressed as: In the formula, gradient grad[·] represents the rate of change of reliability; The dimensionless reliability sensitivity to the mean value τ _i and standard deviation sensitivity η _i of the basic random parameters are expressed as: The dimensionless reliability sensitivity gradient _si is: After standardizing s _{i ,} the sensitivity factor λ _i is: