CN114580292B - Method for analyzing reliability and sensitivity of plastic deformation of rolling bearing - Google Patents
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Abstract
The invention discloses a method for analyzing the reliability and sensitivity of plastic deformation of a rolling bearing, which combines a Newton iteration method, a La Ding Chao cube design, a neural network method and a high-order moment method, considers the randomness of a rolling bearing load parameter, a geometric parameter and a material parameter, obtains a neural network training sample by utilizing the Newton iteration method and Latin hypercube design sampling on the basis of reasonably constructing a neural network structure, and obtains the mapping relation between a function and a random variable by learning and training the sample. And obtaining a limit state function of contact between the rolling body and the roller path according to the stress-intensity interference model, further obtaining the reliability index and the reliability of the rolling bearing by adopting a high-order moment method, and analyzing the reliability sensitivity of plastic deformation of the rolling bearing.
Description
Technical Field
The invention belongs to the technical field of reliability analysis of rolling bearings, and particularly relates to a method for analyzing reliability and sensitivity of plastic deformation of a rolling bearing.
Background
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 the rolling bearing in a stationary state, elastic deformation between the raceway of the rolling bearing and the rolling elements will be converted into plastic deformation. After the plastic deformation generates the indentation in the rolling bearing, vibration, noise, friction torque change and the like are caused during rotation, so that the rolling bearing cannot work normally, and even becomes a cause of early fatigue failure. Therefore, there is an urgent need to improve the reliability of plastic deformation of rolling bearings to prevent catastrophic failure of the machine.
The reliability design of the rolling bearing mainly focuses on the structural reliability design, in the traditional structural design process, the design parameters are usually quantitatively designed, the method is a probability design method, and the reliability measurement index generally relates to the reliability index beta and the reliability R. Rolling bearings are one of the earliest mechanical products that employed reliability designs, such as: lundberg and Palmgren in 1947 gave a rated life and base rated dynamic load calculation method with 90% reliability, which was incorporated into the ISO international standard in 1962 and is in use today. Thus, unlike other mechanical designs, which consider the reliability design as a special design, a modern advanced design, etc., the conventional design of a rolling bearing is the reliability design.
Because the characteristics and parameters (such as strength, stress, physical variables, geometric dimensions and the like) of the mechanical product have inherent randomness, the existing stress-strength interference model is used for establishing a limit state function, and the important design parameters of the rolling bearing are usually processed into basic random variables, so that the reliability analysis of the rolling bearing is further carried out. When processing design parameters as substantially random variables for reliability analysis, a large number of test samples are required to determine the probability distribution of the substantially random variables and their probability statistical characteristics.
In the conventional rolling bearing design process, the external load, the material strength and the geometric dimension of the rolling bearing are regarded as a certain fixed value, and in fact, the parameters have great randomness, so that the calculation result generates great deviation and even error.
The existing reliability analysis of the plastic deformation of the rolling bearing based on the stress-intensity interference model needs to process important design parameters of the rolling bearing into basic random variables, and because the basic conditions of the reliability design of the rolling bearing are weaker, the working condition is complex, the influence factors are numerous and effective data is lacking, the probability distribution of the design parameters of the rolling bearing is difficult to accurately determine in engineering practice. It would be a very challenging matter to conduct reliability analysis and design in the event of a lack of probability information. In recent years, as rolling bearings are developed in a plurality of directions of high speed, heavy load, high reliability, and the like, there is a higher demand for the reliability of the rolling bearings.
Disclosure of Invention
In order to solve the defects existing in the prior art, the invention provides a method for analyzing the reliability and sensitivity of the plastic deformation of a rolling bearing, which can analyze the reliability of the plastic deformation of the rolling bearing under the condition of probability information loss, and the specific technical scheme of the invention is as follows:
a method for analyzing the reliability and sensitivity of plastic deformation of a rolling bearing, comprising the steps of:
S1: establishing a mechanical model of plastic deformation of the rolling bearing, taking the load parameters, the geometric parameters and the material parameters of the rolling bearing as basic random variables, sampling the basic random variables by n groups by using Latin hypercube design, and taking the sampling result groups into a static balance equation of the rolling bearing to obtain n corresponding maximum contact loads Q;
S2: according to the Hertz contact theory, for an elliptic contact area formed by contact of a rolling body and a raceway, the maximum contact stress is Q max, the fourth strength theory is adopted to obtain equivalent stress Q eq, and n equivalent stresses Q eq are obtained through n maximum contact loads Q;
S3: fitting a relation between a basic random variable and equivalent stress q eq through a BP neural network, obtaining the optimal initial weight and threshold value of the network through a particle swarm algorithm (PSO), and establishing a limit state function of plastic deformation of the rolling bearing according to a stress-intensity interference model;
S4: and (3) analyzing the reliability and sensitivity of the plastic deformation of the rolling bearing based on a high-order moment method, determining the reliability index and the reliability of the plastic deformation of the rolling bearing and the variation trend of the reliability along with the basic random variable, and revealing the influence of the variation of the basic random variable on the plastic deformation reliability of the rolling bearing.
Further, the specific process of step S1 is as follows:
S101: the static equilibrium equation of the rolling bearing under the combined load action of the axial force F a, the radial force F r and the moment M is as follows:
Wherein K n is a rigidity coefficient, alpha 0 is an initial contact angle, R i is a radius of a curvature center track of an inner raceway groove, psi is an azimuth angle, d m is a pitch circle diameter, delta a is relative axial displacement, delta r is relative radial displacement, and theta is relative angular displacement;
A is the distance between the curvature centers of the inner groove and the outer groove at any position under the no-load contact state of the rolling bearing:
A＝(fi+fe-1)Dw (4)
Is a dimensionless displacement:
Wherein f i is an inner raceway groove curvature radius coefficient, f e is an outer raceway groove curvature radius coefficient, and D w is a rolling element diameter;
S102: equations (1) - (3) are simultaneous nonlinear equations with delta a、δr and theta as unknown quantities, values of delta a、δr and theta are calculated by adopting a newton iteration method, and the maximum contact load Q born by the rolling element of the rolling bearing is obtained at the position of phi=0:
s103: and sampling the basic random variable by using Latin hypercube design, and taking the sampling result group into a static force balance equation to obtain n corresponding maximum contact loads Q.
Further, the specific process of step S2 is as follows:
S201: according to the hertz basic theory, for an elliptical contact area formed by the contact of the rolling bodies and the rollaway nest, the maximum contact stress q max is:
The elliptic long half shaft a and the elliptic short half shaft b of the contact area are respectively:
Wherein E is the equivalent elastic modulus, Xi 1、ξ2 is poisson ratio of the roller path and the rolling body respectively, and E 1、E2 is elastic modulus of the roller path and the rolling body respectively; Σρ is the sum of the curvatures of the raceways;
Pi (e) is a second type of complete elliptic integral, e is the elliptic eccentricity, in particular,
e2＝1-1/k2 (11)
Wherein R x is the equivalent radius of curvature in the direction of motion at the ball bearing contact point, and R y is the equivalent radius of curvature in the direction of vertical motion at the ball bearing contact point;
S202: under the point contact condition, the stress states of the contact center point are three-way compressive stress states sigma 1、σ2 and sigma 3, and the engineering principal stress symbol is adopted, namely sigma 1≥σ2≥σ3:
σ1＝-qmax[0.505+0.255(b/a)0.6609] (13)
σ2＝-qmax[1.01-0.250(b/a)0.5797] (14)
σ3＝-qmax (15)
wherein, the negative sign indicates that the main stress is compressive stress;
For the static stress state of three-way compression, adopting a fourth intensity theory, the equivalent stress q eq in the elliptical contact area is as follows:
Bringing formulae (13) - (15) into (16) gives:
s203: and (3) bringing the n maximum contact loads Q obtained in the step S1 into the formulas (7) and (17) respectively to obtain corresponding n equivalent stresses Q eq.
Further, the specific process of step S3 is as follows:
S301: the n groups of basic random variables obtained in the step S1 and the corresponding n equivalent stresses q eq obtained in the step S2 form n groups of data sets, and the data sets are divided into a training set, a testing set and a verification set;
s302: normalization and inverse normalization of data;
the random variable data of the training set is normalized by the formula (18), x k is the original data, and y k is the corresponding normalized data:
The random variable data of the test set and the validation set are normalized and inverse normalized by equation (19):
Wherein x min is the minimum value in the data sequence, x max is the maximum value in the data sequence, and the maximum value and the minimum value of the record y k are y max and y min respectively;
S303: assuming that in a D-dimensional space there are n particles, i.e., a population h= (H 1 H2… Hn) of the aforementioned n sets of data sets, the i-th particle is represented as a D-dimensional vector H i＝(hi1,hi2,…,hiD)T, representing the position of the i-th particle in the D-dimensional search space, and also representing 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＝(Vi1,Vi2,…,ViD)T, the individual extremum is P i＝(Pi1,Pi2,…,PiD)T, and all extremum of the population is P g＝(Pg1,Pg2,…,PgD)T;
in each iteration process, the particle updates its own speed and position through the individual extremum and the global extremum, and the update formula is:
Wherein ω is an inertial weight; i=1, 2, …, n, d=1, 2, …, D; k is the current iteration number; v id is the velocity of the particles; 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], and the position and the speed of the particles are limited to be within a certain interval [ -X max,Xmax]、[-Vmax,Vmax ] in order to prevent blind searching of the particles;
According to the initial weight and threshold value of BP neural network obtained by individual, training BP neural network with training data, predicting system output, taking the sum of absolute error values E between predicted output and expected output as individual fitness value F, and calculating the formula as follows:
wherein each particle forms a node, namely n is the number of nodes output by the network in the neural network, y i is the expected output of the ith node of the BP neural network, o i is the predicted output of the ith node, and t is a coefficient;
s303: considering the geometric dimension, acting load and material strength of the rolling bearing as basic random parameters X= (X 1,x2,…,xn), the BP neural network is a unidirectional propagation forward network formed by an input layer, a hidden layer and an output layer, wherein the node number of the input layer is p, the node number of the hidden layer is q, the node number of the output layer is r, and the functional relation of each layer is obtained by neuron transfer:
Wherein h k is the output of the hidden layer, y j is the output of the output layer, x i is the input variable, w ik and v kj are weights, alpha k and beta j are the thresholds of the hidden layer and the output layer, f 1 (·) is an S-type nonlinear function tansig, and f 2 (·) is a linear function purelin;
Fitting the input and output of the data within the error allowable range by the neural network mapping relation shown in the formula (23) and the formula (24), wherein the fitting function relation is as follows:
According to the stress-intensity interference model, the limit state function g (X) of the rolling bearing plastic deformation reliability under the condition of probability information loss is as follows:
where the allowable yield limit stress σ s.
Further, the specific process of step S4 is as follows:
S401: performing reliability analysis on plastic deformation of the rolling bearing;
according to the stress-intensity interference model of the structural reliability, solving the reliability requires establishing a limit state function g (X) of reliability analysis for representing two states of reliability of the rolling bearing in the working process, namely
Wherein, 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 mu X,C2(X),C3 (X) and C 4 (X) respectively represent a mean matrix, a variance and covariance matrix, a third-order moment matrix and a fourth-order moment matrix of the basic random parameter X; whether the basic random parameter X is an independent variable or not, a diagonal term of a variance-covariance matrix C 2 (X) is adopted as a standard deviation vector C 2(X)＝diag(σX)[ρ]diag(σX;
In the case where 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:
In the method, in the process of the invention, Reliability index when the basic random parameter vector X obeys normal distribution,As a bias coefficient of the limit state function g (X)The kurtosis coefficient of the limit state function g (X) is processed to obtain another expression form of the reliability index of the high-order moment method, wherein the expression form is as follows:
Further determining an approximate estimate of reliability R FM is:
RFM＝Φ(βFM) (34)
Wherein Φ (·) represents a standard normal distribution function;
S402: performing sensitivity analysis on plastic deformation of the rolling bearing under the condition of probability information loss;
According to the formulas (33) and (34), the sensitivities of the reliability R FM to the basic random variable vector X-means vector mu X and standard deviation vector sigma X are respectively as follows:
In the method, in the process of the invention,
Wherein φ (-) represents a standard normal probability density function;
The derivatives of the reliability index with respect to the first fourth order moments (μ g,σg,θg and η g) of the limit state function are:
The derivative of the first fourth moment of the limit state function with respect to the basic random variable mean μ X is:
The derivative of the first fourth moment of the limit state function with respect to the standard deviation σ X of the basic random variable is:
Wherein, [ Ρ ] is a correlation coefficient matrix, diag (·) is a diagonal matrix, expressed as:
[ ρ 3 ] is a third-order correlation coefficient matrix;
[ ρ 4 ] is a fourth-order correlation coefficient matrix;
Substituting the known conditions and the related data into the formulas (35) and (36) respectively to obtain the reliability sensitivity AndA numerical value;
The form of the reliability sensitivity gradient is expressed as:
wherein, gradient grad [. Cndot. ] represents the rate of change of reliability;
The mean sensitivity τ i and standard deviation sensitivity η i of the reliability after dimensionless treatment to the basic random parameters are expressed as:
the dimensionless reliability sensitivity gradient s i is:
The sensitivity factor lambda i is obtained by normalizing s i:
The invention has the beneficial effects that:
1. According to the stress-intensity interference model, the invention provides a theoretical method for designing the reliability and analyzing the sensitivity of the plastic deformation of the rolling bearing under the condition of the probability information deficiency of the allowable yield limit stress, and considers the random parameters such as the geometric dimension, the material property and the like of the rolling bearing and the influence of the joint load of the rolling bearing.
2. The invention deduces a matrix form formula of the reliability design of the rolling bearing plastic deformation under the condition of probability information loss, and deduces a reliability differential sensitivity operation formula of the matrix form according to a direct differential method of sensitivity analysis. For reliability and reliability sensitivity in the absence of probability information, equations in the form of matrices are derived. The matrix format has the advantages of clear expression and easy programming.
3. The invention fits the relationship between the basic random variable and the equivalent stress through the BP neural network because the complexity of the hydrostatic equilibrium equation of the rolling bearing can not directly use the high-order moment method. Meanwhile, in order to solve the defects of local optimization and premature convergence in the artificial neural network, a Particle Swarm Optimization (PSO) is introduced into a penalty function to optimize the BP neural network, so that the optimized BP neural network can better predict the output of the function.
Drawings
For a clearer description of an embodiment of the invention or of the solutions of the prior art, reference will be made to the accompanying drawings, which are used in the embodiments and which are intended to illustrate, but not to limit the invention in any way, the features and advantages of which can be obtained according to these drawings without inventive labour for a person skilled in the art. Wherein:
FIG. 1 is a flow chart of a method for analyzing the reliability and sensitivity of plastic deformation of a rolling bearing according to the present invention;
FIG. 2 is a graph of the displacement of the inner ring of a rolling bearing (outer ring fixation) under combined load of radial force, axial force and moment;
FIG. 3 is a graph of relative positions of groove center of curvature trajectories of rolling bearings before and after inner ring displacement;
FIG. 4 is a flow chart of particle swarm optimization;
FIG. 5 is a topological structure diagram of a BP neural network;
Fig. 6 is a regression diagram of a BP neural network, in which fig. 6 (a) is a regression diagram of a training set, fig. 6 (b) is a regression diagram of a verification set, fig. 6 (c) is a regression diagram of a test set, and fig. 6 (d) is a regression diagram of an overall data set;
FIG. 7 is a graph of reliability sensitivity analysis results for a substantially random variable, wherein FIG. 7 (a) is the mean sensitivity of reliability versus the substantially random variable, FIG. 7 (b) is the standard deviation sensitivity of reliability versus the substantially random variable, FIG. 7 (c) is the reliability sensitivity gradient, and FIG. 7 (d) is a sensitivity factor level graph;
FIG. 8 is a graph of reliability versus rolling element diameter;
Fig. 9 is a graph of reliability versus radius of curvature of the inner groove.
Detailed Description
In order that the above-recited objects, features and advantages of the present invention will be more clearly understood, a more particular description of the invention will be rendered by reference to the appended drawings and appended detailed description. It should be noted that, without conflict, the embodiments of the present invention and features in the embodiments may be combined with each other.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, but the present invention may be practiced in other ways than those described herein, and therefore the scope of the present invention is not limited to the specific embodiments disclosed below.
As shown in fig. 1, a method for analyzing reliability and sensitivity of plastic deformation of a rolling bearing includes the steps of:
S1: establishing a mechanical model of plastic deformation of the rolling bearing, taking the load parameters, the geometric parameters and the material parameters of the rolling bearing as basic random variables, sampling the basic random variables by n groups by using Latin hypercube design, and taking the sampling result groups into a static balance equation of the rolling bearing to obtain n corresponding maximum contact loads Q; the specific process is as follows:
S101: the static equilibrium equation of the rolling bearing under the combined load action of the axial force F a, the radial force F r and the moment M is as follows:
Wherein K n is a rigidity coefficient, alpha 0 is an initial contact angle, R i is a radius of a curvature center track of an inner raceway groove, psi is an azimuth angle, d m is a pitch circle diameter, delta a is relative axial displacement, delta r is relative radial displacement, and theta is relative angular displacement; the outer ring of the rolling bearing is fixed under the combined load action of radial force, axial force and moment, the displacement relation of the inner ring is shown in figure 2, and the relative position relation of the groove curvature center tracks of the rolling bearing before and after the displacement of the inner ring is shown in figure 3;
A is the distance between the curvature centers of the inner groove and the outer groove at any position under the no-load contact state of the rolling bearing:
A＝(fi+fe-1)Dw (59)
Is a dimensionless displacement:
Wherein f i is an inner raceway groove curvature radius coefficient, f e is an outer raceway groove curvature radius coefficient, and D w is a rolling element diameter;
S102: equations (56) (1) - (3) (58) are simultaneous nonlinear equations with delta a、δr and theta as unknowns, values of delta a、δr and theta are calculated by adopting a newton iteration method, and the maximum contact load Q of the rolling element of the rolling bearing is obtained at the position of phi=0:
s103: and sampling the basic random variable by using Latin hypercube design, and taking the sampling result group into a static force balance equation to obtain n corresponding maximum contact loads Q.
Among them, latin Hypercube Design (LHD), which has an efficient space filling capability, can be studied in more combinations with the same number of points than in orthogonal experiments. LHD is performed in n-dimensional space with each dimension of coordinate intervalEvenly divided into m intervals, each interval being denoted asM points are randomly selected, so that each level of a factor is only researched once, namely, an n-dimensional space is formed, and a Latin hypercube method with the sample number of m is recorded as m multiplied by n LHD.
S2: according to the Hertz contact theory, for an elliptic contact area formed by contact of a rolling body and a raceway, the maximum contact stress is Q max, the fourth strength theory is adopted to obtain equivalent stress Q eq, and n equivalent stresses Q eq are obtained through n maximum contact loads Q; the specific process is as follows:
S201: according to the hertz basic theory, for an elliptical contact area formed by the contact of the rolling bodies and the rollaway nest, the maximum contact stress q max is:
The elliptic long half shaft a and the elliptic short half shaft b of the contact area are respectively:
Wherein E is the equivalent elastic modulus, Xi 1、ξ2 is poisson ratio of the roller path and the rolling body respectively, and E 1、E2 is elastic modulus of the roller path and the rolling body respectively; Σρ is the sum of the curvatures of the raceways;
pi (e) is a second type of complete elliptic integral, e is elliptic eccentricity, and the approximate calculation formulas are adopted to solve k, e, Γ (e) and pi (e), and compared with the accurate value, the calculated result has an error of not more than 3 percent, specifically,
e2＝1-1/k2 (66)
Wherein R x is the equivalent radius of curvature in the direction of motion at the ball bearing contact point, and R y is the equivalent radius of curvature in the direction of vertical motion at the ball bearing contact point;
S202: under the point contact condition, the stress states of the contact center point are three-way compressive stress states sigma 1、σ2 and sigma 3, and the engineering principal stress symbol is adopted, namely sigma 1≥σ2≥σ3:
σ1＝-qmax[0.505+0.255(b/a)0.6609] (68)
σ2＝-qmax[1.01-0.250(b/a)0.5797] (69)
σ3＝-qmax (70)
wherein, the negative sign indicates that the main stress is compressive stress;
For the static stress state of three-way compression, adopting a fourth intensity theory, the equivalent stress q eq in the elliptical contact area is as follows:
bringing formulae (68) - (70) into (71) yields:
S203: and (3) bringing the n maximum contact loads Q obtained in the step S1 into the formulas (62) and (72) respectively, so as to obtain corresponding n equivalent stresses Q eq.
S3: fitting a relation between a basic random variable and equivalent stress q eq through a BP neural network, obtaining the optimal initial weight and threshold value of the network through a particle swarm algorithm (PSO), and establishing a limit state function of plastic deformation of the rolling bearing according to a stress-intensity interference model; the specific process is as follows:
S301: the n groups of basic random variables obtained in the step S1 and the corresponding n equivalent stresses q eq obtained in the step S2 form n groups of data sets, and the data sets are divided into a training set, a testing set and a verification set;
s302: normalization and inverse normalization of data;
In order to avoid the influence caused by large order of magnitude difference between the input and output dimensional data, the matrix is processed by mapping the minimum value and the maximum value of the rows of the data matrix to [ -1,1] through data normalization processing, so that all indexes are in the same order of magnitude, and the method is suitable for comprehensive comparison evaluation. Secondly, the convergence speed of the model and the precision of the model can be further improved through normalization.
The random variable data of the training set is normalized by the formula (73), x k is the original data, and y k is the corresponding normalized data:
The random variable data for the test set and the validation set are normalized and inverse normalized by equation (74):
Wherein x min is the minimum value in the data sequence, x max is the maximum value in the data sequence, and the maximum value and the minimum value of the record y k are y max and y min respectively;
S303: assuming that in a D-dimensional space there are n particles, i.e., a population h= (H 1 H2… Hn) of the aforementioned n sets of data sets, the i-th particle is represented as a D-dimensional vector H i＝(hi1,hi2,…,hiD)T, representing the position of the i-th particle in the D-dimensional search space, and also representing 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＝(Vi1,Vi2,…,ViD)T, the individual extremum is P i＝(Pi1,Pi2,…,PiD)T, and all extremum of the population is P g＝(Pg1,Pg2,…,PgD)T;
in each iteration process, the particle updates its own speed and position through the individual extremum and the global extremum, and the update formula is:
Wherein ω is an inertial weight; i=1, 2, …, n, d=1, 2, …, D; k is the current iteration number; v id is the velocity of the particles; 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], and the position and the speed of the particles are limited to be within a certain interval [ -X max,Xmax]、[-Vmax,Vmax ] in order to prevent blind searching of the particles;
According to the initial weight and threshold value of BP neural network obtained by individual, training BP neural network with training data, predicting system output, taking the sum of absolute error values E between predicted output and expected output as individual fitness value F, and calculating the formula as follows:
wherein each particle forms a node, namely n is the number of nodes output by the network in the neural network, y i is the expected output of the ith node of the BP neural network, o i is the predicted output of the ith node, and t is a coefficient;
In order to solve the defects of local optimization and premature convergence in an artificial neural network, a Particle Swarm Optimization (PSO) is introduced into a penalty function to optimize the BP neural network, each particle represents the weight and the threshold of the neural network, and 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 flow of the function extremum optimizing algorithm based on PSO algorithm is shown in figure 4. Wherein the particle and velocity initialization imparts random values to the initial particle position and particle velocity. The BP neural network is trained with training data according to the fitness function of equation (77), and the training data error is taken as the individual fitness value F. And determining an individual extremum and a population extremum according to the initial particle fitness value. The particle velocity and position are updated according to equations (75) and (76). And updating the individual extremum and the population extremum according to the particle fitness value in the new population.
S303: considering the geometric dimension, acting load and material strength of the rolling bearing as basic random parameters X= (X 1,x2,…,xn), the BP neural network is a unidirectional propagation forward network formed by an input layer, a hidden layer and an output layer, wherein the node number of the input layer is p, the node number of the hidden layer is q, the node number of the output layer is r, and the functional relation of each layer is obtained by neuron transfer:
Wherein h k is the output of the hidden layer, y j is the output of the output layer, x i is the input variable, w ik and v kj are weights, alpha k and beta j are the thresholds of the hidden layer and the output layer, f 1 (·) is an S-type nonlinear function tansig, and f 2 (·) is a linear function purelin;
Fitting the input and output of the data within the error allowable range by the neural network mapping relation shown in the formula (78) and the formula (79), wherein the fitting function relation is as follows:
According to the stress-intensity interference model, the limit state function g (X) of the rolling bearing plastic deformation reliability under the condition of probability information loss is as follows:
where the allowable yield limit stress σ s.
The stress of the rolling bearing is a complex implicit nonlinear system, and the reliability analysis cannot be directly performed by a high-order moment method. Thus, BP neural networks are used to explicitly express the relationship between a substantially random variable and an equivalent stress of q eq.
S4: based on a high-order moment method, analyzing the reliability and sensitivity of the plastic deformation of the rolling bearing, determining the reliability index and the reliability of the plastic deformation of the rolling bearing and the variation trend of the reliability along with the basic random variable, and revealing the influence of the variation of the basic random variable on the plastic deformation reliability of the rolling bearing; the specific process is as follows:
S401: performing reliability analysis on plastic deformation of the rolling bearing;
according to the stress-intensity interference model of the structural reliability, solving the reliability requires establishing a limit state function g (X) of reliability analysis for representing two states of reliability of the rolling bearing in the working process, namely
Wherein, 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 mu X,C2(X),C3 (X) and C 4 (X) respectively represent a mean matrix, a variance and covariance matrix, a third-order moment matrix and a fourth-order moment matrix of the basic random parameter X; whether the basic random parameter X is an independent variable or not, a diagonal term of a variance-covariance matrix C 2 (X) is adopted as a standard deviation vector C 2(X)＝diag(σX)[ρ]diag(σX;
In the case where 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:
In the method, in the process of the invention, Reliability index when the basic random parameter vector X obeys normal distribution,As a bias coefficient of the limit state function g (X)The kurtosis coefficient of the limit state function g (X) is processed to obtain another expression form of the reliability index of the high-order moment method, wherein the expression form is as follows:
Further determining an approximate estimate of reliability R FM is:
RFM＝Φ(βFM) (89)
Wherein Φ (·) represents a standard normal distribution function;
S402: performing sensitivity analysis on plastic deformation of the rolling bearing under the condition of probability information loss;
According to the formulas (33) and (34), the sensitivities of the reliability R FM to the basic random variable vector X-means vector mu X and standard deviation vector sigma X are respectively as follows:
In the method, in the process of the invention,
Wherein φ (-) represents a standard normal probability density function;
The derivatives of the reliability index with respect to the first fourth order moments (μ g,σg,θg and η g) of the limit state function are:
The derivative of the first fourth moment of the limit state function with respect to the basic random variable mean μ X is:
The derivative of the first fourth moment of the limit state function with respect to the standard deviation σ X of the basic random variable is:
Wherein, [ Ρ ] is a correlation coefficient matrix, diag (·) is a diagonal matrix, expressed as:
[ ρ 3 ] is a third-order correlation coefficient matrix;
[ ρ 4 ] is a fourth-order correlation coefficient matrix;
Substituting the known conditions and the related data into the formulas (35) and (36) respectively to obtain the reliability sensitivity AndA numerical value;
The form of the reliability sensitivity gradient is expressed as:
wherein, gradient grad [. Cndot. ] represents the rate of change of reliability;
The mean sensitivity τ i and standard deviation sensitivity η i of the reliability after dimensionless treatment to the basic random parameters are expressed as:
the dimensionless reliability sensitivity gradient s i is:
The sensitivity factor lambda i is obtained by normalizing s i:
The reliability sensitivity dimensionless avoids that the reliability sensitivity is not compared due to the non-unity of units of random parameters.
In order to facilitate understanding of the above technical solutions of the present invention, the following detailed description of the above technical solutions of the present invention is provided by specific embodiments.
Example 1
Among the various bearing types, the angular contact ball bearing can bear radial force and axial force simultaneously, has good stability and good lubrication characteristic, and can influence the working performance of a rotating system when being widely applied to a machine tool main shaft and an electric main characteristic.
In this embodiment, a certain type of angular contact rolling bearing is selected, the number of rolling bodies z=16, the initial contact angle α=40°, and the poisson ratio and the elastic modulus of the raceway and the rolling bodies are respectively ζ 1＝ξ2＝0.3,E1＝E2 =208 GPa. In the basic random parameter vector x= (d m Dw ri r0 Fa Fr σs)T, the allowable yield limit stress σ s is a random variable whose unique probability distribution is difficult to determine, but the first fourth moment of σ s is known, Other substantially random parametric variables were subject to independent normal distributions, with the mean and standard deviation shown in table 1.
Table 1 basic random variables and statistical characteristics of rolling bearings
D m、Dw、ri、r0、Fa、Fr is sampled according to normal distribution by using Latin hypercube design, sampled data are brought into a static equilibrium equation of the rolling bearing according to groups, equivalent stress q eq is obtained, 500 groups of data are obtained, 60% of data are used for a training set, 20% are used for a verification set, and the rest 20% are used for a test set. The BP neural network with the 3-layer structure is selected, and is a unidirectional propagation forward network formed by an input layer, a hidden layer and an output layer, and the network structure is connected as shown in figure 5. 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 relation, setting the training error as 10 -5, fitting the input and output of the data within the error allowable range, and then the fitting function relation is as follows:
wherein x i is an input variable, w ik and v kj are weights between 3 layers, alpha k and beta j are thresholds of a hidden layer and an output layer, f 1 (·) is an S-shaped nonlinear function sigmoid, and f 2 (·) is a linear function purelin.
As shown in fig. 6 (a) -6 (d), the regression graph of the BP neural network gives the correlation between the equivalent stress output by the neural network and all the data constituting the 3 sets of training set, verification set and test set, and the average value r=0.99995 of the three atlas, which indicates that the correlation degree between the output and the target is 99.995%, and shows that the trained BP neural network has a better fitting effect.
According to the stress-intensity interference model, the reliability limit state function of the rolling bearing plastic deformation in the absence of probability information can be expressed as:
According to the theory deduced previously, the reliability index beta FM and the reliability R FM of the rolling bearing are obtained by substituting the rolling bearing related data into the first fourth moment expressions (83) - (86) of the reliability limit state function and then substituting into the expressions (87) and (89) of the reliability index and the reliability of the plastic deformation of the rolling bearing to perform reliability design calculation, wherein the reliability index beta FM and the reliability R FM are as follows:
βFM＝2.7727
RFM＝0.99722
Monte Carlo Simulation (MCS) has become a target for computational reliability because it is independent of the error of the computation and the dimensionality of the problem, and does not have to be discretized for continuity problems. The reliability obtained by MCS simulation 10 5 times is R MC:
RMC＝0.99805
the calculation result shows that the result obtained by the method has good consistency with the result obtained by Monte Carlo numerical simulation.
TABLE 2 sensitivity results
The sensitivity calculation results are shown in table 2 according to formulas (107) - (110), and the following conclusions are drawn in connection with fig. 8-9:
(1) The analysis result of the reliability average sensitivity is shown in FIG. 7 (a), from It can be seen that the reliability of the plastic deformation of the rolling bearing is positive to the mean sensitivity of d m、Dw、Fa and σ s of the rolling bearing, indicating that the reliability of the rolling bearing increases as their mean increases; that is, the effect is positive. The mean sensitivities of r i、re and F r are negative, indicating that their effects are negative. For example, in the engineering design of the rolling bearing, improving the strength property of the material can effectively reduce the occurrence of plastic deformation and increase the corresponding reliability; as r i and r e increase, the degree of fit of the rolling bearing decreases, and the reliability of plastic deformation also decreases.
(2) The analysis result of the reliability standard deviation sensitivity is shown in FIG. 7 (b), fromIt can be seen that the reliability of the rolling bearing decreases with an increase in the standard deviation of 7 variables, with the result that it tends to fail. Most sensitive are D w and r i, the other variables being less sensitive. The larger the standard deviation of the basic random parameters, the larger the random dispersion of the parameters, the standard deviation of the basic random parameters can negatively affect the reliability of the plastic deformation of the rolling bearing, which is also consistent with the actual engineering design rules.
(3) As shown in fig. 7 (c), the analysis results of the reliability gradient are shown, and the gradient is ordered from high to low in the following order: ① The rolling element diameter D w,② inner groove radius of curvature r i,③ outer groove radius of curvature r e,④ allows for the yield limit stress σ s,⑤ pitch diameter D m,⑥ radial force F r,⑦ axial force F a. On this basis, the degree of influence of the variation of each substantially random parameter on the reliability is evaluated. Can be used as an effective and practical tool for engineering modification design, re-analysis and re-design.
(4) The results of the duty cycle analysis of the reliability gradients for the quantitative rigidification are shown in fig. 7 (D), and as seen from s i and λ i, the three random parameters σ s、Fr and D w are more sensitive, accounting for about 95.5% of the whole, and secondly r i, so that these four factors should be strictly controlled in the design, production and use of the rolling bearing. It can also be seen from the results that it is determined whether the rolling bearing is plastically deformed, and most importantly the bearing material itself and the radial forces acting on the rolling bearing, which also correspond to the actual engineering situation.
The analysis results show that the improvement of the material performance of the rolling bearing material and the reduction of the magnitude of the acting force are the most effective ways to reduce the failure probability. For the geometry of the rolling bearing, the diameters D w and r i of the rolling elements should be strictly controlled to prevent plastic deformation from occurring. Fig. 8 and 9 show the variation of the reliability of the plastic deformation of the rolling bearing around the mean value of D w and r i in order to improve both parameters at the time of design, and it can be seen that the relationship between the reliability and the substantially random variables D w and r i is consistent with the sensitivity analysis result of the reliability.
Theory and practice of reliability engineering show that the reliability index is closely related and very sensitive to the statistical characteristic quantities such as the mean value, standard deviation and the like of the basic random variable. The calculation result obtained from the reliability theoretical model of the rolling bearing plastic deformation under the condition of the probability information deficiency established by the invention is consistent with the actual engineering design rule, and further, the influence of the basic random variable on the rolling bearing plastic deformation reliability is quantitatively given by the theory established by the invention, which is the point that the reliability theoretical model and the sensitivity analysis of the reliability established by the invention embody the value.
The above description is only of the preferred embodiments of the present invention and is not intended to limit the present invention, but various modifications and variations can be made to the present invention by those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (1)
1. A method for analyzing the reliability and sensitivity of plastic deformation of a rolling bearing, comprising the steps of:
S1: establishing a mechanical model of plastic deformation of the rolling bearing, taking the load parameters, the geometric parameters and the material parameters of the rolling bearing as basic random variables, sampling the basic random variables by n groups by using Latin hypercube design, and taking the sampling result groups into a static balance equation of the rolling bearing to obtain n corresponding maximum contact loads Q;
S2: according to the Hertz contact theory, for an elliptic contact area formed by contact of a rolling body and a raceway, the maximum contact stress is Q max, the fourth strength theory is adopted to obtain equivalent stress Q eq, and n equivalent stresses Q eq are obtained through n maximum contact loads Q;
S3: fitting a relation between a basic random variable and equivalent stress q eq through a BP neural network, obtaining the optimal initial weight and threshold value of the network through a particle swarm algorithm (PSO), and establishing a limit state function of plastic deformation of the rolling bearing according to a stress-intensity interference model;
S4: based on a high-order moment method, analyzing the reliability and sensitivity of the plastic deformation of the rolling bearing, determining the reliability index and the reliability of the plastic deformation of the rolling bearing and the variation trend of the reliability along with the basic random variable, and revealing the influence of the variation of the basic random variable on the plastic deformation reliability of the rolling bearing;
the specific process of the step S1 is as follows:
S101: the static equilibrium equation of the rolling bearing under the combined load action of the axial force F a, the radial force F r and the moment M is as follows:
Wherein K n is a rigidity coefficient, alpha 0 is an initial contact angle, R i is a radius of a curvature center track of an inner raceway groove, psi is an azimuth angle, d m is a pitch circle diameter, delta a is relative axial displacement, delta r is relative radial displacement, and theta is relative angular displacement;
A is the distance between the curvature centers of the inner groove and the outer groove at any position under the no-load contact state of the rolling bearing:
A＝(fi+fe-1)Dw (4)
Is a dimensionless displacement:
Wherein f i is an inner raceway groove curvature radius coefficient, f e is an outer raceway groove curvature radius coefficient, and D w is a rolling element diameter;
S102: equations (1) - (3) are simultaneous nonlinear equations with delta a、δr and theta as unknown quantities, values of delta a、δr and theta are calculated by adopting a newton iteration method, and the maximum contact load Q born by the rolling element of the rolling bearing is obtained at the position of phi=0:
s103: sampling n groups of basic random variables by using Latin hypercube design, and taking the sampling result groups into a static force balance equation to obtain n corresponding maximum contact loads Q;
The specific process of the step S2 is as follows:
S201: according to the hertz basic theory, for an elliptical contact area formed by the contact of the rolling bodies and the rollaway nest, the maximum contact stress q max is:
The elliptic long half shaft a and the elliptic short half shaft b of the contact area are respectively:
Wherein, Is equivalent elastic modulus,Xi 1、ξ2 is poisson ratio of the roller path and the rolling body respectively, and E 1、E2 is elastic modulus of the roller path and the rolling body respectively; Σρ is the sum of the curvatures of the raceways;
Pi (e) is a second type of complete elliptic integral, e is the elliptic eccentricity, in particular,
e2＝1-1/k2 (11)
Wherein R x is the equivalent radius of curvature in the direction of motion at the ball bearing contact point, and R y is the equivalent radius of curvature in the direction of vertical motion at the ball bearing contact point;
S202: under the point contact condition, the stress states of the contact center point are three-way compressive stress states sigma 1、σ2 and sigma 3, and the engineering principal stress symbol is adopted, namely sigma 1≥σ2≥σ3:
σ1＝-qmax[0.505+0.255(b/a)0.6609] (13)
σ2＝-qmax[1.01-0.250(b/a)0.5797] (14)
σ3＝-qmax (15)
wherein, the negative sign indicates that the main stress is compressive stress;
For the static stress state of three-way compression, adopting a fourth intensity theory, the equivalent stress q eq in the elliptical contact area is as follows:
Bringing formulae (13) - (15) into (16) gives:
S203: respectively bringing n maximum contact loads Q obtained in the step S1 into formulas (7) and (17) to obtain corresponding n equivalent stresses Q eq;
The specific process of the step S3 is as follows:
S301: the n groups of basic random variables obtained in the step S1 and the corresponding n equivalent stresses q eq obtained in the step S2 form n groups of data sets, and the data sets are divided into a training set, a testing set and a verification set;
s302: normalization and inverse normalization of data;
the random variable data of the training set is normalized by the formula (18), x k is the original data, and y k is the corresponding normalized data:
The random variable data of the test set and the validation set are normalized and inverse normalized by equation (19):
Wherein x min is the minimum value in the data sequence, x max is the maximum value in the data sequence, and the maximum value and the minimum value of the record y k are y max and y min respectively;
S303: assuming that in a D-dimensional space there are n particles, i.e., a population h= (H 1 H2L Hn) of the aforementioned n sets of data sets, the i-th particle is represented as a D-dimensional vector H i＝(hi1,hi2,L,hiD)T, representing the position of the i-th particle in the D-dimensional search space, and also representing 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＝(Vi1,Vi2,L,ViD)T, the individual extremum is P i＝(Pi1,Pi2,L,PiD)T, and all extremum of the population is P g＝(Pg1,Pg2,L,PgD)T;
in each iteration process, the particle updates its own speed and position through the individual extremum and the global extremum, and the update formula is:
Wherein ω is an inertial weight; i=1, 2, l, n, d=1, 2, l, d; k is the current iteration number; v id is the velocity of the particles; 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], and the position and the speed of the particles are limited to be within a certain interval [ -X max,Xmax]、[-Vmax,Vmax ] in order to prevent blind searching of the particles;
According to the initial weight and threshold value of BP neural network obtained by individual, training BP neural network with training data, predicting system output, taking the sum of absolute error values E between predicted output and expected output as individual fitness value F, and calculating the formula as follows:
wherein each particle forms a node, namely n is the number of nodes output by the network in the neural network, y i is the expected output of the ith node of the BP neural network, o i is the predicted output of the ith node, and t is a coefficient;
S303: considering the geometric dimension, acting load and material strength of the rolling bearing as basic random parameters X= (X 1,x2,L,xn), the BP neural network is a unidirectional propagation forward network formed by an input layer, a hidden layer and an output layer, wherein the node number of the input layer is p, the node number of the hidden layer is q, the node number of the output layer is r, and the functional relation of each layer is obtained by neuron transfer:
wherein h k is the output of the hidden layer, y j is the output of the output layer, x i 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, f 1 (g) is an S-type nonlinear function tansig, and f 2 (g) is a linear function purelin;
Fitting the input and output of the data within the error allowable range by the neural network mapping relation shown in the formula (23) and the formula (24), wherein the fitting function relation is as follows:
According to the stress-intensity interference model, the limit state function g (X) of the rolling bearing plastic deformation reliability under the condition of probability information loss is as follows:
Wherein the allowable yield limit stress sigma s;
the specific process of the step S4 is as follows:
S401: performing reliability analysis on plastic deformation of the rolling bearing;
according to the stress-intensity interference model of the structural reliability, solving the reliability requires establishing a limit state function g (X) of reliability analysis for representing two states of reliability of the rolling bearing in the working process, namely
Wherein, 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 mu X,C2(X),C3 (X) and C 4 (X) respectively represent a mean matrix, a variance and covariance matrix, a third-order moment matrix and a fourth-order moment matrix of the basic random parameter X; whether the basic random parameter X is an independent variable or not, a diagonal term of a variance-covariance matrix C 2 (X) is adopted as a standard deviation vector C 2(X)＝diag(σX)[ρ]diag(σX;
In the case where 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:
In the method, in the process of the invention, Reliability index when the basic random parameter vector X obeys normal distribution,As a bias coefficient of the limit state function g (X)The kurtosis coefficient of the limit state function g (X) is processed to obtain another expression form of the reliability index of the high-order moment method, wherein the expression form is as follows:
Further determining an approximate estimate of reliability R FM is:
RFM＝Φ(βFM) (34)
Wherein Φ (·) represents a standard normal distribution function;
S402: performing sensitivity analysis on plastic deformation of the rolling bearing under the condition of probability information loss;
According to the formulas (33) and (34), the sensitivities of the reliability R FM to the basic random variable vector X-means vector mu X and standard deviation vector sigma X are respectively as follows:
In the method, in the process of the invention,
Wherein φ (-) represents a standard normal probability density function;
The derivatives of the reliability index with respect to the first fourth order moments (μ g,σg,θg and η g) of the limit state function are:
The derivative of the first fourth moment of the limit state function with respect to the basic random variable mean μ X is:
The derivative of the first fourth moment of the limit state function with respect to the standard deviation σ X of the basic random variable is:
Wherein, [ Ρ ] is a correlation coefficient matrix, diag (·) is a diagonal matrix, expressed as:
[ ρ 3 ] is a third-order correlation coefficient matrix;
[ ρ 4 ] is a fourth-order correlation coefficient matrix;
Substituting the known conditions and the related data into the formulas (35) and (36) respectively to obtain the reliability sensitivity And (3) withA numerical value;
The form of the reliability sensitivity gradient is expressed as:
wherein, gradient grad [. Cndot. ] represents the rate of change of reliability;
The mean sensitivity τ i and standard deviation sensitivity η i of the reliability after dimensionless treatment to the basic random parameters are expressed as:
the dimensionless reliability sensitivity gradient s i is:
The sensitivity factor lambda i is obtained by normalizing s i:
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