CN115688311A - Uncertainty analysis method and system for planetary roller screw pair - Google Patents
Uncertainty analysis method and system for planetary roller screw pair Download PDFInfo
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Abstract
The invention discloses an uncertainty analysis method and a system of a planetary roller screw pair, which simplify the extreme state function by constructing the extreme state function under the contact fatigue failure mode of a thread of the planetary roller screw pair, introduce an expected risk learning function to construct an active learning agent model of the planetary roller screw pair, generate sampling sample points by adopting a quasi-Monte Carlo method, call the established active learning agent model to calculate the simplified extreme state function corresponding to each group of sample points, and carry out uncertainty analysis on the planetary roller screw pair to obtain the failure probability and the reliability sensitivity of the planetary roller screw pair under any working condition. The method has the advantages that the number of times of calling the function of the extreme state function can be effectively reduced, the calculation cost is reduced, the uncertainty analysis efficiency of the planetary roller screw pair is improved, a designer is helped to find out main factors influencing the reliability of the structure, and further, the theoretical basis is laid for the design optimization of products.
Description
Technical Field
The invention relates to the technical field of precision thread transmission of planetary roller screws, in particular to an uncertainty analysis method and system of a planetary roller screw pair.
Background
The planetary roller screw pair is a precise screw transmission device and is mainly used as an actuating mechanism of an electromechanical actuator. Usually, the screw rod is connected with a servo motor, the nut is connected with a load, and the plurality of rollers perform planetary motion between the screw rod and the nut. The threads on the two sides of the roller are meshed with the screw rod and the nut at the same time, so that the rotary motion of the screw rod can be converted into linear thrust of the nut. The planetary roller screw pair has a large number of contact points and no rolling body circulating device, so that the planetary roller screw pair has the advantages of high bearing capacity, good robustness, high speed and acceleration and the like.
The wide range of uncertainties in the manufacturing, assembly, measurement and operation of planetary roller screw pairs, which may exacerbate the degree of load maldistribution, result in threads that may exhibit excessive contact stresses even within the rated load range, and the same batch of products may have different service lives under the same operating conditions. Therefore, the load distribution and contact characteristics of the multiple thread pairs, taking into account uncertainty, can greatly affect the durability and reliability of the planetary roller screw pair.
Uncertainty analysis is often used as an evaluation criterion for measuring product quality and is considered as an essential part of the design process. However, a reasonable and effective uncertainty analysis model for primarily evaluating the reliability of the planetary roller screw pair is not available at present. The uncertainty analysis of the planetary roller screw pair is helpful for a designer to find out main factors influencing the failure of the planetary roller screw pair before selecting materials of products, so that measures for preventing or delaying the failure are provided, theoretical basis is provided for the design optimization of the planetary roller screw pair, and the method has important theoretical significance and engineering application value for developing the domestic high-performance planetary roller screw pair and improving the comprehensive performance of an electromechanical servo actuating system.
Disclosure of Invention
The invention aims to provide a reasonable and effective uncertainty analysis method for a planetary roller screw pair.
In order to solve the above problems, the present invention provides an uncertainty analysis method of a planetary roller screw pair, including the steps of:
s1, solving load distribution and contact characteristics of the threads of the planetary roller screw pair through structural parameters, material properties and boundary conditions, considering uncertainty existing in the structural parameters, the material properties and the boundary conditions, and constructing a limit state function under a contact fatigue failure mode of the threads of the planetary roller screw pair based on an S-N curve and a stress-intensity interference theory;
s2, introducing a deterministic low-deviation point set Halton sequence, analyzing the sensitivity and contribution degree of uncertainty factors on the influence of the load-bearing capacity and the contact performance of the thread of the planetary roller screw pair by adopting a test design method, and reducing the dimension of a limit state function by using a parameter with the top order of importance as a random variable to obtain a simplified limit state function;
and S3, introducing an expected risk learning function to construct a planetary roller screw pair active learning agent model, generating sampling sample points by adopting a quasi-Monte Carlo method, calling the constructed active learning agent model to calculate simplified limit state functions corresponding to each group of sample points, and carrying out uncertainty analysis on the planetary roller screw pair to obtain the failure probability and reliability sensitivity of the planetary roller screw pair under any working condition.
As a further improvement of the invention, in step S1, solving the thread load distribution and contact characteristics of the planetary roller screw pair through structural parameters, material properties and boundary conditions includes:
solving a load distribution model of the thread of the planetary roller screw pair by adopting an iterative algorithm, which comprises the following steps:
wherein, P S ,P R And P N The screw pitches of the screw rod, the roller and the nut are respectively set, tau is the thread number of the roller participating in contact, z is the number of the roller, the thread is numbered from the fixed end to the free end of the screw rod in sequence, i =1,2, \ 8230, tau and F are axial external loads, F is the axial external load SRi The ith pair of thread upper bearings are arranged at the contact side of the roller and the screw rodAxial load received, F NRi The axial load borne on the ith pair of threads on the contact side of the roller and the nut;
the rigidity of the shaft section of the screw rod is improved,
in order to provide the rigidity of the roller shaft section,
in order to provide rigidity to the shaft section of the nut,
for the rigidity of the screw thread teeth of the screw rod,
in order to provide rigidity to the thread ridge of the roller,
the rigidity of the nut thread;
solving the contact rigidity of the screw and the roller as follows:
wherein the content of the first and second substances,
the contact rigidity of the screw rod and the roller;
the screw rod and the roller are elastically contacted and deformed for the ith pair of thread teeth;
solving the contact rigidity of the nut and the roller thread pair as follows:
wherein the content of the first and second substances,
for the rigidity of the contact between the nut and the roller thread pair,
the nut and the roller are elastically deformed in contact with the i-th pair of screw threads.
As a further improvement of the invention, the screw and the roller are elastically deformed in contact with the i-th pair of threads
The calculation formula of (a) is as follows:
wherein K (e) and L (e) are the first and second class of complete ellipse integrals, a is the major semi-axis of the contact ellipse, b is the minor semi-axis of the contact ellipse,
eccentricity of the contact ellipse, k e ＝b/a，
Is an equivalent modulus of elasticity, E S And v S Is the modulus of elasticity and Poisson's ratio, E, of the screw R And v R The modulus of elasticity and the poisson's ratio of the rollers,
and λ is the contact angle and the helix angle of the roller.
As a further improvement of the invention, an extreme state function under the contact fatigue failure mode of the thread of the planetary roller screw pair is constructed based on an S-N curve and a stress-intensity interference theory, and the extreme state function is as follows:
wherein, x = (x) 1 ,x 2 ,…x i …,x n ) For random variables affecting the extreme state function, x i Is the ith random variable, n is the number of random variables, sigma Hlim To contact fatigue limit, σ SRi For the contact stress on the ith pair of screw threads of the screw rod and the roller,
maximum contact stress between all the threads on the contact side of the roller with the screw spindle, σ NRi For the contact stress on the i-th pair of threads of the nut and the roller,
the maximum contact stress between the roller and all thread teeth on the contact side of the nut; f SRi Axial load on the i-th pair of threads on the contact side of the roller and the screw, F NRi The axial load borne by the ith pair of thread teeth on the contact side of the roller and the nut; a is the major semi-axis of the contact ellipse, b is the minor semi-axis of the contact ellipse,
and λ is the contact angle and the helix angle of the roller.
As a further improvement of the present invention, step S2 includes:
s21, for each random of the planetary roller screw pairsVariable x i In the [0,1 ]]A Halton sequence is generated in the interval (2), and an input parameter matrix obtained from the Halton sequence is X = (X) 1 ,x 2 ,…,x i ,…,x n ) Wherein x is i ＝(x i1 ,x i2 ,…,x ij ,…,x iN ) T Is x i A sample vector of (a);
s22, the response quantity comprises: maximum load distribution coefficient of both contact sides of the roller:
maximum contact stress
And local contact stress sigma on the thread pair SRi ,σ NRi And calculating to obtain a response matrix as follows: y = (Y) 1 ,y 2 ,…,y k ,…,y m ) Wherein y is k ＝(y k1 ,y k2 ,…,y kj ,…,y kN ) T For the k response y k The sample vector of (a);
s23, setting the data in the matrixes X and Y to be [ -1,1]Normalization is carried out within the range, and the data are fitted by a polynomial response surface method:
s24, combining the polynomial coefficients
Expressed in percentage, can reflect the influence degree of each input parameter on the kth response quantity;
and S25, selecting the parameters with the importance degrees ranked in the front to reduce the dimension of the random variable, and obtaining the simplified limit state function of the planetary roller screw pair. Optionally, the top five ranked parameters are selected.
As a further improvement of the present invention, step S3 includes:
s31, randomly generating N in the uncertain domain 0 =20 samples, calculation simplicityIntroducing an expected risk learning function to construct a planetary roller screw pair active learning agent model, and firstly constructing an initial agent model as follows:
u＝F T R -1 r-f(x)
wherein Y = [ g (x) 1 ),g(x 2 ),…,g(x n )] T For a true response function with n sample points, f (x) = [ f = [ f [ ] 1 (x),f 2 (x),…f i (x),…,f p (x)] T Is a vector of regression polynomial basis function, beta = [ beta ] 1 ,β 2 ,…β i ,…,β p ] T Is a regression coefficient vector, p is the number of regression polynomials, r is a correlation vector function between the point to be measured and the sample point, F = [ F (x) 1 ) T ,f(x 2 ) T ,…,f(x n ) T ] T Is an n multiplied by p order spreading matrix, R is an n multiplied by n order symmetrical positive definite correlation matrix,
to approximate the estimate of the extreme state function g (x),
is composed of
G (x) is calculated as follows:
wherein z (x) is subject to N (0, σ) 2 ) Normal scoreAnd (3) in the random process of the cloth, the covariance between any two sample points w and x is:
cov[z(w),z(x)]＝σ 2 R(θ,w,x)
wherein, θ = [ θ = 1 ,θ 2 ,…,θ n ] T For the correlation parameter, R (θ, w, x) is a function representing the correlation of the variables θ, w, x and can be described by a continuous differentiable gaussian correlation function as:
wherein the regression coefficient beta and the variance sigma 2 Can be expressed as:
β＝(F T R -1 F) -1 F T R -1 Y
s32, generating N in the uncertainty domain c ＝10 5 A candidate sample;
s33, calculating N according to the initial agent model c Estimate of a sample point
And estimate of expected risk
The point with the largest ERF value is taken as the training point x,
the calculation method is as follows:
wherein sign (x) is a sign function, when x > 0, sign (x) =1, when x < 0, sign (x) = -1, phi () is a probability density function of a standard normal distribution, and psi () is a cumulative distribution function of the standard normal distribution;
s34, setting a threshold value or a convergence condition as epsilon =10 -5 (ii) a If max (ERF)>Epsilon, increasing x as a new training point, and updating the planetary roller screw pair active learning agent model;
s35, repeating the steps S33-S35 until max (ERF) is less than or equal to epsilon;
s36, generating sampling sample points by using the updated active learning agent model of the planetary roller screw pair and adopting a quasi-Monte Carlo method, calling the established active learning agent model to calculate a simplified limit state function corresponding to each group of sample points, and carrying out uncertainty analysis on the planetary roller screw pair to obtain the failure probability and the reliability sensitivity of the planetary roller screw pair under any working condition.
As a further improvement of the present invention, the generating of the sampling sample points by using the quasi-monte carlo method, calling the established active learning agent model to calculate the simplified extreme state function corresponding to each group of sample points, and performing uncertainty analysis on the planetary roller screw pair to obtain the failure probability and reliability sensitivity of the planetary roller screw pair under any working condition includes:
s361, each random variable xi of the planetary roller screw pair is in [0,1 ]]The Halton sequence is generated in the interval (2), all variables are independent and normally distributed, and the cumulative probability distribution function F (x) of random variables ij ) Can be expressed as:
wherein x is ij Is the jth sample value of the ith random variable, f (x) i ) Is a random variable x i N is the number of sample samples,
is the x (th) of ij Corresponding Halton number, wherein the specific generation process of the Halton sequence is as follows:
if q is any prime number, any natural number j has a unique q-system expression:
j＝j 0 +j 1 q+j 2 q 2 +···+j K q K
j i ∈{0,1,···,q-1}；i＝0,1,···,K
wherein K represents the integer part of lnj/lnq, and the inverse basis function with q as the base is defined as:
for any natural number j > 0, all satisfy
If the first n prime numbers are q 1 ,q 2 ,…,q n Then the n-dimensional Halton sequence can be expressed as:
s362, obtaining the random variable x ij Sample value of
S363, obtaining sample points of allowable contact stress
S364, calculating sample x of each group of random vectors j ＝(x 1j ,x 2j ,…,x nj ) T Load distribution and local contact characteristics under the action of the force are obtained, the maximum contact stress on the contact side of the roller and the screw rod or the nut is obtained, and then the maximum contact stress is further calculatedExtreme state function g (x) j ) If g (x) j ) Less than or equal to 0, status indication function I F (x j ) =1, otherwise I F (x j )＝0；
S365 failure probability
And its coefficient of variation
The calculation is as follows:
wherein N is f The number of failed samples;
s366, the estimated value of reliability sensitivity is as follows:
wherein f is x (x) Is a joint probability density function of the random variables,
is a basic random variable x i K-th distribution parameter of (1), m i Is the ith random variable x i Of normally distributed random variables having a mean value
Sum standard deviation
Two distribution parameters, the reliability sensitivity can be calculated as:
the invention also provides a computer-readable storage medium comprising a stored program, wherein the program performs the method of uncertainty analysis of a planetary roller screw pair as described in any one of the above.
The present invention also provides an electronic device, comprising: one or more processors, memory, and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the one or more programs comprising instructions for performing the method of uncertainty analysis of a planetary roller screw pair of any of the above.
The invention also provides an uncertainty analysis system of the planetary roller screw pair, which comprises the following modules:
the extreme state function building module is used for solving the load distribution and the contact characteristics of the thread of the planetary roller screw pair through the structural parameters, the material performance and the boundary conditions, considering the uncertainty existing in the structural parameters, the material performance and the boundary conditions, and building an extreme state function under the contact fatigue failure mode of the thread of the planetary roller screw pair based on an S-N curve and a stress-intensity interference theory;
the extreme state function simplifying module is used for introducing a deterministic low-deviation point set Halton sequence, analyzing the sensitivity and contribution degree of uncertainty factors on the influence of the load-bearing capacity and the contact performance of the planetary roller screw pair by adopting a test design method, and reducing the dimension of an extreme state function by taking the parameter with the top order of importance as a random variable to obtain a simplified extreme state function;
and the uncertainty analysis module is used for introducing an expected risk learning function to construct a planetary roller screw pair active learning agent model, generating sampling sample points by adopting a quasi-Monte Carlo method, calling the established active learning agent model to calculate a simplified extreme state function corresponding to each group of sample points, and performing uncertainty analysis on the planetary roller screw pair to obtain the failure probability and the reliability sensitivity of the planetary roller screw pair under any working condition.
The invention has the beneficial effects that:
the method comprises the steps of constructing an extreme state function under a contact fatigue failure mode of a thread of the planetary roller screw pair, simplifying the extreme state function, introducing an expected risk learning function to construct an active learning agent model of the planetary roller screw pair, generating sampling sample points by adopting a quasi-Monte Carlo method, calling the established active learning agent model to calculate the simplified extreme state function corresponding to each group of sample points, and carrying out uncertainty analysis on the planetary roller screw pair to obtain the failure probability and reliability sensitivity of the planetary roller screw pair under any working condition. The method has the advantages that the number of times of calling the function of the extreme state function can be effectively reduced, the calculation cost is reduced, the uncertainty analysis efficiency of the planetary roller screw pair is improved, a designer is helped to find out main factors influencing the reliability of the structure, and further, the theoretical basis is laid for the design optimization of products.
The foregoing description is only an overview of the technical solutions of the present invention, and in order to make the technical means of the present invention more clearly understood, the present invention may be implemented in accordance with the content of the description, and in order to make the above and other objects, features, and advantages of the present invention more clearly understood, the following preferred embodiments are described in detail with reference to the accompanying drawings.
Drawings
FIG. 1 is an overall flow chart of a method of analyzing uncertainty of a planetary roller screw pair in an embodiment of the invention;
FIG. 2 is a schematic structural diagram of a planetary roller screw pair according to an embodiment of the present invention;
FIG. 3 is a flowchart of a method for analyzing uncertainty of a planetary roller screw pair according to an embodiment of the present invention;
FIG. 4 is a graph of the importance ranking of uncertainty factors to the load bearing characteristics and contact performance of a planetary roller screw pair in an embodiment of the invention;
FIG. 5 is a comparison graph of the convergence rate and the calculation time for estimating the failure probability of the planetary roller screw pair by using the Monte Carlo method and the quasi-Monte Carlo method in the embodiment of the invention;
FIG. 6 is a three-dimensional mesh surface plot of a limit state function of a planetary roller screw pair in an embodiment of the present invention;
FIG. 7 is a three-dimensional mesh surface diagram of a limit state function of the planetary roller screw pair under different working conditions in the embodiment of the invention.
Description of the labeling: 1. a lead screw; 2. a roller; 3. a nut; 4. an inner gear ring; 5. a planet carrier; 6. a circlip is provided.
Detailed Description
The present invention is further described below in conjunction with the following figures and specific examples so that those skilled in the art may better understand the present invention and practice it, but the examples are not intended to limit the present invention.
As shown in fig. 1, a method for analyzing uncertainty of a planetary roller screw pair in a preferred embodiment of the present invention includes the following steps:
s1, solving load distribution and contact characteristics of the thread teeth of the planetary roller screw pair through structural parameters, material performance and boundary conditions, considering uncertainty existing in the structural parameters, the material performance and the boundary conditions, and constructing a limit state function under a contact fatigue failure mode of the thread teeth of the planetary roller screw pair based on an S-N curve and a stress-intensity interference theory;
as shown in fig. 2, a conventional planetary roller screw pair structure includes: a screw 1, a nut 3, a plurality of rollers 2, a pair of planet carriers 5, a pair of inner gear rings 4, a pair of elastic retainer rings 6 and the like;
the ring gear 4 is installed at nut 3 both ends, planet carrier 5 is installed in the ring gear 4, a plurality of rollers 2 pass through planet carrier 5 evenly distributed is between lead screw 1 and nut 3, 2 both ends of roller have the straight-tooth respectively with the ring gear 4 meshing, lead screw 1 is the trapezoidal external screw thread of bull, nut 3 is the trapezoidal internal thread of bull, lead screw 1 with nut 3 has the same thread head number, roller 2 is the single-end external screw thread, roller tooth profile is arc and the centre of a circle is located the roller axis, lead screw 1 and nut 3 respectively with the thread tooth contact of roller 2 both sides participates in the bearing jointly, the axis of roller 2, lead screw 1 and nut 3 is parallel, roller 2 not only winds self axis rotation but also winds the revolution of lead screw 1 axis, roller 2 with nut 3 keeps the same speed axial displacement, nut 3 does not have circular motion.
In one embodiment, step S1 specifically includes:
an iterative algorithm is adopted to solve a thread load distribution model of the planetary roller screw pair, and the method comprises the following steps:
wherein, P S ,P R And P N The screw pitches of the screw rod, the roller and the nut are respectively set, tau is the thread number of the roller participating in contact, z is the number of the roller, the thread is numbered from the fixed end to the free end of the screw rod in sequence, i =1,2, \ 8230, tau and F are axial external loads, F is the axial external load SRi Axial load on the i-th pair of threads on the contact side of the roller and the screw, F NRi The axial load borne on the ith pair of threads on the contact side of the roller and the nut;
in order to achieve the rigidity of the shaft section of the screw rod,
in order to provide the rigidity of the roller shaft section,
in order to provide the rigidity of the shaft section of the nut,
in order to provide rigidity to the thread teeth of the screw rod,
in order to provide rigidity to the thread ridge of the roller,
the rigidity of the nut thread;
wherein v is S ,h S ,β S ,d S ,
And c S Respectively showing the poisson ratio, thread thickness, flank angle, pitch diameter, root width and crest width of the screw rod.
wherein v is R ,h R ,β R ,d R ,
And c R Showing respectively the poisson ratio, thread thickness, flank angle, pitch diameter, root width and crest width of the roller. On the contact side of the roller and the nut, the calculation method of the rigidity of the thread ridge of the roller is the same, and only the formula F is required SRi By changing to F NRi 。
wherein v is N ,h N ,β N ,d N ,
And c N Respectively showing the poisson ratio, thread thickness, flank angle, pitch diameter, root width and crest width of the nut.
The following calculations are made:
the contact rigidity of the screw and the roller is provided,
the elastic contact deformation of the screw and the ith pair of screw threads of the roller is specifically calculated as follows:
wherein K (e) and L (e) are the first and second class of complete ellipse integrals, a is the major semi-axis of the contact ellipse, b is the minor semi-axis of the contact ellipse,
eccentricity of contact ellipse, k e ＝b/a，
Is an equivalent modulus of elasticity, E S And v S Is the modulus of elasticity and Poisson's ratio of the screw, E R And v R The modulus of elasticity and the poisson's ratio of the roller,
and λ is the contact angle and the helix angle of the roller.
Constructing a limit state function under a contact fatigue failure mode of a thread of the planetary roller screw pair based on an S-N curve and a stress-intensity interference theory, wherein the limit state function is as follows:
wherein, x = (x) 1 ,x 2 ,…x i …,x n ) For random variables affecting the extreme state function, x i Is the ith random variable, n is the number of random variables, sigma Hlim To contact fatigue limit, σ SRi For the contact stress on the ith pair of screw threads of the screw rod and the roller,
maximum contact stress between all the threads on the contact side of the roller with the screw spindle, σ NRi For the contact stress on the i-th pair of threads of the nut and the roller,
is the maximum contact stress between the roller and all the threads on the contact side of the nut.
S2, introducing a deterministic low-deviation point set Halton sequence, analyzing the sensitivity and contribution degree of uncertainty factors on the influence of the load-bearing capacity and the contact performance of the thread of the planetary roller screw pair by adopting a test design method, and reducing the dimension of a limit state function by using a parameter with the top order of importance as a random variable to obtain a simplified limit state function;
in one embodiment, step S2 specifically includes:
step S21, aiming at each random variable x of the planetary roller screw pair i In the [0,1 ]]In the interval (2), a Halton sequence is generated, and an input parameter matrix obtained from the Halton sequence is X = (X) 1 ,x 2 ,…,x i ,…,x n ) Wherein x is i ＝(x i1 ,x i2 ,…,x ij ,…,x iN ) T Is x i The sample vector of (a);
step S22, the response amount includes: maximum load distribution coefficient of both contact sides of the roller:
maximum contact stress
And local contact stress sigma on the thread pair SRi ,σ NRi And calculating to obtain a response matrix as follows: y = (Y) 1 ,y 2 ,…,y k ,…,y m ) Wherein y is k ＝(y k1 ,y k2 ,…,y kj ,…,y kN ) T For the k response y k A sample vector of (a);
step S23, the data in the matrixes X and Y are in [ -1,1 [ ]]Normalization is performed within the range, and the data are fitted by a polynomial response surface method:
step S24, converting the polynomial coefficients
Expressed in percentage form, the influence degree of each input parameter on the kth response quantity can be reflected;
and S25, selecting the parameters with the importance degrees ranked in the front to reduce the dimension of the random variable, and obtaining the simplified limit state function of the planetary roller screw pair.
And S3, introducing an expected risk learning function to construct an Active Learning Kriging (ALK) agent model of the planetary roller screw pair, generating sampling sample points by adopting a Quasi Monte Carlo (QMC), calling the constructed active learning agent model to calculate simplified limit state functions corresponding to each group of sample points, and carrying out uncertainty analysis on the planetary roller screw pair to obtain the failure probability and reliability sensitivity of the planetary roller screw pair under any working condition.
Referring to fig. 3, in one embodiment, step S3 specifically includes:
s31, randomly generating N in the uncertain domain 0 Calculating simplified extreme state function values, introducing an expected risk learning function to construct a planetary roller screw pair active learning agent model, and firstly constructing an initial agent model as follows:
u＝F T R -1 r-f(x)
wherein Y = [ g (x) 1 ),g(x 2 ),…,g(x n )] T For a true response function with n sample points, f (x) = [ f = [ f [ ] 1 (x),f 2 (x),…f i (x),…,f p (x)] T Is a vector of regression polynomial basis function, beta = [ beta ] 1 ,β 2 ,…β i ,…,β p ] T Is a regression coefficient vector, p is the number of regression polynomials, r is a correlation vector function between the point to be measured and the sample point, F = [ F (x) 1 ) T ,f(x 2 ) T ,…,f(x n ) T ] T Is an n multiplied by p order spreading matrix, R is an n multiplied by n order symmetrical positive definite correlation matrix,
to approximate the estimated value of the extreme state function g (x),
is composed of
G (x) is calculated as follows:
wherein z (x) is subject to N (0, σ) 2 ) A random process of normal distribution, the covariance between any two sample points w and x is:
cov[z(w),z(x)]＝σ 2 R(θ,w,x)
wherein, θ = [ θ = 1 ,θ 2 ,…,θ n ] T For the correlation parameter, R (θ, w, x) is a function representing the correlation of the variables θ, w, x and can be described by a continuous differentiable gaussian correlation function as:
wherein the regression coefficient beta and the variance sigma 2 Can be expressed as follows by using a generalized least squares regression method:
β＝(F T R -1 F) -1 F T R -1 Y
s32, generating N in the uncertainty domain c ＝10 5 A candidate sample;
s33, calculating N according to the initial star roller screw pair active learning agent model c Estimate of a sample point
And an estimate of expected risk
The point with the largest ERF value is taken as the training point x,
the calculation method is as follows:
wherein sign (x) is a sign function, when x > 0, sign (x) =1, when x < 0, sign (x) = -1, phi () is a probability density function of a standard normal distribution, psi () is a cumulative distribution function of the standard normal distribution;
s34, setting a threshold value or a convergence condition as epsilon =10 -5 (ii) a If max (ERF)>Epsilon, increasing x as a new training point, and updating the planetary roller screw pair active learning agent model;
s35, repeating the steps S33-S35 until max (ERF) is less than or equal to epsilon;
s36, generating sampling sample points by using the updated active learning agent model of the planetary roller screw pair and adopting a quasi-Monte Carlo method, calling the established active learning agent model to calculate a simplified limit state function corresponding to each group of sample points, and carrying out uncertainty analysis on the planetary roller screw pair to obtain the failure probability and the reliability sensitivity of the planetary roller screw pair under any working condition.
The method comprises the following steps of generating sampling sample points by adopting a quasi-Monte Carlo method, calling an established active learning agent model to calculate a simplified extreme state function corresponding to each group of sample points, and carrying out uncertainty analysis on a planetary roller screw pair to obtain the failure probability and reliability sensitivity of the planetary roller screw pair under any working condition, wherein the method comprises the following steps:
step S361, setting each random variable xi of the planetary roller screw pair to be [0,1]In the interval (2), halton sequences are generated, variablesIndependent of each other and all normally distributed, cumulative probability distribution function F (x) of random variable ij ) Can be expressed as:
wherein x is ij Is the jth sample value of the ith random variable, f (x) i ) Is a random variable x i N is the number of sample samples,
is the x (th) of ij Corresponding Halton number, wherein the specific generation process of the Halton sequence is as follows:
if q is any prime number, any natural number j has a unique q-system expression:
j＝j 0 +j 1 q+j 2 q 2 +···+j K q K
j i ∈{0,1,···,q-1}；i＝0,1,···,K
wherein K represents the integer part of lnj/lnq, and the inverse basis function with q as the base is defined as:
for any natural number j > 0, all satisfy
If the first n prime numbers are q 1 ,q 2 ,…,q n Then the n-dimensional Halton sequence can be expressed as:
step S362, obtaining a random variable x ij Sample value of
Step S363, obtaining sample points of allowable contact stress
Step S364, calculating sample x of each group of random vectors j ＝(x 1j ,x 2j ,…,x nj ) T Load distribution under action and local contact characteristics, obtaining the maximum contact stress on the contact side of the roller and the screw rod or the nut, and further calculating a limit state function g (x) j ) If g (x) j ) Less than or equal to 0, status indicating function I F (x j ) =1, otherwise I F (x j )＝0；
Step S365, failure probability
Estimate of (2) and its coefficient of variation
The calculation is as follows:
wherein N is f The number of failed samples;
step S366, the estimated value of the reliability sensitivity is as follows:
wherein f is x (x) Is a joint probability density function of the random variables,
is a basic random variable x i K-th distribution parameter of (1), m i Is the ith random variable x i Of normally distributed random variables having a mean value
Sum standard deviation
Two distribution parameters, reliability sensitivity can be calculated as:
according to the random structure parameters of the planetary roller screw pair given in table 1, 2000 groups of sample points obtained in the flow shown in fig. 3 were analyzed by a test design method, and the importance of the obtained uncertainty factors on the bearing characteristics and contact performance of the planetary roller screw pair is ranked as shown in fig. 4. The influence of the thread pitch on the maximum contact stress and the maximum load distribution coefficient in all the thread pairs is the largest, and the influence of the half angle and the pitch diameter of the roller thread profile on the contact stress of a single pair of thread pairs is the largest. Taking the first five important parameters P S ,P R ,P N ,β R And d R For constructing the active learning Kriging agent model in step S3.
TABLE 1 random structural parameters of planetary roller screw pairs
By usingThe calculation flow shown in fig. 3 is a ratio of the obtained failure probability estimation value of the planetary roller screw pair to the convergence speed and calculation time of the monte carlo method, as shown in fig. 5. The result shows that the calculation result obtained by adopting the quasi Monte Carlo method has good robustness, high convergence speed and high calculation efficiency. For comparison, 5X 10 is adopted 6 Subsampled Monte Carlo method, 10 5 The results of the planetary roller screw pair uncertainty analysis performed by the subsampled quasi-Monte Carlo method and the method of the present invention are shown in Table 3. In which Table 3 shows only P R And P S Because the reliability of the remaining parameters is sensitive to small.
TABLE 3 comparison of uncertainty analysis calculations
Obviously, the relative error of the failure probability and the reliability sensitivity obtained by the quasi-Monte Carlo method and the Monte Carlo method is less than 1%. Therefore, under the condition of less simulation times, the precision of the Monte Carlo simulation method can be ensured, and the calculation time can be effectively reduced. And the ALK-QMC method can obviously reduce the number of times of calling the extreme state function, thereby further saving the calculation cost. Furthermore, the ALK-QMC method, with an additional 6 training calls, has a relative error of 3.53% compared to the failure probability obtained by the monte carlo method. Because only five random variables are obtained after the ALK-QMC method simplifies the extreme state function, the obtained reliability sensitivity has larger relative error, but the result can still be used for effectively researching the main factors influencing the reliability.
FIG. 6 is a three-dimensional mesh surface plot of the limit state function of a planetary roller screw pair from 1600 Halton sample points showing the relationship between the most significant influencing parameter pitch and reliability.
Referring to FIG. 6, in the graph (a), the gray curved surface indicates the limit stateThe function g (x) =0, which is the interface of the secure domain and the fail domain. The intersection line of the gray plane and the grid curved surface is represented by red, and the black points and the red points are respectively safety sample points and failure sample points. In the (b) diagram, there is no point of failure on the side of the rollers in contact with the nut. In addition, at P R >P S And P R >P N The presence of the ridge structure regions that maximize g (x) indicates that greater reliability can be achieved by optimization, while the distance from the ridge regions increases the probability of failure.
A three-dimensional mesh surface diagram of the pitch versus the extreme state function under different conditions is shown in FIG. 7. Contact fatigue limit σ when external load F is constant Hlim The larger the number of failed sample points; when sigma is Hlim When the working conditions are not changed, the failure probability can be reduced by reducing the F, and no failure point exists under the working condition of 20 kN. Therefore, the reliability of the planetary roller screw pair can be improved by selecting a material with higher contact fatigue limit or optimizing the structural design of the planetary roller screw pair to maximize the value of g (x) under the condition of constant working condition.
The preferred embodiment of the invention also discloses a computer-readable storage medium which comprises a stored program, wherein the program executes the uncertainty analysis method of the planetary roller screw pair of the above embodiment.
The preferred embodiment of the present invention also discloses an electronic device, comprising: one or more processors, memory, and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the one or more programs comprising instructions for performing the method of uncertainty analysis of a planetary roller screw pair of the above-described embodiments.
The invention also discloses an uncertainty analysis system of the planetary roller screw pair, which comprises the following modules:
the extreme state function building module is used for solving the load distribution and the contact characteristics of the thread of the planetary roller screw pair through the structural parameters, the material performance and the boundary conditions, considering the uncertainty existing in the structural parameters, the material performance and the boundary conditions, and building an extreme state function under the contact fatigue failure mode of the thread of the planetary roller screw pair based on an S-N curve and a stress-intensity interference theory;
the extreme state function simplifying module is used for introducing a deterministic low-deviation point set Halton sequence, analyzing the sensitivity and contribution degree of uncertainty factors on the influence of the load-bearing capacity and the contact performance of the planetary roller screw pair by adopting a test design method, and reducing the dimension of an extreme state function by taking parameters with the top-ranked importance degrees as random variables to obtain a simplified extreme state function;
and the uncertainty analysis module is used for introducing an expected risk learning function to construct a planetary roller screw pair active learning agent model, generating sampling sample points by adopting a quasi-Monte Carlo method, calling the established active learning agent model to calculate a simplified extreme state function corresponding to each group of sample points, and performing uncertainty analysis on the planetary roller screw pair to obtain the failure probability and the reliability sensitivity of the planetary roller screw pair under any working condition.
The uncertainty analysis system of the planetary roller screw pair in the embodiment of the present invention is used for implementing the foregoing uncertainty analysis method of the planetary roller screw pair, and therefore, the specific implementation of the system can be seen from the foregoing description of the embodiment of the uncertainty analysis method of the planetary roller screw pair, and therefore, the specific implementation thereof can refer to the description of the corresponding respective embodiment of the portion, and will not be further described herein.
In addition, since the uncertainty analysis system of the planetary roller screw pair of the present embodiment is used for implementing the uncertainty analysis method of the planetary roller screw pair, the function corresponds to that of the above method, and details are not repeated here.
The above embodiments are merely preferred embodiments for fully illustrating the present invention, and the scope of the present invention is not limited thereto. The equivalent substitution or change made by the technical personnel in the technical field on the basis of the invention is all within the protection scope of the invention. The protection scope of the invention is subject to the claims.
Claims (10)
1. An uncertainty analysis method of a planetary roller screw pair is characterized by comprising the following steps of:
s1, solving load distribution and contact characteristics of the thread teeth of the planetary roller screw pair through structural parameters, material performance and boundary conditions, considering uncertainty existing in the structural parameters, the material performance and the boundary conditions, and constructing a limit state function under a contact fatigue failure mode of the thread teeth of the planetary roller screw pair based on an S-N curve and a stress-intensity interference theory;
s2, introducing a deterministic low-deviation point set Halton sequence, analyzing the sensitivity and contribution degree of the uncertain factors on the influence of the load-bearing capacity and the contact performance of the screw thread of the planetary roller screw pair by adopting a test design method, reducing the dimension of a limit state function by using the parameters with the top-ranked importance as random variables, and obtaining a simplified limit state function;
and S3, introducing an expected risk learning function to construct a planetary roller screw pair active learning agent model, generating sampling sample points by adopting a quasi-Monte Carlo method, calling the constructed active learning agent model to calculate simplified limit state functions corresponding to each group of sample points, and carrying out uncertainty analysis on the planetary roller screw pair to obtain the failure probability and reliability sensitivity of the planetary roller screw pair under any working condition.
2. The method for analyzing uncertainty of a planetary roller screw pair according to claim 1, wherein in the step S1, solving the thread load distribution and contact characteristics of the planetary roller screw pair through the structural parameters, material properties and boundary conditions comprises:
solving a load distribution model of the thread of the planetary roller screw pair by adopting an iterative algorithm, which comprises the following steps:
wherein, P S ,P R And P N Respectively a screw rod, a roller and a nutThe screw pitch is that tau is the number of the screw threads of the roller participating in contact, z is the number of the rollers, the screw threads are numbered from the fixed end to the free end of the screw rod in sequence, i =1,2, \8230, tau and F are axial external loads, F SRi Axial load on the i-th pair of threads on the contact side of the roller and the screw, F NRi The axial load borne by the ith pair of thread teeth on the contact side of the roller and the nut;
the rigidity of the shaft section of the screw rod is improved,
the rigidity of the roller shaft section is ensured,
in order to provide the rigidity of the shaft section of the nut,
in order to provide rigidity to the thread teeth of the screw rod,
in order to provide rigidity to the thread ridge of the roller,
the rigidity of the nut thread;
solving the contact rigidity of the screw and the roller as follows:
wherein, the first and the second end of the pipe are connected with each other,
the contact rigidity of the screw and the roller is set;
the i-th pair of screw threads of the screw rod and the roller are elasticDeformation by contact;
solving the contact rigidity of the nut and the roller thread pair as follows:
wherein, the first and the second end of the pipe are connected with each other,
for the rigidity of the contact between the nut and the roller thread pair,
the nut and the roller are elastically deformed in contact with the i-th pair of screw threads.
3. A method of uncertainty analysis of a planetary roller screw set as claimed in claim 2, characterized in that the elastic contact deformation of the i-th pair of threads of the screw and the roller
The calculation formula of (c) is as follows:
wherein K (e) and L (e) are the first and second class of complete ellipse integrals, a is the major semi-axis of the contact ellipse, b is the minor semi-axis of the contact ellipse,
eccentricity of contact ellipse, k e ＝b/a，
Is an equivalent modulus of elasticity, E S And v S Is the modulus of elasticity and Poisson's ratio, E, of the screw R And v R The modulus of elasticity and the poisson's ratio of the roller,
and λ is the contact angle and the helix angle of the roller.
4. The method for analyzing uncertainty of a planetary roller screw pair according to claim 1, wherein the extreme state function in the contact fatigue failure mode of the thread of the planetary roller screw pair is constructed based on the S-N curve and the stress-intensity interference theory as follows:
wherein, x = (x) 1 ,x 2 ,…x i …,x n ) For random variables affecting the extreme state function, x i Is the ith random variable, n is the number of random variables, sigma Hlim To the contact fatigue limit, σ SRi For the contact stress on the ith pair of screw threads of the screw rod and the roller,
maximum contact stress between all the threads on the contact side of the roller with the screw, σ NRi For the contact stress on the i-th pair of threads of the nut and the roller,
the maximum contact stress between the roller and all thread teeth on the contact side of the nut; f SRi Axial load on the i-th pair of threads on the contact side of the roller and the screw, F NRi The axial load borne by the ith pair of thread teeth on the contact side of the roller and the nut; a is the major semi-axis of the contact ellipse, b is the minor semi-axis of the contact ellipse,
and λ is the contact angle and the helix angle of the roller.
5. The method for analyzing the uncertainty of a planetary roller screw pair according to claim 4, wherein the step S2 comprises:
s21, for each random variable x of the planetary roller screw pair i In the [0,1 ]]In the interval (2), a Halton sequence is generated, and an input parameter matrix obtained from the Halton sequence is X = (X) 1 ,x 2 ,…,x i ,…,x n ) Wherein x is i ＝(x i1 ,x i2 ,…,x ij ,…,x iN ) T Is x i The sample vector of (a);
s22, the response quantity comprises: maximum load distribution coefficient of both contact sides of the roller:
maximum contact stress
And local contact stress sigma on thread pair SRi ,σ NRi And calculating to obtain a response matrix as follows: y = (Y) 1 ,y 2 ,…,y k ,…,y m ) Wherein y is k ＝(y k1 ,y k2 ,…,y kj ,…,y kN ) T For the k response y k A sample vector of (a);
s23, setting the data in the matrixes X and Y to be [ -1,1]Normalization is performed within the range, and the data are fitted by a polynomial response surface method:
s24, combining the polynomial coefficients
Expressed in percentage, can reflect the influence degree of each input parameter on the kth response quantity;
and S25, selecting the parameters with the importance degrees ranked in the front to reduce the dimension of the random variable, and obtaining the simplified limit state function of the planetary roller screw pair.
6. The method for analyzing uncertainty of a planetary roller screw set according to claim 5, wherein the step S3 comprises:
s31, randomly generating N in the uncertain domain 0 =20 samples, calculating a simplified extreme function value, introducing an expected risk learning function to construct an active learning agent model of the planetary roller screw pair, and firstly constructing an initial agent model as follows:
u＝F T R -1 r-f(x)
wherein Y = [ g (x) 1 ),g(x 2 ),…,g(x n )] T For a true response function with n sample points, f (x) = [ f = [ f [ ] 1 (x),f 2 (x),…f i (x),…,f p (x)] T Is a vector of regression polynomial basis function, beta = [ beta ] 1 ,β 2 ,…β i ,…,β p ] T Is a regression coefficient vector, p is the number of regression polynomials, r is a correlation vector function between the point to be measured and the sample point,F＝[f(x 1 ) T ,f(x 2 ) T ,…,f(x n ) T ] T is an n multiplied by p order spreading matrix, R is an n multiplied by n order symmetrical positive definite correlation matrix,
to approximate the estimate of the extreme state function g (x),
is composed of
G (x) is calculated as follows:
wherein z (x) is subject to N (0, σ) 2 ) A random process of normal distribution, the covariance between any two sample points w and x is:
cov[z(w),z(x)]＝σ 2 R(θ,w,x)
wherein, θ = [ θ = 1 ,θ 2 ,…,θ n ] T For the correlation parameter, R (θ, w, x) is a function representing the correlation of the variables θ, w, x and can be described by a continuous differentiable gaussian correlation function as:
wherein, the regression coefficient beta and the variance sigma 2 Can be expressed as follows by using a generalized least squares regression method:
β＝(F T R -1 F) -1 F T R -1 Y
s32, generating N in the uncertainty domain c ＝10 5 A candidate sample;
s33, calculating N according to the initial proxy model c Estimate of a sample point
And estimate of expected risk
The point with the largest ERF value is taken as the training point x,
the calculation method is as follows:
wherein sign (x) is a sign function, when x > 0, sign (x) =1, when x < 0, sign (x) = -1, phi () is a probability density function of a standard normal distribution, psi () is a cumulative distribution function of the standard normal distribution;
s34, setting a threshold value or a convergence condition as epsilon =10 -5 (ii) a If max (ERF)>Epsilon, increasing x as a new training point, and updating the planetary roller screw pair active learning agent model;
s35, repeating the steps S33-S35 until max (ERF) is less than or equal to epsilon;
s36, generating sampling sample points by using the updated active learning agent model of the planetary roller screw pair and adopting a quasi-Monte Carlo method, calling the established active learning agent model to calculate a simplified limit state function corresponding to each group of sample points, and carrying out uncertainty analysis on the planetary roller screw pair to obtain the failure probability and the reliability sensitivity of the planetary roller screw pair under any working condition.
7. The method for analyzing the uncertainty of the planetary roller screw pair according to claim 6, wherein the method for analyzing the uncertainty of the planetary roller screw pair to obtain the failure probability and the reliability sensitivity of the planetary roller screw pair under any working condition comprises the following steps of generating sampling sample points by using a quasi-Monte Carlo method, calling the established active learning agent model to calculate the simplified extreme state function corresponding to each group of sample points, and analyzing the uncertainty of the planetary roller screw pair to obtain the failure probability and the reliability sensitivity of the planetary roller screw pair under any working condition:
s361, for each random variable xi of the planetary roller screw pair, setting the value of [0,1 ]]The Halton sequence is generated in the interval, all variables are independent and normally distributed, and the cumulative probability distribution function F (x) of random variables ij ) Can be expressed as:
wherein x is ij Is the jth sample value of the ith random variable, f (x) i ) Is a random variable x i N is the number of sample samples,
is the x (th) of ij Corresponding Halton number, wherein the specific generation process of the Halton sequence is as follows:
if q is any prime number, any natural number j has a unique q-system expression:
j＝j 0 +j 1 q+j 2 q 2 +···+j K q K
j i ∈{0,1,···,q-1}；i＝0,1,···,K
wherein K represents the integer part of lnj/lnq, and the inverse basis function with q as the base is defined as:
for any natural number jGreater than 0, all satisfy
If the first n prime numbers are q 1 ,q 2 ,…,q n Then the n-dimensional Halton sequence can be expressed as:
s362, obtaining the random variable x ij Sample value of
S363, obtaining sample points of allowable contact stress
S364, calculating sample x of each group of random vectors j ＝(x 1j ,x 2j ,…,x nj ) T Load distribution and local contact characteristics under action, obtaining the maximum contact stress on the contact side of the roller and the screw rod or the nut, and further calculating a limit state function g (x) j ) If g (x) j ) Less than or equal to 0, status indicating function I F (x j ) =1, otherwise I F (x j )＝0；
S365 failure probability
Estimate of (2) and its coefficient of variation
The calculation is as follows:
wherein N is f The number of failed samples;
s366, the estimated value of reliability sensitivity is as follows:
wherein f is x (x) Is a joint probability density function of the random variables,
is a basic random variable x i K-th distribution parameter of (1), m i Is the ith random variable x i Of normally distributed random variables having a mean value
And standard deviation of
Two distribution parameters, reliability sensitivity can be calculated as:
8. a computer-readable storage medium, characterized in that the storage medium comprises a stored program, wherein the program performs the method of uncertainty analysis of a planetary roller screw pair according to any one of claims 1 to 7.
9. An electronic device, comprising: one or more processors, memory, and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the one or more programs comprising instructions for performing the method of uncertainty analysis of a planetary roller screw pair as recited in any of claims 1-7.
10. An uncertainty analysis system for a planetary roller screw pair, comprising the following modules:
the extreme state function building module is used for solving the load distribution and the contact characteristics of the thread teeth of the planetary roller screw pair through the structural parameters, the material performance and the boundary conditions, considering the uncertainty existing in the structural parameters, the material performance and the boundary conditions, and building an extreme state function under the contact fatigue failure mode of the thread teeth of the planetary roller screw pair based on an S-N curve and a stress-intensity interference theory;
the extreme state function simplifying module is used for introducing a deterministic low-deviation point set Halton sequence, analyzing the sensitivity and contribution degree of uncertainty factors on the influence of the load-bearing capacity and the contact performance of the planetary roller screw pair by adopting a test design method, and reducing the dimension of an extreme state function by taking parameters with the top-ranked importance degrees as random variables to obtain a simplified extreme state function;
and the uncertainty analysis module is used for introducing an expected risk learning function to construct a planetary roller screw pair active learning agent model, generating sampling sample points by adopting a quasi-Monte Carlo method, calling the established active learning agent model to calculate a simplified extreme state function corresponding to each group of sample points, and performing uncertainty analysis on the planetary roller screw pair to obtain the failure probability and the reliability sensitivity of the planetary roller screw pair under any working condition.
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