Cutting system flutter reliability optimization design method based on first-order control variable

Abstract

The invention discloses a cutting system flutter reliability optimization design method based on a first-order control variable. The method comprises the steps that S1, the cutting depth, the feeding amount and the spindle rotating speed of a cutting system serve as random design variables of cutting flutter reliability optimization design; taking other kinetic parameters of the cutting system as random parameter variables of cutting chatter reliability optimization design; s2, taking the maximum material removal rate as a target function of the flutter reliability optimization design of the cutting system, and establishing a flutter reliability optimization model of the cutting system based on the cutting force constraint and the cutting power constraint; and S3, optimizing the flutter reliability optimization model of the cutting system based on a reliability optimization design method of a first-order control variable to obtain an optimal design point. The method has the advantages that the optimal parameters beneficial to production are obtained by optimizing the cutting parameters, and the purposes of improving the cutting machining efficiency, reducing the machining cost and improving the machining quality are achieved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of mechanical manufacturing processing, and particularly relates to a cutting system chatter reliability optimization design method based on first-order control variables. BACKGROUND

[0002] In actual machining cutting problems, due to the high nonlinearity of the cutting system, the uncertainty of the system parameters will cause great fluctuations in the system performance, which greatly affects the prediction of cutting chatter. In order to deal with the influence of uncertainty factors, the sensitivity of the system can be analyzed to find out the sensitive parameters of the cutting system, and the precision is improved as much as possible and the variation range is reduced. In addition, the system should be optimized and designed to reduce the influence of uncertainty factors on the system and meet the demand of the system for high reliability.

[0003] In traditional optimization design, in order to simplify the design calculation process, it is usually assumed that the load, environment, structure and other parameters in the objective function, constraint condition and design variable are deterministic. However, the uncertainty of the cutting system parameters makes the deterministic optimization result unable to meet the design requirements. Therefore, in order to make up for the shortcomings of deterministic optimization design, it is necessary to carry out reliability optimization design on the cutting system. Under the premise of meeting the reliability requirements, the optimal scheme that makes the machining efficiency of the cutting system highest is obtained, thereby providing reference value for actual production and manufacturing. SUMMARY

[0004] The present application relates to the technical field of mechanical manufacturing processing, and particularly relates to a cutting system chatter reliability optimization design method based on first-order control variables.

[0005] In order to achieve the above-mentioned purpose, the technical scheme adopted by the present application is as follows:

[0006] A cutting system chatter reliability optimization design method based on first-order control variables, comprising the following steps,

[0007] S1, the cutting depth, feed rate and spindle speed of the cutting system are taken as the random design variables of the cutting chatter reliability optimization design; and other dynamic parameters of the cutting system are taken as the random parameter variables of the cutting chatter reliability optimization design;

[0008] S2, the maximum material removal rate is taken as the objective function of the cutting system chatter reliability optimization design, and a cutting system chatter reliability optimization model is established based on the cutting force constraint and the cutting power constraint;

[0009] S3, the chatter reliability optimization model of the cutting system is optimized based on the reliability optimization design method of the first-order control variable, and the optimal design point is obtained.

[0010] Preferably, step S3 specifically comprises the following contents,

[0011] S31, by Nataf transformation, the mean value μ x and μ p of random design variables x and random parameter variables p are transformed into the standard normal space, and are taken as the initial design points of the optimization problem in the first-order control variable method The optimization update obtains

[0012] S32, by each limit state function g i (d, μ x , μ p ), the first-order Taylor expansion at is taken as the strongly correlated limit state function g c,i (d, μ x , μ p ), i = 1, 2, …, n;

[0013] S33, according to each limit state function g i (d, μ x , μ p ) and the corresponding strongly correlated limit state function g c,i (d, μ x , μ p ), a small number of samples are generated by Latin hypercube sampling to estimate the correction coefficient ρ R,i ;

[0014] S34, the mean value μ x and μ p of random design variables x and random parameter variables p are taken as the initial points, and a first cycle is performed, and the optimal point and the objective function value f (1) are obtained by the first cycle optimization model in the sequence optimization and reliability evaluation method;

[0015] S35, by Nataf transformation, the optimal point of the first cycle is transformed into the standard normal space to obtain According to and ρ R,i , the inverse reliability analysis optimization model in the first-order control variable-based inverse reliability analysis method is used to optimize and further through Nataf inverse transformation, the design point corresponding to R i is obtained

[0016] S36, the moving vector of the kth cycle is obtained from the optimal point and the design point of the k-1th cycle The kth cycle is performed by the optimization model in the kth cycle of the sequence optimization and reliability evaluation method to obtain an optimal point and a target function value f (k) ; k = 1, 2, …;

[0017] S37, the optimal point of the kth cycle is converted into a standard normal space by Nataf conversion to obtain According to and p R,i , the inverse reliability analysis optimization model in the inverse reliability analysis method based on the first-order control variable is optimized to obtain and further Nataf inverse conversion is performed to obtain a design point corresponding to R i

[0018] S38, when the optimization result converges and all reliability constraints are within the feasible region, the cycle process is ended, the optimization is completed, and the optimal point and the target function value are output; otherwise, step S36 is returned, and the optimization result is updated.

[0019] Preferably, the sequence optimization and reliability evaluation method sequentially determines the deterministic optimization design and the reliability evaluation of the reliability constraint by a single-layer cycle; in the sequence optimization and reliability evaluation method, the reliability constraint P(g i (d, x, p) > 0) ≥ R i is converted into a constraint function g i (d, x MPP,i , p MPP,i ) ≥ 0, so that the optimization model establishes an equivalent relationship between the reliability optimization and the deterministic optimization, and the optimization model is

[0020]

[0021] Suppose that the initial values of the random design variable x and the random parameter variable p in the reliability optimization model are their mean values μ x and μ p , and the optimal point x is obtained by optimization.

[0022]

[0023] Further, the inverse reliability analysis method is used to evaluate the reliability based on the optimal point x to obtain a design point x of the ith reliability constraint condition.

[0024] If the design point x does not fall within the feasible region, i.e. ​If it is not established, the second cycle is performed and the deterministic constraints of the optimization model are modified, and the deterministic constraint function g in formula (1) is moved i (d,x,p)≥0, so that the design point At least falls on the constraint boundary, that is, at least Then is the deterministic constraint function for vector movement, that is

[0025]

[0026] Among them, the movement vector

[0027] Each deterministic constraint function forms a feasible domain that is narrower than the previous cycle, and optimization is performed on this basis. The optimization model is:

[0028] minf(d,x)

[0029]

[0030] The optimal point obtained by the second cycle Conduct reliability assessment and obtain design points And judge whether the optimization meets the convergence conditions and whether to proceed to the next cycle; similar to the second cycle, the optimization model of the kth cycle is,

[0031]

[0032] in, is the optimal point of the k-1th cycle; is the design point of the k-1th cycle obtained for reliability evaluation.

[0033] Preferably, the first-order control variable method is a control variable method based on Monte Carlo simulation, specifically,

[0034] Assume that the limit state function is g(x), and the failure probability is an n-dimensional random variable x=[x1,x2,…,x n The joint probability density function f X The integral of (x) in the failure domain F = {x:g(x)≤0},

[0035]

[0036] Among them, I F (x) is the failure domain indicator function;

[0037]

[0038] From the probability density function fX (x) is randomly sampled, x i For a random sample of N variables, the integral form of equation (6) is expressed as a mathematical expectation,

[0039] P f =∫ X I F (x)f X (x)dx=E[I F (x)] (8)

[0040] According to the law of large numbers in probability theory, when N is large enough, the sample mean converges to the expected value of the population I F (x)f X with probability 1, therefore, the mathematical expectation in equation (8) can be estimated by the sample mean,

[0041]

[0042] Let g c (x) be the limit state function strongly related to g(x), then the indicator function is a control variable strongly related to the indicator function I F (x), and its corresponding failure domain is expressed as F c ={x:g c (x)≤0}, then the indicator function is,

[0043]

[0044] The new indicator function is,

[0045]

[0046] where ρ is the correction coefficient;

[0047] Substitute equation (11) for I F (x) in equation (6), then the failure probability can be rewritten as,

[0048]

[0049] where, is the failure probability of the limit state function g c (x) strongly related to g(x); let =0, then the correction coefficient ρ is,

[0050]

[0051] At the same time, the failure probability is expressed as,

[0052]

[0053] The reliability R can be expressed as,

[0054] R = p R · R c (15)

[0055] where R c is the reliability of the limit state function g c (x) of strong correlation; p R is a correction coefficient, expressed as,

[0056]

[0057] where I F (x) is the failure domain indicator function of the limit state function g(x), and the failure domain F = {x: g(x) < 0}; is the failure domain indicator function of the limit state function g c (x) of strong correlation, and the failure domain F c = {x: g c (x) < 0}.

[0058] After any random variable x is converted into an independent random variable u in the standard normal space U through Nataf transformation, the limit state function g c (x) of strong correlation is converted into g c (u) in the standard normal space, and it is assumed that g c (u) is the first-order Taylor expansion at the design point u * :

[0059]

[0060] where g * (u) is the first-order partial derivative of g(u) at the design point u c ;

[0061] According to the first-order second-moment method, the reliability R c of the limit state function of strong correlation is obtained from the reliability index β c :

[0062] R c = Φ(β c ) (18)

[0063] And the reliability index β * and the design point u c can be solved by iterative optimization, and the mean value of the random variable u is taken as the initial search point,

[0064] min β = ‖u‖

[0065] s.t.g(u) = 0 (19)

[0066] where ||u|| is the distance from the origin of the standard normal space to the limit state surface;

[0067] Meanwhile, according to the limit state function g(u) and the strong correlation limit state function g c (u) converted into the standard normal space, a small number of samples are generated by Latin hypercube sampling to estimate the correction coefficient p R .

[0068] Preferably, the specific process of estimating the correction coefficient p R by Latin hypercube sampling is as follows,

[0069] A1. By Nataf conversion, n-dimensional random variables x = [x1, x2, …, xn] are converted into independent random variables u = [u1, u2, …, un] in the standard normal space U; n n ;

[0070] A2. For each random variable u i , N independent samples obeying [0, 1] uniform distribution are extracted, and the probability distribution of each random variable u i is divided into N non-overlapping intervals according to equal probability stratification, and then the N samples of the random variable u i are respectively divided into N intervals to obtain the cumulative probability of each sample U i,j ; j = 1, 2, …, N;

[0071] A3. According to the corresponding cumulative distribution function F U (u i ), N samples U i = [U i , U i,1 , …, U i,2 ] of each random variable u i,N are obtained;

[0072] A4. N samples U i of each random variable are randomly combined respectively to obtain the sample U = [U1, U2, …, U j , …, U N ] of N random variables u; where U j = [U 1,j , U 2,j , …, U n,j ];

[0073] A5. The limit state function g(u) and the strong correlation limit state function g c ​(u), and count the number of samples N f for which g(u) < 0 f <20, go to step A2, draw N add samples from the sample set U and add them to the sample set U until N f ≥ 20.

[0074] A6, calculate the correction factor p R from equation (16).

[0075] Preferably, the inverse reliability analysis method based on the first-order control variable is specifically,

[0076] Based on the first-order second-moment method, in the standard normal space U, the corresponding reliability index β can be calculated from the reliability constraint R; according to the geometric meaning of the reliability index, the inverse reliability analysis can be described by an optimization problem, and the specific optimization model is,

[0077] min g(u)

[0078] s.t.‖u‖=β=Φ -1 (R) (20)

[0079] where u=[μ x ,μ p ] is an independent standard normal random variable; Φ(·) is the cumulative distribution function of the standard normal distribution; the optimal solution u * of the optimization problem is the point on the surface with radius β in the standard normal space that minimizes the value of the limit state function, and then through the Nataf inverse transformation, the design point [x MPP , p MPP ] can be obtained.

[0080] According to the first-order control variable method, the reliability constraint R is expressed as,

[0081] R=p R ·R PORM (21)

[0082] where R PORM is the reliability estimated by the first-order second-moment method; therefore, the inverse reliability analysis optimization model based on the first-order second-moment in equation (20) can be converted to,

[0083] min g(u)

[0084] s.t.‖u‖=β=Φ -1 (R PORM )

[0085] where R PORM =R / ρ R (22)

[0086] According to the principle of the first-order control variable method, the correction coefficient p R can eliminate the difference between the reliability constraints R and R PORM ; therefore, in the inverse reliability analysis optimization model, the reliability constraint R is converted into R PORM , and the optimal solution u PORM is solved by optimization. * Thus, the design point [x MPP , p MPP ] corresponding to R is obtained, and the error generated by the inverse reliability analysis optimization model based on the first-order second-moment is eliminated.

[0087] Preferably, in step S2,

[0088] The material removal rate is,

[0089] Q = 10 3 · f0· a p · v c (23)

[0090] Wherein, Q is the material removal rate; f0 is the feed amount; a p is the cutting depth; v c is the cutting speed;

[0091] The cutting speed is,

[0092] v c = 10 -3 · π· d0· Ω (24)

[0093] Wherein, d0 is the rotational cutting diameter of the tool specified point or the workpiece; Ω is the spindle speed;

[0094] The cutting force constraint is,

[0095] L1 = F d -F dmax = k c · a p · f0-F dmax (25)

[0096] Wherein, L1 is the cutting force; F d is the instantaneous cutting force; F dmax is the maximum cutting force; k c is the cutting stiffness coefficient;

[0097] The cutting power constraint is,

[0098] L2 = P c -P cmax = 10 -3 · F d · v c / 60-P cmax (26)

[0099] Wherein, L2 is cutting power;P c is instantaneous cutting power;P cmax is maximum cutting power;

[0100] The cutting system chatter reliability optimization model is,

[0101] minf(x,p)=1 / Q=1 / (π·x1·x2·x3·p1)

[0102]

[0103]

[0104] Wherein, g(x,p) is the limit state function of the cutting system chatter reliability;P(g(x,p)>0) is the reliability model of the cutting system;x=[a p ,f0,Ω] is the random design variable;p is the cutting system dynamics parameter variable;R is the target reliability; And are minimum and maximum allowable cutting depth respectively; And are the lowest and highest allowable feed rate in the cutting process respectively;Ω min And Ω max are the lowest and highest spindle speed in a certain vane range respectively.

[0105] The beneficial effects of the present application are: 1, the present application obtains the optimal parameters conducive to production by optimizing the cutting parameters, achieves the purpose of improving the cutting processing efficiency, reducing the processing cost and improving the processing quality.2, the present application improves the inverse reliability analysis optimization model by combining the first-order control variable method to obtain the inverse reliability analysis method based on the first-order control variable, which can reduce the optimal solution error, ensure that the reliability optimization result meets the design requirements, and the optimization process is easy to converge, the cycle number is less, and the calculation amount is low.3, the present application improves the optimization precision and reduces the operation cost to a certain extent. BRIEF DESCRIPTION OF DRAWINGS

[0106] Figure 1 is the flow chart of the cutting system chatter reliability optimization design method based on the first-order control variable in the embodiment of the present application;

[0107] Figure 2 is the flow chart of the reliability optimization design method based on the first-order control variable in the embodiment of the present application. DETAILED DESCRIPTION

[0108] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0109] Example 1

[0110] Cutting parameter optimization is one of the main ways to improve the stability of the cutting system. By optimizing cutting parameters, the purpose of improving cutting efficiency, reducing processing costs and improving processing quality can be achieved. Considering that the cutting parameters in actual engineering are usually uncertain and have a certain impact on the performance of the cutting system, the traditional optimization design method cannot obtain the optimal parameters that are beneficial to production. Therefore, Figure 1 As shown in the figure, this embodiment provides a cutting chatter reliability optimization design method based on first-order control variables. By optimizing the cutting parameters through reliability optimization design, the optimal solution for achieving the highest system processing efficiency is obtained while meeting the reliability requirements, providing reference value for actual production and manufacturing. Specifically, it includes the following steps:

[0111] 1. The cutting depth, feed rate and spindle speed of the cutting system are used as random design variables for the cutting chatter reliability optimization design; other dynamic parameters of the cutting system are used as random parameter variables for the cutting chatter reliability optimization design.

[0112] 2. The maximum material removal rate is used as the objective function of the cutting system vibration reliability optimization design, and a cutting system vibration reliability optimization model is established based on the cutting force constraint and cutting power constraint.

[0113] 3. Based on the reliability optimization design method of the first-order control variables, the vibration reliability optimization model of the cutting system is optimized to obtain the optimal design point.

[0114] In this embodiment, based on the sequence optimization and reliability assessment method, combined with the first-order control variable method, the inverse reliability analysis optimization model is improved, and then a reliability optimization design method based on the first-order control variable is proposed (the process is as follows Figure 2 In order to maximize the machining efficiency of the cutting system while meeting the reliability requirements, a reliability optimization design method based on first-order control variables was used to optimize the cutting tool system and the cutting machining system. This method formed a cutting system chatter reliability optimization design method based on first-order control variables.

[0115] 1. Sequence Optimization and Reliability Assessment Methods

[0116] The sequence optimization and reliability evaluation method sequentially determines the deterministic optimization design and the reliability evaluation by using the reliability constraint; in the sequence optimization and reliability evaluation method, the reliability constraint P(g i (d,x,p)>0)≥R i is converted into a constraint function g i (d,x MPP,i ,p MPP,i )≥0, so that the optimization model establishes the equivalent relationship between the reliability optimization and the deterministic optimization, and the optimization model is,

[0117]

[0118] where d is the deterministic design variable in the reliability optimization model; x is the random design variable in the optimization model; p is the random parameter variable in the optimization model; g i (·) is the ith limit state function; R i is the ith reliability constraint; n is the number of reliability constraints; L i′ (·) is the ith deterministic constraint function; and r is the number of constraint conditions.

[0119] Let the random design variable x=[x1,x2], x opt =[x opt,1 ,x opt,2 ] be the optimal point, g(x opt,1 ,x opt,2 )=0 be the deterministic constraint boundary without considering the uncertainty factor, and the reliability constraint boundary be P(g(x1,x2)≥0)=x when the uncertainty factor is considered. Since P(g(x1,x2)≥0)=R is equivalent to g(x 1,MPP ,x 2,MPP )=0, where [x 1,MPP ,x 2,MPP ] is the design point, the reliability constraint at the optimal point x opt is equivalent to the deterministic constraint at the design point [x 1,MPP ,x 2,MPP ]. In order to ensure that g(x 1,MPP ,x 2,MPP )=0, the design point [x opt ,x 1,MPP ] corresponding to the optimal point x 2,MPP on the reliability constraint boundary should exactly fall on the deterministic constraint boundary. Therefore, during the process of the reliability optimization design in the sequence optimization and reliability evaluation method, it is necessary to ensure that the design point of each reliability constraint is in the feasible region of the deterministic constraint.

[0120] The sequence optimization and reliability evaluation method sequentially performs deterministic optimization design and reliability evaluation in each cycle. In the current cycle, the design variable in the deterministic optimization design is the design point of the previous cycle. A new design point is obtained through the optimization process, and then reliability is determined through inverse reliability analysis to obtain a new design point and determine whether it is within the feasible region. If the reliability requirement cannot be met, the next cycle is performed by moving the boundary of the deterministic constraint function. The sequence optimization and reliability evaluation method can gradually approach the required reliability through fewer cycles, greatly reducing the computational cost.

[0121] Let the initial values of the random design variable x and the random parameter variable p in the reliability optimization model be their mean values μ x and μ p , and the optimal point is obtained through optimization The optimization model of the first cycle is

[0122]

[0123] Further, the optimal point is obtained through optimization, and the design point of the ith reliability constraint condition is obtained through inverse reliability analysis

[0124] If the design point does not fall within the feasible region, i.e., the inequality is not established, the second cycle is performed, and the deterministic constraint condition of the optimization model is modified, and the deterministic constraint function g i (d,x,p)≥0 in formula (1) is moved to make the design point at least fall on the constraint boundary, i.e., at least make Then, the deterministic constraint function is moved by as a vector, i.e.

[0125]

[0126] where the movement vector

[0127] A narrower feasible region than the previous cycle is formed by each deterministic constraint function, and optimization is performed on this basis, and the optimization model is

[0128]

[0129] The optimal point obtained through the second cycle is subjected to reliability evaluation to obtain the design point and determine whether the optimization meets the convergence condition and whether the next cycle is to be performed; similar to the second cycle, the optimization model of the kth cycle is

[0130]

[0131] where, is the optimal point of the k-1th cycle; is the design point of the k-1th cycle obtained by reliability evaluation.

[0132] The convergence condition of the sequence optimization and reliability evaluation method is: (1) the objective function tends to be stable, that is, the difference between the objective function values of the previous two cycles is less than a specified value, such as |f (k) -f (k-1) |≤10 -6 ; (2) all reliability constraints are satisfied, that is, all deterministic constraint functions are within the feasible region, g i (d,x MPP,i ,p MPP,i )≥0. When the above conditions are met, the cycle process is ended and the optimization is completed.

[0133] II. Inverse reliability analysis method

[0134] In the sequence optimization and reliability evaluation method, the inverse reliability analysis method is usually used to determine the design point (MPP) [x MPP,i ,p MPP,i ] corresponding to each reliability constraint R n , and whether the reliability requirement is met is evaluated by judging whether it is within the feasible region. The inverse reliability analysis optimization model is based on the first-order second-moment method. For complex high nonlinear high-dimensional limit state functions, the optimal solution obtained by the optimization model has large error, which leads to that the reliability optimization result cannot meet the design requirements, and the optimization process is not easy to converge, the number of cycles increases, and the amount of calculation increases. Therefore, combined with the first-order control variable method (First-order control variable method; FOCM), the inverse reliability analysis optimization model is improved, and the inverse reliability analysis method based on the first-order control variable is proposed.

[0135] III. First-order control variable method (control variable method based on Monte Carlo simulation)

[0136] Let the limit state function be g(x), and the failure probability be the integral of the joint probability density function f X (x) of n-dimensional random variable x = [x1, x2, …, x F ] in the failure domain F = {x: g(x)≤0},

[0137]

[0138] where, I X (x) is the failure domain indicator function;

[0139]

[0140] Random sampling is performed on the variable x by the probability density function f X (x), and x i is a random sample of N variables. The integral form of formula (6) is expressed as a mathematical expectation,

[0141] P f =∫ X I F (x)f X (x)dx=E[I F (x)] (8)

[0142] According to the law of large numbers in probability theory, when N is large enough, the sample mean converges to the expected value of the population I F (x)f X with probability 1, so the mathematical expectation in formula (8) can be estimated by the sample mean,

[0143]

[0144] Let g c (x) be a limit state function strongly related to g(x). The indicator function is a control variable strongly related to the indicator function I F (x), and its corresponding failure domain is expressed as F c ={x:g c (x)≤0}, and the indicator function is

[0145]

[0146] The new indicator function is,

[0147]

[0148] where ρ is a correction coefficient.

[0149] Substitute formula (11) for I F (x) in formula (6), and the failure probability can be rewritten as,

[0150]

[0151] where is the failure probability of the limit state function g c (x) strongly related to g(x); let be 0, and the correction coefficient ρ is,

[0152]

[0153] At the same time, the failure probability is expressed as,

[0154]

[0155] The reliability R can be expressed as,

[0156] R = p R · R c (15)

[0157] where R c is the reliability of the strongly correlated limit state function g c (x); p R is a correction factor, expressed as,

[0158]

[0159] where I F (x) is the failure domain indicator function of the limit state function g(x), and the failure domain F = {x: g(x) < 0}; is the failure domain indicator function of the strongly correlated limit state function g c (x), and the failure domain F c = {x: g c (x) < 0}.

[0160] After any random variable x is converted into an independent random variable u in the standard normal space U through Nataf transformation, the strongly correlated limit state function g c (x) is converted into g c (u) in the standard normal space, and let g c (u) be the first-order Taylor expansion at the design point u * :

[0161]

[0162] where g * (u) is the first-order partial derivative of g(u) at the design point u c ;

[0163] According to the first-order second-moment method, the reliability R c of the strongly correlated limit state function is obtained from the reliability index β c :

[0164] R c = Φ(β c ) (18)

[0165] And the reliability index β * and the design point u cIt can be solved by iterative optimization, with the mean of the random variable u as the initial search point.

[0166] minβ=‖u‖

[0167] stg(u)=0 (19)

[0168] Where, ‖u‖ is the distance from the origin of the standard normal space coordinate to the limit state surface;

[0169] At the same time, according to the limit state function g(u) transformed into the standard normal space and the strongly correlated limit state function g c (u), Latin hypercube sampling is used to generate a small number of samples to estimate the correction coefficient ρ R The specific process is as follows:

[0170] (1) Through Nataf transformation, the n-dimensional random variable x=[x1,x2,…,x n ], converted to independent random variables u=[u1,u2,…,u n ];

[0171] (2) For each random variable u i , draw N independent samples that obey the uniform distribution [0,1], and set each random variable u i The probability distribution of is divided into N non-overlapping intervals according to equal probability, and then the random variable u i The N samples are divided into N intervals, and each sample U i,j Cumulative probability; j = 1, 2, ..., N;

[0172] (3) According to the corresponding cumulative distribution function F U (u i ), get each random variable u i N samples U i =[U i,1 ,U i,2 ,…,U i,N ];

[0173] (4) N samples U of each random variable are respectively i Re-randomly combine and obtain N samples of random variables u U=[U1,U2,…,u j ,…U N ]; where U j =[U 1,j ,U 2,j ,…,U n,j ];

[0174] (5) Based on the sample set U, calculate the limit state function g(u) and the strongly correlated limit state function gc (u), and count the number of samples N f for which g(u) < 0 f <20, go to step (2), extract N add more samples and add them to the sample set U until N f ≥ 20

[0175] (6) Calculate the correction factor p R from equation (16)

[0176] Four, the inverse reliability analysis method based on the first order control variable

[0177] Based on the first order second moment (FORM) method, the corresponding reliability index β can be calculated from the reliability constraint R in the standard normal space U; according to the geometric meaning of the reliability index, the inverse reliability analysis can be described by an optimization problem, and the specific optimization model is,

[0178] min g(u)

[0179] s.t.‖u‖=β=Φ -1 (R) (20)

[0180] where u=[μ x ,μ p ] is an independent standard normal random variable; Φ(·) is the cumulative distribution function of the standard normal distribution; the optimal solution u * of the optimization problem is the point on the surface with radius β in the standard normal space that minimizes the value of the limit state function, and then through the Nataf inverse transformation, the design point [x MPP , p MPP ] can be obtained;

[0181] According to the first order control variable method, the reliability constraint R can be expressed as,

[0182] R=ρ R ·R PORM (21)

[0183] where R PORM is the reliability estimated by the first order second moment method; therefore, the inverse reliability analysis optimization model based on the first order second moment in equation (20) can be transformed into,

[0184] min g(u)

[0185] s.t.‖u‖=β=Φ -1 (R PORM )

[0186] where R PORM =R / ρ R (22)

[0187] According to the calculation principle of the first-order control variable method, the correction coefficient ρ R Can eliminate the reliability constraints R and R PORM Therefore, in the inverse reliability analysis optimization model, the reliability constraint R is transformed into R PORM , and by R PORM Optimize and find the optimal solution u * , and thus the design point corresponding to R is obtained [x MPP ,p MPP ], eliminating the error caused by the inverse reliability analysis optimization model based on the first-order second moment.

[0188] 5. Reliability Optimization Design Method Based on First-Order Control Variables

[0189] Based on the sequence optimization and reliability assessment method, combined with the first-order control variable method and the improved inverse reliability analysis optimization model, a reliability optimization design method based on the first-order control variable is proposed. The flow chart of this method is as follows: Figure 2 The specific calculation process is as follows:

[0190] (1) Through Nataf transformation, the mean μ of random design variable x and random parameter variable p is converted to x and μ p Transformed into the standard normal space and used as the initial design point for the optimization problem (Equation 19) in the first-order control variable method Optimized update

[0191] (2) According to the limit state function g i (d,μ x ,μ p ),Will The first-order Taylor expansion at is used as the strongly correlated limit state function g c,i (d,μ x ,μ p ), i=1,2,…,n.

[0192] (3) According to each limit state function g i (d,μ x ,μ p ) and the corresponding strongly correlated limit state function g c,i (d,μ x ,μ p ), Latin hypercube sampling is used to generate a small number of samples to estimate the correction coefficient ρ R,i .

[0193] (4) The mean μ of the random design variable x and the random parameter variable p xand μ p As the initial point, the first cycle is carried out, and the optimal point is obtained by the first cycle optimization model (Formula 2) in the sequence optimization and reliability assessment method. And the objective function value f (1) .

[0194] (5) Through Nataf conversion, the optimal point of the first cycle Transformed into standard normal space, we get according to and ρ R,i , the inverse reliability analysis optimization model (Formula 22) in the inverse reliability analysis method based on the first-order control variable is optimized to obtain Then, the Nataf inverse transformation was used to obtain the i Corresponding design points

[0195] (6) From the optimal point of the k-1th cycle and design points Get the movement vector of the kth cycle The optimal point is obtained by performing the k-th cycle optimization model (Formula 5) of the sequence optimization and reliability assessment method. And the objective function value f (k) ; k=1,2,….

[0196] (7) Through Nataf conversion, the optimal point of the kth cycle Transformed into standard normal space, we get according to and ρ R,i , the inverse reliability analysis optimization model (Formula 22) in the inverse reliability analysis method based on the first-order control variable is optimized to obtain Then, the Nataf inverse transformation was used to obtain the R i Corresponding design points

[0197] (8) When the optimization results converge and all reliability constraints are within the feasible region, the cycle ends and the optimization is completed, and the optimal point and objective function value are output; otherwise, return to step (6) and update the optimization results.

[0198] 6. Optimization design of cutting system chatter reliability

[0199] Modern manufacturing often has higher requirements for cutting accuracy and machining efficiency, so improving productivity is crucial to mechanical processing. In the optimization design of mechanical processing system, the production efficiency or economic benefit is usually taken as the optimization target. For cutting processing, under the premise of ensuring the stability of cutting process, the maximum material removal rate (MRR) is the main purpose of cutting system optimization, that is, the maximum material removal rate under the condition that the cutting system does not vibrate.

[0200] The material removal rate in cutting process is the volume of the processed parts removed per unit time, which is an important standard to measure the processing efficiency. According to the definition of material removal rate, its formula can be expressed as:

[0201] Q=10 3 ·f0·a p ·v c (23)

[0202] Where f0 is the feed rate, which is the relative displacement of the workpiece and the tool along the feed direction per revolution, mm·r -1 ; a p is the cutting depth, mm; v c is the cutting speed, which is the instantaneous speed of the cutting edge specified point relative to the workpiece main motion, m·min -1 . The cutting speed at the maximum diameter of the workpiece is expressed as:

[0203] v c =10 -3 ·π·d0·Ω (24)

[0204] Where d0 is the rotational cutting diameter of the tool specified point or the workpiece, mm; Ω is the spindle speed, r·min -1 .

[0205] In order to improve the cutting processing productivity as much as possible without vibration of the cutting system, shorten the processing time and save production cost, the material removal rate is taken as the optimization objective function of cutting chatter reliability optimization design.

[0206] From the material removal rate formula, it can be seen that the main variables affecting cutting processing are the three elements of cutting parameters, i.e. cutting depth a p , feed rate f0 and spindle speed Ω (or cutting speed v c ), therefore, a p , f0 and Ω are taken as the random design variables of cutting chatter reliability optimization design, i.e. x=[a p , f0, Ω]. Other dynamic parameters of cutting system, such as cutting system natural frequency ω n , equivalent stiffness k, damping ratio ξ, cutting stiffness coefficient kc and so on, are random parameter variables in the optimization design.

[0207] In the actual cutting process, due to the use of machine tool limitations, in the cutting system optimization design, cutting force constraints and cutting power constraints are needed. The cutting force constraint can be expressed as:

[0208] L1=F d -F dmax =k c ·a p ·f0-F dmax (25)

[0209] where k c is the cutting stiffness coefficient, N·mm -2 ; F d is the instantaneous cutting force, N; F dmax is the maximum cutting force, N. And from the cutting power calculation formula, the cutting power constraint is:

[0210] L2=P c -P cmax =10 -3 ·F d ·v c / 60-P cmax (26)

[0211] where P c is the instantaneous cutting power, kw; P cmax is the maximum cutting power, kw; v c is the cutting speed, m·min -1 .

[0212] In the case of considering parameter uncertainty, in order to ensure stable cutting of the cutting system and no chatter in the process, in addition to the boundary constraints of the design variables, chatter reliability constraints are also needed in the cutting system optimization design. Therefore, the specific optimization model expression of cutting chatter reliability optimization design is:

[0213] minf(x,p) = 1 / Q = 1 / (π·x1·x2·x3·p1)

[0214]

[0215] where x = [a p , f0, Ω] is the random design variable; p is the cutting system dynamics parameter variable; and are the minimum and maximum allowable cutting depths, respectively; and are the lowest and highest allowable feed rates during cutting, respectively; Ω min and Ωmax respectively represent the minimum and maximum spindle speed in a certain range of leaflet; is the limit cutting depth; P(g(x, p) > 0) is the reliability model of the cutting system; g(x, p) is the limit state function of the chatter reliability of the cutting system; R is the target reliability, and R = 0.99 is taken.

[0216] Example Two

[0217] In this example, the reliability optimization design of the turning tool system and the turning machining system is respectively carried out by using the reliability optimization design method (SORA-FOCM) based on the first-order control variable provided by the present application and the existing sequence optimization and reliability evaluation method (SORA), and the results of the two are compared.

[0218] 1. Chatter reliability optimization design of the turning tool system

[0219] For the turning tool system, the random design variable is x = [a p , f0, Ω], the mean values of a p and f0are 0.62 mm and 0.05 mm, and the mean values of Ω in different intervals are 3000 r·min -1 , 4000 r·min -1 and 6000 r·min -1 , and the variation coefficients are all 0.01. The random parameter variable p = [d0, f n , ζ, k, φ, α, k t , k r ], and the statistical characteristics are shown in Table 1. According to the stability model of the turning tool system, the limit cutting depth in the optimization model is expressed as:

[0220]

[0221] wherein,

[0222] The corresponding spindle speed Ω is expressed as:

[0223]

[0224] wherein, n = 0, 1, 2, 3, … is the integer wave number generated in the first cutting process.

[0225] Table 1 Statistical characteristics of random parameters of the turning tool system

[0226]

[0227]

[0228] The comparison results of turning tool system reliability optimization before and after are shown in Table 2 when n is 2, 3 and 4 respectively.

[0229] Table 2 Comparison results of turning tool system reliability optimization

[0230]

[0231] 2. Chatter reliability optimization design of turning system

[0232] Considering the turning system failure in the first order mode, let the random design variables be x = [a p , f0, Ω], a p and f0 are 0.054 mm and 0.5 mm, and the mean values of Ω in different intervals are 2500 r·min -1 and 4300 r·min -1 , and the coefficient of variation is 0.01. The random parameter variables The statistical characteristics of the parameter variables of the chatter reliability analysis of the turning system are shown in Table 3. According to the stability model of the turning system, the limit cutting depth in the optimization model is expressed as:

[0233]

[0234] And the corresponding spindle speed Ω is expressed as:

[0235]

[0236] Wherein, and are expressed as:

[0237]

[0238] In the formula, n = 0, 1, 2, 3, … is the integer wave number generated in the first cutting process.

[0239] Table 3 Statistical characteristics of random parameters of turning system

[0240]

[0241]

[0242] When the spindle speed Ω is 2500 r·min -1 and 4300 r·min -1 , i.e. when n = 7 and n = 4, Table 4 shows the comparison results of turning system reliability optimization before and after at each cutting position under different spindle speeds.

[0243] Table 4 turning system reliability optimization comparison results

[0244]

[0245]

[0246]

[0247] As can be seen from Table 2 and Table 4, the material removal rate after optimization by using the reliability optimization design method based on the first-order control variable and the sequence optimization and reliability evaluation method is greatly improved compared with before optimization. In comparison, the result of using the reliability optimization design method based on the first-order control variable is better, which can maximize the material removal rate while basically meeting the requirement of the turning system that the reliability R is greater than 0.99, while the result of the sequence optimization and reliability evaluation method has certain error. At the same time, the cycle number of the reliability optimization design method based on the first-order control variable is slightly less than the cycle number of the sequence optimization and reliability evaluation method, so it can be seen that the reliability optimization design method proposed in the application improves the optimization precision and reduces the operation cost to a certain extent.

[0248] By using the above technical solutions disclosed in the application, the following beneficial effects are obtained:

[0249] The application provides a cutting system chatter reliability optimization design method based on a first-order control variable, which optimizes cutting parameters to obtain optimal parameters conducive to production, so as to improve cutting processing efficiency, reduce processing cost and improve processing quality. The application improves the inverse reliability analysis method by combining the first-order control variable method to improve the inverse reliability analysis optimization model, which can reduce the optimal solution error, ensure that the reliability optimization result meets the design requirement, and the optimization process is easy to converge, the cycle number is small, and the calculation amount is low. The application improves the optimization precision and reduces the operation cost to a certain extent.

[0250] The above only describes the preferred embodiments of the application, and it should be noted that, for those skilled in the art, without departing from the principles of the application, a number of improvements and refinements can be made, which should also be considered as the protection scope of the application.

Claims

1. A cutting system chatter reliability optimization design method based on first-order control variables, characterized by: The following steps are included: S1. The cutting depth, feed rate and spindle speed of the cutting system are used as random design variables for the cutting chatter reliability optimization design; other dynamic parameters of the cutting system are used as random parameter variables for the cutting chatter reliability optimization design; S2. Taking the maximum material removal rate as the objective function of the cutting system chatter reliability optimization design, and establishing a cutting system chatter reliability optimization model based on cutting force constraints and cutting power constraints; S3. Based on the reliability optimization design method of the first-order control variables, the vibration reliability optimization model of the cutting system is optimized to obtain the optimal design point.

2. The cutting system chatter reliability optimization design method based on first-order control variables according to claim 1 is characterized by: Step S3 specifically includes the following contents: S31, through Nataf transformation, the mean μ of random design variable x and random parameter variable p x and μ p Transformed into the standard normal space and used as the initial design point for the optimization problem in the first-order control variable method Optimized update S32, by each limit state function g i (d,μ x ,μ p ),Will The first-order Taylor expansion at is used as the strongly correlated limit state function g c,i (d,μ x ,μ p ), i=1,2,…,n; S33, according to each limit state function g i (d,μ x ,μ p ) and the corresponding strongly correlated limit state function g c,i (d,μ x ,μ p ), Latin hypercube sampling is used to generate a small number of samples to estimate the correction coefficient ρ R,i ; S34, the mean μ of the random design variable x and the random parameter variable p x and μ p As the starting point, the first cycle is carried out, and the optimal point is obtained by the first cycle optimization model in the sequence optimization and reliability assessment method. And the objective function value f (1) ; S35, through Nataf conversion, the optimal point of the first cycle Transformed into standard normal space, we get according to and ρ R,i , the inverse reliability analysis optimization model in the inverse reliability analysis method based on the first-order control variable is optimized to obtain Then, the Nataf inverse transformation was used to obtain the i Corresponding design points S36, from the optimal point of the k-1th cycle and design points Get the movement vector of the kth cycle The optimal point is obtained by performing the k-th cycle optimization model of the k-th cycle in the sequence optimization and reliability assessment method. And the objective function value f (k) ; k = 1, 2, ...; S37, through Nataf conversion, the optimal point of the kth cycle Transformed into standard normal space, we get according to and ρ R,i , the inverse reliability analysis optimization model in the inverse reliability analysis method based on the first-order control variable is optimized to obtain Then, the Nataf inverse transformation was used to obtain the i Corresponding design points S38. When the optimization result converges and all reliability constraints are within the feasible region, the loop process ends to complete the optimization and output the optimal point and objective function value; otherwise, return to step S36 to update the optimization result.

3. The cutting system chatter reliability optimization design method based on first-order control variables according to claim 2, characterized in that: The sequential optimization and reliability assessment method constrains reliability and uses a single-layer cycle to sequentially perform deterministic optimization design and reliability assessment. In the sequence optimization and reliability assessment method, the reliability constraint P(g i (d,x,p)>0)≥R i Transformed into constraint function g i (d,x MPP,i ,x MPP,i )≥0, so that the optimization model establishes an equivalent relationship between reliability optimization and deterministic optimization. The optimization model is, Assume that the initial values ​​of the random design variable x and random parameter variable p in the reliability optimization model are their mean μ x and μ p , and obtain the optimal point through optimization The optimization model for the first cycle is: Furthermore, from the optimal point The inverse reliability analysis method is used to evaluate the reliability and obtain the design point of the i-th reliability constraint condition If the design point It does not fall within the feasible region, that is, If it is not established, the second cycle is performed and the deterministic constraints of the optimization model are modified, and the deterministic constraint function g in formula (1) is moved i (d,x,p)≥0, so that the design point At least falls on the constraint boundary, that is, at least Then is the deterministic constraint function for vector movement, that is Among them, the movement vector Each deterministic constraint function forms a feasible domain that is narrower than the previous cycle, and optimization is performed on this basis. The optimization model is: The optimal point obtained by the second cycle Conduct reliability assessment and obtain design points And judge whether the optimization meets the convergence conditions and whether to proceed to the next cycle; similar to the second cycle, the optimization model of the kth cycle is, in, is the optimal point of the k-1th cycle; is the design point of the k-1th cycle obtained for reliability evaluation.

4. The cutting system chatter reliability optimization design method based on first-order control variables according to claim 3 is characterized by: The first-order control variable method is a control variable method based on Monte Carlo simulation, specifically, Assume that the limit state function is g(x), and the failure probability is an n-dimensional random variable x=[x1,x2,…,x n The joint probability density function f X The integral of (x) in the failure domain F = {x:g(x)≤0}, Among them, I F (x) is the failure domain indicator function; From the probability density function f X (x) Randomly sample the variable x, x i is a random sample of N variables, then the integral form of formula (6) is expressed as mathematical expectation: P f =∫ X I F (x)f X (x)dx=E[I F (x)] (8) According to the law of large numbers in probability theory, when N is large enough, the sample mean Converges to the matrix I with probability 1 F (x)f X Therefore, the mathematical expectation in formula (8) can be estimated by the sample mean. Assume that the limit state function g is strongly related to g(x) c (x), then the indicator function is an indicator function I F (x) is a strongly correlated control variable, and its corresponding failure domain is expressed as F c ={x:g c (x)≤0}, then the indicator function is, The new indicator function for, Where, ρ is the correction coefficient; Substitute formula (11) for I in formula (6) F (x), then the failure probability can be rewritten as, in, is the strongly correlated limit state function g c (x) failure probability; let is 0, and the correction coefficient ρ is obtained as, Meanwhile, the failure probability is expressed as, Reliability R can be expressed as, R=ρ R ·R c (15) Among them, R c is the strongly correlated limit state function g c Reliability of (x); ρ R is the correction coefficient, expressed as, Among them, I F (x) is the failure domain indicator function of the limit state function g(x), and the failure domain F = {x: g(x) ≤ 0}; is a strongly correlated limit state function g c (x) failure domain indicator function, failure domain F c ={x:g c (x)≤0}. Convert any random variable x into an independent random variable u in the standard normal space U through Nataf, then the strongly correlated limit state function g c (x) is transformed into the standard normal space and expressed as g c (u), let g c (u) is the design point u * The first-order Taylor expansion at : in, is g(u) at the design point u * The first-order partial derivative at ; According to the first-order second moment method, the reliability R of the strongly correlated limit state function is c By the reliability index β c To obtain, b c =Φ(β c ) (18) The reliability index β c and design point u * It can be solved by iterative optimization, with the mean of the random variable u as the initial search point. minβ=‖u‖ stg(u)=0 (19) Where, ‖u‖ is the distance from the origin of the standard normal space coordinate to the limit state surface; At the same time, according to the limit state function g(u) transformed into the standard normal space and the strongly correlated limit state function g c (u), Latin hypercube sampling is used to generate a small number of samples to estimate the correction coefficient ρ R .

5. The cutting system chatter reliability optimization design method based on first-order control variables according to claim 4 is characterized in that: Use Latin hypercube sampling to generate a small number of samples to estimate the correction coefficient ρ R The specific process is: A1. Through Nataf transformation, the n-dimensional random variable x=[x1,x2,…,x n ], converted to independent random variables u=[u1,u2,…,u n ]; A2. For each random variable u i , draw N independent samples that obey the uniform distribution [0,1], and set each random variable u i The probability distribution of is divided into N non-overlapping intervals according to equal probability, and then the random variable u i The N samples are divided into N intervals, and each sample U i,j Cumulative probability; j = 1, 2, ..., N; A3. According to the corresponding cumulative distribution function F U (u i ), get each random variable u i N samples U i =[U i,1 ,U i,2 ,…,U i,N ]; A4. Separately transform the N samples U of each random variable i Re-randomly combine and obtain N samples of random variables u U=[U1,U2,…,U j ,…U N ]; where U j =[U 1,j ,U 2,j ,…,U n,j ]; A5. Based on the sample set U, calculate the limit state function g(u) and the strongly correlated limit state function g c (u), and count the number of samples N that make g(u)<0 f , if N f <20, then return to step A2 and extract N add samples are added to the sample set U until N f ≥20; A6. Calculate the correction coefficient ρ using formula (16) R .

6. The cutting system chatter reliability optimization design method based on first-order control variables according to claim 5 is characterized in that: The inverse reliability analysis method based on first-order control variables is as follows: Based on the first-order second moment method, in the standard normal space U, the corresponding reliability index β can be calculated from the reliability constraint R. According to the geometric meaning of the reliability index, the inverse reliability analysis can be described by an optimization problem. The specific optimization model is: min g(u) st‖u‖=β=Φ -1 (R) (20) Where u=[μ x ,μ p ] is an independent standard normal random variable; Φ(·) is the cumulative distribution function of the standard normal distribution; the optimal solution u * is the point on the surface with radius β in the standard normal space that minimizes the limit state function value. Then, through the Nataf inverse transformation, the design point [x MPP ,p MPP ]; According to the first-order control variable method, the reliability constraint R is expressed as, R=ρ R ·R PORM (21) Among them, R PORM is the reliability estimated by the first-order second moment method; therefore, the inverse reliability analysis optimization model based on the first-order second moment in formula (20) can be transformed into, min g(u) st‖u‖=β=Φ -1 (R PORM ) where R PORM =R / ρ R (22) According to the calculation principle of the first-order control variable method, the correction coefficient ρ R Can eliminate the reliability constraints R and R PORM Therefore, in the inverse reliability analysis optimization model, the reliability constraint R is transformed into R PORM , and by R PORM Optimize and find the optimal solution u * , and thus the design point corresponding to R is obtained [x MPP ,p MPP ], eliminating the error caused by the inverse reliability analysis optimization model based on the first-order second moment.

7. The cutting system chatter reliability optimization design method based on first-order control variables according to claim 6, characterized in that: In step S2, The material removal rate is, Q=10 3 ·f0·a p ·v c (23) Where Q is the material removal rate; f0 is the feed rate; a p is the cutting depth; v c is the cutting speed; The cutting speed is, v c =10 -3 ·π·d0·Ω (24) Where d0 is the rotating cutting diameter of the tool at the specified point or workpiece; Ω is the spindle speed; The cutting force constraint is, L1=F d -F dmax =k c ·a p ·f0-F dmax (25) Among them, L1 is the cutting force; F d is the instantaneous cutting force; F dmax is the maximum cutting force; k c is the cutting stiffness coefficient; The cutting power constraint is, L2=P c -P cmax =10 -3 ·F d ·v c / 60-P cmax (26) Among them, K2 is cutting power; P c is the instantaneous cutting power; P cmax is the maximum cutting power; The cutting system chatter reliability optimization model is: minf(x,p)=1 / Q=1 / (π·x1·x2·x3·p1) Among them, g(x,p) is the limit state function of the cutting system vibration reliability; P(g(x,p)>0) is the reliability model of the cutting system; the random design variable x=[a p ,f0,Ω]; p is the cutting system dynamic parameter variable; R is the target reliability; and are the minimum and maximum allowable cutting depths respectively; and are the minimum and maximum allowable feed rates during cutting; Ω min and Ω max are the minimum and maximum spindle speeds within a certain lobe range, respectively; The maximum cutting depth.

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