基于灰色系统理论的车削参数多目标优化

覃孟扬，刘大维，罗永顺，李玉忠广东技术师范学院广东高校数控技术重点实验室，广州 510635

1．Introduction

Turning is an important metal cutting process，and it is commonly used in manufacturing．There are many quality assessment indicators for turning process［1］．Studies have shown that machined quality are affected by many factors［2］，which have very complex effects，even some factors have opposite effect on different quality indicators．At present，rapid development of equipment manufacturing required more comprehensive and higher machined quality than before，choice of turning parameters only based on practical experience can not reach the comprehensive requirements of machining quality．Therefore，multiobjective optimization of turning parameters has come to be very important［3］．

Grey system theory is used to solve problem of uncertainty when the research object has less data and poor information［4］，and it can be developed to extract useful information mainly through the generation of some known information．There are many complex factors contributing to machined quality of turning process and it shows evident grey properties［5］．The gray theory has been applied in manufacturing［6-8］，but turning parameter optimization based on grey theory has not been found before，therefore a turning parameters multi-objective optimization method based on this theory is proposed in this paper．

In turning experiments with various parameters，gray correlation coefficient and correlation were obtained between objective function including cutting rate，dimensional size error and surface roughness，and variables(parameters)including cutting speed，feed rate and cutting depth;and then turning parameters were optimized．A verification experiments showed that the optimized parameters have better results in turning parameters multi-objective optimization，and the result can provide technical support for turning process．

2．Grey system theory

Correlation in grey theory describes to relative changes between different factors during system development process，which have relative changes in the size，direction and speed．When the relative change is same in the basic agreement，correlation between different factors is large;on the contrary，correlation is small．

In grey theory，reference vector sequence in data conversion is X0= {x0(k)，k=1，2，…，n}，the target vector sequence is Xi={xi(k)，k=1，2，…，m}，where，m is target vector sequence number，the correlation coefficients of xifor x0in k points is:

Where，ρis distinguishing coefficient(a given factor，and general is 0．5)，Δ = x0(k)-xi(k )．Correlation of xifor x0is:

Where，γi0is import degree of gray correlation coefficient about xifor x0in i point;λkare the weights．

It is known that largerγi0is closer xiand x0are．When all the γi0(i=1，2，…，m)are known，the degree of similarity between target and reference vector sequence could been found out，and then the optimal parameters can be easily determined．

3．Multi-objective optimization

3．1．Process parameters and objective function

In this paper，cutting speed vc，feed rate f，and cutting depth apare the studied turning parameters，which are variables．Cutting rateη，dimensional size errorεand surface roughness Raare objective functions．An ideal results are the large metal cutting rate，small dimensional accuracy error and surface roughness．

3．2．Experiment

There is a turning experiment with varies parameters to obtain some raw data for gray correlation analysis．Experimental samples are round bar with 60 mm diameter，whose material is 45#steel;cutting tool material is carbide;machine tool is horizontal lathes．In the experiment，experimental factors are the variables(turning parameters)and the results are the objective functions．

The experimental arrangement and results are shown in Tab．1．

Tab．1 Experiment arrangement and results

3．3．Correlation coefficient and correlation

The magnitude of studied factors are so big that they must been treated as dimensionless．When they have roughly the same magnitude，it is accurate to have a quantitative analysis．

If i is the number of experiments，k is the target response，Ei(k)is the original data，xi(k)is the treated data，and E0is a middle looking indicators，formula of initial value treated for a large looking indicator is as follows:

Formula of initial value treated for a small looking indicator is:

Formula of initial value treated for a looking indicator in middle is:

These raw values are initially treated according to formula(3)～(5)，and then correlation coefficient of objective function and variables are obtained according to formula(1)，and results are shown in Tab．2．

When gray correlation coefficients are placed into the formula(2)and weights of objective function are considered equal，correlations can be obtained between objective function and variables，as shown in Tab．2．According to the gray system theory，greater correlation is，greater effect variables have on objective function;Tab．2 shows that bothη andεare af-fected by turning parameters in sequence:vc，f and ap;turning parameters affect Rain sequence:f，vcand ap．The result shows the affected sequence for different objective function is not the same．

Tab．2 Correlation coefficient and correlation between objective function and variables

3．4．Multi-objective optimization

According to the gray system theory，greater correlation，and greater variable response to multiobjective，the parameter which has the greatest correlation with objective functions is the optimal corresponding factor．Large metal cutting rate，small dimensional accuracy error and surface roughness are good turning quality，and they are the optimization goals．If only consider a single-objective optimization，as shown in Tab．3，the optimal parameters for η is:vc=113 m/min，f=0．2 mm/r，and ap=1 mm;the optimal parameters forεis:vc=113 m/min，f=0．05 mm/r，and ap=0．2 mm;the optimal parameters for Rais:vc=113 m/min，f=0．05 mm/r，and ap=0．2 mm．It is easy to know that the optimal parameters for three objectives are not the same，and there is no parameter which causes the optimal result for all objectives．

When three objective functions need be optimized in the same time，correlation for multi-objective is considered．The multi-objective correlation is obtained by the effect of parameter magnitude on all objectives．As shown in Tab．3，turning parameters for multi-objective optimization are as follows:vcis 113 m/min，f is 0．1 mm/r and apis 0．2 mm．

4．Verification experiment

An experiment of comparisons between optimized and non-optimized turning parameters is carried out to find out optimized turning results．Tab．4 shows the turning with optimized parameters reduces cutting rate，but it improves the turning precision and surface quality distinctly，as compared to the non-optimized parameters．Although the result comprehensively illustrates that the optimized turning parameters are more reasonable，it also proves the feasible optimization method adopted．

Tab．3 Correlation between objective function and parameter

Tab．4 Comparison of machined result

In this paper，the weights of three objective functions are the same．If the objective function or their weights are not the same，multi-objective optimization results will change．

5．Conclusions

Turning parameters were optimized based on gray system theory．All the correlation coefficients between turning parameters and multi-objective functions including cutting rate，dimensional size error and surface roughness，and multi-objective correlations are calculated．The optimum turning parameters are obtained through the analysis of variance．Validation Experiment show that the optimized turning parameters can generate the high cutting rate and reduce dimensional size error and surface roughness at the same time．Optimized results provide basis for subsequent experiments，and the gray system theory could help to solve multi-objective optimization problem about turning process．

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