选择最佳工具,几何形状和切削条件

Selection of optimum tool geometry and cutting conditionsusing a surface roughness prediction model for end milling Abstract Influence of tool geometry on the quality of surface produced is well known and hence any attempt to assess the performance of end milling should include the tool geometry. In the present work, experimental studies have been conducted to see the effect of tool geometry (radial rake angle and nose radius) and cutting conditions (cutting speed and feed rate) on the machining performance during end milling of medium carbon steel. The first and second order mathematical models, in terms of machining parameters, were developed for surface roughness prediction using response surface methodology (RSM) on the basis of experimental results. The model selected for optimization has been validated with the Chi square test. The significance of these parameters on surface roughness has been established with analysis of variance. An attempt has also been made to optimize the surface roughness prediction model using genetic algorithms (GA). The GA program gives minimum values of surface roughness and their respective optimal conditions.1 IntroductionEnd milling is one of the most commonly used metal removal operations in industry because of its ability to remove material faster giving reasonably good surface quality. It is used in a variety of manufacturing industries including aerospace and automotive sectors, where quality is an important factor in the production of slots, pockets, precision moulds and dies. Greater attention is given to dimensional accuracy and surface roughness of products by the industry these days. Moreover, surface finish influences mechanical properties such as fatigue behaviour, wear, corrosion, lubrication and electrical conductivity. Thus, measuring and characterizing surface finish can be considered for predicting machining performance.Surface finish resulting from turning operations has traditionally received considerable research attention, where as that of machining processes using multipoint cutters, requires attention by researchers. As these processes involve large number of parameters, it would be difficult to correlate surface finish with other parameters just by conducting experiments. Modelling helps to understand this kind of process better. Though some amount of work has been carried out to develop surface finish prediction models in the past, the effect of tool geometry has received little attention. However, the radial rake angle has a major affect on the power consumption apart from tangential and radial forces. It also influences chip curling and modifies chip flow direction. In addition to this, researchers [1] have also observed that the nose radius plays a significant role in affecting the surface finish. Therefore the development of agood model should involve the radial rake angle and nose radius along with other relevant factors.Establishment of efficient machining parameters has been a problem that has confronted manufacturing industries for nearly a century, and is still the subject of many studies. Obtaining optimum machining parameters is of great concern in manufacturing industries, where the economy of machining operation plays a key role in the competitive market. In material removal processes, an improper selection of cutting conditions cause surfaces with high roughness and dimensional errors, and it is even possible that dynamic phenomena due to auto excited vibrations may set in [2]. In view of the significant role that the milling operation plays in today‟s manufacturing world, there is a need to optimize the machining parameters for this operation. So, an effort has been made in this paper to see the influence of tool geometry(radial rake angle and nose radius) and cutting conditions(cutting speed and feed rate) on the surface finish produced during end milling of medium carbon steel. The experimental results of this work will be used to relate cutting speed, feed rate, radial rake angle and nose radius with the machining response i.e. surface roughness by modelling. The mathematical models thus developed are further utilized to find the optimum process parameters using genetic algorithms.2 ReviewProcess modelling and optimization are two important issues in manufacturing. The manufacturing processes are characterized by a multiplicity of dynamically interacting process variables. Surface finish has been an important factor of machining in predicting performance of any machining operation. In order to develop and optimize a surface roughness model, it is essential to understand the current status of work in this area.Davis et al. [3] have investigated the cutting performance of five end mills having various helix angles. Cutting tests were performed on aluminium alloy L 65 for three milling processes (face, slot and side), in which cutting force, surface roughness and concavity of a machined plane surface were measured. The central composite design was used to decide on the number of experiments to be conducted. The cutting performance of the end mills was assessed using variance analysis. The affects of spindle speed, depth of cut and feed rate on the cutting force and surface roughness were studied. The investigation showed that end mills with left hand helix angles are generally less cost effective than those with right hand helix angles. There is no significant difference between up milling and down milling with regard tothe cutting force, although the difference between them regarding the surface roughness was large. Bayoumi et al.[4] have studied the affect of the tool rotation angle, feed rate and cutting speed on the mechanistic process parameters (pressure, friction parameter) for end milling operation withthree commercially available workpiece materials, 11 L 17 free machining steel, 62- 35-3 free machining brass and 2024 aluminium using a single fluted HSS milling cutter. It has been found that pressure and friction act on the chip – tool interface decrease with the increase of feed rate and with the decrease of the flow angle, while the cutting speed has a negligible effect on some of the material dependent parameters. Process parameters are summarized into empirical equations as functions of feed rate and tool rotation angle for each work material. However, researchers have not taken into account the effects of cutting conditions and tool geometry simultaneously; besides these studies have not considered the optimization of the cutting process.As end milling is a process which involves a large number f parameters, combined influence of the significant parameters an only be obtained by modelling. Mansour and Abdallaet al. [5] have developed a surface roughness model for the end milling of EN32M (a semi-free cutting carbon case hardening steel with improved merchantability). The mathematical model has been developed in terms of cutting speed, feed rate and axial depth of cut. The affect of these parameters on the surface roughness has been carried out using response surface methodology (RSM). A first order equation covering the speed range of 30–35 m/min and a second order equation covering the speed range of 24–38 m/min were developed under dry machining conditions. Alauddin et al. [6] developed a surface roughness model using RSM for the end milling of 190 BHN steel. First and second order models were constructed along with contour graphs for the selection of the proper combination of cutting speed and feed to increase the metal removal rate without sacrificing surface quality. Hasmi et al. [7] also used the RSM model for assessing the influence of the workpiece material on the surface roughness of the machined surfaces. The model was developed for milling operation by conducting experiments on steel specimens. The expression shows, the relationship between the surface roughness and the various parameters; namely, the cutting speed, feed and depth of cut. The above models have not considered the affect of tool geometry on surface roughness.Since the turn of the century quite a large number of attempts have been made to find optimum values of machining parameters. Uses of many methods have been reported in the literature to solve optimization problems for machining parameters. Jain and Jain [8] have used neural networks for modeling and optimizing the machining conditions. The results have been validated by comparing the optimized machining conditions obtained using genetic algorithms. Suresh et al. [9] have developed a surface roughness prediction model for turning mild steel using a response surface methodology to produce the factor affects of the individual process parameters. They have also optimized the turning process using the surface roughness prediction model as the objective function. Considering the above, an attempt has been made in this work to develop a surface roughness model with tool geometry and cutting conditions on the basis ofexperimental results and then optimize it for the selection of these parameters within the given constraints in the end milling operation.3 MethodologyIn this work, mathematical models have been developed using experimental results with the help of response surface methodolog y. The purpose of developing mathematical models relating the machining responses and their factors is to facilitate the optimization of the machining process. This mathematical model has been used as an objective function and the optimization was carried out with the help of genetic algorithms.3.1 Mathematical formulationResponse surface methodology(RSM) is a combination of mathematical and statistical techniques useful for modelling and analyzing the problems in which several independent variables influence a dependent variable or response. The mathematical models commonly used are represented by:where Y is the machining response, ϕ is the response function and S, f , α, r are milling variables and ∈is the error which is normally distributed about the observed response Y with zero mean.The relationship between surface roughness and other independent variables can be represented as follows，where C is a constant and a, b, c and d are exponents.To facilitate the determination of constants and exponents, this mathematical model will have to be linearized by performing a logarithmic transformation as follows:The constants and exponents C, a, b, c and d can be determined by the method of least squares. The first order linear model, developed from the above functional relationship using least squares method, can be represented as follows:where Y1 is the estimated response based on the first-order equation, Y is the measured surface roughness on a logarithmic scale, x0 = 1 (dummy variable), x1, x2, x3 and x4 are logarithmic transformations of cutting speed, feed rate, radial rake angle and nose radius respectively, ∈is the experimental error and b values are the estimates of corresponding parameters.The general second order polynomial response is as given below:where Y2 is the estimated response based on the second order equation. The parameters, i.e. b0, b1, b2, b3, b4, b12, b23, b14, etc. are to be estimated by the method of least squares. Validity of the selected model used for optimizing the process parameters has been tested with the help of statistical tests, such as F-test, chi square test, etc. [10].3.2 Optimization using genetic algorithmsMost of the researchers have used traditional optimization techniques for solving machining problems. The traditional methods of optimization and search do not fare well over a broad spectrum of problem domains. Traditional techniques are not efficient when the practical search space is too large. These algorithms are not robust. They are inclined to obtain a local optimal solution. Numerous constraints and number of passes make the machining optimization problem more complicated. So, it was decided to employ genetic algorithms as an optimization technique. GA come under the class of non-traditional search and optimization techniques. GA are different from traditional optimization techniques in the following ways:1.GA work with a coding of the parameter set, not the parameter themselves.2.GA search from a population of points and not a single point.3.GA use information of fitness function, not derivatives or other auxiliary knowledge.4.GA use probabilistic transition rules not deterministic rules.5.It is very likely that the expected GA solution will be the global solution.Genetic algorithms (GA) form a class of adaptive heuristics based on principles derived from the dynamics of natural population genetics. The searching process simulates the natural evaluation of biological creatures and turns out to be an intelligent exploitation of a random search. The mechanics of a GA is simple, involving copying of binary strings. Simplicity of operation and computational efficiency are the two main attractions of the genetic algorithmic approach. The computations are carried out in three stages to get a result in one generation or iteration. The three stages are reproduction, crossover and mutation.In order to use GA to solve any problem, the variable is typically encoded into a string (binary coding) or chromosome structure which represents a possible solution to the given problem. GA begin with a population of strings (individuals) created at random. The fitness of each individual string is evaluated with respect to the given objective function. Then this initial population is operated on by three main operators – reproduction cross over and mutation– to create, hopefully, a better population. Highly fit individuals or solutions are given the opportunity to reproduce by exchanging pieces of their genetic information, in the crossover procedure, with other highly fit individuals. This produces new “offspring” solutions, which share some characteristics taken from both the parents. Mutation is often applied after crossover by altering some genes (i.e. bits) in the offspring. The offspring can either replace the whole population (generational approach) or replace less fit individuals (steady state approach). This new population is further evaluated and tested for some termination criteria. The reproduction-cross over mutation- evaluation cycle is repeated until the termination criteria aremet.4 Experimental detailsFor developing models on the basis of experimental data, careful planning of experimentation is essential. The factors considered for experimentation and analysis were cutting speed, feed rate, radial rake angle and nose radius.4.1 Experimental designThe design of experimentation has a major affect on the number of experiments needed. Therefore it is essential to have a well designed set of experiments. The range of values of each factor was set at three different levels, namely low, medium and high as shown in Table 1. Based on this, a total number of 81 experiments (full factorial design), each having a combination of different levels of factors, as shown in Table 2, were carried out.The variables were coded by taking into account the capacity and limiting cutting conditions of the milling machine. The coded values of variables, to be used in Eqs. 3 and 4, were obtained from the following transforming equations:where x1 is the coded value of cutting speed (S), x2 is the coded value of the feed rate ( f ), x3 is the coded value of radial rake angle(α) and x4 is the coded value of nose radius (r).4.2 ExperimentationA high precision …Rambaudi Rammatic 500‟ CNC milling machine, with a vertical milling head, was used for experimentation. The control system is a CNC FIDIA-12 compact. The cutting tools, used for the experimentation, were solid coated carbide end mill cutters of different radial rake angles and nose radii (WIDIA: DIA20 X FL38 X OAL 102 MM). The tools are coated with TiAlN coating. The hardness, density and transverse rupture strength are 1570 HV 30, 14.5 gm/cm3 and 3800 N/mm2 respectively.AISI 1045 steel specimens of 100×75 mm and 20 mm thickness were used in the present study. All the specimens were annealed, by holding them at 850 ◦C for one hour and then cooling them in a furnace. The chemical analysis of specimens is presented in Table 3. The hardness of the workpiece material is 170 BHN. All the experiments were carried out at a constant axial depth of cut of 20 mm and a radial depth of cut of 1 mm. The surface roughness (response) was measured with Talysurf-6 at a 0.8 mm cut-off value. An average of four measurements was used as a response value.5 Results and discussionThe influences of cutting speed, feed rate, radial rake angle and nose radius have been assessed by conducting experiments. The variation of machining response with respect to the variables was shown graphically in Fig. 1. It is seen from these figures that of the four dependent parameters, radial rake angle has definite influence on the roughness of the surface machined using an end mill cutter. It is felt that the prominent influence of radial rake angle on the surface generation could be due to the fact that any change in the radial rake angle changes the sharpness of the cutting edge on the periphery, i.e changes the contact length between the chip and workpiece surface. Also it is evident from the plots that as the radial rake angle changes from 4◦to 16◦, the surface roughness decreases and then increases. Therefore, it may be concluded here that the radial rake angle in the range of 4◦to 10◦would give a better surface finish. Figure 1 also shows that the surface roughness decreases first and then increases with the increase in the nose radius. This shows that there is a scope for finding the optimum value of the radial rake angle and nose radius for obtaining the best possible quality of the surface. It was also found that the surface roughness decreases with an increase in cutting speed and increases as feed rate increases. It could also be observed that the surface roughness was a minimum at the 250 m/min speed, 200 mm/min feed rate, 10◦radial rake angle and 0.8 mm nose radius. In order to understand the process better, the experimental results can be used to develop mathematical models using RSM. In this work, a commercially available mathematical software package (MATLAB) was used for the computation of the regression of constants and exponents.5.1 The roughness modelUsing experimental results, empirical equations have been obtained to estimate surface roughness with the significant parameters considered for the experimentation i.e. cutting speed, feed rate, radial rake angle and nose radius. The first order model obtained from the above functional relationship using the RSM method is as follows:The transformed equation of surface roughness prediction is as follows:Equation 10 is derived from Eq. 9 by substituting the coded values of x1, x2, x3 and x4 in terms of ln s, ln f , lnαand ln r. The analysis of the variance (ANOV A) and the F-ratio test have been performed to justify the accuracy of the fit for the mathematical model. Since the calculated values of the F-ratio are less than the standard values of the F-ratio for surface roughness as shown in Table 4, the model is adequate at 99% confidence level to represent the relationship between the machining response and the considered machining parameters of the end milling process.The multiple regression coefficient of the first order model was found to be 0.5839. Thisshows that the first order model can explain the variation in surface roughness to the extent of 58.39%. As the first order model has low predictability, the second order model has been developed to see whether it can represent better or not.The second order surface roughness model thus developed is as given below:where Y2 is the estimated response of the surface roughness on a logarithmic scale, x1, x2, x3 and x4 are the logarithmic transformation of speed, feed, radial rake angle and nose radius. The data of analysis of variance for the second order surface roughness model is shown in Table 5.Since F cal is greater than F0.01, there is a definite relationship between the response variable and independent variable at 99% confidence level. The multiple regression coefficient of the second order model was found to be 0.9596. On the basis of the multiple regression coefficient (R2), it can be concluded that the second order model was adequate to represent this process. Hence the second order model was considered as an objective function for optimization using genetic algorithms. This second order model was also validated using the chi square test. The calculated chi square value of the model was 0.1493 and them tabulated value at χ2 0.005 is 52.34, as shown in Table 6, which indicates that 99.5% of the variability in surface roughness was explained by this model.Using the second order model, the surface roughness of the components produced by end milling can be estimated with reasonable accuracy. This model would be optimized using genetic algorithms (GA).5.2 The optimization of end millingOptimization of machining parameters not only increases the utility for machining economics, but also the product quality toa great extent. In this context an effort has been made to estimate the optimum tool geometry and machining conditions to produce the best possible surface quality within the constraints.The constrained optimization problem is stated as follows: Minimize Ra using the model given here:where xil and xiu are the upper and lower bounds of process variables xi and x1, x2, x3, x4 are logarithmic transformation of cutting speed, feed, radial rake angle and nose radius.The GA code was developed using MATLAB. This approach makes a binary coding system to represent the variables cutting speed (S), feed rate ( f ), radial rake angle (α) and nose radius (r), i.e. each of these variables is represented by a ten bit binary equivalent, limiting the totalstring length to 40. It is known as a chromosome. The variables are represented as genes (substrings) in the chromosome. The randomly generated 20 such chromosomes (population size is 20), fulfilling the constraints on the variables, are taken in each generation. The first generation is called the initial population. Once the coding of the variables has been done, then the actual decoded values for the variables are estimated using the following formula: where xi is the actual decoded value of the cutting speed, feed rate, radial rake angle and nose radius, x(L) i is the lower limit and x(U) i is the upper limit and li is the substring length, which is equal to ten in this case.Using the present generation of 20 chromosomes, fitness values are calculated by the following transformation:where f(x) is the fitness function and Ra is the objective function.Out of these 20 fitness values, four are chosen using the roulette-wheel selection scheme. The chromosomes corresponding to these four fitness values are taken as parents. Then the crossover and mutation reproduction methods are applied to generate 20 new chromosomes for the next generation. This processof generating the new population from the old population is called one generation. Many such generations are run till the maximum number of generations is met or the average of four selected fitness values in each generation becomes steady. This ensures that the optimization of all the variables (cutting speed, feed rate, radial rake angle and nose radius) is carried out simultaneously. The final statistics are displayed at the end of all iterations. In order to optimize the present problem using GA, the following parameters have been selected to obtain the best possible solution with the least computational effort: Table 7 shows some of the minimum values of the surface roughness predicted by the GA program with respect to input machining ranges, and Table 8 shows the optimum machining conditions for the corresponding minimum values of the surface roughness shown in Table 7. The MRR given in Table 8 was calculated bywhere f is the table feed (mm/min), aa is the axial depth of cut (20 mm) and ar is the radial depth of cut (1 mm).It can be concluded from the optimization results of the GA program that it is possible to select a combination of cutting speed, feed rate, radial rake angle and nose radius for achieving the best possible surface finish giving a reasonably good material removal rate. This GA program provides optimum machining conditions for the corresponding given minimum values of the surface roughness. The application of the genetic algorithmic approach to obtain optimal machining conditions will be quite useful at the computer aided process planning (CAPP) stage in the production of high quality goods with tight tolerances by a variety of machining operations, and in the adaptive control of automated machine tools. With the known boundaries of surfaceroughness and machining conditions, machining could be performed with a relatively high rate of success with the selected machining conditions.6 ConclusionsThe investigations of this study indicate that the parameters cutting speed, feed, radial rake angle and nose radius are the primary actors influencing the surface roughness of medium carbon steel uring end milling. The approach presented in this paper provides n impetus to develop analytical models, based on experimental results for obtaining a surface roughness model using the response surface methodology. By incorporating the cutter geometry in the model, the validity of the model has been enhanced. The optimization of this model using genetic algorithms has resulted in a fairly useful method of obtaining machining parameters in order to obtain the best possible surface quality.。