A Fuzzy System Approach of Feed Rate Determination for CNC Milling

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1 A Fuzzy System Approach of Determination for CNC Milling Zhibin Miao Department of Mechanical and Electrical Engineering Heilongjiang Institute of Technology Harbin, China Wei Li Department of Mechanical and Electrical Engineering Heilongjiang Institute of Technology Harbin, China Abstract Determination of optimal machining parameters - spindle rate, feed rate, and depth of cut - has been a research topic for decades. Since the parameters of CNC machining significantly influence part machining time, part surface quality, and tool life, techniques of determining optimal machining parameters are in high demand in manufacturing industry. Usually the depth of cut and spindle rate are determined according to machinist manuals before machining; and the feed rate is determined subjectively either by CNC machine operators or programmers. As a result, the feed rate is not optimal in terms of the machining condition at every cutter location, and it is fixed at a conservative value causing longer machining time or shorter tool life. In this work, a generic and intelligent approach of feed rate determination for CNC profile milling is proposed. First a simplified cutting force model is introduced and an example database of machining parameters is presented. Second a fuzzy rule-based system is established to predict the cutting force based on the radial and axial depths of cut, and the assumed feed rate. Then identify the geometric features of the part, calculate the engagement angles of the geometric features, and find optimal feed rates for them. Finally apply this approach on an example part for profile milling, and the results are simulated with CATIA CAD/CAM system to demonstrate the advantages of this approach. key words Fuzzy Logical System,Machining Parameters Determination, CNC Milling, and Intelligent Machining. I. INTRODUCTION In mechanical parts machining, cutting efficiency, part accuracy and finish, and tool life are the major concerns. Higher cutting efficiency can reduce manufacturing costs, better part accuracy and finish can improve part quality, and extended tool life can benefit part machining. Many researches have been carried out on these machining concerns in terms of machining parameters such as spindle speed, feed rate, and depth of cut. Optimal machining parameters can significantly improve cutting efficiency and part quality. Usually the spindle speed and depth of cut are determined beforehand according to workpiece material and tool material and size; and the feed rate is determined by a machinist based on his/her working experience. Such feed rate is often quite conservative and fixed in machining a whole part due to the concern that if the feed rate is too high, the surface quality is reduced, and the tool wears out severely, or even breaks down ([6]). The conservative feed rate is not optimal since the geometry varies across the part and the cutting force changes accordingly. Therefore, the feed rate should be adjusted for different features. Optimal feed rate could be determined based on part material, tool material and size, cutting force, and depth of cut. The proposed approaches on feed rate determination can be classified as on-line and off-line approaches. The on-line approaches utilize force/torque sensors on tools to measure the cutting forces/torques when machining and adjust the feed rate based on the cutting force/torque ([1]). However, the control system and sensors in the on-line approaches are quite expensive, so these approaches are not widely used in industry. On the other hand, the off-line approaches predict cutting forces according to cutting force models and set different feed rates at different locations ([4], [5]). To determine the optimal feed rate, the current off-line approaches interpolate the experimental machining data to calculate the cutting force and find the proper feed rate. However, the drawback of these methods is that the interpolating surface is usually not smooth and the predicted cutting force is not reasonable and accurate. In this work, a fuzzy system approach of feed rate determination for CNC milling is proposed. First a simplified cutting force model is introduced and an example database of machining parameters is presented. Second a fuzzy rule-based system is established to predict cutting forces based on the radial and axial depth of cut, and the feed rate. Then the geometric features of the part are identified, the engagement angles for the geometric features are calculated, and the optimal feed rates are found. Finally this approach is applied on an example part for profile milling, and the results are simulated with CATIA CAD/CAM system to demonstrate the advantages of this approach /09/$ IEEE 1911 ICIEA 2009

2 II. RELATED WORK The research on cutting force prediction and feed rate determination has been conducted for a long time. An approach proposed by Bae, et al., ([1]) is a simple cutting force regularization method to adjust feed rates for pocket machining. This approach first introduces cutter engagement angle to measure chip load in two dimensions and derives the relationship between the cutter engagement angle and the radial depth of cut. Then it builds up a simplified cutting force model based on the machining data in a cutting experiment with a tool dynamometer. Finally a method of automatic feed rate adjustment is used to find proper feed rates. This approach is more stable than the infinitesimal flute model and more precise than the mean cutting force model. Nevertheless, since the axial depth of cut is not taken into account, the approach could not be applied when different axial depths of cut are used in CNC profile machining. On the other hand, the cutting force model is constructed by interpolating the experiment data with a Bezier surface. Since Bezier surfaces are prone to fluctuation, the model can not represent the correct cutting forces with a local undulating patch. To improve part quality using optimal feed rates, the concept of surface roughness model is adopted in a method proposed by Baek, et al., ([2]), which could plan the optimal feed rates. This method analyzes the effect of feed rate variation on surface roughness and dimension accuracy in a face milling operation, and the optimal feed rate, which provides the maximum material removal rate under the given surface roughness constraint, could be selected using a bisection method. However, it is very difficult to find the optimal feed rates using the bisection method for nonlinear systems. Although the two methods could find the optimal feed rates, they are difficult to use in a generic machining. The major contribution of this work is to find a new, practical, and generic way to determine optimal feed rates in CNC profile machining. With the help of a fuzzy logical system, the approach proposed in this paper takes the axial depth of cut into account and determines the feed rates with intelligence. III. A MACHINING PARAMETER DATABASE FOR CNC PROFILE MILLING In profile milling of a part, flat end mills are often used, and a CNC profile milling is illustrated in Figure 1. The depth of cut includes axial and radial depths of cut; the axial depth of cut (ADC) is the thickness of removed material along the tool axis direction, and the radial depth of cut (RDC) is the thickness of the removed material by the tool side. Due to the geometry variation along the part profile, the chip load changes at different geometries, so does the cutting force. The chip load at a cutter location can be measured with cutter engagement angle γ, and the larger the cutter engagement angle, the greater the cutting force. Meanwhile, the radial depth of cut is represented by the effective cutting depth as δ e, and the effective cutting depth is normalized with respect to the cutter radius (see Figure 1). Thus the cutter engagement angle and the radial depth of cut satisfy the relationship as δ = 1 cosγ. e The basic geometries of a part profile consist of straight lines, concave and convex shape formed by line segments (called concave and convex turns) or concave and convex shape formed by curves (called concave and convex curves). The cutter engagement angle changes when the tool is located at different points. For example, the cutter engagement angle is larger when the tool is in the concave shape than in the convex shape. The cutter engagement angle is likely different from point to point if the tool cuts a curve. With the profile of pre-machined part and the profile of the design part, the engagement angle can be calculated. As a general rule, the cutting force in a machining can not exceed the maximum cutting force the tool can undertake. When the cutting force is high, the feed rate should be reduced accordingly. Thus the cutting force is a function of the depth of cut and the feed rate. Figure 1. Profile Machining Model However, the relationship among the cutting force, the depth of cut, and the feed rate is very complicated depending the workpiece material, cutter material and size, so a generic equation of the relationship is impossible to get. In this work, a machining parameter database is built to represent the relationship. To build this database, first the axial depth of cut is set with a proper value, a part is then machined with different radial depths of cut and different feed rates. The cutting forces are measured in each case of different machining parameters. After the set of data have been obtained, change the axial depth of cut and repeat the procedure for more data. At last a machining parameter database is obtained to represent the cutting force function. In table 1, part of an example machining parameter database is shown. The cutter is a high-speed-steel flat end-mill with four flutes and the diameter of 10 mm, and the workpiece material is pre-hardened steel. TABLE I. Cutting Force (N) δ e MACHINING PARAMETER DATABASE Axial Depth of Cut is 3 mm Feed rate (mm/min)

3 Axial Depth of Cut is 4 mm Cutting Force (N) δ e Cutting Force (N) δ e Feed rate (mm/min) Axial Depth of Cut is 5 mm Feed rate (mm/min) In this database, four different feed rates (50, 100, 150, and 200 mm/min) are used. The six effective radius depths of cutting are selected as (0.1, 0.2, 0.3, 0.4, 0.5, and 1.0), where the maximum radial depth of cut is set as the cutter radius. IV. FUZZY LOGICAL SYSTEM OF CUTTING FORCE PREDICTION To represent the machining parameter database, a fuzzy logical system is employed. A fuzzy logic system consists of fuzzy variable membership functions, a rule base, fuzzy reasoning and defuzzification. The four parts of a fuzzy system are explained in detail in the following. A. Membership Functions for Fuzzy Variables To generate a fuzzy set for a fuzzy variable, some membership functions for the fuzzy variable should be defined at the beginning. The membership functions of the fuzzy variable are chosen among several functions which are defined within the variable interval, and the range of each membership function is between zero and one. Different membership functions result in different output, and the selection of the membership functions relies on individual experience. Specifically, Gaussian function is used to define the three membership functions for the radial depth of cut, and they are plotted in Figure 2. Since the three membership functions reach their maximums at different values of the radial depth of cut, the membership functions are identified as low, medium, and high according to these values ([7]). Similarly, Gaussian function is adopted as the membership functions for both the radial depth of cut and the feed rate. The membership functions for the radial depth of cut are also identified as low, medium, and high; and those for the feed rate as slow, medium, and fast. These membership functions for the radial depth of cut and the feed rate are plotted in Figure 3 and 4, respectively. The feed rate interval is specified between 50 and 200 mm/min, and the interval of the radial depth of cut is from four to seven millimeters. Figure 2. Membership Functions of the Radial Depth of Cut Figure 3. Membership Functions of the Moreover, nine membership functions of the cutting force are defined and shown in Figure 5. With the same way to identify the membership functions of the axial depth of cut, the nine membership functions of the cutting force are identified as level one of small (S1), level two of small (S2), level three of small (S3), level one of medium (M1), level two of medium (M2), level three of medium (M3), level one of large (L1), level two of large (L2), and level three of large (L3). Figure 4. Membership Functions of the Axial Depth of Cut. Figure 5. Membership Functions of the Cutting Force. 1913

4 B. Fuzzy Rules A fuzzy rule base should be set up after the membership functions for the fuzzy variables have been defined. Generally a fuzzy rule of a rule base is expressed in the form of a logical statement, e.g. IF_THEN ([7]). However, this fuzzy system has three inputs and one output, which is called multiple-input-and-single-output (MISO) system, the logical operator AND is used to connect the three inputs to form an aggregated statement as a rule, and all the rules are in the form as IF_AND_AND_THEN. Each fuzzy variable of the input has three different membership functions and the combination of the three fuzzy variables can compose twenty seven rules in this fuzzy system. The combination is listed in Table 2. TABLE II. COMBINATION OF THE MEMBERSHIP FUNCTIONS OF FUZZY VARIABLES Axial Depth of Cut is Low Cutting Force Slow Medium Fast Low S1 S2 S3 Radial Depth of Medium S2 S3 M1 Cut High S2 M1 M2 Axial Depth of Cut is Medium Cutting Force Slow Medium Fast Low S1 S3 M1 Radial Depth of Medium S2 M1 M2 Cut High S3 M3 L1 Cutting Force Radial Depth of Cut Axial Depth of Cut is High Slow Medium Fast Low S2 S3 M2 Medium S3 M3 L1 High M1 M3 L3 Based on the Table 2, the twenty seven fuzzy rules can be composed as followings and form a rule base. IF the axial depth of cut is low AND the radial depth of cut is low AND the feed rate is slow, THEN the cutting force is S1. IF the axial depth of cut is low AND the radial depth of cut is low AND the feed rate is medium, THEN the cutting force is S2. IF the axial depth of cut is high AND the radial depth of cut is high AND the feed rate is fast, THEN the cutting force is L3. Each fuzzy rule in this rule base reflects a specific machining case with different machining parameters in principle and is correct with respect to the machining parameter database. For example, in the first rule, when the radial depth of cut is low, the axial depth of cut is low, and the feed rate is slow, the cutting force is the level one of small. This fuzzy rule describes a portion of the relationship between the cutting force and the fuzzy variables, and it is a qualitative interpolation of some data in the machining parameter database. With all the fuzzy rules, the machining parameter database is interpolated qualitatively and the function of the cutting force is represented completed. The fuzzy rule base is the heart of the fuzzy system and can be used to predict the cutting force according to the three inputs, the axial depth of cut, the radial depth of cut, and the feed rate. C. Cutting Force Prediction With the fuzzy rule base, instead of the close-formed equation of the cutting force, the cutting force can be calculated. Determining the cutting force value using the fuzzy rule base is called fuzzy reasoning. Since the fuzzy rules reflect the actual machining correctly, the predicted cutting force is reasonable and close to reality if some new values of the fuzzy variables are input. In detail, two approaches, the min-max-gravity method and product-sum-gravity method, are popular in the fuzzy reasoning process ([8]). Because of the large amount of machining data, the product-sum-gravity method is employed in this work for more precise cutting force prediction. In this method, the result of the fuzzy reasoning is obtained by calculating the algebraic products of the fuzzy variables and summing the products. However, the result of the fuzzy reasoning is not cutting force itself and can not be used directly. This value should be defuzzified into the predicted cutting force. D. Defuzzification of Cutting Force Defuzzification in this procedure is to map the fuzzy reasoning result to the actual cutting force. The center average defuzzification is commonly used in fuzzy logic systems, and it is computationally simple and intuitively plausible, so the center average defuzzification is used in the approach. As a result, the relationship between the cutting force and the fuzzy variables in this fuzzy system is shown in Figure 6 and Figure 7. Figure 6. Surface of the Fuzzy System; When the Axial Depth of Cut is 5.5 mm 1914

5 shown as the curve in between. Based on the information about the design of the part and the cutter size, the tool path of CNC profile milling is planned and shown as the external curve. Figure 7. When the Axial Depth of Cut is 6 mm V. PROCEDURE OF OPTIMAL FEED RATE DETERMINATION The goal of this work is to determine the optimal feed rate for each geometry feature in the part profile. The criteria of the optimal feed rate is that under this feed rate the cutting force is close but less than the maximum cutting force, which the tool can undertake without severe tool wear or tool breakage. By using the fuzzy system, the cutting force can be estimated when the depth of cut and the feed rate are known, and the cutting force is compared with the maximum cutting force. With several iterations of calculation, the optimal feed rate can be found. The procedure of the optimal feed rate determination is provided in five steps. Find the maximum cutting force based on the workpiece material, cutter size and material. Set the axial depth of cut for each profile machining. Locate the geometry features of the part profile. Calculate the cutter engagement angle and the radial depth of cut for each geometry feature. Initialize a feed rate for each feature. Apply the fuzzy logic system to find the cutting force for a feature. Compare the cutting force with the maximum cutting force; if the cutting force is less than the maximum cutting force, increase the feed rate; if the cutting force is greater than the maximum cutting force, decrease the feed rate. Repeat step 3. Otherwise, go to the next step. After find the optimal feed rate for one feature, move to the next feature, and repeat step 3. Otherwise, end the program. VI. APPLICATIONS To demonstrate the advantages of this approach, it is applied on an example part to plan the feed rate for its CNC profile machining. This example part on a platform is shown in Figure 8. Suppose a stock is prepared for this part, and its width, length, and height are 125 mm, 130 mm, and 20 mm, respectively. A flat end mill with the radial of 10 mm is used to machine this part, and the axial depth of cut in this CNC profile machining is set as 5 mm. In Figure 9, the part profile is the inner curve and the machined part after rough machining is Figure 8. Example Part for CNC Profile Machining Figure 9. Tool Path of the Profile Milling By evaluating the part profile, the convex turn, concave turn, and concave arcs with different radius are located. For each geometry feature, the cutter engagement angle is calculated and shown in Figure 10. It can be seen that the cutter engagement angles are greater at CD, IJ, MN, and RQ segments, and these angles are listed in Table 3. At each corner, the cutter engagement angle drops. For example, at the corner OP segment the cutter engagement angle is only Based on the cutter engagement angles, this approach can find the optimal feed rates for each geometry feature in the part profile machining. TABLE III. Engagement Angle (Degree) (mm/min) FEED RATES FOR DIFFERENT GEOMETRIC FEATURES Tool Path Segments AB CD IJ MN OP RQ

6 In this case, if the feed rate is set as 135 mm/min by an experienced NC operator for the unmanned machining without any tool breakage or impaired surface quality, the total machining time of this profile milling is 2916 seconds. Whereas, using the feed rates determined with this proposed method (see Table 3), the total machining time of the same profile milling is 2374 seconds. Thus, this intelligent feed rate determination method saves the machining time of 18.6% with respect to the conventional method. The profile machining of the part is simulated with the optimal feed rates with CATIA CAD/CAM system (see Figure 11). VII. CONCLUSIONS A generic and intelligent approach of the feed rate determination for CNC profile machining is proposed. The way to build a fuzzy inference system based on any machining parameter database is introduced. The cutting force can be predicted when the axial and radial depth of cut and the feed rate are input. With the maximum cutting force, the optimal feed rate can be found. The work introduces a generic method to predict the cutting force for different type of workpiece and tool material. By identifying the geometric features of the part profile, the proper feed rate can be determined for each geometric features and the CNC profile milling is carried out with different feed rates from place to place. Thus, the cutting efficiency will reach maximum in this cutting. This approach can also be employed for high speed cutting. ACKNOWLEDGMENT The financial support of this work from the Science and Technology Department of Helongjiang is thankfully acknowledged. This work is supported by the Key Science and Technology Project of Heilongjiang of China under Grant No. GZ07A107 and the Key Research Program of Heilongjiang Institute of Technology under Grant No. Z REFERENCES Figure 10. Cutter Engagement Angles for Geometric Features Figure 11. Simulation of CNC Profile Machining with CATIA System [1] X. H. Cong, H. Ning and Z. B. Miao A Fuzzy Logical Application in a Robot Self Navigation, Proceedings of the International Conference on Industrial Electrical Application, Harbin, China [2] Z. Z. Chen and Z. B. Miao A Intelligent Approach of Non-Constant Determination for High-Performance 2D CNC Milling, International Journal of Computer Applications in Technology, Canada [3] S. H. Bae, K. Ko, B. H. Kim, and B. K. Choi: Automatic Feedrate Adjustment for Pocket Machining, Computer-Aided Design, Vol.35, pp , [4] D. K. Baek, T. J. Ko, and H. S. Kim: Optimization of in a Face Milling Operation Using a Surface Roughness Model, International Journal of Machine Tools & Manufacture, Vol.41, pp , [5] J. H. Ko, W. S. Yun, and D. W. Cho: Off-Line Scheduling Using Virtual CNC Based on an Evaluation of Cutting Performance, Computer-Aided Design, Vol.35, pp , [6] B. K Choi, and R. Jerard, Sculptured Surface Machining Theory and Applications, Kluwer Academic Publishers, [7] W. L. R. Ip: A Fuzzy Basis Material Removal Optimization Strategy for Sculptured Surface Machining Using Ball-nosed Cutters, International Journal of Production Researches, Vol.36, No.9, pp , [8] M. Young, The Technical Writer's Handbook. Mill Valley, CA: University Science, [9] J. M. Lee, C. N. Chu, S. Y. Kim, and B. H. Kim: Feed-Rate Optimization of Ball End Milling Considering Local Shape Features, Annals of the CIRP, Vol.46, [10] L. X. Wang, A Course in Fuzzy Systems and Control, Prentice Hall PTR,

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