Empirical Modeling of Cutting Forces in Ball End Milling using Experimental Design

5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India Empirical Modeling of Cutting Forces in Ball End Milling using Experimental Design VenkateswaraSarma M.N.M. 1*,Manu. R. 2 1* M. Tech. Scholar, NIT Calicut, 673601, vsmeduri@yahoo.com 2 Associate Professor, NIT Calicut, 673601, manu@nitc.ac.in Abstract Precision parts with curved surfaces such as dies and molds are required in many manufacturing industries. Ball end milling is one of the most common manufacturing processes for such parts. Force modeling of ball-end milling is important for tool life estimation, chatter prediction, tool condition monitoring and to estimate the tool deflection which affects the quality of the finished parts. This project presents an approach for modeling the cutting forces acting on ball end mill in milling process. The steps used in developing the model are based on mechanistic principles of metal cutting. Initially, the forces acting on the ball end mill are modeled based on the literature, in which the empirical relationships were used to relate the cutting forces to the undeformed chip geometry.these modeling equations governing the cutting forces are programmed in the MATLAB software. A series of slot milling experiments are conducted using a ball end mill by varying the feed and depth of cut and the cutting forces acting on the work piece are measured. An algorithm was developed, to calculate the empirical parameters, by using the deviation between the average forces measured while doing experiments and the force values predicted by the software program. Keywords:Force prediction, ball end mill, Algorithm 1. Introduction The ball-end milling process is one of the most widely used machining processes for the components that are characterized with free-form surfaces like dies, molds and various automotive components. Machining parts like automotive and aerospace components demands a high level of accuracy. Cutting forces acting on the tool are the important factors which governs chatter and surface quality while machining. In general, prediction of forces in flat end milling (or any other conventional milling cutters) is not a complex problem. But, in the case of ball end milling, it is not easy, due to the complex geometry of the ball end mill, which makes it difficult, either for geometric modeling of the tool or simulating using design software. Therefore different approaches are being developed by researchers in this area since many years. Force prediction gives the manufacturer, a clear idea about the process being done. There are two important approacheswhich were proposed in literature for estimating the cutting force coefficients in milling process: orthogonal to oblique cutting transformation approach and analytical fitting of estimated cutting forces to the experimentally measured ones. In case of geometric modeling, cutting edge of the ball-end mill was considered as a series of infinitesimal elements, and the geometry of a cutting edge element was analyzed to calculate the necessary parameters for its oblique cutting process assuming that each cutting edge was straight. Transformation mechanics of orthogonal to oblique cutting define the cutting force coefficients using the shear stress, shear angle and friction angle. The coefficients are determined from orthogonal turning experiments, which are then transformed into oblique cutting edge for prediction of cutting forces in helical end milling forces.specifically, for ball-end milling process,yang and Park (1990)developed a model using orthogonalcutting data obtained from end turning tests and then extended the model for flexible cutting systems. Yucesan and Altintas(1994) evaluated the varying rake face friction and pressure distribution and the chip flowangle in peripheral milling in order to provide accuratecutting force predictions. Subsequent studies by Altintaset al. (1996) further demonstrated that the milling force coefficients could be determined from orthogonalcutting tests with oblique cutting analysis and transformation. In analytical modeling, initially, a force model is developed based on the mechanistic principles of metal cutting. The cutting forces are calculated on the 464-1

Empirical Modeling of Cutting Forces in Ball End Milling using Experimental Design basis of the engaged cut geometry, the undeformed chip thickness distribution along the cutting edges, and the empirical relationships that relate the cutting forces to the undeformed chip geometry. Now these empirical relations should be estimated by real force values obtained by conducting experiments.feng and Menq (1994) designed a cutting force model and determined empirical cutting force coefficients using numerical polynomial fit.wan et al. (2006) did numerical simulations for general end mills utilizing specific data points that do not vary with respect to the cutting force coefficients, as references. However, these proposed methods have either some geometric constraints in conducting the calibration test, which are often different from the intended machining geometries, or part of the analysis needs to be conducted offline, which introduces delay and thus hinder the use of these techniques to monitor and control of the process. In the present work, the force modeling was done according to the mechanistic principles of metal cutting, in which the forces acting on the ball end mill are directly proportional to the undeformed chip geometry. A set of model building experiments was conducted and measured force values are used to estimate the empirical coefficients using numerical fitting procedure. 2.Cutting force model The cutting force model was developed based on Mechanistic principles of metal cutting in which the forces are directly proportional to the undeformed chip geometry. The forces acting on the ball end mill are modeled based on the equations from the paper given by Feng and Menq, in which the empirical relations are used to relate the cutting forces to the undeformed chip geometry. The following simple formula shows the estimation of cutting force with the size effect explicitly considered: (1) where F is the principal cutting force responsible for the total energy consumed, b is the width of cut, t is the undeformed chip thickness, K is the cutting mechanics parameter, and 1 > m > 0 for most metallic materials. In the above expression, K represents the condensed effects of all process parameters except b and t, the undeformed chip geometry parameters. It is clear from equation (1) that a factor essential to the prediction of cutting forces is the undeformed chip geometry along the cutting edges engaged with the work piece. If the cutting speed is much larger than the table feed rate, the circular tooth path can be used and a good approximation is obtained for the instantaneous chip thickness sin (2) where f is the feed rate (mm/tooth) and θ indicates the angular position of the cutting edge. A complete representation of the undeformed chip geometry along the cutting edges is dependent on the geometric design of the ball end milling cutter. Therefore, the angular position of a differential element on the ith cutting edge of an n- fluted cutter at a distance z from the free end can be expressed as:, tan 1 (3) where is the angular position designated to tooth number 1 (arbitrarily selected) at the free end (z = 0), β is the helix angle, and R is the nominal radius of the ball-end mill. Combining equations (2) and (3), the undeformed chip thickness for the differential cutting edge element is, sin, (4) With equations (1) and (4), the general expression for the elemental tangential cutting force and may now be written as, dz(5), dz (6) where dz stands for the width of cut of the differential cutting edge along the z direction. In equations (5) and (6), and, which basically characterize the local cutting mechanics of the differential cutting edges at z, are approximated by polynomial expressions. In order to evaluate the associated empirical parameters with reasonable efficiency, the following expressions were selected to approximate and (7) 464-2

5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India (8) In the above expressions, z is nondimensionalized with respect to R and the two domain representations of and are based on the fact that the differential cutting edges on the cylindrical part are geometrically the same and exhibit the same cutting characteristics, while those on the ball part are geometrically different from each other, which results in varying cutting characteristics along the axial direction. To obtain the total forces acting on the ballend mill at any instant, the elemental tangential and radial cutting forces are resolved into the external x, y coordinate system and summed over all the engaged differential cutting edges. The summation (integration) is done numerically along the z-axis to yield the instantaneous forces in the x, y directions:, cos,, sin, (9), sin,, cos, (10) In case of slot milling experiments, due to the anti-symmetry of the cosine function to the same axis, it can be neglected. It is then clear that slot machining should be used in the model- building experiments so that, in the numerical fitting procedure, the tangential and radial model parameters are decoupled and independent of each other 3. Programming in MATLAB The modelling equations governing the cutting forces were programmed in the MATLAB software. Now by using this MATLAB program, we can calculate the cutting forces, when the empirical and geometric data is available. 4. Experimental Work A series of model building experiments was performed on Agni BMV45 TC24 4-axis vertical machining centre. The carbide ball-end mills with 10 mm diameters, four right-handed flutes and 30 helix angles were used.the ball-end mills were placed in the collet-type tool holders and the work piece isaluminium alloy 2024-T6 block. A total of 12 slot cuts were carried out and the details of the cutting conditions are shown in Table 1. KISTLER dynamometer of Type 9257B was used to measure the cutting forces while machining. Itsmulti-channel charge amplifier Type 5070A for multi component force measurement was used to amplify the charge signal. Table 1 Cutting conditions for model building experiments Depth of cut(mm) Feed rate(mm/min) Spindle speed (rpm) (N) (N) 3 40 600 265.7-178.6 4 40 600 479.6-34.33 5 40 600 653.7 347.11 3 50 600 244.7-125.46 4 50 600 464.11-42.5 5 50 600 818.7 431.7 3 60 800 219.7-202.3 4 60 800 347.5-38.3 5 60 800 735.6 387.5 3 70 800 256.4-236.8 4 70 800 562.5-44.6 5 70 800 859.4 453.7 5. Methodology The MATLAB program for the model equation of the cutting forces is used to estimate the empirical parameters. An iterative numerical algorithm is described in the Fig. 1. The algorithm starts with an assumption that the empirical coefficients obtained from the literature, also holds good for the forces obtained from the slot milling experiments. Those empirical coefficients are used to find the forces with the geometric data available with the slot milling experiments using the program. But due to the difference in the tool, work piece material, geometry of ball end mill and other physical parameters, the predicted force values and experimental values do not match each other. The 464-3

Empirical Modeling of Cutting Forces in Ball End Milling using Experimental Design difference between the forces is used to approximate the new empirical coefficients. Then, the algorithm goes back, to estimate the model parameters again by using the latest obtained empirical coefficients. This procedure repeats itself until the estimates for the model parameters stabilize numerically. 5.1. Study of parameters (12) The approximation of the empirical coefficients should be done in such a way that, a special weightage should be given to each and every coefficient. In order to find the role of each coefficient, the study of parameters is done separately by varying them (keeping the remaining parameters constant) and observing the change in force values accordingly. In order to give the weightage while approximating the model parameters, slopes of the lines representing the parameters individually are calculated. The following are the slopes respectively. = = - = = Fig. 1. An iterative algorithm to find empirical parameters The initial empirical parameters that are taken from the literature review are as follows: (11) An iterative algorithm is followed to find the parameters as explained above. After each and every iterative step, the deviation between the measured and simulated forces was found and the model parameters were modified according to the given weightage. The algorithm stops when the deviation is less than 2%. The following are the conditions for which the parameters are estimated. Depth of cut = 3mm Feed = 40 mm/min Spindle speed = 600 rpm Measured force, = 265.7 N Table 2Iterative algorithm to find parameters in X direction It. No. 1 5682-7507 2 4529-9439 Simulated forces, (N) deviation 3416 225 738 472.4 189-730 368 103 Percentage deviation 178.26% 38.86% 464-4

5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India 3 4077-9883 -451-938 287 22.6 8.52% 4 3821-9975 -496-983 269.9 4.93 1.82% The slopes of the lines representing the effect of model parameters are used to estimate the required results. Initially, the parameters are taken from the literature and the corresponding forces should be found. There will be a deviation between the simulated and measured values. Now, the parameters should be changed in such a way that the deviation reduces after each step. Let us assume that each coefficient contributes equally for the deviation. Since there are four parameters, divide the deviation into four equal parts. Each parameter is modified according to their slope respectively. Since it was assumed that each coefficient contributes equally for the deviation, again there will be deviation but less than the previous one. After a few iterative steps, the deviation decreases and simulated force values tend towards the measured values. The cut-off value for the deviation is 2%. The above iterative algorithm was applied for 5 different conditions and similar results have been obtained. Now the average of all the model parameters at different conditions gives the output model parameters. = 3872 = -514 = -992 5.2. Study of parameters,,, Finding the parameters in Y direction is also done in the same way as explained above. =. =. = -. =. The following are the conditions for which the parameters are estimated. Depth of cut = 3mm Feed = 40 mm/min Spindle speed = 600 rpm Measured force, = -178.6 N = -9872 Table 3 Iterative algorithm to find parameters in Y direction It. No. Simulated forces, (N) Deviation Percentage deviation 1 480 7055-11953 5617-400.44 222.14 124.53% 2 330 6497-13397 2582-220.7 41.7 23.38% 3 301 6392-13668 2012-186.7 8.4 4.7% 4 296 6371-13772 1897-180.51 2.21 1.23% The average of all the model parameters at different conditions gives the output model parameters. The final model parameters of the model equation governing the cutting forces are as follows: = 282 = 6383 = -13705 3872 9872 514 992 (13) = 1913 464-5

Empirical Modeling of Cutting Forces in Ball End Milling using Experimental Design 282 6383 13705 1913 (14) 6. Model validation The above obtained model parameters were used to estimate the forces and then compared with the measured values. The verification experiments indicate that the magnitudes of the predicted forces correspond well with the actual test results. Sl. No. Depth of cut(mm) Table 4 Validation of results in X direction Feed rate(mm/min) Spindle speed(rpm) Simulated force, (N) Measured Force, (N) Deviation percentage 1 3 60 800 224.4 219.7 2.13% 2 4 70 800 546.8 562.6 2.8% 3 5 50 600 800.6 818.7 2.22% Sl. No. Depth of cut(mm) TABLE 5 Validation of results in Y direction Feed rate(mm/min) Spindle speed(rpm) Simulated force, (N) Measured Force, (N) Deviation percentage 1 3 60 800-195.8-202.3 3.21% 2 4 70 800-43.3-44.6 2.91% 3 5 50 600 422.6 431.7 2.11% 7. Conclusion The above model could successfully predict the forces developed during ball end milling operations. This was validated using slot milling operations at different cutting conditions. This model can be extended to develop a more accurate and practical model for the ball-end milling process so as to provide guidance in selecting machining conditions in free form surface machining. References Azeem, A., Feng, H.Y., Wang, L. (2004), Simplified and efficient calibration of a mechanistic cutting force model for ball-end milling, International Journalof Machine Tool and Manufacture, Vol. 44, pp. 291 298. Dhupia, J., Girsang, I. (2012), Correlation-based estimation of cutting force coefficients for ball-end milling, Machining Science and Technology: An International Journal, Vol. 16, No. 2, pp. 287-303. Feng, H.Y., Menq, C.H. (1994), The prediction of cutting forces in the ball-end milling process I, Model formulation and model building procedure, International Journal Machine Tools Manufacturing, Vol 34, No. 5, pp. 697 710. Lee, P., Altintas, Y.(1996), Prediction of ball-end milling forces from orthogonal cutting data, International Journal of Machine Tools and Manufacture, Vol. 36, No.9, pp. 1059 1072. Wan, M., Zhang, W.H., Tan, G., Qin, G.H. (2007), An in-depth analysis of the synchronization between the measured and predicted cutting forces for developinginstantaneous milling force model, International Journal of Machine Tools & Manufacture, Vol. 47, pp. 2018 2030. Yang, M., Park, H. (1991), The prediction of cutting force in ball-end milling, International Journal of Machine Tools and Manufacture, Vol. 31, No. 1, pp. 45 54. Yucesan, G., Altintas, Y., (1994), Improved modeling of cutting force coefficients in peripheral milling, International Journal of Machine Tools and Manufacture, Vol. 34, No. 4, pp. 473-487 Yuwen, S., Ren, F., Guo, D., Jia, Z., (2009), Estimation and experimental validation of cutting forces in ball-end milling of sculptured surfaces, International Journal of Machine Tool and Manufacture, Vol. 49, pp. 1238 1244. 464-6